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 / 72
<latexit sha1_base64="eomrWUK6OaVvc7Eq0kx7vCymfLs=">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</latexit> <latexit sha1_base64="k9Q3pBFeuA5cDIvVUYehe1RYWM=">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</latexit> <latexit sha1_base64="Ujgbdjtnry/PfeSsz6jUyp8ng8=">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</latexit> <latexit sha1_base64="13lg4KIAoBzLJKbzNZh3GUYJ0Q=">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</latexit> Foreword • In this lecture, we focus on parametric optimization, or optimal control: • The shape is described by a set h of parameters, lying in a fixed vector space. • The state equations, accounting for the physical behavior of the shape, depend on h in a “simple” way. • Many key concepts and methods of the course can be exposed in this framework, with a minimum amount of technicality. h ( x ) • x S An elastic plate can be described by its height h : S → R with respect to a fixed cross-section S . 2 / 72
Part II Optimal control and parametric optimization problems 1 Parametric optimization problems Presentation of the model problem Non existence of optimal design Calculation of the derivative of the objective function The formal method of Céa 2 Numerical algorithms 3 / 72
A model problem involving the conductivity equation (I) g • We return to the problem of optimizing the ther- Γ N mal conductivity h : D → R . Γ D • The temperature u h is the solution in H 1 ( D ) to the “state”, conductivity equation: in D , − div ( h ∇ u h ) = f 0 on Γ D , u h = D h ∂ u h on Γ N , = g ∂ n where f ∈ L 2 ( D ) and g ∈ L 2 (Γ N ) . The considered cavity • The set U ad of design variables is: U ad = { h ∈ L ∞ ( D ) , α ≤ h ( x ) ≤ β a.e. x ∈ D } ⊂ L ∞ ( D ) , where 0 < α < β are fixed bounds. 4 / 72
A model problem involving the conductivity equation (II) • We consider a problem of the form: � h ∈U ad J ( h ) , where J ( h ) = min j ( u h ) d x , D and j : R → R is a smooth function satisfying growth conditions: ∀ s ∈ R , | j ( s ) |≤ C ( 1 + | s | 2 ) , and j ′ ( s ) ≤ C ( 1 + | s | ) . • Many variants are possible, e.g. featuring constraints on h or u h . • In this simple setting, • the state u h is evaluated on the same domain D , regardless of the actual value of the design variable h ∈ U ad ; • the design variable h acts as a parameter in the coefficients of the state equation. • Even in this simple case, the optimization problem has no (global) solution in general... 5 / 72
Part II Optimal control and parametric optimization problems 1 Parametric optimization problems Presentation of the model problem Non existence of optimal design Calculation of the derivative of the objective function The formal method of Céa 2 Numerical algorithms 6 / 72
Non existence of optimal design (I) • This counter-example is discussed in details in [All] §5.2. • The defining domain is the unit square D = ( 0 , 1 ) 2 . • We consider two physical situations: − div ( h ∇ u h , 1 ) = 0 in D , − div ( h ∇ u h , 2 ) = 0 in D , ∂ u h , 1 ∂ u h , 2 in Γ N , 1 , and = 0 in Γ N , 1 , h = e 1 · n h ∂ n ∂ n ∂ u h , 1 ∂ u h , 2 = 0 in Γ N , 2 , in Γ N , 2 . = e 2 · n h h ∂ n ∂ n u h, 1 u h, 2 Γ N, 1 D D D Γ N, 2 (Left) Boundary conditions, (middle) boundary data for u h , 1 ; (right) boundary data for u h , 2 . 7 / 72
Non existence of optimal design (II) The optimization problem of interest is: h ∈U ad J ( h ) , min where the considered objective function is: � � J ( h ) = e 1 · n u h , 1 d s − e 2 · n u h , 2 d s , Γ N , 1 Γ N , 2 and the set U ad of admissible designs is augmented with a volume constraint: � � h ∈ L ∞ ( D ) , α < h ( x ) < β a.e. x ∈ D , U ad = � . D h d x = V T In other terms, one aims to • Minimize the temperature difference between the left and right sides in Case 1. • Maximize the temperature difference between the top and bottom sides in Case 2. 8 / 72
Non existence of optimal design (III) Theorem 1. The parametric optimization problem min h ∈U ad J ( h ) does not have a global solution. Hint of the proof: The proof unfolds in three stages: Step 1: One calculates a lower bound m on the values of J ( h ) for h ∈ U ad : ∀ h ∈ U ad , J ( h ) ≥ m . Step 2: One proves that this lower bound is not attained by an element in U ad : ∀ h ∈ U ad , J ( h ) > m . Step 3: One constructs a minimizing sequence of designs h n ∈ U ad : n →∞ J ( h n ) − − − → m . Hence, m is the infimum of J ( h ) over U ad but it is not attained by any h ∈ U ad . 9 / 72
Non existence of optimal design (IV) The minimizing sequence is constructed as a laminate, i.e. a succession of layers with maximum and minimum conductivities. α 1 · · · n β Two elements in a minimizing sequence h n of conductivities. Homogenization effect: To get more optimal, designs tend to create very thin structures, at the microscopic level. 10 / 72
Non existence of optimal design (V) • In general, shape optimization problems, even under their simplest forms, do not have global solutions, for deep physical reasons. • See [Mu] for many such examples of non existence of optimal design in optimal control problems. • To ensure existence of an optimal shape, two techniques are usually employed: • Relaxation: the set U ad of admissible designs is enlarged so that it contains “microscopic designs”: this is the essence of the Homogenization method for optimal design [All2]. • Restriction: the set U ad is restricted to, e.g. more regular designs. • In practice, we shall be interested in the search of local minimizers of such problems, which are e.g. “close” to an initial design inspired by intuition. 11 / 72
Part II Optimal control and parametric optimization problems 1 Parametric optimization problems Presentation of the model problem Non existence of optimal design Calculation of the derivative of the objective function The formal method of Céa 2 Numerical algorithms 12 / 72
Derivative of the objective function (I) Let us return to our (further simplified) problem: h ∈U ad J ( h ) , min where � J ( h ) = j ( u h ) d x , Γ D D the set of admissible designs is: U ad = { h ∈ L ∞ ( D ) , α ≤ h ( x ) ≤ β a.e. x ∈ D } , and the temperature u h is the solution in H 1 0 ( D ) to: D � − div ( h ∇ u h ) in D , = f 0 on ∂ D . = u h Remark Again, for simplicity, we omit constraints on h or u h . 13 / 72
Derivative of the objective function (II) To solve this program numerically, we intend to apply a gradient-based algorithm, of the form: Initialization: Start from an initial design h 0 , For n = 0 , ... convergence: ❶ Calculate the derivative J ′ ( h n ) of the mapping h �→ J ( h ) at h = h n ; ❷ Select an appropriate time step τ n > 0; ❸ Update the design as: h n + 1 = h n − τ n J ′ ( h n ) . The cornerstone in the practice of any such method is the calculation of the derivative of J ( h ) . 14 / 72
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