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

Abstract

In the formulation of shape optimization problems, multiple geometric constraint functionals involve the signed distance function to the optimized shape Ω. The numerical evaluation of their shape derivatives requires to integrate some quantities along the normal rays to Ω, a challenging operation to implement, which is usually achieved thanks to the method of characteristics. The goal of the present paper is to propose an alternative, variational approach for this purpose. Our method amounts, in full generality, to compute integral quantities along the characteristic curves of a given velocity field without requiring the explicit knowledge of these curves on the spatial discretization; it rather relies on a variational problem which can be solved conveniently by the finite element method. The well-posedness of this problem is established thanks to a detailed analysis of weighted graph spaces of the advection operator β ⋅ ∇ associated to a C1 velocity field β. One novelty of our approach is the ability to handle velocity fields with possibly unbounded divergence: we do not assume div(β) ∈ L∞. Our working assumptions are fulfilled in the context of shape optimization of C2 domains Ω, where the velocity field β = ∇dΩ is an extension of the unit outward normal vector to the optimized shape. The efficiency of our variational method with respect to the direct integration of numerical quantities along rays is evaluated on several numerical examples. Classical albeit important implementation issues such as the calculation of a shape’s curvature and the detection of its skeleton are discussed. Finally, we demonstrate the convenience and potential of our method when it comes to enforcing maximum and minimum thickness constraints in structural shape optimization.

Body-fitted topology optimization of 2D and 3D fluid-to-fluid heat exchangers

“…(2.13) see [4,10,44]. Unfortunately, the analytical expression of the scalar field v P (Ω) : Γ → R (which is not reported here for brevity) involves integrals along the normal rays to ∂Ω, as well as the principal curvatures of ∂Ω.…”

“…These issues can be overcome thanks to the variational method from our previous work [44]: instead of evaluating the analytic formula for the function v P (Ω), we actually compute it numerically as the solution to a variational problem. More precisely, we consider the problem…”

“…which is easily solved numerically by means of standard piecewise linear finite elements. In the above formulation, the function ω is a positive weight which is continuous on D\Σ, while V ω is a weighted Hilbert space, whose precise definition and mathematical properties are described in Section 2 of [44].…”

“…Our three-dimensional implementation involves the solution of moderately large-scale finite element systems, computational meshes considered feature 1 to 2 million tetrahedra, which are performed thanks to domain decomposition and preconditioning techniques described in [45]. Finally, the variational method of [44] allows us to enforce conveniently our non-penetration and minimum thickness constraints.…”

“…This raises the need to add a non-mixing constraint between both fluid components, or alternatively, a constraint whereby the minimum distance between them be positive. It is formulated as an averaged constraint functional involving the signed distance functions to both fluid subdomains, whose shape derivatives are efficiently evaluated thanks to the variational method of [44]. Let us emphasize that the implementation of this variational method is rather straightforward in both 2D and 3D.…”