Higher order stabilized finite element method for hyperelastic finite deformation

This paper presents a higher order stabilized finite element formulation for hyperelastic large deformation problems involving incompressible or nearly incompressible materials. A Lagrangian finite element formulation is presented where mesh dependent terms are added element-wise to enhance the stability of the mixed finite element formulation. A reconstruction method based on local projections is used to compute the higher order derivatives that arise in the stabilization terms, specifically derivatives of the stress tensor. Linearization of the weak form is derived to enable a Newton–Raphson solution procedure of the resulting non-linear equations. Numerical experiments using the stabilization method with equal order shape functions for the displacement and pressure fields in hyperelastic problems show that the stabilized method is effective for some non-linear finite deformation problems. Finally, conclusions are inferred and extensions of this work are discussed.