热力耦合方程组的并行多尺度算法

This project will focus on the research of homogenization and multiscale algorithms for the dynamic coupled thermoelastcity equation for materials with lattice structure, which are the new generation of super-lightweight structural materials in aviation and space industries. It is one of the unsolved problems in homogenization of lattice structures. It appears several structural parameters in lattice structures and their physical geometry are rapidly oscillating, so they can't be solved by traditional numerical methods. This project will transfer the dynamic coupled thermoelasticity equation to a stationary coupled equation with complex parameters by integral transformation, then analyse the asymptotic behaviour of the steady coupled equation respect to these structural parameters about the period and the thickness of the plate or beam by multiscale asymptotic expansion method, technique of dilatations and singular perturbation method, to develop a new homogenization theory for the dynamic coupled thermoelasticity equation with lattice structures. It is a new method to deal with the asymptotic analysis of dynamic equations. Base on the progress of theoretical research, taking advantage of advanced FEM and numerical integral transformation techniques, it will develop a time-domain parallel multiscale algorithm which is theoretically reliable and technically feasible. Also, it will develop efficient and robust parallel computational program which can be applied to predict and simulate the coupled thermoelasticity performace and behaviour of matericals with lattice structures.

本项目将围绕航空、航天领域的新一代超轻质结构材料- - 点阵结构材料开展耦合热弹性方程组的渐近均匀化理论和多尺度算法研究，这是点阵结构均匀化理论尚未解决的问题之一。点阵结构具有多个结构小参数，几何物理结构剧烈变化，不能用传统的数值方法模拟。本项目将通过积分变换将动力热弹性耦合方程组转化为复参数的稳态耦合方程，利用多尺度渐近展开方法、扩张技术和奇异摄动技术对复参数稳态耦合方程关于周期结构参数和材料壁/杆厚结构参数作渐近分析，从新的角度发展一套点阵结构动力耦合热弹性方程的均匀化理论。这是动力方程的渐近均匀化分析的一种新尝试。基于理论工作的突破，结合先进有限元和数值积分变换技术，构造一套理论上可靠、计算上可行的时间空间并行的多尺度算法。为点阵结构材料热力耦合性能评价、行为模拟研发一套高效的、健壮的并行多尺度计算程序。为其他多场耦合问题的多尺度算法研究提供参考。

本项目以航空航天领域的周期轻质结构材料热力问题为背景，结合积分变换，将时间域上的热力耦合多尺度渐近分析问题转化至频率域上讨论，频率域上各个方程相互独立，以此思想发展时间方向上也可并行计算的多尺度算法。首先，针对抛物方程，利用Fourier变换提出了一种时间上可并行计算的多尺度算法。其次，对热力耦合方程组，利用Laplace变换将问题提出了一种时间并行多尺度算法。再次，将提出的热力学并行多尺度算法推广至轻质多孔周期结构。对一般的有界凸多边形区域，通过构造边界层方程严格证明了并行多尺度解的强收敛估计式。进一步地，利用工程上常用的孔洞填充思想，研究两套多尺度分析方法的关系，并提出了一套在无孔区域上研究多孔结构的统一的并行多尺度算法。在三维自适应有限元并行程序开发平台PHG上用C++语言实现了相关算法的并行多尺度计算程序。大量的数值模拟结果表明，这种时间并行多尺度算法不仅保持了原有计算格式的良好的计算精度，而且可以方便的实现空间、时间上的并行化，可以大大提高算法的并行度、实现超大规模计算。