In this work, the original boundary condition-enforced immersed boundary method (IBM) [Wu and Shu (2009) [1], (2010) [2]] is improved to efficiently simulate incompressible flows with moving boundaries. The original boundary condition-enforced IBM can accurately interpret the no-slip boundary condition but becomes computationally tedious in simulating moving boundary problems due to the assembly of a large matrix at every time step and the implicit resolving process. The computational complexity of $O\left(N^a\right)$ grows significantly with the number of Lagrangian points N distributed on the immersed boundary. To alleviate these limitations, the conjugate gradient technique and the explicit technique are proposed to improve the efficiency of the boundary condition-enforced IBM. The IBM with the conjugate gradient technique fulfills the boundary condition in an iterative way with computational complexity of $O\left(N^c\right)$, while the IBM with the explicit technique is a non-iterative approach based on error analysis with computational complexity of $O\left(N^d\right)$. We also prove that the multi-direct forcing IBM [Luo et al. (2007) [7]; Wang et al. (2008) [8]] which is another popular IBM, is essentially a gradient descent approach to implement the boundary condition-enforced IBM with computational complexity of $O\left(N^b\right)$. Detailed analyses reveal $2=a>b>c>d=1$, which implies the high efficiency of the improved versions of IBM, especially the explicit technique-based IBM with a linear computational complexity. For validation, the IBMs are coupled with D1Q4 lattice Boltzmann flux solver (LBFS) to simulate two-dimensional and three-dimensional flows with moving boundaries. The results show that the conjugate gradient technique-based IBM and the explicit technique-based IBM have computational complexities of $O\left(N^{1.4}\right)$ and $O\left(N\right)$, respectively. Both of them have 2^nd order accuracy in space.

在这项工作中，对原始边界条件强制浸入边界方法 (IBM) [Wu 和 Shu (2009) [1]、(2010) [2]] 进行了改进，以有效模拟具有移动边界的不可压缩流动。原始边界条件强制 IBM 可以准确地解释无滑移边界条件，但由于在每个时间步长的大型矩阵的组装和隐式求解过程，在模拟移动边界问题时变得计算繁琐。的计算复杂度$哦\left(N^{\mathrm{一种}}\right)$随浸没边界上分布的拉格朗日点数N显着增长。为了减轻这些限制，共轭梯度技术和显式技术被提出来提高边界条件强制 IBM 的效率。具有共轭梯度技术的 IBM 以迭代方式满足边界条件，计算复杂度为$哦\left(N^C\right)$，而采用显式技术的 IBM 是一种基于错误分析的非迭代方法，其计算复杂度为 $哦\left(N^d\right)$. 我们还证明了多直接强迫 IBM [Luo et al. (2007) [7]; 王等人。（2008）[8]]，这是另一种流行的IBM，本质上是一种梯度下降方法，用于实现边界条件强制的IBM，其计算复杂度为$哦\left(N^{乙}\right)$. 详细分析揭示$2=\mathrm{一种}>乙>C>d=1$，这意味着 IBM 改进版本的高效率，尤其是具有线性计算复杂度的基于显式技术的 IBM。为了进行验证，IBM 与 D1Q4 晶格玻尔兹曼通量求解器 (LBFS) 结合使用，以模拟具有移动边界的二维和三维流动。结果表明，基于共轭梯度技术的 IBM 和基于显式技术的 IBM 的计算复杂度为$哦\left(N^{1.4}\right)$ 和 $哦\left(N\right)$， 分别。他们都有2^次太空订单的准确性。