Isogeometric Shape OptimizationŽ‹振培.pdf · Isogeometric Shape Optimization: A brief introduction about shape sensitivity analysis and search direction normalization Wang Zhenpei

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


Outline





5 Research trends


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 ?




Stiffness matrix:

K =∑e

∫Ωe

BCBdΩ =∑e

∫Ωe

BCB|J |dχ

Size optimization: δx ⇒ δC⇒ δK with C = hCTopology optimization: δx ⇒ δC⇒ δK with C = ρC




Stiffness matrix:

K =∑e

∫Ωe

BCBdΩ =∑e

∫Ωe

BCB|J |dχ

Shape optimization: δx ⇒ δB, δJ ⇒ δK


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


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

Outline





5 Research trends


IGA for shape optimization

Advantages:

Seamless integration between CAD and CAE

Direct geometry updatingMeshing and re-meshing is easyCurved features are preserved

Enhanced sensitivity analysis

High order derivativesMore accurate structural responseEasily 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 Gurdal].


Outline





5 Research trends


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

...


Finite difference and direct differential methods

Optimization problem

Obj. Ψ[u[x ji ]] with n design variables: x ji , i = 1, 2, 3, j = 1, 2, · · ·

Finite difference

DΨ

Dx ji=

Ψ[x ji + ∆]−Ψ[x ji ]

∆, by sovling KU = F for n + 1 times

Direct differential method

DΨ

Dx ji= Ψ,UU , by solving KU = F once

U =DUDx ji


Semi-analytical methods

DΨ

Dx ji= Ψ,UU

U =DUDx ji

= K−1[∆F

∆x ji− ∆K

∆x jiU], by sovling K−1 once

Remark: spatial and material design derivatives of strain/stress

ε′[u] = (∇u)′ = ∇(u ′) = ε[u ′];

ε[u] = ∇u = (∇u)′ +∇(∇u)v = ∇u − (∇u)(∇v) = ε[u]− (∇u)(∇v).

U B−→ ∇u


Adjoint method

Optimization problem statement:

Objetive function Ψ

s. t.

c[u] := divC∇u + f = 0 in Ω

(C∇u) n − t = 0 on Γ

u − u = 0 on Γ

or 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 formulationand compute:

Ψ[u], BVP formulation


Adjoint method – discrete approach

Optimization problem statement:

Objetive function Ψs. t. KU = F

Augmented formulation

Ψ = Ψ = Ψ + U∗T(−KU + F )

Note that U∗(KU − F ) = 0,

˚Ψ = Ψ,UU + U∗T(−KU −KU + F )

= (Ψ,U −U∗TK )U + F −U∗TKU

Introducing an adjoint problem with U∗ that satisfies KU∗ = Ψ,U we have

˚Ψ = F −U∗TKU


Adjoint method – discrete approach

Example: minimizing structure compliance

min Ψ := FUs. t. KU = F with F = 0 (Design-independent load)

Adjoint problem

KU∗ = Ψ,U = F⇒ U∗ = U (self-adjoint problem)

Shape sensitivity

˚Ψ = −U∗TKU = −UTKU


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] := divC∇u + f = 0 in Ω

(C∇u) n − t = 0 on Γ

u − u = 0 on Γ

⇓⇓⇓

〈c[u],u∗〉Ωs =−∫

Ωs

C∇u · ∇u∗ dΩ +

∫Ωs

f · u∗ dΩ

+

∫Γs

t

t · u∗ dΓ +

∫Γs

u

t · u∗ dΓ = 0



Material and spatial derivatives

Material/full derivative: h[p; s] := ∂h∂s [p; s]

∣∣∣p

= DhDs

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

∣∣∣x

= ∂h∂s

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

∣∣p



Transport relations

Volumed

ds

∫Ωs

f dΩ =

∫Ωs

f ′dΩ +

∫Γs

f ν · n dΓ

Boundary

d

ds

∫Γs

hdΓ =

∫Γs

(h − κhν · n

)dΓ κ := −divΓn



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

t · u∗ dΓ +

∫Γs

u

t · u∗ dΓ = 0

Augmented function

Ψ := Ψ + 〈c[u],u∗〉Ωs



Derivatives:

dΨ

ds=

∫Ωs

ψω,uu ′ dΩ +

∫Γs

ψωνn dΓ +

∫Γs

(∇ψγ · nνn − ψγκνn) dΓ

+

∫Γs

t

ψγ,uu ′ dΓ +

∫Γs

u

ψγ,u u ′ dΓ +

∫Γs

t

ψγ,t 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

(t ′ · u∗ + t · u∗′

)dΓ +

∫Γsu

(t ′ · u∗ + t · u∗′

)dΓ

+

∫Γs

(∇ (t · u∗) nνn − (t · u∗)κνn

)dΓ



Derivatives:

