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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