A new coupled analysis for nearly incompressible and impermeable saturated porous media on mixed finite element method: I. Proposed method

Abstract

In the general mixed finite element analysis for the porous media, a fluid is assumed to be nearly incompressible; however, the solid may undergo a range of deformations from relatively large to small. If both constituent are assumed to be nearly incompressible and impermeable, the finite element analysis is very complex because of element locking. In order to overcome this difficulty, stable procedures using equal order elements have been introduced with stability analyses. However, these methods have a drawback that critical time steps in the stability analysis have to be determined by modal analysis. In this paper, a new coupled analysis based on a stable algorithm is introduced to remedy the drawback. First, governing equations for saturated porous media in frame of Biot's theory are derived on macroscopic sense, and that are represented into approximation forms which is divided into solid and fluid phase for finite element procedure. Moreover, a new coupled analysis based on a stable algorithm is introduced. In this procedure, the multi time step, the remeshing step and the sub iteration step are proposed for a stable analysis. These steps make it possible to simply solve numerical instabilities such as convergence and compatibility problems.

Computational Coupled Method for Multiscale and Phase Analysis

“…In mixed FEM, because this assumption can bring numerical instability, such as element locking by incompatibility condition between finite elements, we use mixed FEM proposed by Park and Tak [28] and Tak and Park [29]. In this mode, total 808 nodes and 600 finite elements are used.…”

An adaptive order finite element method for poroelastic materials described through the Biot equations

“…Especially displacement‐based formulations are prone to element locking 35‐37 . This mainly happens at low orders where the basis functions are insufficiently rich to capture the physics of the problem, for example, in case of incompressibility 38 . Hereto, specialized mixed elements, taking both displacement and pressure into account 37,39 are required for low order approaches.…”

“…[35][36][37] This mainly happens at low orders where the basis functions are insufficiently rich to capture the physics of the problem, for example, in case of incompressibility. 38 Hereto, specialized mixed elements, taking both displacement and pressure into account 37,39 are required for low order approaches. In order for these mixed formulation elements to be stable, however, specific relations are necessary between the physical variables.…”

The Analysis of Porous Media using the Mixed Finite Element Method and the FETI Method

“…However, the restrictions on the determination of the tome step are quite severe. In order to remedy these problems, Park and Tak [13,14] proposed Staggered method with multi time step method, remeshing scheme and sub-iteration. At the multi time step and the subiteration step, compatibility and the convergence problems are solved with equal order shape functions.…”

“…Indeed, finite element should be satisfied Baduska-Brezzi Condition [10,11,12] or pass to the patch test in which different high order interpolation is required for an element. In order to solve this problem, staggered method is proposed by Park and Tak [13,14]. The nearly incompressible and impermeable saturated porous media using equal order element and staggered method.…”