Identification of Parameters for Thermoelasticity Models

An algorithm for identiication of unknown parameters of a Landau-Devonshire model describing phase transitions in shape memory alloys 1] is proposed. Distributions of the displacement and of the temperature observed with an error are input data of the algorithm. Finite element approximations of the input data and of the model are used. The algorithm utilizes the idea of the direct minimization of the residual of equations written in an appropriate variational form 4, 5]. The convergence of the algorithm output to the set of all parameters compatible with the exact solution is proved. The question of the convergence under local measurements of input data is discussed and suucient conditions for the convergence are given. The paper is illustrated by computer simulations. 1. Formulation of the problem and the algorithm We consider a model of structural phase transitions in shape memory alloys reported in 2, 3]. We denote = (0; 1), I = (0; T) and Q T = I. The model is described by the following system: u tt ? p(t; ; u x) x + u xxxx = f; (t; x) 2 Q T ; c 0 t ? k 0 xx ? 1 u x u xt = g; (t; x) 2 Q T : (1) Our objective is to construct an algorithm for nding the function p on the basis of solutions of (1) observed with an error. Speaking more deenitely, we have functions u ; such that where u; is an exact solution of (1) and is an error of measurements. We are to construct an algorithm that processes the functions u ; and gives on the output a function p such that p ! p as ! 0. We assume that the function p can be found in the following parametric form: