scholarly journals Galerkin's method for boundary integral equations on polygonal domains

1984 ◽  
Vol 26 (1) ◽  
pp. 1-13 ◽  
Author(s):  
G. A. Chandler
Keyword(s):  
Integral Equation ◽  
Smooth Boundary ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Galerkin’S Method ◽  
Layer Potential ◽  
Polygonal Domains ◽  
Galerkin's Method ◽  
Piecewise Polynomials ◽  
Convergence And Superconvergence

AbstractA harmonic function in the interior of a polygon is the double layer potential of a distribution satisfying a second kind integral equation. This may be solved numerically by Galerkin's method using piecewise polynomials as basis functions. But the corners produce singularities in the distribution and the kernel of the integral equation; and these reduce the order of convergence. This is offset by grading the mesh, and the orders of convergence and superconvergence are restored to those for a smooth boundary.

A New Boundary Integral Equation Formulation for Linear Elastic Solids

1992 ◽  
Vol 59 (2) ◽  
pp. 344-348 ◽  
Author(s):  
Kuang-Chong Wu ◽  
Yu-Tsung Chiu ◽  
Zhong-Her Hwu
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Boundary Integral Equations ◽  
Analytic Solutions ◽  
Boundary Integral ◽  
Tangential Displacement ◽  
Infinite Body ◽  
Equation Formulation ◽  
Integral Equation Formulation ◽  
Elasticity Problems

A new boundary integral equation formulation is presented for two-dimensional linear elasticity problems for isotropic as well as anisotropic solids. The formulation is based on distributions of line forces and dislocations over a simply connected or multiply connected closed contour in an infinite body. Two types of boundary integral equations are derived. Both types of equations contain boundary tangential displacement gradients and tractions as unknowns. A general expression for the tangential stresses along the boundary in terms of the boundary tangential displacement gradients and tractions is given. The formulation is applied to obtain analytic solutions for half-plane problems. The formulation is also applied numerically to a test problem to demonstrate the accuracy of the formulation.

Analysis of Clamped Plates on Elastic Foundation by the Boundary Integral Equation Method

1984 ◽  
Vol 51 (3) ◽  
pp. 574-580 ◽  
Author(s):  
J. T. Katsikadelis ◽  
A. E. Armena`kas
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Elastic Foundation ◽  
Boundary Integral Equation ◽  
Numerical Evaluation ◽  
Boundary Integral Equations ◽  
Numerical Technique ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Plates On Elastic Foundation

In this investigation the boundary integral equation (BIE) method with numerical evaluation of the boundary integral equations is developed for analyzing clamped plates of any shape resting on an elastic foundation. A numerical technique for the solution to the boundary integral equations is presented and numerical results are obtained and compared with those existing from analytical solutions. The effectiveness of the BIE method is demonstrated.

Integral equation methods in potential theory. II

1963 ◽  
Vol 275 (1360) ◽  
pp. 33-46 ◽  
Keyword(s):  
Integral Equation ◽  
Potential Theory ◽  
Digital Computer ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Integral Equation Methods ◽  
Representative Selection ◽  
Selection Of

The boundary integral equations of potential theory can be solved to a tolerable accuracy without undue labour by digital computer techniques, and the computed datagenerate numerical values of the potential field wherever required. Tests have been made with a representative selection of two-dimensional problem s, some of which would not be amenable to any other treatment.

Modeling of Wave Propagation in the Unsaturated Soils Using Boundary Element Method

2017 ◽  
Vol 743 ◽  
pp. 158-161
Author(s):  
Andrey Petrov ◽  
Sergey Aizikovich ◽  
Leonid A. Igumnov
Keyword(s):  
Integral Equation ◽  
Wave Propagation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equation ◽  
Boundary Integral Equations ◽  
Water Pressure ◽  
Boundary Integral ◽  
Element Method

Problems of wave propagation in poroelastic bodies and media are considered. The behavior of the poroelastic medium is described by Biot theory for partially saturated material. Mathematical model is written in term of five basic functions – elastic skeleton displacements, pore water pressure and pore air pressure. Boundary element method (BEM) is used with step method of numerical inversion of Laplace transform to obtain the solution. Research is based on direct boundary integral equation of three-dimensional isotropic linear theory of poroelasticity. Green’s matrices and, based on it, boundary integral equations are written for basic differential equations in partial derivatives. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. To approximate the boundary consider its decomposition to a set of quadrangular and triangular 8-node biquadratic elements, where triangular elements are treated as singular quadrangular. Every element is mapped to a reference one. Interpolation nodes for boundary unknowns are a subset of geometrical boundary-element grid nodes. Local approximation follows the Goldshteyn’s generalized displacement-stress matched model: generalized boundary displacements are approximated by bilinear elements whereas generalized tractions are approximated by constant. Integrals in discretized boundary integral equations are calculated using Gaussian quadrature in combination with singularity decreasing and eliminating algorithms.

