The immersed boundary method (IBM) has been frequently utilized to simulate the motion and deformation of biological cells and capsules in various flow situations. Despite the convenience in dealing with flow-membrane interaction, direct implementation of membrane viscosity in IBM suffers severe numerical instability. It has been shown that adding an artificial elastic element in series to the viscous component in the membrane mechanics can efficiently improve the numerical stability in IBM membrane simulations. Recently Li and Zhang (Int J Numer Methods Biomed Eng 35:e3200, 2019) proposed a finite-difference method for calculating membrane viscous stress. In the present paper, two new schemes are developed based on the convolution integral expression of the Maxwell viscoelastic element. We then conduct several tests for the accuracy, stability, and efficiency performances of these three viscous stress schemes. By studying the behavior of a one-dimensional Maxwell element under sinusoidal deformation, we find that a good accuracy can be achieved by selecting an appropriate relaxation time. The twisting sphere tests confirm that, compared to the numerical errors induced by other components in capsule simulations, such as the finite element method for membrane discretization and IBM for flow-membrane interaction, the errors from the viscous stress calculation are negligible. Moreover, extensive simulations are conducted for the dynamic deformation of a spherical capsule in shear flow, using different numerical schemes and various combinations of the artificial spring constants and calculation frequency for the membrane viscous stress calculation. No difference is observed among the results from the three schemes; and these viscous stress schemes require very little extra computation time compared to other components in IBM simulations. The simulation results converge gradually with the increase in the artificial spring stiffness; however, a threshold value exists for the spring stiffness to maintain the numerical stability. The viscous stress calculation frequency has no apparent influence on the calculation results, but a large frequency number can cause the simulation to collapse. We therefore suggest to calculate the membrane viscous stress at each simulation time step, such that a better numerical stability can be achieved. The three numerical schemes have nearly identical performances in all aspects, and they can all be utilized in future IBM simulations of viscoelastic membranes.

浸入边界法（IBM）经常被用来模拟生物细胞和胶囊在各种流动情况下的运动和变形。尽管处理流动-膜相互作用很方便，但在 IBM 中直接实施膜粘度会遭受严重的数值不稳定性。已经表明，在膜力学中的粘性分量中串联添加人工弹性元件可以有效提高 IBM 膜模拟中的数值稳定性。最近 Li 和 Zhang（Int J Numer Methods Biomed Eng 35:e3200, 2019）提出了一种计算膜粘滞应力的有限差分法。在本文中，基于麦克斯韦粘弹性单元的卷积积分表达式开发了两种新方案。然后我们对准确性、稳定性、这三种粘性应力方案的效率和性能。通过研究一维麦克斯韦单元在正弦变形下的行为，我们发现通过选择合适的弛豫时间可以获得良好的精度。扭曲球体测试证实，与胶囊模拟中其他组件引起的数值误差相比，例如用于膜离散化的有限元方法和用于流膜相互作用的 IBM，粘性应力计算的误差可以忽略不计。此外，对剪切流中球形胶囊的动态变形进行了广泛的模拟，使用不同的数值方案和人工弹簧常数的各种组合以及膜粘滞应力计算的计算频率。三个方案的结果之间没有观察到差异；与 IBM 模拟中的其他组件相比，这些粘性应力方案只需要很少的额外计算时间。仿真结果随着人工弹簧刚度的增加逐渐收敛；然而，弹簧刚度存在阈值以保持数值稳定性。粘滞应力计算频率对计算结果没有明显影响，但频率数过大会导致模拟崩溃。因此，我们建议在每个模拟时间步长计算膜粘滞应力，以便获得更好的数值稳定性。这三种数值方案在所有方面都具有几乎相同的性能，它们都可以用于未来的 IBM 粘弹性膜模拟。