A MATLAB-Based Symbolic Approach for the Quick Developing of Nonlinear Solid Mechanics Finite Elements

A symbolic mathematical approach for the rapid early phase developing of finite elements is proposed. The algebraic manipulator adopted is MATLAB® and the applicative context is the analysis of hyperelastic solids or structures under the hypothesis of finite deformation kinematics. The work has been finalized through the production, in an object-oriented programming style, of three MATLAB® classes implementing a truss element, a tetrahedral element and plane element. The approach proposed, starting from the mathematical formulation and finishing with the code implementation, is described and its effectiveness, in terms of minimization of the gap between the theoretical formulation and its actual implementation, is highlighted.

Analytical Comparison of Two Multiscale Coupling Methods for Nonlinear Solid Mechanics

The aim of this work is to compare two existing multilevel computational approaches coming from two different families of multiscale methods in a nonlinear solid mechanics framework. A locally adaptive multigrid method and a numerical homogenization technique are considered. Both classes of methods aim to enrich a global model representing the structure’s behavior with more sophisticated local models depicting fine localized phenomena. It is clearly shown that even being developed with different vocations, such approaches reveal several common features. The main conceptual difference relying on the scale separation condition has finally a limited influence on the algorithmic aspects. Hence, this comparison enables to highlight a unified framework for multiscale coupling methods.