Introduction Topology Optimization Model Order Reduction Applications Conclusion Rapid Nonlinear Topology Optimization using Precomputed Reduced-Order Models Matthew J. Zahr and Charbel Farhat Farhat Research Group Stanford University 17th U.S. National Congress on Theoretical and Applied Mechanics Michigan State University June 15 - 20, 2014 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 (structural displacements) is implicitly defined as a function of χ through the HDM equation f int ( u ) = f ext � 0 , e / ∈ Ω ∗ C e = C e ρ e = ρ e χ e = 0 χ e 0 χ e e ∈ Ω ∗ 1 , General nonlinear setting considered (geometric and material nonlinearities) 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 Approximation of reduced internal force, Φ T f int ( Φy ) For general nonlinear problems, high-dimensional quantities cannot be precomputed since they change at every iteration For polynomial nonlinearities, there is an opportunity for precomputation Approach Approximate f r = Φ T f int ( Φy ) by polynomial via Taylor series We choose a third-order series Exact representation of reduced internal force for St. Venant-Kirchhoff materials Precompute coefficient tensors Online operations will only involve small quantities Remove online bottleneck Pay price in offline phase 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 Taylor Series of Φ T f int ( Φy ) Consider Taylor series expansion of f r ( y ) = Φ T f int ( Φy ) about ¯ y y ) + ∂ f r f r i ( y ) ≈ f r i i (¯ (¯ y ) · ( y − ¯ y ) j ∂ y j ∂ 2 f r + 1 i (¯ y ) · ( y − ¯ y ) j ( y − ¯ y ) k 2 ∂ y j ∂ y k ∂ 3 f r + 1 i (¯ y ) · ( y − ¯ y ) j ( y − ¯ y ) k ( y − ¯ y ) l 6 ∂ y j ∂ y k ∂ y l 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 Derivatives Reduced derivatives computable by: Projection of full order derivatives Directly via finite differences Φ pi f int α i = f r i (¯ y ) = p ( Φ ¯ y ) ∂ f int β ij = ∂ f r p i (¯ y ) = ( Φ ¯ y ) Φ pi Φ qj ∂ y j ∂ u q ∂ f int ∂ 2 f r p i γ ijk = (¯ y ) = Φ pi Φ qj Φ rk ( Φ ¯ y ) ∂ y j ∂ y k ∂ u q ∂ u r ∂ f int ∂ 3 f r p i ω ijkl = (¯ y ) = ( Φ ¯ y ) Φ pi Φ qj Φ rk Φ sl ∂ y j ∂ y k ∂ y l ∂ u q ∂ u r ∂ u s 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 internal force Reduced internal force becomes f r i ( y ) = α i + β ij ( y − ¯ y ) j + 1 2 γ ijk ( y − ¯ y ) j ( y − ¯ y ) k + 1 6 ω ijkl ( y − ¯ y ) j ( y − ¯ y ) k ( y − ¯ y ) l , which only depends on quantities scaling with the reduced dimension. 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 internal force - material dependence As written, the material properties for a given material are baked into the polynomial coefficients For notational simplicity, we consider two material parameters: ρ (density) and η α = α ( ρ, η ) β = β ( ρ, η ) γ = γ ( ρ, η ) ω = ω ( ρ, η ) In the context of 0-1 topology optimization, α , β , γ , ω need to be recomputed at each new distribution of ρ, η Extremely expensive – destroy all speedup potential 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 Recall the material parameters are spatial distributions , i.e. ρ = ρ ( X ) and η = η ( X ) Define admissible distributions: { φ ρ i =1 , { φ η i } n i } n i =1 Require ρ ( X ) = φ ρ i ( X ) ξ i η ( X ) = φ η i ( X ) ξ i Many possible choices admissible distributions Here, collected via configuration snapshots 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 internal force - material dependence Suppose the coefficient matrices depend linearly on material parameters Can be accomplished by carefully choosing parameters (i.e. λ, µ instead of E, ν ) or linearization via Taylor series Use material assumptions in reduced internal force � f r α i ( φ ρ a , φ η i ( y ) = a ) ξ a a � β ij ( φ ρ a , φ η + a ) ξ a ( y − ¯ y ) j a + 1 � γ ijk ( φ ρ a , φ η a ) ξ a ( y − ¯ y ) j ( y − ¯ y ) k 2 a + 1 � ω ijkl ( φ ρ a , φ η a ) ξ a ( y − ¯ y ) j ( y − ¯ y ) k ( y − ¯ y ) l 6 a Quantities in blue can be precomputed offline 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 Φ T f int ( Φy ) = Φ T f ext Advantages Only need to solve small, cubic nonlinear system online Large speedups possible without hyperreduction, O (10 2 ) Amenable to 0-1 material topology optimization Disadvantages 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 ˆ minimize L ( y ( ξ ) , ξ ) ξ ∈ R n subject to ˆ c ( y ( ξ ) , ξ ) ≤ 0 y is implicitly defined as a function of ξ through the ROM equation Φ T f int ( Φy ) = Φ T f ext which can be computed efficiently M. J. Zahr and C. Farhat
Introduction Topology Optimization Model Order Reduction Wing Box Design Applications Conclusion Problem Setup Neo-Hookean material 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 Model Order Reduction Wing Box Design Applications Conclusion Problem Setup Neo-Hookean material 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 Model Order Reduction Wing Box Design Applications Conclusion Problem Setup Neo-Hookean material 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
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