Про один підхід до згладження негладкостей розв’язків крайових задач числовими методами квазіконформних відображень
The problem of modeling the motion of particles (charges, liquids, etc.) in a single-connected quadrangular curvilinear domain bounded by smooth two streamlines and two equipotential lines is formulated. Although if they are not «joined» at right angles and the corresponding medium is isotropic, the...
Збережено в:
| Дата: | 2021 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | Ukrainian |
| Опубліковано: |
Kamianets-Podilskyi National Ivan Ohiienko University
2021
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| Онлайн доступ: | http://mcm-tech.kpnu.edu.ua/article/view/251063 |
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| Назва журналу: | Mathematical and computer modelling. Series: Technical sciences |
Репозитарії
Mathematical and computer modelling. Series: Technical sciences| Резюме: | The problem of modeling the motion of particles (charges, liquids, etc.) in a single-connected quadrangular curvilinear domain bounded by smooth two streamlines and two equipotential lines is formulated. Although if they are not «joined» at right angles and the corresponding medium is isotropic, then, according to the quasiconformal mapping method, there will be singularities in the vicinities of exactly four points of the boundary. To avoid these singularities, an approach to approximation the boundary of the studied domain (by cubic B-splines) with the use of a special procedure of “fictitious orthogonalization” is proposed. The corresponding direct and inverse problems on quasiconformal mappings are formulated. In this, two ways for the formation of orthogonality on smooth sections of the boundary (using some «two-» and «four-point» schemes for comparison; the corresponding difference problems and algorithms for their solving are given) are proposed. An approach to estimating the accuracy of quasiconformity properties fulfilling is proposed, separately calculating the averaged orthogonality residual and the generalized residual of ratio of the lengths of adjacent segments in the small. Numerical experiments were carried out and corresponding results were analyzed. In particular, the distributions of both types of residuals and the number of nodes with singularities when different mesh partitions take place are illustrated in the graphs. As expected, «fictitious orthogonalization» with sufficiently «dense» discretization provides an opportunity to solve the problem of singularity at the points of «junction» of boundary streamlines and equipotential lines, contributes to increasing the accuracy of quasiconformal mappings and improving the «transparency» of solving process of the corresponding problem. Also, as expected, the «five-point» scheme for ensuring orthogonality on smooth boundary lines showed greater efficiency compared to the «two-point» one.
As a prospect for further application of the developed procedure of «fictitious orthogonalization», the mechanism of its adaptation is described on the example of electrical tomography problems. |
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