Relating phase field and sharp interface approaches to structural - - PowerPoint PPT Presentation

▶
relating phase field and sharp interface approaches to
SMART_READER_LITE
LIVE PREVIEW

Relating phase field and sharp interface approaches to structural - - PowerPoint PPT Presentation

Jan 28, 2024 277 likes •790 views

Weierstrass Institute for Applied Analysis and Stochastics Relating phase field and sharp interface approaches to structural topology optimization M.Hassan Farshbaf Shaker joint work with: L. Blank, H. Garcke (Regensburg), V. Styles (Sussex)

slide-1
SLIDE 1

Weierstrass Institute for Applied Analysis and Stochastics

Relating phase field and sharp interface approaches to structural topology optimization

M.Hassan Farshbaf Shaker joint work with: L. Blank, H. Garcke (Regensburg), V. Styles (Sussex)

Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin

slide-2
SLIDE 2

Content

1

Introduction into structural topology optimization

2

Multi-material phase field approach

3

Sharp interface limit of the phase field approximation

4

Numerics

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 2 (32)

slide-3
SLIDE 3

1

Introduction into structural topology optimization

2

Multi-material phase field approach

3

Sharp interface limit of the phase field approximation

4

Numerics

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 3 (32)

slide-4
SLIDE 4

Problemsetting in structural topology optimization

Domain Ω to be designed:

  • 1. solid domain ΩM with

fixed given volume

|ΩM| = m

  • 2. void Ω \ ΩM

Volume forces f given Surface loads g, Traction forces

given

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 4 (32)

slide-5
SLIDE 5

Problemsetting in structural topology optimization

Domain Ω to be designed:

  • 1. solid domain ΩM with

fixed given volume

|ΩM| = m

  • 2. void Ω \ ΩM

Volume forces f given Surface loads g, Traction forces

given Aim of the structural topology optimization: Distribute a limited amount of material ΩM in a design domain Ω such that an objective functional J is minimized.

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 4 (32)

slide-6
SLIDE 6

Structural topology optimization Mathematical formulation: Given a design domain Ω ⊂ Rd and m

min

ΩM ∈Uad

J(ΩM)

Admissible set: Uad = {ΩM ⊂ Ω such that |ΩM| = m}

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 5 (32)

slide-7
SLIDE 7

Structural topology optimization Mathematical formulation: Given a design domain Ω ⊂ Rd and m

min

ΩM ∈Uad

J(ΩM)

Admissible set: Uad = {ΩM ⊂ Ω such that |ΩM| = m} Solve simultaneously the elasticity equation (solid domain modeled as linear elastic material)

−div(CME(u)) = f in ΩM and boundary conditions

elasticity tensor CM linearized strain tensor E(u) = 1

2(∇u + ∇uT )

Displacement field u

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 5 (32)

slide-8
SLIDE 8

Possible objective functionals First choice Maximize stiffness or equivalently Minimize compliance (work done by the load)

J1(ΩM) =

  • ΩM f · u +
  • Γg

g · u .

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 6 (32)

slide-9
SLIDE 9

Possible objective functionals First choice Maximize stiffness or equivalently Minimize compliance (work done by the load)

J1(ΩM) =

  • ΩM f · u +
  • Γg

g · u .

Concrete examples for mean compliance Michell type structure Cantilever beam configuration

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 6 (32)

slide-10
SLIDE 10

Possible objective functionals (tracking-type) Second choice Error compared to target displacement

J2(ΩM) =

  • ΩM c(x)|u − uΩ|2

κ , κ ∈ (0, 1]

c(x): given weighting factor, uΩ: target displacement κ = 1

2 in applications, κ = 1 least square minimization

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 7 (32)

slide-11
SLIDE 11

Possible objective functionals (tracking-type) Second choice Error compared to target displacement

J2(ΩM) =

  • ΩM c(x)|u − uΩ|2

κ , κ ∈ (0, 1]

c(x): given weighting factor, uΩ: target displacement κ = 1

2 in applications, κ = 1 least square minimization

Concrete example for compliant mechanism Minimize error to given target displacement Push configuration

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 7 (32)

slide-12
SLIDE 12

Possible objective functionals Structural optimization problem Minimize J(ΩM) = αJ1(ΩM) + βJ2(ΩM) subject to

