Error estimates for elastic flows

弹性流的误差估计

• 批准号: 514616861
• 负责人: Professor Dr. Sören Bartels
• 金额: --
• 依托单位: Abteilung für Angewandte Mathematik
• 依托单位国家: 德国
• 项目类别: Research Units
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/514616861?language=en
• 关键词: Error; estimates; elastic; flows

Thin elastic structures lead to a variety of interesting phenomenaand applications. Often, two energy stable bending states are used to create a quasistationary switching device or to generate mechanical locomotion. For all of these technical developments large deformations of thin elastic objects are important. Their mathematical modeling leads to constrained bending energies in which the constraint captures the resistance to shearing and stretching effects of thin elastic plates or the inextensibility and twist behavior of an elastic rod with small cross section. Different finite element discretizations of such energies have recently been devised. By following ideas for the numerical approximation of harmonic maps, which describe director fields with values in given manifolds, practical and convergent numerical methods were obtained. In parallel research the derivation of error estimates for classes of geometric evolution problems was addressed, which include the Willmore, the mean curvature, and the harmonic map heat flow. Crucial in those developments was the identification of an equivalent formulation of the evolution problems as a nonlinear parabolic system that includes equations for geometric quantities such as the normal field. In this project we aim at bringing the developments together to obtain error estimates for finite element discretizations of time stepping schemes for gradient flows of various bending energies. The motivation for this is twofold: first, this justifies the numerical simulation of natural, strongly damped relaxation dynamics, and second, it leads to error estimates for the approximation of stationary states obtained via certain evolution processes. In addition, this serves as a step towards simulating general dynamical processes that capture inertial effects, e.g., vibrating elastic rods. Other applications include the determination of stationary states of magnetic elastic films and rods.

薄弹性结构导致了各种有趣的现象和应用。通常，两个能量稳定的弯曲状态被用来创建准静止开关装置或产生机械运动。对于所有这些技术发展，薄弹性物体的大变形是重要的。他们的数学模型导致约束弯曲能，其中约束捕获了薄弹性板的剪切和拉伸效应的阻力或小截面弹性杆的不可扩展和扭转行为。这种能量的不同的有限元离散化最近已被设计出来。利用谐波映射的数值逼近思想，给出了具有给定流形中值的方向域的实用且收敛的数值逼近方法。在平行研究中，讨论了几何演化问题（包括Willmore、平均曲率和调和映射热流）的误差估计的推导。在这些发展中至关重要的是确定了演化问题的等效公式，即非线性抛物系统，其中包括诸如法向场等几何量的方程。在这个项目中，我们的目标是将这些发展结合起来，以获得各种弯曲能量梯度流的时间步进格式的有限元离散化的误差估计。这样做的动机是双重的：首先，这证明了自然的、强阻尼的松弛动力学的数值模拟是合理的；其次，它导致了通过某些进化过程获得的稳态近似的误差估计。此外，这可以作为模拟捕获惯性效应的一般动力学过程的一步，例如，振动弹性杆。其他的应用包括测定磁性弹性薄膜和棒的固定状态。