Note 〈c[u],u∗′〉 = 0,∫Ωs C∇u ′ · ∇u∗ dΩ =

∫Γs t∗ · u ′dΓ−

∫Ωs C∇2u∗ · u ′ dΩ, we have

DΨDs = 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∗) · u ′ dΓ +

∫Γs

t

(ψγ,t + u∗

)· t ′ dΓ



Adjoint model:

Introducing

C∇2u∗ + f ∗ = 0 with f ∗ = ψω,u in Ωs ;

u∗ = u∗ with u∗ = −ψγ,t on Γsu;

t∗ = (C∇u∗)Tn = t∗ with t∗ = ψγ,u on Γst .

such that

Φ1 = 0

Eventually,

DΨ

Ds=

DΨ

Ds= Φ2



Shape sensitivity:

DΨ

Ds= Φ2 =

∫Γs

(ψω − C∇u · ∇u∗ + f · u∗

)νn dΓ +

∫Ωs

f ′ · u∗dΩ

+

∫Γs


)dΓ

+

∫Γs

u

(ψγ,u − t∗) · u ′ dΓ +

∫Γs

t

(ψγ,t + u∗

)· t ′ dΓ

ν = ˚x =∑I

R I dx I [s]

ds

DΨ

Dx I=

∫Γs

(ψω − C∇u · ∇u∗ + f · u∗

)nR I dΓ +

∫Ωs

f ′ · u∗dΩ

+

∫Γs

((∇ψγ · n − ψγκ) +∇ (t · u∗) · n − (t · u∗)κ

)nR I dΓ

+

∫Γs

u

(ψγ,u − t∗) · u ′ dΓ +

∫Γs

t

(ψγ,t + u∗

)· t ′ dΓ



Shape sensitivity:

DΨ

Ds= Φ2 =

∫Γs

(ψω − C∇u · ∇u∗ + f · u∗

)νn dΓ +

∫Ωs

f ′ · u∗dΩ

+

∫Γs


)dΓ

+

∫Γs

u

(ψγ,u − t∗) · u ′ dΓ +

∫Γs

t

(ψγ,t + u∗

)· t ′ dΓ

ν = ˚x =∑I

R I dx I [s]

ds



Shape sensitivity:

DΨ

Dx I=

∫Γs

(ψω − C∇u · ∇u∗ + f · u∗

)nR I dΓ +

∫Ωs

f ′ · u∗dΩ

+

∫Γs

((∇ψγ · n − ψγκ) +∇ (t · u∗) · n − (t · u∗)κ

)nR I dΓ

+

∫Γs

u

(ψγ,u − t∗) · u ′ dΓ +

∫Γs

t

(ψγ,t + u∗

)· t ′ dΓ



Example: minimize structural compliance

Obj: Ψ[s] :=∫

Γstψγ dΓ with ψγ = t · u

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

BVP constraint:c[u] := divC∇u + f = 0 with f = 0 in Ωs

t = t 6= 0 on Γst

u = u = 0 on Γsu

Adjoint model = primary model (self-adjoint)C∇2u∗ + f ∗ = 0 with f ∗ = ψω,u = 0 in Ωs ;

t∗ = (C∇u∗)Tn = t∗ with t∗ = ψγ,u = t on Γst

u∗ = u∗ with u∗ = −ψγ,t = 0 on Γsu.



Example: minimizing the structural compliance

Shape sensitivity:

DΨ

Dx I=−

∫Γs

C∇u · ∇u∗nR I dΓ +

∫Γst

(∇ψγ · n − ψγκ)nR I dΓ

+

∫Γs

(∇ (t · u∗) · n − (t · u∗)κ

)nR I dΓ

=−∫

Γst

C∇u · ∇u∗nR I dΓ + 2

∫Γst

(∇ (t · u) · n − (t · u)κ

)nR I dΓ

Compared with the discrete approach:

Ψ = −U∗TKU = −UTKU

Which one is easier for you to compute??


Adjoint method: Question !

Discrete approach vs Continuous approach

For problems with design-dependentboundary conditions,

which approach is easier ??

The answers can be different for different people.

In general, just choose the one you like.


Adjoint method: Question !

Discrete approach vs Continuous approach

For problems with design-dependentboundary conditions,

which approach is easier ??The answers can be different for different people.

