Shape and Topology optimization of 2-D multiphysics systems

The examples below illustrate the applications of Hadamard’s boundary variation method and our nullspace gradient flow for tackling various shape optimization problems of fluid thermal mechanical systems.

Our implementation relies on a level-set based mesh evolution algorithm implemented with the remeshing software Mmg, and on our null space gradient flow for nonlinear constrained optimization. See the following publications for full details:

[9] Feppon, F., Allaire, G., Bordeu, F., Cortial, J. and Dapogny, C. Shape optimization of a coupled thermal fluid-structure problem in a level set mesh evolution framework (2019). SeMA, 76: 413. HAL preprint hal-01686770. (abstract) (bibtex)

[11] Feppon, F., Allaire, G. and Dapogny, C. Null space gradient flows for constrained optimization with applications to shape optimization (2020). ESAIM: COCV, 26 90 (Open Access). HAL preprint hal-01972915. (abstract) (bibtex)

[PhD] Feppon, F. Shape and topology optimization of multiphysics systems (2019). Thèse de doctorat de l'Université Paris-Saclay préparée à l'École polytechnique. (abstract) (bibtex)

Thermal diffusion

In this example we minimize the average temperature over the domain: \[ \newcommand{\D}{\mathrm{d}} \renewcommand{\div}{\mathrm{div}} \begin{aligned} \min_{\Gamma} & \quad \int_{D} T\D x \\ s.t. & \left\{ \begin{aligned} \int_{\Omega_s} \D x & = V_0 \\ -\div(k_s \nabla T) & = Q_s \textrm{ in }\Omega_s\\ -\div(k_f \nabla T) & = Q_f \textrm{ in }\Omega_f\\ T& = T_0\textrm{ on }\Gamma_D \\ T& \textrm{ continuous on }\Gamma \end{aligned}\right. \end{aligned}.\] where \(\Omega_s\) and \(\Omega_f\) are the black and white (fluid and solid) domain respectively, and \(\Gamma=\Omega_f\cap\Omega_s\) is the common interface.

2D airfoil

In this example we maximize the lift generated by a flow obstacle subject to a constraint on the volume and the position and with a maximum drag value: \[\newcommand{\v}{{\bf v}} \newcommand{\n}{\bf n} \newcommand{\<}{\leqslant} \begin{aligned} \min_{\Gamma} & \quad \int_{\partial\Omega_f} \bf e_y\cdot\sigma_f(\v,p)\cdot \n\D s \\ s.t. & \left\{ \begin{aligned} \int_{\Omega_f} \D x & = V_0 \\ \int_{\Omega_f}\bf x\D x & = \bf x_0 \\ \int_{\Omega_f} \sigma_f(\v,p):\nabla \v\D x & \< C_0\\ -\Delta \v +\rho \nabla \v \v +\nabla p & = 0 \textrm{ in }\Omega_f\\ \v &= \v_0 \textrm{ on }\Gamma_D\\ \v &= 0 \textrm{ on }\Gamma, \end{aligned}\right. \end{aligned}.\] where \(\sigma_f({\bf v}{},p)=2\nu e({\bf v}{})-pI\) is the fluid stress tensor.

Fluid-structure interaction

Here we maximize the rigidity of a structure subject to the force of a left incoming flow and a volume constraint: \[ \begin{aligned} \min_{\Gamma} & \quad \int_{\Omega_s} Ae({\bf{u}}):e({\bf{u}})\D x \\ s.t. & \left\{ \begin{aligned} \int_{\Omega_s} \D x & = V_0 \\ -\Delta {\bf v}+\rho \nabla {\bf v}{\bf v}+\nabla p & = 0 \textrm{ in }\Omega_f\\ \div({\bf v}) & = 0 \textrm{ in }\Omega_f\\ -\div(Ae({\bf{u}})) & = 0 \textrm{ in }\Omega_s\\ {\bf v}&= {\bf v}_0 \textrm{ on }\Gamma_D\\ {\bf v}&= 0 \textrm{ on }\Gamma\\ \sigma_f({\bf v},p)\cdot\bf n& = Ae({\bf{u}})\cdot\bf n\textrm{ on }\Gamma, \end{aligned}\right. \end{aligned}.\] where \(Ae({\bf{u}})=2\mu e({\bf{u}})+\lambda \div({\bf{u}}) I\) is the Hooke’s law. The fluid-structure constraint is enforced by the equality of the normal stresses \(\sigma_f({\bf v}{},p)\cdot\bf n= Ae({\bf{u}}{})\cdot\bf n{}\) on \(\Gamma\) (weak coupling).

Convective heat transfer

The goal of this test case is to maximize the heat transferred by the fluid subject to an upper bound on the output pressure drop and a volume constraint: \[ \begin{aligned} \min_{\Gamma} & \quad J(\Gamma,{\bf v}(\Gamma),T(\Gamma)):=-\int_{\Omega_f} \rho c_p {\bf v}\cdot\nabla T\D x\\ s.t. & \quad \left\{\begin{aligned} \texttt{DP}(p(\Gamma)):=\int_{\partial\Omega_{f}^{D}} p\D s -\int_{\partial\Omega_{f}^{N}} p\D s & \< \texttt{DP}_{static}\\ \mathrm{Vol}(\Omega_f) &= V_{target}. \end{aligned}\right. \end{aligned} \]

The temperature variable \(T\) is determined by a convection-diffusion equation, which itself depends on the fluid variable \({\bf v}\) through the incompressible Navier-Stokes equations: \[ \newcommand{\partialn}[1]{\frac{\partial#1}{\partial \bf n}} \newcommand{\In}{\textrm{ in }} \newcommand{\On}{\textrm{ on }} \left\{ \begin{aligned} -\Delta {\bf v}+\rho \nabla {\bf v}{\bf v}+\nabla p & = 0 \textrm{ in }\Omega_f\\ \div({\bf v}) &= 0 \textrm{ in }\Omega_f\\ -\div(k_f\nabla T_f)+\rho c_p{\bf v}\cdot\nabla T_f& =0 \In\Omega_f\\ -\div(k_s\nabla T_s)& = 0\In\Omega_s\\ T&=T_0 \On\partial\Omega_{T}^D \\ T_f &= T_s\On\Gamma\\ -k_f\partialn{T_f}&=-k_s \partialn{T_s} \On\Gamma,\\ {\bf v}&= {\bf v}_0 \textrm{ on }\Gamma_D\\ {\bf v}&= 0 \textrm{ on }\Gamma\\ \end{aligned}\right. \]

2-D three physics system

This test case is somewhat very academic, however it demonstrates the ability of our methodology to handle all three physics simultaneously: hydraulic, thermic and mechanical properties are coupled through convection, fluid-structure interaction and thermal dilation.
The goal of the test case is to maximize the rigidity of a structure subjected to an input flow and thermal load (see the above publications for the full details), and to a volume constraint: \[ \begin{aligned} \min_{\Omega_s\subset D} & \quad J(\Omega_s,{\bf{u}}{} (\Omega_s)):=\int_{\Omega_s} Ae({\bf{u}}):e({\bf{u}})\D x\\ s.t. & \quad \mathrm{Vol}(\Omega_s):= \int_{\Omega_s} \D x =0.6. \end{aligned} \]