Topological optimization via cost penalization

Applications of implicit parametrizations

“…One question of interest, in this context, is to obtain efficient gradient algorithms, in general shape optimization problems. Certain results of this type are reported in [7], for Dirichlet boundary conditions. Another question is related to the possibility to use just one control in the "extension" (4.5), (4.6) of the state system, while preserving all the other properties.…”

“…This result was proved in [18] and gives the global existence and the periodicity of the solution for the Hamiltonian system (2.9), (2.10). It has an important role in the analysis of shape optimization problems in dimension two, which is a case of interest [18], [7].…”

“…The technique employed in [18] includes as well a modification of {y ε n } outside Ω Φ ε n . In the paper [7], a differentiable variant of this approach is studied. The implicit parametrization theorem gives a global representation of the boundary and allows to compute integrals as in (4.7), (4.8), (4.10).…”

“…This new representation of manifolds (of arbitrary dimension and codimension) was intended for applications in geometric optimization problems and we quote [18,7,21,22] for recent results in this respect. It turns out that it is also useful in mathematical programming and in optimal control as shown in [23,20,25].…”

A Parabolic Shape Optimization Problem

“…We underline that the well known level set method, due to Osher and Sethian [13] (see as well Allaire [2]), is essentially different from our approach. For instance, although we also apply level functions, no Hamilton-Jacobi equation is needed, but ordinary Hamiltonian systems are used instead (see [18], [8]). The final section includes a brief discussion of other problems and methods.…”