RBF-PU方法求解二维非局部扩散问题和近场动力学问题

张尚元; 聂玉峰; 李义强

doi:10.21656/1000-0887.420295

RBF-PU方法求解二维非局部扩散问题和近场动力学问题

doi: 10.21656/1000-0887.420295

西北工业大学数学与统计学院，西安 710129

基金项目: 国家自然科学基金(面上项目)(11971386)；国家重点研发计划(2020YFA0713603)

详细信息

作者简介:
张尚元(1995—)，男，博士生(E-mail：zhangshangyuan@mail.nwpu.edu.cn)

聂玉峰(1968—)，男，教授，博士，博士生导师(通讯作者. E-mail：yfnie@nwpu.edu.cn)

李义强(1984—)，男，讲师，博士(E-mail：liyiqiang@nwpu.edu.cn)

中图分类号: O242.2
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出版历程
- 收稿日期: 2021-09-26
- 修回日期: 2021-10-29
- 网络出版日期: 2022-05-24
- 刊出日期: 2022-06-30

The RBF-PU Method for Solving 2D Nonlocal Diffusion and Peridynamic Equations

School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129, P.R.China

摘要

摘要:
采用单位分解径向基函数(radial basis function partition of unity，RBF-PU)方法，数值求解了二维非局部扩散问题和近场动力学问题。主要思想是对求解区域进行局部划分，在局部子区域上分别进行函数逼近，然后加权得到未知函数的全局逼近。这种基于方程强形式的径向基函数方法在求解非局部问题时，不需要处理网格与球形邻域求交的问题，避免了额外的一层积分计算，实施简便，计算量小。数值实验显示计算结果与解析解吻合较好，RBF-PU方法可以准确有效地求解非局部扩散方程和近场动力学方程。
- 径向基函数插值 /
- 单位分解 /
- 非局部扩散 /
- 近场动力学
Abstract:
The radial basis function partition of unity (RBF-PU) method was applied to obtain the numerical solution of 2D nonlocal diffusion and peridynamic problems. The main idea is to partition the original domain into several patches, use the RBF approximation on each local domain, and then give weighting to obtain the global approximation of the unknown function. The radial basis function method based on the strong form of the equation has many advantages, such as avoiding an additional layer of integral calculation, no need to deal with intersections of neighborhoods with the mesh, and easiness of implementation. The numerical results show that, this method can solve nonlocal diffusion equations and peridynamic equations accurately and efficiently.
- RBF interpolation /
- partition of unity /
- nonlocal diffusion /
- peridynamics

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图 1 无网格点和PU覆盖示意图

Figure 1. A set of regular meshless points and a set of circular PU patches

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图 2 极坐标变换

Figure 2. Polar transformation

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图 3 均匀离散无网格点的分布和非局部扩散方程矩阵结构：(a) 无网格点分布和单位分解划分； (b) 矩阵结构

Figure 3. The distribution of points and the matrix structure of nonlocal diffusion under uniform discretization: (a) the distribution of meshless points and PU patches;(b) the matrix structure

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图 4 Halton离散下点的分布和非局部扩散方程矩阵结构：(a) Halton 点的分布和单位分解划分；(b) 矩阵结构

Figure 4. The distribution of points and the matrix structure of nonlocal diffusion under Halton discretization: (a) the distribution of Halton points and circular PU patches； (b) the matrix structure

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图 5 均匀离散下非局部扩散方程数值解和误差分布：(a) 数值解； (b) 误差分布

Figure 5. The numerical solution and the error distribution for the nonlocal diffusion equation under uniform discretization: (a) the numerical solution; (b) the error distribution

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图 6 Halton离散下非局部扩散方程的数值结果和误差分布图：(a) 数值解； (b) 误差分布

注　为了解释图中的颜色，读者可以参考本文的电子网页版本，后同。

Figure 6. The numerical solution and the error distribution for the nonlocal diffusion equation under Halton discretization: (a) the numerical solution; (b) the error distribution

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图 7 均匀离散下无网格点的分布和近场动力学方程矩阵结构：(a) 无网格点分布和单位分解划分； (b) 矩阵结构

Figure 7. The distribution of points and the matrix structure for the peridynamic equation under uniform distribution: (a) the distribution of meshless points and circular PU patches; (b) the matrix structure

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图 8 近场动力学方程$x$方向数值解和误差分布：(a) 数值解； (b) 误差分布

Figure 8. The numerical solution and the error distribution of the peridynamic equation in the $x$ direction: (a) the numerical solution; (b) the error distribution

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图 9 近场动力学方程$y$方向数值解和误差分布：(a) 数值解； (b) 误差分布

Figure 9. The numerical solution and the error distribution of the peridynamic equation in the $y$ direction: (a) the numerical solution; (b) the error distribution

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表 1 均匀离散非局部扩散方程数值结果($\delta = 0.2$)