−div(CME(u)) = f

in ΩM

(CME(u))n = 0

  • n Γ0

(CME(u))n = g

  • n Γg

u = 0

  • n ΓD
  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 8 (32)

slide-13
SLIDE 13

Possible objective functionals Structural optimization problem Minimize J(ΩM) = αJ1(ΩM) + βJ2(ΩM) subject to

−div(CME(u)) = f

in ΩM

(CME(u))n = 0

  • n Γ0

(CME(u))n = g

  • n Γg

u = 0

  • n ΓD

Problem is not well-posed ! A possible path to well-posedness Add the perimeter: P(ΩM) =

  • (∂ΩM )∩Ω ds:

J(ΩM) = αJ1(ΩM) + βJ2(ΩM) + γP(ΩM)

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 8 (32)

slide-14
SLIDE 14

Approaches used to tackle structural optimization problems

Classical method of shape calculus: Boundary variations based on a parametric approach drawbacks: topology changes difficult / serious remeshing necessary

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 9 (32)

slide-15
SLIDE 15

Approaches used to tackle structural optimization problems

Classical method of shape calculus: Boundary variations based on a parametric approach drawbacks: topology changes difficult / serious remeshing necessary Homogenization methods and variants of it (Allaire, Bendsoe, Sigmund and many others)

(power law materials, Solid Isotropic Material with Penalization (SIMP) method)

Applicability restricted to particular objective functionals

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 9 (32)

slide-16
SLIDE 16

Approaches used to tackle structural optimization problems

Classical method of shape calculus: Boundary variations based on a parametric approach drawbacks: topology changes difficult / serious remeshing necessary Homogenization methods and variants of it (Allaire, Bendsoe, Sigmund and many others)

(power law materials, Solid Isotropic Material with Penalization (SIMP) method)

Applicability restricted to particular objective functionals Level set methods (Sethian, Osher, Allaire, Burger and many others)

(Applicable to a wide range of problems)

drawbacks: difficult to create holes / (topological derivatives would help)

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 9 (32)

slide-17
SLIDE 17

1

Introduction into structural topology optimization

2

Multi-material phase field approach

3

Sharp interface limit of the phase field approximation

4

Numerics

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 10 (32)

slide-18
SLIDE 18

Multi-material phase field approach

another strategy: Phase field method approximate P(ΩM) by Ginzburg-Landau functional allows for topology changes (nucleation or elimination of holes), easy to construct a multi-phase model (more than one material and void possible) In the following

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 11 (32)

slide-19
SLIDE 19

Multi-material phase field approach

another strategy: Phase field method approximate P(ΩM) by Ginzburg-Landau functional allows for topology changes (nucleation or elimination of holes), easy to construct a multi-phase model (more than one material and void possible) In the following

Perimeter functional

Γ−convergence

← −

Ginzburg Landau functional

↓shape deriv. ↓sensitivity analysis

shape sensitivity

asymptotic analysis

← −

first order necessary optimality system

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 11 (32)

slide-20
SLIDE 20

Multi-material phase field approach

another strategy: Phase field method approximate P(ΩM) by Ginzburg-Landau functional allows for topology changes (nucleation or elimination of holes), easy to construct a multi-phase model (more than one material and void possible) In the following

Perimeter functional

Γ−convergence

← −

Ginzburg Landau functional

↓shape deriv. ↓sensitivity analysis

shape sensitivity

asymptotic analysis

← −

first order necessary optimality system

asymptotic analysis: formal rigorous justification by Γ-convergence machinery, still ongoing research

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 11 (32)

slide-21
SLIDE 21

Phase field approach, earlier work

  • B. Bourdin, A. Chambolle, Design-dependent loads in topology optimization, ESAIM Contr. Optim. Calc. Var. 9 (2003) 19-48.

M.Y. Wang, S.W. Zhou, Phase field: A variational method for structural topology optimization. Comput. Model. Eng. Sci. 6(2004) 547-566.

  • B. Bourdin, A. Chambolle, The phase field method in optimal design, in: IUTAM Symposium on Topological Design Optimization of

Structures, Machines and Materials (2006) 207-215.

  • M. Burger, R. Stainko, Phase field relaxation of topology optimization with local stress constraints. SIAM J. Control Optim. 45 (2006)

1447-1466.

