thesis

Partitioned Analysis of

Nonlinear Soil-Structure Interaction

Hamid ZOLGHADR ZADEH JAHROMI

BSc, MSc

August 2009

A thesis submitted in fulfilment

of the requirements for the degree of

Doctor of Philosophy of the University of London

and the Diploma of Imperial College London

Department of Civil and Environmental Engineering

Imperial College London

London, SW7 2AZ

2

Abstract

This work has been primarily motivated by the lack of sophisticated monolithic tools

for modelling nonlinear soil–structure interaction problems, while recognising the

existence of advanced tools for nonlinear analysis of structure and soil in isolation.

Although coupled modelling of soil-structure interaction problems may be achieved

using a monolithic treatment, the partitioned treatment has been advocated as

offering major benefits in the context of coupled modelling of nonlinear soil-

structure interaction. Accordingly, the aim of this work has been to develop

advanced numerical methods for nonlinear coupling of soil-structure interaction

problems, where the partitioned approach is adopted as a framework for coupling

field-specific tools with minimal intrusion into codes.

In this respect, the partitioned approach for soil-structure interaction analysis has

been fully investigated in this work. Various coupling techniques are developed in

the context of soil-structure interaction analysis, and their computational

characteristics are discussed. Novel formulations for coupling soil-structure systems,

based on relaxation coupling methods and also utilizing the tangent stiffness matrix

of the partitioned sub-domains at the interface, are proposed, and their relative

performance is evaluated. The proposed approaches are believed to possess superior

convergence characteristics in comparison with existing coupling methods, rendering

these methods more general procedures for modelling soil-structure interaction

problems using coupled field-specific tools.

Based on the presented coupling algorithms, a novel simulation environment is

developed, utilising discipline-oriented solvers for nonlinear structural and

geotechnical analysis. Using the developed simulation environment in several

examples in which nonlinearity arises in both structure and soil, its applicability and

Abstract

3

high potential are demonstrated. Indeed, it is shown that the partitioned treatment is a

feasible and realistic approach for coupled modelling of nonlinear soil-structure

interaction problems, providing an integrated interdisciplinary computational

approach which combines the advanced features of both structural and geotechnical

modelling for a variety of challenging soil-structure interaction problems.

4

Acknowledgements

First and foremost I would like to express my deep and sincere gratitude to my

supervisors, Prof. Bassam A. Izzuddin and Dr. Lidija Zdravkovic. I owe a great deal

of appreciation for this thesis to Prof. Bassam A. Izzuddin for his insightful

supervision, invaluable guidance and unfailing support from very early stages of this

research, without which this thesis would not have been possible. I thank him for his

suggestions, refinements and constructive comments on full drafts. Likewise, I am

deeply grateful to Dr. Lidija Zdravkovic for her continuous encouragement and

support, substantive guidance, valuable comments on full drafts, and her overall

contribution to the final outcome.

Being part of the research groups in the Systems & Mechanics section and the Soil

Mechanics section at Imperial College has been a great experience. Thanks are due

to all academic staff and students for creating such an enjoyable and unique

environment. Special thanks are due to Mrs. Fionnuala Donavan the Postgraduate

Administrator, whose kindness will always be remembered. Many thanks are owed

to my friends and colleagues for the remarkable moments I have shared with them.

I am very grateful and deeply indebted to my parents for their inseparable love and

support throughout my life. Their support has been instrumental in all my

achievements thus far. I would like to thank my brother for being supportive and

caring.

Words fail me to express my deep appreciation to my wife Ahoura for her dedication

and persistence confidence in me. Without her love and understanding I would not

have been able to complete this thesis.

5

Table of Contents

Abstract 2

Acknowledgements 4

List of Tables 9

List of Figures 11

Notation 19

Chapter 1

1 Introduction 21

1.1 Soil-Structure Interaction 21

1.2 Coupled Systems 23

1.3 Treatment of Coupled Soil-Structure Interaction 24

1.4 Partitioned vs. Monolithic Approaches 26

1.5 Aims and Scope of Research 27

1.6 Layout of Thesis 29

Chapter 2

2 Literature Review 32

2.1 Background 32

2.2 Modelling of Soil-Structure Interaction 34

2.2.1 Field Elimination 35

2.2.2 Finite Element Method (FEM) 40

2.2.3 Integrated Modelling 42

2.2.4 Partitioned Analysis 46

2.2.5 Staggered Approach 48

2.2.6 Iterative Coupling 52

2.3 Concluding Remarks 60

Chapter 3

3 Staggered Approach 63

Table of Content

6

3.1 Introduction 63

3.2 Dynamic Analysis Formulation 65

3.3 Partitioning 67

3.4 Staggered Coupling Procedure 70

3.5 Predictors 71

3.6 Stability Analysis by Amplification Method 72

3.7 Stability Analysis of Staggered Coupling Scheme for a Test System 76

3.8 Accuracy 84

3.9 Example 84

3.10 Conclusion 91

Chapter 4

4 Iterative Coupling 92

4.1 Introduction 92

4.2 Iterative Coupling 96

4.2.1 Sequential Dirichlet-Neumann Iterative Coupling 97

4.2.2 Sequential Neumann-Dirichlet Iterative Coupling 98

4.2.3 Parallel Dirichlet-Neumann Iterative Coupling 100

4.2.4 Parallel Neumann-Dirichlet Iterative Coupling 102

4.2.5 Parallel Dirichlet-Dirichlet Iterative Coupling 103

4.2.6 Parallel Neumann-Neumann Iterative Coupling 105

4.3 Treatment of Interactive Boundary Conditions 106

4.4 Convergence of Iterative Coupling 107

4.4.1 Convergence of Sequential D-N with Trivial Update 107

4.4.2 Example 112

4.5 Simulation Environment 117

4.6 ADAPTIC 117

4.7 ICFEP 119

4.8 INTERFACE 121

4.9 Simulation Environment Architecture 122

4.10 Data Communication 125


Table of Content

7

Chapter 5

5 Interface Relaxation 129

5.1 Introduction 129

5.2 Constant Relaxation 131

5.2.1 General Convergence Analysis 134

5.2.2 Convergence Studies 142

5.2.2.1 Example 1: Dynamic FEM-FEM Coupling 142

5.2.2.2 Example 2: Static FEM-FEM Coupling 154

5.3 Adaptive Relaxation 159

5.3.1 Convergence Analysis 159

5.3.2 Convergence Studies 163

5.3.2.1 SDOF Test System 163



5.4 Soil-Structure Interaction Analysis 172


Chapter 6

6 Reduced Order Method 187


6.2 Condensed Interface Tangent Stiffness 188

6.2.1 Numerical Example 193

6.2.2 Analogy between Interface Relaxation and Condensed Interface Stiffness Approaches 198

6.3 Approximation of the Condensed Tangent stiffness 201

6.3.1 Condensed Interface Secant Stiffness Matrix 202



6.3.1.3 Discussion on Nonlinear Analysis 213

6.3.2 Reduced Order Method 214

6.3.2.1 Singularity of .T

T TU U and .T

B BF F 217

6.3.2.2 Selective Addition or Replacement of the

Displacement/Force Vectors 219

Table of Content

8

6.3.2.3 Singularity of Approximated Stiffness/Flexibility

Matrices 227



6.3.2.6 Example 3: Linear Soil-Structure Interaction 239

6.3.3 Mixed Reduced Order Method 242

6.4 Case Study: Nonlinear Soil-Structure Interaction Problem 249

6.5 Conclusion 253

Chapter 7

7 Case Studies 255


7.2 Nonlinear Behaviour of Pitched-Roof Frame on Flexible Soil 256

7.3 Settlement Analysis of Multi-storey Five-bay Steel Frame 264

7.4 Building Response to an Adjacent Excavation 274

7.5 Coupled Modelling of Retaining Steel Sheet Piles 289

7.6 Conclusion 296

Chapter 8

8 Conclusion 297


8.2 Conclusions 298

8.2.1 Staggered Approach 299

8.2.2 Iterative Coupling 300

8.2.3 Interface Relaxation 301

8.2.4 Reduced Order Method 302

8.2.5 Case Studies 304

8.3 Recommendations for Future Works 304

References 307

Appendix A: Structure of INTERFACE Data File 316

Appendix B: Structure of Communication Data File 317

Appendix C: Iterative Coupling Algorithms 318

Appendix D: Numerical Example 322

9

List of Tables

Table 2.1: Basic solution requirements satisfied by various methods of

geotechnical analysis 41

Table 2.2: Design requirements satisfied by various methods of

geotechnical analysis 41

Table 3.1: Single-step predictors for second-order dynamic systems 72

Table 5.1: Range of suitable and optimal relaxation parameter for different

1

2

m

m 145


T

B

E

E 147


/T BE E 155

Table 5.4: Number of coupling iterations with adaptive and optimum relaxation

for 1000 time-steps ( 0.01t s ) with a tolerance of 10-4 m 165

Table 5.5: Number of required coupling iterations with adaptive and optimum

relaxation 171

Table 5.6: Geometric and material properties of the partitioned soil-structure

system 175

Table 5.7: Convergence characteristics of constant and adaptive relaxation

schemes 177

Table 6.1: Coupling Procedure 193

List of Tables

10

Table 6.2: Required coupling iterations for different coupling schemes 206

Table 6.3: Number of required coupling iterations for 500 time-steps

( 0.01t s ) with a tolerance of 10-4 m 210

Table 6.4: Required coupling iterations for different coupling schemes 229

Table 6.5: Number of required coupling iterations for 500 time-steps

( 0.01t s ) with a tolerance of 1e-4 m 235

Table 6.6: Mixed reduced order method coupling procedures 248


system 250

Table 6.8: Number of required coupling iterations for different coupling

schemes 250


system 266

Table 7.2: Comparison of different coupling methods 267


system 277

Table 7.4: Loading scenarios 278

Table 7.5: Comparison of different coupling methods 285

Table 7.6: Comparison of maximum bending moment in fully coupled

analysis 289

Table 7.7: Loading scenario in different incremental stages of the analysis 290

Table 7.8: Loading scenario in different incremental stages of the analysis 293

11

List of Figures

Figure 1.1: Field elimination treatment of soil-structure interaction 25

Figure 1.2: Partitioned treatment of soil-structure interaction 26

Figure 2.1: Illustrative SSI problem 33

Figure 2.2: Schematic diagram of a Winkler foundation 35

Figure 2.3: Schematic view of a) elasto-plastic b) visco-elastic foundation 38

Figure 2.4: Field elimination model for SSI analysis of lateral vibration of an elevated water tank 39

Figure 2.5: Idealization of jack-up models for SSI 43

Figure 2.6: a) Schematic illustration of the pile and soil model b) Schematic diagram of the structure model 44

Figure 2.7: a) Idealisation of the structure sub-domain b) Idealisation of the foundation system 44

Figure 2.8: a) Soil-structure coupled system b) Idealization of the coupled system 45

Figure 2.9: Idealization of the coupled soil-structure interaction system 45

Figure 2.10: Monolithic treatment of soil-structure interaction problem 46


Figure 2.12: Staggered coupling approach 48

Figure 2.13: Schematic representation of the staggered coupling approach 50

Figure 2.14: Staggered coupling of BEM-FEM for soil-structure interaction simulation of high speed train induced vibrations 50

Figure 2.15: Schematic of sequential iterative coupling approach 52

Figure 2.16: Coupled dam-reservoir-soil system 55

List of Figures

12

Figure 2.17: Flowchart of iterative coupling method 56

Figure 3.1: Staggered approach in dynamic soil-structure interaction 64

Figure 3.2: Partitioning and discretisation 68

Figure 3.3: Z-transformations 75

Figure 3.4: Example: geometric configuration and material response 85

Figure 3.5: Acceleration at the base 85

Figure 3.6: Problem partitioning 86

Figure 3.7: Rotation at the interface 87

Figure 3.8: Horizontal displacement at the interface 87

Figure 3.9: Horizontal displacement at the tip of the cantilever 88

Figure 3.10: Rotation at the interface 89

Figure 3.11: Horizontal displacement at the interface 89


Figure 3.13: Rotation at the tip of the cantilever 90


Figure 4.2: Schematics of iterative coupling algorithms 95

Figure 4.3: Schematics of sequential D-N iterative coupling 98

Figure 4.4: Schematics of sequential N-D iterative coupling algorithms 100

Figure 4.5: Schematics of Parallel D-N iterative coupling 101

Figure 4.6: Schematics of Parallel N-D iterative coupling 103

Figure 4.7: Schematics of Parallel D-D iterative coupling 104

Figure 4.8: Schematics of Parallel N-N iterative coupling 106

Figure 4.9: a) coupled mass-spring system, b) partitioned sub-domain B , and c)

partitioned sub-domain T 113

Figure 4.10: Communication and synchronization between ADAPTIC and ICFEP, via INTERFACE 122

List of Figures

13

Figure 4.11: Schematics of the interaction sequence between the INFERCAE, ADAPTIC and ICFEP 124

Figure 4.12: Data exchange structure 126

Figure 5.1: Variation of error reduction factor against relaxation parameter 133

Figure 5.2: Convergence range and optimum relaxation parameter 141

Figure 5.3: Dynamic FEM-FEM coupling 143


Figure 5.5: Partitioned sub-domains 144

Figure 5.6: Influence the effective mass ratio on convergence 146

Figure 5.7: Influence the effective stiffness ratio on convergence 147

Figure 5.8: Error reduction for different relaxation schemes (Time = 3.6s) 148



Figure 5.11: Rotation at the interface of sub-domain T 150

Figure 5.12: Horizontal displacement at the interface of T 151

Figure 5.13: Rotation at the interface of sub-domain B 151

Figure 5.14: Horizontal displacement at the interface of B 152

Figure 5.15: Rotation at the tip of the cantilever 152


Figure 5.17: Rotation at the tip of the cantilever 5( 10 )Tolerance 153

Figure 5.18: a) Plane strain problem, b) Problem partitioning 154

Figure 5.19: Influence of relaxation parameter on convergence properties 156

Figure 5.20: Error reduction for different relaxation schemes (model M4) 157

Figure 5.21: Different discretatzions of model M4 158

Figure 5.22: Effect of mesh density on convergence 158

List of Figures

14

Figure 5.23: Error reduction at t=3.05s for model S8 166

Figure 5.24: Error reduction at t=3s for model S8 167



Figure 5.27: Error reduction at t=3.05s for model K1 168

Figure 5.28: Error reduction at t=1.85 s for model K1 169



Figure 5.31: Error reduction for adaptive and constant relaxation schemes 171

Figure 5.32: Linear soil-structure interaction 172

Figure 5.33: Monolithic vs. Partitioned Approach 173

Figure 5.34: Plane frame resting on soil 174

Figure 5.35: Plan view of the analysed building frame 174

Figure 5.36: Geometric configuration of considered frame 176

Figure 5.37: Influence of relaxation parameters on convergence properties 177

Figure 5.38: Error reduction for different relaxation schemes for the first load increment 178

Figure 5.39: Error reduction for different relaxation schemes for the fifth load increment 178

Figure 5.40: Error reduction for different relaxation coupling schemes for the last load increment 179

Figure 5.41: Convergence performance over full range of response 180

Figure 5.42: Vertical displacement of the soil surface 181

Figure 5.43: deformed mesh of the soil partitioned sub-domain 182

Figure 5.44: Vector plot of displacements in the soil partitioned sub-domain 182

Figure 5.45: Contours of stress level in soil partitioned sub-domain 183

List of Figures

15

Figure 5.46: Deformed shape and bending moment (kN.m) of the frame (scale=5) 183

Figure 6.1: Coupled spring system 194

Figure 6.2: a) Linear FEM-FEM coupled problem b) Partitioned sub-domain B c)

Partitioned sub-domain T 205

Figure 6.3: Error reduction of different coupling schemes for model A1 206



Figure 6.6: a) Coupled dynamic FEM-FEM problem b) Partitioned sub-domain Bc) Partitioned sub-domain T 209


Figure 6.8: Horizontal displacements at the interface of T 211

Figure 6.9: Rotation at the interface of T 211

Figure 6.10: Comparison between different coupling schemes for Model C1 212

Figure 6.11: Comparison between different coupling schemes for Model C5 213

Figure 6.12: Error reduction for Model A1 230






Figure 6.18: Error reduction of different schemes (Time=2.74s) for Model C1 236





List of Figures

16

Figure 6.23: Error reduction of different scheme (Time=2.74s) for Model C1 238

Figure 6.24: Coupled soil-structure interaction problem 239

Figure 6.25: Error reduction for different coupling schemes 240



Figure 6.28: Plane frame resting on soil 249

Figure 6.29: convergence behaviour over full range of response 251

Figure 6.30: Error reduction in the first load step 251

Figure 6.31: Error reduction in the 5th load step 252

Figure 6.32: Error reduction in the 6th load step 252

Figure 7.1: Pitched-roof steel frame resting on soil 256

Figure 7.2: Vertical displacement of the footing A 258

Figure 7.3: Vertical displacement of the footing E 258

Figure 7.4: Horizontal displacement of node B 259

Figure 7.5: Horizontal displacement of node D 259

Figure 7.6: Horizontal displacement of node C 260

Figure 7.7: Vertical displacement of node C 260

Figure 7.8: Variation of moment at node E 261

Figure 7.9: Variation of moment at node A 261

Figure 7.10: Variation of moment at node C 262

Figure 7.11: Deformed shape (scale=5.0) and bending moment (kN-m) in final load step for a) non-interactive case b) interactive case 263

Figure 7.12: Contours of stress level in soil sub-domain (at final increment) 263

Figure 7.13: Multi-Storey five-bay steel frame 265

Figure 7.14 : vertical settlement of Column C2 268

List of Figures

17

Figure 7.15: Vertical displacement profile of the soil surface 269

Figure 7.16: Displacement vectors in soil partitioned sub-domain 270

Figure 7.17: Contours of stress level in soil partitioned sub-domain 270

Figure 7.18: Bending moment (kN-m) at the base of C1 for different load-steps 271

Figure 7.19: Bending moment at the middle of beam B1 for different load-steps 271

Figure 7.20: Deformed shape (scale=5.0) and bending moment (kN-m) in final load step for a) linear Winkler foundation b) nonlinear partitioned analysis 272

Figure 7.21: Variation of bending moment along beam B1 273

Figure 7.22: Variation of bending moment along Column C1 273

Figure 7.23: Variation of vertical displacement along the beam B1 274

Figure 7.24: Plane frame resting on soil subject to ground excavation 275

Figure 7.25: Plan view of considered building 276

Figure 7.26: Geometric configuration of considered frame 276

Figure 7.27: Vertical settlement of the left footing for different load cases 279

Figure 7.28: Vertical settlement of the middle footing for different load cases 279

Figure 7.29: Vertical settlement of the right footing for different load cases 280

Figure 7.30: Cumulative vertical displacement of the ground surface 280

Figure 7.31: Cumulative horizontal displacement of excavation wall 281

Figure 7.32: Vertical displacement of ground surface for different excavation depths (Case 6) 282

Figure 7.33: Horizontal displacement of the excavation wall for different excavation depths (Case 6) 282

Figure 7.34: Vertical settlement of the left footing for different Le (Case 6) 283

Figure 7.35: Vertical settlement of the right footing for different Le (Case 6) 284

Figure 7.36: Vertical settlement of the right footing for different Le (Case 6) 284

Figure 7.37: Vectors of displacement in soil sub-domain in increment 6 (Case 6) 286

List of Figures

18

Figure 7.38: Contour plots of stress levels and plasticity induced in soil sub-domain in increment 6 (Case 6) 286

Figure 7.39: Vectors of displacement in soil sub-domain in increment 12 (Case 6) 287

Figure 7.40: Contour plots of stress levels and plasticity induced in soil sub-domain in increment 12 (Case 6) 287

Figure 7.41: Deformed shape (scale=5) and bending moment (kN-m) of structure for (a) 1st, (b) 6rd , (c) 7th and (d) 12th increment 288

Figure 7.42: Schematic diagram of the cantilever retaining wall 290

Figure 7.43: Vectors of accumulated displacements in soil sub-domain (final increment) 292

Figure 7.44: Deflection (scale: 5) and bending moment (kN-m/m) distribution of the retaining wall for (a) 1st, (b) 4th, (c) 7th, (d) 8th and (e) 9th increment 292

Figure 7.45: Schematic diagram of the propped retaining wall 293

Figure 7.46: Vectors of accumulated displacements in soil sub-domain (final increment) 294

Figure 7.47: Deflection (scale: 5) and bending moment (kN-m/m) distribution of the retaining wall for (a) 1st, (b) 2nd , (c) 3rd , (d) 4th and (e) 5th increment 295

Figure 7.48: Horizontal reaction at the lateral support 295

19

Notation

All symbols used in this thesis are defined where they first appear. The reader is

cautioned that some symbols denote more than one quantity; in such cases the

meaning should be clear when read in the context.

- Symbols of matrices and vectors are represented by [ ] and operators

respectively, with right hand side superscript and subscript (e.g. CBK , i

XU ).

- Equations are identified by numbers located on the right-most margin and

composed of two entries with the first entry indicating the Chapter in which the

equations appear.

A summary of operators and symbols is given in the following:

ΩT : partitioned structure sub-domain

ΩB : partitioned soil sub-domain

B : right-side subscript, denotes the partitioned sub-domain ΩB

T : right-side subscript, denotes the partitioned sub-domain ΩT

T : right-side superscript, denotes the transpose sign

I : right-side superscript, denotes the iteration number

n : right-side subscript, denotes the time/load increment number

[ ] : encloses terms of a matrix

: encloses terms of a vector

a : encloses absolute value of a

a : encloses second norm of a

I : identity matrix

Notation

20

XXU : displacement vector of the non-interface degrees of freedom in sub-

domain ΩX

XXF : force vector of the non-interface degrees of freedom in sub-domain ΩX

iXU : displacement vector of the interface degrees of freedom in ΩX

iXF : force vector of the interface degrees of freedom in sub-domain ΩX

CTK : condensed tangent stiffness matrices at the interface of the structure

sub-domain

CBK : condensed tangent stiffness matrices at the interface of the soil sub-

domain

Chapter 1

Introduction

1.1 Soil-Structure Interaction

There are numerous problems in Civil Engineering construction that require the use

of realistic models for the structure, the supporting soil and the soil-structure

interface. Examples include the assessment of various structures under earthquake

loading, the analysis of offshore jackup structures under extreme wave loading and

the evaluation of the structural damage due to excavations to name but a few. This

interaction can sometimes modify the stresses and deflections of the whole structural

system significantly. In fact, the structure with its loading conditions imposes

stresses and forces on the ground, which in turn deforms and as a consequence

transmits back additional forces and deformation to the structure. This process

continues until full equilibrium of the whole soil-structure system is satisfied, or

until both the soil and the structure fail in the case of excessive loading and

deformations of the system.

Numerical analysis, typically using the finite element or finite difference method, is

currently the most advanced tool available to facilitate soil and structure analysis.

Chapter 1 Introduction

22

With the development of these numerical tools, there are advanced techniques

employed for addressing the behaviour of structure and soil. Although there are

common techniques for structural and geotechnical modelling, the distinct demands

of the two fields meant that modelling has evolved differently in each domain, thus

leading to two modelling disciplines with distinctive high-level features.

In structural analysis, nonlinear modelling has evolved to address relevant issues

including i) the variety in structural form (e.g. frame, shell, membrane, whole

building), ii) the influence of geometric nonlinearity (e.g. large displacements/

rotations, buckling), iii) the nonlinear constitutive response of structural materials

(e.g. steel, concrete, composites) under serviceability and extreme loading conditions

(e.g. cracking, creep, plasticity, elevated temperature, high strain-rate). In

geotechnical analysis, on the other hand, developments have focused on i) the

constitutive modelling of different soils including pertinent nonlinear phenomena

(e.g. highly nonlinear elasto-plastic soil behaviour), ii) coupling of

mechanical/hydraulic/thermal/chemical processes in soils, iii) modelling of special

boundary conditions (e.g. excavation, construction, pore fluid pressure, non-

reflective boundary), iv) time dependent process such as consolidation and creep,

and v) problem reduction (e.g. Fourier series aided elements).

The above distinct challenges in the two fields have been reflected in the

development of discipline-oriented computational tools, which offer sophisticated

nonlinear modelling for their respective domain but, at best, a crude approximation

of the other, leading to poor representation of soil-structure interaction. This usually

means that structural analysis simplifies soil behaviour, while geotechnical analysis

simplifies structural behaviour using field elimination techniques. It is therefore a

real challenge to achieve the same amount of sophistication in modelling both the

soil and the structure in a single soil-structure interaction analysis.

In this respect, existing advanced discipline-oriented computational tools are

inadequate, on their own, for modelling a soil-structure interaction problem that

involves considerable nonlinearity in both the structure and the soil; rather such a


23

problem demands an integrated interdisciplinary computational model combining the

features of both structural and geotechnical modelling.

1.2 Coupled Systems

Here, the term system is identified as a set or assemblage of interacting or

independent entities, real or abstract, forming an integrated whole. The term

“system” is mainly used in this thesis for “physical systems” and in particular those

of importance in civil engineering, specifically soil-structure interaction problems.

Frequently there are number of functionally distinct physical systems interacting

with each other, where finding a solution of any one system is completely impossible

without considering the interaction with the other systems. Such systems are

categorized as “Coupled Systems” (Zienkiewicz & Taylor, 1991).

In general, coupled systems are analyzed by breakdown of the considered system

(domain) into several subsystems (sub-domains), where each sub-domain is

modelled according to its specific characteristics. This breakdown can be dictated

by: a) physical characteristics, b) functional characteristics and c) computational

characteristics (Fellippa et al., 2001).

One of the generic classes of coupled systems in the field of Civil Engineering

relates to the modelling of soil-structure interaction problems, where due to the

interaction between the physically distinct sub-domains, soil and structure, at the

interface, neither domain can be solved separately from the other without undue

simplifications, especially in the nonlinear range of response.

The multi-physics nature of soil-structure interaction modelling requires choosing

among a wide range of models, algorithms, and implementations to deal with

modelling of the soil and structure sub-domains, as well as implementing a coupling

procedure which can vary significantly in terms of the employed techniques in


24

treatment of the interaction effects at the interface of the coupled soil-structure

system.

In the following, general procedures available for coupled modelling of soil-structure

interaction are presented, considering that models and algorithms already exist for

nonlinear analysis of soil and structure on their own.

1.3 Treatment of Coupled Soil-Structure Interaction

In principle there are three possible approaches for dealing with soil-structure

interaction problems, namely: i) field elimination, ii) direct/monolithic/

simultaneous, and iii) domain decomposition/partitioned analysis (Rugonyi & Bathe,

2001).

In the field elimination treatment, one or more sub-domains of the coupled problem

are eliminated using a simple reduction technique, and the remaining sub-domain(s)

are considered under appropriate boundary conditions representing the eliminated

sub-domain(s). An example of implementing the field elimination technique in

modelling soil-structure interaction problems is illustrated in Figure 1.1, where the

soil sub-domain is replaced by spring type boundary conditions in order to model the

interaction effects.

Field elimination techniques are mostly restricted to simple problems that permit

efficient decoupling. In the context of nonlinear soil-structure interaction problems,

the simplification made in the modelling brings with it a loss of generality and

accuracy, due to the complexity of numerical modelling of this phenomenon.


25

Figure 1.1: Field elimination treatment of soil-structure interaction

In contrast with the field elimination treatment, the monolithic and partitioning

treatments as discussed in the following are general in nature. In the monolithic

treatment of soil-structure interaction, the whole problem is modelled as a single

computational entity and the solution of the complete system of equations of the

coupled problem is attenuated in one analysis scheme. Alternatively, in partitioned

analysis as shown in Figure 1.2, the soil-structure coupled system is physically

partitioned into soil and structure sub-domains. These partitioned sub-domains are

then computationally treated as isolated entities, and the response of each sub-

domain is calculated using already developed soil and structural solvers. In this

procedure, the interaction effects are viewed as force/displacement effects at the

interface of soil-structure system which are communicated between the individually

modelled soil and structure sub-domains using prediction, substitution and

synchronization techniques.


26

Figure 1.2: Partitioned treatment of soil-structure interaction

1.4 Partitioned vs. Monolithic Approaches


using a monolithic treatment (as is likely with developing new software and solution

methods for different soil-structure interaction applications), a partitioned treatment

with different partitioned sub-domains modelled as separate computational entities,

amongst which interaction effects are exchanged, offers major benefits in the context

of nonlinear soil-structure interaction. Such benefits include (Felippa et al., 2001;

Rugonyi & Bathe, 2001; Lai, 1994; Hagen & Estorff, 2005):

Allowing field-specific discretisation and solution procedures that have

proven performance for each partitioned domain. In this respect, the

partitioned soil and structure sub-domains can be modelled and solved to the

desired degree of sophistication and accuracy with techniques and algorithms

that are known to perform well for each individual sub-domain.

Facilitating the reuse of existing nonlinear discipline oriented soil and

structural analysis solvers (private, public or commercial), with all the

resource savings that this brings.


27

Project breakdown advantages in the analysis of complex systems.

Suitable for research environments, where software development is more

likely to be cyclical. In other words, the existing discipline-oriented

programs developed for individual soil and structural sub-domains, can be

reused and developed further, while all these new developments could be

easily incorporated in soil-structure interaction analysis.

Facilitating parallel computing thorough problem partitioning.

However, despite the significant potential benefits of the partitioned approach, this

method should be formulated and implemented with great care, since the stability

and convergence characteristics of coupling algorithms are typically conditional and

problem dependent. Moreover, gain in computational efficiency using the partitioned

approach over monolithic treatment is not guaranteed.

1.5 Aims and Scope of Research

This work aims at developing an advanced generic numerical method for nonlinear

coupling of static and dynamic soil-structure interaction problems. In this respect,

this work is primarily motivated by the lack of sophisticated monolithic tools for

modelling nonlinear soil–structure interaction. Recognising the existence of

advanced tools for nonlinear analysis of structure and soil, in isolation, the

partitioned approach is adopted as a framework for coupling field-specific tools with

minimal intrusion. Accordingly, in adopting the partitioned treatment, the focus is on

providing an advanced capability for modelling nonlinear soil–structure interaction

with existing field-specific codes rather than on achieving superior computational

performance in comparison with the monolithic treatment.

Due to the aforementioned advantages of the partitioned approach in soil-structure

interaction analysis (especially large-scale nonlinear problems), this method is fully

investigated and developed. It is therefore the aim of this thesis to develop and

investigate the mathematical and computational characteristics of various domain


28

decomposition methods in the context of nonlinear soil-structure interaction analysis,

where particular emphasis is given to utilizing the finite element method in

discretising the partitioned sub-domains.

In this regard, novel formulations for coupling soil-structure systems, based on

relaxation coupling methods and also utilizing the tangent stiffness matrix of the

partitioned sub-domains at the interface, are proposed. The proposed approaches are

believed to possess superior convergence characteristics in comparison with existing

coupling methods, rendering these methods more general procedures for modelling

real soil-structure interaction problems using coupled field-specific tools.

Based on the presented coupling algorithms a novel simulation environment,

utilising discipline-oriented solvers for nonlinear structural and geotechnical

analysis, is developed. This is carried out through coupling of two powerful in-house

nonlinear analysis programs developed at Imperial College, ADAPTIC (Izzuddin,

1991) and ICFEP (Potts & Zdravkovic, 1999). Each of the programs has been

developed for sophisticated analysis of nonlinear structural and soil behaviour,

respectively, and has been used both in research and consulting. Although the

developed coupling method will be applied to the coupling of ADAPTIC and ICFEP,

a higher level objective is for this method to be generally applicable to the coupling

of other existing nonlinear soil and structural software. As a result, a generic

numerical tool for coupling nonlinear finite element systems, focusing on nonlinear

soil-structure interaction problems, is developed.

The developed simulation environment is subsequently used to demonstrate the

performance characteristics and merits of the various presented and proposed

algorithms, with particular reference to nonlinear-soil structure interaction problems.

The main focus of the initial numerical studies is on the criteria for algorithmic

stability, accuracy and convergence as well as computational costs, where some of

the presented techniques are investigated first for linear analysis prior to their

exclusion/adoption/generalisation for nonlinear analysis.


29

Finally, the developed tool is used in a number of case studies involving nonlinear

soil-structure interaction in which nonlinearity arises in both the structure and the

soil, thus leading to important conclusions regarding the adequacy and applicability

of the proposed coupling approaches for such problems, as well as the prospects for

further enhancements.

1.6 Layout of Thesis

Chapter 1 (Introduction): This chapter describes the general background and

provides the basic motivation behind investigation and development of the

partitioned treatment of soil-structure interaction problems in this thesis.

Chapter 2 (Literature Review): In this chapter an extensive critical review of

various available modelling approaches, applicable to soil-structure interaction

analysis, is presented. The advantages and disadvantages of different methods are

discussed. Moreover, a comprehensive overview of current coupling algorithms is

provided. Staggered and iterative coupling algorithms and their respective origins are

introduced. The shortcomings and benefits of current iterative coupling techniques

such as interface relaxation and reduced order method are identified. The need for

further enhancement and development of iterative coupling algorithms is established

in the context of soil-structure interaction.

Chapter 3 (Staggered Approach): The applicability of the staggered approach for

coupling dynamic FEM-FEM problems is investigated, with particular focus on soil-

structure interaction. The mathematical procedure for evaluating the stability of the

staggered approach is presented. Stability is conditional for this approach, where the

convergence characteristics may be enhanced by employing modifications in the

formulation of the partitioned sub-domains or change in the governing equations by

augmentations. It is concluded that stability and accuracy requirements of the

staggered approach typically demand excessively small time steps for the types of

problem under consideration, rendering this scheme computationally prohibitive.


30

Chapter 4 (Iterative Coupling): In this chapter, different iterative coupling

methods in partitioned analysis of soil-structure interaction problems are presented.

It is assumed that the overall domain is divided into physical partitions consisting of

soil and structure sub-domains, and the coupling of separately modelled sub-domains

is undertaken via the presented iterative algorithms. It is established that an

important feature of the proposed coupling approaches that needs to be addressed

comprehensively is the convergence behaviour of the scheme, which is directly

dictated by the chosen update technique during successive iterations. The developed

coupling simulation environment, utilising discipline-oriented solvers for nonlinear

structural and geotechnical analysis, is explained in detail. The software architecture

of the developed simulation environment, which is based on Sequential Dirichlet-

Neumann iterative coupling algorithms, is described and the structure of the data

exchange between the field-specific codes is elaborated.

Chapter 5 (Interface Relaxation): The use of interface relaxation coupling

techniques in domain decomposition analysis of soil-structure interaction is

investigated in this chapter. The general mathematical convergence characteristics of

constant and adaptive interface relaxation schemes are established. The applicability

and merits of the presented coupling techniques are highlighted trough various case

studies. It is shown that the adaptive scheme improves the convergence

characteristics in both linear and nonlinear analysis significantly, though the need for

further enhancements is also established.

Chapter 6 (Reduced Order Method): In this chapter, various domain

decomposition methods for nonlinear analysis of soil-structure interaction problems

based on approximating the condensed interface stiffness matrix are proposed. It is

shown that by using the condensed tangent interface stiffness matrices of the

partitioned sub-domains in the update of boundary conditions, superior convergence

characteristics could be achieved. This brings the performance of the proposed

coupling approach close to the monolithic treatment. It is proposed that suitable

approximations for the condensed stiffness matrices can be constructed during

successive coupling iterations to avoid explicit assemblage of the stiffness matrices.


31

In this respect, various novel reduced order method coupling algorithms are

developed and their performance is examined. The applicability of the presented

coupling techniques is demonstrated for nonlinear soil-structure interaction analysis

via an example investigating the interactive behaviour of a plane frame and a

supporting soil system.

Chapter 7 (Case Studies): The applicability of the partitioned treatment and its

great potential towards performing realistic nonlinear soil-structure interaction

analysis are illustrated through various case studies in this chapter.

Chapter 8 (Conclusion): In this chapter, detailed conclusions on the applicability

and efficiency of the developed coupling approaches as well as the benefits of the

developed soil-structure interaction approach are drawn. Several issues are also

identified for future research.

Chapter 2

Literature Review

2.1 Background

Soil-structure interaction problems relate to two-way interactive coupled systems,

where the state of structural deformations and stresses depend on the earth pressures

and movements, while simultaneously the earth pressures depend on the loading and

deformations of the structure. Accordingly, analysing such problems requires

simultaneous modelling of the structure and retaining/retained soil, using reasonably

accurate and computationally efficient techniques. Examples of such problems

include the assessment of various structures under static and dynamic loading,

offshore jackup structures under extreme wave loading, structures subject to ground

movement or excavations, reinforced slopes and embankments, retaining walls, and

tunnels to name but a few.

For example, consider the frame of Figure 2.1 supported on soil. As the loading on

the frame is symmetric, it is clear that there is a higher concentration of load over the

middle column and its support. This results in the soil below the middle footing

sustaining greater vertical settlements than the adjacent footings. In turn, framing

Chapter 2 Literature Review

33

action causes redistribution of the loads to the end columns due to the generated

differential settlements. These differential settlements among various parts of the

structure alter both the axial forces and the moments in the structural members

considerably, where the amount of redistribution of loads depends upon the rigidity

of the structure and the load-settlement characteristics of soil (Dutta & Roy, 2002).

This considerable influence of the structural rigidity on the interactive response has

been qualitatively explained in the literature long back (Taylor, 1964).

Figure 2.1: Illustrative soil-structure interaction (SSI) problem

Indeed, such interaction between structure and soil at their common interface leads

to an actual response for both physical sub-domains, which is considerably different

from what is obtained for each sub-domain in isolation. In view of the

interdependence between the responses of the two sub-domains, the frame structure,

its foundations and the soil on which it rests constitute together a coupled system.

The effect of soil–structure interaction under both static and dynamic loading has

attracted significant research interest over many years due to its important role in the

analysis of real civil engineering problems. In this respect, research by

Noorzaei et al. (1993), Allam et al. (1991), Roy & Dutta (2001) and Potts &

Zdravkovic (2001) on static soil-structure interaction analysis, and also research by


34

Stewart et al. (1999) and Dutta et al. (2004) on dynamic soil-structure interaction

analysis, demonstrates the significant effect of soil-structure interaction and its

importance in predicting the overall coupled response. There are also numerous

reported case histories in the literature which demonstrate the significant influence of

considering soil-structure interaction effects in civil engineering design and practice,

such as the destruction of buildings in Carcas earthquake in 1967 (Dowrick, 1977),

the enormous damage to structures in the Hanshin and Awaji areas during Kobe

earthquake in 1999 (Inaba et al., 2000), the case of leaning tower of Pisa (Burland &

Potts, 1994), and the reported damage to several building structures due to

differential settlements (Charles & Skinner, 2004). All of these researches deliver an

important message that considering soil-structure interaction effects can be essential

for a multitude of real civil engineering construction problems.

The primary expected results from a coupled soil-structure interaction (SSI) analysis

are the stresses and displacements of the structure and the soil sub-domains. SSI

analysis tools can be used in design to calculate the stresses and deformations of the

structure and soil sub-domain to compare with the allowable values and, if

necessary, to modify the system configuration so as to meet specific design criteria

of serviceability, safety and economy. In addition, SSI tools can be used for the

assessment of existing coupled systems, enabling the prediction of potential damage

and the proposal of strengthening and repair measures.

2.2 Modelling of Soil-Structure Interaction

Towards this end, modelling the material behaviour of both structure and soil,

modelling the geometric nonlinearity of structure under extreme events, modelling of

special boundary conditions such as excavation in soil, etc., are amongst the most

central issues that should be considered in a reliable SSI analysis.

Works by Jardine et al. (1986), Wrana (1993), Noorzaei et al. (1993), Noorzaei et al.

(1995a), Inaba et al. (2000), Estorff & Firuzian (2000), Krabbenhoft et al. (2005)

and Bourne-Webb et al. (2007), concentrating on the nonlinear modelling of soil-


35

structure interaction problems, have emphasised the importance of employing

accurate material modelling in nonlinear SSI analysis. Indeed, completely

misleading predictions may be obtained, unless the interactive study of the soil–

structure is conducted by considering accurate models of both soil and structure for

the problem under consideration.

There exist various models, algorithms and implementations for modelling SSI

problems. These all depend on the availability and limitations of computational tools

to carry out the SSI analysis, the physical effects that are to be captured, the desired

degree of accuracy, etc. In the following some of these modelling techniques are

presented and their advantages and disadvantages are briefly discussed.

2.2.1 Field Elimination

For a long time, field elimination techniques have been favoured over fully coupled

analysis of soil-structure interaction problems, largely due to their computational

simplicity. In this respect, numerous idealisation and elimination techniques

representing either soil or structure, depending on their relative significance for the

problem under consideration, have been employed and evaluated. In most cases,

however, the lack of accuracy of field elimination approaches is apparent.

Figure 2.2: Schematic diagram of a Winkler foundation

One of the most common field elimination techniques for idealization of the soil sub-

domain is Winkler model (Winkler, 1867), where the soil medium is removed and

the soil-structure interface is modelled by a system of independent and closely

spaced linear springs as shown in Figure 2.2. Clearly, in this type of modelling the


36

deformation at the interface is only confined to the loaded region, which is simply

incorrect.

Numerous studies for SSI analysis have been carried out using the Winkler

hypothesis, including studies by Terzaghi (1955), Vesic (1961), Brown et al. (1977)

and Bowles (1996). Despite the simplicity and low computational cost of the

Winkler idealization, the fundamental problem is the determination of the stiffness of

the associated elastic springs replacing the soil sub-domain. As a coupled problem,

the value of the sub-grade reaction is not only dependent on the sub-grade but also

on the parameters of the loaded area as well. However, the sub-grade reaction is the

only parameter in Winkler idealization, thus great care is required in determination

of the sub-grade parameter (using plate load test, consolidation test, triaxial test or

CBR tests (Dutta & Roy, 2002)).

