Weighted Sobolev spaces for the advection operator: A variational - PowerPoint PPT Presentation

Weighted Sobolev spaces for the advection operator: A variational method for computing shape derivatives of geometric constraints along rays. Florian Feppon Gr´ egoire Allaire, Charles Dapogny Julien Cortial, Felipe Bordeu S´ eminaire des doctorants – October 17th, 2018

Outline 1. Brief presentation of thickness constraints in Hadamard’s method of shape differentiation 2. A variational method for avoiding integration along rays 3. A few insights regarding the mathematical framework: weighted graph spaces of the advection operator β · ∇ .

Outline 1. Brief presentation of thickness constraints in Hadamard’s method of shape differentiation 2. A variational method for avoiding integration along rays 3. A few insights regarding the mathematical framework: weighted graph spaces of the advection operator β · ∇ .

Outline 1. Brief presentation of thickness constraints in Hadamard’s method of shape differentiation 2. A variational method for avoiding integration along rays 3. A few insights regarding the mathematical framework: weighted graph spaces of the advection operator β · ∇ .

Outline 1. Brief presentation of thickness constraints in Hadamard’s method of shape differentiation 2. A variational method for avoiding integration along rays 3. A few insights regarding the mathematical framework: weighted graph spaces of the advection operator β · ∇ .

♥ 1. Hadamard’s method of boundary variations We rely on the method of Hadamard (figure from [1] ): min Ω J (Ω) Ω θ = ( I + θ )Ω θ with θ ∈ W 1 , ∞ (Ω , R d ) , || θ || W 1 , ∞ ( R d , R d ) < 1 . [1] Charles Dapogny et al. “Geometrical shape optimization in fluid mechanics using FreeFem++”. In: Structural and Multidisciplinary Optimization (2017).

♥ 1. Hadamard’s method of boundary variations We rely on the method of Hadamard (figure from [1] ): min Ω J (Ω) Ω θ = ( I + θ )Ω θ with θ ∈ W 1 , ∞ (Ω , R d ) , || θ || W 1 , ∞ ( R d , R d ) < 1 . | o ( θ ) | J (Ω θ ) = J (Ω) + d J θ → 0 d θ (Ω)( θ ) + o ( θ ) , where − − − → 0 , || θ || W 1 , ∞ (Ω , R d ) [1] Charles Dapogny et al. “Geometrical shape optimization in fluid mechanics using FreeFem++”. In: Structural and Multidisciplinary Optimization (2017).

1. Hadamard’s method of boundary variations We rely on the method of Hadamard (figure from [1] ): min Ω J (Ω) Ω θ = ( I + θ )Ω θ with θ ∈ W 1 , ∞ (Ω , R d ) , || θ || W 1 , ∞ ( R d , R d ) < 1 . | o ( θ ) | J (Ω θ ) = J (Ω) + d J θ → 0 d θ (Ω)( θ ) + o ( θ ) , where − − − → 0 , || θ || W 1 , ∞ (Ω , R d ) � In practice d J ∂ Ω u θ · ♥ d y for u ∈ L 1 ( ∂ Ω). d θ (Ω)( θ ) = [1] Charles Dapogny et al. “Geometrical shape optimization in fluid mechanics using FreeFem++”. In: Structural and Multidisciplinary Optimization (2017).

1. The signed distance function The signed distance function d Ω to the domain Ω ⊂ D is defined by:   − min y ∈ ∂ Ω || y − x || if x ∈ Ω ,  ∀ x ∈ D , d Ω ( x ) =   y ∈ ∂ Ω || y − x || min if x ∈ D \ Ω .

1. The signed distance function ◮ The signed distance function d Ω ( x ) is differentiable at every x ∈ D such that Π ∂ Ω ( x ) = { y ∈ ∂ Ω | || x − y || = d ( x , ∂ Ω) } is a singleton.

