Uncoupling of coupled flows in soil—a finite element method

Abstract

SUMMARYCoupled flow of water, chemicals, heat and electrical potential in soil are of significance in a variety of circumstances. The problem is characterized by the coupling between different flows, i.e. a flow of one type driven by gradients of other types, and by the dual nature of certain flows, i.e. combined convection and conduction. Effective numerical solutions to the problem are challenged due to the coupling and the dual nature.In this paper, we first present a general expression that can be used to represent various types of coupled flows in soil. A finite element method is then proposed to solve the generalized coupled flows of convection-conduction pattern. The unknown vector is first decomposed into two parts, a convective part forming a hyperbolic system and a conductive part forming a parabolic system. At each time step, the hyperbolic system is solved analytically to give an initial solution. To solve the multi-dimensional hyperbolic system, we assume that a common eigenspace exists for the coefficient matrices, so that the system can be uncoupled by transforming the unknown vector to the common eigenspace. The uncoupled system is solved by the method of characteristics. Using the solution of the hyperbolic system as the initial condition, we then solve the parabolic system by a Galerkin finite element method for space discretization and a finite difference scheme for time stepping.The proposed technique can be used for solving multi-dimensional, transient, coupled or simultaneous flows of convection-conduction type. Application to a flow example shows that the technique indeed exhibits optimality in convergence and in stability.

Analytical solution to 1D coupled water infiltration and deformation in unsaturated soils

“…The issue of numerical solutions to combined seepage and deformation problems in unsaturated soil has been previously studied by several researchers [7][8][9][10][11][12][13][14]. Sheng [7] proposed a finite element method to solve generalized coupled flows in soil. Thomas [9] presented a theoretical formulation for the analysis of coupled heat, moisture and air transfer, which is applicable to deformable unsaturated soils.…”

Soluciones numéricas estables de sistemas acoplados mixtos de ecuaciones en derivadas parciales.

“…Los sistemas mixtos difusivos de ecuaciones en derivadas parciales acoplados del tipo (1.1) y (1.2) aparecen en el estudio de muchos campos de la ciencia tales como, geomecánica [42]; física-química; mecánica [36]; en problemas de dispersión de mecánica cuántica [2,31]; en la modelización termoelastoplástica; en problemas de conducción nerviosa [30,17,30]; en modelos del armamento [14]; en el encendido de unaúnica componente de un gas no reactivo en un recipiente cilíndrico cerrado con conservación de masa [27]; o en el estudio de la muerte cardíaca repentina como consecuencia de la fibrilación ventricular [48].…”

“…El acoplamiento entre varios problemas escalares se puede expresar de forma más elegante mediante una formulación matricial. Atendiendo a [42,34,46] podemos modelar diferentes fenómenos de conducción mediante el siguiente sistema general, transcrito en su versión unidimensional:…”

“…Los sistemas hiperbólicos fuertemente acoplados de ecuaciones en derivadas parciales del tipo (1.5) surgen en procesos de calentamiento por microondas [32], [15],óptica [10], cardiología [47], flujos del suelo [42,49], y otros. 0.4pt0.4pt 0pt0.4pt…”

“…Debido a la complejidad que involucra el acoplamiento de los sistemas matriciales, los métodos usuales para la resolución de estos problemas transforman el sistema acoplado en un nuevo sistema desacoplado para poder abordarlo escalarmente [9,42]. Estas técnicas de desacoplamiento, a pesar de ser tan utilizadas, tienen desventajas bien conocidas tales como: asumir hipótesis innecesarias, incrementar el orden de derivabilidad del sistema desacoplado, ser técnicas de aplicación limitada ya que la matriz de coeficientes tiene que ser simétrica, y otras [8].…”