RESEARCH ADVANCES ON THE COLLOCATION METHODS BASED ON THE PHYSICAL-INFORMED KERNEL FUNCTIONS
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摘要: 在过去几十年里, 尽管有限元等传统计算方法已被成功用于众多科学与工程领域, 但是其在数值模拟无限域波传播、大尺寸比结构、工程反演和移动边界问题时仍面临计算量大、计算效率低、网格生成困难等计算难题. 本文介绍一类基于物理信息依赖核函数的无网格配点法及其在上述难点问题中的应用. 物理信息依赖核函数配点法的关键在于构建能反映问题微分控制方程物理信息的基函数. 基于这些物理信息依赖核函数, 该方法无需/仅需少量配点对所求微分控制方程进行离散, 即可有效提高计算效率. 本文首先介绍满足常见齐次微分方程的基本解、调和函数、径向Trefftz函数以及T完备函数等典型物理信息依赖核函数. 接着依次介绍非齐次、非均质、非稳态以及隐式微分方程构造物理信息依赖核函数的方法. 随后, 根据所求问题特点, 选用全域配点或局部配点技术, 建立相应的物理信息依赖核函数配点法. 最后, 通过几个典型算例验证所提物理信息依赖核函数配点法的有效性.Abstract: In the past few decades, although traditional computational methods such as finite element have been successfully used in many scientific and engineering fields, they still face several challenging problems such as expensive computational cost, low computational efficiency, and difficulty in mesh generation in the numerical simulation of wave propagation under infinite domain, large-scale-ratio structures, engineering inverse problems and moving boundary problems. This paper introduces a class of collocation discretization techniques based on physical-informed kernel function to efficiently solve the above-mentioned problems. The key issue in the physical-informed kernel function collocation methods is to construct the related basis functions, which includes the physical information of the considered differential governing equation. Based on these physical-informed kernel functions, these methods do not need/only need a few collocation nodes to discretize the considered differential governing equations, which may effectively improve the computational efficiency. In this paper, several typical physical-informed kernel functions that satisfy common-used homogeneous differential equations, such as the fundamental solutions, the harmonic functions, the radial Trefftz functions and the T-complete functions and so on, are firstly introduced. After that, the ways to construct the physical-informed kernel functions for nonhomogeneous differential equations, inhomogeneous differential equations, unsteady-state differential equations and implicit differential equations are introduced in turn. Then according to the characteristics of the considered problems, the global collocation scheme or the localized collocation scheme is selected to establish the corresponding physical-informed kernel function collocation method. Finally, four typical examples are given to verify the effectiveness of the physical-informed kernel function collocation methods proposed in this paper.
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图 1 配点法采用不同试函数求解齐次微分方程问题式(5)和式(6)的计算结果. PIKF-12表示在计算区域边界上均匀布置12个节点并采用物理信息依赖核函数
