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An Element-Free Galerkin Method for Solving Two-Dimensional Helmholtz Equation
DOI: 10.12677/AAM.2019.88154, PDF, HTML, XML, 下载: 908  浏览: 3,995

Abstract: The element-free Galerkin method is originally an important method for numerical simulation of solid mechanics problems. In this paper, we use this method to solve the Helmholtz equation and verify the numerical simulation results. In the process of solving Helmholtz equation, the shape function is constructed by moving least squares approximation, then the boundary condition is introduced by Lagrange multiplier method. Finally, the general Helmholtz equation on the plane rule domain is numerically simulated. The obtained results tend to be exact solutions. Furthermore, high precision and good convergence could be contained as the nodes increased.

1. 引言

2. 移动最小二乘近似(MLS)

$u\left(x\right)\approx {u}^{h}\left(x,{x}_{i}\right)=\underset{i\in I\left(x\right)}{\sum }{\varphi }_{i}\left(x\right)u\left({x}_{i}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,I\left(x\right)$ (1)

(1)式中 ${\varphi }_{i}\left(x\right)$ 是形函数， $u\left({x}_{i}\right)$ 是在节点 ${x}_{i}$ 上的节点参数， $I\left(x\right)$ 表示在支持域 $x$ 中的节点指标集，定义如下所示：

$I\left(x\right)=\left\{i\in \left\{1,2,\cdots ,N\right\}:{‖x-{x}_{i}‖}_{2}\le \delta \right\}$

$\delta$ 为支持域的半径。

(1)式可由近似函数 ${u}^{h}\left(x\right)$ 在这些节点 $x={x}_{i}$ 处误差的加权平方和推导可得，如下：

${u}^{h}\left(x,{x}_{i}\right)=\underset{i\in I\left(x\right)}{\sum }{p}_{i}\left({x}_{i}\right){a}_{i}\left(x\right)={p}^{\text{T}}\left({x}_{i}\right)a\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\left(x\right)\in {P}_{m}\left({R}^{d}\right)$ (2)

$J\left(a\right)=\underset{i\in I\left(x\right)}{\sum }{\left({u}^{h}\left(x,{x}_{i}\right)-u\left({x}_{i}\right)\right)}^{\text{2}}w\left(x,{x}_{i}\right)$ (3)

$p\left(x\right)$ 为基函数， ${P}_{m}\left({R}^{d}\right)$ 中m指的是基函数的最高次幂，d指的是空间的维数。则函函数的个数为 $Q=\left(\begin{array}{c}m+d\\ d\end{array}\right)$

$\frac{\partial J}{\partial a}=0$

${\Phi }^{\text{T}}\left(x\right):=\left[{\varphi }_{1}\left(x\right),{\varphi }_{2}\left(x\right),\cdots ,{\varphi }_{|I\left(x\right)|}\left(x\right)\right]={p}^{\text{T}}\left(x\right){A}^{-1}\left(x\right)B\left(x\right)$ (4)

$A\left(x\right)=\underset{i\in I\left(x\right)}{\sum }w\left(x,{x}_{i}\right)p\left({x}_{i}\right){p}^{\text{T}}\left({x}_{i}\right)$ (5)

$B\left(x\right)=\left[w\left(x,{x}_{1}\right)p\left({x}_{1}\right)\text{ }w\left(x,{x}_{2}\right)p\left({x}_{2}\right)\text{ }\cdots \text{ }w\left(x,{x}_{I\left(x\right)}\right)p\left({x}_{I\left(x\right)}\right)\right]$ (6)

$w\left(x,{x}_{i}\right)=\left\{\begin{array}{l}\frac{\mathrm{exp}\left(-{\epsilon }^{2}{r}^{2}\right)-\mathrm{exp}\left(-{\epsilon }^{2}\right)}{1-\mathrm{exp}\left(-{\epsilon }^{2}\right)},\text{ }0\le r\le 1\\ 0,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}r>1\end{array}$ (7)

$r:=‖x-{x}_{i}‖/\delta$ ，其中 $\delta$ 为节点 ${x}_{i}$ 的支撑域半径，且 $\epsilon$ 是一个正参数。

3. Helmholtz方程的离散格式

$\left\{\begin{array}{l}{\nabla }^{2}u+{k}^{2}u=f,\text{ }\text{\hspace{0.17em}}\text{ }x在\Omega 内\\ u=\stackrel{¯}{u},\text{ }\text{ }\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}x在{\Gamma }_{u}上\\ \frac{\partial u}{\partial n}=q,\text{ }\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x在{\Gamma }_{t}上\end{array}$ (8)

