AJS - Simona Lo Franco

Title: A mixed–hybrid flux–preserving element for the coupled problem of finite-strain poroelasticity Speaker: Simona Lo Franco (University of Modena and Reggio Emilia) Abstract: Computational modeling of coupled diffusion phenomena in the finite strain regime constitutes a domain of considerable complexity, stemming from the intricate interplay between the deformation of a porous solid skeleton and the flow of a fluid permeating its pore network. Existing finite element formulations predominantly adopt mixed approaches, typically grounded in two-field discretizations involving solid displacement and porefluid pressure, or in more elaborate three–field formulations in which displacement is retained as a primary variable. From a variational standpoint, such formulations give rise to saddle–point problems whose mathematical well–posedness is contingent upon the satisfaction of the inf–sup stability condition [3]. However, even when this condition is satisfied, spurious pressure oscillations may nonetheless arise in critical regimes, often encountered in modeling soft biological tissues, thereby necessitating the adoption of stabilization techniques or, alternatively, the employment of expensively fine meshes. An alternative paradigm [4] involves the adoption of a different set of state variables that inherently enforces local mass conservation and leads to a variational minimization principle, thus guaranteeing inf–sup stability a priori. While the formulation benefits from intrinsic stability properties, it requires the flux variable to belong to the H(div) functional space, thus resulting in high computational costs. In the present contribution, building upon the minimization principle by Miehe et al. , a mixed–hybrid formulation is proposed [5]. Solid displacement and fluid mass flux are retained as primary variables, while the fluid potential is incorporated as a Lagrange multiplier defined on the element boundaries. The resulting mixed-hybrid formulation relaxes the global H(div) conformity requirement while preserving conservation of the fluid mass flux at the element level. The corresponding finite element has been implemented within the open–source software FEAP and assessed through several classical benchmark problems. The numerical results demonstrate accurate resolution of the pressure field, highlighting the advantages of the proposed approach with respect to standard mixed formulations. Finally, prospective directions for future research are outlined. Of particular concern is the enforcement of physical admissible constraints on the porosity field, whose violation may yield solutions that, while appearing well-behaved with respect to the primary variables, hide physically inadmissible states. Existing strategies, which rely on penalty approaches, are prone to solution divergence in the proximity of the compaction point, the accurate modeling of which constitutes an open and practically relevant challenge in regimes involving low-permeability and large deformations, which are characteristic of soft biological tissues.