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1

On evaluation of influence coefficients for edge and intermediate boundary elements in 3D problems involving strong field concentrations

Rybarska-Rusinek L.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2019

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Vol. 67, nr 1

69--76

EN

The paper presents a tool for accurate evaluation of high field concentrations near singular lines, such as contours of cracks, notches and grains intersections, in 3D problems solved the BEM. Two types of boundary elements, accounting for singularities, are considered: (i) edge elements, which adjoin a singular line, and (ii) intermediate elements, which while not adjoining the line, are still under strong influence of the singularity. An efficient method to evaluate the influence coefficients and the field intensity factors is suggested for the both types of the elements. The method avoids time expensive numerical evaluation of singular and hypersingular integrals over the element surface by reduction to 1D integrals. The method being general, its details are explained by considering a representative examples for elasticity problems for a piecewise homogeneous medium with cracks, inclusions and pores. Numerical examples for plane elements illustrate the exposition. The method can be extended for curvilinear elements.

2

Heat conduction in anisotropic medium with perfectly conductive thread-like inclusions

Sulym Heorhiy, Ilchuk Nataliia, Pasternak Iaroslav

Acta Mechanica et Automatica

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2019

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Vol. 13, no. 4

251--254

EN

The paper presents a novel approach for the analysis of steady-state heat conduction of solids containing perfectly conductive thread-like inhomogeneities. Modelling of a thread-like heat conductive inhomogeneity is reduced to determination of density of heat distributed along a spatial curve, which replaces the inclusion. Corresponding boundary integral equations are obtained for anisotropic solids with thread-like inclusions. Non-integral terms are computed in a closed form. It is shown that, nevertheless the singularity of the equation is 1/r, it is hypersingular, since the kernel is symmetric. Boundary element approach is adopted for solution of the obtained equations. Numerical example is presented for a rectilinear conductive thread, which verifies derived boundary integral equations.

3

Supplementing the numerical solution of singular/hypersingular integral equations/inequalities with parametric inequality constraints with applications to crack problems

Ioakimidis N. I.

Computer Assisted Methods in Engineering and Science

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2017

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Vol. 24, no. 1

41--65

EN

Singular and hypersingular integral equations appear frequently in engineering problems. The approximate solution of these equations by using various numerical methods is well known. Here we consider the case where these equations are supplemented by inequality constraints-mainly parametric in equality constraints, but also the case of singular/hypersingular integral inequalities. The approach used here is simply to employ the computational method of quantifier elimination efficiently implemented in the computer algebra system Mathematica and derive the related set of necessary and sufficient conditions for the validity of the singular/hypersingular integral equation/inequality together with the related in equality constraints. The present approach is applied to singular integral equations/inequalities in the problem of periodic arrays of straight cracks under loading- and fracture-related inequality constraints by using the Lobatto-Chebyshev method. It is also applied to the hypersingular integral equation/inequality of the problem of a single straight crack under a parametric loading by using the collocation and Galerkin methods and parametric inequality constraints.

4

On the numerical solution of the initial-boundary value problem with neumann condition for the wave equation by the use of the laguerre transform and boundary elements method

Litynskyy S., Muzychuk Y., Muzychuk A.

Acta Mechanica et Automatica

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2016

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Vol. 10, no. 4

285--290

EN

We consider a numerical solution of the initial-boundary value problem for the homogeneous wave equation with the Neumann condition using the retarded double layer potential. For solving an equivalent time-dependent integral equation we combine the Laguerre transform (LT) in the time domain with the boundary elements method. After LT we obtain a sequence of boundary integral equations with the same integral operator and functions in right-hand side which are determined recurrently. An error analysis for the numerical solution in accordance with the parameter of boundary discretization is performed. The proposed approach is demonstrated on the numerical solution of the model problem in unbounded three-dimensional spatial domain.

5

Solution of 2D hyperbolic equation by means of the BEM using discretization in time

Drozdek J., Lupa M., Ładyga E.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2009

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Vol. 8, nr 1

19-24

EN

The hyperbolic equation (2D problem) supplemented by adequate boundary and initial conditions is considered. To solve the problem the boundary element method using discretization in time is adapted. In the final part of the paper the example of computations is shown.

6

A boundary integral equation for the 2d external potential flow

Nasser M., Murid A., Amin N.

International Journal of Applied Mechanics and Engineering

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2006

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Vol. 11, no 1

61-75

EN

Based on the recently discovered second kind Fredholm integral equation for the exterior Riemann problem, a boundary integral equation is developed in this paper for the two-dimensional, irrotational, incompressible fluid flow around an airfoil without a cusped trailing edge. The solution of the integral equation contains one arbitrary real constant, which may be determined by imposing the Kutta-Joukowski condition. Comparisons between numerical and analytical values of the pressure coefficient on the surface of the NACA 0009 and NACA 0012 airfoils with zero angle of attack show a very good agreement.

7

A solution of 3D Helmholtz equation for boundary geometry modeled by Coons patches using the Parametric Integral Equation System

Zieniuk E., Szerszeń K.

