Problem In Potential Theory Research Articles

Underlying Principles of the Boundary Element Method

3R4. Underlying Principles of the Boundary Element Method. - D Cartwright (Col of Eng, Bucknell Univ PA). WIT Press, Southampton, UK. 2001. 276 pp. ISBN 1-85312-839-2. $149.00. Reviewed by DE Beskos (Dept of Civil Eng, Univ of Patras, Patras, GR-26500, Greece).This is a very well written introductory textbook on the foundations of the direct Boundary Element Method (BEM). It is very useful to both teachers and their undergraduate students in applied mathematics and engineering, as well as those interested in learning the basics of the method. The emphasis is on the principles and the mathematical derivations of the BEM and not on its numerical implementation. In that sense, the book is unique since most of the existing books emphasize the numerical implementation of the method. Detailed mathematical derivations are provided and solved problems are presented in detail in each chapter to help the student understand the subject matter of the book. Applications are described for one-, two- and three-dimensional problems of potential theory and elastostatics in a unified manner. Only constant elements are considered here for which the computation of singular integrals can be done analytically (in closed form). There are two aspects of the book this reviewer considers very important and worth mentioning: i) The concepts of the Green’s function and the fundamental solution are both discussed in detail. It is further shown how one can obtain the latter as a combination of Green’s functions defined for different boundary conditions. In many other books these two concepts are used one for the other and this creates confusion. ii) The boundary integral equation for internal points is derived here through the method of weighted residuals, which is a very powerful and general method for formulating boundary element and finite element methods. The whole book consists of seven chapters, one appendix, a bibliography, and a subject index. More specifically, Chapter 1 deals with a discussion on the derivation of the basic field equations (governing equations) of the scalar or vector type and of the second or fourth order in one or more dimensions. Chapter 2 discusses Green’s functions, their properties, derivation, and use in solving boundary value problems. The concept of the fundamental solution, its relation to Green’s functions, and its derivation are taken up in Chapter 3. The method of weighted residuals is described in Chapter 4, and its use in the derivation of the direct boundary integral equation for one-dimensional, potential, and elastostatic problems are described in Chapters 5, 6, and 7, respectively. In those last three chapters, related problems are solved to illustrate the method and demonstrate its advantages. Finally, the last chapter contains various appendices dealing with details of various mathematical derivations, which were presented in Chapters 3 and 5–7. The book concludes with a list of 20 reference books on the BEM and a short subject index. The author has certainly succeeded in fulfilling his stated aim of providing an introductory textbook emphasizing the underlying principles of the BEM. Underlying Principles of the Boundary Element Method should be purchased by teachers, undergraduate and graduate students, researchers who would like to start working in the field, and certainly by libraries.

Effects of the spherical terrain on gravity and the geoid

The determination of the gravimetric geoid is based on the magnitude of gravity observed at the topographical surface and applied in two boundary value problems of potential theory: the Dirichlet problem (for downward continuation of gravity anomalies from the topography to the geoid) and the Stokes problem (for transformation of gravity anomalies into the disturbing gravity potential at the geoid). Since both problems require involved functions to be harmonic everywhere outside the geoid, proper reduction of gravity must be applied. This contribution deals with far-zone effects of the global terrain on gravity and the geoid in the Stokes–Helmert scheme. A spherical harmonic model of the global topography and a Molodenskij-type spectral approach are used for a derivation of suitable computational formulae. Numerical results for a part of the Canadian Rocky Mountains are presented to illustrate the significance of these effects in precise (i.e. centimetre) geoid computations. Their omission can be responsible for a long-frequency bias in the geoid, especially over mountainous areas. Due to the rough topography of the testing area, these numerical values can be used as maximum global estimates of the effects (maybe with the exception of the Himalayas). This study is a continuation of efforts to model adequately the topographical effects on gravity and the geoid, especially of a comparing the effects of the planar topographical plate and the spherical topographical shell on gravity and the geoid [Vanicek, Novak, Martinec (2001) J Geod 75: 210–215].

The meshless hypersingular boundary node method for three-dimensional potential theory and linear elasticity problems

The Boundary Node Method (BNM) represents a coupling between Boundary Integral Equations (BIEs) and Moving Least Squares (MLS) approximants. The main idea here is to retain the dimensionality advantage of the former and the meshless attribute of the latter. The result is a ‘meshfree’ method that decouples the mesh and the interpolation procedures. The BNM has been applied to solve 2-D and 3-D problems in potential theory and linear elasticity. The Hypersingular Boundary Element Method (HBEM) has diverse important applications in areas such as fracture mechanics, wave scattering, error analysis and adaptivity, and to obtain a symmetric Galerkin boundary element formulation. The present work presents a coupling of Hypersingular Boundary Integral Equations (HBIEs) with MLS approximants, to produce a new meshfree method — the Hypersingular Boundary Node Method (HBNM). Numerical results from this new method, for selected 3-D problems in potential theory and in linear elasticity, are presented and discussed in this paper.

