Okey Oseloka Onyejekwe

Green element simulations of the transient nonlinear unsaturated flow equation

This paper presents numerical solutions to transient one-dimensional (1-D) and 2-D nonlinear flows in unsaturated media which are based on the Green element method (GEM). GEM is a novel numerical scheme based essentially on the singular integral theory of the boundary element method (BEM) but which implements the theory in an element-by-element fashion so that when the discrete equations from all the elements are assembled the resultant global coefficient matrix is banded and easier to invert, in contrast to the fully populated global coefficient matrix commonly associated with conventional BEM. Because GEM implements the singular integral theory of BEM within each element, the elemental integrals of the theory are evaluated analytically, thereby enhancing the accuracy of the method, and medium properties that vary spatially and temporally as a result of their dependence on the flow are readily incorporated into the theory. The transient nonlinear flow problem in unsaturated porous medium with soil constitutive relations that are of practical interest is highly nonlinear and had not been amenable to the boundary element theory, but here, for the first time, such problems are successfully solved by that theory using the Green element approach which incorporates a Picard nonlinear solution algorithm. Two Green element models for transient 1-D and 2-D unsaturated flows are developed and successfully tested on four examples of infiltration flow problems whose soil constitutive relations cover most of those reported in the literature.

Green's function-based integral approaches to nonlinear transient boundary-value problems (II)

Abstract Two Greens function-based numerical formulations are used to solve the time-dependent nonlinear heat conduction (diffusion) equation. These formulations, which are an extension of the first paper, utilize two fundamental solutions and the Greens second identity to achieve integral replications of the governing partial differential equation. The integral equations thus derived are discretized in space and time and aggregated in a finite element sense to give a system of nonlinear discrete equations that are solved by the Newton–Raphson algorithm. The mathematical simplicity of the Greens function of the first formulation facilitates its numerical implementation. The performance of the formulations is assessed by comparing their results with available numerical and analytical solutions. In all cases satisfactory and physically realistic results are obtained.

TRANSIENT 1D TRANSPORT EQUATION SIMULATED BY A MIXED GREEN ELEMENT FORMULATION

New discrete element equations or coefficients are derived for the transient 1D diffusion–advection or transport equation based on the Green element replication of the differential equation using linear elements. The Green element method (GEM), which solves the singular boundary integral theory (a Fredholm integral equation of the second kind) on a typical element, gives rise to a banded global coefficient matrix which is amenable to efficient matrix solvers. It is herein derived for the transient 1D transport equation with uniform and non-uniform ambient flow conditions and in which first-order decay of the containment is allowed to take place. Because the GEM implements the singular boundary integral theory within each element at a time, the integrations are carried out in exact fashion, thereby making the application of the boundary integral theory more utilitarian. This system of discrete equations, presented herein for the first time, using linear interpolating functions in the spatial dimensions shows promising stable characteristics for advection-dominant transport. Three numerical examples are used to demonstrate the capabilities of the method. The second-order-correct Crank–Nicolson scheme and the modified fully implicit scheme with a difference weighting value of two give superior solutions in all simulated examples.

A Green element treatment of isothermal flow with second order reaction

The present work addresses the problem of a tubular reactor in which axial diffusion is accompanied by nonlinear kinetics. The governing equations are solved by a new integral formulation which relies on the Greens second identity. The formulation fundamental solution is applied to the highest order derivative term of the governing equation before implementing the singular integral theory of the Boundary element method (BEM) in an element-by-element fashion. The technique which is essentially domain based provides a friendly interface with the finite element method (FEM) while at the same time retains the second order convergence of BEM. Physically realistic results obtained by applying the technique to flow involving nonlinear kinetics serve to demonstrate the usefulness and accuracy of this technique.

A flux-correct Green element model of quasi three-dimensional multiaquifer flow.

Transient flow in multiply layered aquifers, separated by connecting layers of aquitards which provide hydraulic interactions between the aquifers, is solved by the Green element method (GEM) in a manner that reveals one of its strengths of being able to correctly model the leakage flux without resorting to adjusting the grid representing the one-dimensional (1-D) flow in the aquitards, as done in the finite element method (FEM). The hydraulic approach of approximating the flow is adopted so that flow in the aquifer takes place in two dimensions and that in the aquitards takes place in the one-dimensional vertical direction. The 1-D Green element (GE) model earlier developed for transient diffusion and referred to as the transient GE (TGE) formulation [Taigbenu and Onyejekwe, 1999] is used in modeling the flow in the aquitards, while the 2-D GE model developed for linear and nonlinear transient diffusion [Taigbenu and Onyejekwe, 1998] is used for calculating the flows in the confined and unconfined aquifers. Both models are coupled to solve regional flow problems in multiaquifer systems of arbitrary geometry which receive point and distributed recharge of arbitrary strengths. The solution procedure, which is iterative, provides information on the hydraulic heads and fluxes in the aquifers and aquitards at various specified times. Because GEM is founded on the singular integral theory, singularities that arise from water abstractions at wells (point recharge) are naturally captured in the singular Greens function, thereby making it possible to use a more coarse grid for problems in which there exist active wells. Furthermore, the implementation procedure of GEM achieves sparsity of the coefficient matrix so that less amount of computing resources is required for its decomposition. The superiority of the current approach over FEM in predicting the leakage flux through the aquitards and achieving comparable accuracy for well problems with coarser grid is demonstrated.

