DIRECT COLLOCATION METHOD AND STABILIZED COLLOCATION METHOD BASED ON LAGRANGE INTERPOLATION FUNCTION
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摘要: 由于无网格法中大多数近似函数均为有理式, 不具有Kronecker delta性质, 因此难以精确地施加本质边界条件. 边界误差较大容易导致整个求解域求解结果精度低, 甚至引起数值不稳定现象. 文章在无网格直接配点法和稳定配点法中引入拉格朗日插值函数作为形函数, 构建了拉格朗日插值配点法(LICM)和拉格朗日插值稳定配点法(SLICM). 由于拉格朗日插值具有Kronecker delta性质, 可以像有限元法一样简单而精确地施加本质边界条件, 提高这两种方法的数值求解精度. 稳定配点法基于子域对强形式方程进行积分, 可以满足高阶积分约束, 即可以保证形函数在积分形式下也满足高阶一致性条件, 实现精确积分. 同时, 进行子域积分还可以减少离散矩阵的条件数, 从而提高算法的稳定性. 进一步提高拉格朗日插值稳定配点法的精度和稳定性. 通过数值算例验证这两种方法的精度、收敛性和稳定性, 结果表明基于拉格朗日插值的配点法的精度优于基于重构核近似的配点法, 拉格朗日插值稳定配点法的精度和稳定性均优于拉格朗日插值配点法.
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关键词:
- 拉格朗日插值配点法 /
- 拉格朗日插值稳定配点法 /
- Kronecker delta性质 /
- 精确积分 /
- 精度 /
- 稳定性
Abstract: Since most of the approximation functions in the meshfree method are rational and do not have the Kronecker delta property, it is difficult to accurately impose the essential boundary conditions. Large errors on the boundary can easily lead to low accuracy of the solution in the whole solution domain and may even introduce the numerical instability in solution process. In this paper, the Lagrange interpolation function is introduced as the shape function in the meshfree direct collocation method and the stabilized collocation method, and the Lagrange interpolation collocation method (LICM) and the stabilized Lagrange interpolation collocation method (SLICM) are constructed. Since Lagrange interpolation has the Kronecker delta property, the essential boundary conditions can be imposed as simply and precisely as the finite element method, which promotes the numerical solution accuracy of the two methods. The stabilized collocation method is based on the subdomain integration, which can satisfy the high order integration constraints. That is, it can ensure that the shape function also meets the high-order consistency conditions in the integral form and achieve accurate integration. At the same time, the subdomain integration can also reduce the condition number of the discrete matrix, which improves the stability of the algorithm. By combining the Lagrange interpolation function and the stabilized collocation method, the accuracy and stability of the stabilized Lagrange interpolation collocation method is further improved. Numerical examples validate the accuracy, convergence and stability of the proposed Lagrange interpolation collocation method (LICM) and the stabilized Lagrange interpolation collocation method (SLICM). The results show that the accuracy of the collocation methods based on the Lagrange interpolation function is higher than that of the collocation method based on the reproducing kernel function, and the accuracy and stability of the stabilized Lagrange interpolation collocation method are superior to those of the Lagrange interpolation collocation method. -
无网格法[1-6]由于不需要划分网格, 不存在网格类方法在求解大变形问题时容易出现的网格畸变问题, 而且具有精度高、收敛率高等优点, 近年来受到越来越多的关注, 广泛应用于高速冲击、爆炸等复杂问题. 常用的无网格法主要分为两类: 伽辽金型和配点型. 由于进行区域积分, 伽辽金型无网格法具有较好的精度和稳定性, 然而其缺点是计算效率比较低. 配点型无网格法通常更加简单高效[7-11], 而且在一些简单问题中可以获得较好精度[12], 但是对于一些复杂问题, 其精度和稳定性明显降低. Zhang等[13-14]提出最小二乘配点法(least squares collocation method, LSCM), 通过采用比源点个数更多的配点来构建超定离散方程进行计算, 显著提升计算精度和稳定性, 然而超定方程求解也大幅降低计算效率. 分区配点法[15-17]通过将区域分成若干个子域, 提高了计算效率, 而且降低了离散矩阵的条件数, 提高了结果的稳定性, 但是这种方法也是基于最小二乘求解, 无法避免超定方程计算的缺点. 最近, Wang等[18-20]提出一种基于重构核近似(reproducing kernel, RK)的稳定配点法(stabilized collocation method, SCM), 这种方法将问题域划分为若干个规则子域, 对强形式方程在子域内进行积分, 由于满足积分约束, 可以实现精确积分. 该方法提高了无网格法计算结果的精度和稳定性. 由于积分效率高, 该方法仍然保持配点型无网格法的高效特性.
常用的无网格法采用的形函数通常都是有理式, 不具有插值特性, 难以像有限元法(finite element method, FEM)一样方便准确地施加本质边界条件. 这个问题也是目前无网格法的研究热点之一. 在无网格方法中常见的处理本质边界条件的方法有拉格朗日乘子法[21]、罚函数法[22]和一些修正的方法[23-24]. 然而拉格朗日乘子法会增加未知数的数量, 导致刚度矩阵不对称, 从而增加了计算难度. 罚函数法虽可得到对称的刚度矩阵, 但其计算精度往往取决于罚参数的选取. 另一些学者努力尝试将有限元法与无网格法相结合来施加边界条件. 基于强形式配点法和有限元的单元, Gao等[25]提出单元微分法(element differential method, EDM), 之后改进等参单元的构建形式, 可以由配点和相邻节点构建, 提出自由单元配点法(free element collocation method, FECM)[26-27]. 在此基础上进一步弱化单元, 提出有限线法(finite line method, FLM)[28], 这种方法在求解低维和高维问题时均只需要在一个方向上构建单元, 降低了单元畸变的可能性. 这几种方法都是基于直接配点法, 求解比较简便, 虽然可以利用有限元单元的优势方便地施加本质边界条件, 但是没能避免直接配点法的缺点和完全消除有限元单元畸变的可能性.
为了结合无网格法无单元畸变的优势和有限元法能方便施加本质边界条件的特点, Wang等[29]在稳定配点法的基础上提出一种基于拉格朗日插值的稳定配点法, 称为拉格朗日插值稳定配点法(stabilized Lagrange interpolation collocation method, SLICM). 这种方法继承了稳定配点法精度高和稳定性好的优势, 能够实现精确积分, 同时由于拉格朗日插值形函数具有Kronecker delta性质, 可以像有限元法一样简便准确地直接施加本质边界条件. Wang等[29]考虑了均匀离散点和对应结构化网格的非均匀离散点的离散布置方案, 并没有详细讨论离散点任意布置的情形. 本文通过引入曲线拉格朗日插值形函数, 实现了基于拉格朗日插值的直接配点法和稳定配点法的任意离散, 进一步提升了这两种方法的适用范围.
