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)
Abstract:
Hadamard’s method of shape differentiation is applied to
topology optimization of a weakly coupled three physics problem. The
coupling is weak because the equations involved are solved consecutively,
namely the steady state Navier–Stokes equations for the fluid domain,
first, the convection diffusion equation for the whole domain, second, and
the linear thermo-elasticity system in the solid domain, third. Shape
sensitivities are derived in a fully Lagrangian setting which allows us to
obtain shape derivatives of general objective functions. An emphasis is
given on the derivation of the adjoint interface condition dual to the one
of equality of the normal stresses at the fluid solid interface. The
arguments allowing to obtain this surprising condition are specifically
detailed on a simplified scalar problem. Numerical test cases are presented
using the level set mesh evolution framework of Allaire et al. (Appl Mech
Eng 282:22–53, 2014). It is demonstrated how the implementation enables
to treat a variety of shape optimization problems.
@article{Feppon2019Sep,
author = {Feppon, F. and Allaire, G. and Bordeu, F. and Cortial, J. and Dapogny, C.},
title = {{Shape optimization of a coupled thermal fluid-structure problem in a level set mesh evolution framework}},
journal = {SeMA},
volume = {76},
number = {3},
pages = {413--458},
year = {2019},
month = {Sep},
issn = {2254-3902},
publisher = {Springer International Publishing},
doi = {10.1007/s40324-018-00185-4}
}
Abstract:
The purpose of this article is to introduce a gradient-flow
algorithm for solving generic equality or inequality constrained
optimization problems, which is suited for shape optimization applications.
We rely on a variant of the Ordinary Differential Equation (ODE) approach
proposed by Yamashita for equality constrained problems: the search
direction is a combination of a null space step and a range space step,
which are aimed to reduce the value of the minimized objective function and
the violation of the constraints, respectively. Our first contribution is
to propose an extension of this ODE approach to optimization problems
featuring both equality and inequality constraints. In the literature, a
common practice consists in reducing inequality constraints to equality
constraints by the introduction of additional slack variables. Here, we
rather solve their local combinatorial character by computing the
projection of the gradient of the objective function onto the cone of
feasible directions. This is achieved by solving a dual quadratic
programming subproblem whose size equals the number of active or violated
constraints, and which allows to identify the inequality constraints which
should remain tangent to the optimization trajectory. Our second
contribution is a formulation of our gradient flow in the context
of-infinite-dimensional-Hilbert space settings. This allows to extend the
method to quite general optimization sets equipped with a suitable manifold
structure, and notably to sets of shapes as it occurs in shape optimization
with the framework of Hadamard's boundary variation method. The cornerstone
of this latter setting is the classical operation of extension and
regularization of shape derivatives. Some numerical comparisons on simple
academic examples are performed to illustrate the behavior of our
algorithm. Its numerical efficiency and ease of implementation are finally
demonstrated on more realistic shape optimization problems.
@article{feppon2020optim,
author = {{Feppon, F.} and {Allaire, G.} and {Dapogny, C.}},
doi = {10.1051/cocv/2020015},
journal = {ESAIM: COCV},
pages = {90},
title = {Null space gradient flows for constrained optimization with applications to shape optimization},
url = {https://doi.org/10.1051/cocv/2020015},
volume = 26,
year = 2020
}
[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)
Abstract:
This work is devoted to shape and topology optimization of multiphysics systems
motivated by aeronautic industrial applications. Shape derivatives of arbitrary
objective functionals are computed for a weakly coupled thermal fluid-structure
model. A novel gradient flow type algorithm is then developed for solving generic
constrained shape optimization problems without the need for tuning non-physical
metaparameters. Motivated by the need for enforcing non-mixing constraints in the
design of liquid-liquid heat exchangers, a variational method is developed in order
to simplify the numerical evaluation of geometric constraints: it allows to compute
line integrals on a mesh by solving a variational problem without requiring the
explicit knowledge of these lines on the spatial discretization. All these
ingredients allowed us to implement a variety of 2-d and 3-d multiphysics shape
optimization test cases: from single, double or three physics problems in 2-d, to
moderately large-scale 3-d test cases for structural design, thermal conduction,
aerodynamic design and a fluid-structure interacting system. A final opening chapter
derives high order homogenized equations for the Stokes system in a porous medium.
These high order equations encompass the three classical
homogenized regimes---namely Stokes, Brinkman and Darcy---associated with different
obstacle's size scalings. They could
allow, in future works, to develop new topology optimization methods for the design
of fluid systems characterized by multi-scale patterns such as industrial heat exchangers.
@phdthesis{feppon2020,
author = {Feppon, Florian},
title = {Shape and topology optimization of multiphysics systems},
school = {Th\`{e}se de doctorat de l'Universit'{e} Paris-Saclay pr'{e}par'{e}e
\`a l''{E}cole polytechnique},
year = {2019}
}
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: \[
\begin{aligned}
\min_{\Omega_s\subset D} & \quad J(\Omega_s,{\bf{u}}{} (\Omega_s)):=\int_{\Omega_s} Ae({\bf{u}}):e({\bf{u}})\D x\\
s.t. & \quad \mathrm{Vol}(\Omega_s):= \int_{\Omega_s} \D x =0.6.
\end{aligned}
\]