A fast numerical method for harmonic equation based on natural boundary integral

Based on the coupling of the natural boundary integral method and the finite elements method, we mainly investigate the numerical solution of Neumann problem of harmonic equation in an exterior elliptic. Using our trigonometric wavelets and Galerkin method, there obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. On the other hand, we prove that the numerical solution possesses exponential convergence rate. Especially, examples state that our method still has good accuracy for small j when the solution u ₀(θ) is almost singular.     

The boundary integral equation method applied to shallow membrane shells of positive Gaussian curvature

The boundary integral equation method (BIEM) is developed for the analysis of shallow membrane shells with positive Gaussian curvatures. Shells with constant thickness and constant curvatures are considered. In the infinite domain, fundamental solutions are obtained which correspond to generalized concentrated tangential forces in the x and y coordinate directions. The Betti-Maxwell reciprocal theorem and Green's second identity are used to obtain the boundary integral equations of the solution presented.This approach, which is applied for the first time in membrane shell theory, seems to be a powerful alternative to domain type methods. Shells with various boundary conditions, loadings and arbitrary plan forms can be considered. It is also possible to add the effects of thermal fields and openings in the shells.The potential of the method is demonstrated by means of a worked example.     

On the numerical solution of stochastic quadratic integral equations via operational matrix method

In this paper, stochastic operational matrix of integration based on delta functions is applied to obtain the numerical solution of linear and nonlinear stochastic quadratic integral equations (SQIEs) that appear in modelling of many real problems. An important advantage of this method is that it dose not need any integration to compute the constant coefficients. Also, this method can be utilized to solve both linear and nonlinear problems. By using stochastic operational matrix of integration together collocation points, solving linear and nonlinear SQIEs converts to solve a nonlinear system of algebraic equations, which can be solved by using Newton's numerical method. Moreover, the error analysis is established by using some theorems. Also, it is proved that the rate of convergence of the suggested method is O(h²). Finally, this method is applied to solve some illustrative examples including linear and nonlinear SQIEs. Numerical experiments confirm the good accuracy and efficiency of the proposed method.     

The collocation method for the solution of boundary integral equations

This article is devoted to investigating the approximate solutions to a class of boundary integral equations over a closed, bounded and smooth surface found via the collocation method. This article provides sufficient conditions for the convergence of this method in the space of continuous functions.     

Adaptive meshless Galerkin boundary node methods for hypersingular integral equations

Adaptive refinement techniques are developed in this paper for the meshless Galerkin boundary node method for hypersingular boundary integral equations. Two types of error estimators are derived. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two consecutive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the numerical solution itself and its projection. These error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization scheme is presented to accomodate the non-local property of hypersingular integral operators for the needed computable local error indicators. The convergence of the adaptive meshless techniques is verified theoretically. To confirm the theoretical results and to show the efficiency of the adaptive techniques, numerical examples in 2D and 3D with high singularities are provided.     

Combined boundary integral equations for acoustic scattering‐resonance problems

In this paper, boundary integral formulations for a time‐harmonic acoustic scattering‐resonance problem are analyzed. The eigenvalues of eigenvalue problems resulting from boundary integral formulations for scattering‐resonance problems split in general into two parts. One part consists of scattering‐resonances, and the other one corresponds to eigenvalues of some Laplacian eigenvalue problem for the interior of the scatterer. The proposed combined boundary integral formulations enable a better separation of the unwanted spectrum from the scattering‐resonances, which allows in practical computations a reliable and simple identification of the scattering‐resonances in particular for non‐convex domains. The convergence of conforming Galerkin boundary element approximations for the combined boundary integral formulations of the resonance problem is shown in canonical trace spaces. Numerical experiments confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

A random integral quadrature method for numerical analysis of the second kind of Volterra integral equations