As expected with field elimination techniques, there are serious limitations with

Winkler type idealizations. The major obvious pitfall of such an approach is that it

provides no or very little information regarding the stress and deformation state

within the soil mass. Dutta and Roy (2002) summarises some other primary

limitations of such an idealization:

‘…the basic limitation of Winkler hypothesis lies in the fact that this model

cannot account for the dispersion of the load over a gradually increasing influence

area with increase in depth. Moreover, it considers linear stress–strain behaviour of

soil. The most serious demerit of Winkler model is the one pertaining to the

independence of the springs. So the effect of the externally applied load gets

localized to the sub-grade only to the point of its application. This implies no

cohesive bond exists among the particles comprising soil medium.’ (Dutta & Roy,

2002: p.1582)

Beside the Winklerian approach, there is also a conceptual approach for physical

idealization of the infinite soil sub-domain using the theory of continuum mechanics

(Harr, 1966). The continuum idealization of the soil sub-domain, originally

introduced by Odhe and the research work of Boussinesq (Bowles, 1996), is used to


37

analyse the problem of a semi-infinite, homogeneous, isotropic, linear elastic solid

subjected to a concentrated force acting normal to the plane boundary, using the

theory of elasticity. Although in the continuum idealization, the soil sub-domain is

typically considered as a semi-infinite and isotropic medium, the effect of soil

layering and anisotropy may be conveniently accounted for in the analysis (Carrier &

Christian, 1973; Stavridis, 2002).

Such approaches in principle overcome the inadequacy of the Winkler spring model

to transmit lateral shear stresses, where a load acting on the soil surface cannot

produce any settlement except over its specific loading area. In addition, compared

to the Winklerian approach, the continuum approaches provide much more

information on the stress and deformation state within the soil mass. Furthermore,

these approaches are more realistic as they are based on explicit data of geotechnical

investigations, in contrast to the Winkler method, where the sub-grade reaction

modulus does not represent a soil property.

Nevertheless, there are major drawbacks associated with the elastic continuum

approach such as its inaccuracy in reactions calculated at the edges of the foundation.

Furthermore, although solutions for practical problems idealizing the soil sub-

domain as elastic continuum are available for some limited cases, this more

sophisticated method has never gained popularity among the designers, mainly

because of it leading to various mathematical intricacies and its inherent inability for

a direct analytical implementation for design purposes. Adding to these, the inability

of evaluating complex material behaviours, such as the nonlinear and elasto plastic

behaviour of soil, severely limits the application of this model in practice (Dutta &

Roy, 2002).

The mathematically and computationally attractive but physically inadequate

Winkler hypothesis has attracted several attempts over time to develop modified

models to overcome its shortcomings. Amongst many are Hetenyi’s foundation

(Hetenyi, 1946), Pasternak foundation (Wang et al., 2001), where the continuity in

the soil medium is modelled by introducing some sort of structural elements, and


38

that of Kurian et al. (2001), where the springs are intermeshed so that the

interconnection is automatically achieved. Similarly, different types of continuum

models for foundation modelling have been proposed (Harr et al., 1969; Nogami &

Lain, 1987; Vallabhan & Das, 1991). There are also models that take into account

the elasto-plastic behaviour (Zeevrat, 1972) of foundation soil as illustrated in Figure

2.3a or the visco-elastic behaviour (Noda et al., 2000) of soil as presented by Figure

2.3b. A brief summary of such foundation models could be found in the review paper

of Dutta & Roy (2002).

Figure 2.3: Schematic view of a) elasto-plastic b) visco-elastic foundation

Although conceptually the abovementioned foundation models are useful, little

evidence has been produced to verify the computational accuracy of the various

models in studies representing the soil medium in soil–structure interaction analysis.

Moreover, problems typically occur depending on the choice of the various

parameters as well as the proper adjustment of foundation elements. On the other

hand, field elimination techniques idealising the system more rigorously with fewer

parameters deviate more in predicting the overall response. Most importantly, such

approaches still provide no or very little information regarding the stress and

deformation state within the soil mass.

The search for physically close and mathematically simple models in soil-structure

interaction problems has not been limited to field elimination techniques

representing the soil sub-domain. There are several field elimination techniques

which tend to simplify the structural sub-domain both computationally and


39

mathematically. The work of Standing et al. (1998) falls into this category, where the

movements of the Treasury building in London arising from the construction of twin

tunnels are studied using a finite element model for the soil, while the building

structure is simply modelled as an equivalent raft. Also in the work by Dutta et al.

(2004) on dynamic soil-structure interaction modelling of elevated tanks, as shown

in Figure 2.4, it can be seen that both the structure and soil sub-domains are

simplified.

Figure 2.4: Field elimination model for SSI analysis of lateral vibration of an

elevated water tank (Dutta et al., 2004: p.828)

Field elimination techniques cannot deal accurately with geometric and material

nonlinearity in the replaced sub-domain, hence modelling the nonlinear response of

both soil and structure becomes complex for which more sophisticated modelling

approaches would be required. With the increasing availability of powerful

computers and the wider applicability of numerical methods compared to analytical

approaches, the use of the finite element method has become a common means for

modelling such complex interactive behaviour.


40

2.2.2 Finite Element Method (FEM)

The finite element method is a special form of matrix analysis, where the whole

continuum is discretized into a finite number of elements connected at different

nodal points. The general principles and use of the finite element method are well

documented (e.g. Desai & Abel, 1987; Zienkiewicz et al., 2005).

In structural analysis, nonlinear modelling using finite element analysis has evolved

to address complex issues of various structural forms (Izzuddin, 1991; Zienkiewicz

et al., 2005; Zienkiewicz & Taylor, 2005). In fact, due to the universal nature of the

finite element method in modelling real-life complex conditions including geometric

and material nonlinearity in the response (Izzuddin, 1991), this numerical technique

has been widely used in design and assessment of complex structures (Bull, 1988;

Smith & Coull, 1991; Rombach, 2004).

On the other hand, utilising numerical methods in geotechnical engineering has also

provided geotechnical engineers with an extremely powerful analysis and design tool

(Potts & Zdravkovic, 1999; Potts & Zdravkovic, 2001; Potts, 2003). According to

Potts & Zdravkovic (1999), the numerical methods in modelling the soil medium are

far superior to conventional analytical methods, which tend to relax one or more of

the basic solution requirements (see Table 2.1).


41

Method of Analysis

Solution Requirements

Equilibrium Compatibility Constitutive Behaviour

Boundary Conditions Force Displacement

Closed Form Limit equilibrium Stress Field

S. S. S.

S. N.S. N.S.

Linear elastic Rigid with Failure Criterion

S. S. S.

S. N.S. N.S.

Limit Analysis

Lower Bound Upper Bound

S. N.S.

N.S. S.

Ideal Plasticity with Associated Flow Rule

S. N.S.

N.S. S.

Full Numerical Analysis

S. S. Any S. S.

S. = Satisfied ; N.S. = Not Satisfied

Table 2.1: Basic solution requirements satisfied by various methods of geotechnical

analysis (Potts & Zdravkovic, 1999)

Although the application of numerical methods in geotechnical design is not as

widespread as in structural design, recent work by Gaba et al. (2002) and Ravaska

(2002) has demonstrated that the use of numerical analysis, as opposed to other

conventional methods, can lead to more accurate and economical design. As a result,

the use of this type of analysis in design applications is bound to increase in the

future, where a single simulation can provide all required information for design

purposes (see Table 2.2).

Method of Analysis Design requirements Stability Movements Adjacent structures

Closed form (linear-elastic) Limit equilibrium

Stress field

No Yes Yes

Yes No No

Yes No No

Limit analysis Lower bound Upper bound

Yes Yes

No Crude estimate

No No

Full numerical analysis Yes Yes Yes

Table 2.2: Design requirements satisfied by various methods of geotechnical analysis

(Potts and Zdravkovic, 1999)


42

2.2.3 Integrated Modelling

In view of the above, the finite element method currently stands as the most

powerful and versatile tool for solving complex soil-structure interaction problems.

A finite element procedure for the general problem of soil-structure interaction

involving nonlinearities due to material behaviour, geometric changes and interface

behaviour is presented by Desai et al. (1982). A three-dimensional visco-elastic

finite element formulation has also been proposed by Viladkar et al. (1993) for

studying the interactive behaviour of a space frame with the supporting soil sub-

domain. The interactive behaviour of a plane frame-footing-soil system with elastic-

perfectly plastic soil has been investigated by Viladkar et al. (1991) and Noorzaei et

al. (1995a). The influence of soil strain-hardening on the elasto-plastic soil-structure

interaction of framed structures has also been undertaken by Noorzaei et al. (1995b).

Several studies have been carried out to model the discontinuous behaviour that may

occur at the interface of soil and structure, including the work of Beer (1985) and

Viladkar et al. (1994) who developed interface elements to model this discontinuity.

All of the above mentioned research has two major conceptual characteristics in

common. Firstly, the finite element method is used as a tool to obtain the complex

response of soil-structure interaction, and secondly the soil and structure sub-domain

are modelled simultaneously in a single computational model.

Although numerical analysis, and particularly finite element analysis, is currently the

most advanced approach for modelling soil-structure interaction, the application of

finite element modelling to soil-structure interaction problems is often limited by the

availability of a simulation environment offering advanced modelling capabilities for

both soil and structure sub-domains. Despite the existence of common techniques for

nonlinear structural and geotechnical analysis, the distinct demands of the two fields

in terms of material modelling and solution procedures have led to differently

evolved modelling techniques in each field. In fact, this has led to development of

two modelling disciplines with distinctive high-level features. In turn, this has been

reflected in the development of discipline-oriented computational tools, which offer


43

sophisticated nonlinear modelling for their respective sub-domain (structure/soil)

and, if available, an approximation of the other (soil/structure).

This usually means that structural analysis simplifies soil behaviour, while

geotechnical analysis simplifies structural behaviour using field elimination

techniques. For example, in the work of Vlahos et al. (2006) on soil-structure

interaction of spudcan footings on clay soil subjected to cyclic loading, the structural

model is connected to a hybrid footing model (developed based on available

experimental data) representing the soil medium, as illustrated in Figure 2.5.

Figure 2.5: Idealization of jack-up models for SSI (Vlahos et al., 2006: p.217)

Similarly in the work by Jin et al. (2005) on evaluation of damage to offshore

platform structures, the structure sub-domain is modelled in 3D using FEM, while

the effect of the soil reactions on each pile element is simplified into three kinds of

non-linear springs, as illustrated in Figure 2.6. Further similarity of approach can be

seen in the work of Dutta et al. (2004) on soil-structure interaction analysis of low

rise buildings under seismic ground excitation (Figure 2.7) and that of Bhattacharya


44

et al. (2004) on the assessment of the influence of soil flexibility on the dynamic

behaviour of building frames (Figure 2.8).

Figure 2.6: a) Schematic illustration of the pile and soil model b) Schematic diagram

of the structure model (Jin et al., 2005: p.1324)

Figure 2.7: a) Idealisation of the structure sub-domain b) Idealisation of the

foundation system (Dutta et al., 2004: p.897)


45

Figure 2.8: a) Soil-structure coupled system b) Idealization of the coupled system

(Bhattacharya et al., 2004: p.117)

On the other hand, Tian & Li (2008) undertook work on the dynamic response of

building structures subject to ground shock from a tunnel explosion, where the multi-

story building structure is simplified and represented by a shear lumped mass model

as depicted in Figure 2.9.

Figure 2.9: Idealization of the coupled soil-structure interaction system (Tian & Li,

2008: p.1171)

In summary, with the current discipline-oriented structural and soil solvers, it is a

real challenge to achieve the same amount of sophistication in numerical modelling

of both the soil and the structure in a single soil-structure interaction analysis. There

are different ways to tackle this issue. One possibility is to develop new software or

augment existing software such that both soil and structure may be modelled to an


46

equivalent level of sophistication. Such an approach is referred to as the monolithic

or direct treatment. Example applications of the monolithic approach in modelling

soil-structure interaction are the work of Viladkar et al. (2006) on static soil-

structure interaction response of hyperbolic cooling towers to symmetrical wind

loads, and the work of Noorzaei et al. (2006) on nonlinear interactive analysis of

cooling tower–foundation–soil interaction under unsymmetrical wind load using the

finite element method (Figure 2.10).

As discussed in Chapter 1, the monolithic treatment has the disadvantage that it

requires massive software development resources and it does not allow for

employing models, new techniques and already developed softwares in a modular

fashion. These disadvantages can be addressed by adopting a partitioned treatment,

as reviewed in the following section.

Figure 2.10: Monolithic treatment of soil-structure interaction problem (Noorzaei et

al., 2006: p.1001)

2.2.4 Partitioned Analysis

In the partitioned treatment, the partitioned sub-domains of the coupled mechanical

system are computationally treated as isolated entities, and the response of the


47

coupled system is calculated using already developed solvers. A partitioned

treatment with different partitioned sub-domains modelled as separate computational

entities, amongst which interaction effects are exchanged (see Figure 2.11), can offer

major benefits in the context of nonlinear soil-structure interaction analysis.


Such benefits include i) allowing field-specific discretisation and solution procedures

that have proven performance for each partitioned sub-domain, ii) facilitating the

reuse of existing nonlinear analysis solvers with all the resource savings that this

brings, and iii) enabling parallel computations through problem partitioning (Lai,

1994; Felippa et al., 2001)

According to Felippa et al. (2001), the development and application of partitioned

analysis of coupled systems involving structures goes back to 1970. In their paper

(Felippa et al., 2001) the authors state:

‘The partitioned treatment of coupled systems involving structures emerged

independently in the mid 1970s at three locations: Northwestern University by T.

Belytschko and R. Mullen, Cal Tech University by T.J.R. Hughes and W.K. Liu, and

Lockheed Palo Alto research laboratories (LPARL) by J.A. Deruntz, C.A. Felippa,

T.L. Geers and K.C. Park’ (Fellipa et al, 2001: P. 3255).

The above research groups were focused on different applications and pursued

different types of problem-decomposition methodology. For Instance, Belytschko &

Mullen (1978) studied node by node partitioning and sub-cycling. Hughes & Liu


48

(1978) developed element by element implicit-explicit partitions. While the work of

these groups focused on structure-structure and fluid-structure interaction treated by

FEM discretisation, in LAPRL the above mentioned researchers concentrated on the

coupling of a finite element computational model of submerged structures to a

boundary element model of the exterior fluid (Geers & Felippa, 1980). A staggered

solution procedure was developed for the resulting fluid-structure interaction

problem by Fellippa & Park (1980) and was included in the general class of

partitioned methods (Park, 1980; Park & Felippa, 1980).

2.2.5 Staggered Approach

In general, the partitioned analysis is mainly carried out by using staggered or

iterative sub-structuring (Quarteroni & Valli, 1999) methods. The staggered

approach is suited to transient dynamic analysis only, where the governing equations

of the partitioned domains are solved independently at each time step using predicted

boundary conditions at the interface (force, displacement, velocity or acceleration)

obtained from previous time step(s) by a predictor.

Figure 2.12: Staggered coupling approach

The staggered coupling approach is illustrated in Figure 2.12 for a soil-structure

interaction problem involving two physically partitioned and independent sub-

domains (i.e. soil and structure). In this respect, sub-domain B stands for the soil,

while T represents the structure. The prediction stage typically stands for prediction

of displacements from the soil model at the soil-structure interface, while the


49

substitution stage typically stands for the substitution of the reaction forces from the

structural model into the soil model at the same interface.

This approach is approximate in nature due to the fact that the predicted interface

displacements, obtained using displacements, velocities and accelerations from

previous steps, are invariably different from the displacements evaluated following

force substitution into the soil model, thus violating the compatibility conditions at

the interface.

The application of the staggered approach to coupled mechanical systems has

attracted significant research interest over the past years (Piperno, 1997; Huang &

Zienkiewicz, 1998; Farhat & Lesoinne, 2000). Although the focus of these

researchers has been mainly on fluid-structure interaction, there are a few examples

of employing this technique in soil-structure interaction coupling. One such example

is the work of Rizos & Wang (2002), who developed a partitioned method for soil-

structure interaction analysis in the time domain based on the staggered approach

(Figure 2.13), where a standard Finite Element Method (FEM) model, representing

the structure domain, was coupled to a Boundary Element Method (BEM) model,

representing the soil domain as an elastic half-space. The advantages of this

staggered approach (Rizos & Wang, 2002) are that the derivation of a global system

matrix for the combined FE–BE domain is not required, special solution strategies

do not need to be developed, and smaller systems of simultaneous equations are

encountered.

The aforementioned advantages make the proposed approach appealing for the

solution of large coupled systems. Further to this, O’Brien & Rizos (2005)

considered the application of the staggered approach for the simulation of high speed

train induced vibrations. The Boundary Element Method is used to model the soil-tie

system, while Finite Element Method, along with Newmark’s integration, is used for

the modelling of the rail system. The two methods are coupled at the tie-rail interface

and the solution is obtained following a staggered, time marching scheme

(Figure 2.14).


50

Figure 2.13: Schematic representation of the staggered coupling approach (Rizos &

Wang, 2002: p.878)

Figure 2.14: Staggered coupling of BEM-FEM for soil-structure interaction

simulation of high speed train induced vibrations (O’brien & Rizos, 2005: p.290)

Besides the inapplicability of staggered approach to static analysis, there are major

issues in relation to the staggered approach that need to be addressed in coupled

dynamic analysis, namely stability and accuracy. A staggered solution procedure

should not degrade the numerical stability of any individual system nor the overall


51

coupled problem. The accuracy degradation of the solution procedure is also another

concern in a staggered coupling procedure. In general, for nonlinear mechanical

problems the stability and accuracy of a staggered solution procedure are intertwined

and should be studied concurrently.

Considerable research has been undertaken on the stability and accuracy of the

staggered approach for different coupled problems (Farhat & Park, 1991; Huang &

Zienkiewicz, 1998). For Instance, in the work of Rizos & Wang (2002) on staggered

coupling of soil-structure interaction, it is concluded that in the absence of elastic

(restoring) forces in the structure (e.g. rigid massive foundations on elastic soils), the

FEM solver in their staggered scheme becomes unstable; therefore, a stabilisation

should be augmented into the FEM equations with an equivalent stiffness of the soil

region under the foundation.

In this regard, as rightly discussed by Felippa et al. (2001), achieving stability in a

staggered solution procedure is extremely difficult and in many cases impossible

without reformulation (mainly by augmentation) of the field equations of the

original partitioned sub-domains. Furthermore, a general theory of stability of

discrete and semi-discrete nonlinear coupled systems is yet to be developed.

Although the stability of staggered treatment may be ensured by reformulation of the

partitioned sub-domains’ field equations, this would be a backward step in terms of

modular use of the existing nonlinear structural/soil discipline oriented software. In

fact, such reformulation requires access to the soil/structural solvers solution

procedure codes.

This shortcoming is discussed further in Chapter 3, where the staggered coupling of

linear dynamic FEM-FEM coupled problems is presented, and where it is shown that

without reformulation of the field equations only conditional stability may be

achieved. Indeed in such a scenario, both accuracy and stability will be dependent on

the time-step size, while the choice of the predictor and the partitioning strategy will

also have a significant effect on achieving conditional stability. However, achieving

conditional stability in staggered coupling of linear FEM-FEM problems requires


52

choosing small time-steps which makes such coupling procedures computationally

inefficient. As a result, with only conditional stability, the staggered approach should

be used with great care.

2.2.6 Iterative Coupling

To address the above shortcomings, coupling algorithms have been developed that

are stable and accurate for a wider range of time-step size, which is mainly achieved

by introducing corrective iterations into the staggered approach, hence the name

iterative sub-structuring/coupling methods (Quarteroni & Valli, 1999). In addition to

the enhancement of stability and accuracy, iterative coupling approaches facilitate

parallel computing through problem partitioning which can lead to much greater

computational efficiency. In addition, iterative coupling approaches can be applied to

both static and dynamic problems.

Figure 2.15: Schematic of sequential iterative coupling approach

The general procedure of iterative coupling is illustrated in Figure 2.15, which refers

to a soil-structure interaction coupled system decomposed into the soil and structure

sub-domains (T, B). The governing equations of the partitioned sub-domains are

solved independently at each load increment (or time step in the case of dynamic

analysis), using predicted boundary conditions (either force or displacement) at the

interface. These predicted boundary conditions are then successively updated using


53

corrective iterations until convergence to equilibrium and compatibility is achieved

at the interface and within the partitioned sub-domains.

Unlike the staggered approach, the compatibility and equilibrium conditions at the

interface of partitioned sub-domains are enforced in the iterative coupling approach,

provided convergence is achieved, thus the stability and accuracy concerns

associated with the staggered approach are no longer relevant.

Despite the significant potential benefits of iterative coupling, a major issue relates

to whether convergence to equilibrium and compatibility at the interface can always

be achieved through successive iterative substitutions. In this respect, the utilised

technique in successive update of boundary conditions at the interface of the

partitioned coupled system is the most critical algorithmic stage. In fact, it is the

update technique that dictates the convergence behaviour of the algorithm rather than

the time/load step size.

In this regard, if a trivial update of boundary conditions in FEM-FEM iterative

coupling is utilised, only a conditional convergence to compatibility and equilibrium

could be achieved. Due to this conditional convergence, a relaxation of the updated

boundary conditions is often augmented to the iterative coupling procedure in order

to improve the convergence characteristics, hence the term interface relaxation

(Marini & Quarteroni, 1989).

Interface relaxation is a traditional method in domain decomposition methods which

follows Southwell’s relaxation (Hoffman, 2001). A general framework for solving

composite PDEs based on interface relaxation can be found in the work of Mu

(1999).

Interface relaxation iterative coupling algorithms vary significantly in terms of the

adopted computational procedure. Concerning the computational method, the

algorithms can be categorized into forms of sequential and parallel coupling. Here

the term parallel coupling refers to a form of partitioned computation in which

obtaining the response of each independently modelled soil and structure sub-


54

domains is carried out simultaneously during every coupling iteration. Unlike the

parallel coupling, in sequential coupling the partitioned sub-domains are solved one

after the other at each iterative stage. In addition to the parallel or sequential nature

of iterative coupling procedures, these algorithms also differ in relation to the

treatment of prescribed Dirichlet (Displacement) and Neumann (Force) effects at the

interface of the partitioned sub-domains. Elleithy & Tanaka (2003) presented and

categorized a range of different iterative coupling algorithms according to the

aforementioned characteristics for elasto-static BEM-BEM and FEM-BEM coupling

(for example, Sequential Dirichlet–Neumann coupling algorithm, Parallel Neumann–

Neumann coupling algorithm, Parallel Dirichlet-Neumann coupling algorithms, etc.).

Such algorithms will be described in detail in a more general form in the context of

static and dynamic soil-structure interaction FEM-FEM coupling in Chapter 4.

An example application of parallel Neumann-Neumann and Dirichlet-Dirichlet

iterative coupling algorithms is that of Kamiya et al. (1996). In this research the

authors studied the parallel implementation of the boundary element method using a

domain decomposition method on a cluster computing system. As concluded by the

authors, the main advantage of the proposed domain decomposition technique is that

it avoids the distinct formulation for constructing global matrices for the problems

with different domain decompositions. In the presented method, the global matrices

of the whole domain are not constructed and the boundary element analysis is

performed on individual sub-domains. More recent application of sequential

Dirichlet-Neumann iterative coupling approach for coupling BEM and FEM in

elasto-static analysis is given by Elleithy et al. (2001).

The application of iterative coupling algorithms in partitioned analysis has not been

limited to BEM-FEM coupling or fluid-structure interaction. In fact, these

algorithms are general in nature and with some modification could be applied to any

desired type of discretization technique in partitioned analysis of multi-physics

interaction problems. For instance, Collenz et al. (2004) utilised an FEM-FEM

coupling method based on a sequential approach for their work on modelling the

interactive nonlinear behaviour of the micro-beams under electrostatic loading.


55

The significant potential of the partitioned approach, as highlighted before, was

recognised more recently for coupled modelling of soil-structure interaction

problems. Hagen & Estorff (2005b) presented a domain decomposition approach for

the transient analysis of arbitrary three-dimensional soil–structure interaction

problems where the coupling of the sub-domains is performed in an iterative manner.

Based on such techniques, Hagen & Estorff (2005a) presented a hybrid approach for

transient dynamic investigation of Dam-Reservoir-soil problem, where different

discretisation techniques were utilized for different partitioned sub-domains as

shown in Figure 2.16. Elleithy et al. (2004) presented an interface relaxation FEM–

BEM coupling method for elasto-plastic analysis, which was applied to investigate

the stresses developed in tunnel structures. A schematic representation of their

presented coupling scheme is shown in Figure 2.17.

Figure 2.16: Coupled dam-reservoir-soil system (Hagen & Estorff, 2005a: p.10)


56

Figure 2.17: Flowchart of iterative coupling method by Elliethy et al. (2004: p.852)

Although the superiority of the iterative coupling partitioned approach has been

recognised more recently for coupled modelling of soil-structure interaction, there

remain significant technical challenges related to algorithmic and computational

issues, particularly with reference to convergence.

In all of the abovementioned iterative coupling algorithms, the update technique used

in the successive iterations is the constant interface relaxation scheme. As shown

later in this work, convergence to compatibility and equilibrium at the interface of

the partitioned sub-domains could be guaranteed through the choice of a suitable

value for the relaxation parameter, though the number of required iterations may be

unrealistically large.

Despite the relative simplicity of iterative relaxation schemes, there are several

issues that need to be carefully considered: i) determination of the range of suitable

relaxation parameters for the specific problem under consideration in order to

achieve convergence, and ii) selection of the optimum relaxation parameter in order

to achieve maximum computational efficiency.


57

In this respect, work by El-Gebeily et al. (2002) on convergence analysis of static

coupling of BEM-FEM has shown that the convergence behaviour is very sensitive

to the value of relaxation parameter. Moreover, it has been shown that the value of

relaxation parameter is problem dependent and varies considerably with respect to

the employed combination of the partitioned sub-domains, mesh density, material

properties and the adopted coupling scheme (parallel-sequential).

These results have also been confirmed by the research of Estorff & Hagen (2005)

on dynamic analysis of coupled BEM-FEM problems, where the significant

influence of different problem parameters on the choice of suitable relaxation

parameter is demonstrated. It has also been shown that, depending on the

characteristics of the problem under consideration, there is a range of applicable

relaxation parameters outside of which convergence to compatibility and equilibrium

could not be guaranteed. Moreover, there exists an optimum relaxation parameter in

the convergent range that holds the highest convergence rate and computational

efficiency. In this regard, there is a lack in the literature with regard to establishing

the general mathematical convergence behaviour with relation to the relaxation

parameter. Moreover, the determination of the optimum constant relaxation

parameter would typically require a process of trial and error for every coupling case

under consideration, which is prohibitive for real large scale soil-structure interaction

problems.

To address the above shortcomings with constant relaxation, adaptive relaxation

approaches were proposed by Funaro et al. (1998) for iterative coupling of

partitioned second-order elliptic problems, Wall et al. (2007) and Kuttler & Wall

(2008) focusing on iterative coupling of fluid-structure interaction problems, Elliethy

et al. (2004) for static iterative coupling of BEM-FEM, and Soares (2008) for

iterative coupling of FEM-BEM in dynamic analysis, which avoid the trial and error

process in constant relaxation.

Building on the above, as shown later in this work, the adaptive coupling technique

in the context of nonlinear soil-structure interaction is proposed. In this respect, the


58

need for establishing the general convergence behaviour of the scheme for both

dynamic and static FEM-FEM coupled problems is demonstrated mathematically

and illustrated through various case studies in Chapter 5. In this regard, it is shown

that the performance of interface relaxation in iterative coupling of soil-structure

interaction problems is enhanced significantly through the use of an adaptive

relaxation, using error minimization techniques.

In general, due to the lack of general convergence studies on the application of such

methods and the effects of nonlinearity on the convergence behaviour, specifically in

the context of soil-structure interaction, there remain significant issues to be covered

in relation to the convergence characteristics and performance of constant and

adaptive relaxation methods.

Furthermore, all the presented adaptive relaxation schemes have been employed

within the sequential Dirichlet-Neumann form of coupling algorithm, thus an

adaptive method for evaluating the relaxation parameter in the parallel forms of

coupling algorithms is still largely absent from current research on iterative coupled

modelling via interface relaxation approach.

Albeit, it is possible to extend the existing iterative coupling methods to enhance the

computational efficiency of both adaptive and constant relaxation coupling

algorithms, while overcoming the problematic issues regarding the trial and error

process embedded in evaluation of the constant relaxation parameter.

Kamiya & Iwase (1997), in their research on parallel elasto-static BEM-FEM

coupling, proposed a conjugate gradient scheme to overcome the drawback of the

relaxation scheme in selection of parameters for renewal iterations by users ( which,

as stated by them, require ‘trial and error and deep experience’). The presented

conjugate gradient method was built on the conjugate gradient method as a tool for

solution of simultaneous equations and application in domain decomposition

problems such as potential analysis by Glowinski et al. (1983). In the work by

Kamiya & Iwase (1997) it is shown that the proposed conjugate gradient method


59

coupling scheme is computationally efficient and has a higher convergence rate

compared to constant relaxation coupling algorithms.

In the view of above and considering the need for developing iterative coupling

methods with unconditional convergence characteristics and high convergence rate, a

new approach in coupled nonlinear soil-structure interaction analysis is proposed

later in this thesis. This approach is based on utilizing the condensed interface

tangent stiffness matrices of the structure and soil models, depending on the variant

coupling algorithms under consideration (Chapter 6).

Although the application of adaptive relaxation has been restricted to sequential

Dirichlet-Neumann coupling algorithm, by utilizing the condensed tangent interface

stiffness of the partitioned soil and structure sub-domains, various parallel and

sequential iterative coupling algorithms could be employed effectively without any

need for a trial and error procedure. In this respect, regardless of the employed type

of coupling algorithm (parallel or sequential) and partitioned problem parameters,

superior convergence behaviour compared to relaxation schemes is guaranteed to be

achieved.

Although the condensed tangent stiffness matrix can be determined with current

nonlinear field modelling tools, its implementation for nonlinear soil-structure

interaction coupling using discipline oriented softwares would normally necessitate

direct access to the source codes. In this respect, it is proposed in this work that the

condensed tangent interface stiffness matrices of the partitioned soil and structural

sub-domains may be reasonably approximated by constructing reduced order models

of the structure and soil sub-domains. The benefit of such an approach is that it does

not require the explicit assembly of the stiffness matrices. Indeed, with minimal

intrusion to the source codes of the solvers, a potentially efficient coupling technique

for coupled modelling of soil-structure interaction will be achieved. This builds on a

previous approach presented by Vierendeels (2006) and Vierendeels et al. (2007),

who utilized a procedure for constructing the reduced order model of partitioned sub-

domains throughout the coupling iterations for implicit coupling of fluid–structure


60

interaction problems, although some significant modification has been proposed in

this research for adopting and improving the scheme in the context of soil-structure

interaction.

In this regard, there is a pitfall associated with the traditional reduced order method.

For instance, in FEM-FEM iterative coupling problems, where the number of

coupling iterations exceeds the number of interface degrees of freedoms, due to

existence of linearly dependent displacement/force modes, the reduced order method

approximation for the condensed interface tangent stiffness matrix will be poor. This

shortcoming is discussed and addressed comprehensively in Chapter 6, where a

significant modification in reduced order method formulations is proposed to

overcome this shortcoming. Moreover, an advanced technique for approximating the

reduced order method (mixed reduced order method) which possesses a very high

convergence rate, especially in nonlinear problems, is further proposed.

2.3 Concluding Remarks

A review of various soil-structure interaction modelling techniques, along with their

advantages and disadvantages, has been presented. These include field elimination

techniques, such as the Winkler and continuum approaches, and numerical modelling

techniques, such as the finite element and boundary element methods. The distinct

superiority of numerical methods in SSI analysis, specifically the finite element

method, is discussed.

It is shown that, with the current discipline oriented structural and soil solvers,

achieving the same amount of sophistication in numerical modelling of both soil and

structure in a single soil-structure interaction analysis is still a challenge. Since

sophisticated discipline-oriented structural and geotechnical solvers are readily

available, and continue to be used for either application, the partitioned treatment of

coupled soil-structure interaction problems is introduced as an alternative to the

monolithic approach for tackling interaction problems.


61

The background and origin of the partitioned approach are presented. The staggered

coupling procedure, including its previous application to several transient dynamic

coupled problems, is discussed, and its main disadvantages are highlighted. These

disadvantages will be further demonstrated in Chapter 3, where the staggered

solution procedure for coupled modelling of FEM-FEM problems is

comprehensively examined. To address these shortcomings, iterative coupling

procedures have been introduced as an alternative to the staggered approach. Not

only do these iterative methods address the stability and accuracy issues of the

staggered approach, but they are applicable to both static and dynamic analysis.

Notwithstanding, a major issue associated with iterative coupling algorithms remains

their convergence to compatibility and equilibrium within a single load/time-step. In

this respect, several iterative update techniques such as using interface relaxation,

condensed interface tangent stiffness matrix and reduced order method are discussed.

In the context of relaxation approaches, the shortcomings of constant relaxation and

its enhancement in the form of adaptive relaxation are discussed. Due to the lack of

general convergence studies on the application of such methods and the effects of

nonlinearity on the convergence behaviour, specifically in the context of soil-

structure interaction, the convergence characteristics and performance of constant

and adaptive relaxation methods will be considered in detail in Chapter 5.

It is also pointed out that further enhancement of iterative coupling beyond the use of

relaxation schemes may be achieved through the use of the condensed tangent

stiffness matrices at the soil–structure interface. As a special case of such an

approach, the condensed tangent stiffness matrix may be approximated via reduced

order models of the partitioned sub-domains, where various highly efficient reduced

order methods will be presented in Chapter 6. This builds on previous work by

Vierendeels (2006) and Vierendeels et al. (2007) concerned with implicit coupling of

fluid–structure interaction problems, though major modifications are proposed in this

research for coupled modelling of nonlinear soil-structure interaction.


62

In summary, despite significant research on the partitioned treatment of coupled

systems, there remain significant technical challenges related to algorithmic and

computational issues, especially in the case of nonlinear soil-structure interaction

analysis. The focus of previous research has been mainly on fluid-structure coupling

and FEM-BEM coupled modelling, and where soil-structure interaction problems

were considered the focus has been mainly on the coupling procedure formulation

(again mainly on FEM-BEM coupling) rather than development of an advanced soil-

structure analysis platform. In this work, a novel simulation environment, which

embodies the proposed coupling algorithms and utilises discipline-oriented solvers

for nonlinear structural and geotechnical analysis, is developed and discussed in

Chapter 4. This environment is based on the coupling of two powerful in-house

programs at Imperial College, ADAPTIC (Izzuddin, 1991) and ICFEP (Potts &

Zdravkovic, 1999), for nonlinear structural and geotechnical analysis, respectively.

The developed environment is employed in this work to demonstrate the

performance characteristics and merits of the various presented algorithms, and to

illustrate the practical application of coupled modelling for realistic modelling of

nonlinear soil-structure interaction problems.

Chapter 3

Staggered Approach

3.1 Introduction

One of the earliest approaches proposed for coupling partitioned mechanical systems

is the staggered approach (Felippa & Park, 1980), which is particularly suited to

transient dynamic analysis. In the staggered solution approach, the equations for

each sub-domain are solved once at each time-step and predicted values of the

coupling boundary conditions are used to obtain the response of the individually

modelled sub-domains.

This approach is illustrated in Figure 3.1 for a soil-structure interaction problem

involving two physical partitions. In the context of soil-structure interaction, the

prediction stage typically stands for prediction of displacements from the soil model

at the soil-structure interface, while the substitution stage typically stands for the

substitution of the reaction forces from the structural model into the soil model at the

same interface.

Chapter 3 Staggered Approach

64

A typical approach in the staggered partitioning of soil-structure interaction, as

illustrated in Figure 3.1, would be to fix the contact degrees of freedom of the

structure sub-domain using predicted values obtained by a predictor. The structure

solver then computes the response of the structure as well as the forces at the

interface nodes. Based on these forces, the soil solver evaluates the displacements at

the interface nodes, which are used by the predictor to establish the new initial

conditions for the structure as the solution moves to the next time step.

Figure 3.1: Staggered approach in dynamic soil-structure interaction

The two most important issues of the staggered approach that should be addressed in

any scheme are the stability and the accuracy of the algorithms. The stability of

staggered algorithms depends on the suitable choice of the predictor operator, the

selected time integration scheme in each domain, the employed time-step for the

time integration schemes and the utilized discretization techniques at the interface

and through each field.

Once satisfactory stability is achieved, the next concern is accuracy. The predicted

interface displacements, obtained using velocities/accelerations at the beginning of

the step, are invariably different from the displacements evaluated following force


65

substitution into the soil model, thus leading to compatibility defaults at the

interface. The accuracy of the staggered approach may be improved by reducing the

time-step and using an optimal predictor, though stability requirements may dictate

step sizes that are too small rendering this approach prohibitively expensive for

large-scale soil-structure interaction problems.

In the following, a staggered approach for dynamic soil-structure interaction

problems is discussed, and the stability and accuracy issues are addressed in the

context of linear analysis. The proposed approach employs the Newmark integration

scheme for analysing the dynamic behaviour of the structure and soil sub-domains,

as described in the next section. In this context, the general stability conditions

required for staggered coupling of FEM-FEM are established. It is worth

mentioning, that unconditional stability may be achieved in staggered coupling by

employing significant modifications in the formulation of the partitioned sub-

domains for different interaction applications or by changes to the governing

equations by augmentations, such as those proposed by Farhat & Park (1991) and

Huang & Zienkiewicz (1998) for fluid-structure interaction analysis.

Notwithstanding the benefits of such methods, these are not considered here since

they do not permit the modular use of structural and soil solvers as black box

solvers, as they require major change in formation of the stiffness matrices and

reformulation (mainly by augmentation) of the field equations of the partitioned sub-

domains.

3.2 Dynamic Analysis Formulation

One of the most used techniques in assessing the dynamic response of mechanical

systems is the direct method, where the dynamic equilibrium equations are integrated

directly at the overall structural level, typically using a step-by-step time integration

scheme. This type of analysis is the most general and is readily applicable to

nonlinear analysis.


66

The dynamic equilibrium equations for a discrete linear structural system can be

expressed as:

M U C U K U F (3.1)

where M is the mass matrix, C is the damping matrix, K is the stiffness

matrix, U is the acceleration vector, U is the velocity vector, U is the

displacement vector and F is the applied load vector.

Equation (3.1) represents a semi-discrete problem since only spatial discretization is

undertaken, but not discretization in the time domain. Step-by-step integration

methods deal with the latter issue by considering Equation (3.1) at specific points of

time, and obtaining the solution at time tn+1 utilising the displacement, velocity and

acceleration history over previous discrete times. Consideration is restricted here to

the Newmark method (Zienkiewicz & Taylor, 1991), which is a single-step method

(i.e. only the displacement, velocity and acceleration at time nt are employed in

determining the corresponding entities at time tn+1).

Considering the discrete dynamic problem expressed by:

1 1 1 1n n n nM U C U K U F (3.2)

the Newmark method utilises the following parametric single-step difference

equations for obtaining a solution:

21 1

1

2n n n n nU U t U t U U

(3.3)

1 11n n n nU U t U U (3.4)

in which β and γ are algorithm parameters.


67

The above equations, allow the three unknowns 1nU , 1nU and 1nU

to be

determined as:

1

1 12

1 ˆ ˆn n n nU M C K F M U C U

t t

(3.5)

1 12

1 ˆn n nU U U

t

(3.6)

1 1ˆ

n n nU U Ut

(3.7)

where:

22

1 1ˆ2n n n nU U U t t U

t

(3.8)

ˆ ˆ1n n n nU U t U U (3.9)

3.3 Partitioning

Before developing the general staggered approach framework for coupling, it is

beneficial to illustrate the domain decomposition strategy and discretised

representation of the partitioned sub-domains in coupling algorithms using FEM.

Consider a coupled system governed by the semi-discrete equation of dynamic

equilibrium:

M U C U K U F (3.10)

Assume that the above coupled system is composed of ΩT and ΩB sub-domains with

the interface (ΩΓ= ΩT∩ ΩB), as shown in Figure 3.2 and formulated by:


68

11 12 11 12

21 22 11 12 21 22 11 12

21 22 21 22

11 12

21 22 11 12

21 22

0 0

0 0

0

0

T T T T T TT T

T T B B i T T B B i

B B B B B BB B

T T TT

T T B B i

B B BB

M M U C C U

M M M M U C C C C U

M M U C C U

K K U

K K K K U

K K U

TT

i

BB

F

F

F

(3.11)

In the above, vectors XXU , X

XU , XXU and X

XF correspond to the displacement

vectors and external loads, respectively, for the non-interface degrees of freedom in

sub-domain ΩX, while vectors iU , iU

, iU and iF correspond to

displacement vectors and external loads at the interface degrees of freedom. In

addition, XijM , X

ijC and XijK correspond to mass, damping and stiffness matrices in

sub-domain ΩX, respectively.

Figure 3.2: Partitioning and discretisation

Assuming that the above coupled system is partitioned into two independently

modelled sub-domains (ΩT, ΩB) and discretised by FEM using a step-by-step time


69

integration method, such as the Newmark method, the governing discrete equations

of each independently modelled partitioned sub-domain can be formulated as

follows:

For sub-domain ΩT:

11 12 11 12 11 12

21 22 21 22 21 22

T T T T T T T TT TT TT T

T T T T T T i ii iT TT T

M M C C K K U FU U

M M C C K K U FU U

(3.12)

For sub-domain ΩB:

11 12 11 12 11 12

21 22 21 22 21 22

B B B B B B B BB BB BB T

B B B B B B i ii iB BB B

M M C C K K U FU U

M M C C K K U FU U

(3.13)

In the above, vectors iXU and i

XF are the displacements and external loads,

respectively, for the interface degrees of freedom in sub-domain ΩX, whereas:

11 12

21 22

X X

X X

K K

K K

= stiffness matrix of sub-domain ΩX,

11 12

21 22

X X

X X

M M

M M

= mass matrix of sub-domain ΩX, and

11 12

21 22

X X

X X

C C

C C

= damping matrix of sub-domain ΩX.

