alessandro reali - reluis · reali, 2008, and references therein): many devices, in particular for...

ADVANCED COMPUTATIONAL TOOLS FOR STRUCTURAL

MECHANICS AND EARTHQUAKE ENGINEERING

Alessandro Reali a,b,c,d

a Department of Structural Mechanics, University of Pavia, Italy, [email protected]

b EUCENTRE, Pavia, Italy

c IMATI, CNR, Pavia, Italy

d CeSNA, IUSS, Pavia, Italy

1 INTRODUCTION

This work consists of a presentation of the main research interests and topics studied within

the Group of Computational Mechanics and Advanced Materials of the University of Pavia

(http://www-1.unipv.it/dms/compmech).

In particular, the following topics will be briefly presented:

1. Isogeometric analysis;

2. Shape memory alloy modeling and application;

3. Other interesting research topics such as meshless methods and beam models.

Keywords: finite element methods; isogeometric analysis; shape memory alloys;

collocation/particle methods; beam models.

2 ISOGEOMETRIC ANALYSIS

First introduced by Hughes et al., 2005, isogeometric analysis consists of an isoparametric

analysis approach where basis functions generated from Non-Uniform Rational B-Splines

(NURBS) are employed in order to describe both the geometry and the unknown variables of

the problem. Its name (“isogeometric”) is due to the fact that the use of NURBS leads to an

exact geometric description of the domain, while in standard finite element analysis it is

necessarily approximated by means of a mesh. It is to be noted that, as NURBS are the

standard functions for describing and modeling objects in computer aided design (CAD), the

term “exact” should be read in this context as “as exact as CAD modeling can be”.

The fact that even the coarsest mesh retains exact geometry makes possible a direct

refinement, without any need of going back to the CAD model from which the mesh has been

generated; finite element analysis, instead, needs to interact with the CAD system at every

refinement step, except in case of very simple geometries.

Moreover, besides the equivalents of classical finite element h- and p-refinements, another

higher-continuity refinement strategy, named k-refinement, is possible.

Since 2005, isogeometric analysis has been successfully applied to a number of fields,

ranging from Structural Engineering to Biomechanics, from Wave Propagation to Electro-

magnetism. In particular, our research group focuses on the following topics:

1. Structural vibrations (see Reali, 2006, Cottrell et al., 2006, and Hughes et al. 2008):

the superior isogeometric spectrum approximation properties with respect to standard

finite elements has been proven in many situations (see, e.g., Figure 1). Moreover, the

A. Reali - Earthquake Engineering by the Beach Workshop, July 2-4, 2009, Capri, Italy

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eigenvalue analysis of a NASA structure (Aluminum Testbed Cylinder, see Figure 2)

has been carried out and successfully compared with experimental results.

Figure 1. Isogeometric analysis of structural vibrations. Left: 1D normalized spectra from isogeometric

analysis and finite elements. Right: third eigenmode of a clamped circular thin plate modeled using

isogeometric 3D solid elements.

Figure 2. NASA Aluminum Testbed Cylinder: examples of numerically computed eigenmodes for the

frame only (left) and for the whole structure (frame+skin, right).

2. Refinement and continuity issues in isogeometric structural analysis (see Cottrell et

al., 2007): the implications on structural analysis of the refinement and continuity

peculiar properties of isogeometric analysis have been deeply studied. In particular

some interesting examples of shells modeled by means of 3D solid elements are

analyzed (see Figure 3).

Figure 3. Hemispherical shell with stiffener: full isogeometric model (a) and mesh particular in the case of

single (b) and multiple (c) patches.

Advanced Computational Tools for Structural Mechanics and Earthquake Engineering

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3. Incompressible elastic problems (see Auricchio et al., 2007a, Auricchio et al., 2009a):

a stream function formulation for incompressible elasticity, able of exactly

representing the divergence-free constraint using the higher continuity properties of

isogeometric analysis, has been proposed and successfully employed (see Figure 4).