In general, just choose the one you like.



Some additional references about continuous adjoint method:

[Dems and Mroz(1984)]: Variational approach by means of adjoint systemsto structural optimization and sensitivity analysis—II: Structure shapevariation, IJSS, 1984.

[Choi and Kim(2005)]: Structural sensitivity analysis and optimization 1:Linear systems, 2006.

[Arora(1993)]: An exposition of the material derivative approach forstructural shape sensitivity analysis, CMAME, 1993.

[Tortorelli and Haber(1989)]:First-order design sensitivities for transientconduction problems by an adjoint method, IJNME, 1989.

[Wang and Turteltaub(2015)]: Isogeometric shape optimization forquasi-static processes, IJNME, 2015.

[Wang et al.(2017c)Wang, Turteltaub, and Abdalla]: Shape optimizationand optimal control for transient heat conduction problems using anisogeometric approach, C&S, 2017.


Outline





5 Research trends


Parameterization-dependency of the search directions

10

5

00 5 10 15 20(a) (b)

Design boundary

0 5 10 15 20

10

5

0

Example: volume reduction

Volume: Σ =∫

Ω dΩGradient (continuous):

g = n

Gradient (NURBS discretization):

g Id =

∫ΓnR I dΓ



0 4 8 12 16 20

0

5

10

Updated shape

Original shape

Control points of updated shape

Control points of original shape

0 4 8 12 16 20

0

5

10

Updated shape

Original shape



0 4 8 12 16 20

0

5

10

Updated shape

Original shape


Control points of original shapeCase 3

Case 2Case 1

0 4 8 12 16 20

0

5

10

Updated shape

Original shape



Case 4

0 3 5 7 10 13 15 17 20

0

5

10

Updated shape

Original shape

Nodes of updated shape

Nodes of original shape

Case 5

(x I)(s+1)

=(x I)(s)

+ αd Id =

(x I)(s) − αg I

d



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


A simple example about the quadratic norm induced bydiscretization

A (squared) L2 norm

f[x]

:= x · x ,

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

Quadratic norm induced by discretization x = RTXR is a vector of shape functions, X is a vector of discrete variables x I

f = XTMX , with M = RRT

gradient: g I = f,x I = 2MIJxJ , G = f,X = 2MXsteepest search direction: Dn = −M−1G , Dn = [d 1

n, d 2n, · · · ]

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


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 withthe steepest search direction of a continuous form.


Normalization approaches

1. Standard approach

Dn = −M−1G

2. DLMM normalization approach

The diagonally lumped mapping matrix (DLMM)

M II :=∑J

M IJ = M II =

∫DR I dD, with

∑J

RJ = 1

d In = −

g Id

M II= −

∫D gR I dD∫D R I dD

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



3. B-Spline space (D) normalization

d In ≈ −

∫D gN I dD∫D N I dD

4. Simplified DLMM approach

Unity of integral property of B-spline basis∫N i ,pdξ

ξi+p+1 − ξi=

1

p + 1,

d In = −

(p + 1)∫D gN I dD

ξi+p+1 − ξi,

More information in [Wang et al.(2017a)Wang, Abdalla, and Turteltaub].


Effectiveness of the simplified DLMM approach


Effectiveness of the simplified DLMM approach



References:

[Wang et al.(2017a)Wang, Abdalla, and Turteltaub]: Normalizationapproaches for the descent search direction in isogeometric shapeoptimization, CAD, 2017.

[Kiendl et al.(2014)Kiendl, Schmidt, WWuchner, and Bletzinger]:Isogeometric shape optimization of shells using semi-analyticalsensitivity analysis and sensitivity weighting, CMAME, 2014.

[Boyd and Vandenberghe(2009)]: Convex optimization, CambridgeUniversity Press, 2009.

[Wang and Kumar(2017)]:On the numerical implementation ofcontinuous adjoint sensitivity for transient heat conduction problemsusing an isogeometric approach, SMO, 2017.


Outline





5 Research trends


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, WWuchner, 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]Bezier 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]


Research trends







Research trends







Acknowledgements

Thank you for your attention!

−−With special thanks to

Prof. Qian Xiaoping for his invitation andProf. 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 anymistakes/errors in it. Thank you.


References:

Arora, J. S., 1993. An exposition of the material derivative approachfor structural shape sensitivity analysis. Computer Methods in AppliedMechanics and Engineering 105 (1), 41 – 62.

Bendsøe, M. P., Sigmund, O., 2004. Topology optimization bydistribution of isotropic material. In: Topology Optimization. Springer,pp. 1–69.

