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Solution to the problems of the elasticity theory using -splines Решение задач теории упругости с применением -сплайнов
This article is dedicated to 7 degree S -splines of the class C4 that maintain four continuous derivatives and though remain stable. S -spline is a piecewise-polynomial function. Its coefficients are defined due to two criteria. The first part of coefficients is defined by the smoothness of the spline. The other coefficients are defined by the least-square method. At this moment we have investigated 7 rate S -splines of the class C4.The classic problem the elasticity theory is handled by solving nonhomogeneous biharmonic equation using Galerkin method, where fundamental S -splines are chosen as the system of basic functions. This approach not only provides high accuracy of solution, but also lets determine the required loads easily. It is known, that in the process of determining the loads the obtained potential (which is the solution to biharmonic equation) ought to be differentiated twice, which leads to roundoff accumulation.The methodic of S -splines constructing is given. In the paper the authors introduce the theorems of existence and uniqueness, convergence and stability for constructed S -splines. We described methodics of the problem of space discretization using S -splines. The obtained numerical solution is compared to the known analytic solution to the problem. The approximation error is 0(h8). Taking h = 0,5236, which is equal to 24 grid points, the approximation error is about 0,005. For comparison, it would take 500 first members in order to provide such an error by using a tragicomic function system as basic function of Galerkin method.Described S -splines give an opportunity to use high degree polynomials without fear of stability loss, which provides significant reduction of the grid node quantity. Besides, S -splines provide a simple solution. In order to calculate it in every point the knowledge of only two arithmetic operations is required.
Рассмотрено применение теории полулокальных сглаживающих сплайнов или S -сплайнов высоких степеней к решению задач теории упругости. S -сплайн — кусочно-полиномиальная функция, коэффициенты полиномов которой определяются из двух условий: первая часть коэффициентов определяется условиями гладкой склейки, остальные коэффициенты — методом наименьших квадратов. Мы рассмотрим, каким образом могут быть применены сплайны 7-й степени класса С4 при решении бигармонического уравнения на круге.
Solution to the problems of the elasticity theory using -splines Решение задач теории упругости с применением -сплайнов
This article is dedicated to 7 degree S -splines of the class C4 that maintain four continuous derivatives and though remain stable. S -spline is a piecewise-polynomial function. Its coefficients are defined due to two criteria. The first part of coefficients is defined by the smoothness of the spline. The other coefficients are defined by the least-square method. At this moment we have investigated 7 rate S -splines of the class C4.The classic problem the elasticity theory is handled by solving nonhomogeneous biharmonic equation using Galerkin method, where fundamental S -splines are chosen as the system of basic functions. This approach not only provides high accuracy of solution, but also lets determine the required loads easily. It is known, that in the process of determining the loads the obtained potential (which is the solution to biharmonic equation) ought to be differentiated twice, which leads to roundoff accumulation.The methodic of S -splines constructing is given. In the paper the authors introduce the theorems of existence and uniqueness, convergence and stability for constructed S -splines. We described methodics of the problem of space discretization using S -splines. The obtained numerical solution is compared to the known analytic solution to the problem. The approximation error is 0(h8). Taking h = 0,5236, which is equal to 24 grid points, the approximation error is about 0,005. For comparison, it would take 500 first members in order to provide such an error by using a tragicomic function system as basic function of Galerkin method.Described S -splines give an opportunity to use high degree polynomials without fear of stability loss, which provides significant reduction of the grid node quantity. Besides, S -splines provide a simple solution. In order to calculate it in every point the knowledge of only two arithmetic operations is required.
Рассмотрено применение теории полулокальных сглаживающих сплайнов или S -сплайнов высоких степеней к решению задач теории упругости. S -сплайн — кусочно-полиномиальная функция, коэффициенты полиномов которой определяются из двух условий: первая часть коэффициентов определяется условиями гладкой склейки, остальные коэффициенты — методом наименьших квадратов. Мы рассмотрим, каким образом могут быть применены сплайны 7-й степени класса С4 при решении бигармонического уравнения на круге.
Solution to the problems of the elasticity theory using -splines Решение задач теории упругости с применением -сплайнов
Fedosova Anastasia Nikolaevna (Autor:in) / Silaev Dmitry Alekseevich (Autor:in)
2013
Aufsatz (Zeitschrift)
Elektronische Ressource
Unbekannt
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