Simultaneous determination of the strain hardening exponent, the shear modulus and the elastic stress limit in an inverse problem

• Some important parameters of a class of engineering materials are determined simultaneously in an inverse coefficient problem.
• The inverse problem is solved for both noise free and noisy data.
• A non-local additional condition is used to solve the inverse problem.

This paper is devoted to simultaneous determination of the strain hardening exponent, the shear modulus and the elastic stress limit in an inverse problem. The inverse problem consists of determining the unknown coefficient f=f(T2),T2:=|∇u|2f=f(T2),T2:=|∇u|2 in the nonlinear equation ut−∇.(f(T2)∇u)=2t,ut−∇.(f(T2)∇u)=2t,(x,y,t)∈ΩT:=Ω×(0,T),Ω⊂R2,Ω⊂R2, by measured output data (or additional data) given in the integral form. After we solve direct problem using a semi-implicit finite difference scheme, a numerical method based on discretization of the minimization problem, steepest descent method and least squares method is proposed for the solution of the inverse problem. We use Tikhonov regularization to overcome the ill-posedness of the inverse problem. Numerical examples with noise free and noisy data illustrate applicability and accuracy of the proposed method to some extent.