Showing papers on "Laplace's equation published in 2010"
378 citations
TL;DR: In this article, the authors proposed an analytical method based on the resolution of Laplace's and Poisson's equations (by the separation of variables technique) for each subdomain, i.e., magnets, air gap, and slots.
Abstract: We propose an analytical computation of the magnetic field distribution in a magnetic gear. The analytical method is based on the resolution of Laplace's and Poisson's equations (by the separation of variables technique) for each subdomain, i.e., magnets, air gap, and slots. The global solution is obtained using boundary and continuity conditions. Our analytical model can be used as a tool for design optimization of a magnetic gear. Here, we compare magnetic field distributions and electromagnetic torque computed by the analytical method with those obtained from finite-element analyses.
144 citations
TL;DR: A simple and robust scheme for the determination of a sparse solution to an underdetermined nonlinear optimization problem which replaces the continuation scheme of the previously published works on generalized Gaussian quadratures.
Abstract: We present a new nonlinear optimization procedure for the computation of generalized Gaussian quadratures for a broad class of square integrable functions on intervals. While some of the components of this algorithm have been previously published, we present a simple and robust scheme for the determination of a sparse solution to an underdetermined nonlinear optimization problem which replaces the continuation scheme of the previously published works. The new algorithm successfully computes generalized Gaussian quadratures in a number of instances in which the previous algorithms fail. Four applications of our scheme to computational physics are presented: the construction of discrete plane wave expansions for the Helmholtz Green's function, the design of linear array antennae, the computation of a quadrature for the discretization of Laplace boundary integral equations on certain domains with corners, and the construction of quadratures for the discretization of Laplace and Helmholtz boundary integral equations on smooth surfaces.
125 citations
TL;DR: In this article, the Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise.
Abstract: The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence approach is employed in the proof to establish the Laplace principle, which is equivalent to the large deviation principle in our framework.
120 citations
TL;DR: In this paper, a boundary element analysis approach is presented for solving transient heat conduction problems based on the radial integration method, which makes the representation very simple and having no temperature gradients involved.
Abstract: In this paper, a new boundary element analysis approach is presented for solving transient heat conduction problems based on the radial integration method. The normalized temperature is introduced to formulate integral equations, which makes the representation very simple and having no temperature gradients involved. The Green's function for the Laplace equation is adopted in deriving basic integral equations for time-dependent problems with varying heat conductivities and, as a result, domain integrals are involved in the derived integral equations. The radial integration method is employed to convert the domain integrals into equivalent boundary integrals. Based on the central finite difference technique, an implicit time marching solution scheme is developed for solving the time-dependent system of equations. Numerical examples are given to demonstrate the correctness of the presented approach.
86 citations
01 Jan 2010
TL;DR: In this article, the wave and poisson's equations were solved by double Sumudu transform and the same result can be obtained by double Laplace transform, in particular the wave equation can be solved with double Laplacian transform.
Abstract: In this study, we apply double integral transforms to solve partial differential equation namely double Laplace and Sumudu transforms, in particular the wave and poisson's equations were solved by double Sumudu transform and the same result can be obtained by double Laplace transform.
62 citations
TL;DR: A perturbed compound Poisson risk model with two-sided jumps, where the downward jumps represent the claims following an arbitrary distribution, while the upward jumps are also allowed to represent the random gains.
54 citations
TL;DR: An improved form of the hypersingular boundary integral equation (BIE) for acoustic problems is developed in this paper, which contains only weakly singular integrals and is directly valid for acoustic problem with arbitrary boundary conditions.
Abstract: An improved form of the hypersingular boundary integral equation (BIE) for acoustic problems is developed in this paper. One popular method for overcoming non-unique problems that occur at characteristic frequencies is the well-known Burton and Miller (1971) method [7], which consists of a linear combination of the Helmholtz equation and its normal derivative equation. The crucial part in implementing this formulation is dealing with the hypersingular integrals. This paper proposes an improved reformulation of the Burton–Miller method and is used to regularize the hypersingular integrals using a new singularity subtraction technique and properties from the associated Laplace equations. It contains only weakly singular integrals and is directly valid for acoustic problems with arbitrary boundary conditions. This work is expected to lead to considerable progress in subsequent developments of the fast multipole boundary element method (FMBEM) for acoustic problems. Numerical examples of both radiation and scattering problems clearly demonstrate that the improved BIE can provide efficient, accurate, and reliable results for 3-D acoustics.
