«上一篇
 文章快速检索 高级检索

 哈尔滨工程大学学报  2019, Vol. 40 Issue (4): 683-688  DOI: 10.11990/jheu.201709125 0

### 引用本文

LI Yulong, LIAO Honglie, HU Zhan, et al. 3D partitioned fluid-structure analysis based on non-matching meshes[J]. Journal of Harbin Engineering University, 2019, 40(4), 683-688. DOI: 10.11990/jheu.201709125.

### 文章历史

1. 中山大学 海洋工程与技术学院, 广东 广州 119077;
2. 广州船舶及海洋工程设计研究院, 广东 广州 119077;
3. 中山大学 海洋科学学院, 广东 广州 119077

3D partitioned fluid-structure analysis based on non-matching meshes
LI Yulong 1, LIAO Honglie 2, HU Zhan 3, OU Suying 1
1. College of Marine Engineering and Technology, Sun Yat-sen University;
2. Guangzhou Marine Engineering Corporation Guangzhou 119077, China;
3. College of Marine Science, Sun Yat-sen University, Guangzhou 119077, China
Abstract: To investigate an accurate and reliable numerical calculation method to solve the non-matching mesh problem in fluid-structure coupling, this paper presents a 3D common-refinement method for non-matching meshes between discrete non-overlapping sub-domains of incompressible fluid and nonlinear elastic structure.The incompressible fluid flow was discretized by using a stabilized Petrov-Galerkin finite element method (FEM), and the large deformation structural formulation was discretized by using a continuous Galerkin FEM.An arbitrary Lagrangian-Eulerian formulation was used to process large deformation of fluid-structure mesh, and the fully decoupled implicit partition procedure was applied to solutions of the fluid and solid phases.To satisfy traction equilibrium condition along the fluid-elastic interface with non-matching meshes, this study investigates the accuracy and reliability of the spatial interpolation in the common refinement method (CRM).According to a series of mesh division schemes, this study systematically assessed the accuracy and precision of CRM by varying grid mismatch between the fluid and solid meshes, thereby demonstrating the second-order accuracy of CRM data transmission through uniform refinements of fluid and solid meshes along the interface.This method was further applied to a 3D benchmark problem of cylinder-elastic plate and compared with the standard solution in the literature.Results show that this method is accurate and reliable in solving the fluid-structure coupling problem.
Keywords: fluid-structure coupling    non-matching mesh    common refinement    domain decomposition    finite element method

1 流体-结构控制方程

 $\begin{array}{*{20}{c}} {{\rho ^f}\frac{{\partial {{\mathit{\boldsymbol{\bar u}}}^f}}}{{\partial t}}\left| {_{\hat x}} \right. + {\rho ^f}\left( {{{\mathit{\boldsymbol{\bar u}}}^f} - {{\mathit{\boldsymbol{\bar w}}}^f}} \right) \cdot \nabla {{\mathit{\boldsymbol{\bar u}}}^f} = \nabla \cdot {{\bar \sigma }^f} + }\\ {\nabla \cdot {\mathit{\boldsymbol{\sigma }}^{{\rm{sgs}}}} + {\mathit{\boldsymbol{b}}^f},{\rm{on}}\;{\mathit{\Omega }^{\rm{f}}}\left( {\rm{t}} \right)} \end{array}$ (1)
 $\nabla \cdot {{\mathit{\boldsymbol{\bar u}}}^f} = 0,{\rm{on}}\;{\mathit{\Omega }^f}\left( t \right)$ (2)

 ${t^f}\left( {{x^f}} \right) \approx \sum\limits_{i = 1}^{{m_f}} {N_i^ft_i^f,{t^s}\left( {{x^s}} \right)} \approx \sum\limits_{j = 1}^{{m_s}} {N_j^st_j^s}$ (16)

 $\int_{{{\rm{\Gamma }}^{fs}}} {N_i^s{t^s}{\rm{d}}\mathit{\Gamma }} = \int_{{{\rm{\Gamma }}^{fs}}} {N_i^s{t^f}{\rm{d}}\mathit{\Gamma }}$ (17)