In this paper, a novel meshless technique termed the random integral quadrature (RIQ) method is developed for the numerical solution of the second kind of the Volterra integral equations. The RIQ method is based on the generalized integral quadrature (GIQ) technique, and associated with the Kriging interpolation function, such that it is regarded as an extension of the GIQ technique. In the GIQ method, the regular computational domain is required, in which the field nodes are scattered along straight lines. In the RIQ method however, the field nodes can be distributed either uniformly or randomly. This is achieved by discretizing the governing integral equation with the GIQ method over a set of virtual nodes that lies along straight lines, and then interpolating the function values at the virtual nodes over all the field nodes which are scattered either randomly or uniformly. In such a way, the governing integral equation is converted approximately into a system of linear algebraic equations, which can be easily solved.     

Weak solutions for time‐dependent boundary integral equations associated with the bending of elastic plates under combined boundary data

The existence, uniqueness, stability, and integral representation of distributional solutions are investigated for the equations of motion of a thin elastic plate with a combination of displacement and moment‐stress components prescribed on the boundary. Copyright © 2004 John Wiley & Sons, Ltd.     

A system of boundary integral equations for the transmission problem in acoustics

In this paper, we reduce the classical two-dimensional transmission problem in acoustic scattering to a system of coupled boundary integral equations (BIEs), and consider the weak formulation of the resulting equations. Uniqueness and existence results for the weak solution of corresponding variational equations are established. In contrast to the coupled system in Costabel and Stephan (1985) [4], we need to take into account exceptional frequencies to obtain the unique solvability. Boundary element methods (BEM) based on both the standard and a two-level fast multipole Galerkin schemes are employed to compute the solution of the variational equation. Numerical results are presented to verify the efficiency and accuracy of the numerical methods.     

Boundary integral equations for bending of thermoelastic plates with transmission boundary conditions

The solution of an initial‐boundary value problem for bending of a piecewise‐homogeneous thermoelastic plate with transverse shear deformation is represented as various combinations of single‐layer and double‐layer time‐dependent potentials. The unique solvability of the boundary integral equations generated by these representations is proved in spaces of distributions. Copyright © 2009 John Wiley & Sons, Ltd.     

A fast numerical method for a natural boundary integral equation for the Helmholtz equation

A Neumann boundary value problem of the Helmholtz equation in the exterior circular domain is reduced into an equivalent natural boundary integral equation. Using our trigonometric wavelets and the Galerkin method, the obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. Especially, our method is also efficient when the wave number k in the Helmholtz equation is very large.     

An extrapolation method for a class of boundary integral equations

Boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using Green's formulas. For solving the induced boundary integral equations, a Nyström scheme and its extrapolation method are derived for periodic Fredholm integral equations of the second kind with logarithmic singularity. Asymptotic expansions for the approximate solutions obtained by the Nyström scheme are developed to analyze the extrapolation method. Some computational aspects of the methods are considered, and two numerical examples are given to illustrate the acceleration of convergence.

     

Method of boundary integral equations as applied to the numerical solution of the three-dimensional Dirichlet problem for the laplace equation in a piecewise homogeneous medium

A Dirichlet problem is considered in a three-dimensional domain filled with a piecewise homogeneous medium. The uniqueness of its solution is proved. A system of Fredholm boundary integral equations of the second kind is constructed using the method of surface potentials, and a system of boundary integral equations of the first kind is derived directly from Green’s identity. A technique for the numerical solution of integral equations is proposed, and results of numerical experiments are presented.     

A boundary integral formulation of the Stefan problem

This paper considers a boundary integral formulation of the Stefan problem for two spatial dimensions. This formulation has the advantage that its numerical implementation does not require the discretization of the Stefan condition. Furthermore, the formulation is capable of solving problems with complex boundaries. Several illustrative examples are given.     