A primal mixed formulation for exterior transmission problems in ${\bf R}^2$

We combine a primal mixed finite element approach with a Dirichlet-to-Neumann mapping (arising from the boundary integral equation method) to study the weak solvability and Galerkin approximations of a class of linear exterior transmission problems in potential theory. Our results are mainly based on the Babuska-Brezzi theory for variational problems with constraints. We establish the uniqueness of solution for the continuous and discrete formulations, and show that finite element subspac es of Lagrange type satisfy the discrete compatibility conditions. In addition, we provide the error analysis, including polygonal approximations of the domain, and prove strong convergence of the Galerkin solutions. Moreover, under additional regularity assumptions on the solution of the continuous formulation, we obtain the asymptotic rate of convergence O(h).

Torsional rigidity of reinforced concrete bars with arbitrary sectional shape

In this paper, the torsional rigidity of arbitrarily shape bar made of different materials is studied on the basis of theory of elasticity and finite element approach. With additional boundary conditions for the common boundaries of different materials from the continuous conditions of deformation and traction across the interior boundary, the torsion function can be solved numerically from the second boundary-value problem of potential theory. The traction jump boundary conditions across the interior surfaces are enforced in the alternate finite element approach. Several examples are shown to check the computational approach proposed, and the approach, at last, is applied to calculate the torsional rigidity of reinforced concrete bar and some multiply connected cross sections such as tower leg section of the Tsing Ma Bridge and other engineering structures.

The Method of Fundamental Solutions for axisymmetric elasticity problems

We investigate the use of the Method of Fundamental Solutions (MFS) for the approximate solution of certain problems of three-dimensional elastostatics in isotropic materials. Specifically, we consider problems in which the geometry is axisymmetric and the boundary conditions are either axisymmetric or arbitrary. In each case, the problem reduces to one of solving a two-dimensional problem or a set of such problems in the radial and axial coordinates. As in axisymmetric problems in potential theory and in acoustic scattering and radiation, the fundamental solutions of the governing equations and their normal derivatives required in the formulation of the MFS are expressible in terms of complete elliptic integrals. We present the results of numerical experiments which demonstrate the efficacy of the MFS approach.

Approximation method in problems of potential theory

We investigate the properties of the Dzyadyk approximation method in the case of a binomial function. We study conditions under which an estimate for the relative error of approximation in the uniform metric is exact. The results obtained are applied to certain problems in potential theory.

The boundary node method for three‐dimensional problems in potential theory

The boundary node method for three‐dimensional problems in potential theory

The boundary node method for three-dimensional problems in potential theory

The boundary node method for three-dimensional problems in potential theory

A Problem in Potential Theory and Zero Asymptotics of Krawtchouk Polynomials

In this paper we investigate the asymptotics of the zeros of normalized Krawtchouk polynomials kn(Nx, p, N) when the ratio of the parameters n/N→α as n, N→∞. For this purpose we consider in detail a particular constrained energy problem on the interval [0, 1] in the presence of an external field. We find the support and the density of the constrained extremal measure for all possible values of the parameter α.

Which Eigenvalues Are Found by the Lanczos Method?

When discussing the convergence properties of the Lanczos iteration method for the real symmetric eigenvalue problem, Trefethen and Bau noted that the Lanczos method tends to find eigenvalues in regions that have too little charge when compared to an equilibrium distribution. In this paper a quantitative version of this rule of thumb is presented. We describe, in an asymptotic sense, the region containing those eigenvalues that are well approximated by the Ritz values. The region depends on the distribution of eigenvalues and on the ratio between the size of the matrix and the number of iterations, and it is characterized by an extremal problem in potential theory which was first considered by Rakhmanov. We give examples showing the connection with the equilibrium distribution.

A Green's function numerical method for some inverse boundary value problems

We solve identification problems in potential theory, electrostatics, steady state heat conduction, etc. in conductive plates. A plate with simple shape contains a hole. Either the size, shape, and location of the hole, or the flux across its boundary, is unknown. Boundary conditions are prescribed on part of the plate edge, and voltage or heat is applied to the remainder. An additional measurement is taken on part of the plate edge. From this overspecified data, we determine (1) the structure of the hole, given the flux across its boundary, or (2) the flux across the hole boundary, given the hole. Boundary elements, least squares optimization, and special boundary value properties of Green's functions are exploited in this numerical treatment of the identification problem.

The hunt property and unique solvability of the inverse problem of potential theory for one class of generalized Ukawa potentials

The hunt property and unique solvability of the inverse problem of potential theory for one class of generalized Ukawa potentials

Wavelet approximations on closed surfaces and their application to boundary-value problems of potential theory

Wavelets on closed surfaces in Euclidean space R3 are introduced starting from a scale discrete wavelet transform for potentials harmonic down to a spherical boundary. Essential tools for approximation are integration formulas relating an integral over the sphere to suitable linear combinations of functional values (resp. normal derivatives) on the closed surface under consideration. A scale discrete version of multiresolution is described for potential functions harmonic outside the closed surface and regular at infinity. Furthermore, an exact fully discrete wavelet approximation is developed in case of band-limited wavelets. Finally, the role of wavelets is discussed in three problems, namely (i) the representation of a function on a closed surface from discretely given data, (ii) the (discrete) solution of the exterior Dirichlet problem, and (iii) the (discrete) solution of the exterior Neumann problem.