53 citations
TL;DR: In this article, the Dirichlet-to-Neumann transform was used to solve the boundary value problem (BVP) for linear and integrable nonlinear partial differential equations (PDEs).
Abstract: Integral representations for the solutions of the Laplace and modified Helmholtz equations can be obtained using Green's theorem. However, these representations involve both the solution and its normal derivative on the boundary, and for a well-posed boundary-value problem (BVP) one of these functions is unknown. Determining the Neumann data from the Dirichlet data is known as constructing the Dirichlet-to-Neumann map. A new transform method was introduced in Fokas (1997, Proc. R. Soc. Lond. A, 53, 1411–1443) for solving BVPs for linear and integrable nonlinear partial differential equations (PDEs). For linear PDEs this method can be considered as the analogue of the Green's function approach in the Fourier plane. In this method the Dirichlet-to-Neumann map is characterized by a certain equation, the so-called global relation, which is formulated in the complex k-plane, where k denotes the complex extension of the spectral (Fourier) variable. Here we solve the global relation numerically for the Laplace and modified Helmholtz equations in a convex polygon. This is achieved by evaluating the global relation at a properly chosen set of points in the spectral (Fourier) plane, which is why this method has been called a ‘spectral collocation method’. Numerical experiments suggest that the method inherits the order of convergence of the basis used to expand the unknown functions, namely, exponential for a polynomial basis such as Chebyshev, and algebraic for a Fourier basis. However, the condition number of the associated linear system is much higher for a polynomial basis than for a Fourier one.
52 citations
TL;DR: A unified error analysis of several mixed methods for linear elasticity which impose stress symmetry weakly, including methods where the rotations are approximated by discontinuous polynomials is given.
Abstract: We give a unified error analysis of several mixed methods for linear elasticity which impose stress symmetry weakly. We consider methods where the rotations are approximated by discontinuous polynomials. The methods we consider are such that the approximate stress spaces contain standard mixed finite element spaces for the Laplace equation and also contain divergence free spaces that use bubble functions.
50 citations
TL;DR: Achdou et al. as discussed by the authors considered a newtonian flow in domains with periodic rough boundaries and derived high order boundary layer approximations and rigorously justified their rates of convergence with respect to epsilon.
Abstract: In this work we present new wall-laws boundary conditions including microscopic oscillations. We consider a newtonian flow in domains with periodic rough boundaries that we simplify considering a Laplace operator with periodic inflow and outflow boundary conditions. Following the previous approaches, see [A. Mikelic, W. Jager, J. Diff. Eqs, 170, 96-122, (2001) ] and [Y. Achdou, O. Pironneau, F. Valentin, J. Comput. Phys, 147, 1, 187-218, (1998)], we construct high order boundary layer approximations and rigorously justify their rates of convergence with respect to epsilon (the roughness' thickness). We establish mathematically a poor convergence rate for averaged second-order wall-laws as it was illustrated numerically for instance in [Y. Achdou, O. Pironneau, F. Valentin, J. Comput. Phys, 147, 1, 187-218, (1998)]. In comparison, we establish exponential error estimates in the case of explicit multi-scale ansatz. This motivates our study to derive implicit first order multi-scale wall-laws and to show that its rate of convergence is at least of order epsilon to the three halves. We provide a numerical assessment of the claims as well as a counter-example that evidences the impossibility of an averaged second order wall-law. Our paper may be seen as the first stone to derive efficient high order wall-laws boundary conditions.
01 Jan 2010
TL;DR: In this paper, a combinatory method of the Laplace transform and the homotopy perturbation method is proposed to solve non-homogeneous partial differential equations with variable coefficients.
