Showing papers on "Laplace's equation published in 1999"
TL;DR: An adaptive fast multipole method for the Laplace equation in three dimensions that uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost.
671 citations
TL;DR: In this paper, the authors consider the problem of detecting corrosion damage on an inaccessible part of a metallic specimen and use a simplified model of corrosion appearance to reduce the problem to recovering a functional coefficient in a Robin boundary condition for Laplace's equation.
Abstract: We consider the problem of detecting corrosion damage on an inaccessible part of a metallic specimen. Electrostatic data are collected on an accessible part of the boundary. The adoption of a simplified model of corrosion appearance reduces our problem to recovering a functional coefficient in a Robin boundary condition for Laplace's equation. We review theoretical results and numerical methods based on the thin-plate approximation and the Galerkin method. Moreover, we introduce a numerical algorithm based on the quasi-reversibility method.
209 citations
TL;DR: An infinite-order Boussinesq-type differential equation for wave shoaling over variable bathymetry is derived in this paper, where three scaling parameters are defined: nonlinearity, dispersion, and bottom slope.
Abstract: An infinite-order, Boussinesq-type differential equation for wave shoaling over variable bathymetry is derived Defining three scaling parameters – nonlinearity, the dispersion parameter, and the bottom slope – the system is truncated to a finite order Using Pade approximants the order in the dispersion parameter is effectively doubled A derivation is made systematic by separately solving the Laplace equation in the undisturbed fluid domain and then addressing the nonlinear free-surface conditions We show that the nonlinear interactions are faithfully captured The shoaling and dispersion components are time independent
161 citations
TL;DR: In this article, the signal induced in a readout circuit connected to a pixel electrode in a semiconductor gamma-ray imaging array is calculated by solving the Laplace equation and two approaches are presented that use Green functions in solving the boundary value problem: decomposition into basis functions and construction of an infinite series of image charges.
Abstract: The signal induced in a readout circuit connected to a pixel electrode in a semiconductor gamma-ray imaging array is calculated by solving the Laplace equation. Two approaches are presented that use Green functions in solving the boundary value problem: decomposition into basis functions, and construction of an infinite series of image charges. Another approach is developed based on the Ramo–Shockley theorem, which makes use of weighting potentials. These potentials may be readily calculated in three dimensions using a Fourier-transform propagation technique. An analytic solution is found for the special two-dimensional case of a strip detector. Experiments on CdZnTe square-pixel test structures using alpha radiation confirm the expected trends in pulse shape as a function of pixel size. Signals observed simultaneously on adjacent pixels also follow the predicted division of currents. Trends with pixel size are also confirmed in the shape of pulse-height spectra taken using a 99mTc source.
146 citations
TL;DR: The proposed implementation uses collocation and piecewise constant shape functions to discretise the hypersingular boundary integral equation for crack problems with GMRES (generalised minimum residual method) in connection with FMM (fast multipole method).
Abstract: This paper discusses a three-dimensional fast multipole boundary integral equation method for crack problems for Laplace's equation. The proposed implementation uses collocation and piecewise constant shape functions to discretise the hypersingular boundary integral equation for crack problems. The resulting numerical equation is solved with GMRES (generalised minimum residual method) in connection with FMM (fast multipole method). It is found that the obtained code is faster than a conventional one when the number of unknowns is greater than about 1300.
128 citations
TL;DR: In this paper, a regularization method was proposed and associated error bounds can be derived, and the error between the continuous continuous Cauchy problem and the difference approximation obtained via a suitable minimization problem can be estimated by a discretization and a regularisation term.
Abstract: The standard five-point difference approximation to the Cauchy problem for Laplace's equation satisfies stability estimates---and hence turns out to be a well-posed problem---when a certain boundedness requirement is fulfilled. The estimates are of logarithmic convexity type. Herewith, a regularization method will be proposed and associated error bounds can be derived. Moreover, the error between the given (continuous) Cauchy problem and the difference approximation obtained via a suitable minimization problem can be estimated by a discretization and a regularization term.
111 citations
TL;DR: In this paper, the notion of Laplace invariants is generalized to lattices and discrete equations that are difference analogues of hyperbolic partial differential equations with two independent variables.
