Introduction Topology Optimization Model Order Reduction Applications Conclusion Rapid Topology Optimization using Reduced-Order Models Matthew J. Zahr and Charbel Farhat Farhat Research Group Stanford University 12th U.S. National Congress on Computational Mechanics Raleigh, North Carolina July 22 - 25, 2013 M. J. Zahr and C. Farhat
Introduction Topology Optimization Model Order Reduction Applications Conclusion Motivation For industry-scale design problems, topology optimization is a beneficial tool that is time and resource intensive Large number of calls to structural solver usually required Each structural call is expensive, especially for nonlinear 3D High-Dimensional Models (HDM) Use a Reduced-Order Model (ROM) as a surrogate for the structural model in a material topology optimization loop Large speedups over HDM realized M. J. Zahr and C. Farhat
Introduction Topology Optimization Model Order Reduction Applications Conclusion 0-1 Material Topology Optimization minimize L ( u ( χ ) , χ ) χ ∈ R nel subject to c ( u ( χ ) , χ ) ≤ 0 u is implicitly defined as a function of χ through the HDM equation f int ( u ) = f ext � ∈ Ω ∗ 0 , e / C e = C e ρ e = ρ e χ e = 0 χ e 0 χ e 1 , e ∈ Ω ∗ Assume geometric nonlinearity and linear material law Large deformations of St. Venant-Kirchhoff material M. J. Zahr and C. Farhat
Introduction Projection-Based ROM Topology Optimization Nonlinear ROM Bottleneck Model Order Reduction ROM Precomputations Applications Reduced Topology Optimization Conclusion Reduced-Order Model Model Order Reduction (MOR) assumption State vector lies in low-dimensional subspace defined by a Reduced-Order Basis (ROB) Φ ∈ R N × k u u ≈ Φy k u ≪ N N equations, k u unknowns f int ( Φy ) = f ext Galerkin projection Φ T f int ( Φy ) = Φ T f ext M. J. Zahr and C. Farhat
Introduction Projection-Based ROM Topology Optimization Nonlinear ROM Bottleneck Model Order Reduction ROM Precomputations Applications Reduced Topology Optimization Conclusion NL ROM Bottleneck - Internal Force Φ T f int ( Φy )= Φ T f ext M. J. Zahr and C. Farhat
Introduction Projection-Based ROM Topology Optimization Nonlinear ROM Bottleneck Model Order Reduction ROM Precomputations Applications Reduced Topology Optimization Conclusion NL ROM Bottleneck - Tangent Stiffness Φ T f int ( Φy )= Φ T f ext M. J. Zahr and C. Farhat
Introduction Projection-Based ROM Topology Optimization Nonlinear ROM Bottleneck Model Order Reduction ROM Precomputations Applications Reduced Topology Optimization Conclusion Internal Force The expression for the internal force is � ∂ N L f int jL = d X P ij ∂ X i Ω 0 where N I ( X ) is the shape function corresponding to node I and u i ( X ) = u iI N I ( X ) (FEM discretization) F = I + u ∂ N (Deformation Gradient) ∂ X E = 1 2( F T F − I ) (Green-Lagrange Strain) P = SF T (First Piola-Kirchhoff Stress) S = λ ( X )tr ( E ) I + 2 µ ( X ) E (Second Piola-Kirchhoff Stress) M. J. Zahr and C. Farhat
Introduction Projection-Based ROM Topology Optimization Nonlinear ROM Bottleneck Model Order Reduction ROM Precomputations Applications Reduced Topology Optimization Conclusion Internal Force - Cubic Polynomial in Displacements � ∂ N I f int jL = d X P ij ∂ X i Ω 0 = ¯ A jtIL u tI + ¯ B LI u jI + C LIJj u kI u kJ + � ¯ C ILQt u jQ u tI + ¯ D IJQL u kI u kJ u jQ where, A = ¯ ¯ A (Ω , λ ( X )) B = ¯ ¯ B (Ω , µ ( X )) C = ¯ ¯ C (Ω , λ ( X ) , µ ( X )) C = � � C (Ω , λ ( X ) , µ ( X )) D = ¯ ¯ D (Ω , λ ( X ) , µ ( X )) M. J. Zahr and C. Farhat
Introduction Projection-Based ROM Topology Optimization Nonlinear ROM Bottleneck Model Order Reduction ROM Precomputations Applications Reduced Topology Optimization Conclusion Material Representation Let material distributions be represented with the basis functions: λ ( X ) = φ λ i ( X ) α r i , i = 1 , 2 , . . . , n α µ ( X ) = φ µ i ( X ) α r i , i = 1 , 2 , . . . , n α ρ ( X ) = φ ρ i ( X ) α r i , i = 1 , 2 , . . . , n α . � � Then A = ¯ ¯ Ω , φ λ α r A i i B = ¯ ¯ B (Ω , φ µ i ) α r i C = ¯ ¯ C (Ω , φ λ i , φ µ i ) α r i C = � � i , φ µ C (Ω , φ λ i ) α r i D = ¯ ¯ i , φ µ D (Ω , φ λ i ) α r i M. J. Zahr and C. Farhat
Introduction Projection-Based ROM Topology Optimization Nonlinear ROM Bottleneck Model Order Reduction ROM Precomputations Applications Reduced Topology Optimization Conclusion Pre-computed ROM - cubic nonlinearity HDM jL = ¯ A jtIL u tI + ¯ B LI u jI + ¯ f int C LIJj u kI u kJ + � C ILQt u jQ u tI + ¯ D IJQL u kI u kJ u jQ ROM � � Φ T f int ( Φy ) r = β rp y p + γ rpq y p y q + ω rpqt y p y q y t i , φ µ β = β ( Φ , φ λ i ) α r i i , φ µ γ = γ ( Φ , φ λ i ) α r i i , φ µ ω = ω ( Φ , φ λ i ) α r i Φ T f int ( Φy ) = Φ T f ext M. J. Zahr and C. Farhat
