Unified finite element discretizations of coupled Darcy–Stokes flow

Abstract

The purpose of this paper is to provide analytical and numerical solutions of the formation and evolution of the localized plastic zone in a uniaxially loaded bar with variable cross-sectional area. An energy-based variational approach is employed and the governing equations with appropriate physical boundary conditions, jump conditions, and regularity conditions at evolving elasto-plastic interface are derived for a fourth-order explicit gradient plasticity model with linear isotropic softening. Four examples that differ by regularity of the yield stress and stress distributions are presented. Results for the load level, size of the plastic zone, distribution of plastic strain and its spatial derivatives, plastic elongation, and energy balance are constructed and compared to another, previously discussed non-variational gradient formulation.

Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem

“…[2], [9], [10], [11], [13], [14], [18], [20], [21], [27], [28], [29], [33], [34], [35], [37], and the references therein). This fact has been motivated by the diverse applications of this coupled model (in petroleum engineering, hydrology, and environmental sciences, to name a few), and also by the increasing need of simpler, more accurate, and more efficient procedures to solve it.…”

A Parallel Robin–Robin Domain Decomposition Method for the Stokes–Darcy System

“…Several methods have been developed to numerically solve the Stokes-Darcy problem, including coupled finite element methods [2,8,10,25,32], domain decomposition methods [12][13][14][15][16][17][18]23], Lagrange multiplier methods [20,26], two grid methods [29], discontinuous Galerkin methods [19,31], and boundary integral methods [35]. Many other methods have been developed to solve the Stokes-Brinkman and other similar models; see [1,3,[5][6][7]28,30,34,36,37] and the reference cited therein.…”

A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

“…So far, several numerical methods have been developed to approximate the solution of the Stokes-Darcy coupled problem (see for instance [13,14,18,19,20,22,26,27,28,32,36,37,39,40,41,7]), most of them based on appropriate combinations of stable elements for both media. In this direction, the first theoretical results go back to [39] and [20].…”