Optimal Calculation of Solid-Body Deformations with Prescribed Degrees of Freedom over Smooth Boundaries

Abstract

One of the reasons why the finite element method became the most used technique in Computational Solid Mechanics is its versatility to deal with bodies having a curved shape. In this case method’s isoparametric version for meshes consisting of curved triangles or tetrahedra has been mostly employed to recover the optimal approximation properties known to hold for standard elements in the case of polygonal or polyhedral domains. However isoparametric finite elements helplessly require the manipulation of rational functions and the use of numerical integration. This can be a brain teaser in many cases, especially if the problem at hand is non linear. We consider a simple alternative to deal with boundary conditions commonly encountered in practical applications, that bypasses these drawbacks, without eroding the quality of the finite-element model. More particularly we mean prescribed displacements or forces in the case of solids. Our technique is based only on polynomial algebra and can do without curved elements. Although it can be applied to countless types of problems in Continuum Mechanics, it is illustrated here in the computation of small deformations of elastic solids.