Clearly, the time marching Equations (3.12) and (3.13) cannot be solved

independently. This is due to existence of unknown displacement (i.e. ,i iT BU U )

and force vectors (i.e. ,i iT BF F ) at the interface. However, by employing a

coupling procedure in which the response of the partitioned sub-domains is obtained

for certain interface boundary conditions and united at the interface level, the

partitioned domains can be solved independently. Here this task is achieved using a

staggered coupling procedure as described hereafter.


70

3.4 Staggered Coupling Procedure

Consider the partitioning strategy presented in the previous section, where ΩT refers

to the partitioned structure sub-domain, while ΩB corresponds to the partitioned soil

sub-domain. The general algorithmic framework of staggered coupling in the context

of soil-structure interaction analysis using FEM can be described by the following

procedure, in which n denotes the current time-step number.

For n = 1, 2,...

STEP 1: At the start of each time-step, the ΩT sub-domain (structure) is loaded by

the external forces TT n

F , while the displacements at the interface nodes, iT n

U , are

prescribed by a predictor in accordance with the initial conditions.

STEP 2: The response of ΩT sub-domain (structure) is obtained using Equation

(3.12) for TT n

U and iT nF .

STEP 3: The corresponding interface forces iB nF at ΩB sub-domain (soil) are

determined from equilibrium:

0i iT Bn n

F F (3.14)

STEP 4: Based on the interface forces, iB nF , and the external loading applied to the

soil sub-domain, BB n

F , the soil response iB n

U and BB n

U is obtained using

Equation (3.13).

STEP 5: Before moving to the next time-step, a predictor will evaluate 1

iT n

U

:

1, , ,i i i i

T B B Bn n n nU U U U t

(3.15)


71

while the compatibility of the initial condition for the structure domain and soil

domain is enforced:

i iT Bn n

U U (3.16)

i iT Bn n

U U (3.17)

The above staggered coupling scheme couples the response of the partitioned sub-

domains by enforcing equilibrium condition at the interface level, while the

compatibility default is minimized by choosing a suitable predictor function and

small time step in order to level the response of the coupled sub-domains.

3.5 Predictors

As mentioned before, a suitable choice of the predictor is very important since it

influences the stability and accuracy of the staggered scheme. Emphasis is placed on

Linear Multi Step (LMS) integrators, since they include a large number of

integration formulas used in practice, and both the functions and their derivatives are

evaluated at the same times. The general expression of a LMS predictor for constant

step size t is (Felippa & Park, 1980):

2

11 1 11 1 1

m m mP P P P

i i in in n i n ii i i

U U t U t U

(3.18)

where Pi , P

i and Pi are numeric coefficients of the predictor. Since for second-

order governing equations (as considered here), historical information involves at

most computation of second derivatives, LMS predictors continuing beyond the

1n i

U

terms are not considered. Note also that Equation (3.18) involves m past

terms, where m is the number of steps in the LMS integrator.


72

Trivial Predictors

General Form:

1

Ppredicted nn

U U

Trivial:

1predicted nn

U U

First Order Predictors

General Form:

1

P Ppredicted nn n

U U t U

Central Difference Method:

1predicted nn n

U U t U

Second Order Predictors

General Form:

2

1

P P Ppredicted nn n n

U U t U t U

Central Difference Method:

2

1 2predicted nn n n

tU U t U U

Table 3.1: Single-step predictors for second-order dynamic systems

Some examples of the single-step predictors that can be used here for second-order

dynamic systems are presented in Table 3.1.

3.6 Stability Analysis by Amplification Method

The term “stable” informally means resistant to change. For technical use, the term

has to be defined more precisely in terms of the mathematical model, but the same

connotation applies. In mathematics, stability theory is typically concerned with

whether a given function is sensitive to a small perturbation (Felippa & Park, 2004).

Considering a dynamic problem and supposing that the system is undergoing

periodic motion with a period (T), then:

( ) ( )t T t U U (3.19)

Stability of the system requires that, in studying the behaviour of the system after

application of an arbitrary perturbation in the initial displacement or velocity, the

motion remains within small prescribed limits of the unperturbed motion.


73

One of the widely used methods in stability analysis is the amplification method,

which is also called von Neumann stability analysis (Zienkiewicz & Taylor, 1991;

Felippa & Park, 2004). The approach is based on decomposition of motion into

normal modes and checking the growth or decay of perturbations from one step to

the next, and it can be implemented using standard linear algebra procedures.

Considering the application of a time approximation recurrence algorithm such as

the Newmark method to an initial-value problem, it can be shown that the

homogeneous form (i.e. free vibration) takes the form of (Zienkiewicz & Taylor,

1991):

1n nU U

A (3.20)

in which [A] is known as the amplification matrix. Since methods such as Newmark

represent approximations that are used to derive equations of type (3.20), error is

introduced into the solution 1nU

at each time step. Since the solution 1n

U

at time

1nt depends on the solution nU at time nt the error can grow with time. It can be

clearly observed that any error presented in the solution will of course be subjected

to amplification by precisely the same factor:

k n

k nU U

A (3.21)

The algorithm is considered to be stable if the error introduced in nU does not

grow unbounded as Equation (3.20) is solved repeatedly.

Knowing that the general modal solution of any recurrence algorithm can be written

as (Zienkiewicz & Taylor, 1991):

1n nU U

(3.22)

and by substituting Equation (3.22) into Equation (3.20), it can be observed that is

given by the eigenvalues of the amplification matrix as:


74

0n

U A I (3.23)

Clearly if any eigenvalue, , of the amplification matrix is greater than one in

absolute value, 1 , all initially small errors will increase without limit and the

solution will be unstable. Therefore the stability of the recurrence algorithm requires

that 1 (Zienkiewicz & Taylor, 1991). In the case of complex eigenvalues, the

requirement is modified to the modulus of being less than or equal to one.

The calculation of the eigenvalues in such problems is not trivial. Therefore, two

general procedures are employed here to make the task of stability checks more

practical.

The first procedure is named z transformation (Zienkiewicz & Taylor, 1991). The

determinant equation 0 A I provides the characteristic polynomial yielding

the eigenvalues of the amplification matrix and, as mentioned for stability, it is

sufficient and necessary that the module of all the eigenvalues 1 . By using the

following variable transformation in the obtained characteristic polynomial:

1

1

z

z

(3.24)

where z and are in general complex numbers, it is easy to show that the

requirement of 1 is identical to that demanding the real part of z to be negative,

as illustrated in Figure 3.3 (Zienkiewicz & Taylor, 1991).


75

Figure 3.3: Z-transformations (Zienkiewicz & Taylor, 1991)

The second transformation is the well-known Routh-Hurwitz condition (Routh,

1877; Zienkiewicz & Taylor, 1991) which states that for a polynomial

10 1 ... 0n n

nC z C z C , the real part of all roots will be negative if, for 0 0C ,

1 0C ,

1 3

0 2

0C C

C C

1 3 5

0 2 4

1 3

0

0

C C C

C C C

C C

and so on up to,

1 3 5

0 2 4

1 3

2

...

...

0 ...0

0 0 ...

... ... ... ...

0 n

C C C

C C C

C C

C

C

I

R

R Ii R Iz z iz

Rz

Iz


76

By utilizing the above tools, the stability of the proposed staggered algorithms can

be assessed by implementing the general procedure described in the next section.

3.7 Stability of Staggered Coupling Scheme for a Test System

The stability of the presented staggered approach can be established by considering a

test system where both partitioned sub-domains are assumed to be condensed at the

interface (i.e. all the freedoms are at interface level). The stability analysis consists

of the following steps:

1. Time-discretization of the partitioned sub-domains of a test system, while the

applied force term is dropped.

2. Construction of the amplification matrix through following the algorithmic

steps of the staggered approach under consideration and obtaining the

characteristic equation of the amplification matrix.

3. Converting the characteristic equation of the amplification matrix into a

Routh-Hurwitz polynomial and applying stability check of the system.

Consider a coupled system which is partitioned into the sub-domains (ΩB) and (ΩT),

with each sub-domain discretised in the time domain using Newmark method, and

assume that both domains are condensed to a single degree of freedom at the

interface, neglecting the effects of damping for simplicity. The governing

equilibrium equations for the partitioned sub-domains of the test system are given

by:

Governing equilibrium condition for sub-domain (ΩT):

i i iT T T T TM U K U F (3.25)

Governing equilibrium condition for sub-domain (ΩB):

i i iB B B B BM U K U F (3.26)


77

where iTF and i

BF are the interface forces applied to the structure and soil sub-

domains, respectively.

Following the staggered approach algorithmic steps presented in Section 3.4, the

response of the partitioned sub-domains of the above test system can be coupled as

discussed in the following, yielding to the construction of the amplification matrix:

STEP 1- The initial condition in the ΩB and ΩT sub-domains is prescribed, while at

the interface the initial conditions are:

i iT Bn n

U U (3.27)

i iT Bn n

U U (3.28)

i iT Bn n

U U (3.29)

Choosing the second-order central difference predictor from Table 3.1, the

prescribed displacement in partitioned sub-domain ΩT at time 1nt t is:

2

1 2T T T Tn n n n

tU U t U U

(3.30)

STEP 2 - Solving the ΩT sub-domain for 1

iT n

F

using Newmark method (see

Section 3.2):

2 1 1

ˆ 0i i iTT T T T Tn n

MK U F M U

t

2 2

1

21

ˆi iT T T T Tni

T n

M t K U t M UF

t

(3.31)

where:


78

22

1 1ˆ2

i i i iT T T Tn n n

U U t U t Ut

(3.32)

STEP 3 - Applying equilibrium at the interface:

2 2

1

21 1

î iT T T T Tni i

B Tn n

M t K U t M UF F

t

(3.33)

STEP 4 - Solving the ΩB for 1

iB n

U

using Newmark method:

2 1 1

ˆ 0i i iBB B B B Bn n

MK U F M U

t

2

1

21

î iB B Bni

B nB B

t F M UU

M K t

(3.34)

where:

22

1 1ˆ2

i i i iB B B Bn n n

U U t U t Ut

(3.35)

The velocity and acceleration in sub-domain ΩB at time tn+1 would be:

11

î i iB B Bnn

U U Ut

(3.36)

2 11

1 î i iB B Bnn

U U Ut

(3.37)

where:

ˆ ˆ1i i i iB B B Bn n

U U t U U (3.38)


79

STEP 5 - Prescribing the initial condition of the sub-domain T compatible with

that of sub-domain B for next time step:

1 1

i iT Bn n

U U (3.39)

1 1

i iT Bn n

U U (3.40)

Using a central difference predictor from Table 3.1, the prescribed displacements for

ΩT at time 2nt t is:

2

2 1 1 12T B B Bn n n n

tU U t U U

(3.41)

Substituting Equations (3.35) and (3.33) in Equation (3.34), the displacements of

sub-domain ΩB at time tn+1 takes the form of:

2

21

2

2

2 2

2

2 21

2

i iB TB Bn n

B B

B T iB n

B B

B T T T iB n

B B

M K tU U

M K t

M K t tU

M K t

M M M K t tU

M K t

(3.42)

The velocity at the interface of sub-domain ΩB at time tn+1 can also be calculated by

expanding Equation (3.36), as:

21

2 2

2

2 2 2

2

+

2 2 21 -

2

T Bi iB B nn

B B

B B B T iB n

B B

T B T B B iB n

B B

K K tU U

M K t

M K t K K tU

M K t

K t K t M M K t tU

M K t

(3.43)


80

Similarly the acceleration at the interface of sub-domain ΩB at time tn+1 can also be

expressed in terms of displacement, velocity and acceleration of the previous time

step (tn ) as:

21

2

2 2

2

2 21

2

i iT BB B nn

B B

T B iB n

B B

T B T B iB n

B B

K KU U

M K t

K K tU

M K t

K K t M K tU

M K t

(3.44)

Now, the amplification matrix of the presented test system can be constructed in the

form of 1n n U A U , by knowing Equations (3.42), (3.43) and (3.44), with

A being the amplification matrix of the proposed staggered coupling scheme. The

amplification matrix can be derived as:

11 12 13

21 22 23

31 32 331

B B

B B

B Bn n

U A A A U

U A A A U

U A A A U

(3.45)

The equation 0 A I provides the characteristic polynomial yielding the

eigenvalues of the above amplification matrix as:

2 23 2

2

2 2

2

2

2

2 4 1 (2 2 1) 2 41

2

2 2 1 2 4 1 4 21

2

0

B T T B

B B

B T T B

B B

T T

B B

K t K t M M

M K t

K t K t M M

M K t

M K t

M K t

(3.46)

As mentioned before it is sufficient and necessary for stability that the modulus of all

the eigenvalues is less than or equal to one ( i.e. 1i ).


81

Implementing the variable transformation as discussed earlier, the polynomial of

Equation (3.46) can be converted into a Ruth-Hurwitz polynomial given by Equation

(3.47) through 1

1

z

z

mapping, leading to:

2 2 32

2 2 22

22

22

14 2 4 2 4 4

14 1 4 1 4 4

12

10

B T B TB B

B T T BB B

T BB B

T BB B

K t K t M M zM K t

K t K t M M zM K t

K K t zM K t

K K tM K t

(3.47)

Therefore, the stability condition of 1i will simplify to the following Ruth-

Hurwitz conditions:

0

2

2

0

4 2 4 2 4( )0B T B T

B B

C

K K t M M

M K t

(3.48)

1

2

2

0

4 1 4 1 40T B B T

B B

C

K K t M M

M K t

(3.49)

1 3

0 2

2

22

0

2 1 2 1 2 1 2 10T B B T

B B

C C

C C

K K t M M

M K t

(3.50)


82

1 3

0 2

1 3

2

32

0

0 0

0

2 1 2 1 2 1 2 10T B B T

B B

C C

C C

C C

K K t M M

M K t

(3.51)

Considering Equations (3.48)-(3.51), it can be concluded that the presented

staggered approach is only conditionally stable, as elaborated next.

Consider the above stability conditions under the assumption of very small time

steps (i.e. 0t ):

00

lim( ) 0 0B Tt

C M M

(3.52)

10

lim( ) 0 0B Tt

C M M

(3.53)

1 3

1 30 2

0 00 4

1 3

0

lim 0 lim 0 0 2 1 2 1 0

0B T

t t

C CC C

C C M MC C

C C

(3.54)

The Newmark method is unconditionally stable in linear analysis for all modes

regardless of the choice of t, if:

12 Newmark method unconditionally stable

2 (3.55)

The above conditions for the stability clearly leads to 0B TM M as 0t .

Therefore, it can be concluded that the stability of the presented algorithm is

guaranteed, provided that the mass of the coupled system at the interface level is

distributed in accordance to the previous condition. However, the above sufficient

criterion for stability is valid based on the assumption of a very small time step

( 0t ). In the more general case where this assumption is not accurate, the full


83

conditions given by Equations (3.48)-(3.51) must be satisfied in order to achieve

stability.

The fact that the stability in the presented staggered procedure is conditional and is

guaranteed provided that 0B TM M as 0t , can be further demonstrated by

considering the following lemma:

If [A] is a square n×n matrix with real or complex entries and if ( 1 2, , , n ) the

eigenvalues of [S] are listed according to their multiplicities, then:

1 1det . n A (3.56)

Accordingly, for stability of the system of equations presented by Equation (3.45), it

is sufficient and necessary that the modulus of all the eigenvalues, i , of the

amplification matrix [A] satisfy:

1 ( 1,..., )i i n (3.57)

Considering Equations (3.56) and (3.57), it is clear that a necessary (but not

sufficient) condition for the stability of Equation (3.45) would be:

1 1det . 1n A (3.58)

Using Equation (3.45), it can be shown that the above necessary stability condition

simplifies to:

2

2det 1T T

B B

M K t

M K t

A (3.59)

which confirms the conditional stability of the scheme and shows that conditional

stability may be achieved provided that 0B TM M as 0t . Moreover, it is

important to note in relation to the above discussions that if 0B TM M , then

the use of a small time-step may be counterproductive in relation to stability.


84

3.8 Accuracy

In addition to stability requirements of the staggered scheme, another major issue

related to staggered coupling is achieving the required level of accuracy. Unlike

stability, the lack of which has an obvious effect on progressive escalation of error

with time, assessing accuracy is not so straightforward. While the stability of a

solution is a measure of the boundedness of the approximate solution with time, the

accuracy of the staggered scheme is a measure of the closeness between the

approximate solution obtained by the staggered scheme and the exact solution. This

means that the degradation of the staggered approach with respect to the results

obtained by monolithic approach determines the level of accuracy. The accuracy of a

staggered scheme might be increased by reducing the time step, which as discussed

before also influences the stability. However, in nonlinear problems, stability and

accuracy are interlinked and considered together. In fact, the previously discussed

stability is based on the assumption of a linear dynamic response, and therefore the

derived stability conditions should only be employed as a rough guide. In general,

the accuracy of a staggered scheme can be established by performing a new analysis

that is undertaken with a smaller time step for comparison purposes.

3.9 Example

Here we consider a representative FEM-FEM coupled problem treated both by the

monolithic and staggered approaches. Consider the cantilever column of Figure 3.4

subject to an excitation acceleration signal applied to its bottom support, as given in

Figure 3.5. The length of the cantilever is 20m, and it utilises two rectangular cross

sections. The mass of the system is modelled with two concentrated masses of 2000

kg and 800 kg, in the middle and the free end of the cantilever, respectively. A

bilinear kinematic material model is employed, as illustrated in Figure 3.4, where

9 2210 10E Nm (elastic modulus), 0.01 (strain hardening factor) and

6 2300 10y Nm (yield stress).


85

Figure 3.4: Example: geometric configuration and material response

Figure 3.5: Acceleration at the base

The above system is partitioned into two sub-domains namely T and B , as

shown in Figure 3.6, with three degrees of freedom at the interface (one rotational

and two translational). The partitioned problem of Figure 3.6 is analysed employing

a staggered approach using a central difference predictor, where sub-domain T is


86

treated by Dirichlet boundary conditions (displacement) and sub-domain B is

treated by Neumann boundary conditions (force). The obtained results by the

staggered approach are compared to the corresponding results of the monolithic

problem. The analyses show that the scheme is not stable for 0.01 st and

0.001 st , which makes the staggered algorithm extremely computationally

expensive as the stability requirements dictate very small time steps B TM M .

This would be clearer when compared to the monolithic treatment which is stable for

0.01 st .

Figure 3.6: Problem partitioning

A comparison between the results of the staggered and monolithic approaches is

undertaken, where a time-step of 0.0005 st is considered in the staggered

coupling analysis. Figure 3.7 shows the variation with time of the rotational degree

of freedom at the interface for both the monolithic and staggered approaches.

Despite the very small chosen time step, it can be seen that at around 5 sec, the

staggered approach exhibits high frequency oscillations. The lack of

accuracy/stability can be further demonstrated by comparing the variation with time

of the horizontal (X) displacement at the interface and at the top of the cantilever as

depicted in Figures 3.8 and 3.9, respectively. Again, at around 5 sec the staggered

approach starts to show poor performance.


87

Figure 3.7: Rotation at the interface

Figure 3.8: Horizontal displacement at the interface


88

Figure 3.9: Horizontal displacement at the tip of the cantilever

The stability/accuracy of the staggered approach with the specified time step size

( .0005 st ) is clearly not satisfactory, hence another analysis was carried out

reducing the time step to ( .0001 st ). Although this is an extremely small time

step that makes the staggered approach extremely computational expensive, the

results of staggered analysis with .0001 st confirms that by reducing the time

step size a full coupling of the partitioned sub-domains can be achieved. The fact

that the stability and accuracy are achievable by the staggered coupling approach,

provided that small time steps are employed, is confirmed by comparing the results

against those of the monolithic treatment.

Figures 3.10 and 3.11 show the variation with time of the rotation and horizontal

displacement, respectively, at the interface for both the staggered and monolithic

approaches. The variation with time of the horizontal displacement and rotation at

the free end of the cantilever is also depicted in Figures 3.12 and 3.13, respectively.


89

Figure 3.10: Rotation at the interface

Figure 3.11: Horizontal displacement at the interface


90


Figure 3.13: Rotation at the tip of the cantilever


91

3.10 Conclusion

Coupling of partitioned sub-domains can be achieved using the staggered approach,

though this approach should be used with great care in relation to both stability and

accuracy. In fact, stability and accuracy considerations typically demand excessively

small time steps rendering this scheme computationally prohibitive for many

coupled problems. In general the stability of the staggered scheme is conditional on

the time step and also on the equivalent mass and stiffness on both sides of the

interface, though this depends on the formulation of the partitioned sub-domains and

the employed predictors. Although some progress may be achieved by modifying the

specifics of the staggered scheme, this is not considered here since the ultimate

performance will vary with the coupling application under consideration, and often

its implementation would conflict with the modular use of the structural and soil

solvers as black box solvers.

Clearly therefore, there is a need for enhancement of the performance of the

staggered coupling schemes to be stable and accurate for a wider range of time steps,

whilst maintaining the practical and computational benefits of the partitioned

treatment. This has led to the development of iterative coupling algorithms which

are mainly carried out by introducing corrective iterations in the staggered approach,

hence the name iterative methods. These methods, in addition to the aforementioned

enhancement of the stability and accuracy issues of the staggered approach, are

applicable to static as well as dynamic problems and facilitate parallel computing

through problem partitioning. Such methods are introduced and discussed in detail in

the following chapters.

Chapter 4

Iterative Coupling

4.1 Introduction

As shown in the previous chapter, in staggered coupling the compatibility condition

at the interface of the partitioned sub-domains may be violated, which significantly

affects the stability and accuracy of the coupling scheme. This is due to the inherent

approximation of the prediction stage, which consists of prescribing the predicted

interface displacements at the structure interface level (obtained using displacements,

velocities and accelerations from previous steps), and these are invariably different

from the displacements evaluated following force substitution into the soil model.

Consequently, the staggered approach should be used with great care, since its

stability is conditional and depends greatly on the size of time step. Stability and

accuracy issues related to the staggered approach typically demand excessively small

time steps, rendering this scheme computationally prohibitive for many coupled

problems. This has led to the development of coupling algorithms that are stable and

accurate for a wider range of time step size, which is mainly achieved by introducing

corrective iterations, hence the name iterative sub-structuring methods (Quarteroni &

Valli, 1999). It is worth mentioning that iterative coupling approaches, in addition to

Chapter 4 Iterative Coupling

93

the aforementioned treatment of the stability and accuracy issues of the staggered

approach, facilitate parallel computing through problem partitioning which could

lead to much greater computational efficiency. The other advantage of iterative

coupling algorithms is that, unlike the staggered approach, they could be readily

employed in both static and dynamic analysis of soil-structure interaction problems.

Consider an arbitrary soil-structure system with arbitrary boundary conditions as

shown in Figure 4.1. A common practice, that allows field-specific discretisation and

solution procedures in a partitioned treatment of the system, would be to decompose

the coupled system into two sub-domains according to their physical and material

properties, namely soil and structure sub-domains.


Assuming that each sub-domain is independently discretised by FEM using a step by

step time integration method such as Newmark, the governing equilibrium

conditions for the soil and structure sub-domains can be formulated independently as

given by Equations (4.1) and (4.2), where, without loss of generality, the response of

individual sub-domains is assumed to be linear elastic:

Structure sub-domain:


94

2

1

1 T T TT TT T TT T

Ti i ii iT T TT T TT Tn n n

U F UU UM C K

t t U F UU U

(4.1)

Soil sub-domain:

2

1

1 B B BB BB B BB B

Bi i ii iB B BB B BB Bn n n

U F UU UM C K

t t U F UU U

(4.2)

In the above, XM , X

C and XK are the mass, damping and the stiffness

matrices of the partitioned sub-domain X . XXU , X

XU , XXU and X

XF

correspond to displacement, velocity, acceleration and external load vectors for the

non-interface degrees of freedom in sub-domain X , while iXU , i

XU , iXU and

iXF correspond to displacement, velocity, acceleration and external load vectors

for the interface degrees of freedom in sub-domain X , respectively. and are

Newmark algorithm parameters, and is a function of displacement, velocity and

acceleration at time nt employed for determining the corresponding entities at time

1nt . It should be noted that the suggested discretisation of the partitioned sub-

domains is just for illustration, where the proposed method is in fact also applicable

to different desired types of discretisation techniques. Moreover, the static analysis

of the presented coupled system can be considered as a special case of the presented

dynamic analysis.

Although Equations (4.1) and (4.2) of the partitioned soil and structure sub-domains

cannot be solved independent of each other, by applying an iterative coupling

scheme and coupling the response of the partitioned soil and structure sub-domains

at the interface level, the partitioned sub-domains can be analysed separately. The

proposed solution scheme couples the response of the soil and structure sub-domains

by enforcing explicitly compatibility and equilibrium conditions at the interface.


95

Figure 4.2 shows a soil-structure interaction coupled system, decomposed into the

soil and structure sub-domains treated by iterative coupling algorithms. The

governing equations of the partitioned sub-domains are solved independently at each

time step (or load increment in the case of static analysis), using predicted boundary

conditions (either force or displacement) at the interface. These predicted boundary

conditions are then successively updated using corrective iterations, until

convergence to equilibrium and compatibility is achieved at the interface and within

the partitioned sub-domains. This enables the coupling procedure to have an

effectively similar overall accuracy and stability to the monolithic treatment.

Figure 4.2: Schematics of iterative coupling algorithms

The compatibility and equilibrium requirements that should be satisfied at the

interface of a soil-structure system can be generally defined in view of Equations

(4.1) and (4.2) as:

Compatibility condition:

0i iB Tn n

U U (4.3)


96

Equilibrium condition:

0i iB Tn n

F F (4.4)

Although in the above the assumption is that the soil and the structure always remain

in contact at the interface, the treatment of separation and slip can be a simple

extension through the use of interface elements that may be considered to be either

part of one of the sub-domains or even part of the interface model.

4.2 Iterative Coupling Algorithms

Iterative coupling algorithms, as discussed in Chapter 2, can vary significantly in

terms of the adopted computational procedure. Concerning the computational

method, the algorithms can be categorized into forms of sequential and parallel

coupling. Here, the term parallel coupling refers to a form of partitioned computation

in which obtaining the response of each independently modelled soil and structure

sub-domain is carried out simultaneously during coupling iterations. Unlike parallel

coupling, in sequential coupling the partitioned sub-domains are solved one after the

other at each iteration stage.

In addition to the parallel or sequential nature of the iterative coupling procedures,

these algorithms also differ in relation to the treatment of prescribed Dirichlet

(Displacement) and Neumann (Force) conditions at the interface of the partitioned

sub-domains in order to achieve convergence.

In the following, the various types of iterative coupling algorithms (Elleithy &

Tanaka, 2003) are adapted and categorized in the context of soil-structure interaction

analysis. Throughout this discussion, subscript n and superscript I denote the

time/load increment and the iteration number, respectively.


97

4.2.1 Sequential Dirichlet-Neumann Iterative Coupling

The sequential Dirichlet–Neumann coupling algorithmic steps are described as

follows and illustrated in Figure 4.3.

For n=1,2,…(number of load/time increments)

I=0 (iteration number)

STEP 1: At the start of each increment, the structure domain is loaded by the

external forces, TT n

F , while the displacements (Dirichlet data) at the interface

nodes, IiT n

U , are prescribed in accordance with the initial conditions:

IiT n

U U

STEP 2: The structural solver computes the response of the structure using Equation

(4.1), for non-interface displacements, ITT n

U , and interaction forces at the interface,

IiT n

F .

STEP 3: The corresponding interface forces at the soil domain can be calculated by

applying equilibrium:

0I Ii i

B Tn nF F

STEP 4: Based on these forces, IiB n

F , and the external loading applied to the soil

sub-domain, BB n

F , the soil solver computes the response of the soil domain for

IiB n

U and IBB n

U .

By comparing the initial prescribed interface displacement and the obtained interface

displacements in STEP 4, the following two scenarios can occur:

STEP 5: If convergence to compatibility has not been achieved, the new estimation

of the interface displacements according to the compatibility condition of Equation


98

(4.3) is applied to the structure domain, and iteration continues (I=I+1) from STEP 2

until convergence to compatibility is achieved.

STEP 6: If convergence to compatibility at the interface of partitioned domains has

been achieved, the solution proceeds to the next time/load increment (n=n+1).

Figure 4.3: Schematics of sequential D-N iterative coupling

4.2.2 Sequential Neumann-Dirichlet Iterative Coupling

The sequential Neumann-Dirichlet coupling algorithmic steps are described as

follows and illustrated in Figure 4.4.





F , while the forces (Neumann data) at the interface nodes,

IiT n

F , are prescribed in accordance with the initial conditions:

IiT n

F F

0IiT n

U

IiB n

F 1IiT n

U

0

1

IiT n

U


99



U , and displacements at the interface,

IiT n

U .

STEP 3: The corresponding interface displacements at the soil domain can be

calculated by applying compatibility:

0I Ii i

B Tn nU U

STEP 4: Based on these displacements, IiB n

U , and the external loading applied to

the soil sub-domain, BB n

F , the soil solver computes the response of the soil domain

for IiB n

F and IBB n

U .

By comparing the initial prescribed interface forces and the obtained ones in STEP 4

the following two scenarios can occur:

STEP 5: If convergence to equilibrium has not been achieved, the new estimation of

the interface forces according to the equilibrium condition of Equation (4.4) is

applied to the structure domain, and iteration continues (I=I+1) from STEP 2 until

convergence to equilibrium is achieved.

STEP 6: If convergence to equilibrium at the interface of partitioned domains has

been achieved the solution proceeds to the next time/load increment (n=n+1).


100

Figure 4.4: Schematics of sequential N-D iterative coupling algorithms

4.2.3 Parallel Dirichlet-Neumann Iterative Coupling

The Parallel Dirichlet–Neumann coupling algorithmic steps are described as follows

and illustrated in Figure 4.5.



STEP 1: At the start of each increment, the structure sub-domain is loaded by the


F , while the displacements (Dirichlet data) at the interface

nodes, IiT n

U , are prescribed in accordance with the initial conditions:

IiT n

U U

Concurrently, the soil sub-domain is loaded by the external forces, BB n

F , while the

forces (Neumann data) at the interface nodes, IiB n

F , are prescribed in accordance

with the initial conditions:

IiB n

F F

0IiT n

F

IiB n

U 1IiT n

F

0

1

IiT n

F


101



U , and interaction forces at the interface,

IiT n

F . Simultaneously, the soil solver computes the response of the structure using

Equation (4.2), for non-interface displacements, IBB n

U , and interaction

displacements at the interface, IiB n

U .

By comparing the initial prescribed interface displacements and forces and the

obtained interface displacements and forces in STEP 2, the following two scenarios

can occur:

STEP 3: If convergence to compatibility and equilibrium has not been achieved, the

new estimation of the interface displacements and forces according to the

compatibility and equilibrium conditions of Equations (4.3) and (4.4) is applied to

the structure and soil sub-domains respectively, and iteration continues (I=I+1) from

STEP 2 until convergence to compatibility and equilibrium is achieved.

STEP 4: If convergence to compatibility and equilibrium at the interface of

partitioned sub-domains has been achieved, the solution proceeds to the next

time/load increment (n=n+1).

Figure 4.5: Schematics of Parallel D-N iterative coupling

0IiT n

U

0IiB n

F

1IiB n

F 1Ii

T nU

0

1

IiB n

F

0

1

IiT n

U


102

4.2.4 Parallel Neumann-Dirichlet Iterative Coupling

The Parallel Neumann-Dirichlet coupling algorithmic steps are described as follows




STEP 1: At the start of each increment, the structure sub-domain is loaded by the


F , while the forces (Neumann data) at the interface nodes,

IiT n

F , are prescribed in accordance with the initial conditions:

IiT n

F F

Concurrently, the soil sub-domain is loaded by the external forces, BB n

F , while the

displacements (Dirichlet data) at the interface nodes, IiB n

U , are prescribed in

accordance with the initial conditions:

IiB n

U U



U , and interaction displacements at the

interface, IiT n

U . Simultaneously, the soil solver computes the response of the

structure using Equation (4.2), for non-interface displacements, IBB n

U , and

interaction forces at the interface, IiB n

F .

By comparing the initial prescribed interface displacements and forces and the

obtained interface displacements and forces in STEP 2, the following two scenarios

can occur:


new estimation of the interface displacements and forces according to the

compatibility and equilibrium conditions of Equations (4.3) and (4.4) is applied to


103

the structure and soil sub-domains respectively, and iteration continues (I=I+1) from

STEP 2 until convergence to compatibility and equilibrium is achieved.

STEP 4: If convergence to compatibility and equilibrium at the interface of

partitioned domains has been achieved the solution proceeds to the next time/load

increment (n=n+1).

Figure 4.6: Schematics of Parallel N-D iterative coupling

4.2.5 Parallel Dirichlet-Dirichlet Iterative Coupling

The Parallel Dirichlet–Dirichlet coupling algorithmic steps are described as follows




STEP 1: At the start of each increment, the structure sub-domain and the soil sub-

domain are loaded by the external forces, TT n

F and BB n

F , while the displacements

(Dirichlet data) at the interface nodes of the partitioned sub-domains IiT n

U and

0IiT n

F

0IiB n

U

1IiB n

U 1Ii

T nF

0

1

IiB n

U

0

1

IiT n

F


104

IiB n

U are prescribed in accordance with the initial conditions (preferably enforcing

compatibility):

0 0I Ii iT Bn n

U U U

STEP 2: The structural and soil solver compute the response of the structure and soil

concurrently using Equations (4.1) and (4.2) for non-interface displacements, ITT n

U

and IBB n

U , and interaction forces at the interface, IiT n

F and IiB n

F , simultaneously.

By comparing the obtained forces at the interface of soil and structure sub-domains

in STEP 2, the following two scenarios can occur:

STEP 3: If convergence to equilibrium has not been achieved, a new estimation of

the interface displacements is applied to the structure and soil sub-domains, and

iteration continues (I=I+1) from STEP 2 until convergence to equilibrium is

achieved.

STEP 4: If convergence to equilibrium at the interface of partitioned domains has

been achieved, the solution proceeds to the next time/load increment (n=n+1).

Figure 4.7: Schematics of Parallel D-D iterative coupling

0 0I Ii iT Bn n

U U

1IiB n

U 1Ii

T nU

0 0I Ii iB Tn n

U U

0 0

1 1

I Ii iT Bn n

U U

0 0

1 1

I Ii iT Bn n

U U


105

4.2.6 Parallel Neumann -Neumann Iterative Coupling

The Parallel Neumann–Neumann coupling algorithmic steps are described as follows




STEP 1: At the start of each increment, the structure sub-domain and the soil sub-

domain are loaded by the external forces, TT n

F and BB n

F , while the forces

(Neumann data) at the interface nodes of the partitioned sub-domains IiT n

F and

IiB n

F are prescribed in accordance with the initial conditions (preferably enforcing

equilibrium):

0 0I Ii iT Bn n

F F F

STEP 2: The structural and soil solvers compute the response of the structure and

soil concurrently using Equations (4.1) and (4.2) for non-interface displacements,

ITT n

U and IBB n

U , and interaction displacements at the interface, IiT n

U and IiB n

U ,

simultaneously.

By comparing the obtained displacements at the interface of soil and structure sub-

domains in STEP 2, the following two scenarios can occur:

STEP 3: If convergence to compatibility has not been achieved, a new estimation of

the interface forces is applied to the structure and soil sub-domains respectively, and

iteration continues (I=I+1) from STEP 2 until convergence to compatibility is

achieved.

STEP 4: If convergence to compatibility at the interface of partitioned sub-domains

has been achieved the solution proceeds to the next time/load increment (n=n+1).


106

Figure 4.8: Schematics of Parallel N-N iterative coupling

4.3 Treatment of Interactive Boundary Conditions

The presented sequential and parallel coupling algorithms can be used in partitioned

treatment of the soil-structure interaction problems. However, the algorithms in

which the partitioned structure sub-domain is considered as subjected to Dirichlet

(displacement) boundary conditions at the interface, while the partitioned soil sub-

domain is subjected to a Neumann (force) or Dirichlet (displacement) boundary

condition at the same interface, are more suitable in the context of static soil-

structure interaction analysis. This is purely due to the fact that in static problems

only essential Dirichlet boundary conditions can be imposed on the interface of the

structure sub-domain, where applying the natural Neumann boundary conditions at

this interface results in singularity of the equilibrium equations for the structural sub-

domain. In fact, treating the structure interface with Neumann boundary conditions

in such a problem requires some additional conditions to make the structure sub-

domain solvable.

The above requirement narrows the choice of coupling algorithms in a general soil-

structure interaction analysis to Dirichlet-Neumann or Dirichlet-Dirichlet family of


107

iterative coupling schemes. Therefore the subject of this research is to present

domain decomposition methods for coupled analysis of soil-structure interaction

problems, with particular emphasis on Dirichlet-Neumann types of iterative

schemes, specifically the sequential D-N algorithms.

4.4 Convergence of Iterative Coupling

Despite the significant potential benefits of iterative coupling methods, a major issue

relates to whether convergence to equilibrium and compatibility at the interface can

always be enforced through successive iterations. Thus, besides choosing a suitable

iterative scheme it is important to address the issues related to the convergence

behaviour of the selected iterative scheme.

Achieving convergence and reasonable computational efficiency in the iterative

coupling procedures is directly dictated by the chosen update technique of interface

displacements and/or forces. This is demonstrated by considering the convergence

behaviour of an FEM-FEM coupled test system treated by a sequential D-N iterative

scheme with a trivial update of boundary conditions.

4.4.1 Convergence of Sequential D-N with Trivial Update

The objective of this section is to elaborate the basic procedure used for convergence

analysis of sequential D-N iterative coupling algorithm in the case of trivial update

of boundary conditions. Consider the following iterative coupling solution form:

New Oldcte (4.5)

in which λ and cte are constant parameters.

The subsequent iterations for the above equation, in which the solution process is

started by prescribing some arbitrary initial value 0I , would take the

following form (with variable I denoting the iteration number):


108

1C (4.6)

2 1C (4.7)

3 2C (4.8)

and in general:

1I IC (4.9)

Considering Equations (4.6), (4.7) and (4.8) the solution error, , in different

iteration stages can be written as:

2 2 1 1 (4.10)

3 3 2 2 1 12 (4.11)

Therefore, generally the solution error after I iterations would take the following

form:

11I I (4.12)

Considering Equation (4.12) the convergence of the presented coupling algorithm

can now be related to which can be viewed as the error reduction indicator.

According to Equation (4.12) for the successive approximation 1I Icte

to converge, 0n for any initial value of , it is necessary and sufficient

that is less than one. This can be extended to the matrix form as, for a successive

approximation 1I Icte to converge for any initial vector , it is


109

necessary and sufficient that the moduli of all eigenvalues of the error reduction

matrix, , are less than one.

Using this theorem we can investigate the convergence behaviour of the iterative

coupling algorithms by constructing the matrix during successive iterations. This

can be illustrated by considering a soil-structure interaction SDOF test system,

decomposed into the soil sub-domain ( B ) and the structure sub-domain ( T ),

treated by the sequential D-N iterative coupling procedure. Assume that each sub-

domain is discretised in time using the Newmark step-by-step integration method

and that both sub-domains are condensed at the interface. The governing equilibrium

equations for the partitioned soil and structure domains of the above mentioned

system are given as:

Structure sub-domain ( T ):

i i i iT T T T T T TM U C U K U F (4.13)

Soil sub-domain ( B ):

i i i iB B B B B B BM U C U K U F (4.14)

Following the steps of the sequential D-N coupling algorithm presented in Section

4.2.1 and considering a trivial update of boundary conditions at the interface in

successive iterations ( 1I Ii iT Bn n

U U ), the error reduction factor of this scheme can

be defined by considering a typical iteration:

STEP 1: Set the initial interface displacements, IiT n

U , for the structure sub-domain.

STEP 2-3: Compute the response of the structure forces at the interface and apply

equilibrium:


110

221 1

1 ˆ ˆI Ii i i iT T T T T T T T Tn n n n

F M K t C t U M U C Ut

(4.15)

1 1

I Ii iB Tn n

F F (4.16)

STEP 4: Solving soil domain for displacements at the interface:

2

1

21

ˆ ˆ ˆ ˆIi i i i iT T T T T T B B T T B BnI n n n ni

B nB B B

M K t C t U M U M U C U C U

UM K t C t

(4.17)

STEP 5: If convergence has not been achieved (i.e. 1 1

0I Ii i

B Tn nU U

) the new

estimation of the displacements will be applied to the structure sub-domain:

1

1 1

I Ii iT Bn n

U U

(4.18)

Now, Equation (4.17) can be rewritten in a new form as:

2

21 1

I Ii iT T TB Tn n

B B B

M K t C tU U cte

M K t C t

(4.19)

It can be shown that the compatibility default at the interface after K iterations can

be obtained from the following difference equation:

0 0.

I K I K I II K i i K i iB T B Tn n n n

U U U U U (4.20)

with:

2

2

1/ ( / )

1/ ( / )

T T T

B B B

K t M t C

K t M t C

(4.21)


111

The term 2 21 / ( / )K t M t C in the above equation is usually referred to

as the effective stiffness ( effectiveK ).

Comparing Equations (4.20) and (4.12), the necessary and sufficient condition for

convergence of the presented coupled test system can be written as:

2

2

1/ ( / )1

1/ ( / )

T T T

B B B

K t M t C

K t M t C

(4.22)

In the case of static analysis, this condition would simplify to:

1T

B

K

K (4.23)

Equation (4.22) shows that for the presented sequential D-N iterative coupling

algorithm with trivial update of boundary conditions to converge for any initial

prescribed displacement value, the condensed effective stiffness of the sub-domain

treated by Dirichlet data at the interface should be less than the condensed effective

stiffness of the sub-domain treated by Neumann boundary conditions. Here,

condensation is a process by which some of the degrees of freedom are eliminated

from the overall equilibrium providing a reduced set of equilibrium equations for the

remaining degrees of freedom.

To elaborate on the condensation procedure, consider the following system of

equations in which the condensed stiffness matrix corresponding to the interface

degrees of freedom iU is required:

11 12 x x

21 22 i i

K K U F=

K K U F (4.24)

Expansion of Equation (4.24) gives:

11 x 12 i xK U K U F (4.25)


112

21 x 22 i iK U K U F (4.26)

Substituting Equation (4.25) in (4.26) gives:

* *iK U = F (4.27)

where

* -122 21 11 12K = K - K K K (4.28)

1 *i 21 11 xF F K K F (4.29)

Matrix *K is called the condensed stiffness matrix.

The fact that the convergence behaviour is influenced by the ratio of the condensed

effective stiffness of the partitioned sub-domains can be further established by

constructing the general convergence behaviour of the example presented in the next

section.

4.4.2 Example

Here we examine the convergence behaviour and the effect of mass and stiffness of

coupled domains in a partitioned interaction problem treated by sequential D-N

iterative coupling algorithm using a trivial update of boundary conditions at the

interface. Consider the dynamic problem of the mass-spring system illustrated in

Figure 4.9a, where the problem is partitioned into two sub-domains B and T as

shown in Figures 4.9b and 4.9c respectively.


113

Figure 4.9: a) coupled mass-spring system, b) partitioned sub-domain B , and c)

partitioned sub-domain T

The governing equilibrium condition for the partitioned domain T can be written

in the form of:

1 1

ˆ0

ˆ

TT TTeffective T T

T Ti ii

T Tn n T n

UU FK M

U F U

(4.30)

with

1

2

0

0T

MM

M

(4.31)

1 2 2

2 2T

K K KK

K K

(4.32)

iBU

iBF iTU

iTF


114

2

1effectiveT T TK K M

t

(4.33)

Similarly the governing equilibrium condition for partitioned domain B can be

given by:

1

ˆ

ˆ

ii iBeffective B B

B BB BB

B Bn B n

UU FK M

U F U

0

(4.34)

with

3

4

0

0B

MM

M

(4.35)

3 3

3 3 4B

K KK

K K K

(4.36)

2

1effectiveB B BK K M

t

(4.37)

Following the steps of the sequential Dirichlet-Neumann iterative algorithm for

coupling the above partitioned sub-domains, where the Dirichlet boundary condition

is applied to the sub-domain T and the Neumann boundary condition on sub-

domain B , the following recurrence relationship is obtained:

1 1

I Ii iB Tn n

U U cte (4.38)

where:

2 2 4 23 4 4 2 1 2 2 1 2 2 1 2 1

2 2 4 21 2 1 4 3 3 3 3 4 4 3 4 3

K K t M K K t M K M K M K t M M


(4.39)


115

The essential and sufficient requirement for the above system to converge using the

sequential Dirichlet-Neumann algorithm as demonstrated before would be:

2 2 4 23 4 4 2 1 2 2 1 2 2 1 2 1

2 2 4 21 2 1 4 3 3 3 3 4 4 3 4 3

1K K t M K K t M K M K M K t M M


(4.40)

The convergence condition of Equation (4.40) can also be derived by constructing

the following ratio between the condensed effective stiffness of the partitioned sub-

domains as obtained before:

Condensed effective stiffness of sub-domain at the interface1

Condensed effective stiffness of sub-domain at the interfaceT

B

(4.41)

Considering Equations (4.33) and (4.37), and applying the condensation process of

Equation (4.28) the exact convergence condition of Equation (4.40) can be achieved:

22 2

2 21

1 2 2

23 3

3 24

3 4 2

1

M KK

Mt K Kt

M KK

Mt K Kt

(4.42)

Considering the above convergence condition for very small time steps (i.e. 0t ):

22 3

03

lim 1t

MM M

M

(4.43)

Equation (4.43) shows that if the mass at the interface is assigned such that the mass

of the domain treated by Dirichlet boundary condition is less than that of the sub-

domain treated by Neumann boundary condition, the convergence criteria of the

algorithm is satisfied in the limit as the time step becomes very small. It is important

to point out that the above condition is valid only if 22

0lim 0

tK t

and


116

23

0lim 0

tK t

, which would not be the case if one of the corresponding springs is

rigid. For example, if the spring associated with 2K is rigid 2K , it can be

shown mathematically that:

2

1 21 2 3

03

lim lim 1 ( )t K

M MM M M

M

(4.44)

In fact due to the infinite stiffness 2K , the mass participating at the interface level at

partitioned sub-domain T would be 1 2iTM M M , which according to

Equation (4.43) should be less than 3iBM M (i.e. i i

B TM M ). The same argument

holds for the case of 3K , where in this case 1iTM M should be less than

3 4iBM M M :

3

22 3 4

03 4

lim lim 1 ( )t K

MM M M

M M

Considering the above conditions, it can be concluded that in the sequential D-N

iterative coupling algorithm with trivial update of boundary conditions a conditional

coupling convergence can be achieved. In fact, the convergence can only be

achieved provided that the problem under consideration allows for suitable

partitioning strategies at the interface that satisfy the condition of Equation (4.41).

The conditional convergence behaviour of the scheme employing trivial update of

boundary conditions is unsatisfactory, since the effective stiffness for the structure

domain at the interface is larger than that of the soil domain for numerous real

problems regardless of the time step, while for other problems unrealistically small

time steps would be required to satisfy Equation (4.41). This illustrates the need for

further improvements to the sequential D-N iterative coupling scheme, where

different update techniques are proposed in Chapters 5 and 6 (relaxation and reduced

order methods), and their convergence characteristics are addressed

comprehensively. The assessment of these proposed techniques is carried out


117

through the development and application of an iterative coupling simulation

environment which is described in the next section.

4.5 Simulation Environment

In order to deal with different coupling methods utilizing robust update techniques

for nonlinear-soil structure interaction problems, as proposed in Chapters 5, 6 and 7,

a novel iterative coupling simulation environment is developed, utilizing discipline-

oriented solvers for nonlinear structural and geotechnical analysis. The developed

simulation environment is used to demonstrate the relative performance

characteristics and merits of various presented algorithms. This tool is also applied in

a number of case studies involving nonlinear soil-structure interaction with

nonlinearity in both structure and soil, thus leading to important conclusions

regarding the adequacy and applicability of various coupling techniques for such

problems as well as the prospects for further enhancements.

The simulation of soil-structure interaction via the partitioned approach is carried out

in this work through the coupling of two powerful FEM codes, ADAPTIC (Izzuddin,

1991) and ICFEP (Potts & Zdravkovic, 1999) that have been developed at Imperial

College London for advanced nonlinear structural and geotechnical analysis,

respectively.

It should be emphasised that although the approach developed in this work is applied

to the coupling of ADAPTIC and ICFEP, the related algorithms are general in nature

and equally applicable to the coupling of other existing nonlinear soil and structural

analysis tools.

4.6 ADAPTIC

ADAPTIC (Izzuddin, 1991) is an adaptive static and dynamic structural analysis

program which has been developed to provide an efficient tool for the nonlinear analysis


118

of steel and composite frames, slabs, shells and integrated structures. The program

features are described briefly hereafter.

The development of ADAPTIC was initially driven by the needs of the offshore industry

for accurate and efficient, nonlinear analysis of offshore structures subject to extreme

loading conditions. This motivated the development of pioneering adaptive nonlinear

dynamic analysis techniques for framed structures, accounting for geometric and

material nonlinearity. The program has been extensively developed to deal with different

extreme loading, such as earthquake, fire and blast, as well as numerous additional

structural forms, such as reinforced concrete and steel-decked composite slabs, cable and

membrane structures, and curved shells.

In ADAPTIC, inelastic analysis of steel frames may be performed by either of two

methodologies. The first is an approximate solution using ideal plastic hinge elements,

while the second is a more accurate solution employing elements which account for the

spread of plasticity across the section depth and along the member length. For reinforced

concrete and composite frames, inelastic analysis is performed using the second

approach only. The ADAPTIC library includes a number of uniaxial material models

which can be used to model steel, concrete and other materials with similar behavioural

characteristics.

In ADAPTIC the loading can be either in the terms of applied forces or prescribed

displacements/accelerations at nodal points. The loads can vary proportionally under

static conditions, or can vary independently in the time or pseudo-time domains. The

latter variation can be utilised for static or dynamic analysis. Different types of analysis

in ADAPTIC are namely:

static analysis with proportional loading,

static analysis with time-history loading,

dynamic analysis,

eigenvalue analysis.


119

The following show some of the various assumptions that can be made using

ADAPTIC in structural modelling:

elastic modelling,

geometric and material nonlinearity modelling,

novel frame, plate, membrane and shell elements,

extreme loading modelling (e.g. static, dynamic and fire),

plastic hinge modelling,

elasto-plastic modelling,

adaptive elasto-plastic modelling,

joints and boundary conditions.

4.7 ICFEP

ICFEP (Potts & Zdravkovic, 1999) is a powerful finite element program specifically

written for the analysis of geotechnical engineering problems. It has been used for

many numerical analysis research projects at Imperial College and is continually

being developed. ICFEP has successfully been applied to numerous practical

engineering projects. The comprehensive formulation of ICFEP makes it possible to

consider a very wide range of problems. The most common applications are the

analysis of deep basement excavations, embankments, slopes and tunnelling.

The behaviour of soil is in general highly nonlinear. In many practical situations

some part of the soil is in a state of yield. The global stiffness equation is nonlinear

and therefore its accurate solution is not simple. The ability of a finite element

program to obtain accurate results (in particular accurate failure or ultimate loads) is

very dependent on its solution technique. Commonly used solution methods are the

Tangent stiffness, Visco-plastic, and Newton-Raphson methods. ICFEP uses an


120

accelerated form of the Modified Newton-Raphson method incorporating a sub-

stepping stress point algorithm with automatic error control. This ensures the stress

at any point does not drift from the yield surface and violate the constitutive law. The

solution obtained is independent of the size of the load increment applied.

Convergence of displacements, rotations, pore fluid flux, forces, moments and pore

pressures is required before the solution is accepted.

The following types of analyses can be performed with ICFEP:

two-dimensional plane strain,

two-dimensional plane stress,

axi symmetric,

full 3D (Conventional and Fourier series aided) analysis,

drained and un-drained,

fully coupled consolidation,

large displacement,

cyclic,

partial saturation,

dynamics.

A number of different element types are available in order to model a particular

problem such as: four or eight nodded quadrilateral elements, eight or twenty nodded

hexahedral elements, four or sixteen nodded interface or joint elements and etc. The

followings show some of various constitutive modelling assumptions that can be

made using the ICFEP elements:

linear material behaviour (isotropic, transversely isotropic, anisotropic),

nonlinear and elasto-plastic material behaviour,

Mohr Coulomb - with/without strain hardening /softening,

Tresca - with/without strain hardening /softening,

Von Mises - with/without strain hardening /softening,


121

Drucker Prager,

Cam Clay - with/without Hvorslev surface, general or Mohr Coulomb shape

yield surface,

Modified Cam Clay - with/without Hvorslev surface, general or Mohr

Coulomb shape yield surface,

Dramen Clay - undrained and drained,

2 surface and 3 surface ‘bubble’ models,

Partly saturated soil models,

Boundary surface models for both clays and sand (e.g. MIT-E3, Papadimitriu

and Bouckovalas, respectively).

4.8 INTERFACE

Coupling of the ADAPTIC and ICFEP is carried out using a coupling program called

INTERFACE which utilises a sequential Dirichlet-Neumann type of iterative

coupling algorithms. The INTERFACE program is written in FORTRAN 95.

ADAPTIC and ICFEP run on separate processors as independent black box solvers,

where the task of communication and synchronization between the two individual

codes is achieved via INTERFACE that implements the iterative coupling methods

utilizing various update techniques. In this respect, the interface program manages

the retrieval, manipulation and passing the necessary data between the two field

programs during coupled analysis (See Figure 4.10).


122

Figure 4.10: Communication and synchronization between ADAPTIC and ICFEP,

via INTERFACE

4.9 Simulation Environment Architecture

The developed simulation environment is based on a client-server software

architecture, communicating over a computer network. Client-server architecture

refers to the relationship between a number of computer programs, in which the

client program makes a service request from other programs, the servers, which

execute the request.

The client software, here the INTERFACE program, can send data requests to the

connected servers, ADAPTIC and ICFEP, which in turn accept these requests,

process them, and return the requested information to the client. The important

characteristics of the INTERFACE as a client program are: i) initiation and

synchronisation of the requests to ADAPTIC and ICFEP, and ii) waiting for,

receiving and processing the replies from ADAPTIC and ICFEP. On the other hand,


123

ADAPTIC and ICFEP as server programs never initiate any activity without

receiving a request from the INTERFACE program.

In the developed simulation tool, ADAPTIC and ICFEP run on separate processors

as independent black box solvers and are started separately. After start-up, the two

programs wait until receiving a run request and prescribed interaction data from the

INTERFACE program in a specific sequence which is managed by the INTERFACE

program. Then the programs start to solve their own partitioned sub-domain as black

box solvers, during which the INTERFACE is waiting for the either program to

solve for equilibrium in its respective sub-domain under the prescribed interface

conditions and to return the complementary displacement/force conditions at the

interface boundary.

Using the obtained results, the state of coupling convergence at the interface of the

partitioned sub-domain is checked. If convergence is not achieved at the interface,

the INTERFACE program calculates the new estimates of the prescribed boundary

conditions and initiates the iterative coupling procedure within the same time/load

step in ADAPTIC and ICFEP. If convergence is achieved, the interface sends a

request of updating the initial condition for the next time-load increment to

ADAPTIC and ICFEP and assigns new prescribed initial boundary condition for the

structure partitioned sub-domains, moving the solution to the next time/load

increment. The sequence of interaction between the INTERFACE, ADAPTIC and

ICFEP programs is detailed in Figure 4.11.


124

Figure 4.11: Schematics of the interaction sequence between the INFERFACE,

ADAPTIC and ICFEP


125

4.10 Data Communication

As elaborated before, the communication system is based on client-server

architecture; therefore, only a few small subroutines are needed to link the

INTERFACE program to ADAPTIC and ICFEP. The task of these subroutines is to

receive requests from and send the required data to the interface. These subroutines,

provide a code-independent way for specification of relevant coupling parameters,

configurations and initiation of different procedures. This exchange of data between

the coupled codes is carried out using direct access formatted data files generated by

INTERFACE and accessible by both ADAPTIC and ICFEP.

For a given problem, the user specifies the input data files for ADAPTIC and ICFEP

corresponding to structure and soil partitioned sub-domains, respectively. These

contain the typical relevant information such as the finite element model, material

models, analysis type, etc. In addition to these, an input file for the INTERFACE

program is also specified which defines the coupling region, where the coupling

interaction occurs (soil-structure interface), for both the structure and soil partitioned

models. Moreover, the required number of time/load increments, the coupling

convergence criteria at the interface and the type of update technique required for the

coupling analysis are specified in the same data file. A sample structure of the

INTERFACE data file is given in Appendix A. After launching all three programs,

while ADAPTIC and ICFEP are waiting for the initiation request and the prescribed

interactive values to be posted by the INTERFACE, the INTERFACE starts

generating a formatted direct access FORTRAN data communication file based on

its own input data file. This file is just for communication purposes between the

programs, and is read-write accessible to all programs, where a sample structure is

given in Appendix B. After the communication data file has been generated by the

INTERFACE, the first calls of the augmented subroutines in ADAPTIC and ICFEP

codes identify the coupling regions taking part in the coupling process as described

in Section 4.9 (see Figure 4.12).


126

Figure 4.12: Data exchange structure

The coupling procedure continues until the required time/load steps are completed.

At this stage, both ADAPTIC and ICFEP produce the required results of their own

solved partitioned sub-domain in separate output files, which can be accessed

directly via their own post processing tools for analysing the results. In addition, the

effect of different coupling parameters on the convergence characteristics, including

the achieved compatibility and equilibrium defaults at the interface and the

convergence rate, can be analysed by considering the output file generated by the

INTERFACE program.


127


In this chapter, different iterative coupling methods are presented for partitioned

analysis of soil-structure interaction problems, where it is assumed that the overall

domain is divided into physical partitions consisting of soil and structure sub-

domains. The advantage of the iterative coupling schemes in which the partitioned

structure sub-domain is treated by Dirichlet boundary was discussed, and particular

emphasis is therefore given in this work to D-N iterative coupling methods.

It is shown that an important feature of the proposed approach that needs to be

addressed comprehensively is the convergence behaviour of the scheme, which is

directly dictated by the chosen update technique during successive iterations. This is

further illustrated by means of an example in which a trivial update of boundary

conditions is assumed during the coupling iterations. The obtained convergence

behaviour of the sequential D-N iterative coupling scheme employing trivial update

of boundary conditions demonstrates the need for further improvement in the update

of boundary conditions. Accordingly, the convergence behaviour of D-N iterative

coupling methods is addressed in the following chapters, where more powerful

methods for the update of boundary conditions are discussed and developed.

Finally, a coupling simulation environment that has been developed utilizing

discipline-oriented solvers for nonlinear structural and geotechnical analysis is

described. The software architecture of the developed simulation environment,

which is based on sequential D-N iterative coupling algorithms is outlined, and the

structure of the data exchange between the various codes is elaborated.

The developed tool is used in this research to assess the adequacy and applicability

of various coupling methods in nonlinear soil-structure interaction analysis,

including the new methods proposed in this work. In this context, the developed

approach is believed to offer great potential towards providing an integrated

interdisciplinary computational framework which combines the advanced features of


128

both structural and geotechnical modelling for a variety of challenging problems in

the field of nonlinear soil-structure interaction.

Chapter 5

Interface Relaxation

5.1 Introduction

In Chapter 4 it was discussed that the critical algorithmic stage of an iterative

coupling algorithm such as sequential Dirichlet-Neumann (D-N) is the evaluation of

new estimates of the interface Dirichlet data during the coupling iterations. In fact, it

was shown that this determines the convergence characteristics to compatibility at

the interface (noting that the equilibrium condition at the interface is always satisfied

by prescribing the corresponding interaction forces). It was also demonstrated that in

sequential D-N using a trivial update of boundary conditions for FEM-FEM

coupling, depending on the status of the relative stiffness of the partitioned sub-

domains, only a conditional convergence could be achieved.

In order to improve the unsatisfactory convergence behaviour of using trivial update

of boundary conditions, frequently a relaxation of the iteratively updated Dirichlet

boundary conditions is augmented to the iterative coupling algorithms, hence the

term interface relaxation (Marini & Quarteroni, 1989; Elleithy & Tanaka, 2003).

Employing a relaxation scheme is one of the most common techniques used for

Chapter 5 Interface Relaxation

130

successive update of the boundary conditions in iterative coupling algorithms. In this

method, the convergence to compatibility of the presented D-N coupling algorithm is

accelerated and ensured by employing a suitable relaxation parameter. In this

respect, considering the sequential D-N iterative coupling algorithm presented in

Chapter 4, the update of boundary conditions at the interface level (in STEP 5) using

the relaxation technique can be expressed for a specific iteration I as:

11

I I Ii i iT I T I Bn n n

U U U (5.1)

In the above, I is a real positive parameter, which is called the relaxation parameter

and can improve convergence of the iterative coupling scheme.

It is clear from Equation (5.1) that the interface relaxation method possesses the

benefits of being simple to implement and having an undemanding process of

estimating new updates of boundary conditions in corrective iterations.

Although the superiority of the interface relaxation coupling, as highlighted in

Chapter 2, has been recognised more recently for coupled modelling of soil-structure

interaction (Hagen & Estorff, 2005a; Hagen & Estorff, 2005b; Elleithy et al., 2004)

there remain significant technical challenges related to algorithmic and

computational issues, particularly with reference to convergence issues.

In this chapter, the convergence characteristics of algorithms based on the sequential

Dirichlet-Neumann iterative coupling method are investigated for soil-structure

interaction problems. The efficiency of employing a constant relaxation scheme to

enhance the convergence characteristics and the effect of the partitioned sub-domain

properties on these characteristics are demonstrated through a number of case

studies. Furthermore, it is proposed that the performance of iterative coupling

methods for FEM-FEM coupling may be effectively enhanced through the use of an

adaptive relaxation scheme. Similar approaches have been introduced by Funaro et

al. (1998) for iterative coupling of partitioned second-order elliptic problems and by

Wall et al. (2007) focusing on fluid-structure interaction. In this respect, an


131

analogous procedure for evaluating the adaptive relaxation parameter in the context

of soil-structure interaction using FEM-FEM coupling is proposed here, and its

convergence characteristics is investigated. It is shown that, in contrast to the

traditional relaxation scheme, in which the relaxation parameter is typically

evaluated by trial and error, an adaptive relaxation scheme offers improved prospects

for achieving convergence and computational efficiency in complicated large scale

nonlinear problems. The evaluation of such prospects and the comparison of

different relaxation schemes are therefore primary objectives of this chapter,

particularly considering soil-structure interaction problems with nonlinearity in both

structure and soil.

5.2 Constant Relaxation

As mentioned before, the use of a relaxation scheme as in Equation (5.1) can

enhance convergence. Here, it is assumed that the relaxation parameter is a constant

positive real parameter during the coupling iterations (i.e. 1,..., I ), that should

be specified prior to the start of coupling analysis. Since the constant relaxation

parameter should be determined in advance, there are two major issues regarding the

applicability of such a technique, namely: i) determination of the range of suitable

relaxation parameters for the specific problem under consideration in order to

achieve convergence, and ii) selection of the optimum relaxation parameter in order

to achieve maximum computational efficiency.

Previous convergence analysis of sequential D-N coupling with trivial update of

boundary conditions for a SDOF test system at the interface, as presented in Chapter

4 (Section 4.4.1), demonstrated that convergence to compatibility is guaranteed for

any initial value, if:

2 2

Structure2 2

Soil

1/ ( / ) (Effective Stiffness)1

(Effective Stiffness)1/ ( / )

effectiveT T T T

effectiveBB B B

K t M t C K

KK t M t C

(5.2)


132

It can be shown that by introducing a suitable constant relaxation parameter, not only

an unconditionally convergent algorithm may be achieved, but also significant

improvement in the convergence rate may be attained by prescribing an optimum

relaxation parameter.

This can be illustrated by considering the same SDOF soil-structure interaction test

system and following the sequential D-N coupling algorithmic steps, while

employing the constant interface relaxation:

11

I I Ii i iT T Bn n n

U U U (5.3)

Assume that the following relationship holds at every iteration stage:

I Ii iB Tn n

U U cte (5.4)

with,

2

2

1/ ( / )

1/ ( / )

T T T

B B B

K t M t C

K t M t C

(5.5)

Substituting subsequently Equation (5.4) for K iterations while employing Equation

(5.3) for renewal of boundary conditions, the compatibility error at the interface

would take the following form of:

0 01 ( 1)

K K I IKi i i iB T B Tn n n n

U U U U

(5.6)

Now, it can be easily shown that for this relaxation scheme to converge,

1 1 should be less than one:

1 1 1 1 1 1 1 (5.7)

The above convergence condition restricts the relaxation parameter to be a positive

value less than two, as shown below:


133

1 1 1 0

0 221 1 1

1

(5.8)

Figure 5.1 shows the 1 1 value against the relaxation parameter for

different assumed values. Considering Equations (5.8) and (5.7), it can be shown

that for all possible partitioned soil and structure sub-domains stiffness ratios,

0, , there exists a range of relaxation parameters 0, 2 that

guarantees convergence to compatibility at the interface.

Figure 5.1: Variation of error reduction factor against relaxation parameter

The figure also illustrates that there exists an optimum relaxation value that ensures

convergence at the highest rate:

0 ( ) 1Optimum (5.9)

In fact, the optimum relaxation parameter corresponds to 0 , leading to:


134

1 1 ( )

11

EffectiveB

Effective Effective EffectiveT B TEffectiveB

KOptimum

K K KK

(5.10)

5.2.1 General Convergence Analysis

Discretization of a linear coupled soil-structure system can be described in general

by:

K U = F (5.11)

where K , U and F represent the global stiffness matrix (effective stiffness

matrix for dynamic analysis), the displacement vector and external force vector

(effective force vector for dynamic analysis) of the coupled soil-structure interaction

system.

Assuming that the coupled soil-structure system is composed of soil B and

structure T sub-domains, Equation (5.11) can then be re-written in its general

form as:

11 12

21 22 22 21

12 11

0

0

T T T TT T

T T B B i iext

B B B BB B

K K U F

K K K K U F

K K U F

(5.12)

where vectors XXU and X

XF correspond to displacements and external loads

respectively, for the non-interface degrees of freedom in sub-domain X , while

vectors iU and iextF correspond to displacements and external loads

respectively, for the interface degrees of freedom.


135

Henceforth for the sake of simplicity, but without loss of generality, it is assumed

that there is no external load applied at the interface of the coupled system (i.e.

0iextF ).

Decomposing the coupled system of Equation (5.12) into two sub-domains

according to their physical and material properties, namely soil and structure sub-

domains, the soil and structure sub-domains can be formulated independently,

where, without loss of generality, the response of individual sub-domains are

assumed to be linear elastic:

Governing equilibrium conditions for partitioned structure sub-domain:

11 12

21 22

T T T TT T

T T i iT T

K K U F

K K U F

(5.13)

Governing equilibrium conditions for partitioned soil sub-domain:

11 12

21 22

B B B BB B

B B i iB B

K K U F

K K U F

(5.14)

In the above, vectors iXU and i

XF correspond to displacements and external loads

for the interface degrees of freedom in sub-domain X , respectively.

The response of the above separately modelled sub-domains can now be coupled by

enforcing compatibility and equilibrium conditions at the interface using a sequential

D-N iterative coupling scheme, as follows:


external forces TT n

F , while the displacements at the interface nodes, iT n

U , are

prescribed in accordance with the initial conditions:

IiT n

U U (5.15)


136

where superscript I and subscript n denote iteration and increment numbers,

respectively.

STEP 2: The structural solver computes the response of the structure, using

Equation (5.13), for ITT n

U and IiT n

F :

-1

11 12

I I IT T T T iT T Tn n n

U K F K U (5.16)

21 22

I I Ii T T T iT T Tn n n

F K U K U (5.17)

It should be noted here that although Equations (5.16) and (5.17) are for a linear

response, the same entities can also be readily obtained for a nonlinear response from

the model of the structural sub-domain.

STEP 3: The corresponding interface forces at the soil sub-domain can be calculated

by applying equilibrium:

0I Ii i

B Tn nF F (5.18)

STEP 4: Based on these forces IiB n

F and the external loading applied to the soil

domain IBB n

F , the soil solver computes the response of the soil domain, using

Equation (5.14), for IiB n

U and IBB n

U :

1

11 12

21 22

I IB BB B

B Bn n

B BI Ii iB Bn n

U FK K

K KU F

(5.19)

Again, it is noted that although Equation (5.19) is for a linear response, the same

entities can also be readily obtained for a nonlinear response from the model of the

soil sub-domain.


137

STEP 5: If convergence to compatibility has not been achieved, the new estimation

of the displacements will be applied to the structure domain:

If 2

2

I Ii iB Tn n

U U , then

11


U U U

1I I , go to STEP2

where is a real positive constant relaxation parameter

STEP 6: If convergence to compatibility at the interface of partitioned sub-domains

has been achieved, the solution will proceed to the next increment:

If I Ii iB Tn n

U U , then 1n n , go to STEP 1

The convergence characteristics of the method can be now established by

considering the above algorithmic steps and constructing the error reduction matrix.

Using Equations (5.17), (5.16) and (5.19), the interface displacements of the soil

sub-domain at iteration I , IiB n

U , can be written as:

I Ii iB Tn n

U U cte (5.20)

where,

1B TC CK K

(5.21)

1 1 1 -1

21 11 21 11

I IB B B B B T T TC B C Tcte K K K F K K K F

(5.22)

It is worth mentioning, that [λ] in Equation (5.21) is positive, where λ in Equation

(5.5) is assumed to be negative.


138

It is noted that, as evident from Equations (5.21) and (5.22), cte would be constant

during the coupling iterations at a specific time/load increment, while TCK and

BCK represent the condensed stiffness matrix of the soil and structure sub-

domains, respectively, corresponding to the interface degrees of freedom.

-1

22 21 11 12T T T T TCK K K K K (5.23)

-1

22 21 11 12B B B B BCK K K K K (5.24)

Since the equilibrium condition is automatically satisfied in the above algorithm,

only convergence to compatibility should be checked during the coupling iterations.

Considering Equation (5.20), the interface displacement values at the structure and

soil sub-domains at different iteration stages are obtained as:

0

0 0Iteration no. 0

iT n

i iB Tn n

U U

U U cte

(5.25)

1 0 0

1 1

1Iteration no. 1

i i iT B Tn n n

i iB Tn n

U U U

U U cte

(5.26)

2 1 1

2 2

1Iteration no. 2

i i iT B Tn n n

i iB Tn n

U U U

U U cte

(5.27)

Accordingly, the compatibility defaults at each of the above iteration stages are

given by:

0 0 0i i iB T Tn n n

U U U cte I (5.28)


U U U cte I (5.29)


139


U U U cte I (5.30)

By substituting the value of 1iT n

U from Equation (5.26) in Equation (5.29) the

compatibility default at iteration number 1 takes the following form:

1 1 0 01i i i i

B T B Tn n n nU U U U cte I (5.31)

Expanding Equation (5.31) leads to:

1 1

0 0 0

i iB Tn n

i i iB T Tn n n

U U

U U U cte

I I (5.32)

Substituting Equation (5.28) in the above gives:

1 1

0 0 0 0

i iB Tn n

i i i iB T B Tn n n n

U U

U U U U

I (5.33)

The above equation can be further simplified to:

1 1 0 0i i i iB T B Tn n n n

U U U U I I (5.34)

Similarly it can be shown that the compatibility default at iteration number 2 takes

the following form:

2 2 1 1i i i iB T B Tn n n n

U U U U I I (5.35)

and in general:

1 1I I I Ii i i iB T B Tn n n n

U U U U

I I (5.36)


140

Consequently, at iteration number K using a constant relaxation parameter, the

compatibility default at the interface of the decomposed soil-structure system, can be

obtained by the following difference equation:

0 0KK Ki i i iB T B TU U U U I I (5.37)

In view of Equation (5.37), the convergence of the presented coupling algorithm can

now be related to the eigenvalues of matrix K I I , where the matrix

can be regarded as the kth error reduction factor.

For the above successive iteration process to converge for any initially prescribed

value, it is necessary and sufficient that all eigenvalues of the matrix

I I be less than one in modulus:

1 1 1, 1,...,i i p (5.38)

where, i is the i th eigenvalue of matrix .

Assuming that eigenvalues of are positive, which is true for a positive definite

as would be the case in linear analysis, the above condition simplifies to:

max1 1 1 (5.39)

where, max is the largest eigenvalue of matrix .

Considering Equations (5.39) and (5.21), it can be shown that for all possible

partitioned soil and structure sub-domains, there exists a range of relaxation

parameters that guarantee convergence to compatibility of the coupling algorithm:

max

20 2

1

(5.40)


141

Moreover, the best choice of relaxation parameter, opt , as depicted in Figure 5.2, is

that for which:

max min1 1 1 1opt opt (5.41)

where, min is the smallest eigenvalue of matrix .

Therefore, the optimum relaxation value that ensures the convergence and holds the

best convergence rate can be defined as:

max min

1 10 1

1 12

optav

(5.42)

Figure 5.2: Convergence range and optimum relaxation parameter

The above convergence analysis, illustrated in Figure 5.2, clearly indicates that

convergence to compatibility in sequential D-N iterative coupling algorithms can be

ensured by using the interface relaxation update technique, at least if the coupled

system under consideration is relatively linear. Indeed, by predefining the relaxation

parameter in the range given by Equation (5.40), the reduction of compatibility error

at the interface is guaranteed. Moreover, Equation (5.42) demonstrates the existence

of an optimum relaxation parameter in the convergent range for which not only the


142

convergence to equilibrium is guaranteed, but also the convergence rate is optimum.

Furthermore, the theoretical aspects regarding the convergence behaviour as

discussed above show that the convergence characteristics of the constant relaxation

scheme are highly sensitive to the partitioned domain parameters, specifically the

condensed stiffness of the partitioned sub-domains at the interface.

Notwithstanding, in practice finding the suitable range of relaxation parameters and

the optimum constant relaxation parameter in soil-structure interaction problems

with multi degrees of freedom at the interface is difficult, since it requires first

constructing the matrix and then solving the associated eigenvalue problem, not

to mention additional complications for nonlinear coupled soil-structure systems.

Therefore, in order to avoid this process, the determination of an optimum constant

relaxation parameter must rely on trial and error, which may be applied to the initial

part of the coupled simulation after which full coupling analysis may be carried out.

5.2.2 Convergence Studies

The previous theoretical study has shown that the convergence of the constant

relaxation scheme for D-N iterative coupling is highly sensitive to the partitioned

sub-domain parameters, specifically the condensed stiffness of the partitioned sub-

domains at the interface. Here, these findings are demonstrated through some

illustrative FEM-FEM coupling examples.

5.2.2.1 Example 1: Dynamic FEM-FEM Coupling

The first example is a representative dynamic FEM-FEM coupled problem treated by

D-N iterative coupling scheme with a constant relaxation, where consideration is

given to a cantilever beam (Figure 5.3) under dynamic loading applied to its fixed

end (Figure 5.4). The length of the cantilever beam is 20m (L=10m), with a

rectangular cross section of 20.2 0.2 m . Mass of the system is modelled with two

concentrated masses of M1=2000 kg and M2=1000 kg, in the middle and the free

end of the cantilever respectively. Material modelling of the beam is described in


143

Figure 5.3, where 9 2210 10E Nm , 0.01 and 6 2300 10y Nm . The above

system is partitioned into two sub-domains, namely T and B , as shown in Figure

5.5, with three degrees of freedom at the interface (one rotational and two

translational).

Figure 5.3: Dynamic FEM-FEM coupling


y

y

E

E

E


144

The partitioned problem of Figure 5.5 is analysed employing a constant relaxation

coupling algorithm, where sub-domain T is treated by Dirichlet boundary

conditions (displacements) while sub-domain B is treated by Neumann boundary

conditions (forces).

Figure 5.5: Partitioned sub-domains

Firstly, this problem is analysed to illustrate the trial and error process embedded in

the coupling procedure via constant relaxation scheme, for finding the convergence

range and the optimum relaxation parameter. In this regard, in order to demonstrate

the effect of the condensed effective stiffness matrix on the convergence behaviour,

the proposed problem is analysed for different problem partitioning types. This is

achieved by analysing the same problem with different mass ratios at the interface of

the partitioned sub-domains T and B ( 1 2/m m ). The different analysed models,

including the considered partitioned mass ratios at the interface, range of suitable

relaxation parameters and the optimum relaxation parameter associated with the least

computational cost, are presented in Table 5.1. These values have been obtained by

an initial process of trial and error for each model under consideration for 100 steps

with 0.01st , where the relaxation parameter is chosen as a real value in the range

of ]0, 2[ according to Equation (5.8).


145

Model 1

2

m

m Range of Suitable

Relaxation parameters Optimum Relaxation

S1 200 kg

0. 11800 kg

]0,1.14] [0.8,0.9]

S2 400 kg

0.251600 kg

]0,1.13] [0.7,0.8]

S3 600 kg

0.431400 kg

]0,1.1] [0.6,0.7]

S4 800 kg

0.61200 kg

]0,1] 0.6

S5 1000 kg

1.01000 kg

]0,0.9] 0.5

S6 1200 kg

1.5800 kg

]0,0.7] 0.4

S7 1400 kg

2.3600 kg

]0,0.5] 0.3

S8 1600 kg

4.0400 kg

]0,0.35] 0.2

S9 1800 kg

9.0200 kg

]0,0.15] 0.1

Table 5.1: Range of suitable and optimal relaxation parameter for different 1

2

m

m

Figure 5.6 shows the total number of coupling iterations required for various

constant relaxation parameters and the applicable range of relaxation parameters

ensuring convergence to compatibility. A tolerance of ε = 10-4 m was set for the error

based on the norm of compatibility defaults:

I Ii iB Tn n

U U M (5.43)

where M corresponds to the number of coupled degrees of freedom at the interface,

which is equal to 3 in this case.


146

Figure 5.6: Influence the effective mass ratio on convergence

The results confirm that the convergence characteristics of the D-N iterative scheme

using constant relaxation is very sensitive to the chosen relaxation parameter,

rendering its selection a very difficult task, as evidenced by the significant increase

in number of iterations for model S8, between α0.2 with 126 coupling iterations,

α0.1 with 307 coupling iterations and α0.3 with 331 coupling iterations.

Considering Figure 5.6 more closely, the optimum relaxation parameter for each

model is defined as the value corresponding to the minimum number of iterations

(see Table 5.1). The fact that the optimum relaxation parameter varies and is

sensitive to the partitioned model stiffness ratios is confirmed by the obtained

results.

The dependence of optimum relaxation and convergence rate on the partitioned

model characteristics (effective mass/stiffness) can be further demonstrated by

considering the same problem with different ratios of elastic modulus ( /T BE E )

for model S5 ( i.e. 1 2 1000m m kg ). The different analysed models, including the


147

considered elastic modulus ratios at the interface, range of suitable relaxation

parameters and the optimum relaxation parameter associated with the least

computational cost, are presented in Table 5.2.

Model Range of Suitable Relaxation parameters

Optimum Relaxation

K1 5.0

]0,0.27] 0.2

K2 4.0

]0,0.45] 0.35 K3 2.0

]0,0.75] [0.4,0.5]

K4 1.0 ]0,0.9] 0.5

Table 5.2: Range of suitable and optimal relaxation parameter for different T

B

E

E

Figure 5.7: Influence the effective stiffness ratio on convergence

As shown in Figure 5.7, the sensitivity of the convergence rate and value of the

optimum relaxation parameter to the problem characteristics is even more critical for

cases where the condensed effective stiffness at the interface of the sub-domain

treated by Dirichlet boundary condition, is much greater than that of the other

domain treated by Neumann boundary condition. In general, as this ratio increases


148

the optimum relaxation parameter tends to smaller values, while the range of an

applicable relaxation parameters reduces significantly; moreover, the convergence

rate is also considerably reduced, leading to a significant increase in computational

cost.

The results from the two previous parametric studies clearly indicate that by

employing an optimum relaxation parameter the convergence rate of the coupling

method can be considerably enhanced. This fact is demonstrated in Figures 5.8 , 5.9

and 5.10 for model S7 from Table 5.1, where it is clear that the by using an optimum

relaxation, the prescribed tolerance on compatibility defaults is achieved with far less

iterations than those with non-optimum constant relaxation parameters.

Figure 5.8: Error reduction for different relaxation schemes (Time = 3.6s)

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Compatibility error at Interface (m

)

Iteration Number

Relaxation=0.1

Relaxation=0.2

Optimum Relaxation=0.3

Relaxation = 0.45

Convergence Criterion = 0.00017


149



Despite the shortcomings of the constant relaxation scheme and the difficulties of

determining the optimum relaxation parameter, the superiority of this method over

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Compatibility Error at Interface (m

)

Iteration Number

Relaxation = 0.1

Relaxation = 0.2

Optimum Relaxation = 0.3

Relaxation =0.4

Relaxation = 0.5

Convergence Criterion = 0.00017m

0

0.001

0.002

0.003

0.004

0.005

0.006

0 1 2 3 4 5 6 7 8 9 10

Compatibility Error at Interface (m)

Iteration Number

Relaxation = 0.1

Relaxation = 0.2

Relaxation = 0.3

Relaxation = 0.4

Relaxation = 0.5

Convergence Criterion = 0.00017 m


150

the staggered approach and its verification against the monolithic treatment are

demonstrated next for Model S7. For iterative coupling, the relaxation parameter is

taken at the optimal value of 0.3, and in all cases the dynamic analysis is undertaken

over 10s with a time step of 0.01t s .

Figures 5.11 and 5.12 show the variation with time of the rotation and horizontal

displacement, respectively, at the interface of the sub-domain T for both the

optimum relaxation and monolithic approaches. Similarly Figures 5.13 and 5.14

show the variation with time of the rotation and horizontal displacement,

respectively, at the interface of the sub-domain B for both the optimum relaxation

and monolithic approaches. The variation with time of the rotation and horizontal

displacement at the free end of the cantilever is also depicted in Figures 5.15 and

5.16, respectively. The graphs confirm that the results obtained from coupled

partitioned analysis match very well with those obtained from the monolithic

treatment within the prescribed compatibility tolerance of 410 m .

Figure 5.11: Rotation at the interface of sub-domain T


151

Figure 5.12: Horizontal displacement at the interface of T

Figure 5.13: Rotation at the interface of sub-domain B


152

Figure 5.14: Horizontal displacement at the interface of B

Figure 5.15: Rotation at the tip of the cantilever


153


Figure 5.17: Rotation at the tip of the cantilever 5reduced tolerance to 10


154

Considering Figure 5.15 it can be seen, that there are slight differences between the

values obtained by the partitioned treatment and those obtained via monolithic

approach. This slight inaccuracy is due to the prescribed tolerance for convergence

to compatibility. In fact, by prescribing a smaller convergence tolerance a closer

match to the monolithic approach can be achieved, as demonstrated in Figure 5.17

where a smaller tolerance of 510 is used.

5.2.2.2 Example 2: Static FEM-FEM Coupling

Here, the static plane strain problem of Figure 5.18a is considered and discretised

using 8-noded quadrilateral elements. The presented system is partitioned into two

sub-domains, namely T and B , where each partitioned domain has 5 interface

nodes (10 DOFs) as shown in Figure 5.18b. The resulting partitioned problem is

treated by the D-N iterative coupling technique. As pointed out before, prescribing

Neumann boundary condition on T will result in singularity of the matrices for

static analysis; therefore, the T and B partitioned sub-domains are treated by

Dirichlet and Neumann boundary conditions, respectively.

Figure 5.18: a) Plane strain problem, b) Problem partitioning


155

Iterative coupling is undertaken with constant relaxation for different stiffness ratios

of partitioned sub-domains T and B ( /T BE E ). The results obtained from

coupled partitioned analysis match very well with those obtained from the

monolithic treatment within the prescribed compatibility tolerance.