A MIXED GREEN ELEMENT FORMULATION FOR THE TRANSIENT BURGERS EQUATION

The transient one-dimensional Burgers equation is solved by a mixed formulation of the Green element method (GEM) which is based essentially on the singular integral theory of the boundary element method (BEM). The GEM employs the fundamental solution of the term with the highest derivative to construct a system of discrete first-order non- linear equations in terms of the primary variable, the velocity, and its spatial derivative which are solved by a two-level generalized and a modified time discretization scheme and by the Newton–Raphson algorithm. We found that the two-level scheme with a weight of 0ċ67 and the modified fully implicit scheme with a weight of 1ċ5 offered some marginal gains in accuracy. Three numerical examples which cover a wide range of flow regimes are used to demonstrate the capabilities of the present formulation. Improvement of the present formulation over an earlier BE formulation which uses a linearized operator of the differential equation is demonstrated.

Boundary Integral Procedures for Unsaturated Flow Problems

Abstract. A novel numerical scheme based on the singular integral theory of the boundary element method. (BEM) is presented for the solution of transient unsaturated flow in porous media. The effort in the present paper is directed in facilitating the application of the boundary integral theory to the solution of the highly non-linear equations that govern unsaturated flow. The resulting algorithm known as the Green element method (GEM) presents a robust attractive method in the state-of -the-art application of the boundary element methodology. Three GEM models based on their different methods of handling the non-linear diffusivity, illustrate the suitability and robustness of this approach for solving highly non-linear 1-D and 2-D flows which would have proved cumbersome or too difficult to implement with the classical BEM approach.

A Boundary Element-Finite Element Equation Solutions to Flow in Heterogeneous Porous Media

Abstract. A coupled boundary element-finite element procedure, namely, the Green element method (GEM) is applied to the solution of mass transport in heterogeneous media. An equivalent integral equation of the governing differential equation is obtained by invoking the Greens second identity, and in a typical finite element fashion, the resulting equation is solved on each generic element of the problem domain. What is essentially unique about this procedure is the recognition of the particular advantages and particular features possessed by the two techniques and their effective use for the solution of engineering problems.By utilizing this approach, we observe that the range of applicability of the boundary integral methods is enhanced to cope with problems involving media heterogeneity in a straightforward and realistic manner. The method has been used to investigate problems involving various functional forms of heterogeneity, including head variations in a stream-heterogeneous aquifer interaction and in all these cases encouraging results are obtained without much difficulty.

Green's function-based integral approaches to linear transient boundary-value problems and their stability characteristics (I)

Abstract Two Greens function-based formulations are applied to the governing differential equation which describes unsteady heat or mass transport in an isotropic homogeneous 1-D domain. In this first part of a two series of papers, the linear form of the differential equation is addressed. The first formulation, herein denoted the quasi-steady Green element (QSGE) formulation, uses the Laplace differential operator as auxiliary equation to obtain the singular integral representation of the governing equation, while the second, denoted the transient Green element (TGE), uses the transient heat equation as auxiliary equation. The mathematical simplicity of the Greens function of the first formulation enhances the ease of solution of the integral equations and the resultant discrete equations. From the point of computational convenience, therefore, the first formulation is preferred. The stability characteristics of the two formulations are evaluated by examining how they propagate various Fourier harmonics in speed and amplitude. We found that both formulations correctly reproduce the theoretical speed of the harmonics, but fail to propagate the amplitude of the small harmonics correctly for Courant value of about unity. The QSGE formulation with difference weighting values between 0.67 and 0.75, and the TGE formulation provide optimal performance in numerical stability.

Certain aspects of Green element computational model for BOD–DO interaction

The physical laws governing the interaction of biochemical oxygen demand (BOD), and dissolved oxygen (DO) in a water body are expressed as coupled one-dimensional, transient partial diAerential equations and solved by the Green element method (GEM). The GEM has been developed as a flexible, hybrid numerical approach, that utilizes the finite element methodology to achieve optimum, inter-nodal connectivity in the problem domain, while at the same time retaining the elegant second order accurate formulation of the boundary element method (BEM). While overcoming some of the limitations of classical boundary element approach, GEM guarantees a sparsely populated coeAcient matrix, which is easy to handle numerically. We test the reliability of GEM by solving a one-dimensional mass transport model that simulates BOD‐DO dynamics in a stream. The results compare favorably with those obtained analytically, and by the finite element method (FEM) Galerkin procedure. ” 2000 Elsevier Science Ltd. All rights reserved.