Solutions of 2D and 3D non-homogeneous potential problems by using a boundary element-collocation method

To solve boundary integral equations for potential problems using collocation Boundary Element Method (BEM) on smooth curved 3D geometries, an analytical singularity extraction technique is employed. By adopting the isoparametric approach, curved geometries that are represented by mapped rectangles or triangles from the parametric domain are considered. The singularity extraction on the governing singular integrals can be performed either as an operation of subtraction or division, each having some advantage.A particular series expansion of a singular kernel about a source point is investigated. The series in the parametric coordinates consists of functions of a type Rpxqyr, where R is a square root of a quadratic bivariate homogeneous polynomial with coefficients corresponding to the first fundamental form of a smooth surface, and p,q,r are integers, satisfying p≤−1 and q,r≥0. By extracting more terms from the series expansion of the singular kernel, the smoothness of the regularized kernel at the source point can be increased. Analytical formulae for integrals of such terms are obtained from antiderivatives of Rpxqyr, using recurrence formulae, and by evaluating them at the edges of rectangular or triangular parametric domains.Numerical tests demonstrate that the singularity extraction technique can be a useful prerequisite for a numerical quadrature scheme to obtain high order of convergence for the governing singular integrals in 3D collocation BEM with respect to the number of quadrature nodes or with respect to size of the discretization mesh cells.

The boundary element method (BEM) has been successfully applied to 2D and 3D large scale potential problems and efficiently reduce the computing operations and memory requirements. In this paper, by using Laplace Transformation, a new kind of fast BEM for 2D potential problems is presented. Using this method, the logarithm operation can be converted to product operation, the calculation complexity can be reduced. and the computational speed can be improved.

The boundary element method (BEM) is an efficient numerical method that approximates solutions of various boundary value problems. Despite its success little research has been performed on the conditioning of the linear systems that appear in the BEM. For a Laplace equation with Dirichlet boundary conditions a remarkable phenomenon is observed; the corresponding boundary integral equation (BIE) is singular for a certain critical size of the 2D domain. As a consequence the discrete counterpart of the BIE, the linear system, is singular too, or at least ill-conditioned. This is reflected by the condition number of the system matrix, which is infinitely large, or at least very large. When the condition number of the BEM-matrix is large, the linear system is difficult to solve and the solution of the system is very sensible to perturbations in the boundary data. For a Laplace equation with mixed boundary conditions a similar phenomenon is observed. The corresponding BEM-matrix consists of two blocks; one block originates from the BEM-matrix belonging to the Dirichlet problem, the other block originates from the BEM-matrix belonging to the Neumann problem. The composite matrix inherits the solvability problems from the Dirichlet block. In other words, for the Laplace equation with mixed boundary conditions there exists also a critical size of the 2D domain for which the BEM-matrix has an infinitely large condition number. Hence the size and shape of the domain affects the solvability of the BEM problem. The critical size of the domain for which the BIE becomes singular is related to the logarithmic capacity of the domain. The logarithmic capacity is a positive real number that is a function of the size and shape of the domain. If this logarithmic capacity is equal to one, the domain is a critical domain, and for this domain the BIE becomes singular. Thus by computing the logarithmic capacity we can a-priori determine whether the BIE will be singular or not. The logarithmic capacity depends linearly on the scale of the domain, and thus a domain with logarithmic capacity equal to one can always be found by rescaling the domain. Unfortunately the logarithmic capacity can only be computed analytically for a few simple domains; for more involved domains the logarithmic capacity can be estimated though. There are several possibilities to avoid large condition numbers, i.e. singular BIEs that appear at critical domains. The first option is to rescale the domain such that the logarithmic capacity is unequal to one. One can also add a supplementary condition to the BIE and the linear system. A drawback of this option is that the linear system has more equations than unknowns and different techniques are required to solve the system. A third option is to slightly modify the fundamental solution of the Laplace operator. This fundamental solution directly appears in the BIE and it can be shown that a suitable modification yields BIEs that do not become singular. The critical domains for which the BIEs become singular do not restrict to Laplace equations only. Also for BIEs applied to the biharmonic equation or the elastostatic equations and the Stokes equations such critical domains exist. As the last two equations are vectorial equations, also the corresponding BIE consists of two equations. As a consequence two critical domains can be found for which these BIEs become singular. To obtain nonsingular BIEs techniques similar to the Laplace case can be used. Unfortunately we cannot a-priori determine the sizes for which the BIEs becomes singular, and thus do not know to what size we should rescale the domain to obtain nonsingular BIEs. The existence of critical domains is in essence caused by the logarithmic term in the fundamental solutions for the elliptic boundary value problems in 2D. This logarithmic term does not depend linearly on the size of the domain. When a domain is scaled, i.e. multiplied by a scale factor, the argument of the logarithm is also multiplied by this scale factor, but the logarithm turns this into an additive term. Thus the logarithm transforms multiplication into addition. This affects the BIEs in such a way that critical domains can appear. The fundamental solutions of boundary value problems in 3D do not contain a logarithmic term. Hence scaling of the domain does not affect the fundamental solution, and consequently also the BIE is not affected. Hence we may safely rescale 3D domains without the risk to encounter a critical domain. An example in which a domain takes many different sizes and shapes is the blowing problem. In this problem a viscous fluid is blown to a desired shape. Typically the time is discretised into a set of discrete time steps, and at each step the shape of the fluid is computed by solving the Stokes equations. When attempting to simulate this problem in 2D, we meet a large number of 2D domains, and we risk that one of these domains is equal to or approaches a critical domain. In such a case the BEM will have difficulties with solving the Stokes equations for that particular domain. When simulating the blowing problem in 3D, no critical domains are encountered. It turns out that the BEM is a very efficient numerical method for this particular 3D problem with a moving boundary. As we are merely interested in the shape of the fluid, we only need to know the flow of its boundary. The BEM does exactly that; it does not compute the flow at the interior of the fluid. Furthermore it is rather easy to include other effects from the blowing problem in the model, such as gravity, surface tension and friction from the contact of the fluid with a wall. As only the boundary of the fluid is discretised, the system matrices that appear in the BEM are smaller than the system matrices that appear when solving the problem with a finite element method, for example. Though the BEM-matrices are dense, while the finite element matrices are sparse, the computational effort for the BEM is relatively low. In short, the BEMis a very appropriate numerical method when solving blowing problems.