1. The signed distance function ◮ The signed distance function d Ω ( x ) is differentiable at every x ∈ D such that Π ∂ Ω ( x ) = { y ∈ ∂ Ω | || x − y || = d ( x , ∂ Ω) } is a singleton. ◮ The set Σ on which d Ω is not differentiable is called the skeleton .

♥ ♥ ♥ ♥ 1. The signed distance function ∇ d Ω is a unit extension of the normal constant along rays:

♥ ♥ ♥ 1. The signed distance function ∇ d Ω is a unit extension of the normal constant along rays: ◮ ∀ y ∈ ∂ Ω , ∇ d Ω ( y ) = ♥ ( y ) is the unit outward normal to ∂ Ω.

♥ ♥ 1. The signed distance function ∇ d Ω is a unit extension of the normal constant along rays: ◮ ∀ y ∈ ∂ Ω , ∇ d Ω ( y ) = ♥ ( y ) is the unit outward normal to ∂ Ω. ◮ ∀ s ∈ ( ζ − ( y ) , ζ + ( y )) , d Ω ( y + s ♥ ( y )) = s .

♥ 1. The signed distance function ∇ d Ω is a unit extension of the normal constant along rays: ◮ ∀ y ∈ ∂ Ω , ∇ d Ω ( y ) = ♥ ( y ) is the unit outward normal to ∂ Ω. ◮ ∀ s ∈ ( ζ − ( y ) , ζ + ( y )) , d Ω ( y + s ♥ ( y )) = s . The set ray ( y ) = { y + s ♥ ( y ) | s ∈ ( ζ − ( y ) , ζ + ( y )) } is called the ray emerging from y .

1. The signed distance function ∇ d Ω is a unit extension of the normal constant along rays: ◮ ∀ y ∈ ∂ Ω , ∇ d Ω ( y ) = ♥ ( y ) is the unit outward normal to ∂ Ω. ◮ ∀ s ∈ ( ζ − ( y ) , ζ + ( y )) , d Ω ( y + s ♥ ( y )) = s . The set ray ( y ) = { y + s ♥ ( y ) | s ∈ ( ζ − ( y ) , ζ + ( y )) } is called the ray emerging from y . ◮ For any x ∈ D \ Σ, ||∇ d Ω ( x ) || = 1 and ∀ z ∈ ray ( y ) , ∇ d Ω ( z ) = ♥ ( y ).

1. The signed distance function An example: a meshed subdomain Ω ⊂ D

1. The signed distance function An example: the signed distance function d Ω :

1. The signed distance function An example: the gradient of the signed distance function ∇ d Ω :

1. The signed distance function The signed distance function allows to formulate geometric constraints. ◮ Maximum thickness constraint : ∀ x ∈ Ω , | d Ω ( x ) | ≤ d max / 2 ◮ Minimum thickness constraint: ∀ y ∈ ∂ Ω , | ζ − ( y ) | ≥ d min / 2 .

♥ ♥ Shape derivatives of geometric constraints In practice, one formulates geometric constraints using penalty functionals P (Ω) as follows: � min Ω J (Ω) , s.t. P (Ω) ≤ 0 , where P (Ω) := j ( d Ω ( x )) d x . D \ Σ

Shape derivatives of geometric constraints In practice, one formulates geometric constraints using penalty functionals P (Ω) as follows: � min Ω J (Ω) , s.t. P (Ω) ≤ 0 , where P (Ω) := j ( d Ω ( x )) d x . D \ Σ The shape derivative of P (Ω) reads � P ′ (Ω)( θ ) = j ′ ( d Ω ( x )) d ′ Ω ( θ )( x ) d x D \ Σ with d ′ Ω ( θ )( y + s ∇ ( y )) = − θ ( y ) · ♥ ( y ). θ | θ · ♥ | Ω

