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Abstract

A lot of methods for solving boundary value problems using arbitrary grids, such as SDI (scattered data interpolation) and SPH (smoothed particle hydrodynamics), use families of atomic radial basis functions that depend on parameters to improve the accuracy of calculations. Functions of this kind are commonly called "shape functions". When polynomials or polynomial splines are used as such functions, they are called "basis functions". The term "radial" means that the carrier of the function is a disk or layer. The term "atomic" means that the support of the function is limited, ie the function is finite. In most cases, the term "finite" is used in English-language publications. The article presents an algorithm for constructing such a function, which is the solution of the functional-differential equation where - circle of radius r: , and . The function generated by this equation has two parameters: r and . Variation of these parameters allows to reduce the error in the calculations of the Poisson boundary value problem by several times. The theorem on the existence of such an unambiguous function is proved in the article. The proof of the theorem allows us to construct one-dimensional Fourier transform of this function in the form , where . Previously, function was calculated using its Taylor approximation (at ), and at – using the asymptotic Hankel approximation of the function . Thus in a circle of a point a fairly large error was found. Therefore, the calculation of the function in the range was carried out by Chebyshev approximation of this function in the range . Chebyshev coefficients (calculated in the Maple 18 system with an accuracy of 26 decimal digits) and the range were chosen by an experiment aimed at minimizing the overall error in calculating of the function . Thanks to the use of the Chebyshev approximation, the obtained function has more than twice less error than calculated by the previous algorithm. Arbitrary value of the function is calculated using a six-point Aitken scheme, which can be considered (to some extent) a smoothing filter. The use of Aitken's six-point scheme introduces an error equal to 6% of the total function calculation error , but helps to save a lot of time in the formation of ARBF in solving boundary value problems using the method of collocation.
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一类原子径向基函数的构造的若干特征
许多使用任意网格求解边值问题的方法,如SDI(散射数据插值)和SPH(光滑粒子流体动力学),都使用依赖于参数的原子径向基函数族来提高计算精度。这类函数通常称为“形状函数”。当多项式或多项式样条被用作这样的函数时,它们被称为“基函数”。术语“径向”意味着函数的载体是一个圆盘或层。术语“原子”意味着函数的支持是有限的,即函数是有限的。在大多数情况下,英语出版物中使用术语“finite”。本文给出了构造这样一个函数的算法,该函数是函数微分方程的解,其中-半径为r的圆,和。由这个方程生成的函数有两个参数:r和。这些参数的变化可以使泊松边值问题的计算误差减小几倍。本文证明了这种无二义函数的存在性定理。这个定理的证明允许我们构造这个函数的一维傅里叶变换的形式,其中。以前,函数是使用它的泰勒近似(at)计算的,而at -使用函数的渐近汉克尔近似。这样,在一个点的圆上发现了一个相当大的误差。因此,在值域内的函数的计算是通过对该函数在值域内的切比雪夫近似进行的。切比雪夫系数(在Maple 18系统中计算,精度为26位十进制数字)和范围是通过实验选择的,目的是使计算函数的总体误差最小。由于使用了切比雪夫近似,得到的函数的误差比以前算法计算的误差小两倍以上。函数的任意值使用六点艾特肯方案计算,该方案可以被认为(在某种程度上)是一个平滑滤波器。使用Aitken的六点格式引入了相当于总函数计算误差6%的误差,但在使用配点法求解边值问题时,有助于节省大量ARBF的形成时间。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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