Parametric shape optimization using the support function

Numerical shape optimization among convex sets

“…The definition above shows, in particular, that the support function is well adapted for dealing numerically with width or diameter constraints. This was already observed in the previous works [2], [6], [7] or [24]. In [25,Section 1.7] it is shown that the support function…”

“…It can be noted that the optimal shape for k = 2 presented here is comparable to the one obtained in [4] and that the segments in the boundary are well captured by the parametrization proposed here. In general, the values of the objective function obtained with the current method are better than those in [2] since segments in the boundary are better captured. The minimization of the third eigenvalue gives the disk even without the convexity constraint as shown in [22], [3].…”

“…The support function is particularly useful when dealing with numerical aspects related to the convexity constraint in shape optimization. In the works [1], [2], [6] the authors use truncated Fourier series to parametrize support functions of convex sets. As already underlined in the previous paragraphs, this excludes the possibility of having segments in the boundary.…”

“…The recent work [5] shows how to handle the convexity constraint by working with deformed meshes and constraining the admissible deformations. In [6] and [2] the authors use a truncated spectral decomposition of the support function and handle a wide variety of constraints. One drawback of using truncated spectral decompositions for the support function is the smoothness of the support function in the discrete setting.…”

“…In dimension two, in the case K is of class C 1 , a necessary and sufficient condition for h K ∈ C 1 to be a support function of a convex body is to verify h K (θ) + h K (θ) ≥ 0 (in the sense of distributions) for all θ ∈ [0, 2π] (see [25,Chapter 1] or [6] for example). In higher dimensions the characterization of the constraint becomes more complex as shown in [2], for the three dimensional case.…”