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

1

Structural optimization basics

2

IGA for shape optimization

3

Shape sensitivity analysis methods

4

Search directions related issues with NURBS parametrization

5

Research trends

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 2 / 45

Outline

1

Structural optimization basics

2

IGA for shape optimization

3

Shape sensitivity analysis methods

4

Search directions related issues with NURBS parametrization

5

Research trends

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 =

• e
• Ωe BCBdΩ =
• e
• Ωe BCB|J|dχ

(1) Stiffness matrix variation: δx ⇒ δK ?

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 =

• e
• Ωe BCBdΩ =
• e
• Ωe BCB|J|dχ

Size optimization: δx ⇒ δC ⇒ δK with C = h¯ C Topology optimization: δx ⇒ δC ⇒ δK 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 =

• e
• Ωe BCBdΩ =
• e
• Ωe BCB|J|dχ

Shape optimization: δx ⇒ {δB, δJ} ⇒ δK

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 =

• e
• Ωe BCBdΩ =
• e
• Ωe BCB|J|dχ

Fixed background mesh: δx ⇒ δC ⇒ δK

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 =

• e
• Ωe BCBdΩ =
• e
• Ωe BCB|J|dχ

Shape optimization: δx ⇒ {δB, δJ} ⇒ δK

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 8 / 45

Outline

1

Structural optimization basics

2

IGA for shape optimization

3

Shape sensitivity analysis methods

4

Search directions related issues with NURBS parametrization

5

Research trends

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

1

Structural optimization basics

2

IGA for shape optimization

3

Shape sensitivity analysis methods

4

Search directions related issues with NURBS parametrization

5

Research trends

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[xj

i ]] with n design variables: xj i , i = 1, 2, 3, j = 1, 2, · · ·

Finite difference

DΨ Dxj

i

= Ψ[xj

i + ∆] − Ψ[xj i ]

∆ , by sovling KU = F for n + 1 times

Direct differential method

DΨ Dxj

i

= Ψ,U ˚ U, by solving KU = F once ˚ U = DU Dxj

i

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 13 / 45

Semi-analytical methods

DΨ Dxj

i

= Ψ,U ˚ U ˚ U = DU Dxj

i

= K −1 ∆F ∆xj

i

− ∆K ∆xj

i

U

• ,

by sovling K −1 once

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). ˚ U

B

− → ∇˚ u

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 14 / 45

Adjoint method

Optimization problem statement:

Objetive function Ψ

• s. t.

     c[u] := div C∇u + f = 0 in Ω (C∇u) n − ˆ t = 0

• n Γ

u − ˆ u = 0

• n Γ
• r KU = F

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) Note that ˚ U

∗(KU − F) = 0,

˚ ˜ Ψ = Ψ,U ˚ U + U∗T(−˚ KU − K ˚ U + ˚ F) = (Ψ,U − U∗TK)˚ U + ˚ F − U∗T ˚ KU Introducing an adjoint problem with U∗ that satisfies KU∗ = Ψ,U we have ˚ ˜ Ψ = ˚ F − U∗T ˚ 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 = −UT ˚ 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] :=

• Ωs ψω
• u[x; s]
• dΩ +
• Γs ψγ
• t[x; s], u[x; s]
• dΓ

BVP constraint:

     c[u] := div C∇u + f = 0 in Ω (C∇u) n − ˆ t = 0

• n Γ

u − ˆ u = 0

• n Γ

⇓⇓⇓ c[u], u∗Ωs = −

• Ωs C∇u · ∇u∗ dΩ +
• Ωs f · u∗ dΩ

+

• Γs

t

ˆ t · u∗ dΓ +

• Γs

u

t · u∗ dΓ = 0

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

∂s [p; s]

• p = Dh

Ds

Spatial/partial derivative: h′[x; s] := ∂h

∂s [x; s]

• x = ∂h

∂s

Design velocity: ν[p; s] := ˚ ˆ x[p; s] = ∂ˆ

x ∂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 ds

• Ωs f dΩ =
• Ωs f ′dΩ +
• Γs f ν · n dΓ

Boundary d ds

• Γs hdΓ =
• Γs
• ˚

h − κhν · n

• dΓ

κ := −divΓn

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] :=

• Ωs ψω
• u[x; s]
• dΩ +
• Γs ψγ
• t[x; s], u[x; s]
• dΓ

BVP constraint:

c[u], u∗Ωs = −

• Ωs C∇u · ∇u∗ dΩ +
• Ωs f · u∗ dΩ

+

• Γs

t

ˆ t · u∗ dΓ +

• Γs

u

t · u∗ dΓ = 0

Augmented function

˜ Ψ := Ψ + c[u], u∗Ωs

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 21 / 45

Adjoint method – continuous approach

Derivatives:

