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An Analysis of Fat Boundary Method with Respect to Algorithmic Parameters and an Interpolation Method for Dual Fat Boundary Method
DOI: 10.12677/IJM.2021.101005, PDF, HTML, XML, 下载: 262  浏览: 325

Abstract: Finite Element Method (FEM) has achieved a great success both in the field of academic and engi-neering. However, there are some problems for FEM. For example, the mesh generation procedure consumes a lot of time for an engineering problem to avoid mesh distortion. Some fictitious domain methods have been developed to tackle this problem. Recently, the Fat Boundary Method (FBM) has been proposed and applied to elasticity and as an improvement of FBM, the Dual Fat Boundary Method (DFBM) is proposed. These methods need an iteration procedure. In this article, the algorithmic parameters related to the iteration of FBM are studied and an interpolation method based on DFBM is proposed to reduce computational costs.

1. 引言

2. 弹性力学宽边界法与双宽边界法

$\left\{\begin{array}{l}-\nabla \cdot \sigma =f\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\Omega \\ {u}_{|{\Gamma }_{D}}={g}_{1},\text{\hspace{0.17em}}{u}_{|{\Gamma }_{0}}={g}_{2}\\ {\left(\sigma \cdot n\right)}_{|{\Gamma }_{N}}=t\end{array}$ (1)

$\left\{\begin{array}{l}\text{I}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{\begin{array}{l}-\nabla \cdot \sigma \left({u}_{1}\right)=f\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{\Omega }_{1}\\ {u}_{1|{\Gamma }_{0}}={g}_{2},{u}_{1|{\Gamma }_{1}}={\stackrel{¯}{u}}_{|{\Gamma }_{1}}\end{array}\\ \text{II}.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{\begin{array}{l}-\nabla \cdot \sigma \left(\stackrel{¯}{u}\right)=\stackrel{¯}{f}+\left(\sigma \left({u}_{1}\right)\cdot n-\sigma \left({u}_{0}\right)\cdot n\right){\delta }_{{\Gamma }_{0}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{\Omega }_{fict}\\ {\stackrel{¯}{u}}_{|{\Gamma }_{D}}={g}_{1},{\left(\sigma \left(\stackrel{¯}{u}\right)\cdot n\right)}_{|{\Gamma }_{N}}=t\end{array}\end{array}$ (2)

$\left\{\begin{array}{l}-\nabla \cdot \sigma \left({u}_{0}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{\Omega }_{0}\\ {u}_{0|{\Gamma }_{0}}={g}_{2}\end{array}$ (3)

Figure 1. FBM and DFBM of elasticity: problem definition

${\mathcal{T}}_{\theta }^{h}:\left(\stackrel{¯}{u},{u}_{1}\right)\to \left(\stackrel{¯}{U},{U}_{1}\right)$ (4)

$\left\{\begin{array}{l}\text{I}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{\begin{array}{l}-\nabla \cdot \sigma \left({U}_{1}\right)=f\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{\Omega }_{1}\\ {U}_{1|{\Gamma }_{0}}={g}_{2},{U}_{1|{\Gamma }_{1}}={\left(\theta {u}_{1}+\left(1-\theta \right)\stackrel{¯}{u}\right)}_{|{\Gamma }_{1}}\end{array}\\ \text{II}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{\begin{array}{l}-\nabla \cdot \sigma \left(\stackrel{¯}{U}\right)=\stackrel{¯}{f}+\left(\sigma \left({U}_{1}\right)\cdot n-\sigma \left({u}_{0}\right)\cdot n\right){\delta }_{{\Gamma }_{0}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{\Omega }_{fict}\\ {\stackrel{¯}{U}}_{|{\Gamma }_{D}}={g}_{1},{\left(\sigma \left(\stackrel{¯}{f}\right)\cdot n\right)}_{|{\Gamma }_{N}}=t\end{array}\end{array}$ (5)

$\left\{\begin{array}{l}\text{I}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{\begin{array}{l}-\nabla \cdot \sigma \left({u}_{1}\right)=f\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{\Omega }_{1}\\ {u}_{1|{\Gamma }_{0}}={g}_{2},{u}_{1|{\Gamma }_{1}}={\stackrel{¯}{u}}_{|{\Gamma }_{1}}\end{array}\\ \text{II}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{\begin{array}{l}-\nabla \cdot \sigma \left(\stackrel{¯}{u}\right)=\stackrel{¯}{f}+\left(\sigma \left({u}_{1}\right)\cdot n-\sigma \left({u}_{2}\right)\cdot n\right){\delta }_{{\Gamma }_{0}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{\Omega }_{fict}\\ {\stackrel{¯}{u}}_{|{\Gamma }_{D}}={g}_{1},{\left(\sigma \left(\stackrel{¯}{u}\right)\cdot n\right)}_{\mid {\Gamma }_{N}}=t\end{array}\\ \text{III}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{\begin{array}{l}-\nabla \cdot \sigma \left({u}_{2}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{\Omega }_{2}\\ {u}_{2|{\Gamma }_{0}}={g}_{2},{u}_{2|{\Gamma }_{2}}={\stackrel{¯}{u}}_{|{\Gamma }_{2}}\end{array}\end{array}$ (6)

