Numerical solution of the nonlinear evolutional inverse problem related to elastoplastic torsional problem

摘要

This paper is devoted to the determination of an unknown function that describes elastoplastic properties of a bar under torsion. The mathematical (evolution) model leads to an inverse problem that consists of determining the unknown coefficient g = g(ξ~2), ξ~2 = |{nabla}u|~2, in the nonlinear parabolic equation u_t - {nabla}.(g(|{nabla}u|~2){nabla}u)= 2t, (x, y, t) ∈ (Ω_t)~* := Ω × (0, t~*], Ω {is contained in} R~2, using measured output data given in the integral form. Existence of a quasi-solution of the considered inverse problem is obtained in the appropriate class of admissible coefficients. The direct problem is solved using a semi-implicit finite difference scheme. The inverse problem is solved using the semi-analytic inversion method (also known the fast algorithm). Finally, some examples are presented related to direct and inverse problems.