dΨ ds =

• Ωs ψω,uu′ dΩ +
• Γs ψωνn dΓ +
• Γs(∇ψγ · nνn − ψγκνn) dΓ

+

• Γs

t

ψγ,uu′ dΓ +

• Γs

u

ψγ,u ˆ u′ dΓ +

• Γs

t

ψγ,ˆ

tˆ

t′ dΓ +

• Γs

u

ψγ,tt′ dΓ ∂ ∂s c[u], u∗ =

• Ωs
• −C∇u′ · ∇u∗ − C∇u · ∇u∗′ + f ′ · u∗ + f · u∗′

dΩ −

• Γs
• C∇u · ∇u∗ − f · u∗

νn dΓ +

• Γs

t

• ˆ

t′ · u∗ + ˆ t · u∗′ dΓ +

• Γs

u

• t′ · u∗ + t · u∗′

dΓ +

• Γs
• ∇ (t · u∗) nνn − (t · u∗) κνn
• dΓ

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 22 / 45

Adjoint method – continuous approach

Derivatives:

Note c[u], u∗′ = 0,

• Ωs C∇u′ · ∇u∗ dΩ =
• Γs t∗ · u′dΓ −
• Ωs C∇2u∗ · u′ dΩ, we have

D˜ Ψ Ds = Φ1 + Φ2, where

Φ1 =

• Ωs
• ψω,u + C∇2u∗

· u′ dΩ +

• Γs

t

(ψγ,u − t∗) · u′ dΓ +

• Γs

u

(ψγ,t + u∗) · t′ dΓ Φ2 =

• Γs
• ψω − C∇u · ∇u∗ + f · u∗

νn dΓ +

• Ωs f ′ · u∗dΩ

+

• Γs
• (∇ψγ · nνn − ψγκνn) + ∇ (t · u∗) · nνn − (t · u∗) κνn
• dΓ

+

• Γs

u

(ψγ,u − t∗) · ˆ u′ dΓ +

• Γs

t

• ψγ,ˆ

t + u∗

· ˆ t′ dΓ

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 23 / 45

Adjoint method – continuous approach

Adjoint model:

Introducing C∇2u∗ + f ∗ = 0 with f ∗ = ψω,u in Ωs; u∗ = ˆ u∗ with ˆ u∗ = −ψγ,t

• n

Γs

u;

t∗ = (C∇u∗)Tn = ˆ t∗ with ˆ t∗ = ψγ,u

• n

Γs

t.

such that Φ1 = 0 Eventually, D˜ Ψ Ds = DΨ Ds = Φ2

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 24 / 45

Adjoint method – continuous approach

Shape sensitivity:

DΨ Ds = Φ2 =

• Γs
• ψω − C∇u · ∇u∗ + f · u∗

νn dΓ +

• Ωs f ′ · u∗dΩ

+

• Γs
• (∇ψγ · nνn − ψγκνn) + ∇ (t · u∗) · nνn − (t · u∗) κνn
• dΓ

+

• Γs

u

(ψγ,u − t∗) · ˆ u′ dΓ +

• Γs

t

• ψγ,ˆ

t + u∗

· ˆ t′ dΓ ν = ˚ ˆ x =

• I

RI dxI[s] ds DΨ DxI =

• Γs
• ψω − C∇u · ∇u∗ + f · u∗

nRI dΓ +

• Ωs f ′ · u∗dΩ

+

• Γs
• (∇ψγ · n − ψγκ) + ∇ (t · u∗) · n − (t · u∗) κ
• nRI dΓ

+

• (ψγ,u − t∗) · ˆ

u′ dΓ + ψ

ˆ t + u∗

· ˆ t′ dΓ

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 25 / 45

Adjoint method – continuous approach

Shape sensitivity:

DΨ Ds = Φ2 =

• Γs
• ψω − C∇u · ∇u∗ + f · u∗

νn dΓ +

• Ωs f ′ · u∗dΩ

+

• Γs
• (∇ψγ · nνn − ψγκνn) + ∇ (t · u∗) · nνn − (t · u∗) κνn
• dΓ

+

• Γs

u

(ψγ,u − t∗) · ˆ u′ dΓ +

• Γs

t

• ψγ,ˆ

t + u∗

· ˆ t′ dΓ ν = ˚ ˆ x =

• I

RI dxI[s] ds

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 26 / 45

Adjoint method – continuous approach

Shape sensitivity:

DΨ DxI =

• Γs
• ψω − C∇u · ∇u∗ + f · u∗

nRI dΓ +

• Ωs f ′ · u∗dΩ

+

• Γs
• (∇ψγ · n − ψγκ) + ∇ (t · u∗) · n − (t · u∗) κ
• nRI dΓ

+

• Γs

u

(ψγ,u − t∗) · ˆ u′ dΓ +

• Γs

t

• ψγ,ˆ

t + u∗

· ˆ t′ dΓ

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 27 / 45

Adjoint method – continuous approach

Example: minimize structural compliance

Obj: Ψ[s] :=

• Γs

t ψγ dΓ with ψγ = t · u

t = ˆ t is design-independent, i.e., ˆ t′ = 0.