$ \phi {\text{ = }}{J_0}\left( {\sqrt 2 kr} \right) $ 作为试函数; RBF-16/36表示在计算区域上均匀布置16/36个节点并采用径向基函数$ \phi {\text{ = }}{r^3} $ 作为试函数Figure 1. Numerical results by using collocation method with different trial functions for solving homogeneous differential equation problem Eqs. (5) and (6). PIKF-12 represents
$ \phi {\text{ = }}{J_0}\left( {\sqrt 2 kr} \right) $ is the trial function and 12 nodes are placed at the boundary of computational domain; RBF-16/36 represents$ \phi {\text{ = }}{r^3} $ is the trial function and 16/36 nodes are uniformly placed in computational domain
图 2 物理信息依赖核函数配点法的节点离散示意图: (a) 奇异基本解
$ {G_F} $ 和(b)无奇异物理信息依赖核函数($ {G_H} $ ,$ {G_{RT}} $ ,$ {G_T} $ )Figure 2. Sketch of node discretization of PIKF collocation methods: (a) singular fundamental solutions
$ {G_F} $ and (b) nonsingular PIKFs ($ {G_H} $ ,$ {G_{RT}} $ ,$ {G_T} $ )
图 3 物理信息依赖核函数配点法的节点离散及局部子区域示意图: (a) 节点离散及节点
${\boldsymbol{x}}_1^k$ 的局部子区域${\varXi _k}$ ; (b)无奇异物理信息依赖核函数的局部子区域节点离散; (c)奇异物理信息依赖核函数的局部子区域节点离散Figure 3. Sketch of node discretization and local subdomain of PIKF collocation method: (a) node discretization and local subregion
${\varXi _k}$ for node${\boldsymbol{x}}_1^k$ ; (b) node discretization of local subdomain${\varXi _k}$ for nonsingular PIKFs; (c) node discretization of local subdomain${\varXi _k}$ for singular PIKFs
图 4 球体结构受不同频率简谐激励力作用在点
$\left( {2,0,0} \right)$ 处的声压级变化Figure 4. Sound pressure level results of spherical structure under simple harmonic excitation force on (2, 0, 0) with different frequencies
图 5 受100 Hz简谐激励力下球体结构附近区域的声压级分布
Figure 5. Sound pressure level distribution around the spherical structure under simple harmonic excitation force at frequency 100 Hz
图 7 SR=100的变半径圆环中轴线
${{r}_{{\rm{c}}}}=R-0.475+{0.45\theta }/{\text{π} }\;$ 上不同时刻的温度分布Figure 7. Temperature distributions along with the curve
${{r}_{{\rm{c}}}}=R-0.475+{0.45\theta }/{\text{π} }\;$ inside the FGM under different time instants
图 8 三维类肿瘤区域及可测边界测量点示意图
Figure 8. Schematic configuration of 3D tumor-like domain with measurement points (○).
图 9 物理信息依赖核函数配点法采用可测边界
${\varGamma _1}$ 上不同人工噪音水平的100个测量点数据反演得到位势问题的结果Figure 9. Numerical results obtained by using PIKF collocation method with 100 measurement points under different noise levels on accessible boundary
${\varGamma _1}$ 
图 10 三维数值波浪水槽和梯形潜堤横向剖面示意图
Figure 10. Schematic diagram of 3D numerical wave flume and transverse profile of trapezoidal submerged breakwater
图 11 最后两个周期内三维数值波浪水槽自由表面两个观测点处(G3和G5)的高程演化
Figure 11. Elevation evolutions at two gauges (G3 and G5) of 3D numerical wave flume in the last 2 periods
表 1 Web of Science数据库中各计算方法的论文发表数(搜索日期: 2022-9-5)
Table 1. Bibliographic database search of computational methods based on the Web of Science (search date: September 5, 2022)
