On Extension of Boundary Algebraic Equations to Irregular Lattices

2011 ◽  
Author(s):  
Haifeng Zhao ◽  
Gregory J. Rodin
Keyword(s):  
Boundary Integral Equations ◽  
Computational Effort ◽  
Boundary Integral ◽  
Future Research ◽  
Algebraic Equations ◽  
Infinite Domains ◽  
Multiscale Problems ◽  
Regular Lattices ◽  
Boundary Algebraic Equations ◽  
Interesting Direction

This study is concerned with extending the use of boundary algebraic equations (BAEs) to problems involving irregular rather than regular lattices. Such an extension would be indispensable for solving multiscale problems defined on irregular lattices, as BAEs provide seamless bridging between discrete and continuum models. BAEs share many features with boundary integral equations and are particularly effective for solving problems involving infinite domains. However, it is shown that, BAEs for irregular lattices containing certain terms may require the same amount of computational effort as the original problem for which the equations are formulated. In this paper, we formulate a BAE for a model problem and expose the fundamental obstacle that prevents us from using that BAE as an effective tool. It is shown that, in contrast to regular lattices, BAEs for irregular lattices require a statistical rather than deterministic treatment. This is a very interesting direction for future research.

scholarly journals Boundary algebraic equations for lattice problems

2009 ◽  
Vol 465 (2108) ◽  
pp. 2489-2503 ◽  
Author(s):  
Per-Gunnar Martinsson ◽  
Gregory J. Rodin
Keyword(s):  
Boundary Value Problems ◽  
Spectral Properties ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Boundary Integral ◽  
Linear Boundary ◽  
Algebraic Equations ◽  
Problem Size ◽  
Boundary Algebraic Equations ◽  
Fundamental Boundary

Procedures for constructing boundary integral equations equivalent to linear boundary-value problems governed by partial differential equations are well established. In this paper, it is demonstrated how these procedures can be extended to linear boundary-value problems defined on lattices and governed by algebraic (‘difference’) equations. The boundary equations that arise are then themselves algebraic equations. Such ‘boundary algebraic equations’ (BAEs) are derived for fundamental boundary-value problems defined on both perfect lattices and lattices with defects. It is demonstrated that key advantages of representing a continuum boundary-value problem as an equation on the boundary, such as favourable spectral properties and minimal problem size, are preserved in the lattice environment. Certain spectral properties of BAEs are established rigorously, whereas others are supported by numerical experiments.

Isogeometric analysis of boundary integral equations: High-order collocation methods for the singular and hyper-singular equations

2016 ◽  
Vol 26 (08) ◽  
pp. 1447-1480 ◽  
Author(s):  
Matthias Taus ◽  
Gregory J. Rodin ◽  
Thomas J. R. Hughes
Keyword(s):  
Integral Equations ◽  
Isogeometric Analysis ◽  
Computer Aided Design ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
High Order ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Integration Schemes ◽  
Linear Algebraic Equations

Isogeometric analysis is applied to boundary integral equations corresponding to boundary-value problems governed by Laplace’s equation. It is shown that the smoothness of geometric parametrizations central to computer-aided design can be exploited for regularizing integral operators to obtain high-order collocation methods involving superior approximation and numerical integration schemes. The regularization is applicable to both singular and hyper-singular integral equations, and as a result one can formulate the governing integral equations so that the corresponding linear algebraic equations are well-conditioned. It is demonstrated that the proposed approach allows one to compute accurate approximate solutions which optimally converge to the exact ones.

Calculation of the Magnetization of Permanent Magnets, Based on the Method for Solving the Inverse Problem Using Parallel Calculations

2021 ◽  
Vol 64 (1) ◽  
pp. 13-21
Author(s):  
Petr Denisov ◽  
◽  
Anna Balaban ◽  
Keyword(s):  
Inverse Problem ◽  
Permanent Magnets ◽  
Parallel Implementation ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Tikhonov Regularization Method ◽  
Field Pattern ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
The Matrix

The article proposes the modification of a technique for assessing the magnetization of permanent magnets from the known field pattern. The identification method is based on solving an ill-conditioned system of linear algebraic equations by the Tikhonov regularization method. The method of boundary integral equations based on scalar potentials is used to compile the matrix of coefficients. The article presents the algorithm that uses parallel computations when performing the most time-consuming operations to reduce the time for solving the inverse problem. In order to check the proposed method, a program was developed that allows to simulate the measurement process: to calculate the direct problem and find the magnetic induction at the points of the air gap, then introduce the error into the "measurement results" and solve the inverse problem. The results of nu-merical experiments that allow us to evaluate the advantages of parallel implementation using the capabilities of modern multi-core processors are presented.