1. 拉格朗日插值近似
1.1 拉格朗日插值形函数
将一个一维区域离散为若干个离散点, 基于其中部分离散点${x_1},{x_2}, \cdots ,{x_i}, \cdots ,{x_m}$(这一组离散点也可称为节点), 一维拉格朗日插值多项式可表示为
$$N_I(x)=\prod_{\substack{i=1 \\ i \neq I}}^m \frac{x-x_i}{x_I-x_i}= \frac{x-x_1}{x_I-x_1} \cdots \frac{x-x_{i-1}}{x_I-x_{i-1}} \frac{x-x_{i+1}}{x_I-x_{i+1}} \cdots \frac{x-x_m}{x_I-x_m}, \quad I=1,2,\cdots, m, \quad-1 \leqslant x \leqslant 1 $$ (1) 其中$m$表示离散点个数. 对于所有的$i \ne I$, $N_I^{}\left( {{x_I}} \right)$在$x = {x_i}$处结果为0, 当$x = {x_I}$时, $N_I^{}\left( {{x_I}} \right) = 1$. 因此, 拉格朗日插值具有Kronecker delta性质, 可以表示为以下形式
$$ {N_I}\left( {{x_J}} \right){\text{ = }}{\delta _{IJ}} = \left\{\begin{split} & {0,\quad I \ne J} \\ & {1,\quad \;\;I{\text{ = }}J} \end{split} \right. $$ (2) 图1和图2展示了一维2节点和3节点拉格朗日插值形函数中节点分布情况. 二维和三维形函数可以通过一维形函数在不同方向上的张量积表示为如下形式
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图 1 一维2节点和3节点拉格朗日插值形函数的节点分布图Figure 1. Node distributions of the 1D 2-node and 3-node Lagrange interpolation shape functions![]()
图 2 二维4节点、6节点和9节点拉格朗日插值形函数的节点分布图Figure 2. Node distributions of the 2D 4-node, 6-node and 9-node Lagrange interpolation shape functions$$ {\varphi _L}\left( {\boldsymbol{x}} \right) = {y_I}{N_I}\left( x \right){x_J}{N_J}\left( y \right),\;\; I = 1,2,\cdots, {m_1},\;\; J = 1,2,\cdots, {m_2} $$ (3) $$ \left.\begin{split} &{\varphi _L}\left( {\boldsymbol{x}} \right) = {y_I}{z_I}{N_I}\left( x \right){x_J}{z_J}{N_J}\left( y \right){x_K}{y_K}{N_K}\left( z \right)\\ &I = 1,2,\cdots, {m_1},\quad J = 1,2,\cdots, {m_2},\quad K = 1,2,\cdots, {m_3}\end{split}\right\} $$ (4) 对于二维问题${\boldsymbol{x}} = \left( {x,y} \right)$, 三维问题${\boldsymbol{x}} = \left( {x,y,z} \right)$, 下标$L$由下标$I$, $J$和$K$的顺序排列确定. 二维情况下4节点、6节点和9节点拉格朗日形函数中节点分布如图2所示.
图3 ~ 图5展示了不同阶数下的拉格朗日插值形函数及其导数, 其中$p$表示拉格朗日插值形函数的阶数. 对于一维问题$p = m - 1$, 二维问题$p = \min \left\{ {\left( {{m_1} - 1} \right),\left( {{m_2} - 1} \right)} \right\}$, 三维问题$p = \min \left\{ \left( {{m_1} - 1} \right), \left( {m_2} - 1 \right), \left( {{m_3} - 1} \right) \right\}$.
1.2 拉格朗日插值形函数的导数
一维拉格朗日插值形函数的一阶导数和二阶导数可以表示为以下形式
$$ N_{I, x}(x)=\prod_{\substack{i=1 \\ i \neq I}}^m \frac{1}{x_I-x_i} \sum_{j=1}^m \prod_{\substack{i=1 \\ i \neq j, I}}^m x-x_i=\frac{\left[\left(x-x_1\right)\left(x-x_2\right) \cdots\left(x-x_{m-1}\right)+\cdots+\left(x-x_2\right)\left(x-x_3\right) \cdots\left(x-x_m\right)\right]}{[\left(x_I-x_1\right) \cdots\left(x_I-x_{i-1}\right)\left(x_I-x_{i+1}\right) \cdots\left(x_I-x_m\right)]}, I=1,2,\cdots, m $$ (5) $$\left.\begin{split} &N_{I, x x}(x)=\prod_{\substack{i=1 \\ i \neq I}}^m \frac{1}{x_I-x_i} \sum_{j=1}^m \sum_{\substack{k=1 \\ k \neq j}}^m \prod_{\substack{i=1 \\ i \neq j, k, I}}^m x-x_i=\\ &\qquad \frac{\left[\left(x-x_1\right) \cdots\left(x-x_{m-2}\right)+\cdots+\left(x-x_3\right) \cdots\left(x-x_m\right)\right]}{[\left(x_I-x_1\right) \cdots\left(x_I-x_{i-1}\right)\left(x_I-x_{i+1}\right) \cdots\left(x_I-x_m\right)]}, \\ &\qquad I=1,2,\cdots, m \end{split}\right\} $$ (6) 二维形函数的一阶和二阶导数也可以通过一维形状函数的张量积得到, 可以表示为
$$ \left.\begin{split} & {\varphi _{L,x}}\left( {\boldsymbol{x}} \right) = {y_I}{N_{I,x}}\left( x \right){x_J}{N_J}\left( y \right)\\ &{\varphi _{L,y}}\left( {\boldsymbol{x}} \right) = {y_I}{N_I}\left( x \right){x_J}{N_{J,y}}\left( y \right) \\ & {\varphi _{L,xx}}\left( {\boldsymbol{x}} \right) = {y_I}{N_{I,xx}}\left( x \right){x_J}{N_J}\left( y \right)\\ &{\varphi _{L,yy}}\left( {\boldsymbol{x}} \right) = {y_I}{N_I}\left( x \right){x_J}{N_{J,yy}}\left( y \right) \\ & {\varphi _{L,xy}}\left( {\boldsymbol{x}} \right) = {y_I}{N_{I,x}}\left( x \right){x_J}{N_{J,y}}\left( y \right)\\ &\qquad I = 1,2,\cdots, {m_1},\quad J = 1,2,\cdots, {m_2} \end{split}\right\} $$ (7) 同理, 三维形函数的导数可表示为
