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

Content 1 Introduction into structural topology optimization Multi-material phase field approach 2 3 Sharp interface limit of the phase field approximation Numerics 4 M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin · Page 2 (32)

Introduction into structural topology optimization 1 2 Multi-material phase field approach Sharp interface limit of the phase field approximation 3 Numerics 4 M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin · Page 3 (32)

Problemsetting in structural topology optimization � Domain Ω to be designed: 1. solid domain Ω M with fixed given volume | Ω M | = m 2. void Ω \ Ω M � 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)

Problemsetting in structural topology optimization � Domain Ω to be designed: 1. solid domain Ω M with fixed given volume | Ω M | = m 2. void Ω \ Ω M � Volume forces f given � Surface loads g , Traction forces given Aim of the structural topology optimization: Distribute a limited amount of material Ω M 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)

Structural topology optimization Mathematical formulation: Given a design domain Ω ⊂ R d and m J (Ω M ) min Ω M ∈U ad Admissible set: U ad = { Ω M ⊂ Ω such that | Ω M | = m } M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin · Page 5 (32)

Structural topology optimization Mathematical formulation: Given a design domain Ω ⊂ R d and m J (Ω M ) min Ω M ∈U ad Admissible set: U ad = { Ω M ⊂ Ω such that | Ω M | = m } Solve simultaneously the elasticity equation (solid domain modeled as linear elastic material) − div( C M E ( u )) = f in Ω M and boundary conditions � elasticity tensor C M 2 ( ∇ u + ∇ u T ) � linearized strain tensor E ( u ) = 1 � Displacement field u M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin · Page 5 (32)

Possible objective functionals First choice Maximize stiffness or equivalently Minimize compliance (work done by the load) � � J 1 (Ω M ) = Ω M f · u + g · u . Γ g M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin · Page 6 (32)

Possible objective functionals First choice Maximize stiffness or equivalently Minimize compliance (work done by the load) � � J 1 (Ω M ) = Ω M f · u + g · u . Γ g 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)

Possible objective functionals (tracking-type) Second choice Error compared to target displacement �� � κ J 2 (Ω M ) = Ω M c ( x ) | u − u Ω | 2 κ ∈ (0 , 1] , � c ( x ) : given weighting factor, � u Ω : 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)

Possible objective functionals (tracking-type) Second choice Error compared to target displacement �� � κ J 2 (Ω M ) = Ω M c ( x ) | u − u Ω | 2 κ ∈ (0 , 1] , � c ( x ) : given weighting factor, � u Ω : target displacement � κ = 1 2 in applications, κ = 1 least square minimization Concrete example for compliant mechanism Minimize error to given Push configuration target displacement M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin · Page 7 (32)

Possible objective functionals Structural optimization problem Minimize J (Ω M ) = αJ 1 (Ω M ) + βJ 2 (Ω M ) subject to − div( C M E ( u )) in Ω M = f ( C M E ( u )) n = 0 on Γ 0 ( C M E ( u )) n = g on Γ g = 0 on Γ D u M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin · Page 8 (32)

Possible objective functionals Structural optimization problem Minimize J (Ω M ) = αJ 1 (Ω M ) + βJ 2 (Ω M ) subject to − div( C M E ( u )) in Ω M = f ( C M E ( u )) n = 0 on Γ 0 ( C M E ( u )) n = g on Γ g = 0 on Γ D u Problem is not well-posed ! A possible path to well-posedness � Add the perimeter: P (Ω M ) = ( ∂ Ω M ) ∩ Ω ds : J (Ω M ) = αJ 1 (Ω M ) + βJ 2 (Ω M ) + γP (Ω M ) M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin · Page 8 (32)

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)

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)

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)

Introduction into structural topology optimization 1 2 Multi-material phase field approach Sharp interface limit of the phase field approximation 3 Numerics 4 M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin · Page 10 (32)

Multi-material phase field approach � another strategy: Phase field method � approximate P (Ω M ) 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)

Multi-material phase field approach � another strategy: Phase field method � approximate P (Ω M ) 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 Γ − convergence ← − Perimeter functional Ginzburg Landau functional ↓ shape deriv. ↓ sensitivity analysis asymptotic analysis ← − shape sensitivity first order necessary optimality system M. Hassan Farshbaf-Shaker · SADCO-WIAS Young Research Workshop 2014, January 29-31, Berlin · Page 11 (32)

Multi-material phase field approach � another strategy: Phase field method � approximate P (Ω M ) 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 Γ − convergence ← − Perimeter functional Ginzburg Landau functional ↓ shape deriv. ↓ sensitivity analysis asymptotic analysis ← − shape sensitivity 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)

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