Buckling of Plates by Boundary Elements

1998 ◽  
Author(s):  
Jauhorng Lin ◽  
Roger C. Duffield ◽  
Hui-Ru Shih
Keyword(s):  
Integral Equations ◽  
Weighting Function ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Plane Boundary ◽  
Load Factor ◽  
Algebraic Equations ◽  
Plate Buckling ◽  
Element Method

Abstract This investigation is concerned with the determination of a solution to the onset of instability of elastic plate subjected to non-uniform edge loads and/or displacements using boundary element method. In this study, the in-plane stress distribution was taken to be unknown due to non-uniform external in-plane boundary loading and/or displacements. The fundamental solution which is used to solve plate bending problems was used as the weighting function in the solution to the plate buckling problem. With this approach, the resulting integral equations still retained some domain integral terms which were then converted to boundary integral terms by means of the dual reciprocity method. With the introduction of proper shape functions, the boundary integral equations were transformed into a set of simultaneous algebraic equations expressed in a standard eigenvalue matrix format. A number of examples were studied to obtain the critical load factor and the critical buckling load. Results were then compared with analytical solutions and numerical results obtained by means of the finite element method, to illustrate the accuracy and applicability of the proposed solution procedures for solving plate buckling problems.

scholarly journals Ground Vibration Generated by a Harmonic Load Moving in a Circular Tunnel in a Layered Ground

2003 ◽  
Vol 22 (2) ◽  
pp. 83-96 ◽  
Author(s):  
X Sheng ◽  
C J C Jones ◽  
D J Thompson
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Ground Vibration ◽  
Boundary Integral ◽  
Infinite Length ◽  
Harmonic Load ◽  
Algebraic Equations ◽  
Force Method ◽  
Layered Ground ◽  
Element Method

All modes of transport impact on e environment. Although railways are seen as environmentally advantageous in many ways, the issues of noise and vibration are often seen as their weakness. For trains running in tunnels where direct airborne noise is effectively screened, structure-borne or ‘ground-borne’ noise caused by vibration propagated through the ground is the most important concern. The vibration of interest in this case has frequency components from about 15 Hz to 200 Hz. To understand the mechanisms of vibration propagation from tunnels, a predictive model has been developed for ground vibration generated by a stationary or moving harmonic load applied in a circular lined or unlined tunnel in a layered ground. This study is the first step towards the use of discrete wavenumber methods to model ground vibration from underground trains. Discrete wavenumber methods fall into three categories: the discrete wavenumber fictitious force method, the discrete wavenumber finite element method and the discrete wavenumber boundary element method. This study uses the discrete wavenumber fictitious force method. Based on the moving Green's functions for a layered half-space and those for a cylinder of infinite length, boundary integral equations over the tunnel-soil interface are established. Unlike the conventional boundary integral equation in elastodynamics, the method used here only requires the displacement Green's function. This is achieved by introducing the excavated part of the ground as an extra substructure. The boundary integral equations are further transformed into a set of algebraic equations by expressing each quantity involved in the boundary integral equations in terms of a Fourier series. Results presented in this paper illustrate the effect of a tunnel on vibration propagation at the ground surface and the difference between a lined tunnel and an unlined tunnel.

