Dan Tiba

Fixed domain approaches in shape optimization problems

This work is a review of results in the approximation of optimal design problems, defined in variable/unknown domains, based on associated optimization problems defined in a fixed ?hold-all? domain, including the family of all admissible open sets. The literature in this respect is very rich and we concentrate on three main approaches: penalization?regularization, finite element discretization on a fixed grid, controllability and control properties of elliptic systems. Comparison with other fixed domain approaches or, in general, with other methods in shape optimization is performed as well and several numerical examples are included.

Free boundary conditions for intrinsic shell models

The compactness of trajectories of solutions to various phase-field models is proved. In some cases, the convergence of any strong solution to a single stationary state is also established.

General Optimality Conditions for Constrained Convex Control Problems

In this paper we investigate some optimal convex control problems, with mixed constraints on the state and the control. We give a general condition which allows us to set optimality conditions for nonqualified problems (in the Slater sense). Then we give some applications and examples involving generalized bang-bang results.

Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions

Fixed domain methods have well-known advantages in the solution of variable domain problems including inverse interface problems. This paper examines two new control approaches to optimal design problems governed by general elliptic boundary value problems with Dirichlet boundary conditions. Numerical experiments are also included.

On the Approximation and the Optimization of Plates

This work is devoted to the study of simply supported and of clamped plates together with related variational inequalitiesand optimization problems. We introduce a new unitary approach based on distributed optimal control problems governed by second order elliptic boundary value problems and their penalization. This approach gives the possibility to approximate the solution via piecewise linear continuous finite elements and is simpler than other methods considered in the literature.The convergence with respect to the penalization parameter (ε) is proved under very general assumptions. In order to solve the obtained control problems, optimization procedures of steepest descent type are considered. Relevantnumerical examples illustrate the applicability of the proposed methods.

Existence for Shape Optimization Problems in Arbitrary Dimension

We discuss some existence results for optimal design problems governed by second order elliptic equations with the homogeneous Neumann boundary conditions or with the interior transmission conditions. We show that our continuity hypotheses for the unknown boundaries yield the compactness of the associated characteristic functions, which, in turn, guarantees convergence of any minimizing sequences for the first problem. In the second case, weaker assumptions of measurability type are shown to be sufficient for the existence of the optimal material distribution. We impose no restriction on the dimension of the underlying Euclidean space.

Sur les problèmes d'optimisation structurelle

We discuss existence theorems for shape optimization and material distribution problems. The conditions that we impose on the unknown sets are continuity of the boundary, respectively a certain measurability hypothesis.

A direct algorithm in some free boundary problems

Abstract In this paper we propose a new algorithm for the well known elliptic obstacle problem and for parabolic variational inequalities like one- and two- phase Stefan problem and of obstacle type. Our approach enters the category of fixed domain methods and solves just linear elliptic or parabolic equations and their discretization at each iteration. We prove stability and convergence properties. The approximating coincidence set is explicitly computed and it converges in the Hausdorff-Pompeiu sense to the searched geometry. In the numerical examples, the algorithm has a very fast convergence and the obtained solutions (including the free boundaries) are accurate.

Finite Element Approximation for Shape Optimization Problems with Neumann and Mixed Boundary Conditions

For optimal design problems, defined in domains of class

One Hundred Years Since the Introduction of the Set Distance by Dimitrie Pompeiu

C