Optimal design for 2D wave equations

• Published in: Optimization methods & software Vol. 32; no. 1; pp. 86 - 108
• Main Author: Cea, M.
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 02-01-2017; Taylor & Francis Ltd
• Subjects: adjoint equation; Algorithms; Boundary conditions; complementary-Hausdorff topology; Computer programs; Convergence; Design optimization; Dirichlet problem; Domains; Efficiency; finite elements; Mathematical analysis; Mathematical models; Numerical analysis; optimal control; Optimization; shape optimization; topological derivative; wave equation; Wave equations
• ISSN: 1055-6788; 1029-4937
• DOI: 10.1080/10556788.2016.1201083

In this paper We consider a problem of optimal design in 2D for the wave equation with Dirichlet boundary conditions. We introduce a finite element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. We prove that, as the mesh size tends to zero, any limit, in the sense of the complementary-Hausdorff convergence, of discrete optimal shapes is an optimal domain for the continuous optimal design problem. We work in the functional and geometric setting introduced by V. Šveràk in which the domains under consideration are assumed to have an a priori limited number of holes. We present in detail a numerical algorithm and show the efficiency of the method through various numerical experiments.