An introduction to shape and topology optimization ric Bonnetier - PowerPoint PPT Presentation

An introduction to shape and topology optimization Éric Bonnetier ∗ and Charles Dapogny † ∗ Institut Fourier, Université Grenoble-Alpes, Grenoble, France † CNRS & Laboratoire Jean Kuntzmann, Université Grenoble-Alpes, Grenoble, France Fall, 2020 1 / 94

Foreword: geometric shape optimization We have seen how to optimize shapes when they are parametrized: h J ( h ) s.t. C ( h ) ≤ 0 , min where the design variable h may be: • A set of parameters in a finite-dimensional space (thickness, etc.); • A function h in a suitable, infinite dimensional vector (Banach) space. • • • • • • • • h ( x ) • • • • • • • • • • • x • • • • • • • • • • • • • S • • Description of a mechanical part via the control Parametrization of a plate with cross-section S via points of a CAD model. the thickness function h : S → R . 2 / 94

Foreword: geometric shape optimization (II) Asset: • In the considered examples, the state u h lives in a fixed computational domain, which greatly simplifies the calculation of derivatives with respect to the design. • Efficient methods from mathematical programming (optimization routines, etc.) are readily available in this context. Drawbacks: • This induces a strong bias in the sought shapes. • It may be very difficult, and in practice cumbersome, to find which are the relevant parameters h of shapes. ⇒ It is often desirable to formulate shape optimization problems in terms of the geometry of shapes Ω : min J (Ω) s.t. C (Ω) ≤ 0 . 3 / 94

Part III Geometric optimization problems 1 The method of Hadamard and shape derivatives 2 Shape derivatives of PDE-constrained functionals: the rigorous way, using Eulerian and material derivatives 3 Céa’s method for calculating shape derivatives 4 Numerical aspects of geometric methods 5 The level set method for shape optimization 4 / 94

Differentiation with respect to the domain: Hadamard’s method (I) Ω θ Hadamard’s boundary variation method describes variations of a reference, bounded Lipschitz domain Ω of the form: Ω �→ Ω θ := ( Id + θ )(Ω) , for ‘small’ vector fields θ ∈ W 1 , ∞ ( R d , R d ) . Ω θ Lemma 1. For θ ∈ W 1 , ∞ ( R d , R d ) with norm || θ || W 1 , ∞ ( R d , R d ) < 1 , the mapping ( Id + θ ) is a Lipschitz diffeomorphism. 5 / 94

Differentiation with respect to the domain: Hadamard’s method (II) Definition 1. Given a bounded Lipschitz domain Ω , a function Ω �→ J (Ω) ∈ R is shape differentiable at Ω if the mapping W 1 , ∞ ( R d , R d ) ∋ θ �→ J (Ω θ ) is Fréchet-differentiable at 0 , i.e. the following expansion holds in the vicinity of 0 : o ( θ ) θ → 0 J (Ω θ ) = J (Ω) + J ′ (Ω)( θ ) + o ( θ ) , where → 0 .. − − − || θ || W 1 , ∞ ( R d , R d ) The linear mapping θ �→ J ′ (Ω)( θ ) is the shape derivative of J at Ω . Remark Other spaces are often used in place of W 1 , ∞ ( R d , R d ) , made of more regular deformation fields θ , e.g.: � � θ : R d → R d of class C k , C k , ∞ ( R d , R d ) := x ∈ R d | ∂ α θ ( x ) | < ∞ sup sup . | α |≤ k 6 / 94

First examples of shape derivatives (I) Theorem 2. Let Ω ⊂ R d be a bounded Lipschitz domain, and let f ∈ W 1 , 1 ( R d ) be a fixed function. Consider the functional: � J (Ω) = f ( x ) d x ; Ω then J (Ω) is shape differentiable at Ω and its shape derivative is: � ∀ θ ∈ W 1 , ∞ ( R d , R d ) , J ′ (Ω)( θ ) = f ( θ · n ) d s . ∂ Ω 7 / 94

First examples of shape derivatives (II) Ω θ Ω • x • x + θ ( x ) • Intuition: f takes negative (resp. positive) values on the blue (resp. red) part of the boundary ∂ Ω . The value J (Ω θ ) is minimized from J (Ω) by adding the blue area, (i.e. θ · n > 0 where f < 0 ), and by removing the red area ( θ · n < 0 where f > 0 ), weighted by f . 8 / 94

First examples of shape derivatives (III) Remarks: • This result is a particular case of the Transport (or Reynolds) theorem, used to derive the equations of motion from conservation principles in fluid mechanics (see the Appendix in Lecture 2). • It allows to calculate the shape derivative of the volume functional � 1 d x ; Vol (Ω) = Ω Indeed, it holds: � � ∀ θ ∈ W 1 , ∞ ( R d , R d ) , Vol ′ (Ω)( θ ) = θ · n d s = div θ d x . ∂ Ω Ω In particular, if div θ = 0, the volume is unchanged (at first order) when Ω is perturbed by θ . 9 / 94