Abstract: The notion of Laplace invariants is generalized to lattices and discrete equations that are difference analogues of hyperbolic partial differential equations with two independent variables. The sequence of Laplace invariants satisfies the discrete analogue of the two-dimensional Toda lattice. We prove that terminating this sequence by zeros is a necessary condition for the existence of integrals of the equation under consideration. We present formulas for the higher symmetries of equations possessing such integrals. We give examples of difference analogues of the Liouville equation.
110 citations
TL;DR: In this article, a reduced Navier-Stokes equation for a thin liquid layer falling down a solid wall, either vertical or inclined, is studied by means of a reduced equation developed by the regularized longwave expansion method, which is a combination of the Pade approximation and the long-wave expansion.
Abstract: Waves on a thin liquid layer falling down a solid wall, either vertical or inclined, are studied by means of a reduced equation. This equation is developed by the regularized long-wave expansion method, which is a combination of the Pade approximation and the long-wave expansion. Its numerical solutions are compared with the calculations of the full Navier–Stokes equation, simplified Navier–Stokes equation (the “boundary-layer” equation), and the traditional long-wave equations, as well as with experimental measurements. When the Reynolds number R is as small as unity, the present equation agrees with the Navier–Stokes equation and also with the traditional long-wave equations. For larger values of R, the traditional long-wave equations lose their validity and make a false prediction, while the present equation agrees with the Navier–Stokes equation, as long as the rescaled Reynolds number δ*=R/W1/3 does not exceed unity in the case of vertical films. Unlike the “boundary-layer” equation developed by previous researchers and expected to be valid at moderate and large Reynolds number, the present equation governs the surface evolution alone without postulating to resolve the velocity field. The structure of the present equation, however, has a correspondence to the depth-averaged equations, which facilitates discussing the physical mechanism of the wave dynamics. In particular, the physical origin of the instability mechanism and the wave suppression mechanism are discussed in terms of Whitham’s wave hierarchy theory. The balance of several physical effects such as drag, gravity, and inertia are also discussed in this connection. The analysis of the tail structure of permanent solitary waves predicts that the R dependence of its tail length λ exhibits two distinct regimes in the λ-R diagram. The second regime, which is not predicted by the traditional long-wave equations, arises when the inertia effect becomes dominant.
100 citations
TL;DR: The second-order self-adjoint forms of the transport equation are the even-and odd-parity forms as mentioned in this paper, and a useful alternative to these two forms exists in the form of a second order self-a...
Abstract: The traditional second-order self-adjoint forms of the transport equation are the even- and odd-parity equations. A useful alternative to these equations exists in the form of a second-order self-a...
93 citations
Posted Content•
TL;DR: In this paper, the authors provide two derivations of the Lorentz-Dirac equation, one based on the half-retarded minus half-advanced potential and the other based on energy-momentum conservation.
Abstract: These notes provide two derivations of the Lorentz-Dirac equation. The first is patterned after Landau and Lifshitz and is based on the observation that the half-retarded minus half-advanced potential is entirely responsible for the radiation-reaction force. The second is patterned after Dirac, and is based upon considerations of energy-momentum conservation; it relies exclusively on the retarded potential. The notes conclude with a discussion of the difficulties associated with the interpretation of the Lorentz-Dirac equation as an equation of motion for a point charge. The presentation is essentially self-contained, but the reader is assumed to possess some elements of differential geometry (necessary for the second derivation only).
88 citations
TL;DR: In this paper, the form of topological derivatives of arbitrary shape functionals depending on solutions of the three-dimensional Laplace equation is derived, which can be used for solving shape optimization problems involving diffusion or heat transfer.
Abstract: The form of topological derivatives of arbitrary shape functionals depending on solutions of the three-dimensional Laplace equation is derived. The derivatives can be used for solving shape optimization problems involving diffusion or heat transfer.
TL;DR: In this paper, the heat equation and the sobolev spaces and the laplace's equation in a half space were discussed. But the authors focused on the heat equations in the half space and not on the full space.
Abstract: (1999). Weighted sobolev spaces and laplace's equation and the heat equations in a half space. Communications in Partial Differential Equations: Vol. 24, No. 9-10, pp. 1611-1653.