BVP constraint:

     c[u] := div C∇u + f = 0 with f = 0 in Ωs t = ˆ t = 0

• n Γs

t

u = ˆ u = 0

• n Γs

u

Adjoint model = primary model (self-adjoint)

     C∇2u∗ + f ∗ = 0 with f ∗ = ψω,u = 0 in Ωs; t∗ = (C∇u∗)Tn = ˆ t∗ with ˆ t∗ = ψγ,u = ˆ t

• n

Γs

t

u∗ = ˆ u∗ with ˆ u∗ = −ψγ,t = 0

• n

Γs

u.

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 28 / 45

Adjoint method – continuous approach

Example: minimizing the structural compliance

Shape sensitivity: DΨ DxI = −

• Γs C∇u · ∇u∗nRI dΓ +
• Γs

t

(∇ψγ · n − ψγκ)nRI dΓ +

• Γs
• ∇ (t · u∗) · n − (t · u∗) κ
• nRI dΓ

= −

• Γs

t

C∇u · ∇u∗nRI dΓ + 2

• Γs

t

• ∇ (t · u) · n − (t · u) κ
• nRI dΓ

Compared with the discrete approach:

˚ Ψ = −U∗T ˚ KU = −UT ˚ KU Which one is easier for you to compute??

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 29 / 45

Adjoint method: Question !

Discrete approach vs Continuous approach

For problems with design-dependent boundary conditions, which approach is easier ??

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 30 / 45

Adjoint method: Question !

Discrete approach vs Continuous approach

For problems with design-dependent boundary conditions, which approach is easier ?? The answers can be different for different people. In general, just choose the one you like.

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 30 / 45

Adjoint method – continuous approach

Some additional references about continuous adjoint method: [Dems and Mroz(1984)]: Variational approach by means of adjoint systems to structural optimization and sensitivity analysis—II: Structure shape variation, IJSS, 1984. [Choi and Kim(2005)]: Structural sensitivity analysis and optimization 1: Linear systems, 2006. [Arora(1993)]: An exposition of the material derivative approach for structural shape sensitivity analysis, CMAME, 1993. [Tortorelli and Haber(1989)]:First-order design sensitivities for transient conduction problems by an adjoint method, IJNME, 1989. [Wang and Turteltaub(2015)]: Isogeometric shape optimization for quasi-static processes, IJNME, 2015. [Wang et al.(2017c)Wang, Turteltaub, and Abdalla]: Shape optimization and optimal control for transient heat conduction problems using an isogeometric approach, C&S, 2017.

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 31 / 45

Outline

1

Structural optimization basics

2

IGA for shape optimization

3

Shape sensitivity analysis methods

4

Search directions related issues with NURBS parametrization

5

Research trends

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 32 / 45

Parameterization-dependency of the search directions

10 5 00 5 10 15 20 (a) (b) Design boundary 0 5 10 15 20 10 5

Example: volume reduction

Volume: Σ =

• Ω dΩ

Gradient (continuous): g = n Gradient (NURBS discretization): g I

d =

• Γ

nRI dΓ

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 33 / 45

Parameterization-dependency of the search directions

4 8 12 16 20 5 10 Updated shape Original shape Control points of updated shape Control points of original shape 4 8 12 16 20 5 10 Updated shape Original shape Control points of updated shape Control points of original shape 4 8 12 16 20 5 10 Updated shape Original shape Control points of updated shape Control points of original shape

Case 3 Case 2 Case 1

4 8 12 16 20 5 10 Updated shape Original shape Control points of updated shape Control points of original shape

Case 4 3 5 7 10 13 15 17 20 5 10

Updated shape Original shape Nodes of updated shape Nodes of original shape

Case 5

• xI(s+1) =
• xI(s) + αd I

d =

• xI(s) − αg I

d

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 34 / 45

Parameterization-dependency of the search directions

Parameterization-free approach for FE-based shape optimization [Le et al.(2011)Le, Bruns, and Tortorelli]

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 35 / 45

A simple example about the quadratic norm induced by discretization

A (squared) L2 norm

f

• x
• := x · x,

gradient: g = f,x = 2x steepest search direction: d = −g

Quadratic norm induced by discretization x = RTX

R is a vector of shape functions, X is a vector of discrete variables xI f = X TMX, with M = RRT gradient: g I = f,xI = 2MIJxJ, G = f,X = 2MX steepest search direction: Dn = −M−1G, Dn = [d 1

n, d 2 n, · · · ]

!! Dd = −G is NOT the steepest search direction !!

