Saturation-free numerical scheme for computing the flow past a lattice of airfoils and the determination of separation points in a viscous fluid

A numerical method for computing the potential flow past a lattice of airfoils is described. The problem is reduced to a linear integrodifferential equation on the lattice contour, which is then approximated by a linear system of equations with the help of specially derived quadrature formulas. The quadrature formulas exhibit exponential convergence in the number of points on an airfoil and have a simple analytical form. Due to its fast convergence and high accuracy, the method can be used to directly optimize the airfoils as based on any given integral characteristics. The shear stress distribution and the separation points are determined from the velocity distribution at the airfoil boundary calculated by solving the boundary layer equations. The method proposed is free of laborious grid generation procedures and does not involve difficulties associated with numerical viscosity at high Reynolds numbers.     

Approximation of a fractional power of an elliptic operator

Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis of an integral representation with a singular integrand. In the present article, new integral representations are proposed for operators with fractional powers. Approximations are based on the classical quadrature formulas. The results of numerical experiments on the accuracy of quadrature formulas are presented. The proposed approximations are used for numerical solving a model two‐dimensional boundary value problem for fractional diffusion.     

The numerical solution of an evolution problem of second order in time on a closed smooth boundary

We consider an initial value problem for the second-order differential equation with a Dirichlet-to-Neumann operator coefficient. For the numerical solution we carry out semi-discretization by the Laguerre transformation with respect to the time variable. Then an infinite system of the stationary operator equations is obtained. By potential theory, the operator equations are reduced to boundary integral equations of the second kind with logarithmic or hypersingular kernels. The full discretization is realized by Nyström's method which is based on the trigonometric quadrature rules. Numerical tests confirm the ability of the method to solve these types of nonstationary problems.     

On the convergence of a numerical method for a hypersingular integral equation on a closed surface

We consider a linear integral equation with a hypersingular integral treated in the sense of the Hadamard finite value. This equation arises in the solution of the Neumann boundary value problem for the Laplace equation with a representation of a solution in the form of a double-layer potential. We consider the case in which the interior or exterior boundary value problem is solved in a domain; whose boundary is a smooth closed surface, and an integral equation is written out on that surface. For the integral operator in that equation, we suggest quadrature formulas like the method of vortical frames with a regularization, which provides its approximation on the entire surface for the use of a nonstructured partition. We construct a numerical scheme for the integral equation on the basis of suggested quadrature formulas, prove an estimate for the norm of the inverse matrix of the related system of linear equations and the uniform convergence of numerical solutions to the exact solution of the hypersingular integral equation on the grid.     

On the convergence of the vortex loop method with regularization for the Neumann boundary value problem on a plane screen

We consider a linear integral equation with a supersingular integral treated in the sense of the Hadamard finite value, which arises in the solution of the Neumann boundary value problem for the Laplace equation with the representation of the solution in the form of a doublelayer potential. We consider the case in which the exterior boundary value problem is solved outside a plane surface (a screen). For the integral operator in the above-mentioned equation, we suggest quadrature formulas of the vortex loop method with regularization, which provide its approximation on the entire surface when using an unstructured partition. In the problem in question, the derivative of the unknown density of the double-layer potential, as well as the errors of quadrature formulas, has singularities in a neighborhood of the screen edge. We construct a numerical scheme for the integral equation on the basis of the suggested quadrature formulas and prove an estimate for the norm of the inverse matrix of the resulting system of linear equations and the uniform convergence of the numerical solutions to the exact solution of the supersingular integral equation on the grid.     

Numerical solution of evolutionary integral equations with completely monotonic kernel by Runge–Kutta convolution quadrature

We study the numerical solutions of the initial boundary value problems for the Volterra‐type evolutionary integal equations, in which the integral operator is a convolution product of a completely monotonic kernel and a positive definite operator, such as an elliptic partial‐differential operator. The equation is discretized in time by the Runge–Kutta convolution quadrature. Error estimates are derived and numerical experiments reported. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 105–142, 2015     

Some applications of a galerkin-collocation method for boundary integral equations of the first kind

Here we apply the boundary integral method to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics. These are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the first kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order - 1 which provide certain coercivity properties. The boundary integral equations are approximated by Galerkin's method using B-splines on the boundary curve in connection with an appropriate numerical quadrature, which yields a modified collocation scheme. We present a complete asymptotic error analysis for the fully discretized numerical equations which is based on superapproximation results for Galerkin's method, on consistency estimates and stability properties in connection with the illposedness of the first kind equations in L₂. We also present computational results of several numerical experiments revealing accuracy, efficiency and an amazing asymptotical agreement of the numerical with the theoretical errors. The method is used for computations of conformal mappings, exterior Stokes flows and slow viscous flows past elliptic obstacles.     

