Special Boundary Integral Equations for Approximate Solution of Potential Problems in Three-dimensional Regions with Slender Cavities of Circular Cross-section

1985 ◽  
Vol 35 (3) ◽  
pp. 311-325 ◽  
Author(s):  
M. R. BARONE ◽  
D. A. CAULK
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Cross Section ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Boundary Integral ◽  
Special Boundary ◽  
Potential Problems

Analysis of Elastic Torsion in a Bar With Circular Holes by a Special Boundary Integral Method

1983 ◽  
Vol 50 (1) ◽  
pp. 101-108 ◽  
Author(s):  
D. A. Caulk
Keyword(s):  
Integral Equations ◽  
Cross Section ◽  
Boundary Integral Equations ◽  
Torsional Rigidity ◽  
Circular Cross Section ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Warping Function ◽  
Special Boundary ◽  
Circular Holes

Special boundary integral equations developed in an earlier paper are generalized here for torsion of an elastic bar with circular holes. In this approach, the solution on the boundary of each hole is represented by a series of circular harmonics, and the coefficients in these series are determined by a special system of boundary integral equations. For a cross section with only one hole, the entire system of equations is reduced without approximation to a single integral equation involving only the warping function on the outer boundary. For multiple holes, approximate equations are derived that retain only the first harmonic in the solution representation on each hole. The latter equations are solved analytically for a circular cross section weakened by a concentric ring of circular holes. Simple expressions are derived for torsional rigidity, warping, and maximum stress. The results for torsional rigidity are an improvement over previous ones obtained by another approximate method.

Special Boundary Integral Equations for Potential Problems in Regions With Circular Holes

1984 ◽  
Vol 51 (4) ◽  
pp. 713-716 ◽  
Author(s):  
D. A. Caulk
Keyword(s):  
Integral Equations ◽  
Infinite System ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Special Boundary ◽  
Laplace's Equation ◽  
Special Cases ◽  
Potential Problems ◽  
Circular Holes

An infinite system of special boundary integral equations is derived for the solution of Laplace’s equation in a general two-dimensional region with circular holes. The solution is shown to converge when the number of holes is finite and no two holes are touching. In special cases, these equations are shown to yield the same results as two more restricted methods, which are based on different approaches.

Error analysis of boundary integral methods

Acta Numerica ◽  
1992 ◽  
Vol 1 ◽  
pp. 287-339 ◽  
Author(s):  
Ian H. Sloan
Keyword(s):  
Approximate Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Error Analysis ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Boundary Integral Methods ◽  
Integral Methods

Many of the boundary value problems traditionally cast as partial differential equations can be reformulated as integral equations over the boundary. After an introduction to boundary integral equations, this review describes some of the methods which have been proposed for their approximate solution. It discusses, as simply as possible, some of the techniques used in their error analysis, and points to areas in which the theory is still unsatisfactory.

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.

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