Abstract: In this paper, a combinatory method of the Laplace transform and the homotopy perturbation method is proposed to solve non-homogeneous partial differential equations with variable coefficients. The obtained approximate solutions are compared with exact solutions and those obtained by other analytical methods, showing reliability of the present method. Keyword: Laplace transform, Homotopy Perturbation Method, semi-analytical, non-homogeneous partial differential equation. Biographical Notes: Mahdi Fathizadeh, a chemical engineering doctoral student at Amirkabir University (polytechnic of Tehran, Iran), has published 2 articles in ISI-listed journals, and more than 10 conference papers. His current research interest mainly covers in process modeling, polymer and chemical engineering, membrane technology, and transport phenomena.
TL;DR: In this paper, a semi-analytical solution methodology for the linear hydrodynamic diffraction induced by arrays of elliptical cylinders subjected to incident waves is presented, where the solution of the Laplace equation in elliptic coordinates for both the incident and the diffracted waves is formulated analytically in terms of the even and odd periodic and radial Mathieu functions.
TL;DR: In this article, Γ -convergence results for power-law functionals with variable exponents were obtained for first-failure dielectric breakdown, and connections with the generalization of the ∞ -Laplace equation to the variable exponent setting were explored.
Abstract: Γ -convergence results for power-law functionals with variable exponents are obtained. The main motivation comes from the study of (first-failure) dielectric breakdown. Some connections with the generalization of the ∞ -Laplace equation to the variable exponent setting are also explored.
Journal Article•
TL;DR: In this article, the authors analyzed the set of continuous viscosity solutions of the infinity Laplace equation Nw(x) = f(x), with generally sign-changing right-hand side in a bounded domain.
Abstract: We analyze the set of continuous viscosity solutions of the infinity Laplace equation Nw(x) = f(x), with generally sign-changing right-hand side in a bounded domain. The existence of a least and a greatest continuous viscosity solutions, up to the boundary, is proved through a Perron's construc- tion by means of a strict comparison principle. These extremal solutions are proved to be absolutely extremal solutions.
TL;DR: In this paper, a microdosimetric study on erythrocytes in two parts is described: an assessment of the membrane dielectric model from permittivity measurements of the solution and its uncertainty, and a quasi-static electromagnetic (EM) analysis solving the Laplace equation, both analytically and numerically.
Abstract: This paper describes a microdosimetric study on erythrocytes in two parts: an assessment of the membrane dielectric model from permittivity measurements of erythrocyte solutions and its uncertainty, and a quasi-static electromagnetic (EM) analysis solving the Laplace equation, both analytically and numerically. To evaluate the role of the estimated uncertainty, a stochastic EM solution has been conducted; our results highlight the fundamental role of the dielectric modeling on the reliability of electric field values in the cell membrane. Numerical data, from 3-D cell models, confirm the dependence of the electric field distribution on the extra-cellular field polarization.
TL;DR: In this article, a depth integrated numerical model for the simulation of the generation and propagation of tsunamis generated by submerged landslides is presented, which is able to reproduce at low computational costs the full frequency dispersion of the waves and uses an ad hoc treatment for the incorporation of the effects of the moving seafloor.
TL;DR: In this paper, a set of 3D general solutions for thermoporoelastic media for the steady-state problem is presented, which can serve as a benchmark for various kinds of numerical codes and approximate solutions.
Abstract: This paper presents a set of 3D general solutions for thermoporoelastic media for the steady-state problem. By introducing two displacement functions, the equations governing the elastic, pressure and temperature fields are simplified. The operator theory and superposition principle are then employed to express all the physical quantities in terms of two functions, one of which satisfies a quasi–Laplace equation and the other satisfies a differential equation of the eighth order. The generalized Almansi's theorem is used to derive the displacements, pressure and temperature in terms of five quasi-harmonic functions for various cases of material characteristic roots. To show its practical significance, an infinite medium containing a penny-shaped crack subjected to mechanical, pressure and temperature loads on the crack surface is given as an example. A potential theory method is employed to solve the problem. One integro-differential equation and two integral equations are derived, which bear the same structures to those reported in literature. For a penny-shaped crack subjected to uniformly distributed loads, exact and complete solutions in terms of elementary functions are obtained, which can serve as a benchmark for various kinds of numerical codes and approximate solutions.
