A Conformal Three-Field Formulation for Nonlinear Elasticity: From Differential Complexes to Mixed Finite Element Methods

We introduce a new class of mixed finite element methods for 2D and 3D compressible nonlinear elasticity. The independent unknowns of these conformal methods are displacement, displacement gradient, and the first Piola-Kirchhoff stress tensor. The so-called edge finite elements of the curl operator is employed to discretize the trial space of displacement gradients. Motivated by the differential complex of nonlinear elasticity, this choice guarantees that discrete displacement gradients satisfy the Hadamard jump condition for the strain compatibility. We study the stability of the proposed mixed finite element methods by deriving some inf-sup conditions. By considering 32 choices of simplicial conformal finite elements of degrees 1 and 2, we show that 10 choices are not stable as they do not satisfy the inf-sup conditions. We numerically study the stable choices and conclude that they can achieve optimal convergence rates. By solving several 2D and 3D numerical examples, we show that the proposed methods are capable of providing accurate approximations of strain and stress.