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物理信息依赖核函数配点法的研究进展

傅卓佳 李明娟 习强 徐文志 刘庆国

傅卓佳, 李明娟, 习强, 徐文志, 刘庆国. 物理信息依赖核函数配点法的研究进展. 力学学报, 2022, 54(12): 1-14 doi: 10.6052/0459-1879-22-485
引用本文: 傅卓佳, 李明娟, 习强, 徐文志, 刘庆国. 物理信息依赖核函数配点法的研究进展. 力学学报, 2022, 54(12): 1-14 doi: 10.6052/0459-1879-22-485
Fu Zhuojia, Li Mingjuan, Xi Qiang, Xu Wenzhi, Liu Qingguo. Research advances on the collocation methods based on the physical-informed kernel functions. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 1-14 doi: 10.6052/0459-1879-22-485
Citation: Fu Zhuojia, Li Mingjuan, Xi Qiang, Xu Wenzhi, Liu Qingguo. Research advances on the collocation methods based on the physical-informed kernel functions. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 1-14 doi: 10.6052/0459-1879-22-485

物理信息依赖核函数配点法的研究进展

doi: 10.6052/0459-1879-22-485
基金项目: 国家自然科学基金(12122205), 斯洛文尼亚共和国研究活动公共机构 (ARRS) (P2-0162, L2-3173)资助项目
详细信息
    作者简介:

    傅卓佳, 教授, 主要研究方向: 计算力学与工程仿真. E-mail:paul212063@hhu.edu.cn

  • 中图分类号: O302

RESEARCH ADVANCES ON THE COLLOCATION METHODS BASED ON THE PHYSICAL-INFORMED KERNEL FUNCTIONS

  • 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.

     

  • 图  1  配点法采用不同试函数求解齐次微分方程问题(5)(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 (5)(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} $; (b)无奇异物理信息依赖核函数($ {G_H} $,$ {G_{RT}} $,$ {G_T} $).

    Figure  2.  Sketch of node discretization of PIKF collocation methods: (a) singular fundamental solutions $ {G_F} $; (b) nonsingular PIKFs ($ {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 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  6  功能梯度材料变半径圆环区域示意图

    Figure  6.  Schematic configuration of variable radius FGM ring

    图  7  SR=100的变半径圆环中轴线${{r}_{1}}=R-0.475+{0.45\theta }/{\text{π} }\;$上不同时刻($t=0.4,2,10\;{\rm{s}} $)的温度分布

    Figure  7.  Temperature distributions along with the curve ${{r}_{1}}=R-0.475+{0.45\theta }/{\text{π} }\;$ 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(a) u(x)和(b) q(x).

    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(a)${x_1} = 12.5\;{\rm{m}}$, (b)${x_1} = 14.5\;{\rm{m}}$.

    Figure  11.  Elevation evolutions at two gauges of 3D numerical wave flume in the last 2 periods, (a)${x_1} = 12.5\;{\rm{m}}$, (b)${x_1} = 14.5\;{\rm{m}}$.

    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 methodsSearch phrase in topic fieldNo. of entries
    FEMfinite element(s)585197
    FDMfinite difference(s)102794
    FVMfinite volume(s)39616
    BEMboundary element(s) or boundary integral(s)34105
    DEMdiscrete element(s)19795
    Meshless and Particle methodscollocation method(s) or meshless or meshfree or material point method or element-free or smoothed particle hydrodynamics29807

    (11192)
    下载: 导出CSV

    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 $2D3D
    $ \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}$$\dfrac{ { {{\rm{K}}_0}(\mu r){e^{ - \tfrac{ { {\boldsymbol{v} } \cdot r} }{ {2 D} } } } } }{ {2{\text{π } } } }$$\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 } } } } }$
    下载: 导出CSV

    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 $2D3D
    $ \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}$$\dfrac{1}{ {2{\text{π } } } }{{\rm{I}}_0}(\mu r){e^{ - \tfrac{ { {\boldsymbol{v} } \cdot {\boldsymbol{r} } } }{ {2 D} } } }$$\dfrac{ {\sinh (\mu r)} }{ {4{\text{π} }r} }{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)$
    下载: 导出CSV

    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 $2D3D
    $ \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} $$ {I_0}\left( {k\rho } \right) $${{\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)$
    下载: 导出CSV

    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} $2D3D
    $ {\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} } } }$
    下载: 导出CSV

    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 $2D3D
    $ {\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} } } }$
    下载: 导出CSV

    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 $2D3D
    $ {\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)$
    下载: 导出CSV

    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 $2D3D
    $\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} }$
    下载: 导出CSV

    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 $2D3D
    $\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$$\begin{gathered}\cos \left( { {c_1}\left( {t - \tau } \right)} \right){J_0}(r) + \\ \sin \left( { {c_1}\left( {t - \tau } \right)} \right){J_0}(r) \end{gathered}$$\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}$
    下载: 导出CSV

    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/sSR = 20SR = 60SR = 100
    0.41.19×10−71.19×10−73.10×10−5
    22.35×10−72.35×10−72.35×10−7
    101.32×10−61.32×10−61.32×10−6
    下载: 导出CSV

    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.

    NTCond/1019Rerr(u)/%Rerr(q)/%kSVD
    757.066.509.2942
    1008.645.928.3154
    1255.014.776.4842
    下载: 导出CSV
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  • 收稿日期:  2022-10-11
  • 录用日期:  2022-11-16
  • 网络出版日期:  2022-11-17

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