Continuum Topology Optimization of Buckling-Sensitive Structures Salam Rahmatalla, Ph.D Student Colby C. Swan, Associate Professor Civil & Environmental Engineering Center for Computer-Aided Design The University of Iowa Iowa City, Iowa USA 43 rd AIAA SDM Conference Denver, Colorado USA 24 April 2002 1
Introduction Continuum Design-variable Topology interpolation Optimization options Issue of Structure Sparsity and Instability Problems Formulation Size Reduction Technique Examples Conclusion Presentation Overview Introduction to continuum structural topology optimization Alternative design-variable formulations Issues of structural sparsity and instability Problem formulations for sparse structures Analysis problem size reduction technique Examples Summary 2
What is Structural Topology Optimization Size Optimization Shape Optimization Topology Optimization 3
Essentials of Continuum Structural Topology Optimization Discretization of structural domain Ω d 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 local mechanical properties. Pose and solve an optimization problem to extremize structural performance subject to material usage constraints. 4
The Density or Volume-Fraction Formulation Mostwidelyusedtoday Design variables: b={φ 1, φ 2, φ 3,,φ n } where φ i is either an element or node-based volume fraction of the structural material. Micro-mechanical (or mixing) rule: σ(x) = φ p (X) σ solid (ε)+[1-φ p (X)] σ void (ε) applies more easily to general elastic and inelastic material models than do micro-structure based formulations. Observation: Thelargerp, the stronger the penalty against designs that utilize mixtures. 5
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Design Optimization Algorithm to Solve: min F( b, u) r( b, u) = 0 g i b such that ( b, u) 0, i = 1,2, K,n Initialization b = b 0 Solve for u(b) such that r(b,u) = 0 Sensitivity analysis compute(df/db; dg/db) Optimization step b= b(df/db; g; dg/db) No Yes Optimality? Stop ) 7
Displacement & Design Variable Interpolation Options Q4/U: unstable Q8/U: Slightly unstable Q4/Q4: slightly unstable Q9/U: Slightly unstable Q9/Q4: Stable 8
Established characteristics of continuum structural topology optimization 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 locally optimal design solutions. Here, we attempt to address the first two problems. 9
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 number of 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 10
10 Layering instability appearing in Q4/Q4 formulation 10 30 60 20 60 180 60 50 30 300 60 Cure with uniform mesh refinement. Cure with mesh refinement in direction of layering. 11
Structural Sparsity Issues Typically, large scale civil structures (as bridges) are sparse. occupy only a small fraction of the structure s envelope volume. Structural models must capture the characteristic sparsity to yield realistic performance. This typically requires fine meshing of structural domain Ω d. This adds to the computational expense of the analysis and design process. 12
Alternative Design Formulations to Achieve Stable, Sparse Structures Option 1: Perform structural analysis considering geometrically nonlinear behaviors and instabilities. Design/Optimize the structure to avoid instabilities. Option 2: Perform linearized buckling stability analysis on the structure. Design/Optimize to maximize minimum critical buckling load. 13
Option 1: Analysis/optimization of the structure as a nonlinear hyperelastic system. Constitutive law 1 1 UJ K J J 2 2 2 ( ) = [ ( 1) ln( )] Variational formulation S S h = W τδε dω= ργδudω+ hδudγ ij ij S o j j S j j h Ω Ω Γ 1 - µ [ tr ( ) 3]. 2 θ ' ( ) 2 W τ = JU J 1+ dev. θ d W dl d d d int ( δ ) = δεij v( τij ) Ω S + δετ ji im εjm ΩS, Ω Ω Discrete Equilibrium int A A T A int A ext A r = ( f ) ( f ) = 0 ( ) 1 ( ) : d ext A A A f n+ = B τn + 1 ΩS ( f ) n+ 1= ρonγn + 1dΩ S+ N hn + 1dΓh. n+ 1 n+ 1 n+ 1 K Ω S = B c B dω + N τ N δ dω AB A B A B il ji jk kl s j jk k il s Ω Ω s s S Ω S, S Γ h 14
Option 1 (continued): Solve for the Minimum Critical Instability Load n=0; m=0; t 0 =0; t = t baseline ; t n+1 =t n + t Algorithm for Finding First Point of Instability m=m max? Yes Critical state found P P cr Can r n+1 = 0 be solved, and is K n+1 positive definite? n=n+1 Yes No t = t/4 m=m+1 δ cr δ 15
Option 1: (Continued) Objective function: F =(f crit ) -1 Once the minimum point of instability is found: compute design derivative of minimum critical load. é gk crit = å ê ò B K { } ê g f int df int crit db K = a ( u ) K ù T K τ dωú n g ë K Ω úû g a ( u ) r é ù K T τ å ê ò BK dω + K ú K { n g } ê g ë K b b Ω úû = u ( f int ) crit K 16
17 Option 2: Structure modeled as linear elastic with instability computed by linearized buckling analysis Linear elastic problem: Associated generalized eigenvalue problem: Modified eigenvalue problem Objective function and design derivative. ext L f u K = 0; ψ b G u, ψ b K = + ) ( λ ) L ( ; min(λ) 1 ), F( = b u ø ö ç ç ç ç è æ + ø ö ç è æ + = b f u b K u ψ b K b G ψ b ext L T a L T ) ( λ 1 d dl ψ G ψ ψ K ψ T L T λ = ( ) [ ] ; γ L L 0 ψ K G K = + + ø ö ç è æ = λ 1 λ γ
Potential Problem in Structural Analysis with Option 1: Elements devoid of structural material are highly compliant. These elements can undergo excessive deformation creating difficulty/singularity in solving the structural analysis problem. A strategy to circumvent this problem is needed. 18
Analysis Problem Size Reduction Technique Identifyvoidelements Identify and restrain prime nodes (those surrounded by void elements). Identify and temporarily remove prime elements (those surrounded only by prime nodes). 19
Test Problem #1 for Design of Sparse Buckling Sensitive Structure. Design optimum sparse, elastic structure in the circular domain to carry the design load back to fixed, rigid walls. F F 20
Old Q4/U Design Solutions to Circle Problem Generalized compliance objective function w/ spatial filtering 21
New Q4/Q4 Solutions to Circle Test Problem: (a) Meshusedinanalysis/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) 22
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) 23
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 7 1 Π = 2.3 10 ; λ = 8.3 10 maximizing the minimum buckling load 7 3 Π = 6.5 10 ; λ = 4.9 10 24
Design of Long-Span Over-Deck Bridge Design to minimize elastic compliance 2 Π = 1.65; λ = 3.2 10 Design to maximize minimum linearized buckling eigenvalue Π = 8.20; λ = 1.5 10 4 25
Summary and Conclusions For continuum structural topology optimization methods to be useful in concept design of large-scale civil structures, they must be able to detect potential buckling instabilities. modeling of sparse structures at high mesh refinement. solution of structural analysis problems with geometric nonlinearity. The proposed formulations are promising for concept design of stable, sparse structural systems. 26