$$\left. \begin{split} & {\varphi _{L,x}}\left( {\boldsymbol{x}} \right) = {y_I}{z_I}{N_{I,x}}\left( x \right){x_J}{z_J}{N_J}\left( y \right){x_K}{y_K}{N_K}\left( z \right)\\ &{\varphi _{L,y}}\left( {\boldsymbol{x}} \right) = {y_I}{z_I}{N_I}\left( x \right){x_J}{z_J}{N_{J,y}}\left( y \right){x_K}{y_K}{N_K}\left( z \right) \\ & {\varphi _{L,z}}\left( {\boldsymbol{x}} \right) = {y_I}{z_I}{N_I}\left( x \right){x_J}{z_J}{N_J}\left( y \right){x_K}{y_K}{N_{K,z}}\left( z \right)\\ &{\varphi _{L,xx}}\left( {\boldsymbol{x}} \right) = {y_I}{z_I}{N_{I,xx}}\left( x \right){x_J}{z_J}{N_J}\left( y \right){x_K}{y_K}{N_K}\left( z \right) \\ & {\varphi _{L,yy}}\left( {\boldsymbol{x}} \right) = {y_I}{z_I}{N_I}\left( x \right){x_J}{z_J}{N_{J,yy}}\left( y \right){x_K}{y_K}{N_K}\left( z \right)\\ &{\varphi _{L,zz}}\left( {\boldsymbol{x}} \right) = {y_I}{z_I}{N_I}\left( x \right){x_J}{z_J}{N_J}\left( y \right){x_K}{y_K}{N_{K,zz}}\left( z \right) \\ & {\varphi _{L,xy}}\left( {\boldsymbol{x}} \right) = {y_I}{z_I}{N_{I,x}}\left( x \right){x_J}{z_J}{N_{J,y}}\left( y \right){x_K}{y_K}{N_K}\left( z \right)\\ &{\varphi _{L,xz}}\left( {\boldsymbol{x}} \right) = {y_I}{z_I}{N_{I,x}}\left( x \right){x_J}{z_J}{N_J}\left( y \right){x_K}{y_K}{N_{K,z}}\left( z \right) \\ & {\varphi _{L,yz}}\left( {\boldsymbol{x}} \right) = {y_I}{z_I}{N_I}\left( x \right){x_J}{z_J}{N_{J,y}}\left( y \right){x_K}{y_K}{N_{K,z}}\left( z \right)\\ &\qquad I = 1,2,\cdots, {m_1},\quad J = 1,2,\cdots, {m_2},\quad K = 1 ,2,\cdots, {m_3} \end{split}\right\} $$ (8) 1.3 拉格朗日插值近似
对封闭求解区域$\bar \varOmega$进行离散, 未知变量$u\left( {\boldsymbol{x}} \right)$可以表示成以下形式
$$ u\left( {\boldsymbol{x}} \right) \approx {u^h}\left( {\boldsymbol{x}} \right) = \sum\limits_{I = {\text{1}}}^{\tilde m} {{\varphi _I}\left( {\boldsymbol{x}} \right){a_I}} $$ (9) 其中$\tilde m$为形函数的节点个数. 对于一维问题, $\tilde m = m$; 在二维问题中, $\tilde m = {m_1}{m_2}$, 在三维问题中, $\tilde m = {m_1}{m_2}{m_3}$. ${u^h}\left( {\boldsymbol{x}} \right)$被称为近似函数, ${\varphi _I}\left( {\boldsymbol{x}} \right)$为拉格朗日插值形函数, ${a_I}$是节点系数. 该函数可以满足如下的p阶一致性条件(即可重构p阶多项式)
$$ \sum\limits_{I = 1}^{\tilde m} {{\varphi _I}\left( {\boldsymbol{x}} \right){\boldsymbol{x}}_I^\gamma } = {{\boldsymbol{x}}^\gamma }\;,\quad \left| \gamma \right| \leqslant p $$ (10) 式(10)也可写成如下形式
$$ \sum\limits_{I = 1}^{\tilde m} {{\varphi _I}\left( {\boldsymbol{x}} \right){{\left( {{{\boldsymbol{x}}_I} - {\boldsymbol{x}}} \right)}^\gamma }} = {\boldsymbol{0}},\quad \left| \gamma \right| \leqslant p $$ (11) 其中$\gamma = \left( {{\gamma _{\text{1}}},{\gamma _{\text{2}}}, \cdots, {\gamma _\vartheta }} \right)$表示多维记数法, $\vartheta $表示维度数, $\left| \gamma \right| = \displaystyle\sum\nolimits_{i = {\text{1}}}^\vartheta {{\gamma _i}}$表示$\gamma $的长度. 此外, 在方程(11)的等式两边同时取一阶和二阶微分, 可以得到导数的一致性条件如下
$$ \sum\limits_{I = 1}^{\tilde m} {{\varphi _{I,\alpha }}\left( {\boldsymbol{x}} \right){{\left( {{{\boldsymbol{x}}_I} - {\boldsymbol{x}}} \right)}^\gamma }} = \left| \gamma \right|!{\delta _{\left| \gamma \right|,1}},\quad \left| \gamma \right| \leqslant p $$ (12) $$ \sum\limits_{I = 1}^{\tilde m} {{\varphi _{I,\alpha \beta }}\left( {\boldsymbol{x}} \right){{\left( {{{\boldsymbol{x}}_I} - {\boldsymbol{x}}} \right)}^\gamma }} = \left| \gamma \right|!{\delta _{\left| \gamma \right|,2}},\quad \left| \gamma \right| \leqslant p $$ (13) 其中, $\alpha = x,y,z$,$\beta = x,y,z$, $\gamma {\text{! = }}{\gamma _{\text{1}}}{\text{!}} {\gamma _{\text{2}}}{\text{!}}\cdots {\gamma _\vartheta }{\text{!}}$.
2. 直接配点法(direct collocation method, DCM)
边值问题可以表示为以下的一般形式
$$ {\bf{A}{\boldsymbol{u}}} = {\boldsymbol{f}},\quad {\rm{in}}\;\;\varOmega $$ (14) $$ {{\bf{B}}^h}{\boldsymbol{u}} = {\boldsymbol{h}},\quad {\rm{on}}\;\;\varGamma $$ (15) $$ {{\bf{B}}^g}{\boldsymbol{u}} = {\boldsymbol{g}},\quad {\rm{on}}\;\;\varPi $$ (16) 其中$\varOmega$表示待求问题的开区域, $\varGamma$表示诺伊曼边界, $\varPi$表示狄利克雷边界, 整体求解区域$\bar \varOmega = \varOmega \cup \varGamma \cup \varPi$. ${\bf{A}}$, ${{\bf{B}}^h}$和${{\bf{B}}^g}$分别表示区域$\varOmega$, $\varGamma$和$\varPi$上的微分算子. ${\boldsymbol{u}}$是未知量, ${\boldsymbol{f}}$, ${\boldsymbol{h}}$和${\boldsymbol{g}}$为对应的源项.
