Let (upsilon_{1} x,t)=upsilon x,t k_{1})) and (mathit{upsilon}_{2} x,t)=upsilon x,t k_{2})) be the solutions of direct problem (2), corresponding to the admissible parameters (k_{1}(x inmathcal{nmathcal{K}_{0}).
The following lemma implies the relation between the parameters k 1 ( x ), k 2 ( x ) ∈ K 1 at x = 1 and the corresponding outputs h j ( t ) : = u ( 1, t ; k j ), j = 1, 2. Lemma 2 Let υ 1 ( x, t ) = υ ( x, t ; k 1 ) and υ 2 ( x, t ) = υ ( x, t ; k 2 ) be the solutions of direct problem (2), corresponding to the admissible parameters k 1 ( x ), k 2 ( x ) ∈ K 1.
Conversely, if is a solution of (2.3), direct differentiation of (2.3) implies that, for, (2.15).
In Section 2, we give a representation of the solution of the direct problem using the Laplace transform.
As preparation, we study the solution of the direct problem based on the Banach fixed-point theorem.
The solution of the direct problem is performed on the basis of the BFM (body force method).
The solution of the direct problem is first obtained and the peculiarities of damage detection are examined.
An innovative solution strategy and a computational algorithm are presented for the solution of the direct deformation problem comprised of the kinematic, constitutive, contact and thermal sub-problems.
The main idea of this extension is to design a new "quantity of interest" based on the solution of the direct problem and the given multiple goals.