Details of the analysed models, including the elastic modulus ratios, range of suitable

relaxation parameters and the optimum relaxation parameter associated with the least

computational cost, are presented in Table 5.3. These values have been obtained by a

process of trial and error for each model, where the relaxation parameter is chosen as

a real value in the range of ]0, 2[.

Model /T BE E Range of Relaxation Optimum

Relaxation

M1 8.0 (0-0.14] 0.13

M2 4.0 (0-0.26] 0.2

M3 2.0 (0-0.47] 0.4

M4 1.0 (0-0.75] 0.6

M5 0.5 (0-1.1] [0.6-0.8]

M6 0.2 (0-1.29] [0.8-0.9]

Table 5.3: Range of suitable and optimal relaxation parameter for different /T BE E

Figure 5.19 shows the number of coupling iterations required for various constant

relaxation parameters for each of the considered coupled systems, where

convergence is assumed at a tolerance of 41 10 L for the compatibility defaults

(see Equation (5.43), with L=1m being the characteristic element size. The results

confirm that the convergence behaviour is significantly influenced by the stiffness

ratios of the partitioned sub-domains.


156

Figure 5.19: Influence of relaxation parameter on convergence properties

Considering Figure 5.19, the optimum relaxation parameter for each model can be

easily defined as the one corresponding to the minimum number of iterations. The

fact that the optimum relaxation parameter varies and depends on the partitioned

model stiffness ratio has been illustrated earlier, and is further confirmed by the

currently obtained results. The sensitivity of the convergence rate and value of the

optimum relaxation parameter to the problem parameters is even more critical for

cases where the domain treated by the Dirichlet boundary condition is relatively

stiffer than the other domain treated by the Neumann boundary condition. In general,

as this stiffness ratio increases, the range of convergent relaxation parameters

significantly reduces, and the optimum relaxation parameter tends to smaller values,

leading to deterioration in the convergence rate and a significant increase in

computational cost. The convergence rates of the different relaxation schemes are

illustrated in Figure 5.20 for model M4, which demonstrate that the optimum

relaxation provided a much faster convergence rate than other non-optimum

relaxation schemes.

0

10

20

30

40

50

60

70

80

90

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

Number of Iterations

Relaxation Parameter

M1 M2 M3 M4 M5 M6


157

Figure 5.20: Error reduction for different relaxation schemes (model M4)

The effect of mesh size on convergence characteristics of the relaxation scheme is

investigated next, where different discretisations of the problem are considered with

/ 1.0T BE E . The effect of the different mesh densities shown in Figure 5.21 on

convergence is shown in Figure 5.22.

The results show that as the discretisation of the partitioned sub-domains becomes

finer and as a result more interface degrees of freedom are employed, the range of

applicable relaxation parameter ensuring convergence becomes smaller. Moreover,

as evident from Figure 5.22, the optimum relaxation parameter changes noticeably.

For instance, considering the case of Mesh 0.5 the range of applicable relaxation

parameters is ]0, 1.1] and this would significantly be reduced to ]0, 0.6] for Mesh

2.0. In addition, the value of the optimum relaxation parameter for Mesh 0.5 is about

0.8, which is out of the applicable convergence range of Mesh 2.0 for which the

optimum relaxation parameter is about 0.5, highlighting the significant sensitivity of

the constant relaxation scheme to the partitioned sub-domains parameters.


158

Figure 5.21: Different discretatzions of model M4

Figure 5.22: Effect of mesh density on convergence

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Number of Total Iterations

Relaxation Number

Mesh 0.5

Mesh 1.0

Mesh 2.0


159

5.3 Adaptive Relaxation

As discussed earlier and demonstrated in the examples, finding the optimum

relaxation parameter based on Equation (5.42) will require solving the eigenvalue

problem of the matrix , which is not readily available using black box field

solvers for large multi-degree of freedom problems. Therefore, the optimum constant

relaxation parameter is usually found by a process of trial and error for every case

under consideration. Convergence conditions clearly indicate that convergence of the

Dirichlet-Neumann iterative coupling method depends on the stiffness ratio of the

partitioned soil and structure sub-domains. In linear analysis, the stiffness matrix

remains constant at all computational steps. However, for nonlinear analysis the

effective stiffness depends on the deformation state. This change of the stiffness of

the partitioned domains will have a significant effect on the convergence

characteristics of the coupling method at the interface level, where the optimum

relaxation parameter will change over the load/time increments. Accordingly,

ensuring convergence and computational efficiency in large scale nonlinear

problems, where the optimum relaxation parameter is to be determined over the full

range of response by trial and error, would be very difficult if not impossible.

Instead, such a scenario requires a dynamic change of relaxation parameter to ensure

optimal convergence and associated computational efficiency. This leads to the

concept of adaptive relaxation, where the relaxation parameter is determined during

coupling iterations, using error minimization techniques. With adaptive relaxation,

the use of trial and error for determining the relaxation parameter is avoided, whilst

leading to significant improvement in the convergence rate of iterative coupling, as

discussed hereafter.

5.3.1 Convergence analysis

Consideration is given here to an iterative coupling algorithm utilising an adaptive

relaxation parameter that changes during successive iterations. In this respect the

relaxation parameter is obtained, for a specific coupling iteration, from minimising


160

the compatibility and equilibrium defaults at the next iteration by using the

compatibility and equilibrium default history of the previous iterations.

Considering again Equation (5.20) for a linear coupled system with constant [],

while assuming an adaptive change of relaxation parameter, the interface

displacement values at the structure and soil sub-domains at different iteration stages

can be expressed as:

0

0 0Iteration no. 0

iT n

i iB Tn n

U U

U U cte

(5.44)

1 0 0

1 1

1 1

1Iteration no. 1

i i iT B Tn n n

i iB Tn n

U U U

U U cte

(5.45)

2 1 1

2 2

2 2

1Iteration no. 2

i i iT B Tn n n

i iB Tn n

U U U

U U cte

(5.46)

Accordingly, the compatibility default at each of the above iteration stages is given

by (see Equation (5.36)):

1 1 0 0

1i i i iB T B Tn n n n

U U U U I I (5.47)

Similarly it can be shown that:

2 2 1 1

2i i i iB T B Tn n n n

U U U U I I (5.48)

Therefore, the compatibility default at iteration number 2 can be rewritten as:

2 2 0 0

2 1i i i iB T B Tn n n n

U U U U I I I I (5.49)


161

Consequently at iteration number K, using an adaptive relaxation parameter, the

compatibility default at the interface of the decomposed soil-structure system, can be

obtained by the following difference equation:

0 0

1

K Ki i i iB T K B Tn n n n

U U U U I I I I (5.50)

Accordingly, at any iteration 0I :

1 1 1I I I IE E I I I I (5.51)

with:

I Ii iI B Tn n

E U U (5.52)

Considering Equation (5.51), the proposed method defines the adaptive relaxation

parameter at each coupling iteration stage ( 1I ), based on minimizing the

following:

2

1 1 12

I I IE I I (5.53)

Equation (5.53) is related to the interface compatibility error at previous coupling

iterations (clearly here the assumption is that convergence to compatibility has not

been achieved yet, that is 0I Ii i

B TU U ).

Expanding Equation (5.53) leads to:

2

1 1 1 1 1 1 2I I I I I IE E E (5.54)

Given that:

1I I IE E I I (5.55)


162

it can be shown that:

1 1 1

1 1I I I I

I I

E E E E (5.56)

Substituting Equation (5.56) back into Equation (5.54) leads to the following:

22 2 21

1 1 1 12 2 2

211 12

2

2

II I I I I I

I

II I I

I

E E E E E

E E E

(5.57)

Minimizing 1I with respect to 1I leads to:

11

2

1 12

1 2

1 2

0II

I I I I

I

I I

d

d

E E E

E E

(5.58)

Considering the positive sign of the second derivative:

22

12 21

20I I

II

dE E

d

(5.59)

it is evident that the value of 1I obtained by Equation (5.58) corresponds to the

minimisation of 1I .

Now, defining:

1I I Ii i iT T Tn n n

U U U

(5.60)

1I I Ii i iB B Bn n n

U U U

(5.61)

it can be shown that:


163

1

1

I I Ii i iT T T I In n n

U U U E

(5.62)

1I Ii i

B T I In nU U E E (5.63)

Therefore, Equation (5.58) for the adaptive relaxation parameter can be further

simplified to:

2

21 2

2


II Ii i

T Bn n

U U U

U U

(5.64)

The automatic choice of relaxation parameter from either Equation (5.64) or (5.58)

can be used for successive evaluation of the interface displacements using the

iterative coupling algorithm for 0I . For the first iterative stage (i.e. 0I ), the

relaxation parameter can be chosen as an arbitrary real value to start the coupling

process. Although the choice of the first relaxation parameter does not have

influence on the convergence, a choice which is close to the optimal value of

Equation (5.42) will result in better error reduction for the first coupling iteration,

and hence to a fewer coupling iterations with adaptive relaxation.

5.3.2 Convergence studies

The previous theoretical study has shown that the relaxation parameter in D-N

iterative coupling may be adaptively adjusted for every iteration (I≥1), leading to

enhanced convergence and improved efficiency. Here, these findings are

demonstrated through some illustrative FEM-FEM coupling examples.

5.3.2.1 SDOF Test System

The coupled test system considered in Section 5.2 for constant relaxation is

investigated here with adaptive relaxation. In this case, since the interface of the


164

coupled test system consists of a single degree of freedom, Equation (5.64) can be

rewritten in the form of:

2

21 2

2

I I I Ii i i iT T Bn n n T n

I I II I i ii iT Bn nT Bn n

U U U U

U UU U

(5.65)

Expanding Equation (5.65) using Equations (5.61) and (5.4), it can be shown that:

1 1

1

1 1

11

I I Ii iB Bn n

I Ii iT Tn n

U U

U U

(5.66)

where in the above is defined by Equation (5.5).

Considering Equation (5.66), it is immediately observed that this equation

corresponds to the optimum relaxation parameter introduced by Equation (5.10).

However, in contrast with the constant relaxation approach, the selection of the

relaxation parameter does not require trial and error. In addition, the method holds an

optimum convergence rate as demonstrated by examining the scalar test system

presented in Section 5.2 for convergence analysis.

5.3.2.2 Example 1: Dynamic FEM-FEM Coupling

In this section, the efficiency of the adaptive relaxation scheme is evaluated through

considering the dynamic FEM-FEM coupling problem of Example 1 in Section

5.2.2.1. The problem is analysed for the different models presented in Tables 5.1 and

5.2, where comparison is made between the results of constant and adaptive

relaxation.

As before, the results obtained from partitioned analysis are in excellent agreement

with the monolithic approach, as expected. A comparison between the number of


165

coupling iterations using an optimum constant relaxation and adaptive relaxation is

provided in Table 5.4.

Model Optimum Relaxation

Average Iterations per time-step

Adaptive Relaxation

Average Iterations per time- step

S1 1850 1.8 1824 1.8

S2 1788 1.8 1783 1.7

S3 1657 1.7 1688 1.7

S4 1700 1.7 1698 1.7

S5 1598 1.6 1620 1.6

S6 1806 1.8 1772 1.8

S7 2092 2.1 1883 1.9

S8 3638 3.6 1887 1.9

K1 5230 5.2 2607 2.6

K2 2975 3.0 2546 2.5

K3 2180 2.2 1999 2.0

Table 5.4: Number of coupling iterations with adaptive and optimum relaxation for

1000 time-steps ( 0.01t s ) with a tolerance of 10-4 m

These results clearly indicate that the adaptive relaxation method is far superior to

the constant relaxation scheme, since not only does it avoid the process of trial and

error for finding the relaxation parameter, but it also enhances the convergence rate

of iterative coupling significantly. This fact is further demonstrated in Figures 5.23-

5.30, for the most critical cases with regard to number of coupling iterations required

for convergence with optimum relaxation, namely Model K1 and Model S8. These

figures present the convergence rate of the adaptive and corresponding optimum

relaxation in different randomly selected time steps of the analysis. The automatic

change of the relaxation parameter value is also illustrated in these figures, where it


166

is clear that adaptive relaxation method achieves the prescribed tolerance on

compatibility defaults with less iteration than with optimum or non-optimum

constant relaxation. In some exceptional cases, the optimum relaxation approach

achieves the prescribed compatibility tolerance in less iteration. For example, at

t=4.98s as illustrated in Figure 5.30 the adaptive relaxation converges in 3 iterations

while the optimum relaxation converges within 2 iterations. However it is important

to note that adaptive relaxation can overcome the initial poor convergence rate

arising from a poor initial relaxation parameter, subsequently achieving a superior

convergence rate to that obtained with optimum relaxation.

Figure 5.23: Error reduction at t=3.05s for model S8


167

Figure 5.24: Error reduction at t=3s for model S8



168


Figure 5.27: Error reduction at t=3.05s for model K1


169

Figure 5.28: Error reduction at t=1.85 s for model K1



170


5.3.2.3 Example 2: Static FEM-FEM Coupling

Here the static FEM-FEM coupled problem of example 2 in Section 5.2.2.2 is also

analysed using the adaptive relaxation technique with the same prescribed tolerance,

where a comparison between the number of coupling iterations using optimum

relaxation and adaptive relaxation is provided in Table 5.5.

As before, these results confirm that the adaptive relaxation method is far superior to

the constant relaxation scheme. This fact is further demonstrated in Figure 5.31 for

the most critical case with regard to convergence, namely model M1, where it is

clear that adaptive relaxation method achieves the prescribed tolerance for

compatibility defaults with 10 coupling iterations compared to the optimum constant

relaxation with 31 coupling iterations. Although the compatibility error after the first

iteration is significantly increased, this is only due to the initial choice for the

relaxation parameter at I=0 ( 0 1.0 ). In fact, if the constant relaxation approach is

applied with 1.0 , convergence would not be achieved. However, this is not a


171

serious problem for the adaptive relaxation approach, since the compatibility default

decreases significantly in subsequent iterations.

Model Optimum relaxation Adaptive relaxation

M1 31 10

M2 20 7

M3 11 6

M4 7 4

M5 6 5

M6 4 4

Table 5.5: Number of required coupling iterations with adaptive and optimum

relaxation

Figure 5.31: Error reduction for adaptive and constant relaxation schemes (Model

M1)


172

5.4 Soil-Structure Interaction Analysis

In this section two examples of soil-structure coupling are presented, utilising both

the constant and adaptive relaxation approaches. The first example is a simple linear

soil-structure interaction problem, as shown in Figure 5.32, which is aimed at

verifying the relaxation approaches for iterative coupling against the monolithic

treatment.

Figure 5.32: Linear soil-structure interaction

The problem consists of linear static plane strain analysis of a concrete cantilever

wall, resting on a flexible soil, loaded at the top with a horizontal force. In

partitioned analysis of the problem, the adaptive relaxation shows reasonable

convergence characteristics and converges within 3 coupling iterations to a

prescribed tolerance of 10-4 m. With the same problem also modelled monolithically,

the obtained results from both monolithic and partitioned analysis are in excellent

agreement, as illustrated in Figure 5.33. This figure shows the horizontal

displacement of the beam obtained for the non-interactive case (rigid base) and for

the interactive case by both the partitioned and monolithic approaches. This further

demonstrates that by enforcing convergence to equilibrium and compatibility in the


173

sequential D-N iterative coupling algorithm, a strong coupling of the partitioned sub-

domains can be achieved at the interface.

Figure 5.33: Monolithic vs. Partitioned Approach

The second example represents a typical urban situation, where due to foundation

settlements the force quantities in the structural members often are revised. The

example considers a steel frame resting on soil and subjected to static loading

(Figure 5.34), where nonlinear elasto-plastic constitutive behaviour of the soil as

well as geometric and material nonlinearity of the structure are taken into account.

The considered frame is taken from a building, which is designed for office purposes

and assumed to be loaded equally on each floor, where plan and elevation views are

shown in Figure 5.35 and 5.36, respectively. The soil-structure interaction analysis is

carried out assuming plane strain conditions in the soil using an effective out-of-

plane width of 1m, where the developed domain decomposition approach is

employed utilising ADAPTIC and ICFEP. The main objective of this study is to

establish the applicability and efficiency of the presented coupling algorithm using

the adaptive relaxation technique, highlighting its merits compared to constant

relaxation scheme.


174

Figure 5.34: Plane frame resting on soil

Figure 5.35: Plan view of the analysed building frame


175

According to the partitioned treatment, the considered soil-structure system is

partitioned physically into two sub-domains, soil and structure, where each sub-

domain is discretised separately according to its characteristics as shown in Table

5.6. The frame structure is modelled with ADAPTIC using cubic elasto-plastic

beam-column elements (Izzuddin & Elnashai, 1993), which enable the modelling of

geometric and material nonlinearity. The frame is discretised using 10 elements per

member for both columns and beams, and the material behaviour is assumed to be

bilinear elasto-plastic with kinematic strain hardening. The footings are discretised

using 4 elements per member.

The soil sub-domain is modelled with ICFEP using an elasto-plastic Mohr-Coulomb

constitutive model, with parameters chosen to represent the behaviour of London

clay (Table 5.6). The nonlinear solution procedure employed for analysing the soil

sub-domain is based on the Modified Newton-Raphson technique, with an error

controlled sub-stepping stress point algorithm. The soil continuum is discretised

using 8-noded isoparametric quadrilateral elements. The loading is modelled in the

structure partitioned sub-domain and applied in 10 load increments (5kN/m2). The

interface degrees of freedom are assumed to be at nodes that belong to both the

footings and soil underneath, totalling 30 interface freedoms for this case.

Structure Sub-domain Material Properties All beams and columns (steel)

Steel Grade = S355 Elastic Modules = 210 GPa Strength = 355 MPa Bilinear elasto-plastic with strain hardening factor = 1%

Foundation Beam (concrete)

Elastic Modulus = 30 GPa Linear material

Soil Sub-domain Material Properties Soil Angle of Shear resistance ( 𝛷′) = 22° Dilation angle (𝜈 ) = 11° Effective out of plane depth = 1m Cohesion = 20 kPa

Young’s modulus varies linearly with depth from 10000 kPa at the ground surface (dE/dZ=5000 kPa/m)

Elasto-plastic Mohr-Coulomb constitutive model

Table 5.6: Geometric and material properties of the partitioned soil-structure system


176

Figure 5.36: Geometric configuration of considered frame

To assess the merits of the presented relaxation coupling approaches, both constant

and adaptive relaxation are employed. For the constant relaxation approach, the

considered problem is analysed using different relaxation parameters ( 𝛼 ∈]0,1] ). Figure 5.37 shows the total number of coupling iterations required for various

constant relaxation parameters and the applicable range of relaxation parameters

ensuring convergence to compatibility over the full range of response consisting of

ten increments. A tolerance ε = 10-4 m was set for the compatibility error of each

coupled degree of freedom at the interface (see Equation (5.43)).

The results confirm that the convergence behaviour of the iterative scheme using

constant relaxation is very sensitive to the chosen relaxation parameter, rendering its

selection a very difficult task, as evidenced by the significant increase in number of

iterations between α=0.55 with 64 coupling iterations and α0.65 with 180

coupling iterations.


177

Figure 5.37: Influence of relaxation parameters on convergence properties

Table 5.7 presents the range of constant relaxation parameters which guarantee

convergence, the optimum relaxation parameter and the number of coupling

iterations required for convergence to the prescribed tolerance of 410 m for both

adaptive and constant relaxation schemes.

Relaxation Range

Optimum relaxation

Number of iterations with optimum relaxation

Number of iterations with adaptive relaxation

(0,0.65] 0.55 64 46

Table 5.7: Convergence characteristics of constant and adaptive relaxation schemes

The convergence rates of the three approaches in the first, fifth and last load

increments are illustrated in Figures 5.38, 5.39 and 5.40 respectively. This

demonstrates that the adaptive relaxation achieves a faster convergence rate than the

relaxation scheme, whether optimal or not.


178

Figure 5.38: Error reduction for different relaxation schemes for the first load

increment

Figure 5.39: Error reduction for different relaxation schemes for the fifth load

increment


179

Figure 5.40: Error reduction for different relaxation coupling schemes for the last

load increment

Figure 5.41 shows the number of coupling iterations required in each increment for

the different coupling schemes, which demonstrates that the optimum relaxation

parameter should be determined over the full range of response in nonlinear

problems.

In fact, in nonlinear problems, finding the optimum relaxation parameter by trial and

error is not computationally efficient. This is evidenced by considering the behaviour

of coupling with α0.6, where a relatively good convergence rate is observed in the

first three load increments (even better than the optimum relaxation with α0.55),

but as the coupled system presents more nonlinearity the convergence rate decreases

significantly.


180

Figure 5.41: Convergence performance over full range of response

These results demonstrate the superiority of the adaptive relaxation scheme, which

achieves much faster convergence than the constant relaxation scheme, whether

optimal or not. It is, however, worth observing that the adaptive relaxation scheme

on average still requires about 5 iterations per load step (for 10 increments), which is

relatively large in comparison with what would be necessary in a typical monolithic

treatment, thus highlighting the need for further enhancement in iterative coupling

algorithms.

Notwithstanding, the benefits of the developed simulation environment in the

practical assessment of nonlinear soil-structure interaction problems can now be

demonstrated by considering the results of this case study. The frame structure with

its applied loads transmits the loading to the soil, which in turn deforms due to

cumulative action of these loads. As a consequence, the soil deformation beneath the

structure transmits back additional deformations and corresponding forces to the

structure. This interactive process is continued until the whole coupled system


181

reaches a compatible equilibrium state. In the following, the deformation and stress

states of the coupled problem at the end of load application are briefly described.

The vertical deformation profile of the soil surface, with respect to the distance from

the middle column of the frame, obtained from coupled analysis is given in Figure

5.42, where the three troughs correspond to the locations of the three footings

showing their vertical settlement. Clearly, the generated level of vertical settlement,

of the order of 20cm and vertical differential settlement of the order of 10 cm,

requires the structural analysis model to account for geometric nonlinearity.

The deformed shape of the mesh of the soil partitioned sub-domain is also depicted

in Figure 5.43.

Figure 5.42: Vertical displacement of the soil surface


182

Figure 5.43: deformed mesh of the soil partitioned sub-domain

A vector plot of displacements in the soil sub-domain in the vicinity of the structure

foundations is also shown in Figure 5.44, where the relative magnitude of these

vectors reflects the mechanism of ground deformation.

Figure 5.44: Vector plot of displacements in the soil partitioned sub-domain

Figure 5.45 shows contours of stress level at the end of analysis under the applied

load on the structure. The stress level is the ratio, at the same mean effective stress,

of the current deviatoric stress to the deviatoric stress at failure. It therefore varies

from 0 to 1, where 1 indicates full plasticity and failure. It is evident from Figure

5.45 that the applied loading conditions have mobilised an extensive plastic zone


183

underneath the building. This zone is, however, bigger and deeper under the middle

footing, which is in agreement with the previous figures that show most of the

deformation and load concentration under the middle column.

Figure 5.45: Contours of stress level in soil partitioned sub-domain

Figure 5.46: Deformed shape and bending moment (kN.m) of the frame (scale=5)

Finally, the deformed shape and the bending moment contours of the partitioned

structure sub-domain are shown in Figure 5.46. It is evident from the deformed


184

shape of the structure and also the displacement vectors underneath each of the three

footings in Figure 5.44 that they experience rigid tilting (indicated previously in

Figure 5.42) and significant vertical settlements.


This chapter investigates the use of interface relaxation coupling techniques in FEM-

FEM domain decomposition analysis of soil-structure interaction. In this respect, the

overall domain is divided into physical partitions consisting of soil and structure sub-

domains. Coupling of the separately modelled sub-domains is undertaken with

various interface relaxation algorithms based on the sequential iterative D-N sub-

structuring method, which ensures compatibility and equilibrium at the interface

boundaries of the sub-domains. This enhancement of the convergence characteristics

is achieved by employing a relaxation of the interface Dirichlet entities in successive

iterations. Various mathematical and computational characteristics of the coupling

method, including the governing convergence rate and choice of relaxation

parameter, are studied, where it is demonstrated that the convergence behaviour of

the constant relaxation scheme is very sensitive to the stiffness ratio of the

partitioned sub-domains. This renders the selection of an optimum relaxation

parameter very difficult, leading to considerable computational inefficiency,

especially for realistic large-scale nonlinear problems.

An adaptive relaxation scheme is also considered for enhancing the performance of

iterative coupling algorithms, where the choice of the relaxation parameter is easily

guided by the iterative corrections of Dirichlet entities at the interface. It is shown

that the adaptive scheme improves the convergence characterises in both linear and

nonlinear analysis significantly, where the number of coupling iterations required for

convergence could be reduced by over 50%. Moreover, the adaptive scheme has the

advantage of avoiding trial and error for the selection of an optimum, even adequate,

constant relaxation parameter. In general, adaptive relaxation schemes enjoy great

popularity in multiphysics coupling due to their simplicity and low computational


185

cost of determining the adaptive relaxation parameter, as evidenced by the simple

vector algebra involved.

In addition to demonstrating the key convergence characteristics of the considered

coupling algorithms with interface relaxation, the developed coupling simulation

environment has been used in a number of case studies, and where applicable

comparison with available monolithic treatment has been made. The results verify

the applicability of the developed simulation environment and also demonstrate that

by enforcing convergence to equilibrium and compatibility in the sequential D-N

iterative coupling algorithm, a strong coupling of the partitioned sub-domains can be

achieved at the interface.

It should be emphasised that the considered methods are generally applicable to the

coupling of various computational procedures that are used for nonlinear structural

and geotechnical analysis. In this context, these coupling methods have the potential

to provide an integrated interdisciplinary approach which combines the advanced

features of both structural and geotechnical modelling for a variety of problems in

soil-structure interaction analysis.

Finally, it should be noted that although using the adaptive scheme removes

significant difficulties in the conventional relaxation iterative coupling scheme, there

is a pitfall associated with both constant and adaptive relaxation schemes when the

partitioned sub-domain parameters dictate very small values of the relaxation

parameter for convergence. In such cases, relaxation methods breakdown with poor

convergence rates and significant computational inefficiency.

An example of such problems in the context of soil-structure interaction analysis is

presented in next chapter where both adaptive and constant relaxation techniques

become extremely computationally inefficient. In this respect, it is proposed that the

performance of iterative coupling methods may be effectively enhanced for

nonlinear analysis through the use of the condensed interface stiffness matrices of

the structure and soil partitioned sub-domains, providing an effective first-order

guide to iterative displacements at the soil-structure interface. As discussed in


186

Chapter 6, these techniques require the determination of the condensed stiffness

matrix, which typically involves more interface-related computations than the

relaxation approach. Notwithstanding, their superior convergence rate can be such

that their use is easily justified, if not essential, for iterative coupling of nonlinear

soil-structure interaction problems.

Chapter 6

Reduced Order Method

6.1 Introduction

The performance of interface relaxation iterative coupling schemes can be

effectively enhanced for nonlinear analysis through the use of the condensed

interface tangent stiffness of both the structure and soil models, depending on the

variant coupling algorithm under consideration. The condensed tangent stiffness is

either readily available or easily determined with current nonlinear field modelling

tools, and provides an effective first-order guide to iterative forces/displacements at

the soil-structure interface. Although such an approach involves more interface-

related computations than the relaxation approach, these computations are relatively

minor in comparison with those undertaken in the soil and structure models, and

therefore superior overall computational efficiency should be obtained due to the

enhanced convergence of iterative coupling. In this respect, convergence to

compatibility occurs for linear problems immediately at the first coupling iteration.

Although this immediate convergence does not normally occur for nonlinear

problems, employing this method for the update of interface Dirichlet boundary

conditions ensures a high convergence rate. Indeed, the enhanced approach should

Chapter 6 Reduced Order Method

188

bring the numerical performance of iterative coupling closer to the monolithic

treatment, whilst maintaining the practical and computational benefits of the

partitioned treatment.

Although the condensed tangent stiffness may be readily available with some

nonlinear field modelling tools, a more general approximation for the condensed

stiffness matrices is desirable and can indeed be achieved during the course of

coupling iterations, thus avoiding the need for explicit determination and extraction

of the stiffness matrices. In this respect, the condensed tangent stiffness matrix is

approximated via reduced order models, building on a previous approach by

Vierendeels et al. (2007) concerned with fluid–structure interaction problems.

Various significant modifications to this approach are proposed in this chapter,

leading to a versatile and efficient approach for coupled modelling of nonlinear soil-

structure interaction problems. In this regard, a major pitfall associated with the

original method causing divergence is overcome by a new selective

addition/replacement procedure of force and displacement mode vectors. Moreover,

in order to achieve better approximation of the condensed tangent stiffness matrix in

the initial stages of coupling iterations, a mixed reduced order method is proposed,

which achieves a more robust coupling technique than the conventional reduced

order method. In the following such approaches are discussed in detail.

6.2 Condensed Interface Tangent Stiffness

Consider domain decomposition of a soil-structure interaction problem as presented

below:


11 12

21 22

T T T TT T

T T i iT T

K K U F

K K U F

(6.1)



189

11 12

21 22

B B B BB B

B B i iB B

K K U F

K K U F

(6.2)

In the above, vectors XXU and X

XF correspond to the displacements and external

forces for the non-interface degrees of freedom, while iXU and i

XF correspond to

displacements and forces for the interface degrees of freedom, respectively.

Assume that an iterative coupling method is employed for coupling of the above

partitioned soil and structure sub-domains, where the compatibility and equilibrium

defaults at the interface of the structure and soil sub-domains for iteration number I

of load/time step n take the form:

I II i iU B Tn n n

U U (6.3)

I II i iF B Tn n n

F F (6.4)

Consider the new iterative estimation of the interface displacements and forces in the

successive iteration I+1 expressed incrementally in the general form:


U U U (6.5)


U U U (6.6)


F F F (6.7)


F F F (6.8)

where to a first order:

I Ii C iT T Tn n

F K U (6.9)


190

I Ii C iB B Bn n

F K U (6.10)

in which CTK and C

BK are the condensed tangent stiffness matrices at the

interface of the structure and soil sub-domains.

Clearly in order to achieve convergence to compatibility and equilibrium at the

interface of the partitioned soil-structure system at iteration I+1, the following

compatibility and equilibrium conditions should be satisfied:

1 10

I Ii iB Tn n

U U (6.11)

1 10

I Ii iT Bn n

F F (6.12)

Expanding Equation (6.12) using Equations (6.4), (6.7) and (6.8) gives:

0I II i i

F T Bn n nF F (6.13)

Similarly expanding Equation (6.11) using Equations (6.3), (6.5) and (6.6) gives:

0I II i i

U B Tn n nU U (6.14)

Substituting Equations (6.9) and (6.10) in Equation (6.13) gives:

0I II C i C i

F T T B Bn n nK U K U (6.15)

Now by substituting IiB n

U from Equation (6.14) in Equation (6.15) the following

is obtained:

0I II IC i C i

F T T B T Un nn nK U K U (6.16)

Therefore the incremental value of IiT n

U takes the following form of:


191

1I I Ii C C CT T B B U Fn nn

U K K K

(6.17)

Similarly, by substituting IiT n

U from Equation (6.14) in Equation (6.15) the

incremental value of IiB n

U can be obtained as:

1I I Ii C C CB T B T U Fn nn

U K K K

(6.18)

Substituting Equations (6.13), (6.9) and (6.10) in Equation (6.14) the incremental

value of IiT n

F can be obtained as:

11 1 1I I Ii C C CT B T U B Fn nn

F K K K

(6.19)

Similarly, the incremental value of IiB n

F can be obtained as:

11 1 1I I Ii C C CB B T U T Fn nn

F K K K

(6.20)

The above equations can be used in successive update of boundary conditions of the

different sequential and parallel coupling algorithms presented in Chapter 4 (see

Appendix C). Depending on the utilized coupling algorithm type, the following

equations could provide the new estimation of the iterative Dirichlet/Neumann

boundary conditions ensuring convergence.

11I I I Ii i C C CT T T B B U Fn nn n

U U K K K

(6.21)

11I I I Ii i C C CB B T B T U Fn nn n

U U K K K

(6.22)

11 1 1 1I I I Ii i C C CT T B T U B Fn nn n

F F K K K

(6.23)


192

11 1 1 1I I I Ii i C C CB B B T U T Fn nn n

F F K K K

(6.24)

Although, as evidenced by the above equations, this method requires the access to

the stiffness/flexibility matrices of the partitioned sub-domains and poses more

interface related computations compared to interface relaxation schemes, there are

major advantages associated with the above update technique. More specifically,

these advantages are, i) avoiding the need for using predefined relaxation parameters

and trial and error for ensuring optimal convergence in the coupling procedure ii)

offering a high convergence rate compared to relaxation schemes, and iii)

facilitating the development of different coupling algorithms (see Appendix C). In

this respect, as further shown in Appendix C, this approach provides richer and more

general update technique in iterative coupling algorithms compared to both adaptive

and constant relaxation, leading to a versatile and efficient approach for partitioned

modelling of coupled systems.

The general algorithmic steps of the sequential Dirchlet-Neumann iterative coupling

scheme, using the above update technique are presented in Table 6.1. In the

following sections this coupling algorithm will be examined for illustrative coupled

FEM-FEM examples in order to demonstrate its merits.


193


For I=0,1,…(number of iterations)

STEP 1: At the start of each increment, sub-domain T is loaded by the external

forces, while the displacements at the interface nodes are prescribed in accordance

with the initial conditions: IiT n

U U

STEP 2: The structural solver computes the response of the structure for : IiT n

F


applying equilibrium: 0I Ii i

T Bn nF F

STEP 4: Based on these forces and the external loading applied to the soil domain,

the soil solver computes the response of the soil domain for: IiB n

U

STEP 5: If convergence to compatibility has been achieved the solution proceeds to the next time/load increment (n=n+1) with:

STEP 6: If convergence to compatibility has not been achieved, the following new estimation of the displacements will be applied to the structure sub-domain and the iteration will continue (I=I+1) until convergence:

11I I Ii i C C CT T T B B U nn n

U U K K K

Table 6.1: Coupling Procedure

6.2.1 Numerical Examples

Here, the coupling procedure and convergence behaviour of the sequential Dirichlet-

Neumann algorithm are examined using the condensed interface stiffness matrix for

updating the Dirichlet boundary conditions. Consider the linear coupled spring

system consisting of eight spring elements associated with different stiffness as

illustrated in Figure 6.1. In order to perform the partitioned analysis procedure, the

coupled system is decomposed into two separately modelled partitioned sub-domains

T and B (Note that nodes 5 and 4 are the interface nodes corresponding to 2

interface degrees of freedom).


194

Figure 6.1: Coupled spring system

To facilitate the verification of the various coupling schemes presented hereafter, the

coupled problem is initially modelled and solved monolithically. Performing global

structural analysis using a monolithic approach results in the formation and solution

of the following global system of equations:

2

3

4

5

6

40 10 0 0 0 0

10 40 10 20 0 20

0 10 25 0 15 0

0 20 0 35 15 0

0 0 15 15 40 60

(6.25)

where in the above j denotes the displacement of node j.

Solving Equation (6.25) results in the following nodal displacements:

i i

j j

PK K

PK K


195

2

3

4

5

6

67

231268

231541

231463

231241

77

(6.26)

Clearly any result obtained by the partitioned analysis of the coupled spring system

should mach the results obtained by the monolithic approach given by Equation

(6.26). In order to perform the partitioned analysis using the condensed interface

stiffness approach, the partitioned sub-domains must be modelled in isolation, and

their corresponding condensed interface stiffness matrices must be obtained.

Considering sub-domain T in isolation, its governing equilibrium conditions can be

written in the form of Equation (6.27). In the following iX T

P and iX T

correspond

to the interface forces and displacements of node X in the partitioned sub-domain

ΩT, respectively.

4 4

5 5

6

15 0 15

0 15 15

15 15 40 60

i i

T T

i i

T T

P

P

(6.27)

Applying condensation on interface nodes 4 and 5, the condensed stiffness matrix at

the interface of the partitioned sub-domain T can be determined as:

75 45

15 0 15 1 8 815 150 15 15 45 7540

8 8

CTK

(6.28)


196

Similarly, the governing equilibrium condition for partitioned sub-domain B can be

written as:

2

3

4 4

5 5

040 10 0 02010 40 10 20

0 10 10 0

0 20 0 20

i i

B B

i i

B B

P

P

(6.29)

In the above iX B

P and iX B

correspond to the forces and displacement of the

interface node X in the partitioned sub-domain B , respectively.

Again by applying condensation on interface nodes 4 and 5, the condensed stiffness

matrix at the interface of sub-domain B is obtained as:

122 16

10 0 0 10 40 10 0 0 3 30 20 0 20 10 40 10 20 16 28

3 3

CBK

(6.30)

Utilizing the obtained condensed interface stiffness matrices of Equations (6.28) and

(6.30) in the update of boundary conditions, the partitioned sub-domains can be

effectively coupled at the interface using various coupling algorithms. In the

following, the presented example is coupled using the sequential Dirichlet-Neumann

coupling algorithm which is the main emphasis of this work. Demonstration of the

applicability of the presented update technique for other types of parallel/sequential

coupling algorithms can be found in Appendix D for the same example.

6.2.1.1 Sequential DirichletNeumann Iterative Coupling

Prescribing an initial interface Dirichlet data, 0

4 0Ii

T

and 0

5 0Ii

T

at the

interface of T at iteration (I=0), and solving the partitioned sub-domain T gives:


197

0 0

4 4 6

0 0 0

5 5 4

06

5

30 215 0 15

450 15 15 0

215 15 40 60 45

2

I Ii i

T T

I I Ii i i

T T T

Ii

T

P

P P

P

(6.31)

Prescribing the interaction Neumann data at the interface of B , by applying

equilibrium (i.e. 0 0

4 5

45 45 and

2 2

I Ii i

B BP P

), and solving the partitioned sub-

domain B gives:

2 2

3 3

0 0 0

4 4 4

0 0 0

5 5 5

5

6040 10 0 0 102010 40 10 20 3

45 / 20 10 10 0 67

120 20 0 20 45 / 2107

24

I I Ii i i

B B B

I I Ii i i

B B B

P

P

(6.32)

Comparing 0

4

Ii

B

and 0

5

Ii

B

with 0

4

Ii

T

and 0

5

Ii

T

, it is clear that convergence to

compatibility at the interface is not achieved at this iterative stage. Therefore, new

estimates of Dirichlet data should be calculated according to Equation (6.21) to

enforce convergence in the next iteration:

1

1

4

1

5

75 45 22 16 22 16 670 8 8 3 3 3 3 12.0 45 75 16 28 16 28 107

8 8 3 3 3 3 24

541

231463

231

Ii

T

Ii

T

(6.33)


198

Prescribing the above estimate for interface Dirichlet data at the interface of T , at

iteration I=1, and solving the partitioned sub-domain T :

11

44 6

1 1 1

5 5 4

1

6 5

541 241

231 7715 0 15463 130

0 15 15231 11

15 15 40 60 1300

77

Ii IiT

T

I I Ii i i

T T T

Ii

T

P

P P

P

(6.34)

Prescribing the interaction Neumann data at the interface of B by applying

equilibrium and solving the partitioned sub-domain B gives:

2 2

3 311 1

44 4

1 115 55

670 231

40 10 0 0 2682010 40 10 20 2311300 10 10 0 54111

23113000 20 0 2046377231

II Iii iBB B

I IIi iiB BB

P

P

(6.35)

Clearly the interface displacement values of 1

4

Ii

B

and 1

5

Ii

B

exactly match 1

4

Ii

T

and 1

5

Ii

T

, thus at the first coupling iteration (I=1) convergence to compatibility is

achieved. Moreover, the obtained results by the above coupling procedure given by

Equations (6.34) and (6.35) are identical to those obtained with the monolithic

treatment given by Equation (6.26).

6.2.2 Analogy between Interface Relaxation and Condensed Interface Stiffness

Approaches

Consider the sequential Dirichlet-Neumann coupling of partitioned sub-domains T

and B represented by Equations (6.1) and (6.2). Considering Equation (6.21) and


199

knowing that the equilibrium condition at the interface is already satisfied for each

coupling iteration in STEP 6 (i.e. 0I

F n ), the renewal of the boundary

conditions can be expressed as:

11I I Ii i C C CT T T B B U nn n

U U K K K

(6.36)

Rewriting the above equation in the form of:

11 1I I Ii i C C

T T B T U nn nU U K K

I (6.37)

and substituting Equation (6.3) in the above gives:

1I I Ii i iT B Tn n n

U U U β I β (6.38)

where in the above [I] is the identity matrix, with:

1β I + λ (6.39)

and:

1C CB TK K

λ (6.40)

It can be immediately noted that Equation (6.38) resembles the interface relaxation

scheme presented by Equation (5.1). However, 1β I + λ is a fully populated

relaxation matrix, as opposed to the single relaxation parameter of Equation (5.1),

which ensures convergence to compatibility at the interface of the partitioned sub-

domains while holding an optimum convergence rate. Indeed, the proposed approach

brings the numerical performance of iterative coupling approach close to the

monolithic treatment, whilst maintaining the practical and computational benefits of

the partitioned treatment. Moreover, in contrast to the constant relaxation coupling

algorithms, this method does not require the definition of certain parameters by a


200

process of trial and error for each case under consideration. Although the proposed

approach involves more interface-related computations than the relaxation approach,

overall computational efficiency is envisaged due to the enhanced convergence of

iterative coupling.

The convergence of the presented sequential D-N algorithms can now be established

for a linear case by following the algorithmic steps of sequential D-N, while

assuming that the iteratively updated boundary conditions have the following form:

1I I Ii i iT B Tn n n

U U U X I X (6.41)

where X is an unknown matrix which is to be determined to ensure convergence of

the iterative coupling algorithm.

Using Equation (6.41) for renewal of the interface Dirichlet data, in coupling of

Equations (6.1) and (6.2), the compatibility default after K successive iterations at

the interface of the decomposed soil-structure system, can be obtained by the

following difference equation:

0 0KK Ki i i iB B B BU U U U I - X I + λ (6.42)

with:

1C CB TK K

λ (6.43)

1

22 21 11 12C B B B BBK K K K K

= condensed stiffness matrix of the structure

sub-domain corresponding to the interface degrees of freedom, and

1

22 21 11 12C T T T TTK K K K K

= condensed stiffness matrix of the soil sub-

domain corresponding to the interface degrees of freedom.