未知变量${\boldsymbol{u}}$可采用拉格朗日插值近似表示为
$$ {\boldsymbol{u}}\left( {\boldsymbol{x}} \right) \approx {{\boldsymbol{u}}^h}\left( {\boldsymbol{x}} \right) = {\left[ {u_1^h,u_2^h,u_3^h} \right]^{\rm{T}}} = {{\boldsymbol{\varPhi }}^{\text{T}}}{\boldsymbol{a}} $$ (17) 其中
$$ {\boldsymbol{a}} = {\left[ {{{\boldsymbol{a}}_1},{{\boldsymbol{a}}_2}, \cdots ,{{\boldsymbol{a}}_{{N_s}}}} \right]^{\text{T}}}\;,\quad {{\boldsymbol{a}}_I} = \left[ {{a_{1I}},{a_{2I}},{a_{3I}}} \right] $$ (18) $$ {{\boldsymbol{\varPhi }}^{\text{T}}} = \left[ {{{\boldsymbol{\varPhi }}_1},{{\boldsymbol{\varPhi }}_2}, \cdots ,{{\boldsymbol{\varPhi }}_{{N_s}}}} \right]\;,\quad {{\boldsymbol{\varPhi }}_I} = {\varphi _I}{\boldsymbol{I}} $$ (19) 于是可以得到
$$\left.\begin{split} &{\bf{A}}{{\boldsymbol{\varPhi }}_I}\left( {\boldsymbol{x}} \right) = {\rm{A}}{\varphi _I}\left( {\boldsymbol{x}} \right){\boldsymbol{I}}\\ &{{\bf{B}}^h}{{\boldsymbol{\varPhi }}_I}\left( {\boldsymbol{x}} \right) = {{\rm{B}}^h}{\varphi _I}\left( {\boldsymbol{x}} \right){\boldsymbol{I}}\\ &{{\bf{B}}^g}{{\boldsymbol{\varPhi }}_I}\left( {\boldsymbol{x}} \right) = {{\rm{B}}^g}{\varphi _I}\left( {\boldsymbol{x}} \right){\boldsymbol{I}} \end{split}\right\}$$ (20) 其中${\varphi _I}$表示对应源点${{\boldsymbol{x}}_I}$的拉格朗日插值形函数, ${\boldsymbol{I}}$是单位矩阵. 分别在域内$\varOmega$、诺伊曼边界$\varGamma$和狄利克雷边界$\varPi$上布置${N_p}$, ${N_q}$和${N_r}$个配点, $\left\{ {{{\boldsymbol{p}}_I}} \right\}_{I = 1}^{{N_p}} \subseteq \varOmega$, $\left\{ {{{\boldsymbol{q}}_I}} \right\}_{I = 1}^{{N_q}} \subseteq \varGamma$和$\left\{ {{{\boldsymbol{r}}_I}} \right\}_{I = 1}^{{N_r}} \subseteq \varPi$. 配点的总个数为${N_c} = {N_p} + {N_q} + {N_r}$.
将近似函数式(17)代入方程(14) ~ (16), 并使其在域内和边界上满足配点方程, 可以得到如下离散形式
$$ {\bf{A}}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {{{\boldsymbol{p}}_I}} \right){\boldsymbol{d}} = {\boldsymbol{f}}\left( {{{\boldsymbol{p}}_I}} \right),\quad \forall {{\boldsymbol{p}}_I} \in \;\varOmega \,,\quad I = 1,2, \cdots, {N_p} $$ (21) $$ {{\bf{B}}^h}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {{{\boldsymbol{q}}_I}} \right){\boldsymbol{d}} = {\boldsymbol{h}}\left( {{{\boldsymbol{q}}_I}} \right),\quad \forall {{\boldsymbol{q}}_I} \in \varGamma ,\quad I = 1,2, \cdots, {N_q} $$ (22) $$ {{\bf{B}}^g}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {{{\boldsymbol{r}}_I}} \right){\boldsymbol{d}} = {\boldsymbol{g}}\left( {{{\boldsymbol{r}}_I}} \right),\quad \forall {{\boldsymbol{r}}_I} \in \varPi \,,\quad I = 1,2, \cdots, {N_r} $$ (23) 在直接配点法中, 配点的位置与源点位置相同且数量一致. 当采用拉格朗日插值作为形函数时, 该方法被称为拉格朗日插值配点法(Lagrange interpolation collocation method, LICM). 系数矩阵${\boldsymbol{a}}$可以通过求解离散方程式(21) ~ 式(23)来确定. 离散方程式(21) ~ 式(23)可改写为以下的矩阵形式
$$ {{{{\boldsymbol{K}}}{\boldsymbol{a}}}} = {\boldsymbol{F}} $$ (24) 其中
$$ {\boldsymbol{K}} = \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{K}}^A}} \\ {{{\boldsymbol{K}}^h}} \\ {{{\boldsymbol{K}}^g}} \end{array}} \right], {\boldsymbol{F}} = \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{F}}^A}} \\ {{{\boldsymbol{F}}^h}} \\ {{{\boldsymbol{F}}^g}} \end{array}} \right] $$ (25) 式中子矩阵为
$$\left.\begin{split} & {{\boldsymbol{K}}^A} = \left[ {\begin{array}{*{20}{c}} {{\boldsymbol{A}}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {{{\boldsymbol{p}}_1}} \right)} \\ {{\boldsymbol{A}}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {{{\boldsymbol{p}}_2}} \right)} \\ \vdots \\ {{\boldsymbol{A}}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {{{\boldsymbol{p}}_{{N_p}}}} \right)} \end{array}} \right]\;,\quad {{\boldsymbol{K}}^h} = \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{B}}^h}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {{{\boldsymbol{q}}_1}} \right)} \\ {{{\boldsymbol{B}}^h}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {{{\boldsymbol{q}}_2}} \right)} \\ \vdots \\ {{{\boldsymbol{B}}^h}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {{{\boldsymbol{q}}_{{N_q}}}} \right)} \end{array}} \right]\\ &{{\boldsymbol{K}}^g} = \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{B}}^g}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {{{\boldsymbol{r}}_1}} \right)} \\ {{{\boldsymbol{B}}^g}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {{{\boldsymbol{r}}_2}} \right)} \\ \vdots \\ {{{\boldsymbol{B}}^g}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {{{\boldsymbol{r}}_{{N_r}}}} \right)} \end{array}} \right],\quad {{\boldsymbol{F}}^A} = \left[ {\begin{array}{*{20}{c}} {{\boldsymbol{f}}\left( {{{\boldsymbol{p}}_1}} \right)} \\ {{\boldsymbol{f}}\left( {{{\boldsymbol{p}}_2}} \right)} \\ \vdots \\ {{\boldsymbol{f}}\left( {{{\boldsymbol{p}}_{{N_p}}}} \right)} \end{array}} \right]\\ &{{\boldsymbol{F}}^h} = \left[ {\begin{array}{*{20}{c}} {{\boldsymbol{h}}\left( {{{\boldsymbol{q}}_1}} \right)} \\ {{\boldsymbol{h}}\left( {{{\boldsymbol{q}}_2}} \right)} \\ \vdots \\ {{\boldsymbol{h}}\left( {{{\boldsymbol{q}}_{{N_q}}}} \right)} \end{array}} \right]\;,\quad {{\boldsymbol{F}}^g} = \left[ {\begin{array}{*{20}{c}} {{\boldsymbol{g}}\left( {{{\boldsymbol{r}}_1}} \right)} \\ {{\boldsymbol{g}}\left( {{{\boldsymbol{r}}_2}} \right)} \\ \vdots \\ {{\boldsymbol{g}}\left( {{{\boldsymbol{r}}_{{N_r}}}} \right)} \end{array}} \right] \end{split}\right\} $$ (26) 3. 稳定配点法(stabilized collocation method, SCM)
如图6所示, 在稳定配点法中, 将域内和边界上布置若干个离散点, 以该离散点为中心构建子域${\varOmega _l}/{\varPi _l}/{\varGamma _l}$. 子域的大小对于每个点可以是不同的, 但为了简单起见一般使用相同大小的子域. 子域的形状对于二维问题一般选择正方形, 三维问题选择立方体即可.
