A strong coupling prediction-correction immersed boundary method
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摘要: 为克服传统浸入边界法的质量不守恒缺陷，提出了一种用于可压缩流固耦合问题的强耦合预估-校正浸入边界法。通过阐述一般流固耦合系统的矩阵表示，推导了流固耦合系统的强耦合Gauss-Seidel迭代格式，进一步导出预估-校正格式，提出了预估-校正浸入边界法。该方法使用无耦合边界模型对流体进行预估，将流固耦合边界视为自由面，固体原本占据的空间初始化为零质量的单元，允许流体自由穿过耦合边界。对于流体的计算，使用带有minmod限制器的二阶MUSCL有限体积格式和基于Zha-Bilgen分裂的AUSM+-up方法，配合三阶Runge-Kutta格式推进时间步。在校正步骤中，通过一组质量守恒的输运规则来实现输运过程。输运算法可概括为将边界内侧的流体进行标记，根据标记顺序以均匀方式分割和移动流体，产生一个指向边界外侧的流动，最后在边界附近施加速度校正保证无滑移条件。标记和输运算法避免了繁琐的对截断单元的几何处理，确保了算法易于实现。对于固体的计算，分别采用一阶差分格式和隐式动力学有限元格式求解刚体和线弹性体，并利用高斯积分获得固体表面的耦合力。使用预估-校正浸入边界法计算了一维问题和二维问题。在一维活塞问题中，获得了压力分布、相对质量历史和误差曲线，并与其他方法进行了对比。在二维的激波冲击平板问题中，获得了数值模拟纹影和平板结构的挠度历史，并与实验结果进行了对比。研究表明，该方法区别于传统的虚拟网格方法和截断单元方法，能够精确地维持流场的质量守恒并易于实现，且具有一阶收敛精度，能够较准确地预测激波绕射后的流场以及平板在激波作用下的挠度，为开发流固耦合算法提供了一种新的思路。Abstract: In the traditional immersed boundary methods for solving compressible fluid-structure interaction problems, conservation is one of the problems that must be considered. When the coupling boundary moves on the fixed grid, the structure coverage will change, resulting in many dead elements and emerging elements on the fluid grid. In the ghost-cell immersed boundary method, the reconstructed grid can not maintain the strict mass conservation when the dead elements and emerging elements appear. In order to overcome the shortcomings of traditional methods, a strong coupling prediction-correction immersed boundary method for compressible fluid-structure interaction problems was proposed. Firstly, the matrix representation of a general fluid-structure coupling system was described, and a strong coupling Gauss-Seidel iterative scheme of fluid-structure coupling system was derived. Furthermore, a prediction-correction scheme was derived, and a prediction-correction immersed boundary method was proposed. The fluid-structure coupling boundary was regarded as a free surface, and the space originally occupied by the solid was initialized as zero mass elements, allowing the fluid to pass through the coupling boundary freely. For the calculation of fluid, the second-order MUSCL finite volume scheme with the minmod limiter and the AUSM+-up flux based on Zha-Bilgen splitting were used to advance the time step with the third-order Runge-Kutta scheme. In the correction step, the transport process was realized by a set of mass conservation transport rules. The transport algorithm could be summarized as marking the fluid inside the boundary, dividing and moving the fluid in a uniform way according to the marking order, generating a flow pointing to the outside of the boundary, and finally applying a velocity correction near the boundary to ensure the no-slip condition. The marking and transport algorithm avoided the tedious geometric treatment of the cut-cells, and ensured the easy implementation of the algorithm. For the calculation of solids, the first-order difference scheme and the implicit dynamic finite element scheme were used to solve the rigid body and linear elastic body respectively, and the Gauss quadrature was used to obtain the coupling force on the solid surface. The one-dimensional and two-dimensional problems were calculated by the prediction-correction immersed boundary method. In the one-dimensional piston problem, the accuracy, conservation and convergence of the method were investigated by comparing the results with those in the literature. In the two-dimensional shock wave impact problem, the experimental optical schlieren images were compared with those obtained by the numerical simulation, and the deflection history of the plate structure was investigated. The study shows that this method can accurately maintain the mass conservation of the flow field and has the advantage of easy implementation, which is different from the traditional ghost-cell method and the cut-cell method. This method has the first-order convergence accuracy, and can accurately predict the flow field after shock diffraction and the deflection of plate under shock waves. It provides a new idea for the development of fluid-structure coupling algorithms.