Boyd, S., Vandenberghe, L., 2009. Convex optimization. CambridgeUniversity press.

Cai, S.-Y., Zhang, W. H., Zhu, J. H., Gao, T., 2014. Stressconstrained shape and topology optimization with fixed mesh: AB-spline finite cell method combined with level set function. ComputerMethods in Applied Mechanics and Engineering 278, 361–387.

Cho, S., Ha, S.-H., 2009. Isogeometric shape design optimization:Exact geometry and enhanced sensitivity. Structural andMultidisciplinary Optimization 38 (1), 53–70.


Choi, K. K., Kim, N.-H., 2005. Structural sensitivity analysis andoptimization 1: Linear systems. Vol. 1. Springer-Verlag New York,Inc., New York, NY, USA.

Dems, K., Mroz, Z., 1984. Variational approach by means of adjointsystems to structural optimization and sensitivity analysis—II:Structure shape variation. International Journal of Solids andStructures 20 (6), 527–552.

Kiendl, J., Schmidt, R., WWuchner, R., Bletzinger, K.-U., 2014.Isogeometric shape optimization of shells using semi-analyticalsensitivity analysis and sensitivity weighting. Computer Methods inApplied Mechanics and Engineering 274 (0), 148 – 167.

Le, C., Bruns, T., Tortorelli, D., 2011. A gradient-based,parameter-free approach to shape optimization. Computer Methods inApplied Mechanics and Engineering 200 (9), 985–996.

Nagy, A. P., Abdalla, M. M., Gurdal, Z., 2010. Isogeometric sizing andshape optimisation of beam structures. Computer Methods in AppliedMechanics and Engineering 199 (17), 1216–1230.Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 45 / 45

Nagy, A. P., IJsselmuiden, S. T., Abdalla, M. M., 2013. Isogeometricdesign of anisotropic shells: Optimal form and material distribution.Computer Methods in Applied Mechanics and Engineering 264,145–162.

Qian, X., 2010. Full analytical sensitivities in NURBS basedisogeometric shape optimization. Computer Methods in AppliedMechanics and Engineering 199 (29), 2059–2071.

Seo, Y.-D., Kim, H.-J., Youn, S.-K., 2010. Isogeometric topologyoptimization using trimmed spline surfaces. Computer Methods inApplied Mechanics and Engineering 199 (49-52), 3270–3296.

Tortorelli, D. A., Haber, R. B., 1989. First-order design sensitivities fortransient conduction problems by an adjoint method. InternationalJournal for Numerical Methods in Engineering 28 (4), 733–752.

Wang, C., Xia, S., Wang, X., Qian, X., 2018. Isogeometric shapeoptimization on triangulations. Computer Methods in AppliedMechanics and Engineering 331, 585–622.


Wang, M. Y., Wang, X., Guo, D., 2003. A level set method forstructural topology optimization. Computer methods in appliedmechanics and engineering 192 (1), 227–246.

Wang, Z.-P., Abdalla, M., Turteltaub, S., 2017a. Normalizationapproaches for the descent search direction in isogeometric shapeoptimization. Computer-Aided Design 82, 68–78.

Wang, Z.-P., Kumar, D., 2017. On the numerical implementation ofcontinuous adjoint sensitivity for transient heat conduction problemsusing an isogeometric approach. Structural and MultidisciplinaryOptimization, 1–14.

Wang, Z.-P., Poh, L. H., Dirrenberger, J., Zhu, Y., Forest, S., 2017b.Isogeometric shape optimization of smoothed petal auxetic structuresvia computational periodic homogenization. Computer Methods inApplied Mechanics and Engineering 323, 250–271.

Wang, Z.-P., Turteltaub, S., 2015. Isogeometric shape optimizationfor quasi-static processes. International Journal for Numerical Methodsin Engineering 104 (5), 347–371.Wang Zhenpei (NUS) Isogeometric Shape Optimization: 2018 年 3 月 23 日 45 / 45

Wang, Z.-P., Turteltaub, S., Abdalla, M., 2017c. Shape optimizationand optimal control for transient heat conduction problems using anisogeometric approach. Computers & Structures 185, 59–74.

WANG, Z.-P., WANG, D., ZHU, J.-H., ZHANG, W.-H., 2011.Parametrical fe modeling of blade and design optimization of itsgravity center eccentricity. Journal of Aerospace Power 26 (11),2450–2458.


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Isogeometric Shape OptimizationŽ‹振培.pdf · Isogeometric Shape Optimization: A brief introduction about shape sensitivity analysis and search direction normalization Wang Zhenpei