基于EEP法的杆系结构受迫振动有限元自适应分析的研究

The forced vibrations of structures are widely existing in the mechanics research and the engineering application. At present, the self-adaptive approach based on the Element Energy Projection (EEP) method has achieved full successes in the finite element (FE) solution for various static problems and successively in the free vibrations of structures as well, which has paved the way for its further application in the forced vibrations of structures. The present study is aimed at proposing a general, unified, effective and reliable self-adaptive strategy for the FE solution of the forced vibrations of skeletal structures. Taking the forced vibrations of lumped mass system as the model problems, the self-adaptive FE approach will be proposed for time-domain analysis based on EEP method. This approach adaptively obtain the final FE mesh, on which conventional FE solutions of vibration displacements satisfy the specified tolerance for any mass at any time. The adaptive FE method will be constructed for both the axial vibration and the transverse vibration of bars, which produces displacements satisfying the specified tolerance at any time for any location of the bar. The adaptive approach will be further extended into the forced vibration analysis of skeletal structures and the unified and efficient solution software will be developed. For skeletal structures under some complex dynamic loads, for example, seismic loads, the feasible adaptive scheme and the basic strategy will be proposed. The proposed method in the present study can be further extended to vibration analysis of various two-dimensional problems, plates and shells, and three-dimensional structures. It paves a new way for numerical computation of dynamic responses of structures and shows a wide and bright future.

结构受迫振动分析是力学研究和工程应用广泛面临的问题。基于EEP（单元能量投影）超收敛计算的有限元自适应分析方法在各类静力学问题及结构自由振动问题中已获得广泛成功。本研究旨在针对杆系结构受迫振动问题，基于EEP法构建一套通用、统一、高效、可靠的有限元自适应求解算法：以集中质量体系受迫振动为模型问题，建立时域分析的EEP自适应求解方法，自适应地生成时间网格，该网格上任一集中质量在任一时刻的动位移解均满足给定的误差限；建立单个杆件轴向、横向振动的EEP法有限元自适应求解方法，求得杆件上任一位置任一时刻满足给定误差限的动位移；将其推广至杆系结构的受迫振动分析，编制统一的求解软件；对地震等复杂激励下结构动力响应的自适应求解提出可行方案和基本策略。本研究提出的基本策略和思路可进一步推广至二维平面问题、板壳结构、三维实体结构的受迫振动分析，从而为结构动力响应分析提供了一种新的途径，具有广阔的发展前景。

结构受迫振动分析是力学研究和工程应用广泛面临的问题，自适应分析是数值计算方法的一种新型高效求解方式。本研究对杆系结构受迫振动问题，建立了一种通用、统一、高效、可靠的Galerkin有限元自适应求解方法，空间和时间两个维度均实现自适应分析。具体包括： 对离散系统的线性运动方程组，研究了Galerkin弱型时域有限元法的力学原理、计算格式、数值稳定性，提出了完整的实施策略；对杆件线性强迫振动问题，基于半离散、半解析的思想，提出了一种时-空有限元自适应分析方法，编制程序，求解了杆件轴向振动、Euler梁和Timoshenko梁的横向振动问题；大量数值算例验证了方法的可行性和适用性，其中包括周期荷载、脉冲荷载、地震波等多种输入；将时-空有限元自适应分析方法应用于杆系结构的线性受迫振动求解，编制计算程序，研究了其求解效率；研究并初步建立了非线性时域有限元及其自适应分析方法，该法对材料非线性的粘弹性梁问题，非线性刚度、非线性阻尼引起的运动方程等取得成功；将研究成果应用于工程问题，研究了路面结构静载、动载的自适应分析。