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图 2 从
${t_n}$ 时刻到${t_{n + 1}}$ 时刻边界进行移动，造成红色的失效单元和蓝色的新增单元Figure 2. Boundary motion on a fixed grid from time
${t_n}$ to${t_{n + 1}}$ . Dead (red) and fresh (blue) cells are generated by the motion
图 3 流固耦合系统的子域变化（实线代表
${t_n}$ 时刻的边界，虚线代表${t_{n + 1}}$ 时刻的边界）Figure 3. Changes of the subdomains of the fluid-structure interaction system, where solid lines are boundaries at
${t_n}$ , and dashed lines are boundaries at${t_{n + 1}}$ 
图 4 流体标记和输运方向，曲线为浸入边界
Figure 4. Fluid markers and direction of transportation, the curve is the immersed boundary
图 6 t=0.003时刻的压力分布（网格数为1 440，虚线表示浸入边界）
Figure 6. Pressure distribution at t=0.003 (The number of cells is1 440, and the dashed lines stand for the immersed boundaries.)
图 8 压力和速度的无量纲L2范数误差
Figure 8. Dimensionless L2 norms of error of the pressure and velocity
图 9 两种方法获得的流场密度
$\rho (x,t)$ 云图Figure 9. Mass density
$\rho (x,t)$ contours of the fluid by two methods
图 10 激波管初始条件（蓝色部分为有机玻璃板，灰色部分为静止流场，右侧红色部分为输入边界）
Figure 10. Initial conditions of the shock tube (The PMMA panel is blue, the static fluid is grey, the right boundary in red color is the inflow.)
图 11 激波管实验段（底部的金属方块为基座，竖立薄片为实验测量的平板）
Figure 11. Experimental section of the shock tube (The metal block on the bottom is the base, and the vertical sheet is the tested panel.)
图 12 局部网格（灰色部分为流场，蓝色部分为平板）
Figure 12. Local grids (The fluid is grey, and the panel is blue.)
图 13 不同时刻实验纹影与模拟纹影的对比
Figure 13. Comparison of experimental and simulated shadowgraphs
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[1] PESKIN C S. Flow patterns around heart valves: a numerical method [J]. Journal of Computational Physics, 1972, 10(2): 252–271. DOI: 10.1016/0021-9991(72)90065-4. [2] PESKIN C S. Numerical analysis of blood flow in the heart [J]. Journal of Computational Physics, 1977, 25(3): 220–252. DOI: 10.1016/0021-9991(77)90100-0. [3] 王力, 田方宝. 浸入边界法及其在可压缩流动中的应用和进展 [J]. 中国科学: 物理学 力学 天文学, 2018, 48(9): 094703. DOI: 10.1360/SSPMA2018-00191.WANG L, TIAN F B. Recent progress of immersed boundary method and its applications in compressible fluid flow [J]. Scientia Sinica Physica, Mechanica & Astronomica, 2018, 48(9): 094703. DOI: 10.1360/SSPMA2018-00191. [4] SEO J H, MITTAL R. A high-order immersed boundary method for acoustic wave scattering and low-Mach number flow-induced sound in complex geometries [J]. Journal of Computational Physics, 2011, 230(4): 1000–1019. DOI: 10.1016/j.jcp.2010.10.017. [5] 王力, 田方宝. 弹性拍翼悬停时的流固耦合效应 [J]. 气体物理, 2020, 5(4): 21–30. DOI: 10.19527/j.cnki.2096-1642.0812.WANG L, TIAN F B. Fluid-structure interaction of flexible flapping wing in hovering flight [J]. Physics of Gases, 2020, 5(4): 21–30. DOI: 10.19527/j.cnki.2096-1642.0812. [6] CHENG L, DU L, WANG X Y, et al. A semi-implicit immersed boundary method for simulating viscous flow-induced sound with moving boundaries [J]. Computer Methods in Applied Mechanics and Engineering, 2021, 373: 113438. DOI: 10.1016/j.cma.2020.113438. [7] 赵西增, 付英男, 张大可, 等. 紧致插值曲线方法在流向强迫振荡圆柱绕流中的应用 [J]. 力学学报, 2015, 47(3): 441–450. DOI: 10.6052/0459-1879-14-387.ZHAO X Z, FU Y N, ZHANG D K, et al. Application of a CIP-based numerical simulation of flow past an in-line forced oscillating circular cylinder [J]. Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(3): 441–450. DOI: 10.6052/0459-1879-14-387. [8] 段松长, 赵西增, 叶洲腾, 等. 错列角度对双圆柱涡激振动影响的数值模拟研究 [J]. 力学学报, 2018, 50(2): 244–253. DOI: 10.6052/0459-1879-17-345.DUAN S C, ZHAO X Z, YE Z T, et al. Numerical study of staggered angle on the vortex-induced vibration of two cylinders [J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(2): 244–253. DOI: 10.6052/0459-1879-17-345. [9] 杨明, 刘巨保, 岳欠杯, 等. 