Motivation ROM-Constrained Optimization Numerical Experiments Conclusion Accelerating PDE-Constrained Optimization using Adaptive Reduced-Order Models: Application to Topology Optimization Matthew J. Zahr Farhat Research Group Stanford University Robert J. Melosh Medal Competition, Duke University April 24, 2015 Zahr Topology Optimization with ROMs
Motivation ROM-Constrained Optimization Numerical Experiments Conclusion Overview Finite Element Analysis Reduced Topology Optimization Topology Optimization Model Reduction Optimization Theory Zahr Topology Optimization with ROMs
Motivation ROM-Constrained Optimization Numerical Experiments Conclusion Overview Finite Element Analysis Reduced Topology Optimization Topology Optimization Model Reduction Optimization Theory Zahr Topology Optimization with ROMs
Motivation ROM-Constrained Optimization Numerical Experiments Conclusion Overview Finite Element Analysis Reduced Topology Optimization Topology Optimization Model Reduction Optimization Theory Zahr Topology Optimization with ROMs
Motivation ROM-Constrained Optimization Numerical Experiments Conclusion Overview Finite Element Analysis Reduced Topology Optimization Topology Optimization Model Reduction Optimization Theory Zahr Topology Optimization with ROMs
Motivation ROM-Constrained Optimization Numerical Experiments Conclusion Overview Finite Element Analysis Reduced Topology Optimization Topology Optimization Model Reduction Optimization Theory Zahr Topology Optimization with ROMs
Motivation ROM-Constrained Optimization Numerical Experiments Conclusion Overview Finite Element Analysis Reduced Topology Optimization Topology Optimization Model Reduction Optimization Theory Zahr Topology Optimization with ROMs
Model Order Reduction Motivation Parameter Space Reduction ROM-Constrained Optimization Reduced Topology Optimization Numerical Experiments Reduced Order Basis Adaptivity: Φu Conclusion Reduced Order Basis Adaptivity: Φ µ Problem Formulation Goal: Rapidly solve PDE-constrained optimization problem of the form minimize J ( u , µ ) u ∈ R n u , µ ∈ R n µ subject to c ( u , µ ) ≥ 0 r ( u , µ ) = 0 A µ ≥ b where r : R n u × R n µ → R n u is the discretized (steady, nonlinear) PDE J : R n u × R n µ → R is the objective function c : R n u × R n µ → R n c are the side constraints A ∈ R n A × n µ , b ∈ R n A are linear constraints (independent of u ) u ∈ R n u is the PDE state vector µ ∈ R n µ is the vector of parameters red indicates a large quantity (i.e. scales with size of FE mesh) blue indicates a small quantity (i.e. size independent of size of FE mesh) Zahr Topology Optimization with ROMs
Model Order Reduction Motivation Parameter Space Reduction ROM-Constrained Optimization Reduced Topology Optimization Numerical Experiments Reduced Order Basis Adaptivity: Φu Conclusion Reduced Order Basis Adaptivity: Φ µ Problem Setup 16000 8-node brick elements, 77760 dofs Total Lagrangian form, finite strain, StVK 1 St. Venant-Kirchhoff material 25 Sparse Cholesky linear solver (CHOLMOD 2 ) Newton-Raphson nonlinear solver Minimum compliance optimization problem 40 T u minimize f ext u ∈ R n u , µ ∈ R n µ V ( µ ) ≤ 1 subject to 2 V 0 r ( u , µ ) = 0 Gradient computations: Adjoint method Optimizer: SNOPT [Gill et al., 2002] 1 [Bonet and Wood, 1997, Belytschko et al., 2000] 2 [Chen et al., 2008] Zahr Topology Optimization with ROMs
Model Order Reduction Motivation Parameter Space Reduction ROM-Constrained Optimization Reduced Topology Optimization Numerical Experiments Reduced Order Basis Adaptivity: Φu Conclusion Reduced Order Basis Adaptivity: Φ µ Projection-Based Model Reduction Model Order Reduction (MOR) assumption: state vector lies in low-dimensional subspace u ≈ Φ u u r where ∈ R n u × k u is the reduced basis φ 1 φ k u � � Φ u = · · · u u u r ∈ R k u are the reduced coordinates of u n u ≫ k u Substitute assumption into High-Dimensional Model (HDM), r ( u , µ ) = 0, and apply Galerkin projection T r ( Φ u u r , µ ) = 0 ˆ r r ( u r , µ ) = Φ u Zahr Topology Optimization with ROMs
Model Order Reduction Motivation Parameter Space Reduction ROM-Constrained Optimization Reduced Topology Optimization Numerical Experiments Reduced Order Basis Adaptivity: Φu Conclusion Reduced Order Basis Adaptivity: Φ µ Connection to Finite Element Method S S - infinite-dimensional trial space Zahr Topology Optimization with ROMs