含参数的一类积分方程

根据向量值全纯函数和亚纯函数的理论,由向量值Plemelj公式,讨论一类局部凸空间中具有ζ-函数核的奇异积分方程与边值问题的关系,给出向量值奇异积分方程和边值问题的解及其稳定性.     

An extrapolation method for a class of boundary integral equations

Boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using Green's formulas. For solving the induced boundary integral equations, a Nyström scheme and its extrapolation method are derived for periodic Fredholm integral equations of the second kind with logarithmic singularity. Asymptotic expansions for the approximate solutions obtained by the Nyström scheme are developed to analyze the extrapolation method. Some computational aspects of the methods are considered, and two numerical examples are given to illustrate the acceleration of convergence.

     

含曲线裂纹圆柱扭转问题的新边界元法

研究含曲线裂纹圆柱的Saint-Venant扭转,将问题化归为裂纹上边界积分方程的求解．利用裂纹尖端的奇异元和线性元插值模型,给出了扭转刚度和应力强度因子的边界元计算公式．对圆弧裂纹、曲折裂纹以及直线裂纹的典型问题进行了数值计算,并与用Gauss-Chebyshev求积法计算的直裂纹情形结果进行了比较,证明了方法的有效性和正确性．     

A new interpretation of Cauchy type singular integrals with an application to singular integral equations

Cauchy type integrals were given the interpretation of the principal value for points inside the integration interval. Here this interpretation is modified and generalized in a very simple manner. The new interpretation in general is not equivalent to the classical one. The relationship between the new interpretation and the classical one is investigated and various applications of the new interpretation (to the Plemelj formulas, the Riemann-Hilbert boundary value problem, singular integral equations, the inversion formula, quadrature rules and interface crack problems) are presented.     

The direct method in time-dependent bending of thermoelastic plates

The initial-boundary value problems with Dirichlet and Neumann boundary conditions arising in the theory of bending of thermoelastic plates with transverse shear deformation are reduced to time-dependent boundary integral equations by means of the Somigliana representation formulas. The solvability of these equations is then investigated in Sobolev-type spaces.     

弹性力学问题解唯一的边界积分方程

从积分方程式出发,应用基本解的特性分析,说明在力边值问题中,位移边界积分方程和面力边界积分方程的位移解不唯一.提出了位移解唯一的条件,建立了唯一解的位移边界积分方程和面力边界积分方程.实例计算结果表明唯一解的边界积分方程是有效的.     

Block boundary value methods for solving Volterra integral and integro-differential equations

Reducible quadrature rules generated by boundary value methods are considered in block version and applied to solve the second kind Volterra integral equations and Volterra integro-differential equations. These extended block boundary value methods are shown to possess both excellent stability properties and high accuracy for Volterra-type equations. Numerical experiments are presented and the efficiency, accuracy and stability of the schemes are confirmed.     

Lubich convolution quadratures and their application to problems described by space-time BIEs

In the last years several authors have used Lubich convolution quadrature formulas to discretize space-time boundary integral equations representing time dependent problems. These rules have the fundamental property of not using explicitly the expression of the kernel of the integral equation they are applied to, which is instead replaced by that of its Laplace transform, usually given by a simple analytic function. In this paper, a review of these rules, which includes their main properties, several new remarks and some conjectures, will be presented when they are applied to the heat and wave space-time boundary integral equation formulations. The construction and behavior of the corresponding coefficients are analyzed and tested numerically. When the quadrature is defined by a BDF method, a new approach for the representation of its coefficients is presented.     