A system of boundary integral equations for polygonal plates with free edges

We consider the problem of a polygonal plate with free edges. It is a boundary value problem for the biharmonic operator on a polygon with Neumann boundary conditions. Its resolution is studied via boundary integral equations. A variational formulation of the boundary problem obtained by a double-layer potential is given. Finally, we implement the method and give numerical results. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

The fast multipole method for the symmetric boundary integral formulation

^** Email: of{at}mathematik.uni-stuttgart.de^*** Email: o.steinbach{at}tugraz.at^**** Email: wendland{at}mathematik.uni-stuttgart.de A symmetric Galerkin boundary-element method is used for thesolution of boundary-value problems with mixed boundary conditionsof Dirichlet and Neumann type. As a model problem we considerthe Laplace equation. When an iterative scheme is employed forsolving the resulting linear system, the discrete boundary integraloperators are realized by the fast multipole method. While thesingle-layer potential can be implemented straightforwardlyas in the original algorithm for particle simulation, the double-layerpotential and its adjoint operator are approximated by the applicationof normal derivatives to the multipole series for the kernelof the single-layer potential. The Galerkin discretization ofthe hypersingular integral operator is reduced to the single-layerpotential via integration by parts. We finally present a correspondingstability and error analysis for these approximations by thefast multipole method of the boundary integral operators. Itis shown that the use of the fast multipole method does notharm the optimal asymptotic convergence. The resulting linearsystem is solved by a GMRES scheme which is preconditioned bythe use of hierarchical strategies as already employed in thefast multipole method. Our numerical examples are in agreementwith the theoretical results.     

Meshless Galerkin algorithms for boundary integral equations with moving least square approximations

In this paper, we first give error estimates for the moving least square (MLS) approximation in the Hk norm in two dimensions when nodes and weight functions satisfy certain conditions. This two-dimensional error results can be applied to the surface of a three-dimensional domain. Then combining boundary integral equations (BIEs) and the MLS approximation, a meshless Galerkin algorithm, the Galerkin boundary node method (GBNM), is presented. The optimal asymptotic error estimates of the GBNM for three-dimensional BIEs are derived. Finally, taking the Dirichlet problem of Laplace equation as an example, we set up a framework for error estimates of the GBNM for boundary value problems in three dimensions.     

A consistency analysis for the numerical solution of boundary integral equations

A method is presented for assessing the nature of the error incurred in the boundary integral equation (BIE) solution of both harmonic and biharmonic boundary value problems (BVPs). It is shown to what order of accuracy the governing partial differential equation is actually represented by the approximating numerical scheme, and how raising the order of the boundary ‘shape functions’ affects this representation. The effect of varying both the magnitude and the aspect ratio of the solution domain is investigated; it is found that the present technique may suggest an optimum nondimensional scaling for the BIE solution of a particular harmonic or biharmonic BVP.     

伴有边界摄动的Volterra型积分微分方程组的奇摄动

讨论了伴有边界摄动的二阶非线性Volterra型积分微分方程组的奇摄动.在适当的条件下,利用对角化技巧证明了解的存在性,构造出解的渐近展开式并给出余项的一致有效估计.     

Mechanical quadrature methods and their splitting extrapolations for boundary integral equations of first kind on open arcs

This paper presents the mechanical quadrature methods (MQMs) for solving boundary integral equations (BIEs) of the first kind on open arcs. The spectral condition number of MQMs is only O(h⁻¹), where h is the maximal mesh width. The errors of MQMs have multivariate asymptotic expansions, accompanied with for all mesh widths h_i. Hence, once discrete equations with coarse meshes are solved in parallel, the accuracy order of numerical approximations can be greatly improved by splitting extrapolation algorithms (SEAs). Moreover, a posteriori asymptotic error estimates are derived, which can be used to formulate self-adaptive algorithms. Numerical examples are also provided to support our algorithms and analysis. Furthermore, compared with the existing algorithms, such as Galerkin and collocation methods, the accuracy order of the MQMs is higher, and the discrete matrix entries are explicit, to prove that the MQMs in this paper are more promising and beneficial to practical applications.