Solving hyperelastic problems using mixed LSFEM

The focus of this contribution is the solution of hyperelastic problems using the least-squares finite element method (LSFEM). In particular a mixed least-squares finite element formulation is provided and applied on geometrically nonlinear problems. The basis for the element formulation is a div-grad first-order system consisting of the equilibrium condition and the constitutive equation both written in a residual form. An L₂-norm is adopted on the residuals leading to a functional depending on displacements and stresses which has to be minimized. Therefore the first variations with respect to both free variables have to be zero. The solution can then be found by applying Newton's Method. For the continuous approximation of the displacements in W^1,p with p > 2, standard polynomials are used. Shape functions belonging to a Raviart-Thomas space are applied for the stress interpolation. These vector-valued functions ensure a conforming discretization of the Sobolev space H(div, Ω). Finally the proposed formulation is tested in a numerical example. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)