2D Potential Problems Research Articles

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

A new advance in the Complex Variable Boundary Element Method, or CVBEM, is its extension to three-dimension (3D). This advance breaks down the barrier of limiting CVBEM models to two-dimensional (2D) problems, and also opens the door to solving 3D potential problems with other 2D numerical analogs. In this paper, a 3D analog is developed using 2D basis functions of the complex analytic polynomial type. Thus, 2D complex polynomials are being applied to 3D potential problems. This new advance may be of interest to those involved in applied mathematics, complex variables, boundary elements, and numerical solution of partial differential equations of the Laplace of Poisson type.

A new technique for high-speed boundary element analyses of Laplace equations to obtain solutions in target regions

A new technique for boundary element analyses of 2D potential problems to quickly obtain the solution only in a target region was developed. This technique takes advantage of diffusion effects in Laplace fields. In fields governed by diffusion equations, high-frequency disturbances of boundary conditions on non-target boundaries far from the target region give little influence to the target region. In this technique, boundary conditions on non-target boundaries are transformed into the Fourier series, and only their low-frequency terms are utilized by using special weight functions for boundary integral equations. When the solutions only in a target region are needed, especially in large size boundary value problems, this technique enables us to obtain them quickly and precisely. The present method can be extended not only to Laplace equations but also to any kind of diffusive systems such as elastostatics.

Meshless analysis of potential problems in three dimensions with the hybrid boundary node method

AbstractCombining a modified functional with the moving least‐squares (MLS) approximation, the hybrid boundary node method (Hybrid BNM) is a truly meshless, boundary‐only method. The method may have advantages from the meshless local boundary integral equation (MLBIE) method and also the boundary node method (BNM). In fact, the Hybrid BNN requires only the discrete nodes located on the surface of the domain.The Hybrid BNM has been applied to solve 2D potential problems. In this paper, the Hybrid BNM is extended to solve potential problems in three dimensions. Formulations of the Hybrid BNM for 3D potential problems and the MLS approximation on a generic surface are developed. A general computer code of the Hybrid BNM is implemented in C++. The main drawback of the ‘boundary layer effect’ in the Hybrid BNM in the 2D case is circumvented by an adaptive face integration scheme. The parameters that influence the performance of this method are studied through three different geometries and known analytical fields. Numerical results for the solution of the 3D Laplace's equation show that high convergence rates with mesh refinement and high accuracy are achievable. Copyright © 2004 John Wiley & Sons, Ltd.

Boundary-only and fast meshless analysis of 3D potential problems

Boundary-only and fast meshless analysis of 3D potential problems

A posteriori error approximation in EFG method

AbstractRecently, considerable effort has been devoted to the development of the so‐called meshless methods. Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. One of the paths in the evolution of meshless methods has been the development of the element free Galerkin (EFG) method. In the EFG method, it is obviously important that the ‘a posteriori error’ should be approximated.An ‘a posteriori error’ approximation based on the moving least‐squares method is proposed, using the solution, computed from the EFG method. The error approximation procedure proposed in this paper is simple to construct and requires, at most, nearest neighbour information from the EFG solution. The formulation is based on employing different moving least‐squares approximations. Different selection strategies of the moving least‐squares approximations have been used and compared, to obtain optimum values of the parameters involved in the approximation of the error.The performance of the developed approximation of the error is illustrated by analysing different examples for two‐dimensional (2D) potential and elasticity problems, using regular and irregular clouds of points. The implemented procedure of error approximation allows the global energy norm error to be estimated and also provides a good evaluation of local errors. Copyright © 2003 John Wiley & Sons, Ltd.

Indirect formulations for symmetric Galerkin BEM and space derivative problems

An indirect symmetric Galerkin BEM (SGBEM) is applied to 2D potential problems in this paper. Based on the assumption that solutions from different methods should be the same, the hypersingular matrix appeared in SGBEM is approximately expressed by those matrices appeared in asymmetric Galerkin BEM (AGBEM). As only strong and weak singularities need to be solved, the problem becomes much simpler. The space derivatives of potential are expressed with a set of new meaning distributed flux, which will produce the same potential on the boundary position for Ω in the unbounded domain Ω+Ω′, so that hypersingularity will not appear for boundary points. Therefore, there is no need of C1,α for the spatial interpolation function (no Galerkin integration can be used for this purpose). Formulations for both the steady‐state and time‐domain potential problems are given. Three numerical examples are analyzed to demonstrate the effectiveness and accuracy of the proposed indirect method.