201

For the successive iteration process of Equation (6.42) to converge for any initial

value, it can be clearly shown that X should take the form of:

1 X β I + λ (6.44)

in which case convergence to compatibility occurs for linear problems immediately

at the first coupling iteration as shown in the previous example. It is worth noting

that, in the previous example of Section 6.2.1, the adaptive relaxation update

technique (with an initial relaxation of 1.0 ) requires 7 coupling iterations for

convergence to compatibility with a tolerance of 410 . This highlights the high

convergence rate of the above method compared to both adaptive and constant

relaxation schemes. Of course, this immediate convergence does not normally occur

for nonlinear problems, though use of Equation (6.36) to update the interface

Dirichlet boundary conditions ensures a superior convergence rate to the relaxation

approach.

6.3 Approximation of the Condensed Tangent Stiffness

As evident from Equation (6.37), update of boundary conditions in successive

iterations using the condensed interface stiffness matrices, requires the determination

of these matrices during coupling iterations, which involves more interface-related

computations than the relaxation approach and requires direct access to the field

specific solver codes. Alternatively, suitable approximations for the condensed

stiffness matrices can be obtained, thus avoiding the need for obtaining these

matrices via the field codes. This builds on a previous approach by Vierendeels et al.

(2007) who utilized a procedure for constructing the reduced order model of

partitioned sub-domains for iterative coupling of fluid–structure interaction

problems, though various modifications are proposed here to provide a superior

approach for coupled modelling of nonlinear soil-structure interaction problems.


202

6.3.1 Condensed Interface Secant Stiffness Matrix

Consideration is given here to approximating the condensed interface tangent

stiffness matrix for nonlinear analysis using an initial secant stiffness matrix that

would be exact for linear analysis. In this respect, at the beginning of the coupled

analysis procedure the linear secant condensed stiffness matrices are obtained, and

these are used thereafter throughout the coupling iterations for the update of

boundary conditions. Assume that the partitioned sub-domain T is treated by

Dirichlet boundary conditions at the interface, while the partitioned sub-domain B

is treated by Neumann boundary conditions at the interface. The general

formulations for establishing the initial condensed secant stiffness matrices of the

partitioned sub-domains T and B are presented in the following.

Assume that there are N degrees of freedom at the interface of the coupled problem.

The following N+1 displacement vectors are prescribed at the interface of the sub-

domain treated by Dirichlet boundary conditions ( T ):

1 , 0,...,,..., ,...,Ti

T i ij iNiU i Nu u u (6.45)

where:

0 or 0

and 1iju

i j iu

i j i

(6.46)

and εu is a predefined incremental displacement, which should be relatively small in

comparison with the anticipated interface displacements in nonlinear analysis.

By obtaining the interface force vectors of the partitioned sub-domain T

corresponding to the above prescribed displacement vectors:

1 , 0,...,,..., ,...,Ti

T i ij iNiF i Nf f f (6.47)


203

the initial secant condensed interface stiffness matrix of the above partitioned sub-

domain corresponding to the chosen incremental displacement εu can be obtained as:

1 01 1 01

0 01

, 1,...,

i N

u u

CT

iN N NN

u u

f f f f

K i N

f f f f

(6.48)

Similarly in order to obtain the initial condensed flexibility matrix of the sub-domain

treated by Neumann boundary conditions, the following N+1 force vectors are

prescribed at the interface of the sub-domain ( B ):

1 , 0,...,,..., ,...,Ti

B i ij iNiF i Nf f f (6.49)

where:

0 or 0

and 1fij

i j if

i j i

(6.50)

and f is a predefined incremental force, which should be relatively small in

comparison with the anticipated interface forces in nonlinear analysis.

By obtaining the interface displacement vectors of the partitioned sub-domain B

corresponding to the above prescribed force vectors as:

1 , 0,...,,..., ,...,T

iB i ij iNiU i Nu u u (6.51)

the initial secant condensed flexibility matrix of the above partitioned sub-domain

corresponding to the chosen incremental displacement f can be obtained as:


204

1 01 1 01

1

0 01

, 1,...,

i N

f f

CB

iN N NN

f f

u u u u

K i N

u u u u

(6.52)

Considering Equations (6.48) and (6.52), a robust sequential Dirichlet-Neumann

coupling algorithm with high convergence rate for FEM-FEM coupling problems

can be achieved. Indeed, the use of the secant condensed stiffness matrix offers

convergence within one iteration for linear analysis, and can significantly enhance

convergence in nonlinear analysis in comparison with relaxation methods, depending

on the extent of overall system nonlinearity.

6.3.1.1 Example 1: Static FEMFEM Coupling

This example demonstrates the high performance and applicability of using the

initial secant condensed stiffness matrix in coupling linear static FEM-FEM

problems. Consider the plane strain problem of Figure 6.2a, which is discretized

using four noded elements. The presented system is partitioned into two sub-

domains, namely T and B , where each partitioned sub-domain has 3 interface

nodes at its interface (i.e. a total of 6 degrees of freedom), as shown in Figures 6.2b

and 6.2c.

The resulting partitioned problem is treated by the D-N iterative coupling algorithm.

Since, prescribing Neumann boundary conditions on T results in singularity of the

corresponding partitioned system of equations in static analysis, the T and B

partitioned sub-domains are treated by Dirichlet and Neumann boundary conditions,

respectively. The proposed problem is analysed using the Equations (6.48) and

(6.52) in the update of iterative boundary conditions and the results are compared

with the adaptive relaxation scheme. The problem is analysed for different stiffness


205

ratios of partitioned sub-domains T and B , which is achieved by considering

different ratios of elastic modulus ( /T BE E ).

The results obtained from coupled partitioned analysis match very well with those

obtained from the monolithic treatment within the prescribed compatibility tolerance.

The results corresponding to the analysed models, including the considered elastic

modulus ratios, compatibility tolerance and the number of required coupling

iterations for different considered schemes, are presented in Table 6.2.

Figure 6.2: a) Linear FEM-FEM coupled problem b) Partitioned sub-domain B c)

Partitioned sub-domain T

These results clearly show that using the condensed interface stiffness matrices of

the partitioned sub-domains is a far superior update technique compared to adaptive

relaxation, since it enhances the convergence rate of the coupling method

significantly. In fact, as expected, employing the condensed interface stiffness

matrices for this linear problem leads to convergence after only one iteration. This

fact is further demonstrated in Figures 6.3, 6.4 and 6.5 for models A1, A4 and A6,

respectively.


206

Model T

B

E

E

Adaptive

310 L

Adaptive

410 L

Adaptive

510 L

Adaptive

610 L

Condensed

610 L

A1 8.0 6 10 13 16 1

A2 4.0 5 6 10 13 1

A3 2.0 4 6 6 10 1

A4 1.0 3 5 6 6 1

A5 0.5 3 3 5 6 1

A6 0.2 1 3 3 4 1

Table 6.2: Required coupling iterations for different coupling schemes

Figure 6.3: Error reduction of different coupling schemes for model A1


207




208

6.3.1.2 Example 2: Dynamic FEMFEM Coupling

In this example, a representative dynamic FEM-FEM coupled problem is treated by

a sequential D-N iterative coupling scheme in which the effective condensed

interface stiffness matrices of the partitioned sub-domains are approximated initially

using the procedure outlined in Section 6.3.1 (It is worth noting that this approach

requires some adaptations for dynamic problems where the Dirichlet boundary

conditions are prescribed in terms of acceleration rather than displacements.

Moreover, here it is assumed that the time-step does not change during the analysis,

so it is ensured that the approximated effective stiffness does not change during the

analysis).

Consider a cantilever beam of Figure 6.6a subject to an excitation acceleration signal

applied to its bottom support, as given in Figure 6.7. The length of the cantilever

beam is 20m (L=10m) with a rectangular cross section of 20.1 0.1 m . The mass of

the system is modelled with two concentrated masses of M1=2000 kg and M2=2000

kg, at the middle and the free end of the cantilever respectively. A bilinear material

model is assumed, as depicted in Figure 6.6a, where 9 2210 10E Nm , 0.01

and 6 2300 10y Nm . The above system is partitioned into two sub-domains

namely T and B , as shown in Figures 6.6b and 6.6c, with three degrees of

freedom at the interface (one rotational and two transitional).

The above partitioned problem is coupled with sub-domain T treated by Dirichlet

boundary conditions (displacement) and sub-domain B treated by Neumann

boundary conditions (forces). The proposed problem is analysed for different

problem partitioning, where different mass ratios are assigned at the interface to the

partitioned sub-domains T and B ( 1 2/m m ), as listed in Table 6.3. The problem is

analysed for duration of 5s with 0.01t s using the initial linear elastic condensed

interface stiffness matrices, as outlined in Section 6.3.1.The obtained results for a

prescribed compatibility tolerance of 10-4m, are compared to adaptive relaxation and

summarised in Table 6.3.


209

Figure 6.6: a) Coupled dynamic FEM-FEM problem b) Partitioned sub-domain B

c) Partitioned sub-domain T



210

Model 1

2

m

m

Number of Iterations using

condensed stiffness matrix

Number of iterations using

adaptive relaxation

C1 200 kg

0. 11800 kg

529 1624

C2 400 kg

0.251600 kg

476 1027

C3 600 kg

0.431400 kg

471 879

C4 800 kg

0.61200 kg

456 824

C5 1000 kg

1.01000 kg

451 758

C6 1200 kg

1.5800 kg

448 741

C7 1400 kg

2.3600 kg

448 752

C8 1600 kg

4.0400 kg

446 801

C9 1800 kg

9.0200 kg

444 821

Table 6.3: Number of required coupling iterations for 500 time-steps ( 0.01t s )

with a tolerance of 10-4 m

The results obtained from coupled partitioned analysis match very well with those

obtained from the monolithic treatment within the prescribed compatibility tolerance.

Considering Model C1, Figures 6.8 and 6.9 show the variation with time of the

rotation and horizontal displacement, respectively, at the interface of the sub-domain

T for both the partitioned approach with initial stiffness approximation and the

monolithic approaches.


211

Figure 6.8: Horizontal displacements at the interface of T

Figure 6.9: Rotation at the interface of T


212

The results show that the update of boundary conditions via the condensed interface

stiffness matrix approach provides a much better convergence rate than via the

adaptive relaxation scheme.

Considering the critical analysed model, C1, it is apparent that the update via

condensed interface stiffness matrix achieves superior convergence rate to adaptive

relaxation by more than 50%. In general, it is evident that the proposed procedure

outperforms the adaptive relaxation scheme in linear FEM-FEM coupling, as

expected. Figures 6.10 and 6.11 show the number of coupling iterations required

over the full 5s duration for Models C1 and C5, where it is clear that the condensed

stiffness approach converges typically after one coupling iteration when the response

is linear elastic.

Figure 6.10: Comparison between different coupling schemes for Model C1


213

Figure 6.11: Comparison between different coupling schemes for Model C5

6.3.1.3 Discussion on Nonlinear Analysis

It has been shown in the previous examples that the condensed interface stiffness

approach in iterative coupling enables convergence to compatibility within one

iteration for linear problems. However, this immediate convergence does not

normally occur for nonlinear problems through the use of the secant matrices given

by Equations (6.48) and (6.52). In nonlinear analysis, the effective stiffness depends

on the deformation state. In such cases the rate of convergence depends on the

change in the effective stiffness. This change of the stiffness of the partitioned sub-

domains can have a significant effect on the convergence characteristics of the

coupling method at the interface level, thus requiring the reformation of the tangent

stiffness matrices during iteration in order to ensure convergence to compatibility

using Equation (6.37), as discussed hereafter.


214

6.3.2 Reduced Order Method

Although the condensed tangent stiffness matrix at the interface could be determined

with existing nonlinear field modelling tools, this might require significant

modification of such computational tools, the extent and nature of which would vary

between one tool and another. It is therefore proposed that the condensed tangent

stiffness matrix may be reasonably and generally approximated by constructing

reduced order models of the structure and soil sub-domains. The benefit of such an

approach is that it does not require the explicit assembly of the stiffness matrices,

thus providing a general yet potentially efficient coupling technique.

Considering a sequential Dirichlet-Neumann coupling algorithm, the interface

condensed tangent stiffness matrices of the partitioned soil and structure sub-

domains can be approximated, at each coupling iteration stage 1I of a particular

time/load increment, by constructing the following reduced order models of the

partitioned sub-domains.

For the partitioned structure sub-domain, the following displacement mode matrix

can be constructed at coupling iteration 1I :

0 1,..., ,...,

I J I I Ii i i i i iT T T T T T

M IU U U U U U

TU (6.53)

where M corresponds to the number of coupled degrees of freedom at the interface

and 0,..., 1J I .

Similarly, the variation of the interface forces corresponding to TU can be

constructed as:

0 1,..., ,...,

I J I I Ii i i i i iT T T T T T

M IF F F F F F

TF (6.54)

Considering Equation (6.5), any IiTU can be projected as a linear combination of

the displacement modes given by Equation (6.53):


215

IiT n

U TU .δ (6.55)

where:

0 1,..., ,...,T

J I δ (6.56)

Thus the variation of the interface forces can also be approximated in a similar

manner as:

IiT n

F TF .δ (6.57)

The coefficients 𝛅 which provide the minimum error in approximating any IiTU

according to Equation (6.55) can be obtained as:

1

.IT T i

T nU

T T Tδ U U U (6.58)

Now, the reduced order model of the partitioned structure sub-domain can be

determined from combining Equations (6.57) and (6.58):

1

.I IT Ti i

T Tn nF U

T T T TF U U U (6.59)

Comparing the reduced order model of the structure given by Equation (6.59) with

Equation (6.9), it is evident that the condensed tangent stiffness matrix of the

partitioned structure sub-domain can be approximated as:

1

.T TC

TK

T T T TF U U U (6.60)

In a similar manner, the condensed tangent flexibility matrix of the partitioned soil

sub-domain can be obtained by constructing the following force and displacement

mode matrices at coupling iteration 1I :


216

0 1,..., ,...,

I J I I Ii i i i i iB B B B B B

M IF F F F F F

BF (6.61)

0 1,..., ,...,

I J I I Ii i i i i iB B B B B B

M IU U U U U U

BU (6.62)

where M corresponds to the number of coupled degrees of freedom at soil-structure

interface and 1,..., 1J I .

Now projecting IiB n

F as a linear combination of force modes given by Equation

(6.61):

IiB n

F BF .η (6.63)

where:

0 -1,..., ,...,T

J I η (6.64)

the variation of the interface displacement can also be approximated as:

IiB n

U BU .η (6.65)

Accordingly, the reduced order model of the partitioned soil sub-domain can be

determined as:

1

.I IT Ti i

B Bn nU F

B B B BU F F F (6.66)

Comparing the reduced order model of the soil partitioned sub-domain given by

Equation (6.66) with Equation (6.10), it is evident that the condensed tangent

flexibility matrix of the partitioned soil sub-domain can be approximated as:

11

.T TC

BK

B B B BU F F F (6.67)


217

The abovementioned approximations of the condensed tangent stiffness matrices at

the interface, as given by Equations (6.60) and (6.67), can be constructed through

coupling iterations and used in successive updates of the boundary conditions

utilising Equation (6.37).

It is worth noting that, in the original reduced order scheme proposed by Vierendeels

et al. (2007) for coupling fluid-structure interaction problems, the prescribed

Neumann data are calculated as a result of solving the reduced order models of the

partitioned sub-domains in consecutive iterations, (except for the first two iterations,

where equilibrium is applied). Here, the reduced order models are solved only once

to obtain the Dirichlet data prescribed to the structure sub-domain, while the

Neumann data prescribed to the soil sub-domain is obtained by enforcing

equilibrium at all iterations.

6.3.2.1 Singularity of .T

T TU U and .T

B BF F

Consider both the condensed interface stiffness and flexibility matrices of the

structure and soil partitioned sub-domains as approximated by Equations (6.60) and

(6.67), respectively. These approximations are only valid if and only if the square

matrices .T

T TU U and .T

B BF F are invertible. In fact, if .T

T TU U or

.T

B BF F become singular, Equations (6.60) and (6.67) cannot be used to

approximate the condensed interface stiffness/flexibility matrices during coupling

iterations.

In the following, the general conditions for which .T

T TU U and .T

B BF F are

non-singular are established, by considering that any square matrix n nΤ is

invertible if and only if all its rows and columns are independent.

A column/row vector is said to be independent of other column/row vectors if it

cannot be expressed as a linear combination of these vectors. The number of linearly


218

independent columns/rows is called the column/row rank of a matrix (Horn &

Johnson, 1999). The column rank and row rank are always the same, and therefore

they are simply called the rank of a matrix. In general, the rank of any p q matrix is

at most min( , )p q , and a matrix that has a rank as large as possible is said to have

full rank. If a square matrix has a full rank, none of its column/row vectors can be

expressed as a linear combination of the remaining vectors, and the matrix is

therefore non-singular.

In view of the above, a sufficient condition for non-singularity of n nΤ is for the

square matrix n nΤ to have a full rank (Horn & Johnson, 1999). Since both TU

and BF are M K matrices, with M denoting the number of interface degrees of

freedom and K the current coupling iteration number, it is clear that matrices

.T T TA U U and .

T

B BB = F F would be square K K matrices.

Given that:

( ) ( ) ( )Rank Rank Rank T TX X X X (6.68)

it is evident that:

.T

K K M KRank Rank

T T TΑ U U U (6.69)

.T

K K M KRank Rank

B B BΒ F F F (6.70)

Assuming that K<M, the requirement of .T T TΑ U U and .

T B BΒ F F to be

invertible dictates TU and BF to have rank of K (full column rank). Considering

these matrices as given by Equations (6.53) and (6.61), for TU and BF to have

full column rank, it is essential that in all coupling iterations the newly obtained

displacement and force increments at the interface nodes for T and B are linearly

independent of the previous corresponding incremental vectors.


219

In coupling iterations for which the coupling iteration number is less than the

number of interface freedoms, there is only the possibility of having linearly

dependent incremental displacement/force vectors. However, in problems where the

required number of coupling iterations for achieving convergence exceeds the

number of interface degrees of freedom, any newly obtained incremental

displacement/force vectors is undoubtedly a linear combination of the previous

corresponding independent vectors. As a result, the construction of the condensed

tangent stiffness matrix at the interface would not be possible in this case.

These facts highlight a major shortcoming of the original reduced order method

proposed by Vierendeels et al. (2007). In the following section, a new approach is

proposed for selective addition and replacement of the incremental

displacement/force vectors in TU and BF , thus overcoming the above

shortcoming and offering a robust reduced order model for approximating the

condensed interface tangent stiffness/flexibility matrices.

6.3.2.2 Selective Addition or Replacement of the Displacement/Force

Vectors

As mentioned above, in constructing the reduced order method for approximating the

condensed interface tangent stiffness matrix of the partitioned sub-domains, there is

a need for a procedure to make sure that any newly obtained interface

displacement/force modes added to TU and BF matrices are linearly independent.

In this respect, the procedure proposed in this section ensures that any newly

constructed interface incremental displacement vector, hereafter referred to as

displacement mode, is linearly independent of the previous modes:

1, 1

0 2 with , , , ,I I I Ti

T J IU

TU ζ ζ (6.71)

Consider the reduced order model for partitioned sub-domain T which is treated by

Dirichlet boundary conditions at the interface. Following the algorithmic procedure


220

of sequential D-N coupling method, at the starting coupling iteration (I=0), the

relaxation of the interface boundary conditions is applied. At the first coupling

iteration (I=1), TU and TF matrices would be single column matrices as shown

by Equations (6.72) and (6.73):

0,11I iT TU

U (6.72)

0,11I iT TF

F (6.73)

where:

,I J I Ji i iT T TU U U (6.74)

,I J I Ji i iT T TF F F (6.75)

In the next coupling iteration (I=2), there will be a new set of interface displacement

modes:

0,2 1,22,

I i iT TU U

TU (6.76)

Considering Equation (6.76) it is clear that, in order to avoid the singularity of

.T

T TU U , TU should be a full column rank matrix. In general at any iteration

2I , the column rank of I

TU given by Equation (6.77) should be checked:

0, , 1,,..., ,...,

I J I I II i i iT T TU U U

TU (6.77)

Given that TU of previous iteration (I-1) is of the form:

0, 1 , 1 2, 11,..., ,...,


TU (6.78)


221

adding the currently obtained interface displacement mode, 1,I IiTU

, to all the

previously obtained interface displacement modes given by Equation (6.78) leads to:

1 0, 1 1, 2, 1 1,ˆ ,...,I I I I I I I Ii i i i

T T T TU U U U TU (6.79)

which simplifies to:

1 0, , 2,ˆ ,..., ,...,I I J I I Ii i i

T T TU U U TU (6.80)

Assuming the following normalization of the displacement modes:

,

I Ji iI J T Ti

T I Ji iT T

U UU

U U

(6.81)

1ˆ I TU could be written in the normalised form of:

1 0, , 2,ˆ ,..., ,...,

I I J I I Ii i iT T TU U U

TU (6.82)

At this point, it can be established whether the currently obtained displacement

mode, 1,I IiTU

, is a linear combination of the previous vectors in

1ˆ I TU . If

1,I IiTU

is a linear combination of

1ˆ I TU :

11, ˆ .

II IiTU

TU ζ (6.83)

with

0 2, , , ,T

J I ζ (6.84)

solving Equation (6.83) for ζ gives:


222

1,.

I IiTU

ζ ω (6.85)

where:

1

1 1 1ˆ ˆ ˆT TI I I

T T Tω U U U (6.86)

Substituting (6.85) into (6.83) gives:

11, 1,ˆ . .

II I I Ii iT TU U

TU ω (6.87)

Constructing the following conditions:

1, 1,, 1

I I I II i iT TU U

(6.88)

11, 1,1 ˆ, . .

II I I II i iT TU U

TU ω (6.89)

it can be clearly shown that if 1,I IiTU


1ˆ I TU as

assumed in Equation (6.83), the value of 1I should be equal to I .

Therefore, if in coupling iteration 2I the value of 1I I is greater than a

prescribed tolerance, , 1,I IiTU

cannot be expressed as a linear combination of

1ˆ I TU . In this case, 1,I Ii

TU

can be added to 1ˆ I

TU to construct a new linearly

independent I

TU :

0, , 1,,..., ,...,


TU (6.90)


223

where the corresponding force modes at the interface of sub-domain T take also

the following form:

0, , 1,,..., ,...,

I J I I II i i iT T TF F F

TF (6.91)

On the other hand, when 1I I is less than the prescribed tolerance, 1,I IiTU

is a linear combination of 1ˆ I

TU . In such a case, the previous 1I TU and 1I

TF

can still be used for approximating the condensed stiffness matrix at the interface.

However, a better approach for nonlinear analysis would be to replace one of the

displacement modes in 1ˆ I

TU with the newly obtained 1,I IiTU

to form a new

full rank I

TU .

The replacement procedure considers the outcome of Equation (6.85):

1,

0 2, , , , .I IT i

J I TU

ζ ω (6.92)

It can be shown that if 1,I IiTU

replaces a displacement mode of

1ˆ I TU which

corresponds to the maximum absolute value of ( 0,..., 2)J j I , a full rank matrix

I

TU can be constructed which incorporates the latest incremental mode vectors.

Assuming that 0,..., 2( )J J I mMax , a new I

TU can be constructed where an

intermediate vector is removed from the assembled matrix of Equation (6.90):

0, ,, 1,

,..., ,..., ,...,I J I I II i i i

T T T

Taken

IiT

t

m

ou

U U UU

TU

(6.93)

and the corresponding force mode matrix at the interface of sub-domain T takes

the form:


224

0, ,, 1,

,..., ,..., ,...,I J I I II i i i

T T T

Taken

IiT

t

m

ou

F F FF

TF

(6.94)

In a very similar manner, an enhanced reduced order model is developed for

partitioned sub-domain B which is treated by Neumann boundary conditions at the

interface. In general, at any iteration 2I , the column rank of I

BF given by

Equation (6.95) should be checked:

0, , 1,,..., ,...,

I J I I II i i iB B B BF F F

F (6.95)

Given that BF of previous iteration (I-1) is of the form:

0, 1 , 1 2, 11,..., ,...,


F (6.96)

adding the currently obtained interface force mode, 1,I IiBF

, to all the previously

obtained interface force modes, given by Equation (6.96) leads to:

1 0, , 2,ˆ ,..., ,...,I I J I I Ii i i

B B B BF F F

F (6.97)

Assuming the following normalization of the force modes:

,

I Ji iI J B Bi

B I Ji iB B

F FF

F F

(6.98)

1ˆ I

B

F could be written in the normalised form of:

1 0, , 2,ˆ ,..., ,...,

I I J I I Ii i iB B B BF F F

F (6.99)


225

At this point, it can be established whether the currently obtained displacement

mode, 1,I IiBF

, is a linear combination of the previous vectors in

1ˆ I

B

F . If

1,I IiBF


1ˆ I

B

F :

11, ˆ .

II IiB BF

F ζ (6.100)

With

0 2, , , ,T

J I ζ (6.101)

solving Equation (6.100) for ζ gives:

1,.

I IiBF

ζ ω (6.102)

where:

1

1 1 1ˆ ˆ ˆT TI I I

B B B

ω F F F (6.103)

Substituting (6.102) into (6.100) gives:

11, 1,ˆ . .

II I I Ii iB B BF F

F ω (6.104)

Constructing the following conditions:

1, 1,, 1

I I I II i iB BF F

(6.105)

11, 1,1 ˆ, . .

II I I II i iB B BF F

F ω (6.106)


226

If in coupling iteration 2I the value of 1I I is greater than a prescribed

tolerance, , 1,I IiBF

cannot be expressed as a linear combination of

1ˆ I

B

F . In

this case, 1,I IiBF

can be added to

1ˆ I

B

F to construct a new independent I

BF :

0, , 1,,..., ,...,


F (6.107)

where the corresponding displacement modes at the interface of sub-domain B will

take the following form:

0, , 1,,..., ,...,

I J I I II i i iB B B BU U U

U (6.108)

On the other hand, when 1I I is less than the prescribed tolerance, 1,I IiBF

is a linear combination of 1ˆ I

B

F . In such a case, the previous 1I

B

F and 1I

B

U

can still be used for approximating the condensed stiffness matrix at the interface.

However, a better approach for nonlinear analysis would be to replace one of the

displacement modes in 1ˆ I

B

F with the newly obtained 1,I Ii

BF

to form a new full

rank I

BF .

The replacement procedure considers the outcome of Equation (6.102):

1,

0 2, , , , .I IT i

J I BF

ζ ω (6.109)

It can be shown that if 1,I IiBF

replaces a force mode of

1ˆ I

B

F which corresponds

to the maximum absolute value of ( 0,..., 2)J j I , a full rank matrix I

BF can be

constructed which incorporates the latest incremental mode vectors.


227

Assuming that 0,..., 2( )J J I mMax a new I

BF can be constructed where an

intermediate vector is removed from the assembled matrix of Equation (6.107):

0, , , 1,

,..., ,..., ,...,I J I I II i i i

B B B B

Taken out

m IiBF F FF

F

(6.110)

and the corresponding the corresponding displacement mode matrix at the interface

of sub-domain B takes the following form:

0, , , 1,

,..., ,..., ,...,I J I I II i i i

B B B B

Taken out

m IiBU U UU

U

(6.111)

6.3.2.3 Singularity of Approximated Stiffness/Flexibility Matrices

Considering the presented reduced order method, Equations (6.60) and (6.67) are re-

written as:

1

.T TC

T M K K M M K K MM MK

T T T TF U U U (6.112)

11

.T TC

B M K K M M K K MM MK

B B B BU F F F (6.113)

In view of the discussion in section 6.3.2.2, it is assumed that for coupling iterations

K<M both M KTU and M KBF have full column rank (with M and K denoting the

number of interface degrees of freedom and the current coupling iteration number,

respectively).

Given that for any X and Y matrices:

( ) min ( ), ( )Rank Rank RankX.Y X Y (6.114)

and considering Equations (6.112) and (6.113), it can be easily shown that:


228

CT M M

Rank K K

(6.115)

1CB M M

Rank K K

(6.116)

As previously mentioned, the condition for non-singularity of CTK and

1CBK

requires these square matrices to have a full rank (M). Since K<M, it is clear from

Equations (6.115) and (6.116) that both CTK and

1CBK

approximated by reduced

order method are singular matrices in nature. This is not a problematic issue in the

view of updating the boundary conditions in successive iterations using

Equation (6.37), since there is no need for inverting CTK and

1CBK

. In fact, the

reason for constructing the flexibility condensed interface matrix of the partitioned

sub-domain B directly by the reduced order method in Section 6.3.2 is due to the

singularity of the reduced order method approximation of CBK and not for

computational efficiency. In this context, it is very important to note that in coupling

procedures via reduced order method using any of Equations (6.21) to (6.24), where

an inverse of condensed stiffness matrices is required, the corresponding flexibility

matrix should be constructed instead.

6.3.2.4 Example 1: Static FEMFEM Coupling

Here the static FEM-FEM coupled problem of 6.3.1.1 is also analysed using the

presented reduced order scheme with and without the selective addition and

replacement procedure. A comparison between the number of coupling iterations

using adaptive relaxation and reduced order method is also provided in Table 6.4.

These results clearly show that the reduced order method is far superior to the

adaptive relaxation scheme, since it enhances the convergence rate of the coupling

method significantly. This high convergence rate is further demonstrated in Figures

6.12 to 6.17 for models A1 to A6 respectively.


229

Furthermore, it is shown in Figures 6.12 and 6.13 that without employing the

proposed selective addition/replacement procedure of section 6.3.2.2, if the number

of required coupling iterations for convergence to a prescribed tolerance exceeds the

number of independent modes, the iterative scheme will diverge. This fact highlights

the significance of employing the selective addition/replacement procedure proposed

in Section 6.3.2.2.

Model T

B

E

E


with Add/Rep


without Add/Rep

Adaptive

Relaxation

A1 8.0 5 Not Converged 16

A2 4.0 5 Not Converged 13

A3 2.0 5 7 10

A4 1.0 5 6 6

A5 0.5 5 5 6

A6 0.2 4 4 4

Table 6.4: Required coupling iterations for different coupling schemes


230

Figure 6.12: Error reduction for Model A1



231




232




233

6.3.2.5 Example 2: Dynamic FEMFEM Coupling

The dynamic FEM-FEM coupling example presented in Section 6.3.1.2 is

considered here using the reduced order method, and the obtained results are again

compared with the adaptive relaxation. Two slightly different types of adaptive

relaxation and reduced order method are considered here, depending on the first

coupling iteration update procedure.

In the first set of methods, named as ‘Adaptive’ and ‘Reduced’ in Table 6.5, the

relaxation parameter is initialised at the start of every load/time step to a constant

value, 1.0 , and thereafter the adaptive relaxation or reduced order method is

applied in the normal way.

On the other hand, the second set of coupling methods, named as ‘Adaptive*’ and

‘Reduced*’ in Table 6.5, continue with the parameter or condensed matrices from

the previous step. For the ‘Adaptive*’ approach, the starting relaxation parameter is

considered to be equal to the last computed adaptive relaxation parameter of the

previous time/load step. Similarly in the case of ‘Reduced*’, the last approximation

of the condensed stiffness/flexibility matrices obtained in previous time/load steps is

used instead of a constant relaxation parameter for the update of boundary conditions

in the first iteration.

The dynamic analysis problem is considered for the different problem partitioning

types. This is achieved by analysing the same system with different mass ratios at the

interface of the partitioned sub-domains T and B ( 1 2/m m ). The different

analysed models are presented in Table 6.5 for a 5s duration of response with

0.01t s using different coupling techniques.

The presented results in Table 6.5 again show that the reduced order method

possesses a higher convergence rate than adaptive relaxation. Moreover, it is shown

that the performance of ‘Adaptive*’/‘Reduced*’ techniques are far better than their

corresponding ‘Adaptive’/‘Reduced’ counterparts (see Figure 6.23). However, it

should be noted that, unlike the problem under consideration, in problems where


234

there is a considerable or abrupt change in the state of problem (significant stiffness

change or high nonlinearity), it is not guaranteed that ‘Adaptive*’ and ‘Reduced*’

techniques outperform ‘Adaptive’ and ‘Reduced’ coupling methods.

The higher convergence rate of the reduced order method compared to adaptive

relaxation is further demonstrated in Figures 6.18, to 6.22, where the error reduction

of the two schemes, Adaptive*’/‘Reduced*, are compared in different arbitrarily

chosen time steps.

In all the analyses, the procedure introduced in Section 6.3.2.2 for selective

addition/replacement of the displacement and force vectors is utilised. As discussed

earlier, if such a procedure is not employed, any new obtained displacement/force

modes undoubtedly will be a linear combination of the previous independent

displacement/force modes, since the required number of coupling iterations exceeds

the number of independent modes. As a result, the approximation for the condensed

tangent stiffness matrix at the interface will be poor, if numerically possible due to

round-off errors, thus leading to a very low convergence rate and even divergence.

This shortcoming of the conventional reduced order method is illustrated in Figure

6.23, where it is shown that without employing the selective addition/replacement

procedure the coupling scheme starts to diverge after the third coupling iteration.


235

Model 1

2

m

m

Number of Coupling Iterations

Adaptive Reduced

Method Adaptive*

Reduced

Method*

C1 200 kg

0. 11800 kg

2133 1611 1624 744

C2 400 kg

0.251600 kg

1706 1279 1027 730

C3 600 kg

0.431400 kg

1432 1224 879 696

C4 800 kg

0.61200 kg

1291 1183 824 676

C5 1000 kg

1.01000 kg

1177 1170 758 702

C6 1200 kg

1.5800 kg

1149 1147 741 671

C7 1400 kg

2.3600 kg

1118 1121 752 668

C8 1600 kg

4.0400 kg

1056 1062 801 666

C9 1800 kg

9.0200 kg

972 975 821 644

*using the relaxation parameter/condensed interface stiffness matrix of the last time-

step in which convergence was achieved for the first coupling iteration of the current

time-step

Table 6.5: Number of required coupling iterations for 500 time-steps ( 0.01t s )

with a tolerance of 1e-4 m


236

Figure 6.18: Error reduction of different schemes (Time=2.74s) for Model C1



237




238


Figure 6.23: Error reduction of different scheme (Time=2.74s) for Model C1


239

6.3.2.6 Example 3: Linear SoilStructure Interaction

Consideration is given here to a linear static plane strain analysis problem of a

concrete cantilever wall resting on a flexible soil, which is loaded at the top with a

horizontal force (Figure 6.24). First, the problem is analysed using both the adaptive

relaxation and reduced order methods, while the stiffness of the foundation beam is

partitioned into two and is partly modelled in the structure sub-domain and partly in

soil sub-domain. This would increase the convergence rate of the iterative coupling

schemes, since by adding rigidity to the interface of soil the number of Dirichlet

degrees of freedom at the interface is effectively reduced during coupling iterations.

Using partitioned analysis, both the adaptive relaxation scheme and the reduced

order method show similar convergence characteristics as illustrated in Figure 6.25

Figure 6.24: Coupled soil-structure interaction problem


240

Figure 6.25: Error reduction for different coupling schemes

On the other hand, with the same problem modelled such that the foundation beam is

only modelled in the partitioned structure sub-domain, the convergence

characteristics become poor, as expected and as shown in Figure 6.26. However, it is

clear that the reduced order method with the selective addition/replacement

outperforms adaptive relaxation by far. Moreover, without employing the selective

addition/replacement procedure, the reduced order coupling scheme starts to diverge

after the 10th coupling iteration (noting that there are 10 degrees of freedom at the

soil-structure interface).

According to Equations (6.60) and (6.67), the reduced order method converges when

a good approximation of the condensed tangent stiffness is achieved. Considering

Figure 6.26, it can be seen that this is achieved through eleven cycles of

displacement/force history data. By using the reduced order method, it is guaranteed

that reasonable approximations of the condensed interface stiffness matrices can be

constructed during coupling iterations, and indeed the more displacement/force


241

history data (i.e. the more coupling iterations) the better the approximation of the

tangent stiffness matrix.


It is however worth observing that the required number of coupling iterations for

convergence using the reduced order method is still relatively large in comparison

with what would be necessary in a typical monolithic treatment. Furthermore,

considering the convergence behaviour of the reduced order method, it is clear that

as a result of having a poor approximation of condensed interface stiffness matrices

at the first coupling iterations, the compatibility errors are very big. This could be a

problematic issue in highly nonlinear problems where the high compatibility error of

the starting coupling iterations could cause divergence in the nonlinear solution

procedure. Although in problems such as the one presented this could be avoided by

reallocating the interface stiffness between the two sub-domains, in problems where

such partitioning strategy cannot be employed (such as modelling of a retaining wall)

there is a need for a more robust coupling technique with improved convergence


242

characteristics. Indeed, if a better approximation of the condensed tangent stiffness

matrix could be achieved, the coupling algorithm becomes even more robust, thus

highlighting the potential for further enhancement.

Towards this end, a mixed reduced order method is proposed in the next section. The

approach is based on approximating initial linear condensed stiffness and flexibility

matrices, which are then constantly updated using the iterative displacement and

force modes obtained during successive coupling iterations.

6.3.3 Mixed Reduced Order Method

A further enhancement of the reduced order method is proposed here, which utilises

an initial linear condensed stiffness matrix of the partitioned soil and structure sub-

domains at the beginning of soil-structure interaction coupling procedure. During

subsequent coupling iterations, these initial approximated matrices are continuously

updated based on the iterative history of the interface displacement/force increments,

thus achieving improved approximation in nonlinear analysis. The general

formulation for establishing the initial stiffness matrix is similar to the procedure

presented in Section 6.3.1. In the following, these initial stiffness matrices are

obtained via a reduced order method formulation, which gives identical initial

quantities.

Considering the structure partitioned sub-domain and assuming that there are N

degrees of freedom at the interface of the soil-structure interaction problem, the

following N+1 displacement vectors are prescribed at the interface of the structure

sub-domain:

1,..., ,..., , 0,...,Ti

T i ij iNiU u u u i N (6.117)

where:

0 or 0

and 1iju

i j iu

i j i

(6.118)


243

and εu is a predefined displacement value in the required range of response.

By obtaining the corresponding response of the structure sub-domain to the above

prescribed displacement vectors as:

1,..., ,..., , 0,...,Ti

T i ij iNiF f f f i N (6.119)

the initial condensed stiffness matrix of the structure sub-domain can be formulated

as:

1

0.

T TCTK

T T T TF U U U (6.120)

where:

1

,..., , 0,...,i i i iT T T T Ti N N N

U U U U i N

U (6.121)

1

,..., , 0,...,i i i iT T T T Ti N N N

F F F F i N

F (6.122)

The condensed tangent stiffness matrix of the structure sub-domain at iteration I is

obtained using a correction to the initial stiffness matrix:

0

C C CT T TI

K K K

(6.123)

such that:

I J I JC i i i iT T T T TI n n n n

K U U F F (6.124)

where I is iteration number and J=0,…,I-1.

Taking:

,I J I Ji i iT T TU U U (6.125)


244

,I J I Ji i iT T TF F F (6.126)

Equation (6.124) leads to:

, , ,

0

I J I J I JC i i C iT T T T TK U F K U

(6.127)

By constructing the following incremental displacement and force vectors:

0, , 1,, , , ,

I J I I Ii i iT T T TU U U

U (6.128)

0, , 1,, , , ,

I J I I I

T T T TF F F

F (6.129)

with:

, , ,

0

J I J I J Ii C iT T T TF F K U (6.130)

the corrective stiffness matrix of Equation (6.123) can now be obtained in the normal

way from the reduced order method according to the following:

1 1I IC iT T TF K U

(6.131)

Assuming that the displacement/load increments at iteration I+1 can be

approximated as a linear projection of the calculated displacement/load vectors of

Equations (6.128) and (6.129):

1IiT TU

U δ (6.132)

1I

T TF F δ (6.133)

With:

0 1, , , ,T

J I δ (6.134)


245

and solving Equation (6.132) for δ through minimisation of the error norm, the

following can be obtained from Equation (6.133):

11 1I IT T i

T T T T T TF U F U U U (6.135)

Comparing Equation (6.135) with Equation (6.131), the updated condensed stiffness

matrix of the partitioned structure sub-domain given by Equation (6.123) takes the

following form:

1

0

T TC CT T T T T TI

K K

F U U U (6.136)

In a similar manner, by prescribing the following N+1 force vectors given by

Equation (6.137) at the interface of the soil sub-domain, and obtaining the

corresponding response of the soil sub-domain to the above prescribed force vectors:

1,..., ,..., , 0,...,Ti

B i ij iNiF f f f i N (6.137)

where:

0 or 0

and 1ijf

i j if

i j i

(6.138)

and εf is a predefined incremental force value in the required range of response.

1,..., ,..., , 0,...,Ti

B i ij iNiF f f f i N (6.139)

the initial condensed flexibility matrix of the soil partitioned sub-domain can be

formulated as:

11

0.

T TCBK

B B B BU F F F (6.140)

Now the condensed tangent flexibility matrix of the soil sub-domain at iteration I can

be approximated by using a correction to the initial flexibility matrix:


246

1 1 1

0

C C CB B BI

K K K

(6.141)

such that:

I J I JC i i i iB B B B BI n n n n

K F F U U (6.142)

where I is iteration number and J=0,…,I-1.

Taking:

,I J I Ji i iB B BU U U (6.143)

,I J I Ji i iB B BF F F (6.144)

Equation (6.142) leads to:

, , ,1 1

0

I J I J I JC i i C iB B B B BK F U K F

(6.145)

By constructing the following incremental force and displacement vectors during

coupling iterations:

0, , 1,, , , ,

I J I I Ii i iB B B BF F F

F (6.146)

0, , 1,, , , ,

I J I I I

B B B BU U U

U (6.147)

with:

, , ,

0

J I J I J Ii C iB B B BU U K F (6.148)

the updated condensed flexibility matrix of the partitioned soil sub-domain can be

obtained as:


247

11 1

0

T TC CB B B B B BI

K K U F F F (6.149)

The algorithmic framework of the presented modified reduced order method is

illustrated in Table 6.6.