Moreover, such a technique has been also used in order to study the performance of

mixed finite elements in large strain situations.

Figure 4. Fully constrained quarter annulus with a polynomial body load. Left: y-displacement numerical

solution using a quartic isogeometric stream formulation. Right: convergence plots for displacements.

4. Wave propagation (see Hughes et al. 2008): the superior isogeometric dispersion

properties with respect to standard finite elements have been studied and proven in

many situations (see, e.g., Figure 5).

Figure 5. 1D wave propagation problem solved using quadratic NURBS-based isogeometric analysis and

finite elements. Left: frequency response spectra. Right: numerical solutions for a given wave number.

5. Other intersting topics: variational multiscale turbulence modeling for large eddy

simulation (see Bazilevs et al., 2007); development of suitable quadrature techniques

for isogeometric analysis (see Hughes et al., 2009a).

6. Work in progress: development of isogeometric collocation techniques for structural

dynamics (see some intial ideas and results in Hughes et al., 2009b).

Finally, the main projects that are currently funding our research on isogeometric analysis are

reported in the following:

- January 2006 - December 2009: Office of Naval Research, Isogeometric Analysis for

Naval Ship Structures, coordinator: Prof. T.J.R. Hughes.

- July 2008 - June 2012: FP7 Ideas ERC Starting Grant, GeoPDEs - Innovative

compatible discretization techniques for Partial Differential Equations, coordinator:

Dr. A. Buffa.


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3 SHAPE MEMORY ALLOY MODELING AND APPLICATION

Intelligent, smart or functional materials exhibit special properties that make them a suitable

choice for industrial applications in many branches of engineering. Among different types of

smart materials, shape memory alloys (SMAs) have unique features known as pseudo-

elasticity, one-way and two-way shape memory effects (Duerig et al., 1990; Otsuka and

Wayman, 1998). The interest in the mechanical behavior of SMAs is rapidly growing with the

increasing number of potential industrial applications. Early commercialization activities,

fueled by applications such as rivets, heat engines, couplings, circuit breakers and automobile

actuators, started in the 1970’s, were intense and often highly secretive. However, starting

from the Seventies, the knowledge of SMAs has progressively spread out more and more, up

to the fact that nowadays pseudo-elastic Nitinol is a common and well-known engineering

material in the medical industry. Moreover, recently, the potential of such smart materials is

receiving a big deal of attention for application in a wide range of engineering field, including

Civil, and, in particular, Earthquake Engineering.

The origin of SMA material features is a reversible thermo-elastic martensitic phase

transformation between a high symmetry, austenitic phase and a low symmetry, martensitic

phase. Austenite is a solid phase, usually characterized by a body-centered cubic

crystallographic structure, which transforms into martensite by means of a lattice shearing

mechanism. When the transformation is driven by a temperature lowering, martensite variants

compensate each other, resulting in no macroscopic deformation. However, when the

transformation is driven by the application of a load, specific martensite variants favorable to

the applied stress are preferentially formed, exhibiting a macroscopic shape change in the

direction of the applied stress. Upon unloading or heating, this shape change disappears

through the reversible conversion of the martensite variants into the parent phase (Otsuka and

Wayman, 1998).

Nowadays, many SMA-related research lines are active in different fields of engineering and,

among them, our group mainly focuses on the following topics:

1. SMA macroscopic costitutive modeling (see, e.g., Auricchio and Reali, 2007,

Auricchio et al., 2007b and 2009c): starting from the basis of the model proposed by

Souza et al., 1998, and enhanced by Auricchio and Petrini, 2004, some models, able to

reproduce main and secondary SMA macroscopic behaviors and suitable for finite

element implementation, have been recently developed (Figure 6).

Figure 6. Uniaxial response of advanced SMA models. Left: model including permanent inelasticity and

degradation. Right: model including tension-compression asymmetries and elastic properties depending

on the phase transformation level.

