1

Continuum Topology Optimization of Buckling-SensitiveStructures

Salam Rahmatalla, Ph.D StudentColby C. Swan, Associate Professor

Civil & Environmental EngineeringCenter for Computer-Aided Design

The University of IowaIowa City, Iowa USA

43rd AIAA SDM ConferenceDenver, Colorado USA

24 April 2002

2

• Introduction

• Continuum Topology Optimization

• Issue of Structure Sparsity and Instability

• Problems Formulation

• Size Reduction Technique

• Examples

• Conclusion

Presentation Overview

• Introduction to continuum structural topologyoptimization

• Alternative design-variable formulations• Design-variable interpolation options• Issues of structural sparsity and instability• Problem formulations for sparse structures• Analysis problem size reduction technique• Examples• Summary

3

What is “Structural Topology Optimization”

• Size Optimization

• Shape Optimization

• Topology Optimization

4

Essentials of Continuum Structural TopologyOptimization

• Discretization of structural domain Ωd into a mesh of nodes/volumes.• Use discretized model to describe spatial distribution of design variables.• Specify a micro-mechanical model to relate local design variables to localmechanical properties.• Pose and solve an optimization problem to extremize structural performancesubject to material usage constraints.

5

The “Density” or “Volume-Fraction”Formulation

• Most widely used today

• Design variables: b=φ1, φ2, φ3, …, φn

where φi is either an element or node-based volume fraction ofthe structural material.

• Micro-mechanical (or mixing) rule:

σσσσ(X) = φp(X) σσσσsolid(εεεε) + [1-φp(X)] σσσσvoid(εεεε)

applies more easily to general elastic and inelastic materialmodels than do micro-structure based formulations.

Observation: The larger p, the stronger the penalty againstdesigns that utilize mixtures.

6

7

Design Optimization Algorithm to Solve:

Optimality?

Initializationb = b0

Solve for u(b) such thatr(b,u) = 0

Sensitivity analysiscompute(dF/db; dg/db)

Optimization step∆b= ∆b(dF/db; g; dg/db)

Stop)

YesNo

n,1,2,i0,),(g

),(

such that),(min

i =≤=

ub

0ubr

ubb

F

8

Displacement & Design VariableInterpolation Options

Q8/U: Slightlyunstable

Q9/U: Slightlyunstable

Q9/Q4: Stable

Q4/Q4:slightlyunstable

Q4/U:unstable

9

Established characteristics of continuum structural topologyoptimization as applied to civil structures• The Q4/U element-based numerical formulation has severe

instabilities that result in “checkerboarding” designs.• As modeled, continuum structures are unrealistically “heavy”.

• lack the sparsity of civil structures• continuum joints transmit moments• cannot capture potential buckling behaviours

• The optimization problem admits a large number of locallyoptimal design solutions.

• Here, we attempt to address the first two problems.

10

Element-based design variables (Q4/U):• solid volume-fraction design field can be discontinuous

across element boundaries.• this can lead to the phenomenon of “checkerboarding”design solutions.

• checkerboarding designs can be eliminated via a numberof ad-hoc spatial filtering techniques.

Node-based design variables (Q4/Q4):• forces continuity of solid volume-fraction design

variables across element boundaries.• does not admit “checkerboarding” design solutions.• requires no spatial filtering techniques.• found by Jog & Haber (1996) to be slightly unstable

11

10

300

30

50

60

180

60

30

60

60

10

20

Layering instability appearing in Q4/Q4formulation

Cure with uniform meshrefinement.

Cure with mesh refinement indirection of layering.

12

Structural Sparsity Issues

• Typically, large scale civil structures (as bridges) are sparse.• occupy only a small fraction of the structure’s envelopevolume.

• Structural models must capture the characteristic sparsity toyield realistic performance.

• This typically requires fine meshing of structural domain Ωd.• This adds to the computational expense of the analysisand design process.

13

Alternative Design Formulations to AchieveStable, Sparse Structures

• Option 1:

• Perform structural analysis considering geometricallynonlinear behaviors and instabilities.

• Design/Optimize the structure to avoid instabilities.

• Option 2:

• Perform linearized buckling stability analysis on thestructure.

• Design/Optimize to maximize minimum critical bucklingload.

14

Option 1: Analysis/optimization of the structureas a nonlinear hyperelastic system.

