scholarly journals Predicting the structure of fluids with piecewise constant interactions: Comparing the accuracy of five efficient integral equation theories

2015 ◽  
Vol 91 (4) ◽  
Author(s):  
Kyle B. Hollingshead ◽  
Thomas M. Truskett
Keyword(s):  
Integral Equation ◽  
Piecewise Constant ◽  
Structure Of Fluids

scholarly journals Diagonal scaling of stiffness matrices in the Galerkin boundary element method

2000 ◽  
Vol 42 (1) ◽  
pp. 141-150 ◽  
Author(s):  
Mark Ainsworth ◽  
Bill McLean ◽  
Thanh Tran
Keyword(s):  
Integral Equation ◽  
Stiffness Matrix ◽  
Numerical Experiments ◽  
Degrees Of Freedom ◽  
Boundary Integral ◽  
Piecewise Constant ◽  
Stiffness Matrices ◽  
Diagonal Scaling ◽  
Mesh Ratio ◽  
Galerkin Boundary Element Method

AbstractA boundary integral equation of the first kind is discretised using Galerkin's method with piecewise-constant trial functions. We show how the condition number of the stiffness matrix depends on the number of degrees of freedom and on the global mesh ratio. We also show that diagonal scaling eliminates the latter dependence. Numerical experiments confirm the theory, and demonstrate that in practical computations involving strong local mesh refinement, diagonal scaling dramatically improves the conditioning of the Galerkin equations.

scholarly journals APPLICATION OF COLLOCATION BEM FOR AXISYMMETRIC TRANSMISSION PROBLEMS IN ELECTRO- AND MAGNETOSTATICS

2016 ◽  
Vol 21 (1) ◽  
pp. 16-34 ◽  
Author(s):  
Olga Lavrova ◽  
Viktor Polevikov
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equations ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Meridian Plane ◽  
Piecewise Constant ◽  
Boundary Interpolation ◽  
Transmission Problems ◽  
Polygonal Boundary

This paper considers the numerical solution of boundary integral equations for an exterior transmission problem in a three-dimensional axisymmetric domain. The resulting potential problem is formulated in a meridian plane as the second kind integral equation for a boundary potential and the first kind integral equation for a boundary flux. The numerical method is an axisymmetric collocation with equal order approximations of the boundary unknowns on a polygonal boundary. The complete elliptic integrals of the kernels are approximated by polynomials. An asymptotic kernels behavior is analyzed for accurate numerical evaluation of integrals. A piecewise-constant midpoint collocation and a piecewise-linear nodal collocation on a circular arc and on its polygonal interpolation are used for test computations on uniform meshes. We analyze empirically the influence of the polygonal boundary interpolation to the accuracy and the convergence of the presented method. We have found that the polygonal boundary interpolation does not change the convergence behavior on the smooth boundary for the piecewise-constant and the piecewise-linear collocation.

scholarly journals PIECEWISE CONSTANT SOLUTION OF NON LINEAR VOLTERRA INTEGRAL EQUATION

2014 ◽  
pp. 95-100
Author(s):  
S. CHARAN TEJA ◽  
MONIKA MITTAL
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Volterra Integral Equation ◽  
Computational Efficiency ◽  
Walsh Functions ◽  
Piecewise Constant ◽  
Block Pulse Functions ◽  
Constant Solution ◽  
Non Linear ◽  
Linear Volterra Integral Equations

In this paper, modification in the computational methods, for solving Non-linear Volterra integral equations, is presented. Here, two piecewise constant methods are considered for obtaining the solutions. The first method is based on Walsh Functions (WF) and the second method is via Block Pulse Functions (BPF). Comparison between the two methods is presented by calculating the errors vis-à-vis exact solution. Computational efficiency of BPF is established by profiling the computations for two examples with MATLAB 7.10 Profiler.

scholarly journals Piecewise constant collocation for first-kind boundary integral equations

1994 ◽  
Vol 35 (3) ◽  
pp. 396-398
Author(s):  
I. G. Graham ◽  
Y. Yan
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Parameter Space ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Piecewise Constant ◽  
Break Points ◽  
Image Position ◽  
A Minor ◽  
Closed Polygon

We wish to correct a minor error in the recent paper [2]. That paper was concerned with an integral equation defined on a closed polygon Γ with r corners at the points x0, x2, …, x2r = x0. We parameterized Γ using a mapping γ:[−π,π] → Γ defined as follows. For each l, introduce the mid-point x2l−1 of the side joining x2l—2 to x2l. Then introduce 2r + 1 points in parameter spacewith the property that for each j = 1, …, 2rwhere mj are integers and . Then γ(s) is defined byfor j = 1, …, 2r. The {Sj} are then the preimages of the {xj} under γ. Moreover, in view of (1), a family of uniform meshes can be constructed on [−π, π] which include {Sj} as the break-points. Then γ maps these to meshes which are uniform on each segment joining xj−1 to xj (which we denote Γj). These meshes are used to discretize the integral equation.

scholarly journals Mathematical modeling of monochromatic acoustic wave diffraction on a system of bodies and on flat screens

2021 ◽  
Vol 2142 (1) ◽  
pp. 012002
Author(s):  
S G Daeva ◽  
A L Beskin ◽  
N N Trokhachenkova
Keyword(s):  
Integral Equation ◽  
Acoustic Wave ◽  
Acoustic Pressure ◽  
Pressure Field ◽  
Sound Propagation ◽  
Hypersingular Integral Equation ◽  
Piecewise Constant ◽  
Solid Objects ◽  
Partial Filling ◽  
Complex Shapes

Abstract Some problems of diffraction of a monochromatic acoustic wave on surfaces of complex shapes are considered. To solve such problems, an approach is applied, in which the problem is reduced to a boundary hypersingular integral equation, where the integral is understood in the sense of a finite value according to Hadamard. Such approach allows solving diffraction problems both on solid objects and on thin screens. To solve the integral equation, the method of piecewise constant approximations and collocations, developed in the previous works of the author, is used. In the present study, examples of modeling the diffraction of an acoustic wave by bodies with partial filling are given. It is shown how the filling of bodies influences the acoustic pressure field, and the field direction patterns are given. An example of applying this approach to solving the problem of sound propagation in an urban area is also given: the diffraction of an acoustic wave from a point source on a system of buildings is considered. The presented results demonstrate that this method allows constructing reflected fields and analyze their characteristics on surfaces of complex shapes.

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