Numerical methods Search phrase in topic field No. of entries FEM finite element(s) 585 197 FDM finite difference(s) 102 794 FVM finite volume(s) 39 616 BEM boundary element(s) or boundary integral(s) 34 105 DEM discrete element(s) 19 795 Meshless and particle methods collocation method(s) or meshless or meshfree or material point method or element-free or smoothed particle hydrodynamics 29 807
(11 192)
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表 2 常见微分方程算子的基本解
$ {G_F} $ [12]Table 2. Fundamental solutions
$ {G_F} $ of common-used differential equation operators[12]$ \Re $ 2D 3D $ \Delta $ $ - \dfrac{{\ln \left( r \right)}}{{2{\text{π }}}} $ $\dfrac{1}{ {4{\text{π}}r} }$ $ \Delta + {k^2} $ $\dfrac{ { {\text{i} }{\rm{H}}_0^{\left( 1 \right)}\left( {kr} \right)} }{4}$ $\dfrac{ { {{\rm{e}}^{ - {\text{i} }kr} } } }{ {4{\text{π} }r} }$ $ \Delta - {k^2} $ $\dfrac{ { {{\rm{K}}_0}(kr)} }{ {2{\text{π } } } }$ $\dfrac{ { {{\rm{e}}^{ - kr} } } }{ {4{\text{π} }r} }$ $D\Delta + {\boldsymbol{v} } \cdot \nabla - {k^2}$ $\dfrac{ { { {\rm{K} }_0}(\mu r){{\rm{e}}^{ - \tfrac{ { {\boldsymbol{v} } \cdot r} }{ {2 D} } } } } }{ {2{\text{π } } } }$ $\dfrac{ { {{\rm{e}}^{ - \mu r - \tfrac{ { {\boldsymbol{v} } \cdot r} }{ {2 D} } } } } }{ {4{\text{π} }r} }$ $ {\Delta ^2} $ $ \dfrac{{{r^2}\ln \left( r \right) - {r^2}}}{{8{\text{π }}}} $ $\dfrac{r}{ { { {8\text{π } } } } }$
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表 3 常见微分方程算子的调和函数
$ {G_H} $ 和径向Trefftz函数$ {G_{RT}} $ [12]Table 3. Harmonic functions
$ {G_H} $ and radial Trefftz functions$ {G_{RT}} $ of common-used differential equation operators[12]$ \Re $ 2D 3D $ \Delta $ ${{\rm{e}}^{ - c\left(r_1^2 - r_2^2\right)} }\cos (2 c{r_1}{r_2})$ ${{\rm{e}}^{ - c\left(r_1^2 - r_2^2\right)} }\cos \left(2 c{r_1}{r_2}\right)+ {{\rm{e}}^{ - c\left(r_2^2 - r_3^2\right)} }\cos \left(2 c{r_2}{r_3}\right)+ {{\rm{e}}^{ - c\left(r_3^2 - r_1^2\right)} }\cos \left(2 c{r_3}{r_1}\right)$ $ \Delta + {k^2} $ $\dfrac{1}{ {2{\text{π } } } }{{\rm{J}}_0}(kr)$ $\dfrac{ {\sin (kr)} }{ {4{\text{π} }r} }$ $ \Delta - {k^2} $ $\dfrac{1}{ {2{\text{π} } } }{{\rm{I}}_0}(kr)$ $\dfrac{ {\sinh (kr)} }{ {4{\text{π} }r} }$ $D\Delta + {\boldsymbol{v} } \cdot \nabla - {k^2}$ $\dfrac{1}{ {2{\text{π } } } }{ {\rm{I} }_0}(\mu r){{\rm{e}}^{ - \tfrac{ { {\boldsymbol{v} } \cdot {\boldsymbol{r} } } }{ {2 D} } } }$ $\dfrac{ {\sinh (\mu r)} }{ {4{\text{π} }r} }{{\rm{e}}^{ - \tfrac{ { {\boldsymbol{v} } \cdot {\boldsymbol{r} } } }{ {2 D} } } }$ $ {\Delta ^2} $ ${r^2}{{\rm{e}}^{ - c\left(r_1^2 - r_2^2\right)} }\cos (2 c{r_1}{r_2})$ ${r^2}\left[ { { {\rm{e} }^{ - c(r_1^2 - r_2^2)} }\cos (2 c{r_1}{r_2})} \right. + { {\rm{e} }^{ - c\left(r_2^2 - r_3^2\right)} }\cos (2 c{r_2}{r_3}) + \left. { { {\rm{e} }^{ - c\left(r_3^2 - r_1^2\right)} }\cos (2 c{r_3}{r_1})} \right]$
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表 4 常见微分方程算子的T完备函数
$ {G_T} $ [13]Table 4. T-complete functions
