A Boundary Element Method for Plane Anisotropic Elastic Media

1990 ◽  
Vol 57 (3) ◽  
pp. 600-606 ◽  
Author(s):  
Kyu J. Lee ◽  
A. K. Mal
Keyword(s):  
Integral Equations ◽  
Complex Plane ◽  
Singular Integral Equations ◽  
Boundary Curve ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Numerical Formulation ◽  
Complex Linear ◽  
Anisotropic Elastic ◽  
Improved Accuracy

The general problem of plane anisotropic elastostatics is formulated in terms of a system of singular integral equations with Cauchy kernels by means of the classical stress function approach. The integral equations are represented over the image of the boundary in the complex plane and a numerical scheme is developed for their solution. The boundary curve is discretized and suitable polynomial approximations of the unknown functions in terms of the complex variable are introduced. This reduces the equations to a set of complex linear algebraic equations which can be inverted to yield the stresses in a straightforward manner. The major difference between the present technique and the previous ones is in the numerical formulation. The integral equations are discretized in the complex plane and not in terms of real variables which depend on arc length, resulting in improved accuracy in presence of strong boundary curvature.

scholarly journals A convergence theorem for singular integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 539-552 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Convergence Theory ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

AbstractThe principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

scholarly journals Finite-part singular integral approximations in Hilbert spaces

2004 ◽  
Vol 2004 (52) ◽  
pp. 2787-2793
Author(s):  
E. G. Ladopoulos ◽  
G. Tsamasphyros ◽  
V. A. Zisis
Keyword(s):  
Unit Circle ◽  
Integral Equations ◽  
Singular Integral ◽  
Hilbert Spaces ◽  
Existence Theorems ◽  
Singular Integral Equations ◽  
Finite Part ◽  
Algebraic Equations ◽  
Integration Interval ◽  
Linear Algebraic Equations

Some new approximation methods are proposed for the numerical evaluation of the finite-part singular integral equations defined on Hilbert spaces when their singularity consists of a homeomorphism of the integration interval, which is a unit circle, on itself. Therefore, some existence theorems are proved for the solutions of the finite-part singular integral equations, approximated by several systems of linear algebraic equations. The method is further extended for the proof of the existence of solutions for systems of finite-part singular integral equations defined on Hilbert spaces, when their singularity consists of a system of diffeomorphisms of the integration interval, which is a unit circle, on itself.

scholarly journals Singular Integral Equations of Convolution Type with Cosecant Kernels and Periodic Coefficients

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Pingrun Li
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Complex Analysis ◽  
Boundary Value ◽  
Singular Integral Equations ◽  
Convolution Type ◽  
Algebraic Equations ◽  
Periodic Coefficients ◽  
Linear Algebraic Equations ◽  
Classical Boundary

We study singular integral equations of convolution type with cosecant kernels and periodic coefficients in class L2[-π,π]. Such equations are transformed into a discrete jump problem or a discrete system of linear algebraic equations by using discrete Fourier transform. The conditions of Noethericity and the explicit solutions are obtained by means of the theory of classical boundary value problem and of the Fourier analysis theory. This paper will be of great significance for the study of improving and developing complex analysis, integral equations, and boundary value problems.

scholarly journals Solvability of singular integral equations with rotations and degenerate kernels in the vanishing coefficient case

2014 ◽  
Vol 13 (01) ◽  
pp. 1-21 ◽  
Author(s):  
L. P. Castro ◽  
E. M. Rojas ◽  
S. Saitoh ◽  
N. M. Tuan ◽  
P. D. Tuan
Keyword(s):  
Boundary Value Problems ◽  
Unit Circle ◽  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Degenerate Kernel ◽  
Algebraic Equations ◽  
Explicit Formulas ◽  
Riemann Boundary ◽  
Linear Algebraic Equations

By means of Riemann boundary value problems and of certain convenient systems of linear algebraic equations, this paper deals with the solvability of a class of singular integral equations with rotations and degenerate kernel within the case of a coefficient vanishing on the unit circle. All the possibilities about the index of the coefficients in the corresponding equations are considered and described in detail, and explicit formulas for their solutions are obtained. An example of application of the method is shown at the end of the last section.

scholarly journals Solution of Cauchy-Type Singular Integral Equations of the First Kind with Zeros of Jacobi Polynomials as Interpolation Nodes

2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
G. E. Okecha
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Jacobi Polynomials ◽  
Singular Integral Equations ◽  
Quadrature Rule ◽  
Convergence Result ◽  
Cauchy Type ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Lagrangian Interpolation

Of concern in this paper is the numerical solution of Cauchy-type singular integral equations of the first kind at a discrete set of points. A quadrature rule based on Lagrangian interpolation, with the zeros of Jacobi polynomials as nodes, is developed to solve these equations. The problem is reduced to a system of linear algebraic equations. A theoretical convergence result for the approximation is provided. A few numerical results are given to illustrate and validate the power of the method developed. Our method is more accurate than some earlier methods developed to tackle this problem.

scholarly journals On Solvability Conditions for a Certain Conjugation Problem

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 234
Author(s):  
Vladimir Vasilyev ◽  
Nikolai Eberlein
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Integral Equations ◽  
Mellin Transform ◽  
Algebraic Equations ◽  
Conjugation Problem ◽  
Solvability Conditions ◽  
Linear Algebraic Equations ◽  
Linear Integral ◽  
Linear Integral Equations