在稳定配点法中, 对方程式(14) ~ 式(16)的强形式在相应子域内进行积分, 得到如下积分形式
$$ \int_{{\varOmega _{{\kern 1pt} l}}} {{\bf{A}{\boldsymbol{u}}}\left( {\boldsymbol{x}} \right){\kern 1pt} {\text{d}}} \varOmega = \int_{{\varOmega _{{\kern 1pt} l}}} {{\kern 1pt} {\boldsymbol{f}}\left( {\boldsymbol{x}} \right){\kern 1pt} {\text{d}}\varOmega }, \quad {\text{in}}\;\;{\varOmega _{{\kern 1pt} l}} $$ (27) $$ \int_{{\varGamma _l}} {{{\bf{B}}^h}{\boldsymbol{u}}\left( {\boldsymbol{x}} \right){\kern 1pt} {\text{d}}} {\kern 1pt} \varGamma = \int_{{\varGamma _l}} {{\boldsymbol{h}}\left( {\boldsymbol{x}} \right){\kern 1pt} {\text{d}}{\kern 1pt} \varGamma }, \quad {\text{on}}\;\;{\varGamma _l} $$ (28) $$ \int_{{\varPi _l}} {{{\bf{B}}^g}{\bf{u}}\left( {\boldsymbol{x}} \right){\kern 1pt} {\text{d}}} {\kern 1pt} \varPi = \int_{{\varPi _l}} {{\boldsymbol{g}}\left( {\boldsymbol{x}} \right){\kern 1pt} {\text{d}}{\kern 1pt} \varPi }, \quad {\text{on}}\;\;{\varPi _l} $$ (29) 将近似函数式(17)代入方程式(27) ~ 式(29)可以得到
$$ \int_{{\varOmega _{{\kern 1pt} l}}} {{\bf{A}}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {\boldsymbol{p}} \right){\kern 1pt} {\kern 1pt} {\text{d}}} \varOmega \,{\boldsymbol{a}} = \int_{{\varOmega _{{\kern 1pt} l}}} {{\kern 1pt} {\boldsymbol{f}}\left( {\boldsymbol{p}} \right){\kern 1pt} {\kern 1pt} {\text{d}}\varOmega }, \quad {\text{in}}\;\;{\varOmega _{{\kern 1pt} l}} $$ (30) $$ \int_{{\varGamma _l}} {{{\bf{B}}^h}{{\boldsymbol{\varPhi }}^{\text{T}}}\left( {\boldsymbol{q}} \right){\kern 1pt} {\kern 1pt} {\text{d}}} \varGamma \,{\boldsymbol{a}} = \int_{{\varGamma _l}} {{\kern 1pt} {\boldsymbol{h}}{\kern 1pt} \left( {\boldsymbol{q}} \right){\kern 1pt} {\kern 1pt} {\text{d}}\varGamma }, \quad {\text{on}}\;\;{\varGamma _l} $$ (31) $$ \int_{{\varPi _l}} {{{\bf{B}}^g}{{\boldsymbol{\varPhi}}^{\text{T}}}\left( {\boldsymbol{r}} \right){\kern 1pt} {\text{d}}} {\kern 1pt} \varPi {\boldsymbol{a}} = \int_{{\varPi _l}} {{\boldsymbol{g}}\left( {\boldsymbol{r}} \right){\kern 1pt} {\text{d}}{\kern 1pt} \varPi }, \quad {\text{on}}\;\;{\varPi _l} $$ (32) 其中, ${\boldsymbol{p}}$, ${\boldsymbol{q}}$和${\boldsymbol{r}}$分别表示域内、诺伊曼边界和狄利克雷边界上的离散点. 采用拉格朗日插值作为形函数的稳定配点法称为拉格朗日插值稳定配点法(stabilized Lagrange interpolation collocation method, SLICM). 同时对方程式(30) ~ 式(32)等式两边做数值积分可得
$$ \sum\limits_{i{\text{ = }}1}^\omega {{w_i}{\bf{A}}{{\boldsymbol{\varPhi }}^{\text{T}}}{{\left( {{{{\boldsymbol{\bar p}}}_i}} \right)}_l}} {\kern 1pt} {\boldsymbol{a}} = \sum\limits_{i{\text{ = }}1}^\omega {{w_i}{\boldsymbol{f}}\left( {{{{\boldsymbol{\bar p}}}_i}} \right)} \;,\quad \forall {{\boldsymbol{\bar p}}_i} \in {\varOmega _{{\kern 1pt} l}} $$ (33) $$ \sum\limits_{i{\text{ = }}1}^\omega {{w_i}{{\bf{B}}^h}{{\boldsymbol{\varPhi }}^{\text{T}}}{{\left( {{{{\boldsymbol{\bar q}}}_i}} \right)}_l}} {\kern 1pt} {\boldsymbol{a}} = \sum\limits_{i{\text{ = }}1}^\omega {{w_i}{\boldsymbol{h}}\left( {{{{\boldsymbol{\bar q}}}_i}} \right)} \;,\quad \forall {{\boldsymbol{\bar q}}_i} \in {\varGamma _{{\kern 1pt} l}} $$ (34) $$ \sum\limits_{i{\text{ = }}1}^\omega {{w_i}{{\bf{B}}^g}{{\boldsymbol{\varPhi }}^{\text{T}}}{{\left( {{{{\boldsymbol{\bar r}}}_i}} \right)}_l}} {\kern 1pt} {\boldsymbol{a}} = \sum\limits_{i{\text{ = }}1}^\omega {{w_i}{\boldsymbol{g}}\left( {{{{\boldsymbol{\bar r}}}_i}} \right)} \;,\quad \forall {{\boldsymbol{\bar r}}_i} \in {\varPi _{{\kern 1pt} l}} $$ (35) 其中, $\omega $表示积分点个数, ${w_i}$表示积分权重, ${\left( {{{\bar {\boldsymbol{p}}}_i}} \right)_l}$, ${\left( {{{\bar {\boldsymbol{q}}}_i}} \right)_l}$和${\left( {{{\bar {\boldsymbol{r}}}_i}} \right)_l}$分别表示离散点${{\boldsymbol{p}}_l}$, ${{\boldsymbol{q}}_l}$和${{\boldsymbol{r}}_l}$对应子区域内的积分点. 通常可以使用高斯积分进行数值积分运算. 式(33) ~ 式(35)可以用矩阵形式表示如下