$ {G_T} $ of common-used differential equation operators[13]$ \Re $ 2D 3D $ \Delta $ $ 1 $ ${\rm{P}}_v^0\left( {\cos \varphi } \right)$ $ {\rho ^m}\cos \left( {m\theta } \right) $ ${\rho ^m}{\rm{P}}_v^m\left( {\cos \varphi } \right)\cos \left( {m\theta } \right)$ $ {\rho ^m}\sin \left( {m\theta } \right) $ ${\rho ^m}{\rm{P}}_v^m\left( {\cos \varphi } \right)\sin \left( {m\theta } \right)$ $ \Delta + {k^2} $ ${{\rm{J}}_0}\left( {k\rho } \right)$ ${{\rm{J}}_0}\left( {k\rho } \right){\rm{P}}_v^0\left( {\cos \varphi } \right)$ ${{\rm{J}}_m}\left( {k\rho } \right)\sin \left( {m\theta } \right)$ ${{\rm{J}}_m}\left( {k\rho } \right){\rm{P}}_v^m\left( {\cos \varphi } \right)\cos \left( {m\theta } \right)$ ${{\rm{J}}_m}\left( {k\rho } \right)\sin \left( {m\theta } \right)$ ${{\rm{J}}_m}\left( {k\rho } \right){\rm{P}}_v^m\left( {\cos \varphi } \right)\sin \left( {m\theta } \right)$ $ \Delta - {k^2} $ ${{\rm{I}}_0}\left( {k\rho } \right)$ ${{\rm{I}}_0}\left( {k\rho } \right){\rm{P}}_v^0\left( {\cos \varphi } \right)$ ${{\rm{I}}_m}\left( {k\rho } \right)\cos \left( {m\theta } \right)$ ${{\rm{I}}_m}\left( {k\rho } \right){\rm{P}}_v^m\left( {\cos \varphi } \right)\cos \left( {m\theta } \right)$ ${{\rm{I}}_m}\left( {k\rho } \right)\sin \left( {m\theta } \right) \\$ ${{\rm{I}}_m}\left( {k\rho } \right){\rm{P}}_v^m\left( {\cos \varphi } \right)\sin \left( {m\theta } \right)$ $ {\Delta ^2} $ $ {\rho ^2} $ ${\rho ^2}{\rm{P}}_v^0\left( {\cos \varphi } \right)$ $ {\rho ^{m{\text{ + }}2}}\cos \left( {m\theta } \right) $ ${\rho ^{m{\text{ + } }2} }{\rm{P}}_v^m\left( {\cos \varphi } \right)\cos \left( {m\theta } \right)$ $ {\rho ^{m{\text{ + }}2}}\sin \left( {m\theta } \right) $ ${\rho ^{m{\text{ + } }2} }{\rm{P}}_v^m\left( {\cos \varphi } \right)\sin \left( {m\theta } \right)$
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表 5 常见高阶微分方程算子的基本解
$ G_F^n $ [12]Table 5. Fundamental solutions
$ G_F^n $ of common-used high-order differential equation operators[12]$ {\Re ^n} $ 2D 3D $ {\Delta ^{n{\text{ + }}1}} $ $\dfrac{ { {r^{2 n} } } }{ { { {2\text{π} } } } }\left( { {C_n}\ln r - {B_n} } \right)$ $\dfrac{1}{ {\left( {2 n} \right)!} }\dfrac{ { {r^{2 n - 1} } } }{ { { {4\text{π} } } } }$ $ {\left( {\Delta + {k^2}} \right)^{n{\text{ + }}1}} $ ${A_n}{(kr)^{n + 1 - dim /2} }{\text{i} }{\rm{H} }_{n - 1 + dim /2}^{\left( 1 \right)}\left( {kr} \right)$ $ {\left( {\Delta - {k^2}} \right)^{n{\text{ + }}1}} $ ${A_n}{(kr)^{n + 1 - dim /2} }{ {\rm{K} }_{n - 1 + dim /2} }\left( {kr} \right)$ ${\left( {D\Delta + {\boldsymbol{v} } \cdot \nabla - {k^2} } \right)^{n{\text{ + } }1} }$ ${A_n}{(\mu r)^{n + 1 - dim /2} }{ {\rm{K} }_{n - 1 + dim /2} }\left( {\mu r} \right){ {\rm{e} }^{ - \tfrac{ { {\boldsymbol{v} } \cdot r} }{ {2 D} } } }$