Flux intensity functions for the Laplacian at axisymmetric edges

We derive explicit representation formulas for the computation of flux intensity functions for mixed boundary value problems for the Poisson equation in axisymmetric domains with edges. We rely on the decomposition of the boundary value problems in three dimensions by means of partial Fourier analysis with respect to the rotational angle into boundary value problems in the two‐dimensional meridian domain of . Utilizing smooth cutoff functions, the solutions of the reduced problems are analyzed semi‐analytically near corners of the plane meridian domain, and the edge flux intensity functions are constructed via Fourier synthesis and convergence analysis. The formulas are also applicable in the case of crack fronts. The constructive nature of the formulas provides in a straightforward way an efficient strategy for the accurate computation of edge flux intensity functions in axisymmetric domains. A demonstration example that illustrates the application of the formulas is presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Computation of forces acting on bodies in plane and axisymmetric cavitation flow problems

Plane and axisymmetric cavitation flow problems are considered using Riabouchinsky’s scheme. The incoming flow is assumed to be irrotational and steady, and the fluid is assumed to be inviscid and incompressible. The flow problems are solved by applying the boundary element method with quadrature formulas without saturation. The free boundary is determined using a gradient descent technique based on Riabouchinsky’s principle. The drag force acting on the cavitator is expressed in terms of the Riabouchinsky functional. As a result, for small cavitation numbers, the force is calculated with a fairly high accuracy. Dependences of the drag coefficient are investigated for variously shaped cavitators: a wedge, a cone, a circular arc, and a spherical segment.     

BEM with linear complexity for the classical boundary integral operators

Alternative representations of boundary integral operators corresponding to elliptic boundary value problems are developed as a starting point for numerical approximations as, e.g., Galerkin boundary elements including numerical quadrature and panel-clustering. These representations have the advantage that the integrands of the integral operators have a reduced singular behaviour allowing one to choose the order of the numerical approximations much lower than for the classical formulations. Low-order discretisations for the single layer integral equations as well as for the classical double layer potential and the hypersingular integral equation are considered. We will present fully discrete Galerkin boundary element methods where the storage amount and the CPU time grow only linearly with respect to the number of unknowns.

     

Complex variable equations and the numerical solution of harmonic problems for piecewise-homogeneous media

Complex variable boundary integral equations are derived using of holomorphicity theorems for plane harmonic problems concerning unit structures with inclusions, pores and lines of discontinuity of the potential and/or the flow. Unlike the method of analytical elements, the equations cover problems in which discontinuities in the potential, flow and conductance can simultaneously be encountered at the contact points. Versions of the equations are given for connected half planes and for periodic and biperiodic problems. Formulae are obtained which determine the effective impedance tensor of the equivalent homogeneous medium in cases when the unit structure is biperiodic or when the representative volume of a structured medium is identified with the basic cell of a biperiodic system. Recurrence quadrature formulae are proposed which enable one to solve the resulting equations effectively using the complex variable boundary element method. They indicate the computational advantages of using the complex variable method compared with the real variable method: the three integrals appearing in the resulting equations are evaluated analytically in the case of linear elements (regular and singular) with the densities approximated using algebraic polynomials of arbitrary degree. In the case of elements (regular and singular) in the form of an arc of a circle, only one integral requires numerical integration when the densities are approximated using complex trigonometrical polynomials of arbitrary degree. It is emphasized that the combination of the linear and curved boundary elements which have been developed enables the smooth part of a contour to be approximated while retaining the continuity of the tangent and avoiding the complications which arise when the smoothness of the approximation of a contour is ensured using conformal mapping. Examples are presented which illustrate the computational merits of the method developed. They show a sharp increase in accuracy (by orders of magnitude) when curved elements are used for the curvilinear parts of a contour and when terminal elements are used to calculate the flow intensity coefficient at singular points (the crack tips the vertices of angular notches and the common vertices of the units of the medium).     

Convergence of the piecewise linear approximation and collocation method for a hypersingular integral equation on a closed surface

We study the numerical solution of a linear hypersingular integral equation arising when solving the Neumann boundary value problem for the Laplace equation by the boundary integral equation method with the solution represented in the form of a double layer potential. The integral in this equation is understood in the sense of Hadamard finite value. We construct quadrature formulas for the integral occurring in this equation based on a triangulation of the surface and an application of the linear approximation to the unknown function on each of the triangles approximating the surface. We prove the uniform convergence of the quadrature formulas at the interpolation nodes as the triangulation size tends to zero. A numerical solution scheme for this integral equation based on the suggested quadrature formulas and the collocation method is constructed. Under additional assumptions about the shape of the surface, we prove a uniform estimate for the error in the numerical solution at the interpolation nodes.