Archives of Acoustics

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2006

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Vol. 31, No. 1

99-111

EN

In this paper, the authors propose an algorithm for numerical solution of the 3D Helmholtz equation using the Parametric Integral Equation System (PIES). The PIES, unlike the traditional Boundary Integral Equation (BIE), is characterized by the fact that the boundary geometry has been considered in its mathematical formalism. Polygonal Coons surfaces have been used to describe the 3D domain. This makes it possible to obtain continuous solutions without any discretization of the 3D domain.

8

A regularized Newton method in electrical impedance tomography using shape Hessian information

Eppler K., Harbrecht H.

Control and Cybernetics

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2005

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Vol. 34, no 1

203-225

EN

The present paper is concerned with the identification of an obstacle or void of different conductivity included in a two-dimensional domain by measurements of voltage and currents at the boundary. We employ a reformulation of the given identification problem as a shape optimization problem as proposed by Roche and Sokolowski (1996). It turns out that the shape Hessian degenerates at the given hole which gives a further hint on the ill-posedness of the problem. For numerical methods, we propose a preprocessing for detecting the barycentre and a crude approximation of the void or hole. Then, we resolve the shape of the hole by a regularized Newton method.

9

Integrals of Lipschitz-Hankel type in analysis of magnetostatic fields

Pawluk K.

Journal of Technical Physics

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2003

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Vol. 44, no 2

133-144

EN

A boundary-integral model of the static magnetic field due to cylindrical permanent magnets that is put in free space is considered. Magnetic scalar potential quantities created by a virtual quantity "surface magnetic charge density" is expressed by means of Lipschitz-Hankel integrals that for the considered case are reducible (by the way of hypergeometric series) to some algebraic expressions, in which elliptic integrals of various kinds occur. This approach seems to be more effective than that can be reached by the use of a typical professional software for the field problems in which numerical integration, being not quite conform to the considered case, is common. The magnet subjected to the analysis has to be virtually subdivided in some number of elementary pieces, inside of which the uniform distribution of the inherent magnetization is supposed.

10

nu-spline curves for boundary geometry modeling in two-dimensional potential boundary value problems with singular corner points

Zieniuk E.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2002

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Vol. 6, No 4

641-650

EN

The paper presents a new modeling method of boundary geometry in boundary value-problems by nu-spline curves. To define a smooth boundary geometry both Bezier and B-spline curves are applied. At the segment join points Bezier curves ensure continuity C1, and B-spline curves allow us to maintain continuity C2. However, the curves hinder boundary modeling with corner points. In order to weaken the continuity at segment join points nu-spline curves are proposed. These curves are combined analytically with the Green formula, thus yielding the Parametric Integral Equation System (PIES). To solve the PIES a pseudospectral method is used. The results obtained for the domains with singular corner points are compared with the corresponding non-singular ones as defined by the nu-spline curves.

11

Magneto-static field distribution due to inherent magnetisation of permanent magnets

Pawluk K.

Journal of Technical Physics

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2002

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Vol. 43, no 4

443-452

EN

A boundary-integral model of the static magnetic field due to cylindrical permanent magnets that is put in free space is considered. Magnetic scalar potential quantities created by a virtual quantity "surface magnetic charge density" is expressed by means of Lipschitz-Hankel integrals that for the considered case are reducible (by the way of hypergeometric series) to some algebraic expressions, in which elliptic integrals of various kinds occur. This approach seems to be more effective than that can be reached by the use of a typical professional software for the field problems in which numerical integration, being not quite conform to the considered case, is common. The magnet subjected to the analysis has to be virtually subdivided in some number of elementary pieces, inside of which the uniform distribution of the inherent magnetization is supposed.

12

Optimal Shape Design for Elliptic Equations Via Bie-Methods

Eppler K.

International Journal of Applied Mathematics and Computer Science

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2000

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Vol. 10, no 3

487-516

EN

A special description of the boundary variation in a shape optimization problem is investigated. This, together with the use of a potential theory for the state, result in natural embedding of the problem in a Banach space. Therefore, standard differential calculus can be applied in order to prove the Frechet-differentiability of the cost function for appropriately chosen data (sufficiently smooth). Moreover, necessary optimality conditions are obtained in a similar way as in other approaches, and are expressed in terms of an adjoint state for more regular data.

13

The technique of numerical solution of 2-D direst current modelling problem in inhomogeneous media

Sapuzhak Y., Zhuravchak L.

Acta Geophysica Polonica

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1999

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Vol. 47, nr 2

149-163

EN

Effective numerical technique for solving the direct problem ofd electrical prospecting for homogenous media with inclusions of arbitrary shape and constant or changenable electrical characteristics is suggested. It is based on a simultaneous application of fundamental solution of Laplace's equation for halfplane and principal ideas of the boundary integral equation method and the methods of simple layer potential and collocation, as well as finite-difference correlations inlocal inhomogenous regions with changeable characteristics. Numerical solutions for some mathematical models illustrating the high accurancy and potential abilities of the technique suggested have been considered.