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 36 / 45

Steepest search directions of quadratic and (squared) L2 norms

(a) (b)

d d

Reproduced from [Boyd and Vandenberghe(2009)]: Convex Optimization

Consistency The normalized search direction of a discrete form is consistent with the steepest search direction of a continuous form.

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 37 / 45

Normalization approaches

• 1. Standard approach

Dn = −M−1G

• 2. DLMM normalization approach

The diagonally lumped mapping matrix (DLMM) ¯ MII :=

• J

MIJ = ¯ MII =

• D

RI dD, with

• J

RJ = 1 d I

n = − g I d

¯ MII = −

• D gRI dD
• D RI dD

”Sensitivity weighting” method in [Kiendl et al.(2014)Kiendl, Schmidt, WW¨ uchner, and Bletzinger].

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 38 / 45

Normalization approaches

• 3. B-Spline space ( ¯

D) normalization

d I

n ≈ −

• ¯

D gNI d ¯

D

• ¯

D NI d ¯

D

• 4. Simplified DLMM approach

Unity of integral property of B-spline basis

• Ni,pdξ

ξi+p+1 − ξi = 1 p + 1, d I

n = −(p + 1)

• ¯

D gNI d ¯

D ξi+p+1 − ξi , More information in [Wang et al.(2017a)Wang, Abdalla, and Turteltaub].

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 39 / 45

Effectiveness of the simplified DLMM approach

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 40 / 45

Effectiveness of the simplified DLMM approach

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 41 / 45

Normalization approaches

References: [Wang et al.(2017a)Wang, Abdalla, and Turteltaub]: Normalization approaches for the descent search direction in isogeometric shape

• ptimization, CAD, 2017.

[Kiendl et al.(2014)Kiendl, Schmidt, WW¨ uchner, and Bletzinger]: Isogeometric shape optimization of shells using semi-analytical sensitivity analysis and sensitivity weighting, CMAME, 2014. [Boyd and Vandenberghe(2009)]: Convex optimization, Cambridge University Press, 2009. [Wang and Kumar(2017)]:On the numerical implementation of continuous adjoint sensitivity for transient heat conduction problems using an isogeometric approach, SMO, 2017.

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 42 / 45

Outline

1

Structural optimization basics

2

IGA for shape optimization

3

Shape sensitivity analysis methods

4

Search directions related issues with NURBS parametrization

5

Research trends

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 43 / 45

Research trends

Shape optimization techniques

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 44 / 45

Research trends

Shape optimization techniques Special applications of isogeometric shape optimization, e.g.,

Auxetic structures design [Wang et al.(2017b)Wang, Poh, Dirrenberger, Zhu, and Forest] Curved (laminated) shells [Kiendl et al.(2014)Kiendl, Schmidt, WW¨ uchner, and Bletzinger, Nagy et al.(2013)Nagy, IJsselmuiden, and Abdalla]

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 44 / 45

Research trends

Shape optimization techniques Special applications of isogeometric shape optimization, e.g.,

Auxetic structures design [Wang et al.(2017b)Wang, Poh, Dirrenberger, Zhu, and Forest] Curved (laminated) shells [Kiendl et al.(2014)Kiendl, Schmidt, WW¨ uchner, and Bletzinger, Nagy et al.(2013)Nagy, IJsselmuiden, and Abdalla]

Shape optimization using new analysis techniques, e.g.,

Trimmed spline surface [Seo et al.(2010)Seo, Kim, and Youn] B´ ezier triangle based isogeometric shape optimization [Wang et al.(2018)Wang, Xia, Wang, and Qian] Level set-based topology optimization [Cai et al.(2014)Cai, Zhang, Zhu, and Gao]

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 44 / 45

Acknowledgements

Thank you for your attention! −−

With special thanks to

• Prof. Qian Xiaoping for his invitation and
• Prof. Xu Gang for organizing this webinar.

−− This note may contain errors because of my limited knowledge about related topics. Please feel free to contact me if you find any mistakes/errors in it. Thank you.

Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 45 / 45

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