Two mixed finite element formulations with area bubble functions for tetrahedral elements

We address the solution of geometrically linear elastic problems for three-dimensional solid mechanics using tetrahedral finite elements. Starting from a five field weak formulation involving fields for compatible displacements, incompatible displacements, pressure, enhanced strains and stresses, both, the mixed method of incompatible modes and the mixed method of enhanced strains are considered as special cases. As a key idea, area bubble functions are used for both mixed finite element formulations in order to enrich the displacement field and the enhanced strain field, respectively. Appropriate conditions for satisfaction of the patch test and unique solution of the discrete equation are verified. We also observe that, contrary to a the standard enhanced assumed strain method applied to triangles and tetrahedra our mixed formulations with pressure as an independent variable are not degenerate. In the representative examples firstly a numerical verification of the patch test is obtained. Two additional examples, Cook’s membrane problem and a plate with a ring hole, illustrate the good performance of the presented approaches in comparison to existing finite element formulations.