A New Boundary Integral Equation Method for Cracked Piezoelectric Bodies

2006 ◽  
Vol 306-308 ◽  
pp. 465-470 ◽  
Author(s):  
Kuang-Chong Wu
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Present Method ◽  
Boundary Integral Equations ◽  
Integral Formula ◽  
Electric Displacement ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Equation Method ◽  
Intensity Factors

A novel integral equation method is developed in this paper for the analysis of two-dimensional general piezoelectric cracked bodies. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh’s formalism for anisotropic elasticity in conjunction with Cauchy’s integral formula. The proposed boundary integral equations contain generalized boundary displacement (displacements and electric potential) gradients and generalized tractions (tractions and electric displacement) on the non-crack boundary, and the generalized dislocations on the crack lines. The boundary integral equations can be solved using Gaussian-type integration formulas without dividing the boundary into discrete elements. The crack-tip singularity is explicitly incorporated and the generalized intensity factors can be computed directly. Numerical examples of generalized stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.

scholarly journals NUMERICAL ANALYSIS OF THE DYNAMICS OF THREE-DIMENSIONAL ANISOTROPIC BODIES BASED ON NON-CLASSICAL BOUNDARY INTEGRAL EQUATIONS

2021 ◽  
Vol 83 (1) ◽  
pp. 76-86
Author(s):  
A.A. Belov ◽  
A.N. Petrov
Keyword(s):  
Integral Equation ◽  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Classical Approach ◽  
Classical Boundary ◽  
Nonclassical Formulation

The application of non-classical approach of the boundary integral equation method in combination with the integral Laplace transform in time to anisotropic elastic wave modeling is considered. In contrast to the classical approach of the boundary integral equation method which is successfully implemented for solving three-dimensional isotropic problems of the dynamic theory of elasticity, viscoelasticity and poroelasticity, the alternative nonclassical formulation of the boundary integral equations method is presented that employs regular Fredholm integral equations of the first kind (integral equations on a plane wave). The construction of such boundary integral equations is based on the structure of the dynamic fundamental solution. The approach employs the explicit boundary integral equations. The inverse Laplace transform is constructed numerically by the Durbin method. A numerical solution of the dynamic problem of anisotropic elasticity theory based on the boundary integral equations method in a nonclassical formulation is presented. The boundary element scheme of the boundary integral equations method is built on the basis of a regular integral equation of the first kind. The problem is solved in anisotropic formulation for the load acting along the normal in the form of the Heaviside function on the cube face weakened by a cubic cavity. The obtained boundary element solutions are compared with finite element solutions. Numerical results prove the efficiency of using boundary integral equations on a single plane wave in solving three-dimensional anisotropic dynamic problems of elasticity theory. The convergence of boundary element solutions is studied on three schemes of surface discretization. The achieved calculation accuracy is not inferior to the accuracy of boundary element schemes for classical boundary integral equations. Boundary element analysis of solutions for a cube with and without a cavity is carried out.

Null-Field Integral Equation Approach for Plate Problems With Circular Boundaries

2005 ◽  
Vol 73 (4) ◽  
pp. 679-693 ◽  
Author(s):  
Jeng-Tzong Chen ◽  
Chia-Chun Hsiao ◽  
Shyue-Yuh Leu
Keyword(s):  
Integral Equation ◽  
Fourier Series ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Kernel Functions ◽  
Unknown Boundary ◽  
Null Field ◽  
Circular Holes ◽  
Integral Equation Approach ◽  
Field Integral

In this paper, a semi-analytical approach for circular plate problems with multiple circular holes is presented. Null-field integral equation is employed to solve the plate problems while the kernel functions in the null-field integral equation are expanded to degenerate kernels based on the separation of field and source points in the fundamental solution. The unknown boundary densities of the circular plates are expressed in terms of Fourier series. It is noted that all the improper integrals are transformed to series sum and are easily calculated when the degenerate kernels and Fourier series are used. By matching the boundary conditions at the collocation points, a linear algebraic system is obtained. After determining the unknown Fourier coefficients, the displacement, slope, normal moment, and effective shear force of the plate can be obtained by using the boundary integral equations. Finally, two numerical examples are proposed to demonstrate the validity of the present method and the results are compared with the available exact solution, the finite element solution using ABAQUS software and the data of Bird and Steele.

scholarly journals Influence of viscosity on the scattering of an air pressure wave by a rigid body: a regular boundary integral formulation

2008 ◽  
Vol 464 (2097) ◽  
pp. 2303-2320 ◽  
Author(s):  
Dorel Homentcovschi
Keyword(s):  
Integral Equation ◽  
Rigid Body ◽  
Stress Tensor ◽  
Double Layer ◽  
Pressure Wave ◽  
Direct Proof ◽  
Single Layer ◽  
Boundary Integral ◽  
Fredholm Alternative ◽  
Layer Potential

This paper gives a regular vector boundary integral equation for solving the problem of viscous scattering of a pressure wave by a rigid body. Firstly, single-layer viscous potentials and a generalized stress tensor are introduced. Correspondingly, generalized viscous double-layer potentials are defined. By representing the scattered field as a combination of a single-layer viscous potential and a generalized viscous double-layer potential, the problem is reduced to the solution of a vectorial Fredholm integral equation of the second kind. Generally, the vector integral equation is singular. However, there is a particular stress tensor, called pseudostress, which yields a regular integral equation. In this case, the Fredholm alternative applies and permits a direct proof of the existence and uniqueness of the solution. The results presented here provide the foundation for a numerical solution procedure.

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