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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

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.

3

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.

4

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.

5

Liniowe płaty powierzchniowe Coonsa w modelowaniu wielokątnych obszarów w trójwymiarowych zagadnieniach brzegowych definiowanych równaniem Laplace'a

Zieniuk E., Szerszeń K.

Archiwum Informatyki Teoretycznej i Stosowanej

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2005

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T. 17, z. 2

127-142

PL

W pracy zastosowano liniowe płaty powierzchniowe Coonsa do modelowania trójwymiarowych obszarów w potencjalnych zagadnieniach brzegowych. Do rozwiązywania zagadnień trójwymiarowych otrzymano parametryczny układ równań całkowych (PURC). Sposób otrzymania PURC jest uogólnieniem wcześniejszego sposobu stosowanego w przypadku dwuwymiarowych zagadnień brzegowych. W PURC geometria brzegu została uwzględniona w jego formalizmie matematycznym. Do rozwiązania PURC zaproponowano metodę pseudospektralną (MP). Otrzymano rozwiązania ciągłe na poszczególnych płatach powierzchni brzegu. Zamieszczone przykłady testujące potwierdzają wysoką dokładność i efektywność metody.

EN

This paper presents a method of modeling three-dimensional boundaries in potential boundary value problems using linear Coons surface patches. The boundary definition was connected with the parametric integral equation system (PIES). The PIES are used for numerical solving 3D boundary value problems and was obtained by generalization of the method previous used for 2D boundary problems. The boundary geometry described by Coons patches in the PIES is considered in its mathematical formalism. The pseudospectral method was proposed for solving the PIES that produces continuous solutions on the whole boundary. The efficiency and performance of proposed algorithm was discussed on numerical examples.