涡激诱导并列双圆柱碰撞数值模拟研究 [J]. 力学学报, 2019, 51(6): 1785–1796. DOI: 10.6052/0459-1879-19-224.YANG M, LIU J B, YUE Q B, et al. Numerical simulation on the vortex-induced collision of two side-by-side cylinders [J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(6): 1785–1796. DOI: 10.6052/0459-1879-19-224. [10] 陈威霖, 及春宁, 许栋. 不同控制角下附加圆柱对圆柱涡激振动影响 [J]. 力学学报, 2019, 51(2): 432–440. DOI: 10.6052/0459-1879-18-208.CHEN W L, JI C N, XU D. Effects of the added cylinders with different control angles on the vortex-induced vibrations of a circular cylinder [J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(2): 432–440. DOI: 10.6052/0459-1879-18-208. [11] HOSSEINJANI A A, ROOHI A H. Immersed boundary method for MHD unsteady natural convection around a hot elliptical cylinder in a cold rhombus enclosure filled with a nanofluid [J]. SN Applied Sciences, 2021, 3(2): 270. DOI: 10.1007/s42452-021-04221-3. [12] YE H X, CHEN Y, MAKI K. A discrete-forcing immersed boundary method for turbulent-flow simulations [J]. Proceedings of the Institution of Mechanical Engineers, 2021, 235(1): 188–202. DOI: 10.1177/1475090220927245. [13] SOTIROPOULOS F, YANG X. Immersed boundary methods for simulating fluid-structure interaction [J]. Progress in Aerospace Sciences, 2014, 65: 1–21. DOI: 10.1016/j.paerosci.2013.09.003. [14] YOUSEFZADEH M, BATTIATO I. High order ghost-cell immersed boundary method for generalized boundary conditions [J]. International Journal of Heat and Mass Transfer, 2019, 137: 585–598. DOI: 10.1016/j.ijheatmasstransfer.2019.03.061. [15] 吴晓笛, 刘华坪, 陈浮. 基于浸入边界-多松弛时间格子玻尔兹曼通量求解法的流固耦合算法研究 [J]. 物理学报, 2017, 66(22): 224702. DOI: 10.7498/aps.66.224702.WU X D, LIU H P, CHEN F. A method combined immersed boundary with multi-relaxation-time lattice Boltzmann flux solver for fluid-structure interaction [J]. Acta Physica Sinica, 2017, 66(22): 224702. DOI: 10.7498/aps.66.224702. [16] BOUKHARFANE R, EUGȆNIO RIBEIRO F H, BOUALI Z, et al. A combined ghost-point-forcing/direct-forcing immersed boundary method (IBM) for compressible flow simulations [J]. Computers and Fluids, 2018, 162: 91–112. DOI: 10.1016/j.compfluid.2017.11.018. [17] MAJUMDAR S, IACCARINO G, DURBIN P. RANS solvers with adaptive structured boundary non-conforming grids [J]. Center for Turbulence Research. Annual Research Briefs, 2001: 353–366. [18] 朱祥德, 陈春刚, 肖锋. 一种基于多矩的有限体积浸入边界法 [J]. 计算物理, 2010, 27(3): 342–352. DOI: 10.19596/j.cnki.1001-246x.2010.03.004.ZHU X D, CHEN C G, XIAO F. A multi-moment immersed-boundary finite-volume scheme [J]. Chinese Journal of Computational Physics, 2010, 27(3): 342–352. DOI: 10.19596/j.cnki.1001-246x.2010.03.004. [19] LEE J M, YOU D H. An implicit ghost-cell immersed boundary method for simulations of moving body problems with control of spurious force oscillations [J]. Journal of Computational Physics, 2013, 233(1): 295–314. DOI: 10.1016/j.jcp.2012.08.044. [20] 辛建建, 石伏龙, 金秋. 一种径向基函数虚拟网格法数值模拟复杂边界流动 [J]. 物理学报, 2017, 66(4): 044704. DOI: 10.7498/aps.66.044704.XIN J J, SHI F L, JIN Q. Numerical simulation of complex immersed boundary flow by a radial basis function ghost cell method [J]. Acta Physica Sinica, 2017, 66(4): 044704. DOI: 10.7498/aps.66.044704. [21] XIN J J, LI T Q, SHI F L. A radial basis function for reconstructing complex immersed boundaries in ghost cell method [J]. Journal of Hydrodynamics, 2018, 30(5): 890–897. DOI: 10.1007/s42241-018-0097-3. [22] 石伏龙, 辛建建, 马麟. 梯度增量level set/虚拟网格法模拟波浪结构物相互作用 [J]. 