First examples of shape derivatives (IV) Proof: The formula proceeds from a change of variables in volume integrals: � � J (Ω θ ) = f ( x ) d x = | det ( Id + ∇ θ ) | f ◦ ( Id + θ ) d x . ( Id + θ )(Ω) Ω • The mapping θ �→ det ( Id + ∇ θ ) is Fréchet differentiable, and: o ( θ ) θ → 0 det ( Id + ∇ θ ) = 1 + div θ + o ( θ ) , where → 0 . − − − || θ || W 1 , ∞ ( R d , R d ) • If f ∈ W 1 , 1 ( R d ) , θ �→ f ◦ ( Id + θ ) is also Fréchet differentiable and: f ◦ ( Id + θ ) = f + ∇ f · θ + o ( θ ) . • Combining those three identites and Green’s formula leads to the result. Remark: This idea of ❶ Using the change of variables Ω → ( Id + θ )(Ω) to transport all integrals on the reference domain Ω , ❷ Differentiating with respect to the deformation θ , is the “standard” way to calculate shape derivatives. 10 / 94

First examples of shape derivatives (V) Theorem 3. Let Ω ⊂ R d be a bounded domain of class C 2 , and let g ∈ W 2 , 1 ( R d ) be a fixed function. Consider the functional: � J (Ω) = g ( x ) d s ; ∂ Ω then J (Ω) is shape differentiable at Ω when deformations θ are chosen in C 1 , ∞ ( R d , R d ) := C 1 ( R d , R d ) ∩ W 1 , ∞ ( R d , R d ) , and the shape derivative is: � ∂ g � � J ′ (Ω)( θ ) = ∂ n + κ g ( θ · n ) d s , ∂ Ω where κ is the mean curvature of ∂ Ω . � Example: The shape derivative of the perimeter Per (Ω) = ∂ Ω 1 d s is: � Per ′ (Ω)( θ ) = κ ( θ · n ) d s . ∂ Ω 11 / 94

<latexit sha1_base64="rhq08dkCWcyaVizDALcG4+ToXMQ=">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</latexit> First examples of shape derivatives (VI) • θ Ω • • Ω θ • Intuition: θ = − κ n is a descent direction for Per (Ω) : it is reduced by smearing the bumps of ∂ Ω (i.e. θ · n < 0 when κ > 0 ), and sealing its holes (i.e. θ · n > 0 when κ < 0 ). 12 / 94

Structure of shape derivatives (I) Idea: The shape derivative J ′ (Ω)( θ ) of a “regular” functional Ω �→ J (Ω) only depends on the normal component θ · n of the vector field θ . Ω θ θ Ω At first order, a tangential vector field θ , (i.e. θ · n = 0 ) only results in a convection of the shape Ω , and it is expected that J ′ (Ω)( θ ) = 0 . 13 / 94

Structure of shape derivatives (II) Lemma 4. Let Ω be a domain of class C 1 . Assume that the mapping C 1 , ∞ ( R d , R d ) ∋ θ �→ J (Ω θ ) ∈ R is of class C 1 . Then, for any vector field θ ∈ C 1 , ∞ ( R d , R d ) such that θ · n = 0 on ∂ Ω , one has: J ′ (Ω)( θ ) = 0 . Corollary 5. Under the same hypotheses, if θ 1 , θ 2 ∈ C 1 , ∞ ( R d , R d ) have the same normal component, i.e. θ 1 · n = θ 2 · n on ∂ Ω , then: J ′ (Ω)( θ 1 ) = J ′ (Ω)( θ 2 ) . 14 / 94

Structure of shape derivatives (III) • Actually, the shape derivatives of “many” integral objective functionals J (Ω) can be put under the surface form: � J ′ (Ω)( θ ) = v Ω ( θ · n ) d s , ∂ Ω where the scalar field v Ω : ∂ Ω → R depends on J and on the current shape Ω . • This structure lends itself to the calculation of a descent direction: letting θ = − tv Ω n , for a small enough descent step t > 0 in the definition of shape derivatives yields: � v 2 J (Ω t θ ) = J (Ω) − t Ω ds + o ( t ) < J (Ω) . ∂ Ω • We shall return to this issue during our study of numerical algorithms. 15 / 94

Part III Geometric optimization problems 1 The method of Hadamard and shape derivatives 2 Shape derivatives of PDE-constrained functionals: the rigorous way, using Eulerian and material derivatives 3 Céa’s method for calculating shape derivatives 4 Numerical aspects of geometric methods 5 The level set method for shape optimization 16 / 94

Shape derivatives of PDE constrained functionals • Hitherto, we have studied the shape derivatives of functionals of the form � � f ( x ) d x , and F 2 (Ω) = F 1 (Ω) = g ( x ) d s , Ω ∂ Ω where f , g : R d → R are given, smooth enough functions. • We now intend to consider functions of the form � � j ( u Ω ( x )) d x , or J 2 (Ω) = J 1 (Ω) = k ( u Ω ( x )) d s , Ω ∂ Ω where j , k : R → R are given, smooth enough functions, and u Ω : Ω → R is the solution to a PDE posed on Ω . • Doing so elaborates on the techniques from optimal control theory that we have seen in the parametric optimization context. 17 / 94