Mixed Isogeometric Discretizations for Planar Linear Elasticity

TL;DR

This paper develops two isogeometric analysis-based discretization methods for planar linear elasticity, addressing curved geometries and ensuring stability, with proven convergence and numerical validation.

Contribution

It introduces two novel IGA discretization approaches for linear elasticity, including handling curved boundaries and strong symmetry, with stability and convergence proofs.

Findings

Both methods are proven to converge.

Numerical experiments demonstrate approximation accuracy.

The methods handle curved geometries effectively.

Abstract

In this article we suggest two discretization methods based on isogeometric analysis (IGA) for planar linear elasticity. On the one hand, we apply the well-known ansatz of weakly imposed symmetry for the stress tensor and obtain a well-posed mixed formulation. Such modified mixed problems have been already studied by different authors. But we concentrate on the exploitation of IGA results to handle also curved boundary geometries. On the other hand, we consider the more complicated situation of strong symmetry, i.e. we discretize the mixed weak form determined by the so-called Hellinger-Reissner variational principle. We show the existence of suitable approximate fields leading to an inf-sup stable saddle-point problem. For both discretization approaches we prove convergence statements and in case of weak symmetry we illustrate the approximation behavior by means of several numerical experiments.