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.
Highlights
The recent achievements of shape and topology optimization techniques in predicting very efficient designs beyond the reach of the intuition of engineers have raised a tremendous enthusiasm in industry
The variational formulation (1.4) makes it possible to compute integrals (1.3) along the normal rays to ∂Ω without the need to calculate these rays or the curvatures κi explicitly on a discretization of the ambient space. This feature is especially convenient for shape optimization applications relying on the level set method, as described in Section 4; there, the gradient of the signed distance function ∇dΩ is easy to calculate on an unstructured mesh of the considered hold-all domain D from a P1 approximation of dΩ
We provide a variational theory for the advection operator β · ∇ associated to arbitrary C1 vector fields β on the weighted graph space Vω; in particular, we examine the existence of traces on Γ and we prove an adapted Poincare inequality for functions v ∈ Vω
Summary
The recent achievements of shape and topology optimization techniques in predicting very efficient designs beyond the reach of the intuition of engineers have raised a tremendous enthusiasm in industry. The variational formulation (1.4) makes it possible to compute integrals (1.3) along the normal rays to ∂Ω without the need to calculate these rays or the curvatures κi explicitly on a discretization (i.e. a mesh) of the ambient space This feature is especially convenient for shape optimization applications relying on the level set method, as described in Section 4; there, the gradient of the signed distance function ∇dΩ is easy to calculate on an unstructured mesh of the considered hold-all domain D from a P1 approximation of dΩ. The previous arguments work identically when ∇dΩ is replaced with an arbitrary C1 vector field β: we obtain that a variational formulation analogous to (1.4) (given in (2.5)) allows to compute integrals quantities along the characteristics curves of β, which is subject to offer wider applications than shape optimization With these perspectives in mind, the present article is organized as follows. Our working assumptions are of a different nature: for instance, the hypothesis (H3) essentially requires that the inverse weight α−1 belong to the Muckenhoupt class A1
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