A Level Set Based Shape Optimization Method for an Elliptic Obstacle Problem

Abstract

In this paper we construct a level set method for an elliptic obstacle problem, which can be reformulated as a shape optimization problem. We provide a detailed shape sensitivity analysis for this reformulation and a stability result for the shape Hessian at the optimal shape.Using the shape sensitivities we construct a geometric gradient flow, which can be realized in the context of level set methods. We prove the convergence of the gradient flow to an optimal shape and provide a complete analysis of the level set method in terms of viscosity solutions. To our knowledge this is the first complete analysis of a level set method for a nonlocal shape optimization problem.Finally, we discuss the implementation of the methods and illustrate its behavior through several computational experiments.

A direct algorithm in some free boundary problems

“…Our approach is inspired from shape optimization techniques, but no shape optimization problem is used here although this is a known method in free boundary problems, [5,23]. One may compare the present approach to the recent works [14,24,28].…”

“…Moreover, the coincidence set associated to the perturbed solutions cannot be distinguished graphically from the unperturbed one represented in Figure 2 at the left. Tests 2 In [5], a level set method with artificial time variable is discussed which approximates the steady obstacle problem. We consider the second and the third examples from [5].…”

“…Tests 2 In [5], a level set method with artificial time variable is discussed which approximates the steady obstacle problem. We consider the second and the third examples from [5]. Since the considered boundary data are very close to the given obstacle (just slightly different in certain parts of the boundary) finding the coincidence set becomes a rather difficult task.…”

“…The Algorithm 1 stops after 4 iterations if c = 3. In [5], the parameters c = 1, c = 0.5 are used, but in our algorithms the round-off errors influence the result if c < 3. Test 2.2.…”

“…Test 2.2. The domain D for the third example from [5] is a segment of a circle with radius 1, opening angle π/3 and center (0, 0). As in the Test 2.1, we set f(x) = −1 and ψ(x) = 0.…”