TL;DR: The well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proved and the numerical results show that the proposed numerical methods work effectively.
TL;DR: In this article, it was shown that weak solutions of (1.1) with fixed initial and boundary values converge in any reasonable sense to the solution of the limit problem as p varies.
Abstract: in a cylindrical domain. The main question is that do the weak solutions of (1.1) with fixed initial and boundary values converge in any reasonable sense to the solution of the limit problem as p varies. Apart from mathematical interest, the stability questions is motivated by error analysis in applications: It is desirable that solutions remain stable under small perturbations of the measured parameter p. Equation (1.1) is known as the p-parabolic equation or parabolic pLaplace equation. Sometimes it is also called the non-Newtonian filtration equation which refers to the fact that the equation models the flow of non-Newtonian fluids. For the regularity theory we refer to DiBenedetto’s monograph [4]. See also Chapter 2 of [22]. The equation is singular if 1 < p < 2 and degenerate if p ≥ 2. We shall focus on the degenerate case. The stability turns out to be a rather delicate problem. The main obstruction is that the underlying parabolic Sobolev space changes as p varies and hence the associated energy is not necessarily finite. We give an example of this phenomenon when the lateral boundary of the cylinder is a Cantor type set. In this case, it may also happen that the solutions converge to a solution of a wrong limit problem. These phenomena are already present in the stationary case, see KilpelainenKoskela [8] and Lindqvist [14], but the time dependence offers new challenges. Our main result shows that solutions with varying exponent converge to the solution of the limit problem in the parabolic Sobolev
TL;DR: The present work focuses on time dependent elastic problems, which are indeed not elliptic and the application of the presented fast boundary element formulation on such problems is enabled by introducing the well known Convolution Quadrature Method (CQM) as time stepping scheme.
Abstract: Wave propagation phenomena occur often in semi-infinite regions. It is well known that such problems can be handled well with the boundary element method (BEM). However, it is also known that the BEM, with its dense matrices, becomes prohibitive with respect to storage and computing time. Focusing on wave propagation problems, where a formulation in time domain is preferable, the mentioned limit of the method becomes evident. Several approaches, amongst them the adaptive cross approximation (ACA), have been developed in order to overcome these drawbacks mainly for elliptic problems. The present work focuses on time dependent elastic problems, which are indeed not elliptic. The application of the presented fast boundary element formulation on such problems is enabled by introducing the well known Convolution Quadrature Method (CQM) as time stepping scheme. Thus, the solution of the time dependent problem ends up in the solution of a system of decoupled Laplace domain problems. This detour is worth since the resulting problems are again elliptic and, therefore, the ACA can be used in its standard fashion. The main advantage of this approach of accelerating a time dependent BEM is that it can be easily applied to other fundamental solutions as, e.g., visco- or poroelasticity.
TL;DR: In this paper, it was shown that the radial Schrodinger equation can be derived from the Laplace operator in spherical coordinates and an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.
Abstract: There is much discussion in the mathematical physics literature as well as in quantum mechanics textbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of the radial Schrodinger equation leads to the appearance of a delta function term when the Laplace operator is written in spherical coordinates. As a result, regardless of the behavior of the potential, an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.
01 Jan 2010
TL;DR: In this paper, a unifying procedure to numerically compute enrichment functions for elastic fracture problems with the extended finite element method is presented, which provides flexibility over existing approaches in which each case is treated separately.
Abstract: SUMMARY A unifying procedure to numerically compute enrichment functions for elastic fracture problems with the extended finite element method is presented. Within each element that is intersected by a crack, the enrichment function for the crack is obtained via the solution of the Laplace equation with Dirichlet and vanishing Neumann boundary conditions. A single algorithm emanates for the enrichment field for multiple cracks as well as intersecting and branched cracks, without recourse to special cases, which provides flexibility over existing approaches in which each case is treated separately. Numerical integration is rendered to be simple—there is no need for partitioning of the finite elements into conforming subdivisions for the integration of discontinuous or weakly singular kernels. Stress intensity factor computations for dierent crack configurations are presented to demonstrate the accuracy and versatility of the proposed technique. Copyright 2010 John Wiley & Sons, Ltd.
