Unsaturated-Flow Simulation with Green Element Models

Flows in variably saturated media are of profound interest to numerical analysts, engineers, and scientists because of not only the challenge they pose as a result of their highly nonlinear constitutive relations but also their importance in many fields of engineering such as drainage, irrigation, hydrology, environmental, soil, and petroleum engineering. In this paper, the Picard and Newton-Raphson (N-R) algorithms are incorporated into the Green element method (GEM) to simulate these flows. The GEM offers a viable means of implementing the singular boundary integral theory so that the theory is more generally applicable and computationally efficient. Here GEM discretizes the integro-differential equation in space with suitable polygonal elements and in time with a difference scheme, and the system of nonlinear discretized element equations are linearized by the Picard and N-R algorithms. Calculations carried on three numerical examples of infiltration into unsaturated soils in two spatial dimensions indicate better convergence of the N-R algorithm than the Picard algorithm at comparable computational cost.     

Enhancement of the accuracy of the Green element method: Application to potential problems

The 2-D formulation of the Green element method (GEM) which approximates the internal normal directional fluxes by difference expressions in terms of the field variable had been recognized to be fraught with errors that comprise its accuracy. However, this approach is computational attractive because there is only one degree of freedom at every node, the system matrix is slender, and it does require additional compatibility relationships. There have been attempts to reduce the numerical errors of this original GEM formulation by the use of flux-based formulations which essentially retain the internal fluxes but at the expense of those attractive numerical features. Here the original GEM is revisited and shown that, with difference approximation of the internal normal fluxes whose error is of the order of the square of the size of the element, its accuracy is greatly enhanced to a level comparable to the flux-based formulations. This approach is demonstrated on regular domains with rectangular elements and irregular domains with triangular elements using six examples that cover steady, transient, linear and nonlinear potential flow and heat transfer problems in homogeneous and heterogeneous media.     

An integral formulation applied to the diffusion and Boussinesq equations

A new integral method is proposed here to solve the diffusion equation (confined flow) and the Boussinesq equation (unconfined flow) in a two-dimensional porous medium. The method, based on Green's theorem, derives its integral representation from the portion of the original differential equation with the highest space derivatives so that the resulting kernel of the integral representation is not time dependent. Compared to an earlier integral formulation, namely the direct Green function, based on the same theorem, the kernel is simpler so that the present theory provides a more efficient numerical model without compromising accuracy. An iterative scheme is employed along with the theory to achieve solutions to the non-linear Boussinesq equation. Concepts used in the finite difference and finite element methods enable simplification of the temporal derivative. The method is tested with success on a number of numerical examples from groundwater flow.     

The green element method

This paper discusses an element-by-element approach of implementing the Boundary Element Method (BEM) which offers substantial savings in computing resource, enables handling of a wider range of problems including non-linear ones, and at the same time preserves the second-order accuracy associated with the method. Essentially, by this approach, herein called the Green Element Method (GEM), the singular integral theory of BEM is retained except that its implementation is carried out in a fashion similar to that of the Finite Element Method (FEM). Whereas the solution procedure of BEM couples the information of all nodes in the computational domain so that the global coefficient matrix is dense and full and as such difficult to invert, that of GEM, on the other hand, involves only nodes that share common elements so that the global coefficient matrix is sparse and banded and as such easy to invert. Thus, GEM has the advantage of being more computationally efficient than BEM. In addition, GEM makes the singular integral theory more flexible and versatile in the sense that GEM readily accommodates spatial variability of medium and flow parameters (e.g., flow in heterogeneous media), while other known numerical features of BEM—its second-order accuracy and ability to readily handle problems with singularities are retained by GEM. A number of schemes is incorporated into the basic Green element formulation and these schemes are examined with the goal of identifying optimum schemes of the formulation. These schemes include the use of linear and quadratic interpolation functions on triangular and rectangular elements. We found that linear elements offer acceptable accuracy and computational effort. Comparison of the modified fully implicit scheme against the generalized two-level scheme shows that the modified fully implicit scheme with weight of about 1·25 offers a marginally better approximation of the temporal derivative. The Newton–Raphson scheme is easily incoporated into GEM and provides excellent results for the time-dependent non-linear Boussinesq problem. Comparison of GEM with conventional BEM is done on various numerical examples, and it is observed that, for comparable accuracy, GEM uses less computing time. In fact, from the numerical simulations carried out, GEM uses between 15 and 45 per cent of the simulation time of BEM.