A Direct Method for Boundary Integral Equations on a Contour with a Peak

2003 ◽  
Vol 10 (3) ◽  
pp. 573-593
Author(s):  
V. Maz'ya ◽  
A. Soloviev
Keyword(s):  
Integral Equations ◽  
Potential Theory ◽  
Boundary Integral Equations ◽  
Direct Method ◽  
Logarithmic Potential ◽  
Boundary Integral ◽  
Logarithmic Potential Theory ◽  
A Direct Method

Abstract Boundary integral equations in the logarithmic potential theory are studied by the direct method under the assumption that the contour has a peak.

Completed Double Layer Boundary Element Method

1994 ◽  
Author(s):  
Nhan Phan-Thien ◽  
Sangtae Kim
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Integral Equations ◽  
Mesh Generation ◽  
Boundary Integral Equations ◽  
Direct Method ◽  
Boundary Integral ◽  
Deformation Field ◽  
Spatial Methods ◽  
Element Method

Despite the linearity of the Navier equations, solutions to complex boundaryvalue problems require substantial computing resources, especially in the so-called exterior problems, where the deformation field in the space between the inclusions to infinity must be calculated. In the traditional spatial methods, such as finite difference, finite element, or finite volume, this space must be discretized, perhaps with the help of "infinite" elements or a truncation scheme at a finite but large distance from the inclusions (Beer and Watson, Zienkiewicz and Morgan). There are two important limitations of spatial methods. The first is the mesh generation problem. To be numerically efficient, we must use unstructured mesh and concentrate our effort on where it is needed. Efficient two-dimensional, unstructured, automatic mesh generation schemes exist-one good example is Jin and Wiberg-but unstructured three-dimensional mesh generation is still an active area of research. The second limitation is much more severe: even a moderately complicated problem requires the use of supercomputers (e.g., Graham et al.). Since we are concerned with the large-scaled simulations of particulate composites, with the aim of furnishing constitutive information for modeling purposes, our system will possibly have tens of thousands of particles, and therefore the spatial methods are out of the question. We have seen how the deformation field can be represented by a boundary integral equations, either by a direct method, which deals directly with primitive variables (displacement and trciction) on the surface of the domain, or by the indirect method, where the unknowns are the fictitious densities on the surface of the domain. When the field point is allowed to reside on the surface of the domain, then a set of boundary integral equations results that relates only to the variables on the boundary (displacement and traction, or fictitious densities), and this is the basis of the boundary element method. The boundary is then discretized, and the integrals are evaluated by suitable quadratures; this then leads to a set of algebraic equations to be solved for the unknown surface variables.

Mechanical Quadrature Methods and their Extrapolations for Axisymmetric Flow past the Torus of Anchor Ring

2012 ◽  
Vol 614-615 ◽  
pp. 617-620
Author(s):  
Xin Luo ◽  
Jin Huang
Keyword(s):  
Convergence Rate ◽  
Integral Equations ◽  
Potential Theory ◽  
Flow Problem ◽  
Boundary Integral Equations ◽  
Axisymmetric Flow ◽  
Boundary Integral ◽  
Extrapolation Methods ◽  
Mechanical Quadrature ◽  
Quadrature Methods

By the potential theory, axisymmetric flow problem is converted into boundary integral equations (BIEs). The mechanical quadrature methods (MQMs) are presented to deal with the singularities in the integral kernels, which are simple without any singularity integral computation. In addition, the convergence rate of the MQMs can be improved by using the extrapolation methods (EMs). The efficiency of the algorithms is illustrated by examples.

scholarly journals On the solution of the Helmholtz equation on regions with corners

2016 ◽  
Vol 113 (33) ◽  
pp. 9171-9176 ◽  
Author(s):  
Kirill Serkh ◽  
Vladimir Rokhlin
Keyword(s):  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Potential Theory ◽  
Bessel Functions ◽  
Boundary Integral Equations ◽  
Numerical Algorithms ◽  
Boundary Integral ◽  
Numerical Examples ◽  
Polygonal Domains

In this paper we solve several boundary value problems for the Helmholtz equation on polygonal domains. We observe that when the problems are formulated as the boundary integral equations of potential theory, the solutions are representable by series of appropriately chosen Bessel functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.

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