工程热物理学报, 2018, 39(11): 2420–2428.SHI F L, XIN J J, MA L. A gradient-augmented level set/ghost cell method for the simulation of wave structure interaction [J]. Journal of Engineering Thermophysics, 2018, 39(11): 2420–2428. [23] QU Y G, SHI R C, BATRA R C. An immersed boundary formulation for simulating high-speed compressible viscous flows with moving solids [J]. Journal of Computational Physics, 2018, 354: 672–691. DOI: 10.1016/j.jcp.2017.10.045. [24] HAJI MOHAMMADI M, SOTIROPOULOS F, BRINKERHOFF J. Moving least squares reconstruction for sharp interface immersed boundary methods [J]. International Journal for Numerical Methods, 2019, 90(2): 57–80. DOI: 10.1002/fld.4711. [25] 雷悦, 石伏龙. 虚拟网格法模拟静止或运动并列布置双圆柱绕流现象 [J]. 工程热物理学报, 2020, 41(8): 1974–1983.LEI Y, SHI F L. A ghost cell method for simulating flows around stationary of moving twin cylinders in a side-by-side arrangement [J]. Journal of Engineering Thermophysics, 2020, 41(8): 1974–1983. [26] XIE F T, QU Y G, ISLAM M A, et al. A sharp-interface Cartesian grid method for time-domain acoustic scattering from complex geometries [J]. Computers and Fluids, 2020, 202: 104498. DOI: 10.1016/j.compfluid.2020.104498. [27] CHI C, ABDELSAMIE A, THÉVENIN D. A directional ghost-cell immersed boundary method for incompressible flows [J]. Journal of Computational Physics, 2020, 404: 109122. DOI: 10.1016/j.jcp.2019.109122. [28] ZHENG K Y, ZHAO X Z, YANG Z J, et al. Numerical simulation of water entry of a wedge using a modified ghost-cell immersed boundary method [J]. Journal of Marine Science and Technology, 2020, 25(2): 589–608. DOI: 10.1007/s00773-019-00666-9. [29] CLARKE D K, HASSAN H A, SALAS M D. Euler calculations for multielement airfoils using Cartesian grids [J]. AIAA Journal, 1986, 24(3): 353–358. DOI: 10.2514/3.9273. [30] MEYER M, DEVESA A, HICKEL S, et al. A conservative immersed interface method for large-eddy simulation of incompressible flows [J]. Journal of Computational Physics, 2010, 229(18): 6300–6317. DOI: 10.1016/j.jcp.2010.04.040. [31] MONASSE L, DARU V, MARIOTTI C, et al. A conservative coupling algorithm between a compressible flow and a rigid body using an embedded boundary method [J]. Journal of Computational Physics, 2012, 231(7): 2977–2994. DOI: 10.1016/j.jcp.2012.01.002. [32] SCHNEIDERS L, GÜNTHER C, MEINKE M, et al. An efficient conservative cut-cell method for rigid bodies interacting with viscous compressible flows [J]. Journal of Computational Physics, 2016, 311: 62–86. DOI: 10.1016/j.jcp.2016.01.026. [33] BRADY P T, LIVESCU D. Foundations for high-order, conservative cut-cell methods: stable discretizations on degenerate meshes [J]. Journal of Computational Physics, 2021, 426: 109794. DOI: 10.1016/j.jcp.2020.109794. [34] SEO J H, MITTAL R. A sharp-interface immersed boundary method with improved mass conservation and reduced spurious pressure oscillations [J]. Journal of Computational Physics, 2011, 230(19): 7347–7363. DOI: 10.1016/j.jcp.2011.06.003. [35] 张德良. 计算流体力学教程[M]. 北京: 高等教育出版社, 2010: 279–288.ZHANG D L. A course in computational fluid dynamics [M]. Beijing: Higher Education Press, 2010: 279–288. [36] TORO E F, VÁZQUEZ-CENDÓN M E. Flux splitting schemes for the Euler equations [J]. Computers and Fluids, 2012, 70: 1–12. DOI: 10.1016/j.compfluid.2012.08.023. [37] LIOU M S. A sequel to AUSM, part II: AUSM+-up for all speeds [J]. Journal of Computational Physics, 2006, 214(1): 137–170. DOI: 10.1016/j.jcp.2005.09.020. [38] 王勖成. 有限单元法[M]. 北京: 清华大学出版社, 2003.WANG X C. Finite element method [M]. Beijing: Tsinghua University Press, 2003. [39] 李亭鹤, 阎超. 一种新的分区重叠洞点搜索方法-感染免疫法 [J]. 空气动力学学报, 2001, 19(2): 156–160. DOI: 10.3969/j.issn.0258-1825.2001.02.004.LI T H, YAN C. A new method of hole-point search in grid embedding technique [J]. Acta Aerodynamica Sinica, 2001, 19(2): 156–160. DOI: 10.3969/j.issn.0258-1825.2001.02.004. -
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