Null Space Gradient Flows for Shape Optimization of Multiphysics - PowerPoint PPT Presentation

Null Space Gradient Flows for Shape Optimization of Multiphysics Systems Florian Feppon Gr´ egoire Allaire, Charles Dapogny Julien Cortial, Felipe Bordeu New trends in PDE constrained optimization RICAM (Linz) – October 15th, 2019

Outline 1. Shape derivatives for a weakly coupled multiphysics system 2. Null space gradient flows for constrained optimization 3. Numerical illustrations on 2-d and 3-d test cases

Outline 1. Shape derivatives for a weakly coupled multiphysics system 2. Null space gradient flows for constrained optimization 3. Numerical illustrations on 2-d and 3-d test cases

Outline 1. Shape derivatives for a weakly coupled multiphysics system 2. Null space gradient flows for constrained optimization 3. Numerical illustrations on 2-d and 3-d test cases

Outline 1. Shape derivatives for a weakly coupled multiphysics system 2. Null space gradient flows for constrained optimization 3. Numerical illustrations on 2-d and 3-d test cases

✈ ✉ ✈ Multiphysics shape optimization We are interested in multiphysics systems featuring ◮ fluids: velocity–pressure ( ✈ , p )

✉ ✈ Multiphysics shape optimization We are interested in multiphysics systems featuring ◮ fluids: velocity–pressure ( ✈ , p ) ◮ thermal exchanges: temperature field T , convected by ✈

Multiphysics shape optimization We are interested in multiphysics systems featuring ◮ fluids: velocity–pressure ( ✈ , p ) ◮ thermal exchanges: temperature field T , convected by ✈ ◮ mechanical structures: displacement ✉ , subjected to fluid-structure interaction with ✈ and thermoelasticity with T .

✈ ✉ ✉ ❢ ✉ ♥ ✈ ♥ 1. Shape derivatives for a multiphysics system Proposed system Ω f ∂ Ω D v 0 f Ω s n Γ u 0 ∂ Ω D s ◮ Incompressible Navier-Stokes equations for ( ✈ , p ) in Ω f − div ( σ f ( ✈ , p )) + ρ ∇ ✈ ✈ = ❢ f in Ω f

✉ ✉ ❢ ✉ ♥ ✈ ♥ 1. Shape derivatives for a multiphysics system Proposed system Ω f ∂ Ω D v 0 f Ω s n Γ u 0 ∂ Ω D s ◮ Incompressible Navier-Stokes equations for ( ✈ , p ) in Ω f − div ( σ f ( ✈ , p )) + ρ ∇ ✈ ✈ = ❢ f in Ω f ◮ Steady-state convection-diffusion for T f and T s in Ω f and Ω s : − div ( k f ∇ T f ) + ρ ✈ · ∇ T f = Q f in Ω f − div ( k s ∇ T s ) = Q s in Ω s

1. Shape derivatives for a multiphysics system Proposed system Ω f ∂ Ω D v 0 f Ω s n Γ u 0 ∂ Ω D s ◮ Incompressible Navier-Stokes equations for ( ✈ , p ) in Ω f − div ( σ f ( ✈ , p )) + ρ ∇ ✈ ✈ = ❢ f in Ω f ◮ Steady-state convection-diffusion for T f and T s in Ω f and Ω s : − div ( k f ∇ T f ) + ρ ✈ · ∇ T f = Q f in Ω f − div ( k s ∇ T s ) = Q s in Ω s ◮ Linearized thermoelasticity with fluid-structure interaction for ✉ in Ω s : − div ( σ s ( ✉ , T s )) = ❢ s in Ω s σ s ( ✉ , T s ) · ♥ = σ f ( ✈ , p ) · ♥ on Γ .