${{\mathcal{T}}^{\prime }}_{\theta }^{h}:\left(\stackrel{¯}{u},{u}_{1},{u}_{2}\right)\to \left(\stackrel{¯}{U},{U}_{1},{U}_{2}\right)$ (7)

$\left\{\begin{array}{l}\text{I}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{\begin{array}{l}-\nabla \cdot \sigma \left({U}_{1}\right)=f\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{\Omega }_{1}\\ {U}_{1|{\Gamma }_{0}}={g}_{2},{U}_{1|{\Gamma }_{1}}={\left(\theta {u}_{1}+\left(1-\theta \right)\stackrel{¯}{u}\right)}_{|{\Gamma }_{1}}\end{array}\\ \text{II}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{\begin{array}{l}-\nabla \cdot \sigma \left(\stackrel{¯}{U}\right)=\stackrel{¯}{f}+\left(\sigma \left({U}_{1}\right)\cdot n-\sigma \left({U}_{2}\right)\cdot n\right){\delta }_{{\Gamma }_{0}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{\Omega }_{fict}\\ {\stackrel{¯}{U}}_{|{\Gamma }_{D}}={g}_{1},{\left(\sigma \left(\stackrel{¯}{U}\right)\cdot n\right)}_{\mid {\Gamma }_{N}}=t\end{array}\\ \text{III}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{\begin{array}{l}-\nabla \cdot \sigma \left({U}_{2}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}{\Omega }_{2}\\ {U}_{2|{\Gamma }_{0}}={g}_{2},{U}_{2|{\Gamma }_{2}}={\left(\theta {u}_{2}+\left(1-\theta \right)\stackrel{¯}{u}\right)}_{|{\Gamma }_{2}}\end{array}\end{array}$ (8)

3. 基于双宽边界法的差值方法

4. 数值算例

${e}^{m}=\frac{{‖{u}^{m}-{u}^{a}‖}_{E}}{{‖{u}^{a}‖}_{E}}=\sqrt{\frac{{\int }_{\Omega }\left({ϵ}^{m}-{ϵ}^{a}\right):\left({\sigma }^{m}-{\sigma }^{a}\right)}{{\int }_{\Omega }{ϵ}^{a}:{\sigma }^{a}}}$ (9)

${e}_{rel}^{m+1}=\frac{|{e}^{m+1}-{e}^{m}|}{{e}^{m}}$ (10)

4.1. 孔边固定的开孔板的单轴拉伸问题

$\begin{array}{l}{u}_{r}=\frac{{T}_{x}}{8\mu r}\left\{\left(\chi -1\right)\left({r}^{2}-{R}^{2}\right)+\left[-\frac{2}{\chi }\left(\chi +1\right){R}^{2}+2{r}^{2}+\frac{2{R}^{4}}{\chi {r}^{2}}\right]\mathrm{cos}2\theta \right\}\\ {u}_{\theta }=-\frac{{T}_{x}}{8\mu r}\left\{-\frac{2}{\chi }\left(\chi -1\right){R}^{2}+2{r}^{2}-\frac{2{R}^{4}}{\chi {r}^{2}}\right\}\mathrm{sin}2\theta \end{array}$ (11)

Figure 2. Perforated plate with a fixed hole

Figure 3. Meshes of FBM

Figure 4. Iteration errors for different meshes

Figure 5. Displacement error of global domain and local domain

Figure 6. Comparison of errors of global domain and local domain

Figure 7. Comparison of element numbers of FBM and FEM

Figure 8. Iteration errors for different initializations

4.2. 孔边径向位移的开孔板问题

${u}_{r}=\frac{\epsilon R}{r},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{\theta }=0$ (12)

Figure 9. Perforated plate with a expanded hole

Figure 10. Meshes of DFBM

Figure 11. Meshes of DFBM

Figure 12. Displacement error of global domain and local domain

Figure 13. Comparison of errors of global domain and local domain

5. 结论

NOTES

*通讯作者。

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