The Green Element Method

We may not be properly poised to talk about the Green element method (GEM) without making reference to the boundary element method (BEM) since both methods are founded on the same singular integral theory. The theoretical developments in boundary element (BE) circles can be traced to the eighteenth century when related theories of ideal fluid flow (potential flow) and of integral transforms were established, but it was not until the 1960’s that flurries of research activities on BEM intensified because of its acclaimed advantage of being a boundary-only method. Because the earlier boundary element applications were directed at elliptic boundary-value problems, the free-space Green’s function of the differential operator, which is Inr in 2-D. spatial dimensions and 1/r in 3-D. spatial dimensions, is amenable to Green’s identity which transforms the differential equation into an integral one that can essentially be implemented on the boundary (along a line for 2-D. and on a surface for 3-D.) of the computational domain. In the classical approach, a system of discrete equations is obtained from the integral equation by subdividing the boundary into segments over which distributions are prescribed for the primary variable and the flux (Jaswon [1], Jaswon and Ponter [2], Symm [3], Liggett [4], Liu and Liggett [5], Fairweather et al. [6], Rizzo [7], and others). These research thrusts were considered successful when viewed against the background that contemporary methods, like the finite element method (FEM) solved similar problems by discretizing the entire computational domain. This led to claims by most investigators that BEM was more superior than existing computational methods in terms of accuracy and computational efficiency. It is not in doubt that the integral replication of the differential equation provided by the boundary element theory evolves quite naturally, making use of the response function (free-space Green’s function) to a unit instantaneous input. It is only at the computational stage that the distribution of the dependent variable is approximated by some interpolating polynomial function. It is, thus, expected that for problems where the unit response function can be obtained, BEM achieves secondorder accuracy. The claim that BEM is superior to FEM in computational efficiency is not, in most cases, arrived at after actual CPU comparisons are carried, but it is alluded to on the basis of the boundary-only character of the method.

A good starting point to derive and apply the Green element method is to use a simple second-order ordinary differential equation, and the Laplace/Poisson equation, which is encountered in many engineering applications, serves that purpose.

In the previous chapter, we presented the boundary element and Green element formulations for the linear version of the steady 1-D. second-order differential equation given by eq. (2.1) in which the parameter K assumed a constant value. We used the example of flow in a confined aquifer of uniform thickness to derive the linear form of the differential equation. Here we relax that condition and consider cases where K is a function of the spatial variable x (heterogeneous case), and a function of the dependent variable h (nonlinear case). Nonlinear heterogeneous problems are frequently encountered in many engineering applications — heat flux through a material whose thermal properties are nonlinear, infiltration into unsaturated soils, elastic deformation of a bar of varying section subject to axial loads, etc. We elect to use a heat transfer example to derive the nonlinear equation of the form of eq. (2.1).

Before now, all the problems that have been addressed were steady. In other words neither the primary variable nor the flux had any time history. Steady state solutions may be viewed as states of equilibrium which only become altered when the dynamic forces ensuring equilibrium are altered. With this understanding it means that transient problems could be considered as going through quasi-steady states. Although steady state solutions have their value, especially in evaluating the performance of a system under equilibrium conditions, the real world is dynamic in nature. States of equilibrium are rarely maintained for too long, so that the time history of the performance of the system has to be closely monitored in order to capture its complete picture.

