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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

张尚元,聂玉峰,李义强. RBF-PU方法求解二维非局部扩散问题和近场动力学问题 [J]. 应用数学和力学,2022,43(6):608-618 doi: 10.21656/1000-0887.420295
引用本文: 张尚元,聂玉峰,李义强. RBF-PU方法求解二维非局部扩散问题和近场动力学问题 [J]. 应用数学和力学,2022,43(6):608-618 doi: 10.21656/1000-0887.420295
ZHANG Shangyuan, NIE Yufeng, LI Yiqiang. The RBF-PU Method for Solving 2D Nonlocal Diffusion and Peridynamic Equations[J]. Applied Mathematics and Mechanics, 2022, 43(6): 608-618. doi: 10.21656/1000-0887.420295
Citation: ZHANG Shangyuan, NIE Yufeng, LI Yiqiang. The RBF-PU Method for Solving 2D Nonlocal Diffusion and Peridynamic Equations[J]. Applied Mathematics and Mechanics, 2022, 43(6): 608-618. 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
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

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

  • 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。

  • 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

    图  2  极坐标变换

    Figure  2.  Polar transformation

    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

    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

    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

    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

    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

    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

    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

    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$
    下载: 导出CSV

    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$
    下载: 导出CSV

    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$
    下载: 导出CSV

    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$
    下载: 导出CSV
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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

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