Scattering of Seismic Waves: Applications to the Mexico City Valley

5R55. Scattering of Seismic Waves: Applications to the Mexico City Valley. - E Reinoso (Univ Nacional Autonoma, Mexico). WIT Press, Southampton, UK. 2002. 202 pp. ISBN 1-85312-833-3. $122.00.Reviewed by CS Manohar (Dept of Civil Eng, Indian Inst of Sci, Bangalore, 560 012, India).Amplification of seismic energy, as it filters through subsurface formations and mountainous terrain, is a complex dynamical phenomenon. From an engineering point of view, an understanding of this phenomenon is crucial in appreciating the spatial variability of ground motions and the concomitant damage patterns observed during strong motion earthquakes. Mathematical modeling of this phenomenon has become possible in the recent decades due to the developments in the field of computational solid mechanics and emergence of computing power. This book reports on such modeling efforts as applied to Mexico City valley with special emphasis on delineating the model predictions with observed earthquake amplification patterns. The book presents numerical models to predict the scattering from irregular topographies and alluvial valleys due to plane incident waves and Rayleigh waves. This book forms a part of a series on advances in earthquake engineering. The book consists of eight chapters with a list of references following each chapter. The first chapter summarizes the contents of the book. The basics of linear elastodynamics and wave propagation in infinite media are covered in Chapter 2. The use of the boundary element method for 2D potential problems is detailed in Chapter 3. The problems of 2D and 3D elastodynamics are covered in Chapters 4 and 5. The material covered includes discussion on buried alluvial valleys (made up of two regions: half-space and the valley), canyons of arbitrary shapes, and scattering by mountains with arbitrary geometry. Chapter 6 summarizes the observed amplification patterns in the Mexico City valley. The author here concisely, yet comprehensively, summarizes the features discernable from earthquake accelerographs recorded since the 1985 Michoacan earthquake. In Chapter 7, the gamut of mathematical modeling tools outlined in Chapters 2–5 are brought to bear on the problem of modeling of Mexico City valley. Discussions on comparing observed facts with model predictions are made. While a 1D model is found to be adequate to explain most of the observed amplification, the need for adopting a 2D and 3D model is demonstrated to explain amplification at edges of the valley or when the clay deposits are deeper. The book concludes in Chapter 8 wherein a few suggestions for future research are also provided. The book is well written and achieves well its stated objectives. The book contains ample references to published work. One of the striking features of this book has been the conception and outlay of the figures. This has added immensely to the eloquence of the book. This reviewer considers Scattering of Seismic Waves: Applications to the Mexico City Valley to be an excellent contribution to the literature on earthquake engineering. The book would serve as a valuable starting point for any efforts in mathematical modeling of seismic response of large valleys and mountains that house many cities of the world. The book qualifies to be a reference book that certainly would be a valuable addition to libraries of universities and research laboratories pursuing research and development in the areas of earthquake engineering and engineering seismology.

A BEM formulation for inhomogeneous potential problems by particular integrals

This paper presents particular integral BEM formulations for purely axisymmetric, 2D and 3D inhomogeneous potential problems (Poisson equation). The steady-state potential flow equation (Laplace equation) is used for the complementary solution. 2D and 3D forms of the particular integrals for potential and flux are obtained by introducing the concept of a global shape function to approximate a source term of the inhomogeneous equation. The axisymmetric particular integrals and global shape function are derived by integrating 3D formulation along the circumferential direction leading to elliptic integrals. The numerical results for three example problems are given and compared with their analytical solutions to demonstrate the accuracy of the present formulations. Generally, good agreement among all of those results is obtained.

Spatially Adapted Multiwavelets and Sparse Representation of Integral Equations on General Geometries

In this paper we develop irregular wavelet representations for complex domains with the goal of demonstrating their potential for three-dimensional (3D) scientific and engineering computing applications. We show existence and construction of a large class of continuous spatially adapted multiwavelets in $R^{\eta}$ with vanishing moments over complex geometries. These wavelets share all of the major advantages of conventional wavelets in that they provide an analytical tool for studying data, functions, and operators at different scales. However, unlike conventional wavelets, which are restricted to uniform grids, spatially adapted multiwavelets allow fast transform, localization, and decorrelation on complex meshes, such as those encountered in finite element modeling. We show how these new constructions can be applied to partial differential equations cast in the integral form. We implement the wavelet approach for several model two-dimensional (2D) and 3D potential problems. It is shown that the optimal convergence rate is achieved, with only ${\cal O}( N ( {{\rm log}N} )^{\alpha} )$ entries of the discrete operator matrix, where $\alpha$ is a small number and N is the number of unknowns.