The treatment of lubrication forces in boundary integral equations

2006 ◽  
Vol 462 (2067) ◽  
pp. 855-881 ◽  
Author(s):  
Andrea Alberto Mammoli
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Large Scale ◽  
Boundary Integral Equations ◽  
Hydrodynamic Lubrication ◽  
Integral Equation Method ◽  
Computational Effort ◽  
Boundary Integral ◽  
Spherical Particles ◽  
Fluid Behaviour

Despite their many advantages over other numerical methods, boundary integral formulations still fail to provide accurate predictions of mesoscale motion in dense suspensions of rigid particles because the nearly singular flow between surfaces in close proximity cannot be resolved accurately. A procedure for incorporating analytical solutions for the lubrication flow within a large-scale boundary integral equation method is shown. Although the method is applied to the case of spherical particles, in conjunction with the completed double layer boundary integral equation, it can be developed further to treat more complex geometries and can be adapted to other numerical techniques. In contrast to other apparently similar approaches, the present method does not resort to effective medium approximations, and in principle retains all the advantages typical of boundary integral approaches. The framework also allows for forces other than those due to hydrodynamic lubrication between particles, provided that they are a linear function of the relative velocity or at least can be linearized; for example, forces due to sub-continuum fluid behaviour or forces resulting from surface chemistry. It is shown using several benchmarks that the relative motion between two particles in various flows is captured accurately, both statically and dynamically, in situations where uncorrected simulations fail. Moreover, the computational effort is reduced substantially by the application of the method.

Numerical solution of integral equations for the three-dimensional scalar diffraction problems

2018 ◽  
Author(s):  
С.И. Смагин ◽  
А.А. Каширин
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Acoustic Waves ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Special Method ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Wide Range

Рассматриваются задачи дифракции (трансмиссии) стационарных акустических волн на трехмерных однородных включениях. Методами теории потенциала для них получены два слабо сингулярных граничных интегральных уравнения Фредгольма первого рода с одной неизвестной функцией, каждое из которых эквивалентно исходной задаче. Интегральные уравнения аппроксимируются системами линейных алгебраических уравнений, которые затем решаются численно итерационным методом обобщенных минимальных невязок GMRES. При дискретизации этих уравнений используется специальный метод осреднения интегральных операторов со слабыми особенностями в ядрах, позволяющий получать системы с легко вычисляемыми коэффициентами. Метод допускает эффективное распараллеливание и позволяет проводить расчеты в широком диапазоне волновых чисел. Приводятся результаты вычислительных экспериментов, позволяющие судить о возможностях предлагаемого подхода. Purpose. The purpose of the article is to develop efficient algorithms for numerical solution of the diffraction (transmission) problem of stationary acoustic waves on threedimensional homogeneous inclusions. Methods. By using the combinations of simple and double layer potentials, two Fredholm boundary integral equations of the first kind with one unknown function are obtained for these potentials, each of which is equivalent to the original problem. When sampling these equations, a special method of averaging integral operators with weak singularities in the kernels is applied. Outcomes. The obtained integral equations are approximated by systems of linear algebraic equations with easily-calculated coefficients, which are then solved numerically by means of the generalized method of minimal residuals (GMRES). A series of computing experiments for numerical solution of particular stationary three-dimensional diffraction problems of acoustic waves has been conducted. Conclusions. Computing experiments have shown that the proposed numerical method possesses high accuracy in finding approximate solutions of these problems. It allows both effective parallelization and ability to perform calculations in a wide range of wave numbers and can be used to solve other problems of mathematical physics, formulated in the form of boundary integral equations.

scholarly journals Solution of a Problem Linear Plane Elasticity with Mixed Boundary Conditions by the Method of Boundary Integrals

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Nahed S. Hussein
Keyword(s):  
Boundary Conditions ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Stress Component ◽  
Mixed Boundary Conditions ◽  
Plane Elasticity ◽  
Boundary Integral ◽  
Unknown Boundary ◽  
Algebraic Equations ◽  
Boundary Values

A numerical boundary integral scheme is proposed for the solution to the system of…eld equations of plane. The stresses are prescribed on one-half of the circle, while the displacements are given. The considered problem with mixed boundary conditions in the circle is replaced by two problems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way, and the problem at this stage is reduced to the solution to a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution and the unknown boundary values of stresses or displacements on proper parts of the boundary. On the basis of the obtained results, it is inferred that a stress component has a singularity at each of the two separation points, thought to be of logarithmic type. The results are discussed and boundary plots are given. We have also calculated the unknown functions in the bulk directly from the given boundary conditions using the boundary collocation method. The obtained results in the bulk are discussed and three-dimensional plots are given. A tentative form for the singular solution is proposed and the corresponding singular stresses and displacements are plotted in the bulk. The form of the singular tangential stress is seen to be compatible with the boundary values obtained earlier. The efficiency of the used numerical schemes is discussed.

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