♥ ♥ 1. Shape derivatives of geometric constraint Change of variable x = y + s ♥ ( y ) with y ∈ ∂ Ω, s ∈ ( ζ − ( y ) , ζ + ( y )): � � � ζ + ( y ) P ′ (Ω)( θ ) = j ′ ( d Ω ( x )) d ′ j ′ ( s ) | D η | ( y , s ) d s θ ( y ) · ♥ ( y ) d y Ω ( θ )( x ) d x = − D \ Σ ∂ Ω ζ − ( y )

1. Shape derivatives of geometric constraint Change of variable x = y + s ♥ ( y ) with y ∈ ∂ Ω, s ∈ ( ζ − ( y ) , ζ + ( y )): � � � ζ + ( y ) P ′ (Ω)( θ ) = j ′ ( d Ω ( x )) d ′ j ′ ( s ) | D η | ( y , s ) d s θ ( y ) · ♥ ( y ) d y Ω ( θ )( x ) d x = − D \ Σ ∂ Ω ζ − ( y ) � | D η | ( y , s ) = (1 + κ i ( y ) s ) is the Jacobian of 1 ≤ i ≤ n − 1 η : ( y , s ) �→ y + s ♥ ( y ) The κ i ( y ) are the eigenvalues of the tangential gradient ∇ ♥ ( y ).

1. Shape derivatives of geometric constraint Change of variable x = y + s ♥ ( y ) with y ∈ ∂ Ω, s ∈ ( ζ − ( y ) , ζ + ( y )): � � � ζ + ( y ) P ′ (Ω)( θ ) = j ′ ( d Ω ( x )) d ′ j ′ ( s ) | D η | ( y , s ) d s θ ( y ) · ♥ ( y ) d y Ω ( θ )( x ) d x = − D \ Σ ∂ Ω ζ − ( y ) � | D η | ( y , s ) = (1 + κ i ( y ) s ) is the Jacobian of 1 ≤ i ≤ n − 1 η : ( y , s ) �→ y + s ♥ ( y ) The κ i ( y ) are the eigenvalues of the tangential gradient ∇ ♥ ( y ). The function � ζ + ( y ) � j ′ ( s ) ∀ y ∈ ∂ Ω , u ( y ) = − (1 + κ i ( y ) s ) d s ζ − ( y ) 1 ≤ i ≤ n − 1 is “explicit” and does not involve θ .

1. Shape derivatives of geometric constraints � ζ + ( y ) � j ′ ( s ) ∀ y ∈ ∂ Ω , u ( y ) = − (1 + κ i ( y ) s ) d s . ζ − ( y ) 1 ≤ i ≤ n − 1 Computing u requires:

1. Shape derivatives of geometric constraints � ζ + ( y ) � j ′ ( s ) ∀ y ∈ ∂ Ω , u ( y ) = − (1 + κ i ( y ) s ) d s . ζ − ( y ) 1 ≤ i ≤ n − 1 Computing u requires: 1. Integrating along rays on the discretization mesh:

1. Shape derivatives of geometric constraints � ζ + ( y ) � j ′ ( s ) ∀ y ∈ ∂ Ω , u ( y ) = − (1 + κ i ( y ) s ) d s . ζ − ( y ) 1 ≤ i ≤ n − 1 Computing u requires: 1. Integrating along rays on the discretization mesh: 2. Estimating the principal curvatures κ i ( y ).

Outline 1. Brief presentation of thickness constraints in Hadamard’s method of shape differentiation 2. A variational method for avoiding integration along rays 3. A few insights regarding the mathematical framework: weighted graph spaces of the advection operator β · ∇ .

♥ 2. A variational method for avoiding integration along rays More precisely, the shape derivative of P (Ω) reads � � P ′ (Ω)( θ ) = j ′ ( d Ω ( x )) d ′ Ω ( θ )( x ) d x = u θ · ♥ d y D \ Σ ∂ Ω with d ′ Ω ( θ ) satisfying � ∇ d ′ Ω ( θ ) · ∇ d Ω = 0 in D \ Σ d ′ Ω ( θ )= − θ · ♥ on ∂ Ω . θ | θ · ♥ | Ω