In linear poroelasticity so far only collocation boundary element methods have been available. However, in some applications, e.g., when coupling with finite elements is desired, a symmetric formulation is preferable. Choosing a Galerkin approach which involves the second boundary integral equation, such a formulation is possible. Here, a previously presented integration by part technique for the regularization of the first boundary integral equation is extended to the second boundary integral equation as well. While the weakly singular representation of the double layer operator has been presented before, the emphasis lies here on the so called hyper‐singular boundary integral operator. Due to the regularization, this operator can be evaluated numerically and, hence, be used within a numerical scheme for the first time. Different numerical studies will be presented to show the behavior of the established symmetric Galerkin boundary element method, also comparing it with collocation boundary element methods. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Approximating three-dimensional steady-state potential flow problems using two-dimensional complex polynomials

In electrocardiographic imaging, epicardial potentials are reconstructed computationally from electrocardiographic measurements. The reconstruction is typically done with the help of the boundary element method (BEM), using the point collocation weighting and constant or linear basis functions. In this paper, we evaluated the performance of constant and linear point collocation and Galerkin BEMs in the epicardial potential problem. The integral equations and discretizations were formulated in terms of the single- and double-layer operators. All inner element integrals were calculated analytically. The computational methods were validated against analytical solutions in a simplified geometry. On the basis of the validation, no method was optimal in all testing scenarios. In the forward computation of the epicardial potential, the linear Galerkin (LG) method produced the smallest errors. The LG method also produced the smallest discretization error on the epicardial surface. In the inverse computation of epicardial potential, the electrode-specific transfer matrix performed better than the full transfer matrix. The Tikhonov 2 regularization outperformed the Tikhonov 0. In the optimal modeling conditions, the best BEM technique depended on electrode positions and chosen error measure. When large modeling errors such as omission of the lungs were present, the choice of the basis and weighting functions was not significant.

An adaptive cell-based domain integration method for treatment of domain integrals in 3D boundary element method for potential and elasticity problems

PurposeTwo-dimensional (2D) problems are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are considered. The problems are transformed into a boundary-only integral equation which can be solved numerically using a standard boundary element method (BEM). Some examples are solved to show the validity of the analysis and examine the accuracy of the numerical method.Design/methodology/approach The 2D problems which are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are solved using a combined BEM and Laplace transform. The time–space dependent coefficient equation is reduced to a time-dependent coefficient equation using an analytical transformation. Then, the time-dependent coefficient equation is Laplace transformed to get a constant coefficient equation, which can be written as a boundary-only integral equation. By utilizing a BEM, this integral equation is solved to find numerical solutions to the problems in the frame of the Laplace transform. These solutions are then inversely transformed numerically to obtain solutions in the original time–space frame.FindingsThe main finding of this research is the derivation of a boundary-only integral equation for the solutions of initial-boundary value problems governed by a modified-Helmholtz equation of time–space dependent coefficients for anisotropic functionally graded materials with time-dependent properties.Originality/valueThe originality of the research lies on the time dependency of properties of the functionally graded material under consideration.