$$ {{\hat {\boldsymbol{K}}}}{\boldsymbol{a}}: = \left[ {\begin{array}{*{20}{c}} {{{{{\hat {\boldsymbol{K}}}}}^A}} \\ {{{{{\hat {\boldsymbol{K}}}}}^h}} \\ {{{{{\hat {\boldsymbol{K}}}}}^g}} \end{array}} \right]{\boldsymbol{a}} = \left[ {\begin{array}{*{20}{c}} {{{{{\hat {\boldsymbol{F}}}}}^A}} \\ {{{{{\hat {\boldsymbol{F}}}}}^h}} \\ {{{{{\hat {\boldsymbol{F}}}}}^g}} \end{array}} \right]: = {{\hat {\boldsymbol{F}}}} $$ (36) 其中
$$ \left.\begin{split} & {{{{\hat {\boldsymbol{K}}}}}^A} = \left[ {\sum\limits_{i = {\text{1}}}^\omega {{w_i}} {\kern 1pt} {\bf{A}}{{\boldsymbol{\varPhi }}^{\text{T}}}{{\left( {{{{{\bar {\boldsymbol{p}}}}}_i}} \right)}_l}} \right]\;,\quad {{{{\hat {\boldsymbol{F}}}}}^A} = \left[ {\sum\limits_{i = {\text{1}}}^\omega {{w_i}} {\kern 1pt} {\boldsymbol{f}}{{\left( {{{{{\bar {\boldsymbol{p}}}}}_i}} \right)}_l}} \right]\\ &\qquad \forall {\left( {{{{{\bar {\boldsymbol{p}}}}}_i}} \right)_l} \in \;{\varOmega _l}\,,\quad \forall {{\boldsymbol{p}}_l} \in \;\varOmega \\ & {{{{\hat {\boldsymbol{K}}}}}^h} = \left[ {\sum\limits_{i = {\text{1}}}^\omega {{w_i}} {\kern 1pt} {{\bf{B}}^h}{{\boldsymbol{\varPhi }}^{\text{T}}}{{\left( {{{{{\bar {\boldsymbol{q}}}}}_i}} \right)}_l}} \right]\;,\quad {{{{\hat {\boldsymbol{F}}}}}^h} = \left[ {\sum\limits_{i = {\text{1}}}^\omega {{w_i}} {\kern 1pt} {\boldsymbol{h}}{{\left( {{{{{\bar {\boldsymbol{q}}}}}_i}} \right)}_l}} \right]\\ &\qquad \forall {\left( {{{{{\bar {\boldsymbol{q}}}}}_i}} \right)_l} \in \;{\varGamma _l} \forall {{\boldsymbol{q}}_l} \in \;\varGamma \\ & {{{{\hat {\boldsymbol{K}}}}}^g} = \left[ {\sum\limits_{i = {\text{1}}}^\omega {{w_i}} {\kern 1pt} {{\bf{B}}^g}{{\boldsymbol{\varPhi }}^{\text{T}}}{{\left( {{{{{\bar {\boldsymbol{r}}}}}_i}} \right)}_l}} \right]\;,\quad {{{{\hat {\boldsymbol{F}}}}}^g} = \left[ {\sum\limits_{i = {\text{1}}}^\omega {{w_i}} {\kern 1pt} {\boldsymbol{g}}{{\left( {{{{{\bar {\boldsymbol{r}}}}}_i}} \right)}_l}} \right]\\ &\qquad \forall {\left( {{{{{\bar {\boldsymbol{r}}}}}_i}} \right)_l} \in \;{\varPi _l} \forall {{\boldsymbol{r}}_l} \in \;\varPi \end{split}\right\} $$ (37) 4. 数值算例
本节采用一维、二维和三维3个算例来评估基于拉格朗日插值的无网格直接配点法和稳定配点法的精度和稳定性. 拉格朗日插值形函数的阶数与影响域内源点个数有对应关系, 3个算例均采用二阶形函数, 其影响域均为二阶形函数所对应的影响域范围, 即$p$阶形函数在单个方向上对应$p + 1$个源点. 根据文献[18]的研究结论, 积分域大小和特征节点间距相距较近时, 可以得到比较高的精度. 因此, 本文算例中的积分域大小均参照特征节点间距.
4.1 一维直杆问题
考虑如图7所示左端固支的一维直杆的二阶微分方程问题, 其控制方程为
$$ EA\frac{{{{\rm{d}}^2}u}}{{{\rm{d}}{x^2}}} + f\left( x \right) = 0 $$ (38) $$ u\left( 0 \right) = 0, \quad EA\frac{{{\rm{d}}u\left( L \right)}}{{{\rm{d}}x}} = 0 $$ (39) 其中$EA = 1$, $L = 1$, $f\left( x \right) = 3{x^2}$, 解析解为$u\left( x \right) = - \dfrac{{{x^4}}}{4} + x$.
图8为随机布点的一维直杆离散图, 源点和配点布置在相同位置. 图9展示了拉格朗日插值配点法、拉格朗日插值稳定配点法和传统重构核近似配点法(reproducing kernel collocation method, RKCM)[30]在一维直杆问题中的数值计算结果和相应的误差, 结果表明基于拉格朗日插值方法(LICM和SLICM)的求解精度明显优于基于重构核近似方法(RKCM)的求解精度, 拉格朗日插值稳定配点法相比拉格朗日插值配点法有更好的计算精度, 图10的收敛性分析更清晰地展示了这一结论(图标中的数字为收敛率). 由于源点和配点均随机布置, 收敛率和理论值略有差距, 理论收敛率可参考文献[29]. 图11比较了3种方法的刚度矩阵条件数, 可以看出稳定配点法的矩阵条件数更低, 可以获得更好的稳定性.
4.2 二维泊松问题
考虑二维空间中的泊松问题, 控制方程和边界条件如下
$$ \frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}} = \left( {{x^2} + {y^2}} \right){{\rm{e}}^{xy}}, {\text{in }}\varOmega $$ (40) $$\begin{split} &u\left( { - 1,y} \right) = {{\rm{e}}^{ - y}}, u\left( {1,y} \right) = {{\rm{e}}^y}, u\left( {x, - 1} \right) = {{\rm{e}}^{ - x}} \\ &\quad u\left( {x,1} \right) = {{\rm{e}}^x} \end{split}$$ (41) 其中$\varOmega = \left( { - 1,1} \right) \times \left( { - 1,1} \right)$, 解析解为$u\left( {x,y} \right) = {{\rm{e}}^{xy}}$. 文献[29]中研究了均匀离散和对应结构化网格布点的两种离散方式, 本文讨论如图12所示任意非均匀离散模式. 源点和配点布置在相同位置.
