Liberge Erwan, Béghein Claudine
LaSIE, UMR CNRS 7356, La Rochelle Université, La Rochelle, France.
Phys Rev E. 2022 May;105(5-2):055307. doi: 10.1103/PhysRevE.105.055307.
In this paper, a lattice Boltzmann method (LBM) for the fluid-structure interaction of an incompressible neo-Hookean medium is proposed. The objective is to use the lattice Boltzmann method to model the fluid and the solid domains at the same time, and thus avoid coupling two solvers, one for the fluid and one for the structure. The specific case of a neo-Hookean incompressible medium allows us to use a Eulerian formulation for the structure problem, which resembles the Navier-Stokes equation. Then, a macroscopic multiphase equation can be used to model the fluid and structure problems together. Next, the LBM approach is deduced from this macroscopic multiphase formulation of the fluid-structure interaction problem. It consists in extending the LBM to the solid domain by adding a tensor term in the equilibrium function of the collision operator. The effect of the added tensor term is to cancel in the solid domain the viscous fluid constraints and add neo-Hookean constraints. Thus, only the third moment of the LBM is modified, the first two being conserved, and only the constraints in the macroscopic Navier-Stokes equation are changed. The LBM scheme obtained can then model the interaction of a fluid and a structure composed of an incompressible neo-Hookean medium with a single solver for the fluid and the solid. Two additional equations are used, one to track the fluid and solid domains with the Cahn-Hilliard equation, and the other to compute the solid displacement field by a finite-difference scheme. The proposed method is applied successfully on three cases from the literature.
本文提出了一种用于不可压缩新胡克介质流固相互作用的格子玻尔兹曼方法（LBM）。目的是使用格子玻尔兹曼方法同时对流体域和固体域进行建模，从而避免耦合分别用于流体和结构的两个求解器。新胡克不可压缩介质的具体情况使我们能够对结构问题采用类似于纳维 - 斯托克斯方程的欧拉公式。然后，可以使用宏观多相方程一起对流体和结构问题进行建模。接下来，从流体 - 结构相互作用问题的这种宏观多相公式推导出LBM方法。它包括通过在碰撞算子的平衡函数中添加一个张量项将LBM扩展到固体域。添加的张量项的作用是在固体域中消除粘性流体约束并添加新胡克约束。因此，仅修改LBM的第三矩，前两矩保持不变，并且仅改变宏观纳维 - 斯托克斯方程中的约束。然后，得到的LBM格式可以用一个用于流体和固体的单一求解器对由不可压缩新胡克介质组成的流体和结构的相互作用进行建模。使用了另外两个方程，一个用相场Cahn - Hilliard方程追踪流体和固体域，另一个用有限差分格式计算固体位移场。所提出的方法在文献中的三个案例上成功应用。
