scholarly journals Integral equations for free-molecule ow in MEMS: recent advancements

2017 ◽  
Vol 8 (1) ◽  
pp. 67-80
Author(s):  
Patrick Fedeli ◽  
Attilio Frangi
Keyword(s):  
Integral Equation ◽  
Computer Graphics ◽  
Integral Equations ◽  
Boundary Integral Equation ◽  
Singular Integrals ◽  
Boundary Integral ◽  
Free Molecule ◽  
Micro Electro Mechanical Systems ◽  
Radiosity Equation ◽  
Analytical Formulas

Abstract We address a Boundary Integral Equation (BIE) approach for the analysis of gas dissipation in near-vacuum for Micro Electro Mechanical Systems (MEMS). Inspired by an analogy with the radiosity equation in computer graphics, we discuss an efficient way to compute the visible domain of integration. Moreover, we tackle the issue of near singular integrals by developing a set of analytical formulas for planar polyhedral domains. Finally a validation with experimental results taken from the literature is presented.

Analysis of Clamped Plates on Elastic Foundation by the Boundary Integral Equation Method

1984 ◽  
Vol 51 (3) ◽  
pp. 574-580 ◽  
Author(s):  
J. T. Katsikadelis ◽  
A. E. Armena`kas
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Elastic Foundation ◽  
Boundary Integral Equation ◽  
Numerical Evaluation ◽  
Boundary Integral Equations ◽  
Numerical Technique ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Plates On Elastic Foundation

In this investigation the boundary integral equation (BIE) method with numerical evaluation of the boundary integral equations is developed for analyzing clamped plates of any shape resting on an elastic foundation. A numerical technique for the solution to the boundary integral equations is presented and numerical results are obtained and compared with those existing from analytical solutions. The effectiveness of the BIE method is demonstrated.

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.

Proposal of Extended Boundary Integral Equation Method for Rupture Dynamics Interacting With Medium Interfaces

2012 ◽  
Vol 79 (3) ◽  
Author(s):  
Nobuki Kame ◽  
Tetsuya Kusakabe
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Boundary Integral Equation ◽  
Boundary Integral Equations ◽  
Integral Equation Method ◽  
Seismic Wave Propagation ◽  
Boundary Integral ◽  
Boundary Integral Equation Method ◽  
Equation Method ◽  
Rupture Dynamics

The boundary integral equation method (BIEM) has been applied to the analysis of rupture propagation of nonplanar faults in an unbounded homogeneous elastic medium. Here, we propose an extended BIEM (XBIEM) that is applicable in an inhomogeneous bounded medium consisting of homogeneous sub-regions. In the formulation of the XBIEM, the interfaces of the sub-regions are regarded as extended boundaries upon which boundary integral equations are additionally derived. This has been originally known as a multiregion approach in the analysis of seismic wave propagation in the frequency domain and it is employed here for rupture dynamics interacting with medium interfaces in time domain. All of the boundary integral equations are fully coupled by imposing boundary conditions on the extended boundaries and then numerically solved after spatiotemporal discretization. This paper gives the explicit expressions of discretized stress kernels for anti-plane nonplanar problems and the numerical method for the implementation of the XBIEM, which are validated in two representative planar fault problems.

Treatment of singularities in the integral equation approach

Geophysics ◽  
1979 ◽  
Vol 44 (12) ◽  
pp. 2004-2006 ◽  
Author(s):  
Masayuki Okabe
Keyword(s):  
Integral Equation ◽  
Numerical Integration ◽  
Integral Equations ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Equation Approach ◽  
Element Level ◽  
Integral Equation Approach ◽  
Electrical Prospecting ◽  
Made In

Boundary integral equation techniques are becoming increasingly popular in geophysics as well as in other engineering fields. Excellent contributions have been made in electrical prospecting (Alfano, 1959; Dieter et al., 1969; Barnett, 1972; Snyder, 1976). Through the formulation of the integral equations, we always encounter some singularities. This note presents a method of treating such singularities in the integral equation approach governed by Poisson’s equation. The problems encountered in the numerical integration in each element level also are discussed by reconsideration of previous formulations.

The Calculation of Potential Derivatives by Using NBIE for Anisotropic Potential Problems

2016 ◽  
Vol 33 (2) ◽  
pp. 183-191
Author(s):  
H.-L. Zhou ◽  
B.-X. Bian ◽  
Y. Tian ◽  
B. Yu ◽  
Z.-R. Niu
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Potential Problem ◽  
Singular Integrals ◽  
Integral Kernel ◽  
Anisotropic Media ◽  
Transformation Method ◽  
Boundary Integral ◽  
Potential Problems ◽  
Interior Points

AbstractThe natural boundary integral equation (NBIE) is developed to calculate potential derivatives for potential problems with anisotropic media. Firstly, the governing equation of the two-dimensional anisotropic potential problem is transformed into standard Laplace equation by a coordinate transformation method. Then a potential derivative boundary integral equation named as NBIE is extended to solve the anisotropic potential problem. The most important virtue of the NBIE is that the singularity of the integral kernel function is reduced by one order in comparison with the conventional potential derivative boundary integral equation(CDBIE). Therefore the new potential derivative boundary integral equation only contains strongly singular integrals rather than hyper-singular integrals. Thus the NBIE can calculate more accurate potential derivative results for both boundary nodes and interior points. Moreover, in combination with the analytical integral regularization algorithm of nearly singular integrals, the NBIE can obtain more accurate potential derivatives of interior points very close to the boundary than the CDBIE. Numerical examples on heat conduction in anisotropic media demonstrate the accuracy and efficiency of the NBIE.

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