弹性Cosserat杆的低马赫数-细长极限
Low-Mach-number--slenderness limit for elastic Cosserat rods
AI总结：
本文研究动态弹性Cosserat杆模型与Kirchhoff梁方程的关联，证明低马赫数-细长极限下无角惯性Kirchhoff梁是Cosserat杆的渐近极限，构建四阶精度渐近框架，并采用空间Gauss-Legendre配点结合时间α-方法的格式开展数值研究。
摘要：
本文研究动态弹性Cosserat杆模型与Kirchhoff梁方程之间的关联。研究表明，当细长参数（杆直径与长度的比值）和马赫数（杆速度与典型声速的比值）趋近于零时，即处于低马赫数-细长极限下，无角惯性的Kirchhoff梁是Cosserat杆的渐近极限。该渐近框架在小参数的四阶精度内是精确的，揭示了一种可统一处理两类模型之间过渡区域的数学结构。为通过数值方法研究该过渡区域，本文采用了一种基于空间Gauss-Legendre配点法和时间α-方法的数值格式。
英文摘要：
This paper deals with the relation of the dynamic elastic Cosserat rod model and the Kirchhoff beam equations. We show that the Kirchhoff beam without angular inertia is the asymptotic limit of the Cosserat rod, as the slenderness parameter (ratio between rod diameter and length) and the Mach number (ratio between rod velocity and typical speed of sound) approach zero, i.e. low-Mach-number--slenderness limit. The asymptotic framework is exact up to fourth order in the small parameter and reveals a mathematical structure that allows a uniform handling of the transition regime between the models. To investigate this regime numerically, we apply a scheme that is based on a spatial Gauss-Legendre collocation and an $α$-method in time.
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