2. Validation of SMA numerical modeling versus experimental results (see Auricchio et

al., 2009b): techniques for parameter identification and validation of model prediction


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properties versus experimental results have been studied, also taking advantage of a

collaboration with SAES Getters, the SMA material world leading producer (see, e.g.,

Figure 7).

Figure 7. Model parameter identification and evaluation of prediction properties versus experimental

results. Left: strain-temperature curves. Right: stress-strain curves.

3. Design and finite element simulation of biomedical devices (see, e.g., Auricchio and

Reali, 2008, and references therein): many devices, in particular for biomedical

applications, such as, for instance, stents, spinal spacers, and micro-actuators, have

been simulated using SMA costitutive models implemented within finite element

codes (see, e.g., Figure 8). The use of such a tool allows also the development of new

design strategies.

Figure 8. Simulation and design of SMA-based devices. Left: numerical simulation of a stent. Right:

numerical simulation of a micro-gripper (check of design conditions).

4. SMA applications for Earthquake Engineering (see, among others, Auricchio et al.,

2006a and 2006b, McCormick et al., 2006, Attanasi et al., 2009a and 2009b): SMA

applications to frame braces and for vibration control have been studied, as well as

SMA damping properites. More recently, research is focused on simulation and design

of SMA-based isolation bearings (see Figure 9).

Figure 9. Numerical simulation of SMA isolators, compared with traditional lead rubber bearing systems.


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Finally, the main projects that are currently funding our research on shape memory alloys are

reported in the following:

- December 2006 - December 2009: ESF Eurocores Programme, SMARTeR: Shape

Memory Alloys to Regulate Transient Responses in civil engineering, coordinator:

Prof. M. Fremond.

- September 2008 - October 2013: FP7 Ideas ERC Starting Grant, BioSMA -

Mathematics for Shape Memory Technologies in Biomechanics, coordinator: Dr. U.

Stefanelli.

4 OTHER INTERESTING RESEARCH WORKS IN PROGRESS

In this final section, we now briefly mention other interesting research topics we are dealing

with in the framework of Computational Structural Mechanics, i.e.:

1. Development of meshless particle methods based on collocation techniques. Besides

the aforementioned isogeometric collocation techniques, we are currently studying

also methods obtained as an evolution of Smoothed Particle Hydrodynamics (SPH)

approaches, with the aim of a reliable simulation of fast dynamics, impact problems

and blasts (this is a work in progress and very preliminary results may be found in

Asprone et al., 2008).

2. Study and development of advanced finite element beam formulations. In particular, a

good deal of attention has been devoted to the study of geometrically nonlinear beam

elements (see Auricchio et al., 2008) and currently the focus is on the development of

advanced theories for homogeneous and multi-layered beams, for which some

encouraging preliminary results have been obtained.

5 ACKNOWLEDGEMENTS

The author would like to acknowledge the following co-workers (in alphabetical order):

Jamal Arghavani (Sharif University of Technology);

Domenico Asprone (University of Naples “Federico II”);

Gabriele Attanasi (ROSE School, Pavia);

Ferdinando Auricchio (University of Pavia);

Giuseppe Balduzzi (University of Pavia);

Lourenco Beirao da Veiga (University of Milan);

Annalisa Buffa (IMATI/CNR, Pavia);

Thomas JR Hughes (University of Texas at Austin);

Carlo Lovadina (University of Pavia);

Gaetano Manfredi (University of Naples “Federico II”);

Andrea Prota (University of Naples “Federico II”);

Giancarlo Sangalli (University of Pavia);

Ulisse Stefanelli (IMATI/CNR, Pavia);

as well as the whole research groups led by Prof. Auricchio and Prof. Hughes.


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6 REFERENCES

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Attanasi, Auricchio, Fenves (2009a). “Feasibility Assessment of an Innovative Isolation Bearing System

with Shape Memory Alloys”, Journal of Earthquake Engineering, 13(s.i.1): 18-39.

Attanasi, Auricchio, Fenves (2009b). “Feasibility Investigation of Superelastic Effect Devices for

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