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表 6 常见高阶微分方程算子的调和函数
$ G_H^n $ 和径向Trefftz函数$ G_{RT}^n $ [12]Table 6. Harmonic functions
$ G_H^n $ and radial Trefftz functions$ G_{RT}^n $ of common-used high-order differential equation operators[12]$ \Re $ 2D 3D $ {\Delta ^{n{\text{ + }}1}} $ ${r^{2 n} }{{\rm{e}}^{ - c\left(r_1^2 - r_2^2\right)} }\cos (2 c{r_1}{r_2})$ $\begin{gathered}{r^{2 n} }\left[ { { {\rm{e} }^{ - c(r_1^2 - r_2^2)} }\cos (2 c{r_1}{r_2})} \right. + \\ { {\rm{e} }^{ - c(r_2^2 - r_3^2)} }\cos (2 c{r_2}{r_3}) \left. { + { {\rm{e} }^{ - c(r_3^2 - r_1^2)} }\cos (2 c{r_3}{r_1})} \right]\end{gathered}$ $ {\left( {\Delta + {k^2}} \right)^{n{\text{ + }}1}} $ ${A_n}{(kr)^{n + 1 - dim /2} }{ {\rm{J} }_{n - 1 + dim /2} }\left( {kr} \right)$ $ {\left( {\Delta - {k^2}} \right)^{n{\text{ + }}1}} $ ${A_n}{(kr)^{n + 1 - dim /2} }{ {\rm{I} }_{n - 1 + dim /2} }\left( {kr} \right)$ ${\left( D\Delta + {\boldsymbol{v} } \cdot \nabla - {k^2} \right)^{n{\text{ + } }1} }$ ${A_n}{(\mu r)^{n + 1 - dim /2} }{ {\rm{I} }_{n - 1 + dim /2} }\left( {\mu r} \right){ {\rm{e} }^{ - \frac{ { {\boldsymbol{v} } \cdot r} }{ {2 D} } } }$
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表 7 常见高阶微分方程算子的T完备函数
$ G_T^n $ [23]Table 7. T-complete functions
$ G_{RT}^n $ of common-used high-order differential equation operators[23]$ \Re $ 2D 3D $ {\Delta ^{n{\text{ + }}1}} $ $ {\rho ^{2 n}} $ ${\rho ^{2 n} }{\rm{P}}_v^0\left( {\cos \varphi } \right)$ $ {\rho ^{m{\text{ + }}2 n}}\cos \left( {m\theta } \right) $ ${\rho ^{m{\text{ + } }2 n} }{\rm{P}}_v^m\left( {\cos \varphi } \right)\cos \left( {m\theta } \right)$ $ {\rho ^{m{\text{ + }}2 n}}\sin \left( {m\theta } \right) $ ${\rho ^{m{\text{ + } }2 n} }{\rm{P}}_v^m\left( {\cos \varphi } \right)\sin \left( {m\theta } \right)$ $ {\left( {\Delta + {k^2}} \right)^{n{\text{ + }}1}} $ ${D_n}{{\rm{J}}_n}\left( {k\rho } \right)$ ${D_n}{{\rm{J}}_n}\left( {k\rho } \right){\rm{P}}_v^0\left( {\cos \varphi } \right)$ ${D_n}{{\rm{J}}_{m + n} }\left( {k\rho } \right)\cos \left( {m\theta } \right)$ ${D_n}{{\rm{J}}_{m + n} }\left( {k\rho } \right){\rm{P}}_v^m\left( {\cos \varphi } \right)\cos \left( {m\theta } \right)$ ${D_n}{{\rm{J}}_{m + n} }\left( {k\rho } \right)\sin \left( {m\theta } \right)$ ${D_n}{{\rm{J}}_{m + n} }\left( {k\rho } \right){\rm{P}}_v^m\left( {\cos \varphi } \right)\sin \left( {m\theta } \right)$ $ {\left( {\Delta - {k^2}} \right)^{n{\text{ + }}1}} $ ${D_n}{{\rm{I}}_n}\left( {k\rho } \right)$ ${D_n}{{\rm{I}}_n}\left( {k\rho } \right){\rm{P}}_v^0\left( {\cos \varphi } \right)$ ${D_n}{{\rm{I}}_{m + n} }\left( {k\rho } \right)\cos \left( {m\theta } \right)$ ${D_n}{{\rm{I}}_{m + n} }\left( {k\rho } \right){\rm{P}}_v^m\left( {\cos \varphi } \right)\cos \left( {m\theta } \right)$ ${D_n}{{\rm{I}}_{m + n} }\left( {k\rho } \right)\sin \left( {m\theta } \right)$ ${D_n}{{\rm{I}}_{m + n} }\left( {k\rho } \right){\rm{P}}_v^m\left( {\cos \varphi } \right)\sin \left( {m\theta } \right)$
