Mixed spectral/hp element formulation for nonlinear elasticity

We present a mixed formulation for high-order spectral/hp element methods and investigate its stability in large strain analysis of nearly incompressible solids. Specifically, we employ the displacement/pressure formulation and use a pressure projection method along with general Jacobi polynomial bases to derive the discrete equations. We first study the convergence of the method for smooth and singular solutions of elastic and hyperelastic problems for various combinations of mixed elements. Subsequently, we investigate numerically the validity of ellipticity and inf-sup conditions for various test problems including cases with large deformation. We find that the inf-sup condition is satisfied even when the pressure interpolation order Op is equal to the displacement order Ou, but to resolve the locking-free behavior it is required that Op = Ou − 1, and this choice also gives the best convergence rate. In large deformation analysis, surprisingly, the results are more stable for Ou ⩾ Op ⩾ Ou − k because the ellipticity in kernel is better satisfied, with higher k as Ou increases. Compared to the displacement-only formulation, the mixed formulation seems to have superior accuracy and it is overall more efficient, especially for problems with high bulk modulus.