Shape optimization of a coupled thermal fluid-structure problem in a - PowerPoint PPT Presentation

Shape optimization of a coupled thermal fluid-structure problem in a level set mesh evolution framework Florian Feppon Gr´ egoire Allaire, Charles Dapogny Julien Cortial, Felipe Bordeu ECCM – June 12, 2018

Outline 1. Hadamard’s boundary variation method for a simplified three-physics setting 2. Numerical implementation of various test cases with a mesh evolution algorithm

✈ ✉ ✉ ❢ ✉ ♥ ✈ ♥ Simplified weakly coupled three-physics setting Ω f ∂ Ω D v 0 f min Γ J (Γ , ✈ (Γ) , p (Γ) , T (Γ) , ✉ (Γ)) . 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

✉ ✉ ❢ ✉ ♥ ✈ ♥ Simplified weakly coupled three-physics setting Ω f ∂ Ω D v 0 f min Γ J (Γ , ✈ (Γ) , p (Γ) , T (Γ) , ✉ (Γ)) . 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

Simplified weakly coupled three-physics setting Ω f ∂ Ω D v 0 f min Γ J (Γ , ✈ (Γ) , p (Γ) , T (Γ) , ✉ (Γ)) . 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 .

Hadamard’s method of boundary variations Ω s Ω f θ Γ θ min J (Γ) Γ Γ Γ θ = ( I + θ )Γ , where θ ∈ W 1 , ∞ (Ω , R d ) , || θ || W 1 , ∞ ( R d , R d ) < 1 . 0 | o ( θ ) | J (Γ θ ) = J (Γ) + d J θ → 0 d θ ( θ ) + o ( θ ) , where − − − → 0 , || θ || W 1 , ∞ (Ω , R d )

♥ Hadamard’s method of boundary variations Ω s Ω f θ Γ θ min J (Γ) Γ Γ A descent direction θ ∈ H 1 ( D ) is obtained by solving an identification problem ∀ θ ′ ∈ H 1 ( D ) , a ( θ , θ ′ ) = d J d θ ( θ ′ ) .

Hadamard’s method of boundary variations Ω s Ω f θ Γ θ min J (Γ) Γ Γ A descent direction θ ∈ H 1 ( D ) is obtained by solving an identification problem ∀ θ ′ ∈ H 1 ( D ) , a ( θ , θ ′ ) = d J d θ ( θ ′ ) . Hadamard’s structure theorem: if Γ, θ , and J are smooth enough, then there exists v ∈ L 1 (Γ) such that d J � d θ ( θ ) = v θ · ♥ d s Γ

✇ r ✈ ✉ ❢ ✇ ✈ ✇ ♥ ✇ ✈ ♥ ♥ ✈ ✇ ♥ ♥ ♥ ✉ r ❢ r ♥ r ✉ ♥ ♥ ✉ r ♥ ♥ Analytical shape derivative calculations Outcomes: ◮ We propose a “pedestrian” method to compute shape derivatives in volumetric or surfacic form of general objective functionals in terms of its partial derivatives.

✇ r Analytical shape derivative calculations Outcomes: ◮ We propose a “pedestrian” method to compute shape derivatives in volumetric or surfacic form of general objective functionals in terms of its partial derivatives. 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 ∂ n + 2 k f ( θ · ♥ ) d s ∂ n ∂ n ∂ n Γ � + ( σ s ( ✉ , T s ) : ∇ r − ❢ s · r − ♥ · Ae ( r ) ∇ ✉ · ♥ − ♥ · σ s ( ✉ , T s ) ∇ r · ♥ ) ( θ · ♥ ) d s Γ

Analytical shape derivative calculations Outcomes: ◮ We propose a “pedestrian” method to compute shape derivatives in volumetric or surfacic form of general objective functionals in terms of its partial derivatives. ◮ Adjoint variables ✇ , q , S f , S s , r are solved in a reversed cascade. 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 ∂ n + 2 k f ( θ · ♥ ) d s ∂ n ∂ n ∂ n Γ � + ( σ s ( ✉ , T s ) : ∇ r − ❢ s · r − ♥ · Ae ( r ) ∇ ✉ · ♥ − ♥ · σ s ( ✉ , T s ) ∇ r · ♥ ) ( θ · ♥ ) d s Γ

① ✈ ① r ① ✇ r ✇ ✈ ✇ ✇ ✇ ✇ ✈ ✇ ✈ ✇ ✇ ① ✇ ① ✇ ✈ ✇ r ✉ ♥ ✈ ♥ Adjoint system � Ae ( r ) : ∇ r ′ d ① = ∂ J ∀ r ′ ∈ V ✉ (Γ) . ✉ ( r ′ ) ∂ ˆ Ω s

✇ r ✇ ✈ ✇ ✇ ✇ ✇ ✈ ✇ ✈ ✇ ✇ ① ✇ ① ✇ ✈ ✇ r ✉ ♥ ✈ ♥ Adjoint system � Ae ( r ) : ∇ r ′ d ① = ∂ J ∀ r ′ ∈ V ✉ (Γ) . ✉ ( r ′ ) ∂ ˆ Ω s � � � α div ( r ) S ′ d ① + ∂ J ∀ S ′ ∈ V T (Γ) . k s ∇ S ·∇ S ′ d ① + ( k f ∇ S ·∇ S ′ + ρ c p S ✈ ·∇ S ′ ) d ① = ( S ) ∂ ˆ T Ω s Ω f Ω s

