We present and analyse a numerical method for understanding the low-inertia dynamics of an open, inextensible viscoelastic rod - a long and thin three dimensional object - representing the body of a long, thin microswimmer. Our model allows for both elastic and viscous, bending and twisting deformations and describes the evolution of the midline curve of the rod as well as an orthonormal frame which fully determines the rod's three dimensional geometry. The numerical method is based on using a combination of piecewise linear and piecewise constant functions based on a novel rewriting of the model equations. We derive a stability estimate for the semi-discrete scheme and show that at the fully discrete level that we have good control over the length element and preserve the frame orthonormality conditions up to machine precision. Numerical experiments demonstrate both the good properties of the method as well as the applicability of the method for simulating undulatory locomotion in the low-inertia regime.

中文翻译：

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三维低惯量波动运动的一种稳定有限元方法

我们提出并分析了一种数值方法，用于理解开放的、不可伸展的粘弹性杆（一种细长的三维物体）的低惯性动力学，代表细长的微型游泳者的身体。我们的模型允许弹性和粘性、弯曲和扭曲变形，并描述了杆中线曲线的演变以及完全确定杆的三维几何形状的正交框架。数值方法基于使用分段线性函数和分段常数函数的组合，基于对模型方程的新颖重写。我们推导出半离散方案的稳定性估计，并表明在完全离散的水平上，我们对长度元素有很好的控制，并保持帧正交条件达到机器精度。