Integral Equations Of Potential Theory Research Articles

FEATURES OF BOUNDARY ELEMENT METHOD APPLICATION FOR SOLUTIONS OF BOUNDARY-VALUE PROBLEMS IN MOVING BOUNDARY DOMAINS

Summary. The motion of a fluid with a free surface is the subject of a separate section of theoretical hydromechanics. The development of this direction is stimulated by numerous applications to modern technologies, especially for the design and operation of hydraulic structures, shipbuilding and navigation, the study of ebbs and flows, tsunamis and other natural phenomena. The complexity of setting and solving this class of problems lies in the presence of a moving boundary of the solution area, called the free surface. From a mathematical point of view, the peculiarity of the considered class of problems lies in a specific non-linearity: a solution-dependent change in time of the shape of the solution area. Assuming the fluid to be incompressible and in viscid, under the conditions of the potentiality of the field of body forces and the potentiality of the flow at the initial moment of time, we obtain that the flow is described by the Laplace equation for the velocity potential with the traditional boundary conditions of non-flow at the solid boundaries of the flow region, as well as special boundary conditions at the free boundary, one of which, the Cauchy-Lagrange integral, contains a quadratic nonlinearity in the flow velocity. The assumption made about the potentiality of the alternating field of body forces greatly simplifies the formulation of the problem and allows us to consider the problem formulated above as a model one for a wide class of fluid flows with a free surface. For the numerical solution of the problem described above, a regular version of the boundary element method was used. The use of regular algorithms of the boundary element method makes it possible to bypass numerous computational limitations and improve in one way or another the approximations of the shape of the moving boundary, known and unknown functions on it. To do this, the Laplace equation was transformed into a boundary integral equation of potential theory, and the time interval in which the flow field is studied was divided into the corresponding number of sufficiently small intervals. At each time step, the resulting boundary integral equation was solved using the standard algorithm of the boundary element method. The calculation of the new position of the free boundary of the flow was carried out using the Euler scheme.

On the solution of the Helmholtz equation on regions with corners

In this paper we solve several boundary value problems for the Helmholtz equation on polygonal domains. We observe that when the problems are formulated as the boundary integral equations of potential theory, the solutions are representable by series of appropriately chosen Bessel functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.

Расчет гидродинамического взаимодействия в сверхтекучей жидкости методами вычислительной теории потенциала

The problems of superfluid liquid hydrodynamics are quite relevant today. Previously uninvestigated problem of multiphase flow of superfluid liquid is considered in the paper. The tasks of streamline of groups of objects of different shapes by a flat stationary flow of superfluid liquid is formulated in terms of velocity potential in the form of differential equations in specific derived and boundary integral equations of potential theory. It was shown that for such a flow, the main force, affecting objects, is the strength of hydrodynamic interaction. Calculations were made using a standard algorithm for the boundary element method. The dependences of hydrodynamic interaction forces on the distance between the streamlined objects were obtained for circular and elliptical shaped objects. The cases of a large number of regularly situated circular objects were considered. It was shown that such a system is stable relating to perturbations along the flow velocity vector, and is unstable with respect to perturbations perpendicular to the vector. The considered problems illustrate the features of multiphase flow of superfluid liquid and can serve as model problems for studying such flows.

An adaptive fast direct solver for boundary integral equations in two dimensions

We describe an algorithm for the rapid direct solution of linear algebraic systems arising from the discretization of boundary integral equations of potential theory in two dimensions. The algorithm is combined with a scheme that adaptively rearranges the parameterization of the boundary in order to minimize the ranks of the off-diagonal blocks in the discretized operator, thus obviating the need for the user to supply a parameterization r of the boundary for which the distance ‖ r ( s ) − r ( t ) ‖ between two points on the boundary is related to their corresponding distance | s − t | in the parameter space. The algorithm has an asymptotic complexity of O ( N log 2 N ) , where N is the number of nodes in the discretization. The performance of the algorithm is illustrated with several numerical examples.

On solvability of boundary integral equations of potential theory for a multidimensional cusp domain

The Dirichlet and the Neumann problems for the Laplace equation on a multidimensional cusp domain are considered. The unique solvability of the boundary integral equation for the internal Dirichlet problem for harmonic double layer potential is established. We also prove the unique solvability of the boundary integral equation for the external Neumann problem for harmonic single layer potential. Bibliography: 13 titles.