✇ r ✉ ♥ ✈ ♥ Adjoint system � Ae ( r ) : ∇ r ′ d ① = ∂ J ∀ r ′ ∈ V ✉ (Γ) . ✉ ( r ′ ) ∂ ˆ Ω s � � � α div ( r ) S ′ d ① + ∂ J ∀ S ′ ∈ V T (Γ) . k s ∇ S ·∇ S ′ d ① + ( k f ∇ S ·∇ S ′ + ρ c p S ✈ ·∇ S ′ ) d ① = ( S ) ∂ ˆ T Ω s Ω f Ω s ✇ = r on Γ and ∀ ( ✇ ′ , q ′ ) ∈ V ✈ , p (Γ) � � σ f ( ✇ , q ) : ∇ ✇ ′ + ρ ✇ · ∇ ✇ ′ · ✈ + ρ ✇ · ∇ ✈ · ✇ ′ − q ′ div ( ✇ ) � d ① = Ω f � ∂ J − ρ c p S ∇ T · ✇ ′ d ① + ∂ ( ✈ ′ , p ′ ) ( ✇ ′ , q ′ ) , Ω f

Adjoint system � Ae ( r ) : ∇ r ′ d ① = ∂ J ∀ r ′ ∈ V ✉ (Γ) . ✉ ( r ′ ) ∂ ˆ Ω s � � � α div ( r ) S ′ d ① + ∂ J ∀ S ′ ∈ V T (Γ) . k s ∇ S ·∇ S ′ d ① + ( k f ∇ S ·∇ S ′ + ρ c p S ✈ ·∇ S ′ ) d ① = ( S ) ∂ ˆ T Ω s Ω f Ω s ✇ = r on Γ and ∀ ( ✇ ′ , q ′ ) ∈ V ✈ , p (Γ) � � σ f ( ✇ , q ) : ∇ ✇ ′ + ρ ✇ · ∇ ✇ ′ · ✈ + ρ ✇ · ∇ ✈ · ✇ ′ − q ′ div ( ✇ ) � d ① = Ω f � ∂ J − ρ c p S ∇ T · ✇ ′ d ① + ∂ ( ✈ ′ , p ′ ) ( ✇ ′ , q ′ ) , Ω f ✇ = r on Γ : “strange” boundary condition dual to the equality of normal stresses σ s ( ✉ , T s ) · ♥ = σ f ( ✈ , p ) · ♥ on Γ.

Outline 1. Hadamard’s boundary variation method for a simplified three-physics setting 2. Numerical implementation of various test cases with a mesh evolution algorithm

Numerical implementation : mesh evolution algorithm We consider the algorithm proposed by Allaire, Dapogny, Frey (2013): 1. Given a mesh of Ω = Ω s ∪ Ω f and a moving vector field θ

Numerical implementation : mesh evolution algorithm We consider the algorithm proposed by Allaire, Dapogny, Frey (2013): 2. A level-set function φ associated to Ω = Ω s ∪ Ω f is computed on the mesh.

Numerical implementation : mesh evolution algorithm We consider the algorithm proposed by Allaire, Dapogny, Frey (2013): 3. The level-set function is avected on the computational domain which is then adaptively remeshed:

Numerical implementation : mesh evolution algorithm We consider the algorithm proposed by Allaire, Dapogny, Frey (2013): 3. The level-set function is avected on the computational domain which is then adaptively remeshed: Advection of a level set for Ω on the computational mesh.

Numerical implementation : mesh evolution algorithm We consider the algorithm proposed by Allaire, Dapogny, Frey (2013): 3. The level-set function is avected on the computational domain which is then adaptively remeshed: Breaking the zero isoline of the level set.

Numerical implementation : mesh evolution algorithm We consider the algorithm proposed by Allaire, Dapogny, Frey (2013): 3. The level-set function is avected on the computational domain which is then adaptively remeshed: Remeshing adaptively the computational mesh.

A numerical test case : shape optimization of an airfoil Maximization of the lift and minimization of the viscous forces: � � J (Γ) = − ω ❡ y · σ f ( ✈ , p ) · ♥ d s + (1 − ω ) 2 ν e ( ✈ ) : e ( ✈ ) d x ∂ Ω f Ω f

A numerical test case : fluid structure interaction problem Minimization of the compliance: � J (Γ) = Ae ( ✉ ) : e ( ✉ ) d x Ω s

A numerical test case : fluid structure interaction problem Minimization of the compliance: � J (Γ) = Ae ( ✉ ) : e ( ✉ ) d x Ω s

A numerical test case : fluid structure interaction problem

Heat transfer problem Maximization of heat transfer and minimization of viscous dissipation. � � J (Γ) = ω 2 ν e ( ✈ ) : e ( ✈ ) d x − (1 − ω ) ρ c p T f ✈ · ♥ d s ∂ Ω N Ω f f

Heat transfer problem

Heat transfer problem