Finite element method for localization analysis

A methodology has been developed for solving problems involving interaction between micro-scale phenomena and the overall macroscopic behavior of the structure. These methodologies are aimed at problems such as localization in a shear band or crack propagation, where the computational model must simultaneously treat phenomena at several scales. This is accomplished by developing several numerical schemes which can accurately resolve the structure of strain fields with high gradients. A common feature of all these methods is that the interpolated strain field is composed of two parts: a lower order strain field resulting from a C;In the first technique, the shape of the interpolated "micro" field is assumed to be C;In the second technique, the structure of the interpolated "micro" strain field can be arbitrary and it is approximated by piecewise constant interpolants, which are used to "enrich" the element. This technique is useful for problems involving complex micro-mechanical behavior, where the deformation field cannot be estimated a priori.;In the third method, spectral interpolants are used to approximate the "micro" strain field. The spectral approximation is superimposed on the finite element approximation over the spectral patch which is placed over the region on high gradients. This method originates from the desire to combine the generality of the finite element method with regard to complex boundary conditions with the so-called infinite order convergence of the spectral methods.