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

On the Interaction at Anti-Flat Deformation of Stress Concentrators of the Type of Cracks and Stringers with Regard to the Layer Manufactured from Miscellaneous Materials

2019 ◽  
Vol 828 ◽  
pp. 81-88
Author(s):  
Nune Grigoryan ◽  
Mher Mkrtchyan
Keyword(s):  
Integral Transform ◽  
Composite Layer ◽  
Singular Integral Equations ◽  
Original System ◽  
Fourier Integral ◽  
Quadrature Formulas ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Tangential Forces

In this paper, we consider the problem of determining the basic characteristics of the stress state of a composite in the form of a piecewise homogeneous elastic layer reinforced along its extreme edges by stringers of finite lengths and containing a collinear system of an arbitrary number of cracks at the junction line of heterogeneous materials. It is assumed that stringers along their longitudinal edges are loaded with tangential forces, and along their vertical edges - with horizontal concentrated forces. In addition, the cracks are laden with distributed tangential forces of different intensities. The case is also considered when the lower edge of the composite layer is free from the stringer and rigidly clamped. It is believed that under the action of these loads, the composite layer in the direction of one of the coordinate axes is in conditions of anti-flat deformation (longitudinal shift). Using the Fourier integral transform, the solution of the problem is reduced to solving a system of singular integral equations (SIE) of three equations. The solution of this system is obtained by a well-known numerical-analytical method for solving the SIE using Gauss quadrature formulas by the use of the Chebyshev nodes. As a result, the solution of the original system of SIE is reduced to the solution of the system of systems of linear algebraic equations (SLAE). Various special cases are considered, when the defining SIE and the SLAE of the task are greatly simplified, which will make it possible to carry out a detailed numerical analysis and identify patterns of change in the characteristics of the tasks.

THE SCATTERING OF A PLANE WAVE BY A ROW OF SMALL CYLINDERS

1960 ◽  
Vol 38 (2) ◽  
pp. 272-289 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Boundary Condition ◽  
Plane Wave ◽  
Integral Equations ◽  
Cross Sections ◽  
Simultaneous Equations ◽  
Wave Number ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Higher Order Terms ◽  
Scattering Cross Sections

Consideration is given to the scattering of a plane wave by N cylinders equispaced in a row. The problems associated with scatterers, both "soft" and "hard" in the acoustical sense, are treated. An application of Green's theorem together with the appropriate boundary condition on the cylinders leads to a set of simultaneous integral equations in the unknown function on the cylinders.Solutions in the form of series in powers of a small parameter δ (essentially the ratio of cylinder dimension to wavelength) are assumed. In the case of elliptic cylinders, the integral equations are reduced to sets of linear algebraic equations. Only for the first term in the solution for "soft" cylinders is it necessary to solve N simultaneous equations in N unknowns; all other equations involve essentially only one unknown. Far-fields and scattering cross sections are calculated. The case of two "soft" cylinders is given particular attention.Conditions for justification of the neglect of higher-order terms are discussed. It is found that all terms but the first (in either problem) may be neglected if [Formula: see text] and (N–1)/(ka) is sufficiently small. (Here a is the spacing between centers of adjacent cylinders, and k is the wave number.) For this reason these solutions are most useful when the number of cylinders is small.

Isogeometric analysis of boundary integral equations: High-order collocation methods for the singular and hyper-singular equations

2016 ◽  
Vol 26 (08) ◽  
pp. 1447-1480 ◽  
Author(s):  
Matthias Taus ◽  
Gregory J. Rodin ◽  
Thomas J. R. Hughes
Keyword(s):  
Integral Equations ◽  
Isogeometric Analysis ◽  
Computer Aided Design ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
High Order ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Integration Schemes ◽  
Linear Algebraic Equations

Isogeometric analysis is applied to boundary integral equations corresponding to boundary-value problems governed by Laplace’s equation. It is shown that the smoothness of geometric parametrizations central to computer-aided design can be exploited for regularizing integral operators to obtain high-order collocation methods involving superior approximation and numerical integration schemes. The regularization is applicable to both singular and hyper-singular integral equations, and as a result one can formulate the governing integral equations so that the corresponding linear algebraic equations are well-conditioned. It is demonstrated that the proposed approach allows one to compute accurate approximate solutions which optimally converge to the exact ones.

scholarly journals Numerical Solution for the 2D Linear Fredholm Functional Integral Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Neda Khaksari ◽  
Mahmoud Paripour ◽  
Nasrin Karamikabir
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Unknown Function ◽  
Numerical Solutions ◽  
Functional Integral ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Linear Integral ◽  
Linear Integral Equations

In this work, a numerical method is applied for obtaining numerical solutions of Fredholm two-dimensional functional linear integral equations based on the radial basis function (RBF). To find the approximate solutions of these types of equations, first, we approximate the unknown function as a finite series in terms of basic functions. Then, by using the proposed method, we give a formula for determining the unknown function. Using this formula, we obtain a numerical method for solving Fredholm two-dimensional functional linear integral equations. Using the proposed method, we get a system of linear algebraic equations which are solved by an iteration method. In the end, the accuracy and applicability of the proposed method are shown through some numerical applications.

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