【作者】 刘恒；

【导师】 廖振鹏；

【作者基本信息】 中国地震局工程力学研究所 ， 防灾减灾工程及防护工程， 2009， 博士

【摘要】 如何更精确高效地模拟大型、复杂系统内域的波动是发展和完善近场波动数值模拟技术的一个重要研究课题。内域波动的数值模拟通常采用计算量较小的显式方法,但现有的时空解耦显式有限元方法的精度只有二阶;低精度不仅影响数值模拟的精度,而且制约着计算效率的提高。鉴于此,本文旨在探索内域波动数值模拟具有更高精度且稳定的显式方法。作者发展了内域波动数值模拟的现有显式有限元解耦技术,提出了一种高精度且稳定的显式数值模拟方法。第一,依据波速有限的概念,从波动方程的精确解出发提出了一种新的显式数值模拟方法。此方法与现有有限元技术相似之处在于:适于处理非规则网格节点,且节点递推公式是具有显式和时空解耦特征。其不同之处在于:新方法可给出时空离散精度皆为2M阶的稳定格式,M为正整数。本文通过一维模型详细论述了这一方法的可行性:建立了非规则网格节点递推公式,详细分析了均匀网格标量波动数值模拟的精度和稳定性,提出了构建时空精度皆为2M阶( M为正整数)的稳定递推公式的技术途径,并以构建二阶( M = 1)和四阶( M = 2)公式为例予以说明。第二,将一维情形研究结果推广到高维情形,分别建立了二维、三维非规则网格节点的递推公式。针对二维正方形均匀网格详细论述了时空离散精度皆为2M阶的稳定递推公式的构建方法,给出了二阶和四阶递推公式的具体结果;并对三维立方体均匀网格模型作了简要的讨论。第三,本文通过一维、二维模型算例检验了波动方程数值模拟显式方法的精度和稳定性等理论结果,指出了高阶公式对提高计算效率的价值。第四,从空间解耦有限元常微分方程组(结构动力学方程)出发,通过被积函数的拉格朗日多项式内插和分部积分导出了一组具有高阶精度的时域显式积分格式,并将其推广为适用于一般数学物理方程的显式积分格式。作者以一个简单的线性时不变系统为例,初步考察了此积分格式的稳定性。更多还原

【Abstract】 How to simulate the wave motion in the interior of a complicated system of large scale accurately and effectively is a very important research subject for developing and improving techniques for numerical simulation of near-field wave motion. Traditionally, the explicit method with small computational cost is employed for the numerical simulation of wave motion within the computational region. However, the existing time-space decoupling explicit finite element method is only second order accurate. The low order of accuracy not only affects precision of the numerical simulation, but also restricts improving efficiency of the computation. Accordingly, this paper is devoted to explore stable explicit methods with high order of accuracy for the numerical simulation of wave motion within the computational region. The paper further develops the existing time-space decoupling explicit numerical simulation technique, and proposes a new highly accurate stable explicit method.First, a new explicit method of numerical simulation is proposed based on the exact solution of wave equations. The method shares similarity with the lumped mass FEM, which is suitable for dealing with irregular grids and the recursion formulas derived are explicit and decoupling both in time and space. But, the recursion formulas developed by the method are 2M-order (M is the positive integer) accurate both in time and space. In this paper, the feasibility of the method is demonstrated via 1-D model: the recursion formulas at nodal points of an irregular grid are constructed, meanwhile, the stability and accuracy of the recursion formulas for a uniform grid are discussed in detail; according, an approach is proposed to construct the stable formulas which are of 2M-order of accuracy both in time and space with M being a positive integer and illustrated by constructing formulas of the second order (M=1) and the fourth order (M=2).Second, the method is generalized to the multi-dimensional cases and the corresponding recursion formulas for an irregular grid in 2-D and 3-D are constructed respectively. For a uniform quadrate grid, the approach used to construct the stable formulas of 2M-order of accuracy is deliberated in detail, and the stable formulas of the second order (M=1) and the fourth order (M=2) are presented; for a 3-D cubic grid, the recursion formulas are discussed briefly.Third, the stability and accuracy of the method are demonstrated by a series of 1-D and 2-D numerical tests, and the value of high order formulas for the improvement of calculation efficiency is pointed out. Last, starting from the space decoupling FEM ordinary differential equation system (the equations of motion of the structure), a group of explicit time integration schemes are derived via the Lagrange polynomial interpolation and integration by parts, which are of high order of accuracy; furthermore, the expanding form of the schemes are given for any mathematical- physical model; the stability of the schemes are investigated preliminarily by a simple linear time-invariant system.更多还原