  • L. Blank, H. Garcke, L. Sarbu, T. Srisupattarawanit, V. Styles, A. Voigt, Phase field approaches to structural topology optimization. 2010

(to be published in: K.H. Hoffmann and G. Leugering (Eds.): Optimization with Partial Differential Equations, ISNM, Birkhäuser Verlag)

  • A. Takezawa, S. Nishiwaki, M. Kitamura, Shape and topology optimization based on the phase field method and sensitivity analysis.

Journal of Computational Physics 229 (7) (2010), 2697-2718. P . Penzler, M. Rumpf, B. Wirth, A phase field model for compliance shape optimization in nonlinear elasticity, ESAIM Control Optim.

  • Calc. Var. 18 (2012), no. 1, 229-258.
  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 12 (32)

slide-22
SLIDE 22

Phase field approach, earlier work

  • B. Bourdin, A. Chambolle, Design-dependent loads in topology optimization, ESAIM Contr. Optim. Calc. Var. 9 (2003) 19-48.

M.Y. Wang, S.W. Zhou, Phase field: A variational method for structural topology optimization. Comput. Model. Eng. Sci. 6(2004) 547-566.

  • B. Bourdin, A. Chambolle, The phase field method in optimal design, in: IUTAM Symposium on Topological Design Optimization of

Structures, Machines and Materials (2006) 207-215.

  • M. Burger, R. Stainko, Phase field relaxation of topology optimization with local stress constraints. SIAM J. Control Optim. 45 (2006)

1447-1466.

  • L. Blank, H. Garcke, L. Sarbu, T. Srisupattarawanit, V. Styles, A. Voigt, Phase field approaches to structural topology optimization. 2010

(to be published in: K.H. Hoffmann and G. Leugering (Eds.): Optimization with Partial Differential Equations, ISNM, Birkhäuser Verlag)

  • A. Takezawa, S. Nishiwaki, M. Kitamura, Shape and topology optimization based on the phase field method and sensitivity analysis.

Journal of Computational Physics 229 (7) (2010), 2697-2718. P . Penzler, M. Rumpf, B. Wirth, A phase field model for compliance shape optimization in nonlinear elasticity, ESAIM Control Optim.

  • Calc. Var. 18 (2012), no. 1, 229-258.

Common: focus on numerical aspects and formal analysis,

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 12 (32)

slide-23
SLIDE 23

Phase field approach: Introduction

phase field ϕ := (ϕi)N

i=1; phase-fraction ϕi; void ϕN; ε > 0 interface width

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 13 (32)

slide-24
SLIDE 24

Phase field approach: Introduction

phase field ϕ := (ϕi)N

i=1; phase-fraction ϕi; void ϕN; ε > 0 interface width

Gibbs-Simplex G = {v ∈ RN | vi ≥ 0, N

i=1 vi = 1}

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 13 (32)

slide-25
SLIDE 25

Phase field approach: Introduction

phase field ϕ := (ϕi)N

i=1; phase-fraction ϕi; void ϕN; ε > 0 interface width

Gibbs-Simplex G = {v ∈ RN | vi ≥ 0, N

i=1 vi = 1}

G := {v ∈ H1(Ω, RN) | v(x) ∈ G a.e. in Ω}

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 13 (32)

slide-26
SLIDE 26

Phase field approach: Introduction

phase field ϕ := (ϕi)N

i=1; phase-fraction ϕi; void ϕN; ε > 0 interface width

Gibbs-Simplex G = {v ∈ RN | vi ≥ 0, N

i=1 vi = 1}

G := {v ∈ H1(Ω, RN) | v(x) ∈ G a.e. in Ω} Gm := {v ∈ G |

  • Ω

− v = m}

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 13 (32)

slide-27
SLIDE 27

Phase field approach: Introduction

phase field ϕ := (ϕi)N

i=1; phase-fraction ϕi; void ϕN; ε > 0 interface width

Gibbs-Simplex G = {v ∈ RN | vi ≥ 0, N

i=1 vi = 1}

G := {v ∈ H1(Ω, RN) | v(x) ∈ G a.e. in Ω} Gm := {v ∈ G |

  • Ω

− v = m}

Exapmle: 3 materials

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 13 (32)

slide-28
SLIDE 28

Phase field approach: Introduction mechanical properties of the system

linear elasticity elasticity tensors in material for each phase Ci, i ∈ {1, . . . , N − 1}

