First-Order Compatible-Strain Mixed Quadrilateral Finite Elements for $2$D Nonlinear Elasticity

Abstract

Compatible-strain mixed finite elements (CSMFEs) use the differential complex of nonlinear elasticity to construct discretizations that preserve the underlying topological structure. Existing developments of CSMFEs have focused on simplicial meshes, for which efficient formulations have been developed for first- and second-order elements and successfully applied to compressible and incompressible nonlinear elasticity problems. In this paper, we develop two compatible-strain mixed formulations for quadrilateral elements applicable to compressible and incompressible nonlinear elasticity. For quadrilateral elements, the Piola transformation preserves vector fields tangent and normal to element edges only for special classes of elements, such as rectangles, parallelograms, and trapezoids. Consequently, it cannot be used to construct compatible shape functions for general quadrilateral elements. To overcome this limitation, the shape functions are computed directly in the physical space using numerical integration. N\'ed\'elec shape functions of the first kind are used to construct the interpolation of the displacement gradient. We also introduce a new class of shape functions for the stress tensor that is compatible with the displacement and displacement-gradient discretizations. The formulation for compressible elasticity follows the framework previously developed for first- and second-order simplicial CSMFEs, whereas a new formulation is proposed for incompressible elasticity. The incompressible formulation employs element-level condensation of the pressure field and therefore does not increase the number of global degrees of freedom relative to the corresponding compressible formulation. These developments extend the compatible-strain mixed finite element framework from simplicial to general quadrilateral meshes for both compressible and incompressible nonlinear elasticity. Numerical examples demonstrate that the proposed quadrilateral elements can successfully solve problems that require second-order simplicial CSMFEs while using fewer degrees of freedom.