Semi-analytic integration of hypersingular Galerkin BIEs for three-dimensional potential problems

An accurate and efficient semi-analytic integration technique is developed for three-dimensional hypersingular boundary integral equations of potential theory. Investigated in the context of a Galerkin approach, surface integrals are defined as limits to the boundary and linear surface elements are employed to approximate the geometry and field variables on the boundary. In the inner integration procedure, all singular and non-singular integrals over a triangular boundary element are expressed exactly as analytic formulae over the edges of the integration triangle. In the outer integration scheme, closed-form expressions are obtained for the coincident case, wherein the divergent terms are identified explicitly and are shown to cancel with corresponding terms from the edge-adjacent case. The remaining surface integrals, containing only weak singularities, are carried out successfully by use of standard numerical cubatures. Sample problems are included to illustrate the performance and validity of the proposed algorithm.

A fast spectral Galerkin method for hypersingular boundary integral equations in potential theory

This research is focused on the development of a fast spectral method to accelerate the solution of three-dimensional hypersingular boundary integral equations of potential theory. Based on a Galerkin approximation, the fast Fourier transform and local interpolation operators, the proposed method is a generalization of the precorrected-FFT technique to deal with double-layer potential kernels, hypersingular kernels and higher-order basis functions. Numerical examples utilizing piecewise linear shape functions are included to illustrate the performance of the method.

Explicit expressions for 3D boundary integrals in potential theory

AbstractOn employing isoparametric, piecewise linear shape functions over a flat triangular domain, exact expressions are derived for all surface potentials involved in the numerical solution of three‐dimensional singular and hyper‐singular boundary integral equations of potential theory. These formulae, which are valid for an arbitrary source point in space, are represented as analytic expressions over the edges of the integration triangle. They can be used to solve integral equations defined on polygonal boundaries via the collocation method or may be utilized as analytic expressions for the inner integrals in the Galerkin technique. In addition, the constant element approximation can be directly obtained with no extra effort. Sample problems solved by the collocation boundary element method for the Laplace equation are included to validate the proposed formulae. Published in 2008 by John Wiley & Sons, Ltd.

Fourier series and integral equation method for the exterior Stokes problem

AbstractThe exterior Stokes problem between two parallel planes that are separated by a prismatic cylinder is extended to the interior of the prism by requiring the continuity of the velocity across the lateral faces. The well‐posedness of the exterior–interior problem is proved in suitable weighted Sobolev spaces. The solution is represented by Fourier series in the z‐variable. The Fourier coefficients, solutions of auxiliary two‐dimensional exterior–interior problems, are analyzed by viewing them as boundary integral equations of potential theory and global regularity of the densities, is established in weighted Sobolev spaces of traces. A boundary element method, with suitably refined mesh size, is implemented for the numerical treatment of the Fourier coefficients. This provides optimal convergent semi‐ and fully‐discrete spectral methods of Fourier–Galerkin type. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Combining Kronecker Product Approximation with Discrete Wavelet Transforms to Solve Dense, Function-Related Linear Systems

A new solution technique is proposed for linear systems with large dense matrices of a certain class including those that come from typical integral equations of potential theory. This technique combines Kronecker product approximation and wavelet sparsification for the Kronecker product factors. The user is only required to supply a procedure for computation of each entry of the given matrix. The main sources of efficiency are the incomplete cross approximation procedure adapted from the mosaic-skeleton method of the second author and data-sparse preconditioners (the incomplete LU decomposition with dynamic choice of the fill-in structure with a prescribed threshold and the inverse Kronecker product preconditioner) constructed for the sum of Kronecker products of sparsified finger-like matrices computed by the discrete wavelet transform. In some model, but quite representative, examples the new technique allowed us to solve dense systems with more than 1 million unknowns in a few minutes on a personal computer with 1 Gbyte operative memory.

Numerical solution of a first kind integral equation of potential theory

A boundary integral equation reformulation of the exterior Dirichlet problem for Laplace's equation yields an integral equation of the first kind, in which the unknown is the normal derivative on the boundary of the unknown harmonic function. The integral equation is solved with a collocation method based on piecewise quadratic interpolation. The discretized linear system is solved by using iterative Krylov-subspace methods such as Conjugate Gradient-like methods. Numerical examples are presented.

Numerical quadratures for singular and hypersingular integrals

We present a procedure for the design of high-order quadrature rules for the numerical evaluation of singular and hypersingular integrals; such integrals are frequently encountered in solution of integral equations of potential theory in two dimensions. Unlike integrals of both smooth and weakly singular functions, hypersingular integrals are pseudo-differential operators, being limits of certain integrals; as a result, standard quadrature formulae fail for hypersingular integrals. On the other hand, such expressions are often encountered in mathematical physics (see, for example, [1]), and it is desirable to have simple and efficient “quadrature” formulae for them. The algorithm we present constructs high-order “quadratures” for the evaluation of hypersingular integrals. The additional advantage of the scheme is the fact that each of the quadratures it produces can be used simultaneously for the efficient evaluation of hypersingular integrals, Hilbert transforms, and integrals involving both smooth and logarithmically singular functions; this results in significantly simplified implementations. The performance of the procedure is illustrated with several numerical examples.

