Li Yusong, LeBoeuf Eugene J, Basu P K
Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37325, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Oct;72(4 Pt 2):046711. doi: 10.1103/PhysRevE.72.046711. Epub 2005 Oct 21.
A numerical model of the lattice Boltzmann method (LBM) utilizing least-squares finite-element method in space and the Crank-Nicolson method in time is developed. This method is able to solve fluid flow in domains that contain complex or irregular geometric boundaries by using the flexibility and numerical stability of a finite-element method, while employing accurate least-squares optimization. Fourth-order accuracy in space and second-order accuracy in time are derived for a pure advection equation on a uniform mesh; while high stability is implied from a von Neumann linearized stability analysis. Implemented on unstructured mesh through an innovative element-by-element approach, the proposed method requires fewer grid points and less memory compared to traditional LBM. Accurate numerical results are presented through two-dimensional incompressible Poiseuille flow, Couette flow, and flow past a circular cylinder. Finally, the proposed method is applied to estimate the permeability of a randomly generated porous media, which further demonstrates its inherent geometric flexibility.
开发了一种格子玻尔兹曼方法（LBM）的数值模型，该模型在空间上采用最小二乘有限元法，在时间上采用克兰克-尼科尔森法。该方法能够通过利用有限元法的灵活性和数值稳定性，同时采用精确的最小二乘优化，来求解包含复杂或不规则几何边界的区域中的流体流动。对于均匀网格上的纯平流方程，推导得出了空间四阶精度和时间二阶精度；而通过冯·诺依曼线性化稳定性分析表明具有高稳定性。通过一种创新的逐个单元方法在非结构化网格上实现，与传统的格子玻尔兹曼方法相比，该方法所需的网格点更少，内存占用更少。通过二维不可压缩泊肃叶流、库埃特流和绕圆柱流动给出了精确的数值结果。最后，将该方法应用于估计随机生成的多孔介质的渗透率，这进一步证明了其固有的几何灵活性。