1. Shape derivatives for a multiphysics system The method of Hadamard Ω s Ω f θ Γ θ min J (Γ) Γ Γ

1. Shape derivatives for a multiphysics system The method of Hadamard Ω s Ω f θ Γ θ min J (Γ) Γ Γ Γ θ = ( I + θ )Γ , where θ ∈ W 1 , ∞ ( D , R d ) , || θ || W 1 , ∞ ( R d , R d ) < 1 . 0

1. Shape derivatives for a multiphysics system The method of Hadamard Ω s Ω f θ Γ θ min J (Γ) Γ Γ Γ θ = ( I + θ )Γ , where θ ∈ W 1 , ∞ ( D , R d ) , || θ || W 1 , ∞ ( R d , R d ) < 1 . 0 J (Γ θ ) = J (Γ) + d J | o ( θ ) | θ → 0 d θ ( θ ) + o ( θ ) , where − − − → 0 . || θ || W 1 , ∞ ( D , R d )

✈ ✉ ❢ ✇ ✈ ✇ ♥ ✇ ✈ ♥ ♥ ✈ ✇ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ✉ r ❢ r ♥ r ✉ ♥ ♥ ✉ r ♥ ♥ Shape derivatives for a multiphysics system Shape derivative of arbitrary functionals Proposition Let J (Γ , ✉ , T , ✈ , p ) an arbitrary functional with continuous partial derivatives and ✉ (Γ) , T (Γ) , ✈ (Γ) , p (Γ) the above state variables. Then, if these are smooth enough, Γ �→ J (Γ , ✉ (Γ) , T (Γ) , ✈ (Γ) , p (Γ)) is shape differentiable and the derivative reads:

Shape derivatives for a multiphysics system Shape derivative of arbitrary functionals Proposition Let J (Γ , ✉ , T , ✈ , p ) an arbitrary functional with continuous partial derivatives and ✉ (Γ) , T (Γ) , ✈ (Γ) , p (Γ) the above state variables. Then, if these are smooth enough, Γ �→ J (Γ , ✉ (Γ) , T (Γ) , ✈ (Γ) , p (Γ)) is shape differentiable and the derivative reads: � � d J (Γ θ , ✈ (Γ θ ) , p (Γ θ ) , T (Γ θ ) , ✉ (Γ θ )) ( θ ) d θ � = ∂ J ∂ θ ( θ ) + ( ❢ f · ✇ − σ f ( ✈ , p ) : ∇ ✇ + ♥ · σ f ( ✇ , q ) ∇ ✈ · ♥ + ♥ · σ f ( ✈ , p ) ∇ ✇ · ♥ )( θ · ♥ ) d s Γ � � � ∂ T s ∂ S s ∂ T f ∂ S f + k s ∇ T s · ∇ S s − k f ∇ T f · ∇ S f + Q f S f − Q s S s − 2 k s ∂ ♥ + 2 k f ( θ · ♥ ) d s ∂ ♥ ∂ ♥ ∂ ♥ Γ � + ( σ s ( ✉ , T s ) : ∇ r − ❢ s · r − ♥ · Ae ( r ) ∇ ✉ · ♥ − ♥ · σ s ( ✉ , T s ) ∇ r · ♥ ) ( θ · ♥ ) d s Γ

Shape derivatives for a multiphysics system Shape derivative of arbitrary functionals Proposition Let J (Γ , ✉ , T , ✈ , p ) an arbitrary functional with continuous partial derivatives and ✉ (Γ) , T (Γ) , ✈ (Γ) , p (Γ) the above state variables. Then, if these are smooth enough, Γ �→ J (Γ , ✉ (Γ) , T (Γ) , ✈ (Γ) , p (Γ)) is shape differentiable and the derivative reads: � � d J (Γ θ , ✈ (Γ θ ) , p (Γ θ ) , T (Γ θ ) , ✉ (Γ θ )) ( θ ) d θ � = ∂ J ∂ θ ( θ ) + ( ❢ f · ✇ − σ f ( ✈ , p ) : ∇ ✇ + ♥ · σ f ( ✇ , q ) ∇ ✈ · ♥ + ♥ · σ f ( ✈ , p ) ∇ ✇ · ♥ )( θ · ♥ ) d s Γ � � � ∂ T s ∂ S s ∂ T f ∂ S f + k s ∇ T s · ∇ S s − k f ∇ T f · ∇ S f + Q f S f − Q s S s − 2 k s ∂ ♥ + 2 k f ( θ · ♥ ) d s ∂ ♥ ∂ ♥ ∂ ♥ Γ � + ( σ s ( ✉ , T s ) : ∇ r − ❢ s · r − ♥ · Ae ( r ) ∇ ✉ · ♥ − ♥ · σ s ( ✉ , T s ) ∇ r · ♥ ) ( θ · ♥ ) d s Γ J is a “transported” functional: p , ˆ ✈ ◦ ( I + θ ) − 1 , ˆ p ◦ ( I + θ ) − 1 , ˆ T ◦ ( I + θ ) − 1 , ˆ ✉ ◦ ( I + θ ) − 1 ) . J ( θ , ˆ ✈ , ˆ T , ˆ ✉ ) := J (Γ θ , ˆ

