Evaluation of near-singular integrals for quadrilateral basis in integral equation solver

2016 ◽  
Author(s):  
Chong Luo ◽  
Giacomo Bianconi ◽  
Swagato Chakraborty
Keyword(s):  
Integral Equation ◽  
Singular Integrals

scholarly journals An integral equation–based numerical method for the forced heat equation on complex domains

2020 ◽  
Vol 46 (5) ◽  
Author(s):  
Fredrik Fryklund ◽  
Mary Catherine A. Kropinski ◽  
Anna-Karin Tornberg
Keyword(s):  
Integral Equation ◽  
Heat Equation ◽  
Helmholtz Equation ◽  
Singular Integrals ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Elliptic Pdes ◽  
Time Step ◽  
Modified Helmholtz Equation ◽  
High Regularity

Abstract Integral equation–based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.

scholarly journals Neutral coated inclusions of finite conductivity

2011 ◽  
Vol 468 (2140) ◽  
pp. 954-970 ◽  
Author(s):  
Paweł Jarczyk ◽  
Vladimir Mityushev
Keyword(s):  
Integral Equation ◽  
Conformal Mapping ◽  
Singular Integrals ◽  
Smooth Boundary ◽  
Analytical Form ◽  
Closed Curve ◽  
Finite Conductivity ◽  
The Core ◽  
Fixed Curve ◽  
Shaped Inclusion

We discuss the conductivity of two-dimensional media with coated neutral inclusions of finite conductivity. Such an inclusion, when inserted in a matrix, does not disturb the uniform external field. We are looking for shapes of the core and coating in terms of the conformal mapping ω ( z ) of the unit disc onto coated inclusions. The considered inverse problem is reduced to an eigenvalue problem for an integral equation containing singular integrals over a closed curve L 1 on the transformed complex plane. The conformal mapping ω ( z ) is constructed via eigenfunctions of the integral equation. For each fixed curve L 1 , the boundary of the core is given by the curve ω ( L 1 ). The boundary of the coating is obtained by the mapping of the unit circle. It is justified that any shaped inclusion with a smooth boundary can be made neutral by surrounding it with an appropriate coating. Shapes of the neutral inclusions are obtained in analytical form when L 1 is an ellipse.

Motion of a sphere near planar confining boundaries in a Brinkman medium

1998 ◽  
Vol 375 ◽  
pp. 265-296 ◽  
Author(s):  
J. FENG ◽  
P. GANATOS ◽  
S. WEINBAUM
Keyword(s):  
Integral Equation ◽  
Singular Integrals ◽  
Solid Wall ◽  
Boundary Integral ◽  
Numerical Results ◽  
Membrane Receptors ◽  
Brinkman Equation ◽  
Local Analysis ◽  
Channel Height ◽  
Brinkman Medium

A general numerical method using the boundary integral equation technique of Pozrikidis (1994) for Stokes flow in an axisymmetric domain is used to obtain the first solutions to the Brinkman equation for the motion of a particle in the presence of planar confining boundaries. The method is first applied to study the perpendicular and parallel motion of a sphere in a fibre-filled medium bounded by either a solid wall or a planar free surface which remains undeformed. By accurately evaluating the singular integrals arising from the discretization of the resulting integral equation, one can efficiently and accurately treat flow problems with high α defined by rs/K1/2p in which rs is the radius of the sphere and Kp is the Darcy permeability. Convergence and accuracy of the new technique are tested by comparing results for the drag with the solutions of Kim & Russell (1985a) for the motion of two spheres perpendicular to their line of centres in a Brinkman medium. Numerical results for the drag and torque exerted on the particle moving either perpendicular or parallel to a confining planar boundary are presented for ε[ges ]0.1, in which εrs is the gap between the particle and the boundary. When the gap width is much smaller than rs, a local analysis using stretched variables for motion of a sphere indicates that the leading singular term for both drag and torque is independent of α provided that α = O(1). These results are of interest in modelling the penetration of the endothelial surface glycocalyx by microvilli on rolling neutrophils and the motion of colloidal gold and latex particles when they are attached to membrane receptors and observed in nanovid (video enhanced) microscopy. The method is then applied to investigate the motion of a sphere translating in a channel. The drag and torque exerted on the sphere are obtained for various values of α, the channel height H and particle position b. These numerical results are used to describe the diffusion of a spherical solute molecule in a parallel walled channel filled with a periodic array of cylindrical fibres and to assess the accuracy of a simple multiplicative formula proposed in Weinbaum et al. (1992) for diffusion of a solute in the interendothelial cleft.

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.

scholarly journals The volume integral equation method in magnetostatic problem

2019 ◽  
Vol 27 (1) ◽  
pp. 60-69
Author(s):  
Pavel G Akishin ◽  
Andrey A Sapozhnikov
Keyword(s):  
Integral Equation ◽  
Large Scale ◽  
Singular Integrals ◽  
Integral Equation Method ◽  
Equation Method ◽  
Matrix Elements ◽  
System Matrix ◽  
Volume Integral ◽  
Volume Integral Equation ◽  
Volume Integral Equation Method

This article addresses the issues of volume integral equation method application to magnetic system calculations. The main advantage of this approach is that in this case finding the solution of equations is reduced to the area filled with ferromagnetic. The difficulty of applying the method is connected with kernel singularity of integral equations. For this reason in collocation method only piecewise constant approximation of unknown variables is used within the limits of fragmentation elements inside the famous package GFUN3D. As an alternative approach the points of observation can be replaced by integration over fragmentation element, which allows to use approximation of unknown variables of a higher order.In the presented work the main aspects of applying this approach to magnetic systems modelling are discussed on the example of linear approximation of unknown variables: discretisation of initial equations, decomposition of the calculation area to elements, calculation of discretised system matrix elements, solving the resulting nonlinear equation system. In the framework of finite element method the calculation area is divided into a set of tetrahedrons. At the beginning the initial area is approximated by a combination of macro-blocks with a previously constructed two-dimensional mesh at their borders. After that for each macro-block separately the procedure of tetrahedron mesh construction is performed. While calculating matrix elements sixfold integrals over two tetrahedra are reduced to a combination of fourfold integrals over triangles, which are calculated using cubature formulas. Reduction of singular integrals to the combination of the regular integrals is proposed with the methods based on the concept of homogeneous functions. Simple iteration methods are used to solve non-linear discretized systems, allowing to avoid reversing large-scale matrixes. The results of the modelling are compared with the calculations obtained using other methods.

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