Table 1. Numerical results of the nonlocal diffusion equation under uniform discretization ($\delta = 0.2$)

h	${\varepsilon _{\max } }$	${\varepsilon _{ {\rm{RMSE} } } }$	${\varepsilon _{ {\rm{RE} } } }$	$N$	$t/{\rm{s}}$
$1/10$	$2.392\;362{\rm{E}}-2$	$5.340\;460{\rm{E}}-3$	$1.596\;423{\rm{E}}-1$	$5.103\;764{\rm{E}}+2$	$0.46$
$1/20$	$3.503\;622{\rm{E}}-4$	$1.012\;859{\rm{E}}-4$	$3.298\;962{\rm{E}}-3$	$2.028\;138{\rm{E}}+7$	$3.43$
$1/30$	$8.066\;769{\rm{E}}-6$	$2.375\;277{\rm{E}}-6$	$7.919\;804{\rm{E}}-5$	$8.778\;054{\rm{E}}+10$	$50.01$
$1/40$	$4.408\;203{\rm{E}}-6$	$9.530\;515{\rm{E}}-7$	$3.211\;906{\rm{E}}-5$	$5.302\;566{\rm{E}}+10$	$172.58$
$1/50$	$4.405\;356{\rm{E}}-6$	$1.227\;837{\rm{E}}-6$	$4.163\;255{\rm{E}}-5$	$2.130\;636{\rm{E}}+11$	$605.16$
$1/60$	$3.365\;385{\rm{E}}-5$	$6.696\;457{\rm{E}}-6$	$2.279\;511{\rm{E}}-4$	$4.492\;236{\rm{E}}+11$	$1\;481.48$

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表 2 离散非局部扩散方程的数值结果($\delta = 0.2$)

Table 2. Numerical results of the nonlocal diffusion equation under Halton discretization ($\delta = 0.2$)

$h$	${\varepsilon _{\max } }$	${\varepsilon _{ {\rm{RMSE} } } }$	${\varepsilon _{ {\rm{RE} } } }$	$N$	$t/{\rm{s}}$
$1/10$	$2.679\;438{\rm{E}}-2$	$6.772\;052{\rm{E}}-3$	$2.370\;566{\rm{E}}-1$	$3.206\;995{\rm{E}}+4$	$0.52$
$1/20$	$7.466\;535{\rm{E}}-4$	$1.795\;671{\rm{E}}-4$	$6.252\;326{\rm{E}}-3$	$1.842\;649{\rm{E}}+8$	$5.49$
$1/30$	$5.920\;469{\rm{E}}-6$	$1.175\;443{\rm{E}}-6$	$4.070\;005{\rm{E}}-5$	$1.805\;258{\rm{E}}+11$	$64.71$
$1/40$	$1.246\;481{\rm{E}}-5$	$2.040\;146{\rm{E}}-6$	$7.077\;887{\rm{E}}-5$	$8.713\;394{\rm{E}}+11$	$173.02$
$1/50$	$3.738\;273{\rm{E}}-6$	$1.029\;311{\rm{E}}-6$	$3.570\;744{\rm{E}}-5$	$2.862\;032{\rm{E}}+11$	$599.20$
$1/60$	$1.798\;596{\rm{E}}-6$	$4.251\;399{\rm{E}}-7$	$1.472\;884{\rm{E}}-5$	$7.453\;951{\rm{E}}+10$	$1\;511.76$

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表 3 模型的$x$方向位移的数值解$u_{1}$($\delta = 0.2$)

Table 3. Numerical results of displacement in the $x$ direction for the peridynamic model $u_{1}$ ($\delta = 0.2$)

h	${\varepsilon _{\max } }$	${\varepsilon _{ {\rm{RMSE} } } }$	${\varepsilon _{ {\rm{RE} } } }$	$N$	$t/{\rm{s}}$
$1/10$	$9.242\;8{\rm{E}}-1$	$3.507\;7{\rm{E}}-1$	$1.048\;5{\rm{E}}+1$	$3.56{\rm{E}}+4$	$0.74$
$1/20$	$3.084\;2{\rm{E}}-2$	$6.767\;5{\rm{E}}-3$	$2.204\;2{\rm{E}}-1$	$1.00{\rm{E}}+6$	$3.79$
$1/30$	$4.183\;5{\rm{E}}-4$	$7.338\;0{\rm{E}}-5$	$2.446\;7{\rm{E}}-3$	$1.89{\rm{E}}+10$	$34.05$
$1/40$	$3.455\;7{\rm{E}}-4$	$5.556\;9{\rm{E}}-5$	$1.872\;7{\rm{E}}-3$	$5.38{\rm{E}}+11$	$75.71$
$1/50$	$3.072\;6{\rm{E}}-5$	$5.480\;6{\rm{E}}-6$	$1.858\;3{\rm{E}}-4$	$8.93{\rm{E}}+10$	$188.79$

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表 4 模型的$y$方向位移数值解$u_{2}$ ($\delta = 0.2$)

Table 4. Numerical results of displacement in the $y$ direction for the peridynamic model $u_{2}$ ($\delta = 0.2$)

h	${\xi _{\max } }$	${\xi _{ {\rm{RMSE} } } }$	${\xi _{ {\rm{RE} } } }$	$N$	$t/{\rm{s}}$
$1/10$	$2.674\;1{\rm{E}}-1$	$7.270\;6{\rm{E}}-2$	$1.551\;5{\rm{E}}-1$	$3.56{\rm{E}}+4$	$0.74$
$1/20$	$3.168\;0{\rm{E}}-2$	$7.372\;5{\rm{E}}-3$	$1.583\;9{\rm{E}}-2$	$1.00{\rm{E}}+6$	$3.79$
$1/30$	$3.069\;2{\rm{E}}-4$	$5.738\;3{\rm{E}}-5$	$1.235\;5{\rm{E}}-4$	$1.89{\rm{E}}+10$	$34.05$
$1/40$	$1.803\;9{\rm{E}}-4$	$3.851\;6{\rm{E}}-5$	$8.302\;3{\rm{E}}-5$	$5.38{\rm{E}}+11$	$75.71$
$1/50$	$3.037\;0{\rm{E}}-5$	$6.134\;8{\rm{E}}-6$	$1.323\;2{\rm{E}}-5$	$8.93{\rm{E}}+10$	$188.79$

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