$E=\frac{\text{1}}{\text{2}}{\int }_{\Omega }{u}_{x}^{2}+{u}_{y}^{2}+{k}^{2}{u}^{2}\text{d}\Omega -{\int }_{\Omega }{u}^{\text{T}}f\text{d}\Omega -{\int }_{{\Gamma }_{t}}{u}^{\text{T}}q\text{d}\Gamma$ (9)

$\delta E=\delta \left(\frac{\text{1}}{\text{2}}{\int }_{\Omega }{u}_{x}^{2}+{u}_{y}^{2}+{k}^{2}{u}^{2}\text{d}\Omega -{\int }_{\Omega }{u}^{\text{T}}f\text{d}\Omega -{\int }_{{\Gamma }_{t}}{u}^{\text{T}}q\text{d}\Gamma \right)$ (10)

$\delta E={\int }_{\Omega }{\left(\delta u\right)}_{x}{u}_{x}+{\left(\delta u\right)}_{y}{u}_{y}+\left(\delta u\right){k}^{2}u\text{d}\Omega -{\int }_{\Omega }\delta {u}^{\text{T}}f\text{d}\Omega -{\int }_{{\Gamma }_{t}}\delta {u}^{\text{T}}q\text{d}\Gamma$ (11)

${\int }_{\Omega }{\left(\delta u\right)}_{x}{u}_{x}+{\left(\delta u\right)}_{y}{u}_{y}+\left(\delta u\right){k}^{2}u\text{d}\Omega -{\int }_{\Omega }\delta {u}^{\text{T}}f\text{d}\Omega -{\int }_{{\Gamma }_{t}}\delta {u}^{\text{T}}q\text{d}\Gamma =0$ (12)

$\delta u=\Phi \left(x\right)$ ，将(1)式代入(12)式可得：

$\underset{i\in I\left(x\right)}{\sum }\underset{j=1}{\overset{N}{\sum }}\left(-{\int }_{\Omega }\nabla {\varphi }_{i}\cdot \nabla {\varphi }_{j}\text{d}\Omega +{\int }_{\Omega }{k}^{2}{\varphi }_{i}\cdot {\varphi }_{j}\text{d}\Omega \right){u}_{i}=\underset{i\in I\left(x\right)}{\sum }{\int }_{\Omega }{\varphi }_{i}^{\text{T}}f\text{d}\Omega +\underset{i\in I\left(x\right)}{\sum }{\int }_{{\Gamma }_{t}}{\varphi }_{i}^{\text{T}}q\text{d}\Gamma$ (13)

$K{U}_{N}=F$ (14)

${\int }_{{\Gamma }_{u}}{\lambda }^{\text{T}}\left(u-\stackrel{¯}{u}\right)\text{d}\Gamma$

$\begin{array}{l}{\int }_{\Omega }{\left(\delta u\right)}_{x}{u}_{x}+{\left(\delta u\right)}_{y}{u}_{y}+\left(\delta u\right){k}^{2}u\text{d}\Omega -{\int }_{\Omega }\delta {u}^{\text{T}}f\text{d}\Omega -{\int }_{{\Gamma }_{t}}\delta {u}^{\text{T}}q\text{d}\Gamma \\ \text{ }-{\int }_{{\Gamma }_{u}}\delta {\lambda }^{\text{T}}\left(u-\stackrel{¯}{u}\right)\text{d}\Gamma -{\int }_{{\Gamma }_{u}}\delta {u}^{\text{T}}\lambda \text{d}\Gamma =0\end{array}$ (15)

$\delta \lambda =N\left(s\right)$ ，以及将(1)代入式(15)中，可得：

$\left[\begin{array}{cc}K& G\\ {G}^{\text{T}}& 0\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]=\left[\begin{array}{c}F\\ Q\end{array}\right]$ (16)

$G={\left[-{\int }_{{\Gamma }_{u}}{N}_{i}^{\text{T}}{\varphi }_{j}\text{d}\Gamma \right]}_{1\le j\le N,1\le i\le m}$$Q={\left[-{\int }_{{\Gamma }_{u}}{N}_{i}^{\text{T}}\stackrel{¯}{u}\text{d}\Gamma \right]}_{1\le j\le N,1\le i\le m}$

4. 数值算例

${\nabla }^{\text{2}}u+{k}^{2}u=f$ (17)

$u=\mathrm{sin}\sqrt{{k}^{2}/2}x\mathrm{cos}\sqrt{{k}^{2}/2}y$ (18)

Figure 1. Discrete nodes in regular domain

Figure 2. Meshless method for numerical solution of cloud maps

$erru=\frac{\sqrt{\underset{i=1}{\overset{N}{\sum }}{\left({u}^{h}\left({x}_{i}\right)-u\left({x}_{i}\right)\right)}^{2}}}{\sqrt{\underset{i=1}{\overset{N}{\sum }}{\left(u\left({x}_{i}\right)\right)}^{2}}}$

Table 1. Error analysis of different nodes

5. 结语

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