Introduction Projection-Based ROM Topology Optimization Nonlinear ROM Bottleneck Model Order Reduction ROM Precomputations Applications Reduced Topology Optimization Conclusion ROM Pre-computation Approach Advantages Only need to solve small, cubic nonlinear system online Large speedups possible without hyperreduction, O (10 3 ) Amenable to 0-1 material topology optimization α r provide control over material distribution α r can be used as optimization variables Disadvantages Currently limited to StVK material, Lagrangian elements Offline cost scales as O ( n α · n el · k 4 u ) Offline storage scales as O ( n α · k 4 u ) Online storage scales as O ( k 4 u ) Can only vary material distribution in the subspace defined by the material snapshot vectors M. J. Zahr and C. Farhat
Introduction Projection-Based ROM Topology Optimization Nonlinear ROM Bottleneck Model Order Reduction ROM Precomputations Applications Reduced Topology Optimization Conclusion Reduced Topology Optimization minimize L ( y ( α r ) , α r ) α r ∈ R nα subject to c ( y ( α r ) , α r ) ≤ 0 y is implicitly defined as a function of α r through the ROM equation Φ T f int ( Φy ) = Φ T f ext M. J. Zahr and C. Farhat
Introduction Topology Optimization Cantilever Weight Minimization Model Order Reduction Wing Box Design Applications Conclusion Structural Simulation St. Venant-Kirchhoff 66,191 tetrahedral elements 13,110 nodes, 38,664 dof Static simulation with load applied in 10 increments Loads: Bending, Twisting, Self-Weight ROM size: k u = 5 M. J. Zahr and C. Farhat
Introduction Topology Optimization Cantilever Weight Minimization Model Order Reduction Wing Box Design Applications Conclusion Simulation Results Offline (s) Online (s) Speedup Error (%) HDM - 750 - - ROM 0.38 170 3.96 0.003 ROM-precomp 5,171 0.37 2,051 0.003 M. J. Zahr and C. Farhat
Introduction Topology Optimization Cantilever Weight Minimization Model Order Reduction Wing Box Design Applications Conclusion Optimization Setup Minimize structural weight Constraint on maximum vertical displacement 46 Material Snapshots 45 possible voids volume surrounding all possible voids Material Snapshots M. J. Zahr and C. Farhat
Introduction Topology Optimization Cantilever Weight Minimization Model Order Reduction Wing Box Design Applications Conclusion Optimization Results Optimization Iterates (Location of Voids) M. J. Zahr and C. Farhat
Introduction Topology Optimization Cantilever Weight Minimization Model Order Reduction Wing Box Design Applications Conclusion Optimization Results Deformed Configuration (Optimal Solution) Initial Guess Optimal Solution 2 . 776 × 10 6 2 . 588 × 10 6 Structural Weight 9 . 96 × 10 − 2 1 . 34 × 10 − 10 Constraint Violation M. J. Zahr and C. Farhat
Introduction Topology Optimization Cantilever Weight Minimization Model Order Reduction Wing Box Design Applications Conclusion Problem Setup St. Venant-Kirchhoff 90,799 tetrahedral elements 29,252 nodes, 86,493 dof Static simulation with load applied in 10 increments Loads: Bending (X- and Y- axis), Twisting, NACA0012 Self-Weight ROM size: k u = 5 M. J. Zahr and C. Farhat
Introduction Topology Optimization Cantilever Weight Minimization Model Order Reduction Wing Box Design Applications Conclusion Problem Setup St. Venant-Kirchhoff 90,799 tetrahedral elements 29,252 nodes, 86,493 dof Static simulation with load applied in 10 increments Loads: Bending (X- and Y- axis), Twisting, 40 Ribs Self-Weight ROM size: k u = 5 M. J. Zahr and C. Farhat
Introduction Topology Optimization Cantilever Weight Minimization Model Order Reduction Wing Box Design Applications Conclusion Problem Setup St. Venant-Kirchhoff 90,799 tetrahedral elements 29,252 nodes, 86,493 dof Static simulation with load applied in 10 increments Loads: Bending (X- and Y- axis), Twisting, Self-Weight ROM size: k u = 5 M. J. Zahr and C. Farhat
Introduction Topology Optimization Cantilever Weight Minimization Model Order Reduction Wing Box Design Applications Conclusion Simulation Results Offline (s) Online (s) Speedup Error (%) HDM - 811 - - ROM 1.01 376 2.16 0.002 ROM-precomp 9,603 1.51 538 1.73 M. J. Zahr and C. Farhat
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