地震波动数值模拟及稳定性研究
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摘要
本文针对显式有限元.有限差分法进行了系统的研究,包括理论分析,算法改进以及实际应用。通过理论推导定量地研究了采用这种数值模拟方法对波动传播规律的影响,以及其在结合局部透射人工边界进行计算时的稳定条件。分别就采用该方法结合局部透射人工边界求解波源问题和散射问题的情况分别提出了改进计算的措施,并通过数值试验验证其有效性。提出了一种有限元和显式有限元.有限差分法耦合方法,并采用其对可控震源激震过程进行了数值模拟。利用在汶川地震中获得的数据,进行场地效应分析。主要工作如下：
1.根据由李小军等人提出的一种求解动力方程的显式积分格式应用于显式有限元-有限差分法进行波动数值模拟时的递推形式,给出了波动传递函数的形式；以一维均匀离散体系为模型,推导了该情况下数值波动的传递函数,并进行数值试验,验证了传递函数求解的正确性；并通过分析求解所得的传递函数,探讨了这种显式积分格式对离散网格中波动传播规律包括截止频率、频散现象和能量耗散等方面的影响。
2.针对显式有限元一有限差分法结合局部透射边界进行波动数值模拟的情况,推导并计算了循环系数,给出了计算过程的稳定条件,并通过数值试验对该判定准则进行验证；在给出的循环系数的基础上,定量地分析了积分格式的能耗特性对由边界条件引入的高频失稳的抑制和消除作用；并探讨了如何更有效地利用能耗特性保证计算稳定性,即实际计算模型中时间离散步距以及空间模型大小等参数如何取值既能确保计算稳定又能最大程度减小计算量。
3.分别针对采用显式有限元一有限差分法结合局部透射人工边界求解波源问题和散射问题的情况分别提出了改进措施。对于波源问题,建议针对每个边界节点可以采用不同的人工波速,该人工波速可以通过试算一定时间长度,然后通过相关处理获得。通过数值实验证明了,在波源问题的求解中采用这种方法选择人工波速可以在不增加计算失稳风险的同时提高数值计算精度。对于散射问题,则给出了一种关于波动输入方法的改进措施,即在分区分离波场中入射波不再采用解析解而是采用离散数值解代替,数值实验证明了采用该方法可以有效地抑制漂移失稳。
4.给出了一种基于有限元和显式有限元.有限差分法相耦合的区域分裂法。将计算模型分为两个区域,对于广义结构及其附近区域采用有限元法模拟,而土体其它部分采用显式有限元.有限差分法,两个区域的模拟通过搭接区域实现数据的交换。分别代表波源问题和散射问题的数值试验都验证了该方法的有效性。采用提出的耦合方法针对可控震源激振过程建立的数值计算模型,并对不同的工况进行计算,研究分析了各因素包括接触土体密度、接触土体刚度、接触土层厚度、软、硬夹层以及底平板半径对可控震源能量输入大地体系的影响,为如何能高效地将激震能量输入土体提供了参考。
5.利用自贡地形影响台阵在2008年汶川地震中获得的主震强震记录,采用水平/垂直谱比法(HVSR)对该山脊地形场地对地震动的放大效应进行了分析,并探讨了所获记录结果的可靠性；采用显式有限元一有限差分法结合局部透射人工边界的二维场地影响分析模型与方法,模拟了该场地的地震动响应。对模拟结果与HVSR法分析结果进行对比分析,表明数值模拟结果与观测记录分析结果具有较好的一致性,利用数值计算方法建立的简单二维分析模型基本上可以反映山脊地形对地震动的影响。
In this paper, explicit finite element-finite difference method is studied systematically, including theoretical analysis, algorithm improvement, and practical application. The effect of the numerical simulation method on wave propagation and the calculation stability of the method combination of local artificial boundary conditions are studied quantitatively through theoretical derivation. Improvements about simulating source problems and scattering problems are presented, and these approaches are approved by the numerical simulation. A finite element and explicit finite element-finite difference method coupled method is presented, and using this method to simulate process of vibrator working. In the earthquake using the data obtained, the site effect is analyzed with the Wenchuan earthquake data. Main tasks are as follows:
1. Based on the recursion formula of the integration method which can be used to solve dynamic equation of structural system, the transfer function of wave is presented. For one-dimensional uniform discrete model, the transfer function of numerical wave is put forward and numerical experiments are provided to demonstrate the correctness of the result. The effect of the explicit integration formula on cut-off frequency, frequency dispersion and numerical dissipation of wave propagation in finite element meshes is studied. The present paper aims to provide academic reference for the explicit integration formula application in infinite domain wave problem.
2.Considering the numerical simulation of wave with explicit finite element-finite deference method and Local Transmitting boundary, the circulation coefficient is derived and calculated, the stability condition of numerical simulation is presented and tested by the numerical experiment; Based on the circulation coefficient, the effect of the explicit integration depressing and eliminating the high-frequency induced by local transmitting boundary is studied quantitatively; And how to use this effect to calculate steadily is discussed, that is how to take the time-discrete step and size of the space model values to ensure stability of calculation and decrease calculation amount to the full.