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图 12 二维泊松问题源点离散图Figure 12. Discrete points of source points for the 2D Poisson problem图13和图14分别展示了采用拉格朗日插值配点法和拉格朗日插值稳定配点法在域内和边界上的计算精度, 边界计算结果达到了计算机精度, 表明这两种方法都能够准确地施加本质边界条件. 结合图15的收敛性分析可以看出稳定配点法对于离散点非均匀分布的二维泊松问题也能获得更高的精度. 图16比较了两种方法的刚度矩阵条件数, 稳定配点法的条件数更小, 表示该方法具有更好的稳定性.
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图 14 二维泊松问题边界位移解Figure 14. Numerical solutions of boundary for the 2D Poisson problem4.3 三维亥姆霍兹问题
考虑三维空间下亥姆霍兹方程, 表示如下
$$ \frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}} + \frac{{{\partial ^2}u}}{{\partial {z^2}}} + ku = 0, {\text{in }}\varOmega $$ (42) $$ u(x,y,z) = \sin x + \sin y + \sin z, {\text{on }}{\varGamma _g} $$ (43) 其中${{\varOmega }} = \left[ { - 1,{\text{1}}} \right] \times \left[ { - 1,1} \right] \times \left[ { - 1,1} \right]$, $k = 1$, 解析解为$u(x,y,z) = \sin x + \sin y + \sin z$.
将两种方法进一步扩展至三维空间计算上述亥姆霍兹方程, 可以得到类似的计算结果. 图17展示了三维空间下的离散点布置图, 图18给出了域内$x = 0$上的计算结果. 图19展示三维空间下的边界计算结果, 可以看出两种方法对于边界的计算精度都非常好. 图20和图21分别展示直接配点法和稳定配点法在三维亥姆霍兹问题上的收敛性分析和矩阵条件数分析. 如前面两个算例得到的结论相同, 稳定配点法有更好的精度和稳定性.
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图 17 三维亥姆霍兹问题源点离散图Figure 17. Discrete points of source points for the 3D Helmholtz problem![]()
图 18 三维亥姆霍兹问题域内位移解Figure 18. Numerical solutions of the inner domain for the 3D Helmholtz problem![]()
图 19 三维亥姆霍兹问题边界位移解Figure 19. Numerical solutions of boundary for the 3D Helmholtz problem![]()
19 三维亥姆霍兹问题边界位移解 (续)19. Numerical solutions of boundary for the 3D Helmholtz problem (continued)![]()
图 21 三维亥姆霍兹问题刚度矩阵条件数Figure 21. Condition number comparisons for the 3D Helmholtz problem5. 总结
本文将曲线拉格朗日插值多项式引入无网格直接配点法和稳定配点法中, 构建了离散点可任意布置的拉格朗日插值配点法和拉格朗日插值稳定配点法. 这两种方法的形函数与有限元法一样具有Kronecker delta性质, 能够简单准确地施加本质边界条件, 相比较传统无网格方法在边界处理方面有较大优势, 提升了无网格法在边界上的求解精度. 拉格朗日插值稳定配点法将问题域划分为若干个子域, 在子域内能够实现精确积分, 积分降低了刚度矩阵的条件数, 进一步提高了算法的精度和稳定性, 解决了传统无网格法难以实现精确积分的难题. 该两种方法由于采用曲线拉格朗日插值, 离散点可任意布置, 不再局限于结构化网格布点, 使得这两种方法可以适用于各类复杂区域问题的求解. 基于拉格朗日插值的直接配点法和稳定配点法的计算成本都比较低, 边界求解精度高, 稳定配点法可以通过精确积分进一步提高精度和稳定性, 未来可以将这两种方法应用于更多力学问题和实际工程问题的数值分析.
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图 1 一维2节点和3节点拉格朗日插值形函数的节点分布图
Figure 1. Node distributions of the 1D 2-node and 3-node Lagrange interpolation shape functions
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图 2 二维4节点、6节点和9节点拉格朗日插值形函数的节点分布图
Figure 2. Node distributions of the 2D 4-node, 6-node and 9-node Lagrange interpolation shape functions
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图 12 二维泊松问题源点离散图
Figure 12. Discrete points of source points for the 2D Poisson problem
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图 14 二维泊松问题边界位移解
Figure 14. Numerical solutions of boundary for the 2D Poisson problem
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图 17 三维亥姆霍兹问题源点离散图
Figure 17. Discrete points of source points for the 3D Helmholtz problem
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图 18 三维亥姆霍兹问题域内位移解
Figure 18. Numerical solutions of the inner domain for the 3D Helmholtz problem
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图 19 三维亥姆霍兹问题边界位移解
Figure 19. Numerical solutions of boundary for the 3D Helmholtz problem
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19 三维亥姆霍兹问题边界位移解 (续)
19. Numerical solutions of boundary for the 3D Helmholtz problem (continued)
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[1] Qin X, Shen YJ, Chen W, et al. Bending and free vibration analyses of circular stiffened plates using the FSDT mesh-free method. International Journal of Mechanical Sciences, 2021, 202: 106498
[2] Chen JS, Hillman M, Chi SW. Meshfree methods: progress made after 20 years. Journal of Engineering Mechanics, 2017, 143(4): 04017001 doi: 10.1061/(ASCE)EM.1943-7889.0001176
[3] 张雄, 刘岩. 无网格法. 北京: 清华大学出版社有限公司, 2004 Zhang Xiong, Liu Yan. Meshless Methods. Beijing: Tsinghua University Press, 2004 (in Chinese))
[4] 陈健, 王东东, 刘宇翔等. 无网格动力分析的循环卷积神经网络代理模型. 力学学报, 2022, 54: 1-14 (Chen Jian, Wang Dongdong, Liu Yuxiang, et al. A recurrent convolutional neural network surrogate model for dynamic meshfree analysis. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54: 1-14 (in Chinese) doi: 10.6052/0459-1879-17-020 Chen Jian, Wang Dongdong, et al. A recurrent convolutional neural network surrogate model for dynamic meshfree analysis. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54: 1-14 (in Chinese)) doi: 10.6052/0459-1879-17-020
[5] Zhang X, Song KZ, Lu MW, et al. Meshless methods based on collocation with radial basis functions. Computational Mechanics, 2000, 26(4): 333-343 doi: 10.1007/s004660000181
[6] Huang TH. Stabilized and variationally consistent integrated meshfree formulation for advection-dominated problems. Computer Methods in Applied Mechanics and Engineering, 2023, 403: 115698 doi: 10.1016/j.cma.2022.115698
[7] Li ML, Nikan O, Qiu WL, et al. An efficient localized meshless collocation method for the two-dimensional Burgers-type equation arising in fluid turbulent flows. Engineering Analysis with Boundary Elements, 2022, 144: 44-54 doi: 10.1016/j.enganabound.2022.08.007
[8] Wang DD, Wang JR, Wu JC. Superconvergent gradient smoothing meshfree collocation method. Computer Methods in Applied Mechanics and Engineering, 2018, 340: 728-766 doi: 10.1016/j.cma.2018.06.021
[9] 王莉华, 阮剑武. 配点型无网格法理论和研究进展. 力学季刊, 2021, 42(4): 613 Wang Lihua, Ruan Jianwu. Theory and research progress of collocation-type meshless method. Chinese Quarterly of Mechanics, 2021, 42(4): 613 (in Chinese))