TL;DR: In this paper, a representation for Green's functions for Laplace' equation in domains with infinite boundaries is obtained by integrating solutions to appropriate heat conduction problems with respect to time.
Abstract: Representations for Green's functions for Laplace' equation in domains with infinite boundaries are obtained by integrating solutions to appropriate heat conduction problems with respect to time. By using different representations for these heat equation solutions for small and large times, the changeover being determined by an arbitrary positive parameter a, a one-parameter family of formulae for the required Green' function is derived and by varying a the convergence characteristics of this new representation can be altered. Letting a zero results in known eigenfunction expansions and, in those situations in which they exist, letting a recovers known image series representations. The method, which is essentially equivalent to Ewald summation, is applied to two types of problem. First, it is applied to potential flow between parallel planes and in a rectangular channel, and, second, to two- and three-dimensional water-wave problems in which the depth is constant. In all cases the results of computations ...
TL;DR: In this paper, an implicit analytic solution for two-dimensional groundwater flow through a large number of non-intersecting circular inhomogeneities in the hydraulic conductivity is presented.
TL;DR: In this paper, the Laplace equation is solved using a finite difference method to generate sensitivity maps, and both linear back-projection and an iterative algorithm have been implemented for image reconstruction.
Abstract: Electrical capacitance tomography (ECT) with circular sensors has previously been investigated. For some industrial applications such as circulating fluidised beds, square sensors are required. Research into this specific area has been carried out for the first time. To generate sensitivity maps, the Laplace equation is solved using a finite difference method. Both the linear back-projection algorithm and an iterative algorithm have been implemented for image reconstruction. Experimental results are promising.
TL;DR: In this paper, an implicit analytic solution for 3D groundwater flow through a large number of non-intersecting spheroidal inhomogeneities in the hydraulic conductivity is presented.
TL;DR: In this article, a computational procedure based on the Trefftz method for the solution of non-linear Poisson problems is proposed, where the problem is solved by finding an approximate particular solution to the Poisson equation and using boundary collocation to solve the resulting Laplace equation.
TL;DR: An approach to grid optimization for a second order finite-difference scheme for elliptic equations using the Pade--Chebyshev approximation of the inverse square root increases the convergence order of the Neumann-to-Dirichlet map from second to exponential without increasing the stencil of the finite-Difference scheme and losing stability.
Abstract: We suggest an approach to grid optimization for a second order finite-difference scheme for elliptic equations. A model problem corresponding to the three-point finite-difference semidiscretization of the Laplace equation on a semi-infinite strip is considered. We relate the approximate boundary Neumann-to-Dirichlet map to a rational function and calculate steps of our finite-difference grid using the Pade--Chebyshev approximation of the inverse square root. It increases the convergence order of the Neumann-to-Dirichlet map from second to exponential without increasing the stencil of the finite-difference scheme and losing stability.
TL;DR: In this article, it is shown that the macroscopic transfer properties through a system of arbitrary shape are determined by the characteristics of a first-passage interface-interface random walk operator, which is the distribution of the harmonic measure on the eigenmodes of this linear operator that controls the transfer.
Abstract: The flux across resistive irregular interfaces driven by a force deriving from a Laplacian potential is computed on a rigorous basis. The theory permits one to relate the size of the active zone to the derivative of the spectroscopic impedance with respect to the surface resistivity r through: It is shown that the macroscopic transfer properties through a system of arbitrary shape are determined by the characteristics of a first-passage interface-interface random walk operator . More precisely, it is the distribution of the harmonic measure (or normalized primary current) on the eigenmodes of this linear operator that controls the transfer. In addition, it is also shown that, whatever the dimension, the impedance of a weakly polarizable electrode for any irregular geometry scales under a homothety transformation as Ld-1, L being the size of the system and d its topological dimension. In this new formalism, the question addressed in the title is transformed in a open mathematical question: “Knowing the distribution of the harmonic measure on the eigenmodes of the self-transport operator, can one retrieve the shape of the interface?”
TL;DR: In this article, a new and more accurate set of deterministic evolution equations for the propagation of fully dispersive, weakly nonlinear, irregular, multidirectional waves are derived directly from the Laplace equation with leading order nonlinearity in the surface boundary conditions.