 $\int_{{{\rm{\Gamma }}^{fs}}} {N_i^sN_j^s\tilde t_j^s{\rm{d}}\mathit{\Gamma }} = \int_{{{\rm{\Gamma }}^{fs}}} {N_i^sN_j^f\tilde t_j^f{\rm{d}}\mathit{\Gamma }}$ (18)

 $\tilde t_j^s = \mathit{\boldsymbol{M}}_{ij}^s{}^{ - 1}\mathit{\boldsymbol{f}}_i^s$ (19)

 $\mathit{\boldsymbol{M}}_{ij}^s = \int_{{{\rm{\Gamma }}^{fs}}} {N_i^sN_j^s{\rm{d}}\mathit{\Gamma }}$ (20)

 $\mathit{\boldsymbol{f}}_i^s = \sum\limits_{j = 1}^{{m_f}} {\mathit{\boldsymbol{\tilde t}}_j^f\int_{{{\rm{\Gamma }}^{fs}}} {N_j^fN_i^s{\rm{d}}\mathit{\Gamma }} }$ (21)

 ${x^f} \approx \sum\limits_{i = 1}^{{m_f}} {N_i^f\left( x \right)x_i^f\;{\rm{on}}\;\mathit{\Gamma }_{\rm{h}}^f}$ (22)
 ${x^s} \approx \sum\limits_{j = 1}^{{m_s}} {N_j^s\left( x \right)x_f^s\;{\rm{on}}\;\mathit{\Gamma }_{\rm{h}}^s}$ (23)

Common-Refinement方法生成的网格拓扑由两侧输入的界面网格的交线来定义，或者被称作子单元。

 $f_j^s = \sum\limits_{i = 1}^{{e_c}} {\int_{\sigma _l^c} {N_j^s{{\tilde t}^f}{\rm{d}}\mathit{\Gamma }} }$ (24)

4 非匹配网格误差分析

 ${A_s} = \frac{{{\rm{ \mathsf{ π} }}DH}}{{{w_z}{N_s}}},{A_f} = \frac{{{\rm{ \mathsf{ π} }}DH}}{{{w_z}{N_f}}}$ (25)

 $\begin{array}{*{20}{c}} {{t^s} = {t^s}\left( {\theta ,z} \right) = }\\ { - \left( {\frac{1}{2}{\rho ^f}U_\infty ^2\left( {1 - 4{{\sin }^2}\theta } \right) + {\rho ^f}gz} \right)\left[ {\begin{array}{*{20}{c}} {0.5\cos \theta }\\ {0.5\sin \theta }\\ 0 \end{array}} \right]} \end{array}$ (26)

 Download: 图 2 误差分析圆柱体算例示意 Fig. 2 Presribed load transferred from fluid to solid across a cylinder surface
 ${\varepsilon _1} = \frac{{\sum\limits_j {T_j^s - {t^s}{{\left( {\theta _j^s,z_j^s} \right)}_2}} }}{{\sum\limits_j {{t^s}{{\left( {\theta _j^s,z_j^s} \right)}_2}} }}$ (27)

5 圆柱体-弹性板问题

 ${u^f}\left( {0,y} \right) = 1.5\bar U\frac{{y\left( {H - y} \right)}}{{{{\left( {H/2} \right)}^2}}} = 1.5\bar U\frac{{4.0}}{{0.1681}}y\left( {0.41 - y} \right)$ (28)

 $\mathit{Re} = \frac{{{\rho ^f}\bar UD}}{{{\mu ^f}}},{K_B} = \frac{{E{w^3}}}{{12\left( {1 - {{\left( {{\nu ^s}} \right)}^2}} \right){\rho ^f}{{\bar U}^2}{L^3}}},{\rho ^r} = \frac{{{\rho ^s}}}{{{\rho ^f}}}$ (29)

