The boundary integral equation (boundary element) method in engineering stress analysis

1983 ◽  
Vol 18 (4) ◽  
pp. 199-205 ◽  
Author(s):  
R T Fenner
Keyword(s):  
Finite Element Method ◽  
Integral Equation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
The Finite Element Method ◽  
Advantages And Disadvantages ◽  
Equation Boundary ◽  
Element Method

The principles of the boundary integral equation (BIE) or boundary element method (BEM) are discussed in a non-mathematical way. The technique is compared with other numerical methods, particularly the finite element method (FEM), on the basis of computational efficiency, and the main advantages and disadvantages of the BIE approach are outlined.

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.

Shape Optimal Design of Structures Based on Recent Developments of the Boundary Element Method

1994 ◽  
Author(s):  
A. Portela ◽  
C. A. Mota Soares
Keyword(s):  
Integral Equation ◽  
Boundary Element Method ◽  
Optimal Design ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Material Derivative ◽  
Displacement Boundary ◽  
Stress Boundary ◽  
Element Method

Abstract This paper is concerned with the numerical implementation of the boundary element method for shape optimal design of two-dimensional linear elastic structures. The design objective is to minimize the structural compliance, subject to an area constraint. Sensitivities of objective and constraint functions, derived by means of Lagrangean approach and the material derivative concept with an adjoint variable technique, are computed through analytical expressions that arise from optimality conditions. The boundary element method, used for the discretization of the state problem, applies the stress boundary integral equation for collocation on the design boundary and the displacement boundary integral equation for collocation on other boundaries. The use of the stress boundary integral equation, discretized with discontinuous quadratic elements, allows an efficient and accurate computation of stresses on the design boundary. This discretization strategy not only automatically satisfies the necessary conditions for the existence of the finite-part integrals, which occur naturally in the stress boundary integral equation, but also circumvents the problem of collocation at kinks and corners. The perturbation field is described with linear continuous elements, in which the position of each node is defined by a design variable. Continuity along the design boundary is assured by forcing the end points of each discontinuous boundary element to be coincident with a design node. The optimization problem is solved by the modified method of feasible directions available in the program ADS. Examples of a plate with a hole are analyzed with the present method, for different loading conditions. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of shape optimal design of structures.

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