In the previous chapter (chapter 6) we derived the Green element equations for the mathematical statement that describes the storage and movement of species or contaminants in a fluid. The transported specie can be considered to be the momentum of the flow, and in that case the relevant differential equations become the Navier-Stokes equations [1]. Simplifications of the Navier-Stokes equations can be carried out to achieve easier versions of the equations. One of such second-order simplifications yields the transient one-dimensional form of the equations which provides a useful model for many physical fluid flow phenomena — nonlinear propagating shock wave with viscous dissipation, turbulence, propagating shock waves in gases, propagating flame in a combustion chamber, etc. As a result of the extensive research works carried out by Burgers in modeling of turbulence, the simplified transient nonlinear momentum transport equation in one spatial dimension is popularly referred to as Burgers equation [2,3]. The nonlinear nature of Burgers equation has been exploited as a useful prototype differential equation for modeling many divers and rather unrelated phenomena such as shock flows, wave propagation in combustion chambers, vehicular traffic movement, acoustic transmission, etc. In fact, Burgers equation can be considered applicable to any flow phenomenon in which there exist the balancing effects of viscous and inertia or convective forces. It is probably one of the simplest nonlinear transient partial differential equations which exhibits some very unique features. When inertia or convective forces are dominant, its solution resembles that of the kinematic wave equation which displays a propagating wave front and boundary layers. In that case Burgers equation essentially behaves as a hyperbolic partial differential equation. In contrast, when viscous forces are dominant, it behaves as a parabolic equation, and any propagating wave front is smeared and diffused due to viscous action. Because of these different forms that Burgers equation can assume, coupled with its nonlinear characteristics, it has become a model equation for assessing and evaluating the performance of many computational techniques. It is for the same reason that we have devoted this chapter to the development of a number of schemes of the Green element method for the solution of the Burgers equation.

We pay attention in this chapter to the solution to the problem of flow in variably saturated soils because of its importance in many fields of engineering such as drainage, irrigation, environmental, soil and petroleum engineering. The flow in unsaturated soils is essentially a two-phase flow of two immiscible fluids — air and water. The physical processes that give rise to this kind of flow are infiltration of surface water through the upper layers of soil which enriches the soil moisture, and subsurface flow through soils which are partially filled with air. The interaction of roots of plants with this flow, and the advection and dispersion of fertilizers and pesticides within the unsaturated zone make this flow phenomenon of considerable interest to soil scientists, agronomists, and irrigation engineers. In petroleum explorations of underground reservoirs where immiscible fluid flows are also encountered, the fluids involved are water, oil, and gas. Although the partial differential equations that govern the two-phase immiscible flow in unsaturated formations are similar to those that govern the three-phase immiscible flow in underground reservoirs, it is usual in the case of the former to assume that the dynamics of flow within the air phase play an insignificant role in determining water movement and storage in the unsaturated zone. In the absence of this assumption, the solution would have required a simultaneous solution of two partial differential equations, each of which describes the flow in each fluid phase.

In the earlier seven chapters, we solved by the Green element method a variety of engineering problems in 1-D. spatial dimensions ranging from steady to transient, linear to nonlinear, and from those which apply in homogeneous to heterogeneous media. For all these problems, line segments were used to discretize the physical domain, and functional quantities were approximated by linear interpolation polynomials within each line segment (element). These interpolating polynomials have zero-order continuity in the sense that only the functional quantity is continuous, while its first derivative is discontinuous across elements. Where the variation of the functional quantities with respect to the spatial dimension is marginal, the use of linear interpolation can be adequate in approximating these functional quantities. However, there are certain problems, earlier encountered in chapters 6, 7, and 8, whose solutions have significant spatial gradients, and for which the use of linear interpolation is inadequate. In chapter 6, such a problem is the advection-dominant transport one that exhibits the unique feature of retaining the initial concentration profile as time progresses, so that steep gradients in the initial concentration profile persist throughout the solution history. The same can be said of the momentum transport problem of chapter 7 when viscous effects are negligible. For the unsaturated flow problem which we encountered in chapter 8, we are faced with steep gradients of the soil constitutive relations and that of the solution variable.

The additional dimension of time which is incorporated into the two-dimensional steady problems addressed in the earlier chapter allows for solutions that provide information on the time history of the behaviour of the primary scalar or vector variable. Such solutions do, in most cases, give better representation of the behaviour of engineering systems which are usually under dynamic forces that alter equilibrium states from time to time.

Earlier chapters provided the theoretical basis of the Green element formulation for steady and unsteady problems in one and two spatial dimensions, and its solution capabilities were demonstrated on various examples which are of interest in many fields of engineering. Hereafter, we may wish to refine the numerical aspects of the method to achieve higher accuracy and/or savings in computing resources, and extend the applicability of the method to three-dimensional problems and those that were not addressed in this text.