Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis

2017 ◽  
Vol 34 (5) ◽  
pp. 1527-1550 ◽  
Author(s):  
Jeng-Tzong Chen ◽  
Shyh-Rong Kuo ◽  
Yu-Lung Chang ◽  
Shing-Kai Kao
Keyword(s):  
Integral Equation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Arbitrary Shape ◽  
Boundary Integral ◽  
Degenerate Kernel ◽  
Content Type ◽  
Degenerate Scale ◽  
Circular Case ◽  
Plate Analysis

Purpose The purpose of this paper is to detect the degenerate scale of a 2D bending plate analytically and numerically. Design/methodology/approach To avoid the time-consuming scheme, the influence matrix of the boundary element method (BEM) is reformulated to an eigenproblem of the 4 by 4 matrix by using the scaling transform instead of the direct-searching scheme to find degenerate scales. Analytical degenerate scales are derived from the boundary integral equation (BIE) by using the degenerate kernel only for the circular case. Numerical results of the direct-searching scheme and the eigen system for the arbitrary shape are also considered. Findings Results using three methods, namely, analytical derivation, the direct-searching scheme and the 4 by 4 eigen system, are also given for the circular case and arbitrary shapes. Finally, addition of a constant for the kernel function makes original eigenvalues (2 real roots and 2 complex roots) of the 4 by 4 matrix to be all real. This indicates that a degenerate scale depends on the kernel function. Originality/value The analytical derivation for the degenerate scale of a 2D bending plate in the BIE is first studied by using the degenerate kernel. Through the reformed eigenproblem of a 4 by 4 matrix, the numerical solution for the plate of an arbitrary shape can be used in the plate analysis using the BEM.

A state space boundary element method for elasticity of functionally graded materials

2017 ◽  
Vol 34 (8) ◽  
pp. 2614-2633
Author(s):  
Changzheng Cheng ◽  
Zhilin Han ◽  
Zhongrong Niu ◽  
Hongyu Sheng
Keyword(s):  
Integral Equation ◽  
Boundary Element Method ◽  
State Space ◽  
Boundary Element ◽  
Present Method ◽  
Boundary Integral ◽  
Functionally Graded ◽  
The State ◽  
Content Type ◽  
Element Method

Purpose The state space method (SSM) is good at analyzing the interfacial physical quantities in laminated materials, while it has difficulty in calculating the mechanical quantities of interior points, which can be easily evaluated by the boundary element method (BEM). However, the material has to be divided into many subdomains when the traditional BEM is applied to analyze the functionally graded material (FGM), so that the computational amount will be increased enormously. This study aims to couple these two methods to strengthen their advantages and overcome their disadvantages. Design/methodology/approach Herein, a state space BEM in which the SSM is coupled by the BEM is proposed to analyze the elasticity of FGMs, where one BEM domain is set and the others belong to SSM domains. The discretized elements occur only on the boundary of the BEM domain and at the interfaces between different SSM domains. In SSM domains, the horizontal interfaces of FGMs are discretized by linear elements and the variables along the vertical direction are yielded by the precise integration method. Findings The accuracy of the proposed method is verified by comparing the present results with the ones from the finite element method (FEM). It is found that the present method can provide accurate displacements and stresses in the FGMs by fewer freedom degrees in comparison with the FEM. In addition, the present method can provide continuous interfacial stresses at the interfaces between different material domains, while the interfacial stresses by the FEM are discontinuous. Originality/value The system equations of the state space BEM are built by combining the boundary integral equation with the state equations according to the continuity conditions at the interfaces. The mechanical parameters of any inner point can be evaluated by the boundary integral equation after the unknowns on the boundaries and interfaces are determined by the system equation.

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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