A boundary element method (BEM) approach has been developed to solve the time‐dependent 1D advection‐diffusion equation. The 1D solution is part of a 3D numerical scheme for solving advection‐diffusion (AD) problems in fractured porous media. The full 3D scheme includes a 3D solution for the porous matrix, which is coupled with a 2D solution for fractures and a 1D solution for fracture intersections. As the hydraulic conductivity of the fracture intersections is usually higher than the hydraulic conductivity of the fractures and by at least one order of magnitude higher than the hydraulic conductivity of the porous matrix, the fastest flow and solute transport occurs in the fracture intersections. Therefore it is important to have an accurate and stable 1D solution of the transient AD problems. This article presents two different 1D BEM formulations for solution of the AD problems. The particular advantage of these formulations is that they provide one of the most straightforward and simplest ways to couple multiple intersecting 2D Boundary Element problems discretized with linear discontinuous elements. Both formulations are tested and compared for accuracy, stability, and consistency. The analysis helps to select the more suitable formulations according to the properties of the problem under consideration. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004

The boundary element method (BEM) is now a versatile and powerful tool of computational mechanics which has become a popular alternative to the well-established finite element method (FEM). Recently, some authors indicated that the presence of any domain integrals undermines most advantages of the BEM. We believe that this is not true if the domain integral contains known quantities (such as body sources or initial values). The actual sources of advantages of the BEM formulation are not determined by the question of if it is necessary to discretize the domain or not. The BEM formulation for the solution of a boundary value problem exhibits pure boundary character if the solution at any internal point can be expressed in terms of the boundary integrals of relevant quantities and the domain integrals of known body sources and initial values, if any. In other words, the solution can be expressed at an internal point without the need to know the solution at any other internal point. If the boundary values of the relevant quantities were known exactly and the integrations were performed with absolute accuracy, the integral representation would present the exact solution of the boundary value problem. That is why in the BEM solution, extreme emphasis is put on the accuracy of the approximations of boundary values and computation of boundary integrals. Since in the domain discretization techniques (such as FEM, finite difference method, collocation methods) the unknowns at all nodal points are computed simultaneously, the computational error is accumulated.

The consistent boundary element method for potential and elasticity: Part II — machine-precision numerical evaluations for 2D problems

Jacobian weighted element differential method for solid mechanics

Poroelasticity is concerned with heterogeneous media consisting of an elastic solid skeleton saturated by a diffusing pore fluid. Its range of application covers a variety of important real-life problems, such as design of nuclear reactor cores, exploitation of oil or gas deposits, simulation of living bone behaviour for orthopedical surgery, control of filtration leakage from reservoirs, and manufacturing process design for composite materials. Poroelasticity is now the subject of a fairly abundant literature, stemming from Terzaghi’s concept of effective stress and Biot’s linear consolidation theory. From a computational mechanics point of view, poroelastic analysis has been conducted using either the finite element method or the traditional (collocation) boundary element method [1–3].

The boundary element method (BEM) divides only the boundaries of the region under investigation into elements, so it diminishes the dimensionality of the problem, e.g., the 3D problem becomes a 2D problem, and the 2D problem becomes a 1D problem. This simplifies inputting the model into a computer and greatly reduces the number of algebraic equations. The advantage of this is even more evident for some 3D and infinite regional problems that often are encountered in geophysics. Originally published in China, this well-organized book is likely the most comprehensive work on the subject of solving applied geophysical problems. Basic mathematical principles are introduced in Chapter 1, followed by a general yet thorough discussion of the BEM in Chapter 2. Chapters 3 through 7 introduce the applications of BEM to solve problems of potential-field continuation and transformation, gravity and magnetic anomalies modeling, electric resistivity and induced polarization field modeling, magnetotelluric modeling, and various seismic modeling problems. Finally, in Chapter 8, a brief discussion is provided on how to incorporate the BEM and the finite-element method (FEM) together. In each chapter, detailed practical examples are given, and comparisons to both analytic and other numerical solutions are presented. This is an excellent book for numerically oriented geophysicists and for use as a textbook in numerical-analysis classes.

This paper describes a combination of the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a mesh‐free numerical method (MFS‐DRM model) to solve 2D and 3D nonhomogeneous diffusion problems. Using our method, the homogeneous solutions of the diffusion equations are solved by the MFS, and the DRM, based on the radial basis functions (RBF) of the thin plate splines (TPS), is employed to solve for particular solutions. The present scheme is free from the frequently used Laplace transform and the finite difference discretization method to deal with the time derivative term in the governing equation. By properly placing the source points in the time‐space domain, the solution is advanced in time until a steady state solution (if one exists) is reached. Since the present method does not need mesh discretization and nodal connectivity, the computational effort and memory storage required are minimal as compared to other domain‐oriented numerical schemes such as FDM, FEM, FVM, etc. Test results obtained for 2D and 3D diffusion problems show good comparability with analytical solutions and other numerical solutions, such as those obtained by the MFS‐DRM model based on the modified Helmholtz fundamental solutions. Thus the present numerical scheme has provided a promising mesh‐free numerical tool to solve nonhomogeneous diffusion problems with space‐time unification for diffusion fundamental solutions.