TL;DR: In this paper, the wave equation with a potential was studied and the wave equations with potentials were analyzed in the context of Partial Differential Equations (PDE) and Wave Equations with Potentials (WEE).
Abstract: (1999). On the wave equation with a potential. Communications in Partial Differential Equations: Vol. 25, No. 7-8, pp. 1549-1565.
30 May 1999
TL;DR: In this paper, Wu and Eatock Taylor (1994a, 1994b) used a finite element method to determine the velocity potential, which satisfies the Laplace equation for Neumann and Dirichlet boundary conditions.
Abstract: Nonlinear transient waves are numerically calculated in time domain and validated by laboratory data. The simulation is based on potential flow theory and performed in a two dimensional numerical wave tank with piston-type wave generator. A finite element method developed by Wu and Eatock Taylor (1994a, 1994b) is used to determine the velocity potential, which satisfies the Laplace equation for Neumann and Dirichlet boundary conditions. To develop the solution in time domain the fourth-order Runge-Kutta method is applied. The paper presents numerical results of the potential and velocity distributions of a transient wave train. The calculated and measured surface elevation history at different positions and the related Fourier spectra are compared. Orbital tracks of particles as well as the development of the maximum and minimum surface elevation are shown. Good agreement of numerical and experimental results with wave heights up to 1.1 m is observed.
TL;DR: In this article, the authors consider the subset E of all points whose first and second components coincide with the first eigenvalues of the Laplace operator $-\Delta with zero boundary conditions on domains of $\RR^N$ with prescribed measure.
Abstract: We consider the subset E of $\RR^2$ of all points whose first and second components, respectively, coincide with the first and second eigenvalues of the Laplace operator $-\Delta$ with zero boundary conditions on domains of $\RR^N$ with prescribed measure. We show that the set E is closed in $\RR^2$.
TL;DR: In this paper, an effective model for the evaluation of the electric field dynamics inside electrical components, using nonlinear stress grading materials, is presented, implemented in a numerical procedure, which permits the solution of the Laplace equation and the diffusion equation by adopting the Galerkin method.
Abstract: An effective model is presented for the evaluation of the electric field dynamics inside electrical components, using nonlinear stress grading materials. The model, implemented in a numerical procedure, permits the solution of the Laplace equation and the diffusion equation by adopting the Galerkin method. Two-dimensional domains, even of very complex shapes and the finite thickness of the grading materials are properly taken into account, allowing an accurate evaluation of the electric field distribution and a sound understanding of the influence of the different types of nonlinearities on the stress grading efficiency. The proposed technique has been applied to study the field distributions inside a cable termination equipped with a stress control tube and in a suspension cap-and-pin glass insulator covered with an anti-corona layer. Numerical results for sinusoidal power frequency and standard impulse voltages elucidate the different role of resistive and capacitive contributions in determining the overall potential maps.
TL;DR: In this paper, a strong unique continuation for the laplace operator and its powers is discussed, and some remarks on strong unique continuoustime continuation for laplace operators are given.
Abstract: (1999). Some remarks on strong unique continuation for the laplace operator and its powers. Communications in Partial Differential Equations: Vol. 24, No. 5-6, pp. 1079-1094.
TL;DR: Six error indicators obtained from dual boundary integral equations are used for local estimation, which is an essential ingredient for all adaptive mesh schemes in BEM.
Abstract: In this paper, six error indicators obtained from dual boundary integral equations are used for local estimation, which is an essential ingredient for all adaptive mesh schemes in BEM. Computational experiments are carried out for the two-dimensional Laplace equation. The curves of all these six error estimators are in good agreement with the shape of the error curve. The results show that the adaptive mesh based on any one of these six error indicators converges faster than does equal mesh discretization.
TL;DR: In this article, a convergent Galerkin method for approximating the third-kind boundary condition is proposed and tested in numerical experiments, and the convergence of the method is shown to be a nonlinear inverse problem in the field of non-destructive evaluation.