Using the above presented mixed reduced order method coupling technique, the

same problem of Section 6.3.2.6 is considered with the foundation beam completely

modelled in the structure sub-domain, which is more onerous than the case where the

foundation beam is apportioned between the two sub-domains. Figure 6.27,

compares the convergence rate of the mixed reduced order method with the

conventional reduced order method, highlighting its potential superiority in iterative

coupling of FEM-FEM coupled problems.



248

STEP 0: Calculate the initial linear stiffness/flexibility matrices at the beginning of

soil-structure coupling: 1

0 0,C C

T BK K


For I=0,1,…(number of iterations)

STEP 1: At the start of each increment, the structure domain is loaded by the external forces, while the displacements at the interface nodes are prescribed in

accordance with the initial conditions: IiT n

U U

STEP 2: The structural solver computes the response of the structure for : IiT n

F

IF I ≥1 then :

i. Selective addition/replacement of displacement/force vectors ii. Construct TU and T

F

iii. 1

0

T TC CT T T T T TI

K K

F U U U


applying equilibrium: 0I Ii i

T Bn nF F

STEP 4: Based on these forces and the external loading applied to the soil domain,

the soil solver computes the response of the soil domain for: IiB n

U

IF I ≥1 then :

i. Selective addition/replacement of displacement/force vectors ii. Construct BF and B

U

iii. 11 1

0

T TC CB B B B B BI

K K U F F F

STEP 5: If convergence to compatibility has been achieved the solution proceeds to the next time/load increment (n=n+1) with:

1 1

0 0,C C C C

T T B BI IK K K K

STEP 6: If convergence to compatibility has not been achieved, the following new estimation of the displacements will be applied to the structure sub-domain and the iteration will continue (I=I+1) until convergence:

11 1I I I Ii i C C i iT T B T B TI In n n n

U U K K U U

I

Table 6.6: Mixed reduced order method coupling procedures


249

6.4 Case Study: Nonlinear Soil-Structure Interaction Problem

In this section the presented nonlinear soil-structure interaction problem in Chapter 5

(Section 5.4), which was analysed using the relaxation scheme, is modelled and

analysed using the various reduced order formulations proposed in this chapter. The

example considers a steel frame resting on a soil under static loads, where nonlinear

elasto-plastic constitutive behaviour of the soil as well as geometric and material

nonlinearity of the structure are taken into account. The problem under consideration

is detailed in Figure 6.28 and Table 6.7.

Figure 6.28: Plane frame resting on soil

The building is designed for office purposes and is assumed to be loaded equally on

each floor. The footing is partly modelled in soil sub-domain and partially in the

structure sub-domain and the interface degrees of freedom are assumed to be at

nodes that belong to both of the partially modelled footings with a total number of

degrees of freedom equal to 30. The soil-structure interaction analysis is carried out

assuming plane strain conditions, where the developed domain decomposition

approach is employed utilising ADAPTIC and ICFEP.


250

Structure Sub-domain Material Properties

All beams and columns (steel)

Steel Grade = S355

Elastic Modules = 210 GPa

Strength = 355 MPa

Bilinear elasto-plastic with strain Hardening Factor = 1%


Elastic Modulus = 30 GPa

Linear material

Soil Sub-domain Material Properties

Soil Angle of Shear resistance ( 𝛷′) = 22°

Dilation angle (𝜈 ) = 11°

Cohesion = 20 kPa




Table 6.8 presents the required number of coupling iterations for convergence to the

prescribed tolerance for the reduced order method, mixed reduced order method and

adaptive relaxation scheme. Figure 6.29 shows the number of coupling iterations

required in each increment for the different coupling schemes. The convergence rates

of the three approaches in the first, fifth and sixth load increments are also illustrated

in Figures 6.30 to 6.32, respectively. This demonstrates that the mixed reduced order

method achieves a faster convergence rate than other coupling schemes.

Number of iterations

with adaptive relaxation


with reduced order


with mixed reduced order

46 31 22

Table 6.8: Number of required coupling iterations for different coupling schemes


251

Figure 6.29: convergence behaviour over full range of response for different schemes

Figure 6.30: Error reduction in the first load step


252

Figure 6.31: Error reduction in the 5th load step

Figure 6.32: Error reduction in the 6th load step


253

6.5 Conclusion

In this Chapter various domain decomposition methods for nonlinear analysis of

soil-structure interaction problems, based on approximating the condensed interface

stiffness matrix, are proposed. The overall domain is divided into physical partitions

consisting of soil and structure sub-domains, and coupling of the separately

modelled sub-domains is undertaken via sequential iterative Dirichlet-Neumann sub-

structuring method.

It is shown that by using the condensed tangent interface stiffness matrices of the

partitioned sub-domains in the update of boundary conditions, superior convergence

characteristics could be achieved. In this respect, convergence to compatibility

occurs for linear problems immediately at the first iteration. Although this immediate

convergence does not normally occur for nonlinear problems, employing this method

for the update of interface Dirichlet boundary conditions ensures a high convergence

rate. This brings the performance of the proposed coupling approach close to the

monolithic treatment.


nonlinear field modelling tools, a more general approximation for the condensed

stiffness matrices is desirable and can indeed be achieved during the course of

coupling iterations, thus avoiding the need for explicit determination and extraction

of the stiffness matrices. In this respect, the condensed tangent stiffness matrix is

approximated via reduced order models, building on a previous approach by

Vierendeels et al. (2007) concerned with fluid–structure interaction problems.

Various significant modifications to this approach are proposed here, leading to a

versatile and efficient approach for coupled modelling of nonlinear soil-structure

interaction problems. In this regard, a major pitfall associated with the original

method causing divergence is overcome by a new selective addition/replacement

procedure of force and displacement mode vectors, where the applicability and

advantages of this modification are demonstrated by means of several examples.

Moreover, different to the method proposed by Vierendeels et al. (2007), the reduced


254

order method proposed here is solved only once to obtain the prescribed Dirichlet

data to structure sub-domain, while the prescribed Neumann data to the soil sub-

domain is obtained by enforcing equilibrium at all iterations.

In order to achieve better approximation of the condensed tangent stiffness matrix in

the initial stages of coupling iterations, a mixed reduced order method is proposed,

which achieves a more robust coupling technique than the conventional reduced

order method,. The approach is based on approximating an initial linear condensed

stiffness and flexibility matrix, which is then continuously updated using the

iterative displacement and force modes during successive coupling iterations.

The applicability of the presented coupling techniques is demonstrated for nonlinear

soil-structure interaction analysis via a case study consisting of a plane frame

supported on soil foundations. In this context, it is shown that the mixed reduced

order method achieves a faster convergence rate than the other coupling schemes,

demonstrating a great potential towards providing an integrated interdisciplinary

computational approach for nonlinear soil-structure interaction problems.

Chapter 7

Case Studies

7.1 Introduction

In order to illustrate the application of the proposed coupling techniques in soil-

structure interaction analysis and provide a better grasp of the involved concepts,

several illustrative static nonlinear soil-structure interaction problems are presented

and discussed in this chapter.

The problems under consideration are treated by partitioned analysis, where the

coupling is carried out through coupling of discipline-oriented solvers, ADAPTIC

and ICFEP, for nonlinear structural and geotechnical analysis as outlined in

Chapter 4.

Using the developed simulation environment for a number of problems in which

nonlinearity arises in both the structure and the soil, the advantages of the proposed

partitioned analysis are demonstrated. Moreover, a comparison between various

proposed coupling algorithms is performed in some cases to further highlight their

relative performance characteristics.

Chapter 7 Case Studies

256

7.2 Nonlinear Behaviour of Pitched-Roof Frame on Flexible Soil

In this example, the in-plane nonlinear behaviour of the unbraced single-bay pitched-

roof steel frame of Figure 7.1, resting on a flexible soil is modelled. The effect of

soil-structure interaction at the foundation level is taken into account using the

partitioned treatment. The effect of the soil-structure interaction on the overall

response is demonstrated by comparing the results obtained by the partitioned

treatment to those of a non-interactive case (i.e. rigid soil base).

Figure 7.1: Pitched-roof steel frame resting on soil

The considered steel frame (Figure 7.1) utilises the following material characteristics

and member cross-sections: E= 210 GPa (Young’s modulus), fy=350 MPa (yield

strength), HEB280 columns, and HEB240 rafters. The frame is modelled with

ADAPTIC using elasto-plastic cubic beam-column elements (Izzuddin & Elnashai,

1993), which enable the modelling of geometric and material nonlinearity, while the

material behaviour is assumed to be bilinear elasto-plastic with kinematic strain

hardening of 1%. The soil sub-domain is modelled with ICFEP using an associated

elasto-plastic Mohr-Coulomb constitutive model and discretised using 8-noded

isoparametric quadrilateral elements. Young’s Modulus E varies with depth z


257

according to (E = 10,000 + 5,000z kN/m2), Poisson’s ratio μ=0.2, the bulk unit

weight γ = 20 kN/m3, the angle of shearing resistance, φ′=20° and cohesion c′=20

kPa.

The interface degrees of freedom are assumed to be at nodes that belong to the

footings with a height of 0.5 m and an effective out of plane depth of 1m. There are

20 degrees of interface freedoms in total. The loading system consists of a constant

horizontal force of H=22 kN and a varying vertical load of magnitude 0.25w ,

which are modelled in the structure partitioned sub-domain and applied in 34 load

increments.

The results from the partitioned treatment are presented in Figures 7.2 and 7.3, which

depict the nonlinear variation of the vertical settlements under footings A and E of

the steel frame with respect to the vertical load factor. Although the generated level

of the vertical settlements is relatively small, different settlements of the left end and

right end of each foundation generate significant rigid foundation tilting (Figures 7.2,

7.3). This can provide different equilibrium paths and change the force distribution in

the structure compared to non-interactive case. As a result the response of the

interactive case is significantly different compared to the non-interactive case, for

which the rotation of the base is not taken into account. This is demonstrated in

Figures 7.4 and 7.5, showing the horizontal displacement of nodes B and D of the

steel frame with respect to the load factor, for both interactive and non-interactive

cases, respectively. The horizontal and vertical displacements of the roof top of the

frame (node C) is also depicted in Figures 7.6 and 7.7, showing the significant

difference in the response of the interactive case compared to the non-interactive

case. It is worth noting that due to the small indeterminacy degree of the steel frame

and high distance between the foundations, the horizontal and vertical reactions at

the foundations in both interactive and non-interactive case are almost similar.

However, due to the rotation of the foundations the generated moment at the base as

shown in Figures 7.8 and 7.9 varies significantly. The variation of the moment at

node C with respect to the load factor for both interactive and non-interactive case is

also depicted in Figure 7.10.


258

Figure 7.2: Vertical displacement of the footing A

Figure 7.3: Vertical displacement of the footing E


259

Figure 7.4: Horizontal displacement of node B

Figure 7.5: Horizontal displacement of node D


260

Figure 7.6: Horizontal displacement of node C

Figure 7.7: Vertical displacement of node C


261

Figure 7.8: Variation of moment at node A

Figure 7.9: Variation of moment at node E


262

Figure 7.10: Variation of moment at node C

In Figure 7.11, the deformed shape and the bending moment contours of the

partitioned structure sub-domain in the final load step are shown and compared for

both interactive and non interactive case.

Finally, Figure 7.12 shows contours of stress level at the end of analysis under the

applied load on the structure (load step = 34). The stress level is the ratio, at the same

mean effective stress, of the current deviatoric stress to the deviatoric stress at

failure. It therefore varies from 0 to 1, where 1 indicates full plasticity and failure. It

is evident from Figure 7.12 that the applied loading conditions have mobilised a

plastic zone underneath the footings.


263

Figure 7.11: Deformed shape (scale=5.0) and bending moment (kN-m) in final load

step for a) non-interactive case b) interactive case

Figure 7.12: Contours of stress level in soil sub-domain (at final increment)


264

The spread of plasticity under both footings (Figure 7.12) shows the significant rigid

tilting of both footings A and E. This is further evident from the deformed shape of

the structure which shows the footings experiencing rigid tilting. As a consequence,

there is no plastic hinge formation at the base of right column, as depicted in

Figure 7.11, compared to the non-interactive case.

An alternative treatment to partitioned approach in problems such as the above,

where there is no significant interaction between the influence zones of the soil

beneath the foundations due to the significant distance between them, is to capture

this behaviour using a field elimination treatment. This involves modelling the soil

beneath the foundation sub-grade as a beam on Winkler foundation with a system of

discrete, mechanistic, uncoupled springs. This type of model is useful in engineering

practice due to its simplicity and ease of implementation in a general purpose finite

element platform. However, the nonlinear soil behaviour underneath the shallow

footings cannot be captured by linear spring elements. Moreover, even if nonlinear

springs were to be used, it would involve careful calibration of both the rotational

and horizontal nonlinear springs against field tests. On the other hand, by using a

partitioned treatment, as demonstrated above, a fully coupled nonlinear soil-structure

interaction analysis can be readily carried out, where both the behaviour of the

structure sub-domain and soil sub-domain are evaluated using proven field specific

analysis tools.

7.3 Settlement Analysis of Multi-storey Five-bay Steel Frame

Assessment of the potential construction and settlement-induced damage to building

structures is of significant importance in civil engineering practice (Boone, 1996;

Boone, 2001; Charles & Skinner, 2004), and this can require fully coupled models

for capturing the behaviour of both soil and structure. Not only do settlements

impinge on the performance of structure and soil, but also they are typically

considered in the design and damage assessment of non-structural elements, such as

infill panels.


265

In this section, the response of a multi-storey five-bay steel frame (Figure 7.13)

resting on a flexible soil is modelled using the partitioned analysis. The benefits of

utilising a fully coupled soil-structure analysis via the partitioned approach are

demonstrated, and the results are compared to those from a field elimination

technique where the soil sub-domain is modelled with linear transitional and

rotational springs (Winkler foundation).

Figure 7.13: Multi-Storey five-bay steel frame

The considered soil-structure system is partitioned physically into two sub-domains,

soil and structure, where each sub-domain is discretised separately according to its

characteristics as shown in Table 7.1. The framed structure is modelled with

ADAPTIC using cubic elasto-plastic beam-column elements, which enable the

modelling of geometric and material nonlinearity. The frame is discretised using 10

elements per member for both columns and beams, and the material behaviour is

assumed to be bilinear elasto-plastic with kinematic strain hardening. The footings

are discretised using 4 elements per member. The soil sub-domain is modelled with

ICFEP using an elasto-plastic Mohr-Coulomb constitutive model, with parameters

chosen to represent the behaviour of London clay (Table 7.1). The loading system is

modelled in the structure partitioned sub-domain and applied in 10 load increments

(uniformly distributed load of 5kN/m applied on all beams). The interface degrees of


266

freedom are assumed to be at nodes that belong to both the footings and soil

underneath. The total number of interface degrees of freedom is 60 for this case.

Structure Sub-domain Material Properties Columns Beams All beams and columns (steel)

UC 203×203×46 UB 305×102×25 Steel Grade = S355 Elastic Modules = 210 GPa Strength = 355 MPa Bilinear elasto-plastic with strain Hardening Factor = 1%


Elastic Modulus = 30 GPa Linear material Size: 2m×0.5m

Soil Sub-domain Material Properties Soil Angle of Shear resistance ( 𝛷′) = 22° Dilation angle (𝜈 ) = 11° Effective out of plane depth = 1m Cohesion = 20 kPa




To further assess the merits of the various coupling algorithms presented in this

work, the above problem is analysed using different update techniques for a

tolerance of 410 m, where the total number of required coupling iterations (for 10

increments) is listed in Table 7.2. As expected, the constant relaxation scheme has

the worst convergence rate, while the proposed mixed reduced order coupling

scheme has the highest convergence rate.

However, the mixed reduced order scheme requires a greater number of calls to soil

and structural solvers in this case, since it requires the determination of the initial

condensed stiffness matrix prior to the coupling iterations, which involves more

interface-related computations than the reduced order method. Hence, one might

consider the trade-off for different types of problems under consideration, where the


267

proposed mixed reduced order scheme attains even greater superiority for problems

involving more load steps and/or greater nonlinearity.

Coupling method Number of required coupling iterations

Increase in convergence rate compared to optimum

relaxation Optimum constant relaxation 85 - Adaptive relaxation 64 25% Reduced order method 51 40% Mixed reduced order method 43 50%

Table 7.2: Comparison of different coupling methods

Notwithstanding, the benefits of the developed simulation environment in the

practical assessment of nonlinear soil-structure interaction problems can be

demonstrated by considering the results of this example.

The frame structure with its applied loads transmits the loading to the soil, which in

turn deforms due to cumulative action of these loads. As a result, the soil below the

footings goes under vertical and differential settlements. Consequently, the

deformation of soil surface beneath the foundation could cause significant

redistribution of the loads in the frame structure. Moreover, in cases where the

differential settlements are considerable this could cause significant damage to the

infill walls in the frame structure.

As mentioned before, a common approach in capturing this behaviour, benefitting

from simplicity and ease of implementation, is to model the foundation sub-grade as

a beam-on Winkler foundation with a system of discrete and uncoupled springs.

Clearly, however, this type of modelling requires careful calibration of the spring

elements using different experimental test results. Moreover, due to the uncoupled

nature of the spring elements beneath the foundations, the soil sub-domain is often

modelled with gross inaccuracy.

On the other hand, it is shown here that by using a fully coupled partitioned

approach, both the structure and soil behaviour could be effectively captured to the


268

desired degree of accuracy. This is demonstrated in the following, where the

deformation and stress states of the coupled problem are briefly described.

Comparison is made between the fully coupled partitioned approach and the field

elimination approach, where linear springs with an approximated stiffness of 10,000

kPa are used in the latter approach to represent the soil domain.

The results of both approaches for the variation of the vertical settlement under

column C2 (where the maximum vertical settlement is observed) with respect to the

load factor are presented in Figure 7.14. Clearly, except for the first increment where

the response of both models is linear, the real settlement of the footing is expected to

be much higher (up to 500%) than predicted by the simple field elimination approach

when the nonlinearity of the soil is taken into account.

Figure 7.14 : Vertical displacement (m) of Column C2

The vertical deformation profile of the soil surface, for different load-steps, obtained

from coupled analysis is given in Figure 7.15, where it is clear that the six troughs

correspond to the locations of the footings showing their vertical settlement.

Considering the generated level of vertical settlement and the rigid tilting of the


269

footings, it is evident that geometric nonlinearity should be considered in the

structural analysis.

Figure 7.15: Vertical displacement profile of the soil surface

A vector plot of displacements in the soil sub-domain in the vicinity of the structure

is also shown in Figure 7.16. The absolute magnitudes of these vectors are not

important, though their relative magnitude shows the mechanism of ground

deformation. Contours of stress level in the soil partitioned sub-domain for the final

load-step are also depicted in Figure 7.17, where it is evident that the applied loading

conditions have mobilised an extensive plastic zone underneath the building. This

zone is, however, smaller and shallower under the right hand side footing, which is

in agreement with the previous figures that show most of the deformation and load

concentration nearer the left side footings.


270

Figure 7.16: Displacement vectors in soil partitioned sub-domain

Figure 7.17: Contours of stress level in soil partitioned sub-domain

In Figure 7.18, the variation of bending moment at the base of column C1 for

different load levels is shown and compared with that of the field elimination

technique. This shows the significant effect of taking into account the nonlinear

behaviour of soil sub-domain in soil-structure interaction analysis. It can be clearly

observed that the bending moment of the structural elements in the fully coupled

analysis is significantly higher than that of field elimination analysis. This fact is

further demonstrated in Figure 7.19, where the bending moment in the middle of

beam B1 obtained by partitioned approach is compared with that of field elimination

for different load-steps.


271

Figure 7.18: Bending moment (kN-m) at the base of C1 for different load-steps

Figure 7.19: Bending moment at the middle of beam B1 for different load-steps


272

The deformed shape and the bending moment contours of the partitioned structure

sub-domain at the final load step are shown in Figure 7.20b. The same quantities are

also obtained and presented in Figure 7.20a using the field elimination approach. It is

evident from the deformed shape of the structure in fully coupled interaction analysis

and from the vectors underneath each of the three footings in Figure 7.16 that these

experience rigid tilting and significant vertical settlements.

Figure 7.20: Deformed shape (scale=5.0) and bending moment (kN-m) in final load

step for a) linear Winkler foundation b) nonlinear partitioned analysis


273

The significant support settlements predicted by the more realistic partitioned

approach lead to much greater bending moments in the beams and columns than

predicted by the field elimination approach. This is illustrated in Figures 7.21 and

7.22 in which the bending moment diagram of beam B1 and column C1 at the final

load-step are respectively shown.

Figure 7.21: Variation of bending moment along beam B1

Figure 7.22: Variation of bending moment along Column C1


274

Finally, the deformed shapes of beam B1 at the final load-step are presented in

Figure 7.23 for both partitioned and field elimination approaches, where

significantly larger deflections are confirmed with the partitioned approach.

Figure 7.23: Variation of vertical displacement along the beam B1

7.4 Building Response to an Adjacent Excavation

Open cuts and excavations in a limited urban space are gradually increasing in

frequency because of the development and upgrade of infrastructures and the

construction of new buildings. At the same time, public concerns have risen over the

effects of excavation-induced ground movements on adjacent structures and utilities.

Excavation inevitably results in deformation of the adjacent ground and settlement of

adjacent buildings behind an excavation wall, causing problems such as loss of

invaluable historic property, third party impact, construction delay, and substantial

increase of project cost (Son et al., 2005; Aye et al., 2006; Boone et al., 1999; Seok

et al., 2001).


275

The case study under consideration in this section represents a typical urban

situation, where ground excavation can often induce significant movements and

damage to the nearby structures. It is shown that by utilizing a fully coupled soil-

structure interaction model using the partitioned treatment, such nonlinear behaviour

of both structure and soil could be accurately captured. This shows the high potential

of using a fully coupled soil-structure model towards providing reliable assessment

and minimizing the associated damage in such problems. The example considers a

steel frame resting on a soil subjected to ground excavation, where nonlinear elasto-

plastic constitutive behaviour of the soil, as well as geometric and material

nonlinearity of the structure, are taken into account. Figure 7.24 depicts the problem,

where the left hand side boundary is assumed to be consistent with an axis of

symmetry. The plan view of the analysed building frame is also shown in Figure

7.25.

Figure 7.24: Plane frame resting on soil subject to ground excavation


276

Figure 7.25: Plan view of considered building

Figure 7.26: Geometric configuration of considered frame


277

The soil-structure interaction analysis is carried out assuming plane strain conditions

in the soil using an effective out-of-plane width of 1m, where the developed domain

decomposition approach is employed utilising ADAPTIC and ICFEP. The

considered soil-structure system is partitioned physically into two sub-domains, soil

and structure, where each sub-domain is discretised separately according to its

characteristics as shown in Table 7.3.

Structure Sub-domain Material Properties All beams and columns (steel)

Steel Grade = S355 Elastic Modules = 210 GPa Strength = 355 MPa Bilinear elasto-plastic with strain Hardening Factor =1%


Elastic Modulus = 30 GPa Linear material Size: 2m×0.5m

Soil Sub-domain Material Properties Soil and excavation Angle of Shear resistance ( 𝛷′) = 22° Dilation angle (𝜈 ) = 11° Effective out of plane depth = 1m Cohesion = 20 kPa


Excavation width=20m He (excavation depth) (=1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (m) Le (distance from the structure =2, 4, 6, 8, 10 (m)



The frame structure is modelled with ADAPTIC using cubic elasto-plastic beam-

column elements (Izzuddin & Elnashai, 1993) using 10 elements per member for

both columns and beams, and the material behaviour is assumed to be bilinear elasto-

plastic with kinematic strain hardening. The footings are discretised using 4 elements

per member.

The soil sub-domain and the un-braced excavation are modelled with ICFEP using

an elasto-plastic Mohr-Coulomb constitutive model, with parameters chosen to

represent the behaviour of London clay (Table 7.3). The nonlinear solution

procedure employed for analysing the soil sub-domain is based on a Modified


278

Newton-Raphson technique, with an error controlled sub-stepping stress point

algorithm. The soil continuum is discretised using 8-noded isoparametric

quadrilateral elements. The interface degrees of freedom are assumed to be at nodes

that belong to both the footings and soil underneath. The total number of interface

degrees of freedom is 30 for this case.

The above problem is analysed for various scenarios with respect to the loading

applied to the structure (which is assumed to be loaded equally on each floor with a

total gravity load equal to λ×5 kN/m2), the excavation depth (He) and the distance of

the structure from the excavation wall (Le).

Table 7.4 lists various loading scenarios considered for analysing the above problem

with respect to the load factor (λ) applied in structure sub-domain, and the

excavation depth (He) in the soil sub-domain. Considering Table 7.4, the loads on

the structure are applied in the first six increments, and from increment 7 to 16 the

soil is excavated while the loading in the structure is assumed to be constant.

Model Case

Increment number of the coupled analysis (16 increments in total) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

values for λ (no excavation)

Values for He (m) (λ=constant)

Case 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10Case 2 1 2 2 2 2 2 1 2 3 4 5 6 7 8 9 10Case 3 1 2 3 3 3 3 1 2 3 4 5 6 7 8 9 10Case 4 1 2 3 4 4 4 1 2 3 4 5 6 7 8 9 10Case 5 1 2 3 4 5 5 1 2 3 4 5 6 7 8 9 10Case 6 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10

Table 7.4: Loading scenarios

Figures 7.27, 7.28 and 7.29, show the vertical settlement at the centre of the left,

middle and right footings of the analysed frame (see Figure 7.24), respectively, for

the various loading scenarios, where it is assumed Le=2m. The vertical displacement

of the ground surface is also depicted in Figure 7.30.


279

Figure 7.27: Vertical settlement of the left footing for different load cases

Figure 7.28: Vertical settlement of the middle footing for different load cases


280

Figure 7.29: Vertical settlement of the right footing for different load cases

Figure 7.30: Cumulative vertical displacement of the ground surface (last increment)


281

A common practice in estimating the ground movement adjacent to an excavation is

to regard both the settlement of the ground and the building as identical to avoid

complications in geotechnical analysis. Considering the footing settlements depicted

for various cases in Figures 7.27, 7.28 and 7.29, it is clear from Figure 7.30 that

these are considerably underestimated by the free field response.

The effect of the structural loads on the horizontal displacement of the excavation

wall is also depicted in Figure 7.31. It is a consequence of these lateral ground

movements that the structure undergoes additional settlements due to its weight.

Figure 7.31: Cumulative horizontal displacement of excavation wall (last increment)

The effect of the excavation depth on the vertical deformation profile of the soil

surface and the horizontal displacement of the excavation wall is also depicted in

Figures 7.32 and 7.33, respectively, for Case 6 (Table 7.3).


282

Figure 7.32: Vertical displacement of ground surface for different excavation depths

(Case 6)

Figure 7.33: Horizontal displacement of the excavation wall for different excavation

depths (Case 6)


283

The effect of distance of the structure from the excavation (Le) on the vertical

settlements of the footings is also depicted in Figures 7.34, 7.35 and 7.36 for model

Case 6.

As expected by increasing Le the additional settlements of the structure due to the

excavation decreases. The results further emphasise the importance of using a fully

coupled soil-structure interaction analysis for such cases.

Figure 7.34: Vertical settlement of the left footing for different Le (Case 6)


284

Figure 7.35: Vertical settlement of the right footing for different Le (Case 6)

Figure 7.36: Vertical settlement of the right footing for different Le (Case 6)


285

All the above coupled analyses were carried out using reduced order method with a

tolerance of 10-4 m. However, to further assess the merits of different coupling

algorithms presented in this thesis, Case 6 is analysed using different update

techniques for the same tolerance, where the corresponding total number of required

coupling iterations is listed in Table 7.5. As expected, the constant relaxation

scheme has the worst convergence rate, while the proposed mixed reduced order

coupling scheme outperforms the other coupling schemes.

Coupling method Number of required coupling

iterations Constant relaxation 158 Adaptive relaxation 96

Reduced order method 88 Mixed reduced order method 67

Table 7.5: Comparison of different coupling methods

The benefits of the developed simulation environment in the practical assessment of

nonlinear soil-structure interaction problems is further demonstrated by considering

the results obtained for Case 6. In this regard, vector plots of displacements in the

soil sub-domain in the vicinity of the structure and excavation for increment number

6 (before excavation) and increment 12 (He=6m) are shown in Figures 7.37 and

7.39, respectively.

Figure 7.38 and 7.40 show contours of stress level before and after the full

excavation to 6m depth under the applied load on the structure. It is evident from

Figure 7.40 that the applied loading conditions have mobilised an extensive plastic

zone underneath the building. This zone is, however, smaller and shallower under

the right hand side footing, which is in agreement with the previous figures that

show most of the deformation and load concentration nearer the excavation.


286

Figure 7.37: Vectors of displacement in soil sub-domain in increment 6 (Case 6)

Figure 7.38: Contour plots of stress levels and plasticity induced in soil sub-domain

in increment 6 (Case 6)


287

Figure 7.39: Vectors of displacement in soil sub-domain in increment 12 (Case 6)

Figure 7.40: Contour plots of stress levels and plasticity induced in soil sub-domain

in increment 12 (Case 6)


288

The deformed shape and bending moment variation in the structure sub-domain for

different increments are shown in Figure 7.41. It can be clearly observed that the

maximum bending moments of the structural elements after the excavation are

significantly higher than before the excavation. The comparison of the bending

moment values for four selected regions A, B, C and D, as shown in Figure 7.41d, is

presented for both the 6th and 12th load increments in Table 7.6. It is evident from the

deformed shape of the structure after excavation and also from the vectors

underneath each of the three footings (Figure 7.39), that after 6m of excavation the

footings experience rigid tilting and significant vertical settlements. However, the

footing nearest to the excavation has the smallest tilting, as its deformation is also

dominated by the horizontal movement towards the unsupported excavation.

Figure 7.41: Deformed shape (scale=5) and bending moment (kN-m) of structure for

(a) 1st, (b) 6rd , (c) 7th and (d) 12th increment


289

RegionMz (kN-m)Increment 6

Mz (kN-m)Increment 12

% Increase

A 120 215 80% B -120 -191 60% C -21 -72 340% D 21 153 720%

Table 7.6: Comparison of maximum bending moment in fully coupled analysis

7.5 Coupled Modelling of Retaining Steel Sheet Piles

With the new Eurocodes coming into force, and their introduction of unified

procedures for limit state design of structural elements, plastic bending of steel sheet

piles will be allowed. According to Bourne-Webb et al. (2007), the limit state design

approach is well established for conventional building structures but has not

generally been applied to the design of earth-retaining structures — at least not in a

unified manner. One area of interest is the use of limit state principles in the design

of steel sheet pile retaining walls and, in particular, whether it is safe to allow the

formation of plastic hinges at the ultimate limit state and, if so, how to verify that the

behaviour of the wall zone undergoing plastic deformation is within acceptable

limits. In this section, the soil-structure interaction problem under consideration is a

2D simulation of a cantilever steel sheet pile retaining wall, as depicted in Figure

7.42, where the left hand side boundary of the problem is assumed to be consistent

with an axis of symmetry.

Following the partitioned treatment, the considered soil-structure system is

partitioned physically into two sub-domains, soil and structure, where each sub-

domain is discretised separately according to its characteristics. The wall is modelled

with ADAPTIC using cubic elasto-plastic 2D beam-column elements, which enable

the modelling of geometric and material nonlinearity, while the material behaviour is

assumed to be bilinear elasto-plastic with kinematic strain hardening. The steel sheet

pile retaining wall is modelled (I section) with a second moment of area I = 2346

cm4/m, plastic section modulus of Wpl = 410 cm3/m, and a yield stress of fy=270

MPa. The soil sub-domain is modelled with ICFEP using an associated elasto-plastic


290

Mohr-Coulomb constitutive model and discretised using 8-noded isoparametric

quadrilateral elements with a Young’s Modulus E varying with depth z according to

(E = 4000 + 5000z kN/m2), Poisson’s ratio μ=0.2, the bulk unit weight γ = 20 kN/m3,

the angle of shearing resistance, φ′=20°. The retaining wall is assumed to be

embedded 2 m into the ground, and the retained surface is surcharged by a load W.

The problem is analysed in nine increments as shown in Table 7.7.

Figure 7.42: Schematic diagram of the cantilever retaining wall

Increment Number

1 2 3 4 5 6 7 8 9

He (m) W(kPa)

He=8 W=20

He=8 W=40

He=8 W=60

He=8 W=80

He=8 W=100

He=8 W=120

He=8 W=140

He=8 W=160

He=8 W=180

Table 7.7: Loading scenario in different incremental stages of the analysis


291

In this example, employing the relaxation scheme is very expensive since the

relaxation parameter tends to very small values, in order of 10-3, to ensure

convergence, and as a result both constant and adaptive schemes become extremely

computationally inefficient. Moreover, by employing the standard reduced order

method in this case, appropriate approximation of the condensed stiffness matrix,

and thus convergence, can only be achieved when the coupling iterations exceed the

number of interface degrees of freedom which in this case is 42.

In contrast, by utilizing the proposed mixed reduced order method, superior

convergence characteristics are achieved. In fact, an average of 10 coupling

iterations per increment is sufficient to achieve convergence to a tolerance of 10-3 m

set for the compatibility error of each coupled degree of freedom at the interface.

Indeed, the mixed reduced order method presented in Chapter 4, outperforms other

coupling techniques in numerous problems where the interaction effects at the

interface become more pronounced and the number of interface degrees of freedoms

is large.

The applicability and benefits of the developed coupling technique in nonlinear soil-

structure interaction analysis can be established by considering the results of this

case study. These results show that the excavation has mobilized a significant plastic

deformation near the retaining wall.

A vector plot of displacements in the soil sub-domain in the vicinity of the

excavation is shown in Figure 7.43 for the final increment. The relative magnitude of

these vectors shows the mechanism of ground deformation.

Importantly in the final increment, the soil deformations due to excavation have

caused the retaining wall to reach its plastic moment as illustrated in Figure 7.44,

where the bending moment distribution and the deflected shape of the retaining wall

are depicted for the different incremental stages of the analysis given in Figure 7.44.


292

Figure 7.43: Vectors of accumulated displacements in soil sub-domain (final

increment)

Figure 7.44: Deflection (scale: 5) and bending moment (kN-m/m) distribution of the

retaining wall for (a) 1st, (b) 4th, (c) 7th, (d) 8th and (e) 9th increment


293

The same soil-structure interaction problem of Figure 7.42 is now simulated as a

propped steel sheet pile retaining wall, as depicted in Figure 7.45. Again, the

retaining wall is assumed to be embedded 2 m into ground and the retained surface is

surcharged by a load of W. The above problem is analysed in five increments as

given in Table 7.8.

Increment number

1 2 3 4 5

He (m) W(kPa)

He=8 W=20

He=8 W=40

He=8 W=60

He=8 W=80

He=8 W=100

Table 7.8: Loading scenario in different incremental stages of the analysis

Figure 7.45: Schematic diagram of the propped retaining wall

Similarly, by utilizing the mixed reduced order method, superior convergence

performance is achieved, and an average of 10 coupling iteration per increment is

sufficient to achieve convergence to the same tolerance of 10-3 m.


294

The obtained results for the proposed wall show that the excavation has mobilized a

significant plastic deformation near the retaining wall. The vector plot of

displacements in the soil sub-domain in the vicinity of the excavation is presented in

Figure 7.46, which shows a completely different mechanism in the partitioned soil

sub-domain compared to the previous case.

Importantly in the final increment, the soil deformations due to excavation cause the

retaining wall to reach its plastic moment as illustrated in Figure 7.47, where the

bending moment distribution and the deflected shape of the retaining wall are

depicted for different incremental stages of the analysis outlined in Table 7.8.

Finally, the change in the horizontal force in the lateral support is also depicted in

Figure 7.48.

Figure 7.46: Vectors of accumulated displacements in soil sub-domain (final

increment)


295

Figure 7.47: Deflection (scale: 5) and bending moment (kN-m/m) distribution of the

retaining wall for (a) 1st, (b) 2nd , (c) 3rd , (d) 4th and (e) 5th increment

Figure 7.48: Horizontal reaction at the lateral support


296

7.6 Conclusion

The applicability of the partitioned treatment of nonlinear soil-structure interaction

analysis and its great potential towards capturing the nonlinear behaviour of both soil

and structure are illustrated in this chapter through various case studies. Towards this

end, iterative coupling is carried out using various existing and proposed procedures

using the discipline-oriented solvers, ADAPTIC and ICFEP, for nonlinear structural

and geotechnical analysis, respectively. It is shown that the proposed mixed reduced

order method provides superior convergence characteristics in comparison with

existing coupling methods.

Using the developed simulation environment in a number of case studies in which

nonlinearity arises in both structure and soil, the applicability and high potential of

the developed simulation environment is demonstrated towards providing an

integrated interdisciplinary computational approach which combines the advanced

features of both structural and geotechnical modelling for a variety of challenging

soil-structure interaction problems.

Chapter 8

Conclusion

8.1 Introduction

This work has been primarily motivated by the lack of sophisticated monolithic tools

for modelling nonlinear soil–structure interaction problems, while recognising the

existence of advanced tools for nonlinear analysis of structure and soil in isolation.


using a monolithic treatment, the partitioned treatment has been advocated as

offering major benefits in the context of coupled modelling of nonlinear soil-

structure interaction.

Accordingly, the aim of this work has been to develop advanced numerical methods

for nonlinear coupling of static and dynamic soil-structure interaction problems,

where the partitioned approach is adopted as a framework for coupling field-specific

tools with minimal intrusion into codes.

The partitioned approach in soil-structure interaction analysis is fully investigated in

this work. In this respect, various coupling techniques in the context of soil-structure

interaction analysis are developed, and their computational characteristics are

Chapter 8 Conclusion

298

discussed. In this regard, novel formulations for coupling soil-structure systems,

based on relaxation coupling methods and also utilizing the tangent stiffness matrix

of the partitioned sub-domains at the interface are proposed, and their relative

performance is evaluated.

Based on the presented coupling algorithms, a novel simulation environment,

utilising discipline-oriented solvers for nonlinear structural and geotechnical

analysis, has been developed, which couples two advanced nonlinear finite element

solvers, ADAPTIC (Izzuddin, 1991) and ICFEP (Potts & Zdravkovic, 1999), for

structural and geotechnical analysis respectively. Although the developed methods

are applied to the coupling of ADAPTIC and ICFEP, they are also generally

applicable to the coupling of other existing nonlinear soil and structural software.

In this respect, the developed approach is believed to offer great potential towards

providing an integrated interdisciplinary computational framework for coupled

modelling of soil-structure interaction problems.

The developed simulation environment is used in this work to demonstrate the

performance characteristics and merits of the various presented algorithms.

Accordingly, the developed tool is employed for a number of numerical examples

involving nonlinear soil-structure interaction analysis in which nonlinearity arises in

both the structure and the soil, leading to important conclusions regarding the

adequacy and applicability of the alternative coupling methods as well as the

prospects for further enhancements.

8.2 Conclusions

This work has shown that the partitioned treatment is a feasible and realistic

approach for coupled modelling of nonlinear soil-structure interaction problems.

Unlike the monolithic approach, the partitioned approach offers major benefits,

including i) allowing field-specific discretisation and solution procedures that have

proven performance for each partitioned domain, and ii) facilitating the reuse of


299

existing nonlinear analysis software with all the resource savings that this brings.

The success of the partitioned approach, however, has been shown to hinge on the

adopted coupling algorithm. Indeed, a principal contribution of this work has been

the investigation of the performance of existing coupling algorithms, and the

development of more powerful algorithms that address convergence issues,

particularly in the context of nonlinear soil-structure interaction problems. Towards

this end, this work has successfully developed novel iterative coupling approaches,

which are based on the reduced order method, yet offering major improvements of

the convergence characteristics and computational performance. Furthermore, the

applicability of the developed partitioned approaches to soil-structure interaction

problems, exhibiting significant nonlinearity in both structure and soil, has been

demonstrated by means of several case studies.

Hereafter, the main conclusions from various parts of this work are summarised.

8.2.1 Staggered Approach

It has been shown that coupling of partitioned sub-domains may be achieved using

the staggered approach, though this approach should be used with great care in

relation to both stability and accuracy. The main conclusions in this regard are:

The staggered approach is applicable only to transient dynamic problems.

The stability of the staggered scheme is conditional on the time step and also

on the equivalent mass and stiffness on both sides of the interface, though

this depends on the formulation of the partitioned sub-domains and the

employed predictors.

Stability and accuracy considerations typically demand excessively small

time steps rendering this scheme computationally prohibitive for typical

coupled problems.

In general, achieving stability in a staggered solution procedure is extremely difficult

and in many cases impossible without reformulation of the field equations of the


300

original partitioned sub-domains. Although some further progress may be achieved

by modifying the specifics of the staggered scheme, these are not considered here,

since the ultimate performance will vary with the application problem under

consideration, and often their implementation would conflict with the modular use of

the structural and soil solvers as black box solvers.

8.2.2 Iterative Coupling

Iterative coupling methods have been shown to offer major enhancement over the

staggered approach in the context of partitioned analysis of soil-structure interaction

problems. Considering the applicability requirements for both dynamic and static

analysis, the algorithms that are considered to be more suitable for soil-structure

interaction coupling are:

Sequential/Parallel Dirichlet-Neumann.

Parallel Dirichlet-Dirichlet.

Particular emphasis has been placed in this work on Dirichlet-Neumann (D-N) type

of iterative schemes, specifically the sequential D-N algorithms.

Within the family of sequential D-N algorithms, the convergence performance over a

load/time step has been identified as the most important feature. In this respect, it has

been demonstrated that:

Convergence is directly dictated by the chosen update technique during

successive iterations.

By employing trivial update of boundary conditions, only conditional

convergence could be achieved, which is unsatisfactory.

