Using the method of boundary integral equations in solving of the geological mapping problems with faults

2020 ◽  
Vol 6 (2) ◽  
pp. 110-126
Author(s):  
Andrei A. Sidorov
Keyword(s):  
Integral Equations ◽  
Laplace Equation ◽  
Complex Geometry ◽  
Boundary Integral Equations ◽  
A Priori ◽  
Spline Approximation ◽  
Arbitrary Point ◽  
Boundary Integral ◽  
Geological Mapping ◽  
Elastic Membrane

Methods for creating digital grid models of geological surfaces based on approximation by bicubic B-splines are widely used in solving problems of mathematical geology. The variational-gridding method of geological mapping is a flexible and powerful tool that allows to use a large amount of source data for mapping, as well as a priori information about the spatial distribution of the mapped parameter. The smoothness of the basis functions does not allow the direct use of this effective method for mapping geological surfaces complicated by faults. This fact requires adaptation of the variational-gridding method to mapping with faults. The article discusses the technology of separate construction of the fault and plicative (smooth) components of the structural map. The fault component is represented in the form of an antiplane shear field of an elastic membrane described in a stationary two-dimensional Laplace equation. The fault network is modeled by narrow contours, at the boundaries of which the values of tectonic displacements are set. The Laplace equation is solved by the method of boundary integral equations, which allows one to calculate the displacement field at an arbitrary point of the mapped region, as well as to most accurately approximate the complex geometry of faults. Modeling of the plicative component of the structural surface takes place on the basis of a variational-gridding approach with adjustment of tectonic displacements. The approach combines the advantages of the spline approximation method and the accuracy of the semi-analytical solution for the fault component. It does not impose restrictions on the geometry of faults, and also allows for more efficient mathematical operations with structural surfaces, and their differential and integral characteristics.

scholarly journals Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in R3

2020 ◽  
Vol 22 (3) ◽  
pp. 319-332
Author(s):  
Aleksandr N. Tynda ◽  
Konstantin A. Timoshenkov
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Spline Approximation ◽  
Boundary Integral ◽  
Double Layers ◽  
Fredholm Integral Equations ◽  
Fredholm Type ◽  
Graded Meshes

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.

scholarly journals Circular Slit Maps of Multiply Connected Regions with Application to Brain Image Processing

2020 ◽  
Vol 44 (1) ◽  
pp. 171-202
Author(s):  
Ali W. K. Sangawi ◽  
Ali H. M. Murid ◽  
Khiy Wei Lee
Keyword(s):  
Image Processing ◽  
Integral Equations ◽  
Complex Geometry ◽  
Boundary Integral Equations ◽  
Fast Multipole Method ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Brain Image ◽  
Multiply Connected ◽  
Multiply Connected Regions

AbstractIn this paper, we present a fast boundary integral equation method for the numerical conformal mapping and its inverse of bounded multiply connected regions onto a disk and annulus with circular slits regions. The method is based on two uniquely solvable boundary integral equations with Neumann-type and generalized Neumann kernels. The integral equations related to the mappings are solved numerically using combination of Nyström method, GMRES method, and fast multipole method. The complexity of this new algorithm is $$O((M + 1)n)$$ O ( ( M + 1 ) n ) , where $$M+1$$ M + 1 stands for the multiplicity of the multiply connected region and n refers to the number of nodes on each boundary component. Previous algorithms require $$O((M+1)^3 n^3)$$ O ( ( M + 1 ) 3 n 3 ) operations. The numerical results of some test calculations demonstrate that our method is capable of handling regions with complex geometry and very high connectivity. An application of the method on medical human brain image processing is also presented.

scholarly journals Hypersingular boundary integral equations for plane orthotropic elasticity in terms of first-order stress functions

2020 ◽  
Vol 15 (2) ◽  
pp. 185-207
Author(s):  
Sándor Szirbik
Keyword(s):  
Stress Distribution ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Arbitrary Point ◽  
Boundary Integral ◽  
Computational Technique ◽  
First Order ◽  
Orthotropic Elasticity ◽  
Stress Functions ◽  
Order Stress

This paper is intended to present an implementation of the hypersingular boundary integral equations in terms of first-order stress functions for stress computations in plane orthotropic elasticity. In general, the traditional computational technique of the boundary element method used for computing the stress distribution on the boundary and close to it is not as accurate as it should be. In contrast, the accuracy of stress computations on the boundary is greatly increased by applying the hypersingular integral equations. Contrary to the method in which the solution is based on an approximation of displacement field, here the first-order stress functions and the rigid body rotation are the fundamental variables. An advantage of this approach is that the stress components can be obtained directly from the stress functions, there is, therefore, no need for Hooke's law, which should be used when they are computed from displacements. In addition, the computational work can be reduced when the stress distribution is computed at an arbitrary point on the boundary. The numerical examples presented prove the efficiency of this technique.

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