Isogeometric Shape Optimization: A brief introduction about shape - PowerPoint PPT Presentation

Isogeometric Shape Optimization: A brief introduction about shape sensitivity analysis and search direction normalization Wang Zhenpei NUS 2018 年 3 月 23 日 GAMES Webinar 2018 (37) on IGA Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 1 / 45

Table of contents Structural optimization basics 1 IGA for shape optimization 2 Shape sensitivity analysis methods 3 Search directions related issues with NURBS parametrization 4 Research trends 5 Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 2 / 45

Outline Structural optimization basics 1 IGA for shape optimization 2 Shape sensitivity analysis methods 3 Search directions related issues with NURBS parametrization 4 Research trends 5 Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 3 / 45

Size, shape and topology optimization [Bendsøe and Sigmund(2004)] Stiffness matrix: � � � � K = Ω e B C B dΩ = Ω e B C B | J | d χ (1) e e Stiffness matrix variation: δ x ⇒ δ K ? Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 4 / 45

Size, shape and topology optimization [Bendsøe and Sigmund(2004)] Stiffness matrix: � � � � K = Ω e B C B dΩ = Ω e B C B | J | d χ e e Size optimization: δ x ⇒ δ C ⇒ δ K with C = h ¯ C Topology optimization: δ x ⇒ δ C ⇒ δ K with C = ρ ¯ C Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 5 / 45

Size, shape and topology optimization [Bendsøe and Sigmund(2004)] Stiffness matrix: � � � � K = Ω e B C B dΩ = Ω e B C B | J | d χ e e Shape optimization: δ x ⇒ { δ B , δ J } ⇒ δ K Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 6 / 45

Topology optimization using shape optimization techniques [Wang et al.(2003)Wang, Wang, and Guo] Stiffness matrix: � � � � K = Ω e B C B dΩ = Ω e B C B | J | d χ e e Fixed background mesh: δ x ⇒ δ C ⇒ δ K Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 7 / 45

Shape optimization by changing size parameters [WANG et al.(2011)WANG, WANG, ZHU, and ZHANG] Stiffness matrix: � � � � K = Ω e B C B dΩ = Ω e B C B | J | d χ e e Shape optimization: δ x ⇒ { δ B , δ J } ⇒ δ K Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 8 / 45

Outline Structural optimization basics 1 IGA for shape optimization 2 Shape sensitivity analysis methods 3 Search directions related issues with NURBS parametrization 4 Research trends 5 Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 9 / 45

IGA for shape optimization Advantages: Seamless integration between CAD and CAE Direct geometry updating Meshing and re-meshing is easy Curved features are preserved Enhanced sensitivity analysis High order derivatives More accurate structural response Easily accessible geometry informations such as normal vector, curvature... Double levels discretization for design and analysis e.g., coarse mesh for design & refined mesh for analysis References: [Cho and Ha(2009)], [Qian(2010)], [Nagy et al.(2010)Nagy, Abdalla, and G¨ urdal]. Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 10 / 45

Outline Structural optimization basics 1 IGA for shape optimization 2 Shape sensitivity analysis methods 3 Search directions related issues with NURBS parametrization 4 Research trends 5 Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 11 / 45

Basic modules in a shape optimization problem Optimizer: Update design variables GA, Steepest descent, SQP, MMA, GCMMA, ... Sensitivity analysis: Compute the derivatives of the obj./cons. w.r.t. design variables Finite difference, Direct difference, Semi-analytical, adjoint method... Supplementary processing: Search direction regularization/normalization Mesh updating Mesh regularization/smoothing ... Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 12 / 45

Finite difference and direct differential methods Optimization problem Obj. Ψ[ u [ x j i ]] with n design variables: x j i , i = 1 , 2 , 3 , j = 1 , 2 , · · · Finite difference = Ψ[ x j i + ∆] − Ψ[ x j i ] DΨ , by sovling KU = F for n + 1 times D x j ∆ i Direct differential method DΨ = Ψ , U ˚ U , by solving KU = F once D x j i U = D U ˚ D x j i Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 13 / 45

Semi-analytical methods DΨ = Ψ , U ˚ U D x j i = K − 1 � ∆ F U = D U − ∆ K � by sovling K − 1 once ˚ U , D x j ∆ x j ∆ x j i i i Remark: spatial and material design derivatives of strain/stress ǫ ′ [ u ] = ( ∇ u ) ′ = ∇ ( u ′ ) = ǫ [ u ′ ]; ǫ [ u ] = ˚ ∇ u = ( ∇ u ) ′ + ∇ ( ∇ u ) v = ∇ ˚ ˚ u − ( ∇ u )( ∇ v ) = ǫ [˚ u ] − ( ∇ u )( ∇ v ) . B ˚ − → ∇ ˚ U u Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 14 / 45