Abstract: Let u be harmonic in the interior of a rectangle Ω and satisfy the third-kind boundary condition u n + γu = Φ where Φ ≥ 0, γ ≥ 0 with supports included in the bottom and in the top side of Ω, respectively Recovering γ from a knowledge of Φ and of the trace of u on the bottom of Ω is a nonlinear inverse problem of interest in the field of nondestructive evaluation A convergent Galerkin method for approximating γ is proposed and tested in numerical experiments
01 Jan 1999
TL;DR: In this article, a generalised function is defined and properties and properties of a generalized function are discussed, including the separation of the Variables, separation of variables in other coordinate systems, and the Discrete representation of the Delta Function.
Abstract: 1. Mathematical Preliminaries.- 1.1 Introduction.- 1.2 Characteristics and Classification.- 1.3 Orthogonal Functions.- 1.4 Sturm-Liouville Boundary Value Problems.- 1.5 Legendre Polynomials.- 1.6 Bessel Functions.- 1.7 Results from Complex Analysis.- 1.8 Generalised Functions and the Delta Function.- 1.8.1 Definition and Properties of a Generalised Function.- 1.8.2 Differentiation Across Discontinuities.- 1.8.3 The Fourier Transform of Generalised Functions.- 1.8.4 Convolution of Generalised Functions.- 1.8.5 The Discrete Representation of the Delta Function.- 2. Separation of the Variables.- 2.1 Introduction.- 2.2 The Wave Equation.- 2.3 The Heat Equation.- 2.4 Laplace's Equation.- 2.5 Homogeneous and Non-homogeneous Boundary Conditions.- 2.6 Separation of variables in other coordinate systems.- 3. First-order Equations and Hyperbolic Second-order Equations.- 3.1 Introduction.- 3.2 First-order equations.- 3.3 Introduction to d'Alembert's Method.- 3.4 d'Alembert's General Solution.- 3.5 Characteristics.- 3.6 Semi-infinite Strings.- 4. Integral Transforms.- 4.1 Introduction.- 4.2 Fourier Integrals.- 4.3 Application to the Heat Equation.- 4.4 Fourier Sine and Cosine Transforms.- 4.5 General Fourier Transforms.- 4.6 Laplace transform.- 4.7 Inverting Laplace Transforms.- 4.8 Standard Transforms.- 4.9 Use of Laplace Transforms to Solve Partial Differential Equations.- 5. Green's Functions.- 5.1 Introduction.- 5.2 Green's Functions for the Time-independent Wave Equation.- 5.3 Green's Function Solution to the Three-dimensional Inhomogeneous Wave Equation.- 5.4 Green's Function Solutions to the Inhomogeneous Helmholtz and Schrodinger Equations: An Introduction to Scattering Theory.- 5.5 Green's Function Solution to Maxwell's Equations and Time-dependent Problems.- 5.6 Green's Functions and Optics: Kirchhoff Diffraction Theory.- 5.7 Approximation Methods and the Born Series.- 5.8 Green's Function Solution to the Diffusion Equation.- 5.9 Green's Function Solution to the Laplace and Poisson Equations.- 5.10 Discussion.- A. Solutions of Exercises.
Book•
28 Sep 1999
TL;DR: In this paper, a classification of Boundary Integral Equation Methods (BIM) is presented, based on the Betti and Somigliana Formulae Uniqueness Theorems and the Elastic Potentials Properties of the Boundary Operators.
Abstract: Introduction THE LAPLACE EQUATION Notation and Prerequisites The Fundamental Boundary Value Problems Green's Formulae Uniqueness Theorems The Harmonic Potentials A Classification of Boundary Integral Equation Methods The Classical Indirect Method The Alternative Indirect Method The Modified Indirect Method The Refined Indirect Method The Direct Method The Substitute Direct Method PLANE STRAIN Notation and Prerequisites The Fundamental Boundary Value Problems The Betti and Somigliana Formulae Uniqueness Theorems The Elastic Potentials Properties of the Boundary Operators The Classical Indirect Method The Alternative Indirect Method The Modified Indirect Method The Refined Indirect Method The Direct Method The Substitute Direct Method BENDING OF ELASTIC PLATES Notation and Prerequisites The Fundamental Boundary Value Problems The Betti and Somigliana Formulae Uniqueness Theorems The Plate Potentials Properties of the Boundary Operators Boundary Integral Equation Methods WHICH METHOD? Notation and Prerequisites Connections between the Indirect Methods Connections between the Direct and Indirect Methods Overall View and Conclusions APPENDIX