Hybrid-Trefftz stress and displacement elements for axisymmetric incompressible biphasic media

The stress and displacement models of the hybrid-Trefftz finite element formulation are applied to the solution of incompressible axisymmetric saturated porous media. The use of a Trefftz approximation basis ensures that all domain conditions of the problem are satisfied in a strong form, namely the equilibrium, the constitutive and the strain–displacement relations and the mixture incompressibility condition. The alternative stress and displacement models are fully complementary in terms of approximation criteria. The stress (displacement) model is derived from the direct approximation of the stress and pressure fields (the displacements in the solid and fluid phases) in the domain of the element. The displacements of the solid phase and the normal displacement of the fluid phase are approximated independently on the boundary of the stress element and used to enforce in a weak form the inter-element and boundary equilibrium conditions on the forces in the solid phase and on the fluid pressure. The boundary approximation used in the displacement element is on the solid surface forces and the fluid pressure, and is used to enforce on average the inter-element and boundary displacement continuity conditions. The resulting finite element governing systems are sparse, well-suited to adaptive refinement and parallel processing, and their coefficients are defined by boundary integral expressions. The energy statements associated with the formulation are recovered and sufficient conditions for the uniqueness of the finite element solutions are stated. Benchmark tests on hydrated soft tissue modelling are used to assess the performance of the alternative stress and displacement models of the hybrid-Trefftz formulation.