近场波动数值模拟的高精度显式方法研究

The seismic analysis of structures can be regarded as a near-field wave motion problem. And the numerical simulation in time domain is a promising technique for solving such wave motion problems. The development and perfection of the technique aim at improving the computation efficiency while guaranting precision and stability. And the high accurate method gains in importance for the sake of pursuing high efficiency. In this project, according to the concept that wave speed is finite, a new explicit method will be proposed based on the analytic solution of wave equations, and the corresponding recursion formula is constructed, which is suitable for dealing with both irregular and regular grids. Then the explicit method will be generalized for the numerical simulation of wave motion in inhomogeneous media with interface, which puts emphasis on the stable implementation with proper stability criterion. Furthermore, in light of the Helmholtz＇s theorem, the Navier equation will be decomposed into a scalar equation group，and thus the explicit method can be applied to elastic wave motion problems. Finally, combining with "The nonlinear earthquake response of Fuzhou, Quanzhou and Zhangzhou Basin" engineering example, the accuracy, stability and efficieny of the explicit method are verified by numerical comparsion, which shows the theoretical interest and great relevance to applications for the seismic design and seismic hazard assessment for a complicated and open system of large scale.

结构地震反应分析本质上是近场波动问题，求解此类问题最具前景的技术是在时域内直接进行波动数值模拟。在保证稳定性的前提下提高计算精度和计算效率是发展和完善近场波动数值模拟技术的目标。本项目拟从波速有限的物理概念出发，基于标量波动方程的解析解，发展一种具有时空解耦特性的高精度显式数值模拟方法；并针对均匀介质中波动问题，构建出规则和非规则网格均适用的高精度显式递推公式。同时，将此方法推广用于具有交界面的非均匀介质中波动的数值模拟，研究配套的交界面显式数值模拟技术及其稳定实现。进而，借助Helmholtz分解将矢量弹性波方程分解为标量波动方程组，用此显式方法实现弹性波波动问题的数值模拟。最后，结合"福州、泉州、漳州盆地非线性地震效应研究"工程实例，对比验证该显式方法的精度和稳定性，揭示高精度对提高计算效率的价值。本项目工作对完善和发展大型复杂开放体系的近场波动数值模拟技术具有重要理论意义和实用价值。

本项目致力发展近场波动数值模拟的高效时域方法，对其中涉及的理论问题和关键技术进行了深入研究。依据显式集中质量有限元的解耦思想，基于波动方程初值问题解析解，提出了一种在空间域上采用Lagrange多项式内插、时间域上精确积分的新的显式方法；并分别针对一、二、三维波动模型，构建出系列具有2M（M为正整数）阶精度、稳定的显式递推格式，实现了均匀介质标量波动的显式数值模拟技术的架构；将非均匀介质视为由间断面相互连接而成的不同均匀介质的分域组合，研究了分界面上波动处理技术，构建出与均匀介质格式相匹配（公式相容、精度一致）的分界面上节点运动的显式递推公式，并结合透射人工边界，实现了全域的标量波动数值模拟；通过数值算例验证了该项显式波动数值模拟技术的精确性、稳定性和高效性等理论结果，并通过工程实例展现了此项技术的发展空间和应用前景。