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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

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• 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

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What is “Structural Topology Optimization”

• Size Optimization

• Shape Optimization

• Topology Optimization

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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.

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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.

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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

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Displacement & Design VariableInterpolation Options

Q8/U: Slightlyunstable

Q9/U: Slightlyunstable

Q9/Q4: Stable

Q4/Q4:slightlyunstable

Q4/U:unstable

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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.

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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

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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.

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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.

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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.

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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

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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

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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

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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

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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.

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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).

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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

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Old Q4/U Design Solutions to Circle Problem

• Generalized compliance objective function w/ spatialfiltering

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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)

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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)

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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

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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

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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.