An optimal design problem in wave propagation

Abstract

In this paper we consider an optimal design problem in wave propagation proposed in [1] SIGMUND O. and JENSEN J.S. Systematic Design of Phononic Band-Gap Materials and Structures by Topology Optimization, The Royal Society of London, Philosophical Transactions, Series A, Vol. 361, pp. 1001–1019, 2003 in the one-dimensional situation: Given two materials at our disposal with different Young’s modulus and different density, the problem consists of finding the best distributions of the two initial materials in a rod in order to minimize the vibration energy in the structure under periodic loading of driving frequency Ω. We comment on relaxation and optimality conditions, and perform numerical simulations of the optimal configurations. We also prove the existence of classical solutions in certain cases.