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表 8 时间依赖微分方程算子的基本解
$ G_F^{} $ Table 8. Fundamental solutions
$ G_F^{} $ of time-dependent differential equation operators$ {{{\partial ^m}} \mathord{\left/ {\vphantom {{{\partial ^m}} {\partial {t^m}}}} \right. } {\partial {t^m}}} - \Re $ 2D 3D $\dfrac{\partial }{ {\partial t} } - k\Delta$ $\dfrac{ {\varTheta (t - \tau ){ {\rm{e} }^{ {r^2}/[4 k(t - \tau )]} } } }{ {4\text{π} k(t - \tau )} }$ $\dfrac{ {\varTheta (t - \tau ){ {\rm{e} }^{ {r^2}/[4 k(t - \tau )]} } } }{ { { {\left[ {4\text{π} k(t - \tau )} \right]}^{3/2} } } }$ $\dfrac{ { {\partial ^2} } }{ {\partial {t^2} } } - {c_1}\Delta$ $\dfrac{ { { {\varTheta } }\left( {t - {r \mathord{\left/ {\vphantom {r { {c_1} } } } \right. } { {c_1} } } } \right)} }{ {2\text{π} {c_1}\sqrt {c_1^2{t^2} - {r^2} } } }$ $\dfrac{ { { {\varTheta } }\left( {t - {r \mathord{\left/ {\vphantom {r { {c_1} } } } \right. } { {c_1} } } } \right)} }{ {4\text{π} r} }$
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表 9 时间依赖微分方程算子的径向Trefftz函数
$ G_{RT}^{} $ Table 9. Radial Trefftz functions
$ G_{RT}^{} $ of time-dependent differential equation operators$ {{{\partial ^m}} \mathord{\left/ {\vphantom {{{\partial ^m}} {\partial {t^m}}}} \right. } {\partial {t^m}}} - \Re $ 2D 3D $\dfrac{\partial }{ {\partial t} } - k\Delta$ ${ {\rm{e} }^{ - k(t - \tau )} }{{\rm{J}}_0}(r)$ ${{\rm{e}}^{ - k(t - \tau )} }\frac{ {\sin (r)} }{r}$ $\dfrac{ { {\partial ^2} } }{ {\partial {t^2} } } - {c_1}\Delta$ $\begin{gathered}\cos \left[ { {c_1}\left( {t - \tau } \right)} \right]{{\rm{J}}_0}(r) + \\ \sin \left[ { {c_1}\left( {t - \tau } \right)} \right]{{\rm{J}}_0}(r) \end{gathered}$ $\begin{gathered}\frac{ {\cos \left[ { {c_1}\left( {t - \tau } \right)} \right]\sin \left( r \right)} }{r} +\\ \frac{ {\sin \left[ { {c_1}\left( {t - \tau } \right)} \right]\sin \left( r \right)} }{ { {c_1}r} }\end{gathered}$
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表 10 物理信息依赖核函数配点法求解多个时刻不同尺寸比情况下的计算结果(Merr)
Table 10. Numerical results (Merr) obtained by using PIKF collocation method at varied time instants under different SRs
t/s SR = 20 SR = 60 SR = 100 0.4 1.19×10−7 1.19×10−7 3.10×10−5 2 2.35×10−7 2.35×10−7 2.35×10−7 10 1.32×10−6 1.32×10−6 1.32×10−6
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表 11 物理信息依赖核函数配点法采用可测边界上含5%人工噪音数据反演得到的结果
Table 11. Numerical results obtained by using PIKF collocation method with different boundary measurement points on the accessible boundary under 5% noise level
NT Cond/1019 Rerr(u)/% Rerr(q)/% kSVD 75 7.06 6.50 9.29 42 100 8.64 5.92 8.31 54 125 5.01 4.77 6.48 42
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