scholarly journals Boundary Element Methods in Solid Mechanics - a Tribute to Frank Rizzo

2007 ◽  
Vol 1 (1) ◽  
Author(s):  
Subrata Mukherjee
Keyword(s):  
Integral Equation ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Linear Elasticity ◽  
Solid Mechanics ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Seminal Paper ◽  
Good Fortune ◽  
The Subject

Frank Rizzo proposed the first Boundary Integral Equation (BIE) formulation for linear elasticity in a seminal paper in 1967. I have had the very good fortune to call Frank a friend of mine for many years. The present paper describes some of my experiences with the BIE over nearly thirty years, some of it in the context of Frank’s pioneering contributions to the subject and my interactions with him.

scholarly journals Divergent Integrals in Elastostatics: General Considerations

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
V. V. Zozulya
Keyword(s):  
Integral Equation ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Hypersingular Integrals ◽  
Integral Equation Methods ◽  
Weakly Singular ◽  
Easy Calculation ◽  
2D And 3D

This article considers weakly singular, singular, and hypersingular integrals, which arise when the boundary integral equation methods are used to solve problems in elastostatics. The main equations related to formulation of the boundary integral equation and the boundary element methods in 2D and 3D elastostatics are discussed in details. For their regularization, an approach based on the theory of distribution and the application of the Green theorem has been used. The expressions, which allow an easy calculation of the weakly singular, singular, and hypersingular integrals, have been constructed.

scholarly journals Research on Error Estimations of the Interpolating Boundary Element Free-Method for Two-Dimensional Potential Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Jufeng Wang ◽  
Fengxin Sun ◽  
Ying Xu
Keyword(s):  
Integral Equation ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Potential Problems ◽  
Element Free Method ◽  
Error Estimations ◽  
Element Free

The interpolating boundary element-free method (IBEFM) is a direct solution method of the meshless boundary integral equation method, which has high efficiency and accuracy. The IBEFM is developed based on the interpolating moving least-squares (IMLS) method and the boundary integral equation method. Since the shape function of the IMLS method satisfies the interpolation characteristics, the IBEFM can directly and accurately impose the essential boundary conditions, which overcomes the shortcomings of the original boundary element-free method in enforcing the essential boundary approximately. This paper will study the error estimations of the IBEFM for two-dimensional potential problems and the relationship between the errors and the influence radius and the condition number of the coefficient matrix. Two numerical examples are presented to verify the correctness of the theoretical results in this paper.

A Novel Displacement Gradient Boundary Element Method for Elastic Stress Analysis With High Accuracy

1988 ◽  
Vol 55 (4) ◽  
pp. 786-794 ◽  
Author(s):  
H. Okada ◽  
H. Rajiyah ◽  
S. N. Atluri
Keyword(s):  
Integral Equation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Numerical Differentiation ◽  
Boundary Integral ◽  
Displacement Gradient ◽  
Direct Integral ◽  
Displacement Boundary ◽  
Element Method

The boundary element method (BEM) in current usage, is based on the displacement boundary integral equation. The current practice of computing stresses in the BEM involves the use of a two-tier approach: (i) numerical differentiation of the displacement field at the boundary, and (ii) analytical differentiation of the displacement integral equation at the source point in the interior. A new direct integral equation for the displacement gradient is proposed here, to obviate this two-tier approach. The new direct boundary integral equation for displacement gradients has a lower order singularity than in the standard formulation, and is quite tractable from a numerical view point. Numerical results are presented to illustrate the advantages of the present approach.

Application of the Boundary Element Method to Acoustic Cavity Response and Muffler Analysis

1987 ◽  
Vol 109 (1) ◽  
pp. 15-21 ◽  
Author(s):  
A. F. Seybert ◽  
C. Y. R. Cheng
Keyword(s):  
Integral Equation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Transmission Loss ◽  
Radiation Condition ◽  
Boundary Integral ◽  
Exterior Problems ◽  
Integral Equation Formulation ◽  
Element Method

This paper is concerned with the application of the Boundary Element Method (BEM) to interior acoustics problems governed by the reduced wave (Helmholtz) differential equation. The development of an integral equation valid at the boundary of the interior region follows a similar formulation for exterior problems, except for interior problems the Sommerfeld radiation condition is not invoked. The boundary integral equation for interior problems does not suffer from the nonuniqueness difficulty associated with the boundary integral equation formulation for exterior problems. The boundary integral equation, once obtained, is solved for a specific geometry using quadratic isoparametric surface elements. A simplification for axisymmetric cavities and boundary conditions permits the solution to be obtained using line elements on the generator of the cavity. The present formulation includes the case where a node may be placed at a position on the boundary where there is not a unique tangent plane (e.g., at an edge or a corner point). The BEM capability is demonstrated for two types of classical interior axisymmetric problems: the acoustic response of a cavity and the transmission loss of a muffler. For the cavity response comparison data are provided by an analytical solution. For the muffler problem the BEM solution is compared to data obtained by a finite element method analysis.

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.

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