[10] Li JC, Cheng AHD, Chen CS. A comparison of efficiency and error convergence of multiquadric collocation method and finite element method. Engineering Analysis with Boundary Elements, 2003, 27(3): 251-257 doi: 10.1016/S0955-7997(02)00081-4
[11] Wang LH, Liu YJ, Zhou YT, et al. Static and dynamic analysis of thin functionally graded shell with in-plane material inhomogeneity. International Journal of Mechanical Sciences, 2021, 193: 106165 doi: 10.1016/j.ijmecsci.2020.106165
[12] Hillman M, Chen JS. Performance comparison of nodally integrated Galerkin meshfree methods and nodally collocated strong form meshfree methods//Advances in Computational Plasticity, 2018: 145-164
[13] Zhang X, Liu XH, Song KZ, et al. Least-squares collocation meshless method. International Journal for Numerical Methods in Engineering, 2001, 51(9): 1089-1100 doi: 10.1002/nme.200
[14] Liu Y, Zhang X, Lu MW. Meshless least-squares method for solving the steady-state heat conduction equation. Tsinghua Science and Technology, 2005, 10(1): 61-66 doi: 10.1016/S1007-0214(05)70010-9
[15] Chen JS, Wang LH, Hu HY, et al. Subdomain radial basis collocation method for heterogeneous media. International Journal for Numerical Methods in Engineering, 2009, 80(2): 163-190 doi: 10.1002/nme.2624
[16] Wang LH, Chen JS, Hu HY. Subdomain radial basis collocation method for fracture mechanics. International Journal for Numerical Methods in Engineering, 2010, 83(7): 851-876 doi: 10.1002/nme.2860
[17] 王莉华, 李溢铭, 褚福运. 基于分区径向基函数配点法的大变形分析. 力学学报, 2019, 51(3): 743-753 Wang Lihua, Li Yiming, Chu Fuyun. Finite subdomain radial basis collocation method for the large deformation analysis. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(3): 743-753 (in Chinese))
[18] Wang LH, Qian ZH. A meshfree stabilized collocation method (SCM) based on reproducing kernel approximation. Computer Methods in Applied Mechanics and Engineering, 2020, 371: 113303 doi: 10.1016/j.cma.2020.113303
[19] Qian ZH, Wang LH, Gu Y, et al. An efficient meshfree gradient smoothing collocation method (GSCM) using reproducing kernel approximation. Computer Methods in Applied Mechanics and Engineering, 2021, 374: 113573 doi: 10.1016/j.cma.2020.113573
[20] 王莉华, 刘义嘉, 钟伟等. 无网格稳定配点法及其在弹性力学中的应用. 计算力学学报, 2021, 38(3): 305-312 (Wang Lihua, Liu Yijia, Zhong Wei, et al. Meshfree stabilized collocation method in elasticity. Chinese Journal of Computational Mechanics, 2021, 38(3): 305-312 (in Chinese) Wang Lihua, Liu Yijia, Zhong Wei, et al. Meshfree stabilized collocation method in elasticity. Chinese Journal of Computational Mechanics, 2021, (3): 305-312 (in Chinese))
[21] Belytschko T, Lu YY, Gu L. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229-256 doi: 10.1002/nme.1620370205
[22] Zhu T, Atluri SN. A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method. Computational Mechanics, 1998, 21(3): 211-222 doi: 10.1007/s004660050296
[23] Liu DJ, Cheng YM. The interpolating element-free Galerkin (IEFG) method for three-dimensional potential problems. Engineering Analysis with Boundary Elements, 2019, 108: 115-123 doi: 10.1016/j.enganabound.2019.08.021
[24] Qian ZH, Wang LH, Zhang CZ, et al. A highly efficient and accurate Lagrangian–Eulerian stabilized collocation method (LESCM) for the fluid−rigid body interaction problems with free surface flow. Computer Methods in Applied Mechanics and Engineering, 2022, 398: 115238 doi: 10.1016/j.cma.2022.115238
[25] Gao XW, Huang SZ, Cui M, et al. Element differential method for solving general heat conduction problems. International Journal of Heat and Mass Transfer, 2017, 115: 882-894 doi: 10.1016/j.ijheatmasstransfer.2017.08.039
[26] Gao XW, Gao LF, Zhang Y, et al. Free element collocation method: A new method combining advantages of finite element and mesh free methods. Computers & Structures, 2019, 215: 10-26
[27] 高效伟, 徐兵兵, 吕军等. 自由单元法及其在结构分析中的应用. 力学学报, 2019, 51(3): 703-713 (Gao Xiaowei, Xu Bingbing, Lü Jun, et al. Free element method and its application in structural analysis. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(3): 703-713 (in Chinese) Gao Xiaowei, Xu Bingbing, Lü Jun, et al. Free element method and its application in structural analysis. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(3): 703-713(in Chinese))
[28] 高效伟, 丁金兴, 刘华雩. 有限线法及其在流固域间耦合传热中的应用. 物理学报, 2022, 71(19): 190201 (Gao Xiaowei, Ding Jinxing, Liu Huayu. Finite line method and its application in coupled heat transfer between fluid-solid domains. Acta Physica Sinica, 2022, 71(19): 190201 (in Chinese) doi: 10.7498/aps.71.20220833 Gao Xiaowei, Ding Jinxing, Liu Huayu. Finite line method and its application in coupled heat transfer between fluid-solid domains. Acta Physica Sinica, 2022, 71(19): 190201 (in Chinese)) doi: 10.7498/aps.71.20220833
[29] Wang LH, Hu MH, Zhong Z, et al. Stabilized Lagrange interpolation collocation method: A meshfree method incorporating the advantages of finite element method. Computer Methods in Applied Mechanics and Engineering, 2023, 404: 115780 doi: 10.1016/j.cma.2022.115780
[30] Hu HY, Lai CK, Chen JS. A study on convergence and complexity of reproducing kernel collocation method. Interaction and Multiscale Mechanics, 2009, 2(3): 295-319
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