Constitutive law

Variational formulation

Discrete Equilibrium

21 1( ) [ ( 1) ln( )]

2 2U J K J J= − − 1

[ ( ) 3].2

W trµ−

= −-

θ ' ( ) 2 .W

JU J devτ−

∂= +∂

1θ

S S h

ij ij S o j j S j j hd u d h u dτ δε ργ δ δΩ Ω Γ

Ω = Ω + Γ int( ) ( ) ,

S S

ij v ij S ji im jm Sd W dL d d dδ δε τ δε τ εΩ Ω

= Ω + Ω

int1 1 1( ) ( ) 0A A ext A

n n n+ + += − =r f fint

1 1( ) ( ) :S

A A Tn n Sd+ +

Ω

= Ω τf B 1 1 1( ) .S h

ext A A An o n S n hN d N dρ γ+ + +

Ω Γ

= Ω + Γ f h

,s s

AB A B A Bil ji jk kl s j jk k il sB c B d N N dτ δ

Ω Ω

= Ω + Ω K

15

P

δ

Pcr

δcr

Option 1 (continued):

Solve for the Minimum Critical Instability Load

n=0; m=0;t0 = 0;∆t = ∆tbaseline;

tn+1 = tn + ∆t

Can r n+1 = 0 be solved, andis K n+1 positive definite?

n = n + 1

∆t = ∆t/4m = m + 1

Yes No

m = mmax?Critical state

found

Yes

Algorithm for FindingFirst Point of Instability

16

Option 1: (Continued)

• Objective function:

• F = (fcrit)-1

• Once the minimum point of instability is found:

• compute design derivative of minimum critical load.

f int

crit ∈ Ω

Ω⋅⋅−=

gnK

TK

K

K dτBgg

( ) d

df intcrit

∈ Ω

∂∂⋅+Ω

∂∂⋅⋅−=

gnK

aK

TK

K

K dbr

ubτ

Bgg

b

( ) ( )f intcrit

uuK

∂∂

−=⋅ KaK

17

Option 2: Structure modeled as linear elastic withinstability computed by linearized buckling analysis

Linear elastic problem:

Associated generalized eigenvalue problem:

Modified eigenvalue problem

Objective function and design derivative

.

texL fuK =⋅

;0ψbu,GψbK =+ )(λ)(L

;)λmin(

1),F( =bu

∂∂−⋅

∂∂+

∂∂+

∂∂−=

bf

ub

Kuψ

bK

bG

ψb

extLTaLT )(

λ

1

d

dL

ψGψ

ψKψ

TL

T

λ −=

( )[ ] ;γ LL 0ψKGK =⋅++

−=λ

1λγ

18

Potential Problem in Structural Analysis withOption 1:

• Elements devoid of structural material are highlycompliant.

• These elements can undergo excessive deformationcreating difficulty/singularity in solving thestructural analysis problem.

• A strategy to circumvent this problem is needed.

19

Analysis Problem Size Reduction Technique

Identify void elements

Identify and restrain “prime” nodes(those surrounded by void elements).

Identify and temporarily remove“prime” elements (those surrounded only

by prime nodes).

20

Test Problem #1 for Design of SparseBuckling Sensitive Structure.

• Design optimum sparse, elastic structure in the circulardomain to carry the design load back to fixed, rigid walls.

F F

21

Old Q4/U Design Solutions to Circle Problem

• Generalized compliance objective function w/ spatialfiltering

22

New Q4/Q4 Solutions to Circle Test Problem:

(a) Mesh used in analysis/design

(b) Design solution [20% material, option #1](c ) Deformed shape of (b)

(d) Design solution [5% material, option #1](e) Deformed shape of (d)

(f) Design solution [5% material, option #2](g) Deformed shape of (f)

23

Fix-end Beam Problem

Option #1:(a) Mesh, restraints, load case(b) Design solution(c) Design at onset of instability

Option #2:(d) Mesh, restraints, load case(e) Design solution(f) Buckling mode (local)

24

Sparse, Long Span Canyon Bridge Concept Design Problem:• span = 1000m; max height = 400m;• design can occupy only 10% of envelope volume• 16000 elements, 16281 design variables used.

Problem description

minimizing linear elastic compliance maximizing the minimum buckling load37 104.9λ;105.6 ⋅=⋅=Π

17 103.8λ;103.2 ⋅=⋅=Π

25

Design of Long-Span Over-Deck Bridge

Design to minimize elasticcompliance

2102.3;65.1 ⋅==Π λ

Design to maximize minimumlinearized buckling eigenvalue

4105.1;20.8 ⋅==Π λ

26

Summary and Conclusions

• For continuum structural topology optimization methods to be usefulin concept design of large-scale civil structures, they must be able todetect potential buckling instabilities.

• modeling of sparse structures at high mesh refinement.

• solution of structural analysis problems with geometricnonlinearity.

• The proposed formulations are promising for concept design of stable,sparse structural systems.