A New Spectral Boundary Integral Collocation Method for Three-Dimensional Potential Problems

In this work we propose and analyze a new global collocation method for classical second-kind boundary integral equations of potential theory on smooth simple closed surfaces $\Gamma \subset {\Bbb R}^3$. Under the assumption that $\Gamma$ is diffeomorphic to the unit sphere $\partial B$, the original equation is transferred to an equivalent one on $\partial B$ which is solved using collocation onto a nonstandard set of basis functions. The collocation points are situated on lines of constant latitude and longitude. The interpolation operator used in the collocation method is equivalent to a certain discrete orthogonal (pseudospectral) projection, and this equivalence allows us to establish the fundamental properties of the interpolation process and subsequently to prove that our collocation method is stable and super-algebraically convergent. In addition, we describe a fast method for computing the weakly singular collocation integrals and present some numerical experiments illustrating the use of the method. These show that at least for model problems the method attains an exponential rate of convergence and exhibits a good accuracy for very small numbers of degrees of freedom.

Effective construction of integral equations of potential theory for basic boundary problems of the theory of elasticity. Report 1

Integral identities, which can be used in direct methods of constructing resolving equations of the method of boundary elements, are obtained for basic boundary problems of the theory of elasticity. As compared with familiar methods, numerical realization of integral identities is simplified and makes it possible to employ with flexibility various functions of stresses and displacements as unknowns.

Existence and evaluation of the two free terms in the hypersingular boundary integral equation of potential theory

Hypersingular boundary integral equations (HBIE) have been studied very intensively in recent years especially because of their application for precise computations of potential gradient and stresses on the boundary. One free term which in general does not vanish for non-smooth boundary points with adjacent curved boundary parts has been omitted in previous formulations of HBIE. This paper demonstrates the presence of this new free term in 2D and 3D HBIE of potential gradient. Simple formulae for coefficients of both this new free term related to potential value and that related to potential gradient value, suitable for evaluation in the boundary element method, are derived in detail.

Preconditioned, Adaptive, Multipole-Accelerated Iterative Methods for Three-Dimensional First-Kind Integral Equations of Potential Theory

This paper presents a preconditioned, Krylov-subspace iterative algorithm, where a modified multipole algorithm with a novel adaptation scheme is used to compute the iterates for solving dense matrix problems generated by Galerkin or collocation schemes applied to three-dimensional, first-kind, integral equations that arise in potential theory. A proof is given that this adaptive algorithm reduces both matrix-vector product computation time and storage to order N, and experimental evidence is given to demonstrate that the combined preconditioned, adaptive, multipole-accelerated (PAMA) method is nearly order N in practice. Examples from engineering applications are given to demonstrate that the accelerated method is substantially faster than standard algorithms on practical problems.

Iterative Solution of Linear Systems Arising from the Boundary Integral Method

In this paper the behavior of some standard two-grid iterative schemes for the solution of linear systems arising from discretizations of second-kind boundary integral equations of potential theory is studied. When the boundary of the domain has corners, these schemes converge slowly, and in some cases even diverge. New iterative schemes are derived which always converge, and estimates for the accuracy of the solution to which they converge are given. Extensive numerical experiments are reported and discussed.

Isolation of singularities in the density and the kernel of a two-dimensional integral equation for complex boundary surfaces

We consider the solution of integral equations in problems of three-dimensional electromagnetic fields in piecewise-homogeneous media. An algorithm is proposed for analytical isolation of singularities in the kernel and the density of the integral equation of potential theory for the case of complex boundary surfaces.

Special integral equations of potential theory

Special integral equations are proposed for the Dirichlet problem on a system of open contours and the mixed boundary-value problem. For a two-dimensional elliptic differential operator of second order. It is shown that the corresponding integral operators are selfadjoint and positive relative to a suitably defined scalar product. A numerical method is proposed for solving the integral equations, based on the use of adaptive grids.

Direct and iterative methods for solving the integral equations of potential theory

Direct and iterative methods are given for solving numerically the integral equations of potential theory on realistic meshes. The methods are aimed at achieving a given accuracy, regardless of the curvature of the boundary curve.