Model Order Reduction Motivation Parameter Space Reduction ROM-Constrained Optimization Reduced Topology Optimization Numerical Experiments Reduced Order Basis Adaptivity: Φu Conclusion Reduced Order Basis Adaptivity: Φ µ Connection to Finite Element Method S h S S - infinite-dimensional trial space S h - (large) finite-dimensional trial space Zahr Topology Optimization with ROMs
Model Order Reduction Motivation Parameter Space Reduction ROM-Constrained Optimization Reduced Topology Optimization Numerical Experiments Reduced Order Basis Adaptivity: Φu Conclusion Reduced Order Basis Adaptivity: Φ µ Connection to Finite Element Method S k h S h S S - infinite-dimensional trial space S h - (large) finite-dimensional trial space S k h - (small) finite-dimensional trial space S k h ⊂ S h ⊂ S Zahr Topology Optimization with ROMs
Model Order Reduction Motivation Parameter Space Reduction ROM-Constrained Optimization Reduced Topology Optimization Numerical Experiments Reduced Order Basis Adaptivity: Φu Conclusion Reduced Order Basis Adaptivity: Φ µ Reduced Basis Construction Method of Snapshots [Sirovich, 1987] Collect state snapshots by sampling parameter space: u ( µ ) � u ( µ 1 ) u ( µ n ) � · · · X = Proper Orthogonal Decomposition (POD) [Sirovich, 1987, Holmes et al., 1998] Compress snapshot matrix using POD, or truncated Singular Value Decomposition (SVD) Φ u = POD( X ) Trial subspace selection Finite element method: polynomial basis; local support Rayleigh-Ritz: analytical, empirical basis functions; global support POD: data-driven, empirical basis functions; global support Zahr Topology Optimization with ROMs
Model Order Reduction Motivation Parameter Space Reduction ROM-Constrained Optimization Reduced Topology Optimization Numerical Experiments Reduced Order Basis Adaptivity: Φu Conclusion Reduced Order Basis Adaptivity: Φ µ Restriction of Parameter Space Parameter restriction: restrict parameter to a low-dimensional subspace µ ≈ Φ µ µ r � � ∈ R n µ × k µ is the reduced basis k µ φ 1 Φ µ = · · · φ µ µ µ r ∈ R k µ are the reduced coordinates of µ n µ ≫ k µ Substitute restriction into Reduced-Order Model, ˆ r r ( u r , µ ) = 0 to obtain T r ( Φ u u r , Φ µ µ r ) = 0 r r ( u r , µ r ) = Φ u Related work: [Maute and Ramm, 1995, Lieberman et al., 2010, Constantine et al., 2014] Zahr Topology Optimization with ROMs
Model Order Reduction Motivation Parameter Space Reduction ROM-Constrained Optimization Reduced Topology Optimization Numerical Experiments Reduced Order Basis Adaptivity: Φu Conclusion Reduced Order Basis Adaptivity: Φ µ Restriction of Parameter Space Parameter space Cantilever mesh Zahr Topology Optimization with ROMs
Model Order Reduction Motivation Parameter Space Reduction ROM-Constrained Optimization Reduced Topology Optimization Numerical Experiments Reduced Order Basis Adaptivity: Φu Conclusion Reduced Order Basis Adaptivity: Φ µ Restriction of Parameter Space Parameter space Macroelements Zahr Topology Optimization with ROMs
Model Order Reduction Motivation Parameter Space Reduction ROM-Constrained Optimization Reduced Topology Optimization Numerical Experiments Reduced Order Basis Adaptivity: Φu Conclusion Reduced Order Basis Adaptivity: Φ µ Standard Difficulty: Binary Solutions (a) Without penalization Zahr Topology Optimization with ROMs
Model Order Reduction Motivation Parameter Space Reduction ROM-Constrained Optimization Reduced Topology Optimization Numerical Experiments Reduced Order Basis Adaptivity: Φu Conclusion Reduced Order Basis Adaptivity: Φ µ Standard Difficulty: Binary Solutions Relaxed, Penalized Problem Setup T u minimize f ext u ∈ R n u , µ ∈ R n µ V ( µ ) ≤ 1 subject to 2 V 0 r ( u , µ p ) = 0 (a) Without penalization µ ∈ [0 , 1] k µ Effect of Penalization K e ← ( µ e ) p K e K e : e th element stiffness matrix Zahr Topology Optimization with ROMs
Model Order Reduction Motivation Parameter Space Reduction ROM-Constrained Optimization Reduced Topology Optimization Numerical Experiments Reduced Order Basis Adaptivity: Φu Conclusion Reduced Order Basis Adaptivity: Φ µ Standard Difficulty: Binary Solutions Relaxed, Penalized Problem Setup T u minimize f ext u ∈ R n u , µ ∈ R n µ V ( µ ) ≤ 1 subject to 2 V 0 r ( u , µ p ) = 0 (a) Without penalization µ ∈ [0 , 1] k µ Effect of Penalization K e ← ( µ e ) p K e K e : e th element stiffness matrix (b) With penalization Zahr Topology Optimization with ROMs
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