The Green Element Method

Abstract

SUMMARYThis 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 timedependent 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.

Green element method solutions to steady inverse contaminant transport problems

“…Although GEM is amenable to heterogeneous media [15], only the homogenous case is addressed in this paper and hence eqn. (6) becomes…”

“…The small singular values cause instability in the solution of w, and this can be ameliorated by using, for instance, the zero order Tikhonov regularization technique which requires the minimization of the Euclidean norm ║Aw-b║ 2 +α 2 ║w║ 2 that yields the solution for w ) ( σ α σ α (15) where α is the regularization parameter, and the factor σ i /(α 2 +σ i 2 ) dampens the contribution of the small singular values. The choice of α is critical as too small values may retain the instability in the numerical solution or too large values will result in smooth solutions that do not reflect the physics of the problem.…”

“…Generally, inverse problems emerge when available measurements of concentration at some points and fluxes along a part of a boundary are used to predict medium parameters or concentration and/or fluxes or to locate pollution sources. In this paper, the Green element method [15,16] is applied to steady inverse contaminant transport problems in which concentration and fluxes are determined at parts of the computational domain using available data.…”

An Element-Driven Boundary Integral Treatment for Nonlinearity: A Coupled Nonlinear Two-Dimensional Burger’s Equation Test Case

“…Hence there is always a need to modify BEM application for those problems that unambivalently incorporate the problem domain into problem formulation such as is found with body-force terms, nonlinearity, transience, heterogeneity, source and sink terms etc. Current efforts to prevail over this challenge have resulted in a variety of hybrid BEM techniques Grigoriev [23], Taigbenu [35], Onyejekwe [36], Onyejekwe [37], Taigbenu and Onyejekwe [38], Hibersek and Skerget [39], Onyejekwe [40], Grigoriev and Dargush [41], Perata and Popov [42], Portapilla and Power [43], Sladeck et al [44], Toutip [25].…”

Green’s function-stochastic methods framework for probing nonlinear evolution problems: Burger’s equation, the nonlinear Schrödinger’s equation, and hydrodynamic organization of near-molecular-scale vorticity

“…A number of papers have used Green's function approaches to tackle nonlinear evolution problems [11,18,24]. However, with the exception of a limited collection of systematic techniques, e.g.,…”