6 结论

1) 当前的实现过程可以灵活的运用在不同领域的流固耦合问题中，同时因为可以任意划分和使用非匹配网格，因此具有足够的灵活性和稳健性。

2) 提出的三维数据传输方法基于L2模最小化了流体和结构网格之间的数据传输的误差，提供了一个准确的耦合方式。

3) 通过系统的误差分析，能够发现Common-Refinement方法对三维数据传输的一致性，不同网格比率的结算结果能够证明Common-Refinement方法具有的空间二阶精度。

4) 通过三维低密度比的基于非匹配网格的流固耦合问题的一系列数值实验，以及和标准算例的对比和验证，证明了这种方法的准确性、可靠性。

 [1] HRON J, TUREK S. A monolithic FEM/Multigrid solver for an ALE formulation of fluid-structure interaction with applications in Biomechanics[M]//BUNGARTZ H J, SCHÄFER M. Fluid-Structure Interaction. Berlin, Heidelberg: Springer, 2006: 146-170. (0) [2] FELIPPA C, PARK K. Staggered transient analysis procedures for coupled mechanical systems:formulation[J]. Computer methods in applied mechanics and engineering, 1980, 24(1): 61-111. DOI:10.1016/0045-7825(80)90040-7 (0) [3] GURUGUBELLI P, JAIMAN R K. Self-induced flapping dynamics of a flexible inverted foil in a uniform flow[J]. Journal of fluid mechanics, 2015, 781: 657-694. DOI:10.1017/jfm.2015.515 (0) [4] JAIMAN R K, SEN S, GURUGUBELLI P S. A fully implicit combined field scheme for freely vibrating square cylinders with sharp and rounded corners[J]. Computers & fluids, 2015, 112: 1-18. (0) [5] JAIMAN R, GEUBELLE P, LOTH E, et al. Combined interface boundary condition method for unsteady fluid-structure interaction[J]. Computer methods in applied mechanics and engineering, 2011, 200(1/2/3/4): 27-39. (0) [6] De BOER A, VAN ZUIJLEN A H, BIJL H. Review of coupling methods for non-matching meshes[J]. Computer methods in applied mechanics and engineering, 2007, 196(8): 1515-1525. DOI:10.1016/j.cma.2006.03.017 (0) [7] DE BOER A, VAN ZUIJLEN A H, BIJL H. Comparison of conservative and consistent approaches for the coupling of non-matching meshes[J]. Computer methods in applied mechanics and engineering, 2008, 197(49/50): 4284-4297. (0) [8] JAIMAN R K, JIAO X, GEUBELLE P H, et al. Conservative load transfer along curved fluid-solid interface with non-matching meshes[J]. Journal of computational physics, 2006, 218(1): 372-397. DOI:10.1016/j.jcp.2006.02.016 (0) [9] JAIMAN R, JIAO X, GEUBELLE P H, et al. Assessment of conservative load transfer for fluid-solid interface with non-matching meshes[J]. International journal for numerical methods in engineering, 2005, 65(15): 2014-2038. (0) [10] JANSEN K E, WHITING C, HULBERT G. A generalized-α method for integrating the filtered Navier-Stokes equations with a stabilized finite element method[J]. Computer methods in applied mechanics and engineering, 2000, 190(3/4): 305-319. (0) [11] JANDRON M A, HURD R, BELDEN J, et al. Modeling of hyperelastic water-skipping spheres using abaqus/explicit[C]//Proceedings of the 2014 SIMULIA Community Conference. 2014. (0) [12] ZEMERLI C, LATZ A, ANDRÄ H. Constitutive models for static granular systems and focus to the Jiang-Liu hyperelastic law[R]. Fraunhofer (ITWM): Fraunhofer-Institut für Techno-und Wirtschaftsmathematik, 2012. (0) [13] TUREK S, HRON J. Proposal for numerical benchmarking of fluid-structure interaction between an elastic object and laminar incompressible flow[M]//BUNGARTZ H J, SCHÄFER M. Fluid-Structure Interaction. Berlin, Heidelberg: Springer, 2006: 371-385. (0)