In this work it is shown how the immersed boundary method of (Peskin2002) for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations. A stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps. An important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved, partially resolved, or fully resolved while retaining a good level of accuracy. Error estimates in each of these regimes are given for the method. A number of theoretical and numerical checks are furthermore performed to assess its physical fidelity. For a conservative force, the method is found to simulate particles with the correct Boltzmann equilibrium statistics. It is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters. The method is also shown to reproduce a well-known hydrodynamic effect of a Brownian particle in which the velocity autocorrelation function exhibits an algebraic tau^(-3/2) decay for long times. A few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method.
Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method
在简单剪切流中自由悬浮的初始球形胶囊的变形可以在小变形的极限下解析计算[D。Barthes-Bianca，J. M. Rallison，在线性剪切流中自由悬浮的胶囊的时间相关变形，J. Fluid Mech.113（1981）251-267]。这些解析近似被用来研究的网格镶嵌方法，空间分辨率，和离散δ函数的浸入边界法的数值结果上得到的耦合浸入边界格子Boltzmann有限元方法的影响。对于囊膜的描述，采用有限元法和Skalak本构模型[R。Skalak等人，红细胞膜的应变能函数。J. 13（1973）245-264]已经被使用。我们的主要目标是调查所提出的模型为小分辨率提供了一个良好的基础，有效而准确的模拟多个可变形的粒子浸没在流体中。我们得出的结论是，膜网格的细节，镶嵌方法和分辨率，只发挥次要作用。流体动力学分辨率，即，离散δ函数的宽度可以显著地影响模拟的精度。δ函数的离散化引入了人工长度尺度，其有效地改变了胶囊的半径和可变形性。我们讨论的可能性，减少计算时间的变形物体浸没在流体中的模拟，同时保持高精度。
