基于复合时间积分的结构动力学数值算法研究

结构动力学问题的求解方法研究是目前国际上计算力学领域的一个前沿基础课题，随着应用需求的增长、结构复杂性的增加以及非线性问题的凸显，发展稳定、可靠和高效的数值求解算法成为了迫切需求。本项目重点研究基于复合时间积分思想的结构动力学数值求解算法，通过复合两种或多种具有不同数值性能的时间积分算法，进而设计出综合性能增强的数值算法。首先，将复合隐式时间积分方案的子步算法范围扩展到线性多步法，并引入数值耗散可控性能，旨在保证求解的稳定性和计算效率的同时提高求解精度；然后，发展多子步复合时间积分算法的精度分析方法，并对数值耗散性能可控的复合时间积分方案进行严格的精度分析和参数设计；最后，结合典型线性和非线性结构动力学问题优化复合时间积分算法的数值耗散性能，建立一套求解精度、稳定性和计算效率综合性能优异的求解算法。本项目研究将为复合时间积分算法的发展奠定理论基础，也为结构动力学问题提供有效的数值求解方法。

Research on solutions for structural dynamics has been a basic and frontier issue in computational mechanics in the world, and the developing of stable, reliable, and high computational efficient numerical solution algorithms has been an urgent demand with the growing of application requirements, the increasing of structures complexity and the arising of nonlinearity. The present proposal focuses on the numerical solution algorithms for structural dynamics based on the composite time integration scheme, in which two or more time integration algorithms with different numerical properties are composited together to design new time integration algorithms with strengthen synthetical properties. Firstly, the choice of sub-step algorithms for the composite implicit time integration scheme will be extended to the linear multistep methods, and the controllable numerical dissipation will be introduced into the composite time integration scheme to increase the accuracy of solution when the stability and the computational efficiency are guaranteed. Secondly, the accuracy analysis method for the multi-sub-step composite time integration algorithms will be developed, and the rigorous accuracy analysis and parameters design for the proposed composite time integration scheme with controllable numerical dissipation will be performed. Finally, the numerical dissipation property of the composite time integration algorithm will be optimized by analyzing typical linear and nonlinear structural dynamics problems, and a numerical algorithm with excellent performance in accuracy, stability and efficiency will be obtained. The research not only can lay a theory foundation for the studying and developing of the composite time integration scheme, but also can provide an effective numerical solution approach for the structural dynamics problems.