时空有限元精细算法的研究及应用

发布时间：2018-05-26 00:10

本文选题：时空有限元 + 精细积分　；参考：《合肥工业大学》2015年博士论文

【摘要】：动力学问题是土木工程领域中的重要课题之一,如结构的振动控制、结构时程分析、优化控制等过程中都要求较精确地求出系统动力响应,为结构提供良好的响应数据,实现结构设计、检测、控制所要求的功能。合理、有效、计算精度高、耗时少,且易于编制通用程序的数值分析以及计算方法的研究一直是动力学研究领域的热点。现有的大部分数值分析方法均奠定于连续时间系统,但应用力学有限元、控制与信号处理、瞬态场的分析等需要离散时间系统,且需要考虑不同时间的位移向量,因此对时间坐标运用有限元,将时间域有限元和空间域有限元联合离散成为一个需要进一步研究和发展的问题。本文时空有限元算法可以大大提高计算速度,精简计算量,易程序化,在大型结构计算中将更具优势。该方法可以解决移动荷载的结构振动问题,波动问题,接触问题,几何非线性问题,热传导问题,现代控制问题,损伤问题等,在这些领域时空有限元既优于隐式算法又优于显式算法,并且可以解决非线性问题。本文算例表明,与传统的有限单元法相比,在求解与时间有关的由时间依赖性问题所形成的常微分方程或偏微分方程时,时空有限元法在精度、收敛性和稳定性方面均具有一定的优越性。本文其主要研究内容如下：通过Hamilton变作用定律,推导了时空有限元；并利用时间的一维性构造了时空传递矩阵；参考精细积分法给出了精细的时空传递矩阵,保证解的稳定性和缩减了求解规模；本文进一步给出了子结构时空有限元法,解决了大规模计算的问题,同时也解决了传统子结构内部节点与外部节点的状态耦合问题。通过该方法进行了结构损伤分析、结构施工中的吊装动力分析等一些强非线性问题,证明该方法具有良好的效果。最后本文采用了时空有限元并结合Openinventor编制了一套具有3维显示便于操控的施工吊装仿真软件。目前国内外对于空间域应用有限单元的研究已相当完善,而对时空有限元的研究,虽取得了很大的进展,但仍有广阔的发展空间,本文将进行进一步的探索。
[Abstract]:Dynamic problem is one of the most important problems in the field of civil engineering, such as vibration control of structure, structural time history analysis, optimization control and so on. It is necessary to find out the dynamic response of the system accurately and provide good response data for the structure. Achieve structural design, detection, control of the required functions. It is reasonable, effective, high precision, less time consuming, and easy to program the numerical analysis of the general program and the research of calculation methods has been a hot topic in the field of dynamics research. Most of the existing numerical analysis methods are established in the continuous time system, but the application of mechanical finite element method, control and signal processing, transient field analysis and so on need the discrete time system, and need to consider the displacement vector of different time. Therefore, the finite element method is used to discretize the time domain finite element and the space domain finite element, which needs further research and development. In this paper, the space-time finite element method can greatly improve the calculation speed, simplify the calculation amount, and be easy to program, which will be more advantageous in the large-scale structure calculation. This method can solve the structural vibration problem, wave problem, contact problem, geometric nonlinear problem, heat conduction problem, modern control problem, damage problem and so on. In these fields, the spatio-temporal finite element is superior to both implicit and explicit algorithms, and can solve nonlinear problems. The numerical examples show that compared with the traditional finite element method, the time-space finite element method is accurate in solving ordinary differential equations or partial differential equations formed by time-dependent problems. Both convergence and stability have some advantages. The main research contents of this paper are as follows: the finite element of time and space is derived by Hamilton's law of variable action, and the space-time transfer matrix is constructed by using the one-dimensional property of time, and the fine space-time transfer matrix is given by reference precise integration method. The stability of the solution is guaranteed and the solution scale is reduced. In this paper, a substructure space-time finite element method is proposed to solve the problem of large-scale computation and the state coupling problem between the internal node and the external node of the traditional substructure. Some strong nonlinear problems, such as structural damage analysis and dynamic analysis of hoisting in construction, are carried out by this method, which proves that the method has a good effect. Finally, a 3D display simulation software for construction hoisting is developed by using space-time finite element method and Openinventor. At present, the research on the application of finite element in space domain has been quite perfect at home and abroad. Although great progress has been made in the study of space-time finite element, there is still wide room for development.
【学位授予单位】：合肥工业大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：TU311.3

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