Shape derivatives for a multiphysics system Shape derivative of arbitrary functionals Proposition Let J (Γ , ✉ , T , ✈ , p ) an arbitrary functional with continuous partial derivatives and ✉ (Γ) , T (Γ) , ✈ (Γ) , p (Γ) the above state variables. Then, if these are smooth enough, Γ �→ J (Γ , ✉ (Γ) , T (Γ) , ✈ (Γ) , p (Γ)) is shape differentiable and the derivative reads: � � d J (Γ θ , ✈ (Γ θ ) , p (Γ θ ) , T (Γ θ ) , ✉ (Γ θ )) ( θ ) d θ � = ∂ J ∂ θ ( θ ) + ( ❢ f · ✇ − σ f ( ✈ , p ) : ∇ ✇ + ♥ · σ f ( ✇ , q ) ∇ ✈ · ♥ + ♥ · σ f ( ✈ , p ) ∇ ✇ · ♥ )( θ · ♥ ) d s Γ � � � ∂ T s ∂ S s ∂ T f ∂ S f + k s ∇ T s · ∇ S s − k f ∇ T f · ∇ S f + Q f S f − Q s S s − 2 k s ∂ ♥ + 2 k f ( θ · ♥ ) d s ∂ ♥ ∂ ♥ ∂ ♥ Γ � + ( σ s ( ✉ , T s ) : ∇ r − ❢ s · r − ♥ · Ae ( r ) ∇ ✉ · ♥ − ♥ · σ s ( ✉ , T s ) ∇ r · ♥ ) ( θ · ♥ ) d s Γ Partial derivative for J with respect to the shape.

Shape derivatives for a multiphysics system Shape derivative of arbitrary functionals Proposition Let J (Γ , ✉ , T , ✈ , p ) an arbitrary functional with continuous partial derivatives and ✉ (Γ) , T (Γ) , ✈ (Γ) , p (Γ) the above state variables. Then, if these are smooth enough, Γ �→ J (Γ , ✉ (Γ) , T (Γ) , ✈ (Γ) , p (Γ)) is shape differentiable and the derivative reads: � � d J (Γ θ , ✈ (Γ θ ) , p (Γ θ ) , T (Γ θ ) , ✉ (Γ θ )) ( θ ) d θ � = ∂ J ∂ θ ( θ ) + ( ❢ f · ✇ − σ f ( ✈ , p ) : ∇ ✇ + ♥ · σ f ( ✇ , q ) ∇ ✈ · ♥ + ♥ · σ f ( ✈ , p ) ∇ ✇ · ♥ )( θ · ♥ ) d s Γ � � � ∂ T s ∂ S s ∂ T f ∂ S f + k s ∇ T s · ∇ S s − k f ∇ T f · ∇ S f + Q f S f − Q s S s − 2 k s ∂ ♥ + 2 k f ( θ · ♥ ) d s ∂ ♥ ∂ ♥ ∂ ♥ Γ � + ( σ s ( ✉ , T s ) : ∇ r − ❢ s · r − ♥ · Ae ( r ) ∇ ✉ · ♥ − ♥ · σ s ( ✉ , T s ) ∇ r · ♥ ) ( θ · ♥ ) d s Γ Three “adjoint” terms for each of the three physics.