Mathematical Simulation of Coupled Elastic Deformation and Fluid Dynamics in Heterogeneous Media

Mathematical simulation of deformation processes occurring in fluid-saturated media requires solving multiphysical problems. We consider a multiphysical problem as a system of differential equations with special conjugation conditions for the physical fields on the interfragmentary surfaces. The interfragmentary contact surface between solid and liquid phases is a 1-connected contact surface. Explicit discretization of the interfragmentary contact surfaces leads to an increase in the degrees of freedom. To treat the problem, we propose a hierarchical splitting of physical processes. At the macro-level, the process of elastic deformation is simulated, taking into account the pressure on the inner surface of fluid-saturated pores. At the micro-level, to determine the fluid pressure inside the pores, the Navier-Stokes equations are numerically solved with the external mechanical loading. For coupling the physical fields, we use the matching conditions for the normal components of the stress tensor on the interfragmentary surfaces. Mathematical simulation of the coupled processes of elastic deformation and fluid dynamics is a resource-intensive procedure. In addition, a computational scheme has to take into account the specifics of the multiphysical problem. We propose modified computational schemes of multiscale non-conforming finite element methods. To discretize the mathematical model of the elastic deformation process, we apply a heterogeneous multiscale finite element method with polyhedral supports (macroelements). To discretize the Navier-Stokes equations, the non-conforming discontinuous Galerkin method with the tetrahedral supports (microelements) is used. This strategy makes it possible to apply a parallel algorithm to solve the elastic deformation and fluid dynamics problems under the assumption of the hydrophobicity of macroelements surfaces. In computational experiments, we deal with idealized models of heterogeneous natural media. The developed computational schemes make it possible to accelerate the solution of problems more than five times.