Shape optimization of a coupled thermal fluid–structure problem in a level set mesh evolution framework

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 [4]. It is demonstrated how the implementation enables to treat a variety of shape optimization problems.

Null space gradient flows for constrained optimization with applications to shape optimization

“…In the implementation, we use the mesh evolution technique of our previous works [5,32]. In a few words, at every iteration n, the current shape Ω n is explicitly discretized as a submesh of a triangulated mesh T n of D as a whole (see e.g.…”

“…A common strategy in the literature (see for instance [10,14,21,24,32,45]) consists in taking simply H 1 (D, R d ) as for the Hilbert space V , equipped with the inner product ∀θ, θ ∈ V, θ, θ V = D (γ 2 ∇θ : ∇θ + θ · θ )dx, (5.5) where γ > 0 is a user-defined parameter which can physically be interpreted as a length-scale for the regularity of deformations θ (typically, γ = 3 hmin where hmin is the minimum edge length of the mesh discretization). Note that this choice of V is an abuse of the above framework since H 1 (D, R d ) is not a subspace of W 1,∞ (D, R d ).…”

“…An open-source implementation of our null space algorithm is made freely available online at https://gitlab.com/safrandrti/null-space-project-optimizer. The repository includes demonstrative pedagogical examples for the resolution of optimization problems in R d , but can also serve as a basis for the resolution of more complicated applications; it has been used as is for the realisation of the shape optimization test cases of this article and other of our related works [31,32].…”

A variational formulation for computing shape derivatives of geometric constraints along rays

“…The resolution of the constrained optimization program (4.2) is carried out by the discretization of a specific gradient flow in the spirit of [56], see the details of our algorithm in our recent work [37]. When it comes to representing shapes and their evolution in the course of the iterative resolution of (4.2), the level set based mesh evolution method of [4,36] is used, as a convenient combination of the "classical" level set method for shape and topology optimization [2,61] and the mmg open-source mesh modification algorithm [21].…”

First and Second Order Shape Optimization Based on Restricted Mesh Deformations

“…Over the past 10 years, a range of various techniques have been proposed to circumvent this major obstacle in computational shape optimization. A natural choice is to remesh the computational domain; see for instance Wilke, Kok, Groenwold, 2005;Morin et al, 2012;Sturm, 2016;Dokken et al, 2018;Feppon et al, 2018. Remeshing can be carried out either in every iteration or whenever some measure of mesh quality falls below a certain threshold. Drawbacks of remeshing include the high computational cost and the discontinuity introduced into the history of the objective values.…”