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 14 (32)

slide-29
SLIDE 29

Phase field approach: Introduction mechanical properties of the system

linear elasticity elasticity tensors in material for each phase Ci, i ∈ {1, . . . , N − 1} void is modeled as a very soft ”ersatz material” with a very small elasticity tensor

CN = CN(ε) = ε2 ˜ CN

(quadratic rate in ε accelerates the convergence in the void as ε → 0 and is hence chosen in the numerical computations)

elasticity tensor in the interfacial region: C(ϕ) = C(ϕ) + CN(ε)ϕN, ∀ϕ ∈ G,

C(ϕ) :=

N−1

  • i=1

Ciϕi

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 14 (32)

slide-30
SLIDE 30

Phase field approach

Ginzburg-Landau functional

Eε(ϕ) :=

  • Ω

ε 2|∇ϕ|2 + 1 ε Ψ(ϕ)

  • dx,

ε > 0,

Prototype examples for Ψ are given by

Ψ(ϕ) := 1 2(1 − ϕ · ϕ)

and

Ψ(ϕ) := 1 2ϕ · Wϕ

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 15 (32)

slide-31
SLIDE 31

Phase field approach

Ginzburg-Landau functional

Eε(ϕ) :=

  • Ω

ε 2|∇ϕ|2 + 1 ε Ψ(ϕ)

  • dx,

ε > 0,

Prototype examples for Ψ are given by

Ψ(ϕ) := 1 2(1 − ϕ · ϕ)

and

Ψ(ϕ) := 1 2ϕ · Wϕ

mean complinace J1 and compliant mechanism J2

J1(u, ϕ) =

  • Ω

(1 − ϕN)f · u +

  • Γg

g · u, J2(u, ϕ) :=

  • Ω

c (1 − ϕN)|u − uΩ|2 κ , κ ∈ (0, 1]

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 15 (32)

slide-32
SLIDE 32

Similarities to optimal control problem, control: ϕ, state: u

(Pε)                            min Jε(u, ϕ) := αJ1(u, ϕ) + βJ2(u, ϕ) + γEε(ϕ),

  • ver

(u, ϕ) ∈ H1

D(Ω, Rd) × H1(Ω, RN),

s.t.

(SE)            −∇ · [C(ϕ)E(u)] =

  • 1 − ϕN

f

in Ω,

u =

  • n ΓD,

[C(ϕ)E(u)] n = g

  • n Γg,

[C(ϕ)E(u)] n =

  • n Γ0,

ϕ ∈ Gm ∩ U c, U c := {ϕ ∈ H1(Ω, RN) | ϕN = 0 a.e. on S0 and ϕN = 1 a.e. on S1}. Gm := {v ∈ G |

  • Ω

− v = m}, G := {v ∈ H1(Ω, RN) | v(x) ∈ G a.e. in Ω}, G = {v ∈ RN | vi ≥ 0, N

i=1 vi = 1}

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 16 (32)

slide-33
SLIDE 33

“Control-to-state” operator and reduced functional “Control-to-state” operator Lemma:

S : L∞(Ω, RN) → H1

D(Ω, Rd),

S(ϕ) := u

The control-to-state operator S : L∞(Ω, RN) → H1

D(Ω, Rd) is Fréchet-differentiable.

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 17 (32)

slide-34
SLIDE 34

“Control-to-state” operator and reduced functional “Control-to-state” operator Lemma:

S : L∞(Ω, RN) → H1

D(Ω, Rd),

S(ϕ) := u

The control-to-state operator S : L∞(Ω, RN) → H1

D(Ω, Rd) is Fréchet-differentiable.

Reduced cost-functional Lemma:

j(ϕ) := Jε(S(ϕ), ϕ) = αJ1(S(ϕ), ϕ) + βJ2(S(ϕ), ϕ) + γEε(ϕ)

The reduced cost-functional j : H1(Ω, RN) ∩ L∞(Ω, RN) → R is Fréchet-differentiable.

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 17 (32)

slide-35
SLIDE 35

the optimal control problem (Pε) can be reformulated to

min

ϕ∈Gm∩Uc j(ϕ).