315 ポテンシャル問題に対する注目領域解の境界要素高速解法

A new technique of boundary element methods for 2D potential problems to obtain the solution only in a target region quickly was developed. This technique takes advantage of diffusion effects in Laplace fields. In fields governed by diffusion equations, high frequency disturbances of boundary conditions on non-target bounds far from the target region give little infection to the target region. In this technique, boundary conditions on non-target bound are transformed into Fourier series, and their only low frequency terms are utilized by using special weight functions for boundary integral equations. When the solutions only in a target region are needed especially in large size boundary value problems, this technique enable us to obtain them quickly and precisely with solving non-target solutions roughly.

New approaches for error estimation and adaptivity for 2D potential boundary element methods

AbstractThis work presents two new error estimation approaches for the BEM applied to 2D potential problems. The first approach involves a local error estimator based on a gradient recovery procedure in which the error function is generated from differences between smoothed and non‐smoothed rates of change of boundary variables in the local tangential direction. The second approach involves the external problem formulation and gives both local and global measures of error, depending on a choice of the external evaluation point. These approaches are post‐processing procedures. Both estimators show consistency with mesh refinement and give similar qualitative results. The error estimator using the gradient recovery approach is more general, as this formulation does not rely on an ‘optimal’ choice of an external parameter. This work presents also the use of a local error estimator in an adaptive mesh refinement procedure. This r‐refinement approach is based on the minimization of the standard deviation of the local error estimate. A non‐linear programming procedure using a feasible‐point method is employed using Lagrange multipliers and a set of active constraints. The optimization procedure produces finer meshes close to a singularity and results that are consistent with the problem physics. Copyright © 2002 John Wiley & Sons, Ltd.

Hypersingular kernel integration in 3D Galerkin boundary element method

We consider Cauchy singular and Hypersingular boundary integral equations associated with 3D potential problems defined on polygonal domains, whose solutions are approximated with a Galerkin boundary element method, related to a given triangulation of the boundary. In particular, for constant and linear shape functions, the most frequently used basis functions, we give explicit results of the analytical inner integrations and suggest suitable quadrature schemes to evaluate the outer integrals required to form the Galerkin matrix elements. These numerical indications are given after an analysis of the singularities arising in the whole integration process, which is valid also for shape functions of higher degrees.

Application of a direct Trefftz method with domain decomposition to 2D potential problems

This article presents an application of a direct Trefftz method with domain decomposition method to the two-dimensional potential problem. In the direct Trefftz methods, regular T-complete functions satisfying the governing equations are taken as the weighting functions and then, the boundary integral equations are derived from the weighted residual expressions of the governing equations. Since the T-complete functions are regular, the final equations are also regular and therefore, much simpler than the ordinary boundary element methods employing the singular fundamental solutions. Their computational accuracy, however, is dependent on the condition number of the coefficient matrices of the algebraic system of equations. So, for improving the accuracy, we introduce the domain decomposition method to the direct Trefftz methods. The present method is applied to the two-dimensional potential problem in order to confirm the validity.

An adaptive algorithm for boundary element method postprocessing

This paper deals with a novel algorithm for drawing contour lines in the postprocessing of the boundary element method calculations for 2D potential problems. The basic idea is to use an adaptive approach to produce a triangular mesh of higher order interpolation. Based on this mesh, the smooth equipotential map can be generated. The advantage is that the CPU time in the postprocessing can be greatly reduced and a grid for drawing the equipotential map has not to be prescribed.

On the application of 2D potential theory to electrostatic simulation

AbstractComputing the charge density on the surfaces of conductors is fundamental to many electrostatic applications. Difficulties arise in two‐dimensional simulation due to the O(log|r|) potential. Fully understanding the 2D potential problem, and reasonably connecting the mathematical formulation with physical interpretation, are key to handling this difficulty properly. In the absence of these factors it is easy for one to propose an ill‐posed problem. In this paper, a complete investigation of potential theory, from 3D to 2D, is made for computing charge density on the surfaces of conductors. Various integral formulations are derived ‐ these are applicable in different situations. It is shown that, in general, the electrostatic potential need not be finite valued at infinity for 2D problems. Numerical examples for the 2D case are constructed to show that the formulations are consistent. Criteria for choosing the most appropriate formulation for a given problem are suggested.