Development of Space-Time Finite Elements for Nonlinear Transient Response of Flexible Structures

该项目主要研究受冲击载荷作用的结构中速度场和位移场中与陡峭梯度相关的计算困难。传统的有限元方法建议在具有大梯度的区域周围创建更精细的单元网格，但由于梯度在结构中移动，这样的过程效率不高。这可以通过增加整个结构中的有限元数量来解决，但这样做在计算上是不切实际的。另一种办法将要求用模式分析的方式来表述问题。因此，对结构进行建模所需的大量模式意味着类似的计算困难。第三种选择是空间和时间上的有限差分，在面向设计的分析中使用时计算效率很低。本文所采用的方法，即在时空域中建立的有限元，提供了克服这些计算问题的可能性。将分析三类物理系统，包括恒定张力的弦、非均匀张力的弦和线性Timoshenko梁。对于弦模型，将解决初值问题，包括波沿弦向下传播的问题，以及涉及移动荷载和脉冲荷载的问题。对于Timoshenko梁，将考虑移动荷载和移动约束问题。该项目包括三个阶段：(1)确定跨越时空的有限元的形函数和收敛准则；(2)计算弦问题的解和研究收敛性质，建立Timoshenko梁的时空有限元列式的通解；(3)确定具有移动荷载和移动约束的梁问题的解。这项研究将对具有移动载荷或约束、展开问题和灵活机械臂的系统的分析方法产生影响。

This project focuses on the computational difficulties associated with steep gradients in the velocity and displacement fields in structures subjected to impulsive loadings. The conventional finite element approach would suggest creating a finer element mesh around the areas with large gradients, but since the gradients move through the structure, such a procedure is not efficient. This could be solved by increasing the number of finite elements throughout the structure, but it would be computationally impractical to do so. An alternative approach would call for formulation of the problem in terms of a modal analysis. The large number of modes thereby required to model the structure implies similar computational difficulties. A third alternative, finite differences in space and time is computationally inefficient for use in design oriented analysis. The approach adopted herein, finite elements formulated in the space time domain, offers the possibility of overcoming these computational problems. Three classes of physical systems will be analyzed including a string under constant tension, a string under nonuniform tension, and a linear Timoshenko beam. For the string models, initial value problems including those for which a wave travels down the string will be solved as well as those involving moving load and impulsive load problems. For the Timoshenko beam, moving loads and moving constraint problems will be considered. The project comprises three phases including: (1) determining shape functions and convergence criteria for finite elements that cut across space and time; (2) computing solutions for string problems and study of convergence properties, and set up of the general solution for a space time finite element formulation for Timoshenko beams; and (3) determining solutions for beam problems with moving loads and moving constraints. The impact of this research will be on the methods of analysis for systems with moving loads or constraints, deployment problems, and flexible manipulator arms.