Then the following variational inequality is fulfilled:

j′(ϕ)(˜ ϕ − ϕ) ≥ 0 ∀˜ ϕ ∈ Gm ∩ U c. j′(ϕ)(˜ ϕ − ϕ) = Jε

′u(u, ϕ)

u∗

  • S′(ϕ)(˜

ϕ−ϕ)

+ Jε

′ϕ(u, ϕ)(˜

ϕ − ϕ) Jε

′u(u, ϕ)u∗ = −E(p), E(u)C′(ϕ) ˜

ϕ−ϕN ) −

  • Ω

( ˜ ϕN − ϕN)f · p

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 18 (32)

slide-36
SLIDE 36

Theorem (optimality system) The functions (u, ϕ, p) ∈ H1 D(Ω, Rd) × (Gm ∩ Uc) × H1 D(Ω, Rd) fulfill (SE)

         −∇ · [C(ϕ)E(u)] =

  • 1 − ϕN

f

in Ω,

u =

  • n ΓD,

[C(ϕ)E(u)] n = g

  • n Γg,

[C(ϕ)E(u)] n =

  • n Γ0,

the adjoint equations (AE) (AE)

               −∇ · [C(ϕ)E(p)] = α

  • 1 − ϕN

f+ +2βκJ0(u, ϕ)

κ−1 κ

c(1 − ϕN )(u − uΩ)

in Ω,

p =

  • n ΓD,

[C(ϕ)E(p)] n = αg

  • n Γg,

[C(ϕ)E(p)] n =

  • n Γ0

and the gradient inequality (GI) (GI)

             γε

  • Ω ∇ϕ : ∇(˜

ϕ − ϕ) + γ

ε

  • Ω Ψ′

0(ϕ) · (˜

ϕ − ϕ) −βκJ0(u, ϕ)

κ−1 κ

  • Ω c( ˜

ϕN − ϕN )|u − uΩ|2 −

  • Ω( ˜

ϕN − ϕN )f · (αu + p) − E(p), E(u)C′(ϕ)( ˜

ϕ−ϕ) ≥ 0,

∀˜ ϕ ∈ Gm ∩ Uc.

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 19 (32)

slide-37
SLIDE 37

1

Introduction into structural topology optimization

2

Multi-material phase field approach

3

Sharp interface limit of the phase field approximation

4

Numerics

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 20 (32)

slide-38
SLIDE 38

Asymptotic expansion

Interfacial regions have thickness ε Structure of phase field across interface

Use rescaled variable z =

dist(x,{ϕε= 1 2 }) ε

Matched asymptotic expansions

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 21 (32)

slide-39
SLIDE 39

Sharp interface limit as ε → 0

Domain Ω is partitioned into N regions Ωi separated by interfaces Γij, [w]j

i jump across interface.

We obtain for i, j ∈ {1, . . . , N − 1}: (material-material)

(SE)i

     −∇ ·

  • CiE(u)
  • = f

in Ωi,

[u]j

i

= 0

  • n Γij,

[CE(u)ν]j

i

= 0

  • n Γij,

(AE)i

     −∇ ·

  • CiE(p)
  • =

αf + 2βκJ0(u, ϕ)

κ−1 κ c(u − uΩ) in Ωi,

[p]j

i

=

  • n Γij,

[CE(p)ν]j

i

=

  • n Γij,

and we have (CiEi(u))ν = (CiEi(p))ν = 0 on ΓiN (material-void).

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 22 (32)

slide-40
SLIDE 40

Sharp interface limit as ε → 0 For all i, j = N (material-material) interface

0 = γσijκ−[CE(u) : E(p)]j

i + [CE(u)ν · (∇p)ν]j i

+[CE(p)ν · (∇u)ν]j

i−[λ1]j i

Term −[CE(u) : E(p)]j

i + [CE(u)ν · (∇p)ν]j i + [CE(p)ν · (∇u)ν]j i generalizes the

Eshelby traction (compare materials science)

0 = γσiNκ + CiEi(u) : Ei(p) −βκJ0(u, ϕ)

κ−1 κ c |u − uΩ|2 − f · (αu + p) + (λ1)i − (λ1)N.

