Comparison of geometrically nonlinear LSFEM formulations based on different hyperelastic models

Abstract

This contribution deals with the solution of geometrically nonlinear elastic problems solved by the least-squares mixed finite element method (LSFEM). The degrees of freedom (displacements and stresses) will be approximated using suitable spaces, namely W 1,p with p > 4 and H(div, Ω). In order to define the stress response of the material, different hyperelastic free energy functions will be presented. The residual forms R i of the balance of momentum and the constitutive equation build a system of differential equations of first order. Choosing suitable weighting operators and applying L 2 -norms lead to a leastsquares functional F (P ,u). The interpolation of the unknowns is accomplished using a standard polynomial interpolation for the displacements and vector-valued Raviart-Thomas functions for the approximation of the stresses. The formulations presented will be compared considering a uni-axial spatial tension test.

Weighted overconstrained least‐squares mixed finite elements for hyperelasticity

“…STEEGER ET AL. [4]. To fulfill the solution condition δF := 0, we seek the solution (u, P ) ∈ V × X with…”

An improved mixed finite element based on a modified least‐squares formulation for hyperelasticity

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“…Basis for the element formulation is a weak form resulting from a least-squares method, see e.g. [1]. The L2-norm minimization of the residuals of the given first-order system of differential equations leads to a functional depending on displacements and stresses.…”