The Tikhonov regularization method in elastoplasticity

The numerical simulation of the mechanical behavior of industrial materials is widely used for viability verification, improvement and optimization of designs. Elastoplastic models have been used to forecast the mechanical behavior of different materials. The numerical solution of most elastoplastic models comes across problems of ill-condition matrices. A complete representation of the nonlinear behavior of such structures involves the nonlinear equilibrium path of the body and handling of singular (limit) points and/or bifurcation points. Several techniques to solve numerical problems associated to these points have been disposed in the specialized literature. Two examples are the load-controlled Newton-Raphson method and displacement controlled techniques. However, most of these methods fail due to convergence problems (ill-conditioning) in the neighborhood of limit points, specially when the structure presents snap-through or snap-back equilibrium paths. This study presents the main ideas and formalities of the Tikhonov regularization method and shows how this method can be used in the analysis of dynamic elastoplasticity problems. The study presents a rigorous mathematical demonstration of existence and uniqueness of the solution of well-posed dynamic elastoplasticity problems. The numerical solution of dynamic elastoplasticity problems using Tikhonov regularization is presented in this paper. The Galerkin method is used in this formulation. Effectiveness of Tikhonov's approach in the regularization of the solution of elastoplasticity problems is demonstrated by means of some simple numerical examples.