(void-material) interface

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 23 (32)

slide-41
SLIDE 41

Choose angle at triple junction Example computation

Typically angles at void should be 180◦ or

close to it (otherwise cracks are possible)

Angles in phase field model are determined

by energy Ψ (Bronsard, Reitich; Bronsard, Garcke, Stoth)

Important: Asymptotic analysis shows

elasticity does not influence equilibrium angles

possible configuration at triple junction

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 24 (32)

slide-42
SLIDE 42

1

Introduction into structural topology optimization

2

Multi-material phase field approach

3

Sharp interface limit of the phase field approximation

4

Numerics

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 25 (32)

slide-43
SLIDE 43

Solving the gradient inequality

In order to solve the gradient inequality (GI), we use a L2-gradient flow dynamic. Allen-Cahn (2.order)

ε(∂tϕ, ˜ ϕ − ϕ)L2 + j′(ϕ)(˜ ϕ − ϕ) ≥ 0 ∀˜ ϕ ∈ Gm ∩ Uc.

(4.1) Setting ∂ϕ

∂t = 0 in (4.1) we obtain a solution of (GI).

Cahn-Hilliard dynamics possible

H−1-gradient flow dynamics Leads to different evolution, fourth-order PDE (compare Voigt et al.) Computationally more ”expensive“

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 26 (32)

slide-44
SLIDE 44

Numerical solution techniques

finite element approximation of ϕ, u, p semi-implicit in time: pseudo-time stepping approach to gradient inequality solve variational inequality with primal dual active set method (PDAS) (Hintermüller, Ito,

Kunisch)

Analyzed for Cahn-Hilliard variational inequality by Blank, Butz, Garcke Analyzed for Allen–Cahn systems by Blank, Garcke, Sarbu, Styles More efficient methods for the overall optimization problem Blank, Rupprecht

H1-gradient projection and SQP method

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 27 (32)

slide-45
SLIDE 45

Michell construction d = 2, N = 2, β = 0 (red: material, blue: void)

Deformed state

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t = 0.0 t = 0.0002 t = 0.0003 t = 0.0005 t = 0.001 t = 0.006 Typical computation starting with checker-board Topology changes

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 28 (32)

slide-46
SLIDE 46

Cantilever beam construction d = 2, N = 3, β = 0 Here: two materials and void

Deformed state

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t = 0.000 t = 0.0015 t = 0.01 t = 0.02 t = 0.04 t = 0.3

(red: hard material, green: soft material, blue: void )

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 29 (32)

slide-47
SLIDE 47

Cantilever beam construction d = 3, N = 2, β = 0 (red: hard material, blue: void ) final state boundary of material region

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 30 (32)

slide-48
SLIDE 48

Push construction (red: material, blue: void)

t=0.0 t = 0.004 t = 0.01 t=0.02 t = 0.035 t = 2.25 Computation of adjoint state necessary

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 31 (32)

slide-49
SLIDE 49

Summary / Conclusions

Introduced a phase field method for multi-material structural topology optimization

(based on earlier work by Bourdin/Chambolle, Wang/Zhou, Burger/Stainko)

First order optimality conditions were rigorously derived Sharp interface limit was analyzed with the help of formally matched asymptotic

expansions In the limit we obtained classical shape derivatives and some new shape derivatives for the multi-material situation

With the help of asymptotics at the triple junction Numerical computations demonstrated the applicability of the approach

  • L. Blank, M.H. Farshbaf-Shaker, H. Garcke, V. Styles, Relating phase field and sharp

interface approaches to structural topology optimization. to be published in ESAIM Contr.

  • Optim. Calc. Var.
  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 32 (32)

slide-50
SLIDE 50

Summary / Conclusions

Introduced a phase field method for multi-material structural topology optimization

(based on earlier work by Bourdin/Chambolle, Wang/Zhou, Burger/Stainko)

First order optimality conditions were rigorously derived Sharp interface limit was analyzed with the help of formally matched asymptotic

expansions In the limit we obtained classical shape derivatives and some new shape derivatives for the multi-material situation

With the help of asymptotics at the triple junction Numerical computations demonstrated the applicability of the approach

  • L. Blank, M.H. Farshbaf-Shaker, H. Garcke, V. Styles, Relating phase field and sharp

interface approaches to structural topology optimization. to be published in ESAIM Contr.

  • Optim. Calc. Var.

Thank you for your Attention!

  • M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin ·

Page 32 (32)