From Potential Theory to Matrix Iterations in Six Steps

The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, Bi-CGSTAB, and so on) is reviewed. For a computation of this kind, an estimated asymptotic convergence factor $\rho \le 1$ can be derived by solving a problem of potential theory or conformal mapping. Six approximations are involved in relating the actual computation to this scalar estimate. These six approximations are discussed in asystematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming.

The third boundary value problem in potential theory for domains with a piecewise smooth boundary

The paper investigates the third boundary value problem $$\frac{{\partial u}}{{\partial n}}{ } + { \lambda }u{ } = { }\mu $$ for the Laplace equation by the means of the potential theory. The solution is sought in the form of the Newtonian potential (1), (2), where ν is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure $$T\nu . { Denote by }T:{ }\nu { } \to { }T\nu $$ the corresponding operator on the space of signed measures on the boundary of the investigated domain G. If there is α ≠ 0 such that the essential spectral radius of $$\left( {\alpha I - T} \right)$$ is smaller than |α| (for example, if G ⊂ R 3 is a domain “with a piecewise smooth boundary” and the restriction of the Newtonian potential $$U{ \lambda }$$ on ∂G is a finite continuous functions) then the third problem is uniquely solvable in the form of a single layer potential (1) with the only exception which occurs if we study the Neumann problem for a bounded domain. In this case the problem is solvable for the boundary condition $$\mu { } \in { }C'$$ for which μ(∂G) = 0.

Variational approach to three-dimensional inverse problems of potential theory

A method of regularization and solving problems of flow around bodies according to a desired distribution of the characteristics of the fields on the surface of the bodies themselves is described. Examples of solutions of such problems for potential flows are given, and a method of reducing inverse problems of the fluid mechanics of an effectively inviscid liquid to potential problems is presented.

SOLUTION OF LARGE LINEAR SYSTEMS ON PIPELINED SIMD MACHINES

We developed a direct out-of-core solver for dense non-symmetric linear systems of ‘arbitrary’ size N×N. The algorithm fully employs the Basic Linear Algebra Subprograms (BLAS), and can therefore easily be adapted to different computer architectures by using the corresponding optimized routines. We used blocked versions of left-looking and right-looking variants of LU decomposition to perform most of the operations in Level 3 BLAS, to reduce the number of I/O operations and to minimize the CPU time usage. The storage requirements of the algorithm are only 2N×NB data elements where NB≪N. Depending on the sustained floating point performance and the sustained I/O rate of the given hardware, we derived formulas that allow for choosing optimal values of NB to balance between CPU time and I/O time. We tested the algorithm by means of linear systems derived from 3D-BEM for strongly and weakly singular integral equations and from interpolation problems for scattered data on closed surfaces in ℝ3. It took only about 2⋅5 CPU minutes on a 5 GFLOPS vector computer SNI S600/20 to solve a linear system of size 10000, which corresponds to a performance of 4⋅3 GFLOPS; a value of NB=650 gives a reasonable I/O time and the necessary main storage size is about 13 Mwords. In addition, we compared the algorithm with (1) an out-of-core version of GMRES and (2) a wavelet transform followed by in-core GMRES after thresholding. At least for boundary integral equations of classical boundary value problems of potential theory, the out-of-core version of GMRES is superior to the direct out-of-core solver and the wavelet transform since the algorithm converged after at most 5 iteration steps. It took about 17 s to solve a system with 8192 unknowns compared with 146 s for direct out-of-core and 402 s for wavelet transform followed by in-core GMRES. © 1997 by John Wiley & Sons, Ltd.

Hermite-Pade approximants for systems of Markov-type functions

The Hermite-Pade approximants are studied for systems of Markov functions (introduced in this paper) with structure described by a graph. Results of an asymptotic nature are stated in terms of certain equilibrium problems of potential theory concerning vector potentials.

Mapping EEG-potentials on the surface of the brain: a strategy for uncovering cortical sources.

This paper describes a uniform method for calculating the interpolation of scalp EEG potential distribution, the current source density (CSD), the cortical potential distribution (cortical mapping) and the CSD of the cortical potential distribution. It will be shown that interpolation and deblurring methods such as CSD or cortical mapping are not independent of the inverse problem in potential theory. Not only the resolution but also the accuracy of these techniques, especially those of deblurring, depend greatly on the spatial sampling rate (i.e., the number of electrodes). Using examples from simulated and real (64 channels) data it can be shown that the application of more than 100 EEG channels is not only favourable but necessary to guarantee a reasonable accuracy in the calculations of CSD or cortical mapping. Likewise, it can be shown that using more than 250 electrodes does not improve the resolution.