Alternative Boundary Element Method Research Articles

An Alternative BEM for Simulating the Flow Behavior of a Leaky Confined Fractured Aquifer With the Use of the Semianalytical Approach

AbstractThis study developed an alternative boundary element method (BEM) to simulate the transient flow behavior of groundwater induced by well extraction in a confined fractured aquifer containing a network of discrete or connected fractures. The matrix flow, network‐fracture flow, and matrix‐fracture fluid exchange were considered. The aquifer was treated as a heterogeneous whole that consisted of fracture and matrix blocks with locally homogeneous hydraulic properties. The fractures were explicitly represented to be of true finite volume rather than nonrepresentational line sources. A semianalytical solution was developed based on the theory of a BEM in the Laplace transform domain, but the analytical Green's function was used for the bounded domain rather than the free‐space Green's function in a conventional BEM. Case studies were presented in order to investigate the flow exchange behavior between the matrix and fractures and the corresponding transient drawdown response. The results showed that (1) exchange flux distribution calculated with the classical infinitesimal fracture model was consistent with the difference of the normal drawdown derivative values on both sides of the fracture body in our model. (2) When the well was in the matrix, the fractures acted as both highly conductive conduits and leaky faults, and the drawdown derivative behaviors resembled the characteristics of a dual‐porosity reservoir model. (3) When the well was in the network fracture and when the volume of fracture was of the same order of magnitude as the matrix, the drawdown derivative might exhibit the look‐alike behavior of a dual‐porosity model.

COMPARISON BETWEEN THE CLASSICAL SUB REGIONS TECHNIQUE AND A NEW APPROACH WITH DOMAIN SUPERPOSITION TO SOLVE SECTORIAL INHOMOGENEOUS LAPLACE’S PROBLEMS

Abstract. The Boundary Element Method (BEM) has excellent performance in applications where the variable field is scalar and stationary. However, there is a wide range of issues in science and engineering that are difficult to solve by the BEM. Among these issues, there are the constitutive non-homogeneous problems, where the physical properties vary sectorally. In these kind of problems, the domain techniques, such as Finite Element Method (FEM), Finite Volume Method (FVM) or Finite Difference Method (FDM), present considerable advantages. However, even for these cases, there is a consistent BEM formulation, the classic sub-region technique. This work presents numerical comparisons between the classic subregion technique and an alternative BEM technique that is not based on a partition of the domain. Results are compared with analytical results and other achieved by domain methods using finer meshes. Keywords: Boundary Element Method, Inhomogeneous Laplace’s Problems, Sub-regions Technique

Cohesive crack growth modelling based on an alternative nonlinear BEM formulation

Fracture mechanics and crack propagation problems have been widely studied by the scientific community in recent years, because crack growth phenomenon can explain the failure of structures. In order to model accurately the structural behaviour of complex engineering structures numerical techniques are required. In this regard, the boundary element method (BEM) has been widely used to solve complex engineering problems, especially those where its mesh dimension reduction includes advantages on the modelling. This paper addresses the analysis of crack propagation in quasi-brittle materials using an alternative BEM formulation. In this type of problem, the damaged zone ahead the crack tip is modelled based on the fictitious crack model. Therefore, the residual resistance of the damaged zone is represented by cohesive stresses, which tends to close the crack faces. The alternative BEM formulation proposed aims at modelling the cohesive stresses using the domain term of the direct integral representation. This term is degenerated, in order to be non null only at the fictitious crack path. As a result of the domain term manipulation, appears a dipole of stresses, which will govern the cohesive stresses. It leads a nonlinear formulation, where the cohesive stresses are determined according the crack opening displacement. Linear, bi-linear and exponential cohesive laws are used to model the residual resistance of the cohesive zone. The results achieved by the proposed formulation are compared with experimental and numerical ones available in literature, in order to validate and prove its robustness and accuracy.

Multiple pole residue approach for 3D BEM analysis of mathematical degenerate and non‐degenerate materials

AbstractIn this paper we develop an alternative boundary element method (BEM) formulation for the analysis of anisotropic three‐dimensional (3D) elastic solids. Our implementation is based on the derivation of explicit expressions for the fundamental solution displacements and tractions, of general validity for any class of anisotropic materials, by means of Stroh formalism and Cauchy's residue theory. The resulting fundamental solution remains valid for mathematical degenerate cases when Stroh's eigenvalues are coincident, meanwhile it does not exhibit numerical instabilities for quasi‐degenerate cases when Stroh's eigenvalues are nearly equal. A multiple pole residue approach is followed, leading to general explicit expressions to evaluate the traction fundamental solution for poles of m‐multiplicity. Despite the existence of general displacement solutions in the literature, and for the sake of completeness, the same approach as for the traction solution is considered to derive the displacement fundamental solution as well. Based on these solutions, an explicit BEM approach for the numerical solution of 3D linear elastic problems for solids with general anisotropic behavior is presented. The analysis of cracked anisotropic solids is also considered. Details on the numerical implementation and its validation for degenerate cases are discussed. Copyright © 2010 John Wiley & Sons, Ltd.

An alternative multi-region BEM technique for three-dimensional elastic problems

The main objective of this work is to present an alternative boundary element method (BEM) formulation for the static analysis of three-dimensional non-homogeneous isotropic solids. These problems can be solved using the classical boundary element formulation, analyzing each subregion separately and then joining them together by introducing equilibrium and displacements compatibility. Establishing relations between the displacement fundamental solutions of the different domains, the alternative technique proposed in this paper allows analyzing all the domains as one unique solid, not requiring equilibrium or compatibility equations. This formulation also leads to a smaller system of equations when compared to the usual subregion technique, and the results obtained are even more accurate.

Boundary Element Method For The Analysis OfFlow And Concentration In A Water Reservoir

In an earlier study, a boundary element method (BEM) and a weighted finite difference method (WFDM) were combined to analyse the flow in a water reservoir. In this paper, the BEM is developed to obtain numerically stable and convergent results for the water flow and concentration distribution in the water reservoir. In order to apply the BEM to the problem, three techniques were used. (1) The first is the penalty method, in which the pressure terms are eliminated in the Navier-Storks equations for the BEM. (2) The second involves taking the uniform velocity into the basic solution in both the flow and concentration distribution analyses of our BEM. (3) The third is to identify the kinetic and dynamic conditions for pressure against the vertical wall or the bottom and on the free surface of the reservoir because it is necessary to ascertain the pressure conditions in case that the penalty method is not used. Referring to the velocity vectors of the water flow that are calculated by the WFDM and observed in the model simulation of a water reservoir constructed in our laboratory, the effect and accuracy of the alternative BEM are estimated.

Integral representations in elastostatics and their application to an alternative boundary element method

AbstractWe obtain a new representation for derivatives and anti‐derivatives of any order of the displacement and stress fields for elastostatics problems when the boundary data is given in terms of polynomials of arbitrary degree. The result includes, as a special case, Somigliana's theorem. Based on this identity, we propose an alternative algorithm for the boundary element method that uses polynomial approximations of arbitrary order. The method provides accurate results for two‐dimensional elastostatics boundary value problems and has the advantage that it is easy to implement. The formula can also be used to accurately compute stresses and strains. For domains bounded by polygons, we provide closed‐form analytical expressions for the terms that appear in the stiffness matrix and the load vector for polynomials of arbitrary degree, thus avoiding numerical integration. We analyse the accuracy of the numerical solution as a function of the degree of the polynomial approximation by solving a representative boundary value problem. Copyright © 2004 John Wiley & Sons, Ltd.