Dynamic analysis of porous materials: Numerical modeling with a space‐time FEM

Abstract

In the current work, we investigate the dynamic analysis of a two-phase porous material using the space-time discontinuous Galerkin method. The physical model is based on the Theory of Porous Media (TPM). The finite element approximation consists of continuous approximations in space but discontinuous ones in time. The continuity condition between the adjacent time intervals is weakly enforced by the upwind flux treatment. No artificial penalty function is involved. Moreover, the Embedded Velocity Integration technique is applied to reduce the second-order equation in time into a first order one without introducing an additional constraint.

A EVI-space-time Galerkin method for dynamics at finite deformation in porous media

“…We denote that in the community of the TPM, it is usual to introduce the kinematic relation v s = u s (35) as an extra auxiliary equation to construct a four-field equation system with the primary variables {u s , v s , Q, θ} and solved the resulting first-order system with an implicit numerical approach, e.g., the backward Euler method or others, [43]. Alternatively, it is also possible to solve a three-fields formulation with the primary variables {u s , Q, θ} with the Newmark scheme [44].…”

“…In our previous work, we have investigated a space-time discontinuous Galerkin (DGT) approach for the solution strategy of incompressible porous mixtures with a four-field formulation [33]. However, due to the restriction of the flux treatment in the DGT method to time-dependent systems of first-order, a four-field formulation including the kinematic relation (35) as an extra governing equation is applied. It is obvious that by introducing the extra equation (35) in the governing set of equations, a larger system of algebraic equation is resulted, which increases the computational burden.…”

“…The solid displacement is subsequently computed according to a consistent integration of the velocities. Furthermore, beside the exact integration scheme proposed in [34,35], a generalized EVI integration scheme complemented with a stabilization factor α is developed. The overall numerical stability of the solution can be enhanced by the proper choice of the stabilization parameter α.…”

Dynamic analysis of porous materials: Numerical simulation with an adaptive space‐time FEM

“…In the current work, we solved the governing set of equations with the high-order accurate time-discontinuous Galerkin (DGT) method [1]. Moreover, due to the employment of the Embedded Velocity Integration (EVI) technique [2], we are able to solve a three-fields formulation with the primary variables in the solid velocity v s , the seepage velocity w f and the pore pressure p. The solid displacement u s is achieved in a post-processing step according to consistent integration of the velocities v s . The finite element variational form of the three-fields formulation solved by the DGT method has been presented in [3], which will not be repeated here.…”