Model order reduction for hyperelastic materials

In this paper, we develop a novel algorithm for the dimensional reduction of the models of hyperelastic solids undergoing large strains. Unlike standard proper orthogonal decomposition methods, the proposed algorithm minimizes the use of the Newton algorithms in the search of non-linear equilibrium paths of elastic bodies.The proposed technique is based upon two main ingredients. On one side, the use of classic proper orthogonal decomposition techniques, that extract the most valuable information from pre-computed, complete models. This information is used to build global shape functions in a Ritz-like framework.On the other hand, to reduce the use of Newton procedures, an asymptotic expansion is made for some variables of interest. This expansion shows the interesting feature of possessing one unique tangent operator for all the terms of the expansion, thus minimizing the updating of the tangent stiffness matrix of the problem.The paper is completed with some numerical examples in order to show the performance of the technique in the framework of hyperelastic (Kirchhoff-Saint Venant and neo-Hookean) solids.