Adjoint method Optimization problem statement: Objetive function Ψ  c [ u ] := div C ∇ u + f = 0 in Ω   ( C ∇ u ) n − ˆ s. t. t = 0 on Γ or KU = F  u − ˆ u = 0 on Γ  Discrete approach Discretize the problem first, then derive the formulation: Ψ[ U ] , KU = F Continuous approach Derive the formulation first as a continuum, then distretize the formulation and compute: Ψ[ u ] , BVP formulation Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 15 / 45

Adjoint method – discrete approach Optimization problem statement: Objetive function Ψ s. t. KU = F Augmented formulation ˜ Ψ = Ψ = Ψ + U ∗ T ( − KU + F ) ∗ ( KU − F ) = 0, Note that ˚ U ˚ Ψ = Ψ , U ˚ ˜ U + U ∗ T ( − ˚ KU − K ˚ U + ˚ F ) F − U ∗ T ˚ = (Ψ , U − U ∗ T K )˚ U + ˚ KU Introducing an adjoint problem with U ∗ that satisfies KU ∗ = Ψ , U we have ˚ F − U ∗ T ˚ Ψ = ˚ ˜ KU Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 16 / 45

Adjoint method – discrete approach Example: minimizing structure compliance min Ψ := FU s. t. KU = F with ˚ F = 0 (Design-independent load) Adjoint problem KU ∗ = Ψ , U = F ⇒ U ∗ = U (self-adjoint problem) Shape sensitivity ˚ Ψ = − U ∗ T ˚ KU = − U T ˚ ˜ KU Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 17 / 45

Adjoint method – continuous approach Objective function[Wang and Turteltaub(2015)]: � � � � � � Ψ[ s ] := Ω s ψ ω u [ x ; s ] dΩ + Γ s ψ γ t [ x ; s ] , u [ x ; s ] dΓ BVP constraint:  c [ u ] := div C ∇ u + f = 0 in Ω   ( C ∇ u ) n − ˆ t = 0 on Γ  u − ˆ u = 0 on Γ  ⇓⇓⇓ � � Ω s C ∇ u · ∇ u ∗ dΩ + Ω s f · u ∗ dΩ � c [ u ] , u ∗ � Ω s = − � � t · u ∗ dΓ + t · u ∗ dΓ = 0 ˆ + Γ s Γ s t u Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 18 / 45

Adjoint method – continuous approach Material and spatial derivatives � Material/full derivative: ˚ h [ p ; s ] := ∂ h p = D h ∂ s [ p ; s ] � D s � � Spatial/partial derivative: h ′ [ x ; s ] := ∂ h x = ∂ h ∂ s [ x ; s ] � ∂ s � Design velocity: ν [ p ; s ] := ˚ x [ p ; s ] = ∂ ˆ x � ˆ ∂ s [ p ; s ] � p Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 19 / 45

Adjoint method – continuous approach Transport relations Volume d � � � Ω s f dΩ = Ω s f ′ dΩ + Γ s f ν · n dΓ d s Boundary d � � � � ˚ Γ s h dΓ = h − κ h ν · n dΓ κ := − div Γ n d s Γ s Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 20 / 45

Adjoint method – continuous approach Objective function[Wang and Turteltaub(2015)]: � � � � � � Ψ[ s ] := u [ x ; s ] dΩ + t [ x ; s ] , u [ x ; s ] dΓ Ω s ψ ω Γ s ψ γ BVP constraint: � � Ω s C ∇ u · ∇ u ∗ dΩ + Ω s f · u ∗ dΩ � c [ u ] , u ∗ � Ω s = − � � t · u ∗ dΓ + t · u ∗ dΓ = 0 ˆ + Γ s Γ s t u Augmented function ˜ Ψ := Ψ + � c [ u ] , u ∗ � Ω s Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 21 / 45

Adjoint method – continuous approach Derivatives: dΨ � � � Ω s ψ ω, u u ′ dΩ + d s = Γ s ψ ω ν n dΓ + Γ s ( ∇ ψ γ · n ν n − ψ γ κν n ) dΓ � � � � t ′ dΓ + ψ γ, u u ′ dΓ + u ′ dΓ + ψ γ, t t ′ dΓ t ˆ ψ γ, u ˆ + ψ γ, ˆ Γ s Γ s Γ s Γ s t u t u ∂ � − C ∇ u ′ · ∇ u ∗ − C ∇ u · ∇ u ∗′ + f ′ · u ∗ + f · u ∗′ � � ∂ s � c [ u ] , u ∗ � = dΩ Ω s � C ∇ u · ∇ u ∗ − f · u ∗ � � − ν n dΓ Γ s � � t ′ · u ∗ + ˆ � t · u ∗′ � � t ′ · u ∗ + t · u ∗′ � ˆ + dΓ + dΓ Γ s Γ s t u � � ∇ ( t · u ∗ ) n ν n − ( t · u ∗ ) κν n � + dΓ Γ s Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 22 / 45