On optimization problems with prescribed rearrangements

In this paper, a PDE-constrained optimization problem corresponding to the bi-Laplacian operator is investigated. The optimization problem arises when we search for the maximum load potential in a uniform elastic plate hinged at its boundary with a given total amount of force. It is established that there is a bang–bang solution to the optimization problem, and an optimality condition is derived. Based on the optimality condition, we obtain some topological and geometrical properties of the maximizer. Although analytical solutions for PDE-constrained problems are very rare, an analytical solution is determined if the plate is circular and its thickness has a particular form. In those cases, it is proved that the solution is unique. For plates of general shape, an efficient numerical algorithm is developed to solve the PDE-constrained optimization problem, and its convergence is studied. According to the results, one can determine the profile of the regions in a plate where the largest and smallest external force should be exerted to have the maximum load potential. According to the research results, the thickness of the plate plays a crucial role in determining the shape of those regions. Therefore, disregarding the impact of plate thickness in our studies for the sake of simplicity would be impractical and might result in inaccurate outcomes. By utilizing our findings, engineers and designers can determine the optimal external force pattern that yields the maximum load potential for plates. This, in turn, enables them to create plates that are more efficient and effective, tailored to meet their specific needs and requirements.

The present chapter is devoted to introduce the reader to the applications of the homogenization theory to the resolution of optimal material design problems. With this aim, we consider a model problem consisting in distributing m diffusion materials in a given body with prescribed amounts in order to minimize a cost functional depending on the state function. It is known that this type of problems has no solution in general and thus the need of work with a relaxed formulation which can be obtained by means of the homogenization theory. The idea is to replace the original materials by composite materials obtained as microscopic mixtures of them. From the numerical point of view, besides the existence of solution, the relaxed problem has better convexity and derivability properties which allow us to use more efficient algorithms. Here we get the relaxed formulation, we show the convergence of a discrete version of the problem based on the finite element method, we provide a gradient descent method adapted to the problem and we carry out some numerical experiments.