TL;DR: This paper proposes the new approaches for deriving the asymptotes of Cond, and applies them for the Dirichlet problem of Laplace’s equation, to provide the sharp bound of Cond for disk domains.
Abstract: Since the stability of the method of fundamental solutions (MFS) is a severe issue, the estimation on the bounds of condition number Cond is important to real application. In this paper, we propose the new approaches for deriving the asymptotes of Cond, and apply them for the Dirichlet problem of Laplace’s equation, to provide the sharp bound of Cond for disk domains. Then the new bound of Cond is derived for bounded simply connected domains with mixed types of boundary conditions. Numerical results are reported for Motz’s problem by adding singular functions. The values of Cond grow exponentially with respect to the number of fundamental solutions used. Note that there seems to exist no stability analysis for the MFS on non-disk (or non-elliptic) domains. Moreover, the expansion coefficients obtained by the MFS are oscillatingly large, to cause the other kind of instability: subtraction cancelation errors in the final harmonic solutions.
TL;DR: A new approach to boundary extraction by a level set model that is embedded in several scalar functions that is flexible in handling complex topological changes and concise in extracting object boundaries despite of deep depression is presented.
TL;DR: In this article, lower bounds for the eigenvalues of the Stokes operator and the Laplace operator were shown to be equivalent to the lower bounds of the Berezin-Li-Yau lower bound.
Abstract: We prove Berezin-Li-Yau-type lower bounds with additional
term for the eigenvalues of the Stokes operator and
improve the previously known estimates for the Laplace
operator. Generalizations to higher-order operators are
given.
TL;DR: In this paper, renormalized solutions for the problems of the kind β(u)+(−)s/2u f in Rn,======¯¯¯¯f ∈ L1(Rn) were defined.
TL;DR: In this article, it was shown that the spectrum of the Laplace equation with the Steklov spectral boundary condition can have a nontrivial essential component even in case of a bounded basin with a horizontal water surface.
Abstract: We show that the spectrum of the Laplace equation with the Steklov spectral boundary condition, in the connection of the linearized theory of water-waves, can have a nontrivial essential component even in case of a bounded basin with a horizontal water surface. The appearance of the essential spectrum is caused by the boundary irregularities of the type of a rotational cusp or a cuspidal edge. In a previous paper the authors have proven a similar result for the Steklov spectral problem in a bounded domain with a sharp peak.
TL;DR: In this article, the transient response of a layered magnetoelectroelastic medium of finite size with an interface crack is analyzed using a generalized singularity integral equation in the Laplace transform domain.
TL;DR: Numerical tests with and without noise are conducted based on the methodology proposed in Younga, Fana, Hua, and Atluri (2009), and the results show larger stability of the local versions of the method in comparison with the global ones.
Abstract: This paper focuses on the comparative study of global and local mesh- less methods based on collocation with radial basis functions for solving two di- mensional initial boundary value diffusion-reaction problem with Dirichlet and Neumann boundary conditions. A similar study was performed for the boundary value problem with Laplace equation by Lee, Liu, and Fan (2003). In both global and local methods discussed, the time discretization is performed in explicit and implicit way and the multiquadric radial basis functions (RBFs) are used to inter- polate diffusion-reaction variable and its spatial derivatives. Five and nine nodded sub-domains are used in the local support of the local method. Uniform and non- uniform space discretization is used. Accuracy of global and local approaches is assessed as a function of the time and space discretizations, and value of the shape parameter. One can observe the convergence with denser nodes and with smaller time-steps in both methods. The global method is prone to instability due to ill- conditioning of the collocation matrix with the increase of the number of the nodes in cases N t 3000. On the other hand, the global method is more stable with re- spect to the time-step length. Numerical tests with and without noise are conducted based on the methodology proposed in Younga, Fana, Hua, and Atluri (2009). The results show larger stability of the local versions of the method in comparison with the global ones. The accuracy of the local method is comparable with the accuracy of the global method. The local method is more efficient because we solve only a small system of equations for each node in explicit case and a sparse system of equations in implicit case. Hence the local method represents a preferable choice to its global counter part.