Based on a generalised sequential Dirichlet-Neumann iterative coupling algorithm, a

novel simulation has been developed which utilises the discipline-oriented solvers,

ADAPTIC and ICFEP, for nonlinear structural and geotechnical analysis. The

software architecture of the developed simulation environment has been outlined,


301

where the structure of the data exchange between the various codes is also

elaborated.

8.2.3 Interface Relaxation

The use of interface relaxation update techniques in FEM-FEM domain

decomposition analysis of soil-structure interaction has been studied, where coupling

of the separately modelled sub-domains is undertaken on the sequential iterative D-N

sub-structuring method. In this respect, the convergence characteristics of iterative

coupling algorithms are enhanced by employing a relaxation of the interface

Dirichlet entities in successive iterations.

Due to the lack of general convergence analysis for the relaxation update technique

in FEM-FEM coupling, various mathematical and computational characteristics of

the coupling method, including the governing convergence rate and choice of

constant relaxation parameter, have been established. The work undertaken has

shown that:

Convergence to compatibility in sequential D-N iterative coupling algorithms

can be ensured by employing an appropriate constant relaxation parameter.

There exists a range of relaxation parameters that guarantee convergence to

compatibility.

There exists an optimum relaxation parameter in the convergent range for

which not only is convergence guaranteed, but also the convergence rate is

optimum.

The choice of a suitable constant relaxation scheme is problem dependent

and highly sensitive to the parameters of the partitioned sub-domains,

specifically the condensed stiffness at the interface of the partitioned sub-

domains.

In practice, the determination of an optimum constant relaxation parameter

must rely on trial and error, which may be applied to the initial part of the

coupled simulation after which full coupling analysis may be carried out.


302

In nonlinear problems, finding the optimum relaxation parameter by trial and

error is computationally prohibitive, since it should be determined over the

full range of response.

An adaptive relaxation scheme has been developed for enhancing the performance of

iterative coupling algorithms, where the choice of the relaxation parameter is guided

by the iterative corrections of Dirichlet entities at the interface. It has been shown

that the adaptive relaxation scheme:

Avoids the trial and error procedure for the selection of an optimum, even

adequate, constant relaxation parameter.

Improves the convergence rate of constant relaxation in both linear and

nonlinear analysis significantly.

Finally, although using the adaptive scheme removes significant difficulties in the

conventional relaxation iterative coupling scheme, there is a pitfall associated with

both constant and adaptive relaxation schemes when the partitioned sub-domain

parameters dictate very small values of the relaxation parameter for convergence. In

such cases, relaxation methods break down with poor convergence rates and

significant computational inefficiency.

8.2.4 Reduced Order Method

It has been proposed in this work that the performance of iterative coupling methods

may be effectively enhanced for nonlinear analysis through the use of the condensed

interface stiffness matrices of the structure and soil partitioned sub-domains,

providing an effective first-order guide to iterative displacements at the soil-structure

interface. This would bring the performance of the proposed coupling approach very

close to the monolithic treatment.

Various domain decomposition methods for nonlinear analysis of soil-structure

interaction problems based on approximating the condensed interface stiffness

matrix have been considered and proposed. It has been shown that by using the


303

condensed tangent interface stiffness matrices of the partitioned sub-domains in the

update of boundary conditions:

Unconditional and problem independent convergence characteristics could

be achieved.

Much greater convergence rate could be achieved compared to adaptive

relaxation scheme.

Convergence to compatibility occurs for linear problems immediately at the

first iteration. Although this immediate convergence does not normally occur

for nonlinear problems, employing this method ensures a high convergence

rate.

Moreover, in the absence of an adaptive method for evaluating the relaxation

parameter in parallel forms of coupling algorithms, utilising the above method could

be employed within various iterative coupling algorithms.


nonlinear field modelling tools, more general approximation for the condensed

stiffness matrices is desirable. Towards this end, the condensed tangent stiffness

matrix has been approximated in this work via reduced order models, building on a

previous approach by Vierendeels et al. (2007). Nevertheless, major modifications to

this approach have been proposed in this work, leading to a more versatile and

efficient approach for coupled modelling of nonlinear soil-structure interaction

problems. In this respect:

A major pitfall associated with the originally introduced reduced order

method causing divergence is overcome by a new selective

addition/replacement procedure of force and displacement mode vectors,

where the applicability and advantages of this modification are demonstrated

by means of several examples.

In order to achieve better approximation of the condensed tangent stiffness

matrix in the initial stages of coupling iterations, a mixed reduced order


304

method is proposed, which has a much higher convergence rate than the

conventional reduced order method.

The applicability of the presented coupling techniques has been demonstrated for

nonlinear soil-structure interaction, where the superior convergence rate of the

presented reduced order schemes, particularly the proposed mixed reduced order

method, compared to relaxation scheme is highlighted.

8.2.5 Case Studies

The applicability of the partitioned treatment of nonlinear soil-structure interaction

analysis and its great potential towards capturing the nonlinear behaviour of both soil

and structure have been illustrated through various examples. Iterative coupling has

been carried out using various existing and proposed procedures employing the

discipline-oriented solvers, ADAPTIC and ICFEP. It has been shown that the

proposed mixed reduced order method provides superior convergence characteristics

in comparison with existing coupling methods. The successful application of the

partitioned approach with iterative coupling based on the proposed coupling method

has demonstrated the applicability and great potential of the developed simulation

environment and the underlying methods towards providing an integrated

interdisciplinary computational tool for nonlinear soil-structure interaction analysis.

8.3 Recommendations for Future Works

Although iterative coupling method proposed in this work, utilising the condensed

tangent stiffness matrices of the partitioned sub-domains, possesses a high

convergence rate, its performance may be further enhanced through future research

in the following areas:

Combined coupling/field iterations: Since the field models for structure and

soil are nonlinear, iterations are typically performed in such models to

determine the state of the corresponding sub-domains. Therefore, there are


305

potentially considerable computational benefits to be gained from combining

the field and coupling iterations. Further research is suggested on the

development of a generalised approach, which could allow combined

coupling/field iterations at arbitrary ratios of iteration (e.g. 1 (coupling):3

(structure):2 (soil)), where the optimal ratio would be established adaptively

during the specific soil-structure simulation.

Interface boundary conditions: The emphasis of this work has been on

iterative coupling methods based on a Dirichlet-Neumann algorithm. It is

suggested that other coupling algorithms could be considered in detail as part

of future research. While, the application of such algorithms with the

relaxation approach is fraught with problems, the proposed approach of

utilising the condensed interface tangent stiffness could readily transform

displacement incompatibilities at the interface to boundary forces, and is in

fact applicable to both Dirichlet-Neumann and Dirichlet-Dirichlet algorithms.

Indeed, the use of the latter algorithm with the proposed approach using a

single coupling/field iterative loop should be identical in terms of numerical

performance to the standard monolithic treatment.

Parallel processing: The partitioned treatment offers a natural framework for

parallel computations, provided the coupling algorithm is parallel over the

various partitions. It remains to be established for soil-structure interaction

problems whether parallelisation over multi-processor machines, including

optimal load balancing, is most effectively undertaken through low-level

procedural parallelisation of the field models with only two physical

partitions or through coupling-level parallelisation with additional

computational partitions. In both cases, it would be important to employ an

effective communication protocol between interface and sub-domain models,

which is equally applicable to single processor, multi-processor and

distributed computing simulations.


306

In addition to the above, the following extensions to the originally developed

simulation environment for coupled modelling of soil-structure interaction,

specifically the INTERFACE code, would enhance its applicability:

Interface modelling: The coupling operations undertaken at the interface in

the current work assume that the adjacent sub-domains have matching

meshes, in which case the INTERFACE operations consist mainly of

transferring displacements/forces from one side of the soil/structure interface

to the other. In practice, it would be very useful to allow for non-matching

soil and structural meshes, either due to different element sizes or different

element types, thus requiring more involved INTERFACE models that

address nonlinear compatibility and equilibrium between the non-matching

sides. Another related issue is the treatment of phenomena such as contact,

sliding and friction between adjacent sub-domains, which can potentially add

further demands on the INTERFACE model.

Multiple sub-domains: Attention has been focussed in this work on coupled

modelling of soil-structure interaction problems that involve two sub-

domains only, often dictated by physical partitioning. However, the iterative

coupling algorithms could be generalized to multiple sub-domains

partitioning, to realise the additional benefits of computational, physical and

functional partitioning, including parallelisation at the coupling level.

References

Allam, M. M., Subba-Rao, K. S. & Subramanya, B. V. 1991. Frame soil interaction and winkler model Proceedings of Institution of Civil Engineers, 91, 477-494.

Aye, Z. Z., Karki, D. & Schulz, C. 2006. Ground movement prediction and building damage risk-assesment for the deep excavations and tunneling works in bangkok subsoil. International Symposium on Underground Excavation and Tunneleing. Bangkok, Thailand.

Beer, G. 1985. An isoparametric joint/interface element for finite element analysis International Journal for Numerical Methods in Engineering, 21, 585-600.

Belytschko, T. & Mullen, R. 1978. Stability of explicit-implicit mesh partitions in time integration. International Journal for Numerical Methods in Engineering, 12,1575-1586.

Bhattacharya, K., Dutta, S. C. & Dasgupta, S. 2004. Effect of soil-flexibility on dynamic behaviour of building frames on raft foundation. Journal of Sound and Vibration, 274, 111-135.

Boone, S. J. 1996. Ground-movement-related building damage. Journal of Geotechnical Engineering, 122, 886-896.

Boone, S. J. 2001. Assesing construction and settlement-induced building damage: A return to fundamental principles. In: Proceedings, Underground Construction, Institution of Mining and Metalurgy, London 559-570.

References

308

Boone, S. J., Westland, J. & Nusink, R. 1999. Comparative evaluation of building responses to an adjacent braced excavation. Canadian Goetechnical Journal, 36, 210-223.

Bourne-Webb, P. J., Potts, D. M. & Rowbottom, D. 2007. Plastic bending of steel sheet piles. Geotechnical Engineering, 160, 129-140.

Bowles, J. E. 1996. Foundation analysis and design, McGraw-Hill International Editions.

Brown, C. B., Tilton, J. R. & Laurent, J. M. 1977. Beam-plate system on winkler foundation. Journal of the Engineering Mechanics, ASCE, 103, 589-600.

Bull, J. W. 1988. Finite element analysis of thin-walled structures, Taylor & Francis.

Burland, J. B. & Potts, D. M. 1994. Development and application of a numerical model for the leaning tower of pisa. In: SHIBUYA, S., MITACHI, T. & MIURE, S. (eds.) In Prefailure deformation of geomaterials. Rotterdam: Balkema.

Carrier, W. D. & Christian, J. T. 1973. Rigid circular plate resting on a non-homogeneous elastic half space. Geotechnique, 23, 67-84.

Charles, J. A. & Skinner, H. D. 2004. Settlement and tilt of low-rise buildings. Proceedings of the Institution of Civil Engineers: Geotechnical Engineering, 157, 65-75.

Collenz, A., Bona, F. D., Gugliotta, A. & Soma, A. 2004. Large deflections of microbeams under electrostatic loads. Journal of Micromechanics and Microengineering 14, 365-373.

Desai, C. S. & Abel, J. F. 1987. Introduction to the finite element method: A numerical method for engineering analysis., CBS Publisher and Distributors.

Desai, C. S., Phan, H. V. & Perumpral, J. V. 1982. Mechanics of threedimensional soil–structure interaction. Journal of Engineering Mechanics, ASCE, 108, 731-747.

Dowrick, D. J. 1977. Earthquake resistance design: A manual for engineers and architects, New York, John Wiley and Sons Ltd.

Dutta, S., Mandal, A. & Dutta, S. C. 2004. Soil–structure interaction in dynamic behaviour of elevated tanks with alternate frame staging configurations. Journal of Sound and Vibration, 277, 825-853.

Dutta, S. C. & Roy, R. 2002. A critical review on idealization and modeling for interaction among soil–foundation–structure system. Computers & structures, 80, 1579-1594.

References

309

Dutta, S. C., Bhattacharya, K. & Roy, R. 2004. Response of low-rise buildings under seismic ground excitation incorporating soil–structure interaction. Soil Dynamics and Earthquake Engineering, 24, 893-914.

El-Gebeily, M., Elleithy, W. M. & Al-Gahtani, H. J. 2002. Convergence of the domain decomposition finite element–boundary element coupling methods. Computer Methods in Applied Mechanics and Engineering, 191, 4851-4867.

Elleithy, W. M., Al-Gahtani, H. J. & El-Gebeily, M. 2001. Iterative coupling of BE and FE methods in elastostatics. Engineering Analysis with Boundary Elements, 25, 685-695.

Elleithy, W. M. & Tanaka, M. 2003. Interface relaxation algorithms for BEM–BEM coupling and FEM–BEM coupling. Computer Methods in Applied Mechanics and Engineering, 192, 2977-2992.

Elleithy, W. M., Tanaka, M. & Guzik, A. 2004. Interface relaxation FEM–BEM coupling method for elasto-plastic analysis. Engineering Analysis with Boundary Elements, 28, 849-857.

Estorff, O. V. & Firuziaan, M. 2000. Coupled BEM/FEM approach for nonlinear soil/structure interaction. Engineering Analysis with Boundary Elements, 24, 715-725.

Estorff, O. V. & Hagen, C. 2005. Iterative coupling of FEM and BEM in 3D transient elastodynamics. Engineering Analysis with Boundary Elements, 29, 775-787.

Farhat, C. & Lesoinne, M. 2000. Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems. Computer Methods in Applied Mechanics and Engineering, 182, 499-515.

Farhat, C. & Park, K. C. 1991. An unconditionally stable staggered algorithm for transient finite element analysis of coupled thermoelastic problems. Computer Methods in Applied Mechanics and Engineering, 85, 349-365.

Felippa, C. A. & Park, K. C. 1980. Staggered transient analysis procedures for coupled-field mechanical systems: Formulation. Computer Methods in Applied Mechanics and Engineering, 24, 61-111.

Felippa, C. A. & Park, K. C. 2004. Synthesis tools for structural dynamics and partitioned analysis of coupled systems. Multi-Physics and Multi-Scale Computer Models in Nonlinear Analysis and Optimal Design of Engineering Structures under Extreme Conditions. Ljubliana, Slovenia: Proceedings NATO-ARW PST ARW980268.

References

310

Felippa, C. A., Park, K. C. & Farhat, C. 2001. Partitioned analysis of coupled mechanical systems. Computer Methods in Applied Mechanics and Engineering, 190, 3247-3270.

Funaro, D., Quarteroni, A. & Zanolli, P. 1988. An iterative procedure with interface relaxation for domain decomposition methods. SIAM Journal on Numerical Analysis, 25, 1213-1236.

Gaba, A. R., Simpson, B., Powrie, W. & Beadman, D. R. 2002. Embedded retaining walls: Guidance for economic design, research project: 629. London: Construction Industry Information and Research Association.

Geers, T. L. & Felippa, C. A. 1980. Doubly asymptotic approximations for vibration analysis of submerged structures. Journal of the Acoustical Society of America, 73, 1152-1159.

Glowinski, R., Dinh, Q. V. & Periaux, J. 1983. Domain decomposition methods for nonlinear problems in fluid dynamics. Computer Methods in Applied Mechanics and Engineering, 40, 27-109.

Hagen, C. & Estorff, O. V. 2005a. Transient dynamic investigation of 3D dam-reservoir-soil problems using an iterative coupling approach. International Conference on Computational Methods for Coupled Problems in Science and Engineering, COUPLED PROBLEMS 2005. Barcelona: M. Papadrakakis, E. Onate and B. schrefler ( Eds)

Hagen, C. & Estorff, O. V. 2005b. Transient dynamic investigation of nonlinear 3D soil–structure interaction problems using an iterative coupling approach. PAMM · Proc. Appl. Math. Mech, 5, 399-400.

Harr, M. E. 1966. Foundation of theoretical soil mechanics, New York, Mcgraw Hill book co.

Harr, M. E., Davidson, J. L., Ho, H. O., Pombo, L. E., Ramaswamy, S. V. & Rosner, J. C. 1969. Euler beams on a twoparameter elastic foundation model. Journal of Geotechnical and Geoenvironmental Engineering, 95,933-948.

Hetenyi, M. 1946. Beams on elastic foundation University of Michigan Press.

Hoffman, J. D. 2001. Numerical methods for engineers and scientists, CRC Press.

Horn, R. A. & Johnson, C. R. 1999. Matrix analysis, Cambridge University Press.

Huang, M. & Zienkiewicz, O. C. 1998. New unconditionally stable staggered solution procedures for coupled soil-pore fluid dynamics problems. International Journal for Numerical Methods in Engineering, 43, 1029-1052.

References

311

Hughes, T. J. R. & Liu, W. K. 1978. Implicit-explicit finite elements in transisent analysis: I. Stability theory; II. Implementations and numerical examples. Journal of Applied Mechanics, 45, 371-378.

Inaba, T., Dohi, H., Okuta, K., Sato, T. & Akagi, H. 2000. Nonlinear response of surface soil and NTT building due to soil-structure interaction during the 1995 Hyogo-ken Nanbu (Kobe) earthquake. Soil Dynamics and Earthquake Engineering, 20, 289-300.

Izzuddin, B. A. 1991. Nonlinear dynamic analysis of framed structures. Ph.D., Imperial College, University of London.

Izzuddin, B. A. & Elnashai, A. S. 1993. Adaptive space frame analysis. 2. A. Distributed plasticity approach. ICE Proceedings, Structures and Buildings, 99, 317-326.

Jardine, R. J., Potts, D. M., Fourie, A. B. & Burland, J. B. 1986. Studies of the influence of non-linear stress-strain characteristics in soil-structure interaction. Geotechnique, 36, 377-396.

Jin, W.-L., Song, J., Gong, S.-F. & Lu, Y. 2005. Evaluation of damage to offshore platform structures due to collision of large barge. Engineering Structures, 27, 1317-1326.

Kamiya, N. & Iwase, H. 1997. BEM and FEM combination parallel analysis using conjugate gradient and condensation. Engineering Analysis with Boundary Elements, 20, 319-326.

Kamiya, N., Iwase, H. & Kita, E. 1996. Parallel implementation of boundary element method with domain decomposition. Engineering Analysis with Boundary Elements, 18, 209-216.

Krabbenhoft, K., Damkilde, L. & Krabbenhoft, S. 2005. Ultimate limit state design of sheet pile walls by finite elements and nonlinear programming. Computers & structures, 83, 383-393.

Kurian, N. P. & Manjokumar, N. G. 2001. A new continuous winkler model for soil-structure interaction. Journal of Structural Engineering, 27, 269-276.

Kuttler, U. & Wall, W. A. 2008. Fixed-point fluid–structure interaction solvers with dynamic relaxation. Computational Mechanics, 43, 61-72.

Lai, C. H. 1994. Diakoptics, domain decomposition and parallel computing. The Computer Journal, 37, 841-846.

Marini, L. D. & Quarteroni, A. 1989. A relaxation procedure for domain decomposition methods using finite elements. Numerische Mathematik, 55, 575-598.

References

312

Mu, M. 1999. Solving composite problems with interface relaxation. SIAM Journal on Scientific Computing, 20, 1394-1416.

Noda, T., Fernado, G. S. K., Asaoka, A. 2000. Delayed failure in soft clay foundations. Journal of the Japanese Geotechnical Society of Soils and Foundations, 40, 85-97.

Nogami, T. & Lain, Y. C. 1987. Two parameter layer model for analysis of slab on elastic foundation. Journal of Engineering Mechanics, ASCE, 1131, 1279-1291.

Noorzaei, J., Godbole, P. N. & Viladkar, M. N. 1993. Nonlinear soil-structure interaction of plane frames-a parametric study. Computers & Structures, 49, 561-566.

Noorzaei, J., Naghshineh, A., Kadir, M. R. A., Thanoon, W. A. & Jaafar, M. S. 2006. Nonlinear interactive analysis of cooling tower–foundation–soil interaction under unsymmetrical wind load. Thin-Walled Structures, 44, 997-1005.

Noorzaei, J., Viladkar, M. N. & Godbole, P. N. 1995a. Elasto-plastic analysis for soil–structure interaction in framed structures. Computers & Structures, 55, 797-807.

Noorzaei, J., Viladkar, M. N. & Godbole, P. N. 1995b. Influence of strain hardening on soil-structure interaction of framed structures. Computers & Structures, 55, 789-795.

O’brien, J. & Rizos, D. C. 2005. A 3D BEM-FEM methodology for simulation of high speed train induced vibrations. Soil Dynamics and Earthquake Engineering, 25, 289-301.

Park, K. C. 1980. Partitioned transient analysis procedures for coupled-field problems: Stability analysis. Journal of Applied Mechanics, 47, 370-376.

Park, K. C. & Felippa, C. A. 1980. Partitioned transient analysis procedures for coupled-field problems: Accuracy analysis. Journal of Applied Mechanics, 47, 919-926.

Piperno, S. 1997. Explicit/implicit fluid/structure staggered procedures with a structural predictor and fluid subcycling for 2D inviscid aeroelastic simulations. International Journal for Numerical Methods in Fluids, 25, 1207-1226.

Potts, D. M. 2003. Numerical analysis: A virtual dream or practical reality? Géotechnique, 53, 535-573.

References

313

Potts, D. M. & Zdravkovic, L. 1999. Finite element analysis in geotechnical engineering: Theory, London, Thomas Telford.

Potts, D. M. & Zdravkovic, L. 2001. Finite element analysis in geotechnical engineering: Application, London, Thomas Telford.

Quarteroni, A. & Valli, A. 1999. Domain decomposition methods for partial differential equations, Oxford, Clarendon press.

Ravaska, O. 2002. A sheet pile wall design according to Eurocode 7 and Plaxis. In: MESTAT, P. (ed.) Numerical methods in geotechnical engineering. Paris: Presses de l’ENPC/LCPC.

Rizos, D. C. & Wang, Z. 2002. Coupled BEM–FEM solutions for direct time domain soil–structure interaction analysis. Engineering Analysis with Boundary Elements, 26, 877-888.

Rombach, G. A. 2004. Finite element design of concrete structures: Practical problems and their solution, Thomas Telford.

Routh, E. J. 1877. A treatise on the stability of a given state or motion, London, Macmillan.

Roy, R. & Dutta, S. C. 2001. Differential settlement among isolated footings of building frames: the problem, its estimation and possible measures. International Journal of Applied Mechanics and Engineering 6, 165-186.

Rugonyi, S. & Bathe, K. J. 2001. On finite element analysis of fluid flows fully coupled with structural interactions. Computer Modelling in Engineering & Sciences, 2, 195-212.

Seok, J. W., Kim, O. Y., Chung, C. K. & Kim, M. M. 2001. Evaluation of ground and building settlement near braced excavation sites by model testing. Canadian Goetechnical Journal, 38, 1127-1133.

Smith, B. S. & Coull, A. 1991. Tall building structures: Analysis and design, Wiley-Interscience.

Soares-Jr., D. 2008. An optimised fem-bem time-domain iterative coupling algorithm for dynamic analyses. Computers & Structures, 86, 1839-1844.

Son, M. & Cording, E. J. 2005. Estimation of building damage due to excavation-induced ground movements. Journal of Geotechnical and Geoenvironmental Engineering 131, 162-177.

Standing, J. R., Farina, M. & Potts, D. M. 1998. The prediction of tunnelling buidling settlements-a case study. In: NEGRO, A. & FERREIRA, A. (EDS),

References

314

Proceedings of the world tunnel congress 98 on tunnels and metropolises. Sao Paulo, Brazil.

Stavridis, L. T. 2002. Simplified analysis of layered soil-structure interaction. Journal of Structural engineering, 128, 224-230.

Stewart, J. P., Fenres, G. L. & Seed, R. B. 1999. Seismic soil-structure interaction in buildings. I: Analytical methods. Journal of Geotechnical and Geoenvironmental Engineering 125, 26-37.

Taylor, D. W. 1964. Fundamentals of soil mechanics, New York, John Wiley and Sons.

Terzaghi, K. V. 1955. Evaluation of coefficient of subgrade reaction. Geotechnique, 5, 297-326.

Tian, L. & Li, Z.-X. 2008. Dynamic response analysis of a building structure subjected to ground shock from a tunnel explosion. International Journal of Impact Engineering, 35, 1164-1178.

Vallabhan, C. V. G. & Das, Y. C. 1991. Modified vlasov model for beams on elastic founadtions. Journal of Geotechnical Engineering, 117, 956-966.

Vesic, A. B. 1961. Bending on beams resting on isotropic elastic solid. Journal of Engineering Mechanic, ASCE, 2, 35-53.

Vierendeels, J. 2006. Implicit coupling of partitioned fluid–structure interaction solvers using reduced-order models. In: BUNGARTZ, S. (ed.) Fluid–structure interaction, modelling, simulation, optimisation, lecture notes in computational science and engineering. Springer.

Vierendeels, J., Lanoye, L., Degroote, J. & Verdonck, P. 2007. Implicit coupling of partitioned fluid–structure interaction problems with reduced order models. Computers & Structures, 85, 970-976.

Viladkar, M. N., Godbole, P. N. & Noorzaei, J. 1991. Soil–structure interaction in plane frames using coupled finite infinite elements. Computers & Structures, 39, 535-546.

Viladkar, M. N., Godbole, P. N. & Noorzaei, J. 1994. Modelling of interface for soil-structure interaction studies. Computers & Structures, 52, 765-779.

Viladkar, M. N., Karisiddappa, Bhargava, P. & Godbole, P. N. 2006. Static soil–structure interaction response of hyperbolic cooling towers to symmetrical wind loads. Engineering Structures, 28, 1236-1251.

Viladkar, M. N., Ranjan, G. & Sharma, R. P. 1993. Soil–structure interaction in time domain. Computers & Structures, 46, 429-442.

References

315

Vlahos, G., Cassidy, M. J. & Byrne, B. W. 2006. The behaviour of spudcan footings on clay subjected to combined cyclic loading Applied Ocean Research, 28, 209-221.

Wall, W. A., Genkinger, S. & Ramm, E. 2007. A strong coupling partitioned approach for fluid structure interaction with free surfaces. Computers & Fluids, 36, 169-183.

Wang, C. M., Xiang, Y. & Wang, Q. 2001. Axisymmetric buckling of reddy circular plates on pasternak foundation. Journal of Engineering Mechanics, ASCE, 127, 254-259.

Winkler, E. 1867. Die lehre von der elasticitaet und festigkeit [the systematic teaching of elasticity and strength], Prag,Czechoslovakia, H. Domenicus, ed.

Wrana, B. 1993. Nonlinear elastic-plastic model of soil-structure interaction in time domain Transactions on the Built Environment.

Zeevrat, L. 1972. Foundation engineering for difficult sub-soil conditions, New York Van Nostrand Reinold Company.

Zienkiewicz, O. C., Taylor R. L., Jhu J. Z. 2005. The finite element method: Its basis and fundamentals, Butterworth-Heinemann.

Zienkiewicz, O. C. & Taylor, R. L. 1991. Finite element method solid and fluid mechanics: Dynamics and nonlinearity, New York, MCGraw-Hill.

Zienkiewicz, O. C. & Taylor, R. L. 2005. The finite element method for solid and structural mechanics, Butterworth-Heinemann.

316

Appendix A

Structure of INTERFACE Data File

The INTERFACE data file includes the following information:

1- NINOD : Total number of nodes at the interface of soil-structure system

2- STGSTP: Total number of load-time increments

3- DTIM: Time step size (for dynamic analysis)

4- CV : Convergence tolerance

5- RELAX: Predefined relaxation parameter

6- TYPE: 1-Adaptive*/Condensed* 2- Adaptive/Condensed

7- METHOD : 1- Constant relaxation 2- Adaptive relaxation 3- Reduced Order

Method

8- REDTYPE: 1- Secant approximation 2- Reduced Order Method 3- Mixed


9- LCV: Tolerance for the ADD/REP procedure

10- EPSU: Predefined incremental displacement

11- EPSF: Predefined incremental force

12- INODA: Interface node numbers of the structure partitioned sub-domain

13- INODI: Interface node numbers of the soil partitioned sub-domain

317

Appendix B

Structure of Communication Data File

A sample structure of the communication data file is presented in the following:

Record Number Variable Description 1 TGSTP Total number of load/time increments 2 TIME Current time (for dynamic analysis) 3 DTIM Time step (for dynamic analysis) 4 NINOD Total number of nodes at the interface 5 RUNA Run switch for ADAPTIC (structure solver) 6 RUNI Run switch for ICFEP (soil solver) 7 EQA Equilibrium switch (ADAPTIC) 8 EQI Equilibrium switch (ICFEP) 9 CONA Interface convergence switch (ADAPTIC) 10 CONI Interface convergence switch (ICFEP) 11 to 20

- Reserved space

21 to 20+NINOD

INODA Node numbers at the interface of structure sub-

domain 21+NONOD to 20+2×NINOD

INODI Node numbers at the interface of soil sub-domain

21+2×NINOD to 20+3×NINOD

UA Prescribed Dirichlet data at the interface of

structure 21+3×NINOD to 20+4×NINOD

FA Obtained Neumann data at the interface of

structure sub-domain 21+4×NINOD to 20+5×NINOD

FI Prescribed Neumann data at the interface of soil

sub-domain 21+5×NINOD to 20+6×NINOD

UI Obtained Dirichlet data at the interface of soil

sub-domain

Appendix C

Iterative Coupling algorithms

In the following the critical algorithmic step (the update of iterative boundary

conditions at the interface) of various iterative coupling schemes (presented in

Chapter 4) are presented, where the condensed interface tangent stiffness matrices

of partitioned sub-domains are used for the update of boundary conditions in

successive iterations enforcing convergence to compatibility/equilibrium at the

interface.

Consider domain decomposition of a soil-structure interaction problem, into structure T and soil B partitioned sub-domains, as presented below:


11 12

21 22

T T T TT T

T T i iT T

K K U F

K K U F

(C.1)


11 12

21 22

B B B BB B

B B i iB B

K K U F

K K U F

(C.2)

Appendix C

319

In the above, vectors XXU and X

XF correspond to the displacements and external

forces for the non-interface degrees of freedom, while iXU and i

XF correspond to

displacements and forces for the interface degrees of freedom, respectively.

Assume that the compatibility and equilibrium defaults at the interface of the

structure and soil sub-domains for iteration number I of load/time step n take the

form:

I II i iU B Tn n n

U U (C.3)

I II i iF B Tn n n

F F (C.4)

Assume that CTK and C

BK are the condensed tangent stiffness matrices at the

interface of the structure and soil sub-domains.

C.1 Sequential Neumann-Dirichlet Iterative Coupling

This algorithm is presented in Chapter 4 (Section 4.2.2).

Using the aforementioned update technique, the update of boundary conditions in

successive iterations takes the following form of:

STEP 5: If convergence to equilibrium has not been achieved, the new estimation of

the interface forces according to the Equation (C.5) is applied to the structure

domain, and iteration continues (I=I+1) from STEP 2 until convergence to

equilibrium is achieved.


F F K K K

(C.5)

C.2 Parallel Dirichlet-Neumann Iterative Coupling


Appendix C

320




new estimation of the interface displacements and forces according to Equations

(C.6) and (C.7) is applied to the structure and soil sub-domains respectively, and

iteration continues (I=I+1) from STEP 2 until convergence to compatibility and



U U K K K

(C.6)


F F K K K

(C.7)

C.3 Parallel Neumann-Dirichlet Iterative Coupling





new estimation of the interface displacements and forces according to Equations

(C.8) and (C.9) is applied to the soil and structure sub-domains respectively, and

iteration continues (I=I+1) from STEP 2 until convergence to compatibility and



U U K K K

(C.8)


F F K K K

(C.9)

Appendix C

321

C.4 Parallel Dirichlet-Dirichlet Iterative Coupling




STEP 3: If convergence to equilibrium has not been achieved, a new estimation of

the interface displacements, according to Equations (C.10) and (C.11) is applied to

the structure and soil sub-domains, and iteration continues (I=I+1) from STEP 2

until convergence to equilibrium is achieved.


U U K K K

(C.10)


U U K K K

(C.11)

C.5 Parallel Neumann -Neumann Iterative Coupling




STEP 3: If convergence to compatibility has not been achieved, a new estimation of

the interface forces according to Equations (C.12) and (C.13) is applied to the

structure and soil sub-domains respectively, and iteration continues (I=I+1) from

STEP 2 until convergence to compatibility is achieved.


F F K K K

(C.12)


F F K K K

(C.13)

Appendix D

Numerical Example

In the following, the presented example in Chapter 4 (Figure D.1) is coupled using

various coupling algorithms presented in Appendix C.

Figure E.1: Coupled spring system

i i

j j

PK K

PK K

Appendix D Numerical Example

323

To facilitate the verification of the various coupling schemes presented hereafter, the

coupled problem is initially modelled and solved monolithically. Performing global

structural analysis using a monolithic approach results in the formation and solution

of the following global system of equations:

2

3

4

5

6

40 10 0 0 0 0

10 40 10 20 0 20

0 10 25 0 15 0

0 20 0 35 15 0

0 0 15 15 40 60

(D.1)

Where in the above j denotes the displacement of node j.

Solving Equation (D.1) results in the following nodal displacements:

2

3

4

5

6

67

231268

231541

231463

231241

77

(D.2)

In order to perform the partitioned analysis using the condensed interface stiffness

approach, the partitioned sub-domains must be modelled in isolation, and their

corresponding condensed interface stiffness matrices must be obtained.

Considering sub-domain T in isolation, its governing equilibrium conditions can be

written in the form of Equation (D.3). In the following iX T

P and iX T

correspond

to the interface forces and displacements of node X in the partitioned sub-domain T

, respectively.


324

4 4

5 5

6

15 0 15

0 15 15

15 15 40 60

i i

T T

i i

T T

P

P

(D.3)

Applying condensation on interface nodes 4 and 5, the condensed stiffness matrix at

the interface of the partitioned sub-domain T can be determined as:

75 45

15 0 15 1 8 815 150 15 15 45 7540

8 8

CTK

(D.4)

Similarly, the governing equilibrium condition for partitioned sub-domain B can be

written as:

2

3

4 4

5 5

040 10 0 02010 40 10 20

0 10 10 0

0 20 0 20

i i

B B

i i

B B

P

P

(D.5)

In the above iX B

P and iX B

correspond to the forces and displacement of the

interface node X in the partitioned sub-domain B , respectively.

Again by applying condensation on interface nodes 4 and 5, the condensed stiffness

matrix at the interface of sub-domain B is obtained as:

122 16

10 0 0 10 40 10 0 0 3 30 20 0 20 10 40 10 20 16 28

3 3

CBK

(D.6)


325

D.1 Parallel Dirichlet-Neumann

Prescribing an initial guess for the interface Dirichlet data at the interface of T ,

0

4 0Ii

T

and 0

5 0Ii

T

, for the first iteration (I=0):

0 0

4 4

0 0

5 5

6

015 0 15

0 15 15 0

15 15 40 60

I Ii i

T T

I Ii i

T T

P

P

6

0

4

0

5

3

245

245

2

Ii

T

Ii

T

P

P

(D.7)

Prescribing an initial guess for the interface Neumann data at the interface of B (

0

4 0Ii

BP

and 0

5 0Ii

BP

), for the first iteration (I=0):

2 2

3 3

0 0 0

4 4 4

0 0 0

5 5 5

040 10 0 0 2 / 3

2010 40 10 20 8 / 3

00 10 10 0 8 / 3

0 20 0 20 8 / 30

I I Ii i i

B B B

I I Ii i i

B B B

P

P

(D.8)

Comparing 0

4

Ii

BP

and 0

5

Ii

BP

with 0

4

Ii

TP

and 0

5

Ii

TP

, and comparing 0

4

Ii

B

and

0

5

Ii

B

with 0

4

Ii

T

and 0

5

Ii

T

, it is clear that convergence to either equilibrium or

compatibility at the interface is not achieved. Therefore new estimates of Neumann

and Dirichlet data should be calculated enforce convergence in the next iteration:


326

1

4

1

5

1

0

0

75 45 22 16 22 16 8 54145

8 8 3 3 3 3 3 2312.45 75 16 28 16 28 8 45 463

8 8 3 3 3 3 3 2 231

Ii

T

Ii

T

(D.9)

1

4

1

5

11 1 1

0

0

75 45 22 16 8 75 45 13045

8 8 3 3 3 8 8 112.130045 75 16 28 8 45 75 45

8 8 3 3 3 8 8 2

Ii

B

Ii

B

P

P

77

(D.10)

Prescribing the new estimate for the interface Dirichlet data at the interface of T at

iteration I=1, and solving the partitioned sub-domain T gives:

11

44 6

1 1 1

5 5 4

1

6 5

541 241

231 7715 0 15463 130

0 15 15231 11

15 15 40 60 1300

77

Ii IiT

T

I I Ii i i

T T T

Ii

T

P

P P

P

(D.11)

Prescribing the new estimate for the interface Neumann data at the interface of B

at iteration I=1, and solving the partitioned sub-domain B gives:


327

2 2

3 311 1

44 4

1 115 55

670 231

40 10 0 0 2682010 40 10 20 2311300 10 10 0 54111

23113000 20 0 2046377231

II Iii iBB B

I IIi iiB BB

P

P

(D.12)

It is clear that at the first coupling iteration (I=1) convergence to equilibrium and

compatibility is achieved. Moreover, the obtained results by the above coupling

procedure are identical to those obtained by the monolithic treatment.

D.2 Parallel Dirichlet-Dirichlet

Prescribing an initial guess for the interface Dirichlet data at the interface of T (

0

4 0Ii

T

and 0

5 0Ii

T

), for the first iteration (I=0) gives:

0 0

4 4 6

0 0 0

5 5 4

06

5

30 215 0 15

450 15 15 0

215 15 40 60 45

2

I Ii i

T T

I I Ii i i

T T T

Ii

T

P

P P

P

(4.13)

Prescribing an initial guess for the interface Dirichlet data at the interface of B (

0

4 0Ii

B

and 0

5 0Ii

B

), for the first iteration (I=0):


328

2 2

3 3

0 0 0

4 4 4

0 0 0

5 5 5

2

15040 10 0 0 82010 40 10 20 15

00 10 10 0 16

30 20 0 20 032

3

I I Ii i i

B B B

I I Ii i i

B B B

P P

P P

(D.14)

Comparing 0

4

Ii

BP

and 0

5

Ii

BP

with 0

4

Ii

TP

and 0

5

Ii

TP

, it is clear that convergence

to equilibrium at the interface is not achieved. Therefore new estimates of Neumann

data at the interface of T and B should be calculated to enforce convergence in

the next coupling iteration:

1

1

4

1

5

22 16 75 45 103 5410 3 3 8 8 6 2310 16 28 45 75 71 463

3 3 8 8 6 231

Ii

T

Ii

T

(D.15)

Prescribing the new estimate for the interface Dirichlet data at the interface of T at

iteration I=1, and solving the partitioned sub-domain T gives:

00

44 6

0 0 0

5 5 4

0

6 5

541 241

231 7715 0 15463 130

0 15 15231 11

15 15 40 60 1300

77

Ii IiT

T

I I Ii i i

T T T

Ii

T

P

P P

P

(4.16)

Prescribing the new estimate for the interface Dirichlet data, by applying

compatibility, at the interface of B at iteration I=1 and solving the partitioned sub-

domain B gives:


329

22

33

0 0 04 4 4

0 005 55

67

231040 10 0 0 2682010 40 10 20 2315410 10 10 0 130231

110 20 0 20 4631300231

77

Ii I Ii iB B B

I II i iiB BB

P P

P P

(4.17)

It is clear that at the first coupling iteration (I=1) convergence to equilibrium is

achieved. Moreover, the obtained results by the above coupling procedure given by

are identical to those obtained by the monolithic treatment.

D.3 Parallel Neumann-Neumann

Prescribing an initial guess for the interface Neumann data at the interface of T (

0

4 0Ii

TP

and 0

5 0Ii

TP


0 0 0

4 4 4

0 0 0

5 5 5

6 6

015 0 15 6

0 15 15 0 6

15 15 40 660

I I Ii i i

T T T

I I Ii i i

T T T

P

P

(D.18)

Prescribing an initial guess for the interface Neumann data at the interface of B (

0

4 0Ii

BP

and 0

5 0Ii

BP



330

2 2

3 3

0 0 0

4 4 4

0 0 0

5 5 5

2

3040 10 0 0 82010 40 10 20 3

00 10 10 0 8

30 20 0 20 08

3

I I Ii i i

B B B

I I Ii i i

B B B

P

P

(D.19)

Comparing 0

4

Ii

B

and 0

5

Ii

B

with 0

4

Ii

T

and 0

5

Ii

T

, it is clear that convergence to

compatibility at the interface is not achieved. Therefore new estimates of Neumann

data should be calculated to enforce convergence in the next iteration:

11 11

4

1

5

22 16 75 45 26 1300 3 3 8 8 13 11

13000 16 28 45 75 26

773 3 8 8 13

Ii

T

Ii

T

P

P

Prescribing the new estimate for the interface Neumann data at the interface of T

at iteration I=1, and solving the partitioned sub-domain T gives:

10 1

44 4

0 1 1

5 5 5

6 6

541130

2311115 0 151300 463

0 15 1577 231

15 15 4060 241

77

IiI Ii iT

T T

I I Ii i i

T T T

P

P

(D.20)

Prescribing the new estimate for the interface Neumann data at the interface of B

at iteration I=1, and solving the partitioned sub-domain B gives:


331

2 2

3 311 1

44 4

1 115 55

670 231

40 10 0 0 2682010 40 10 20 2311300 10 10 0 54111

23113000 20 0 2046377231

II Iii iBB B

I IIi iiB BB

P

P

(D.21)

It is clear that at the first coupling iteration (I=1) convergence to equilibrium is

achieved. Moreover, the obtained results by the above coupling procedure are

identical to those obtained by the monolithic treatment.

thesis

Documents

soilstructure interaction21

iterative coupling

coupledmodelling of

staggered coupling procedure70

existing coupling methods

coupled systems

partitioned treatment

coupled fieldspecific