3. In this chapter, improvements about simulating source problems and scattering problems using explicit finite element-finite difference method combined Local Transmitting boundary are presented respectively. For wave source problems,the artificial wave velocity is suggested to be different value, which can be calculated through correlation processing of data. Numerical experiments show that solving wave source problems with this approach can improve numerical accuracy without increasing the risk of calculation instability. For wave scattering problems, a improvement about wave inputting is presented, that is incident wave in the wave field separating use the discrete numerical solution instead of using the analytical solution, numerical experiments approve the measure can eliminate the drift instability.
4. This chapter gives an explicit finite element and finite element-finite difference method coupling method. The calculation model is divided into two regions, the general structure and the surrounding area is simulated with finite element method, and the other part of the soil using explicit finite element finite difference method, the data of the two region communicate in the overlap region, This method is approved by numerical experiments representing the wave source problem and the scattering problem. Proposed coupling method used for establishing numerical model to simulate the vibrator working process, the effect of various factors, including soil density and stiffness, surface layer thickness, soft, hard interlayer, and the radius of the bottom plate of the vibrator on energy is studied, providing references for how to input energy to earth.
5. Based on the strong motion accelerations recorded by the topographical effect observation array in Xishan park, Zigong city from the mainshock of Wenchuan earthquake,2008, the site effects of the ridge in the observation array site are studied through Horizontal-to-Vertical Spectral Ratio (HVSR) method, and the reliability of results is discussed. The strong motion accelerations of the observation array site are also simulated by the two-dimensional site model and analysis method, which are based on the explicit finite element-finite deference method and local transmitting boundary. This study indicates that the results of the numerical simulation and the observation data are strong consistent, and the simple two-dimensional site model is suitable for simulating the topographical effects on ground motions.
1.根据由李小军等人提出的一种求解动力方程的显式积分格式应用于显式有限元-有限差分法进行波动数值模拟时的递推形式,给出了波动传递函数的形式；以一维均匀离散体系为模型,推导了该情况下数值波动的传递函数,并进行数值试验,验证了传递函数求解的正确性；并通过分析求解所得的传递函数,探讨了这种显式积分格式对离散网格中波动传播规律包括截止频率、频散现象和能量耗散等方面的影响。
2.针对显式有限元一有限差分法结合局部透射边界进行波动数值模拟的情况,推导并计算了循环系数,给出了计算过程的稳定条件,并通过数值试验对该判定准则进行验证；在给出的循环系数的基础上,定量地分析了积分格式的能耗特性对由边界条件引入的高频失稳的抑制和消除作用；并探讨了如何更有效地利用能耗特性保证计算稳定性,即实际计算模型中时间离散步距以及空间模型大小等参数如何取值既能确保计算稳定又能最大程度减小计算量。
3.分别针对采用显式有限元一有限差分法结合局部透射人工边界求解波源问题和散射问题的情况分别提出了改进措施。对于波源问题,建议针对每个边界节点可以采用不同的人工波速,该人工波速可以通过试算一定时间长度,然后通过相关处理获得。通过数值实验证明了,在波源问题的求解中采用这种方法选择人工波速可以在不增加计算失稳风险的同时提高数值计算精度。对于散射问题,则给出了一种关于波动输入方法的改进措施,即在分区分离波场中入射波不再采用解析解而是采用离散数值解代替,数值实验证明了采用该方法可以有效地抑制漂移失稳。
4.给出了一种基于有限元和显式有限元.有限差分法相耦合的区域分裂法。将计算模型分为两个区域,对于广义结构及其附近区域采用有限元法模拟,而土体其它部分采用显式有限元.有限差分法,两个区域的模拟通过搭接区域实现数据的交换。分别代表波源问题和散射问题的数值试验都验证了该方法的有效性。采用提出的耦合方法针对可控震源激振过程建立的数值计算模型,并对不同的工况进行计算,研究分析了各因素包括接触土体密度、接触土体刚度、接触土层厚度、软、硬夹层以及底平板半径对可控震源能量输入大地体系的影响,为如何能高效地将激震能量输入土体提供了参考。
5.利用自贡地形影响台阵在2008年汶川地震中获得的主震强震记录,采用水平/垂直谱比法(HVSR)对该山脊地形场地对地震动的放大效应进行了分析,并探讨了所获记录结果的可靠性；采用显式有限元一有限差分法结合局部透射人工边界的二维场地影响分析模型与方法,模拟了该场地的地震动响应。对模拟结果与HVSR法分析结果进行对比分析,表明数值模拟结果与观测记录分析结果具有较好的一致性,利用数值计算方法建立的简单二维分析模型基本上可以反映山脊地形对地震动的影响。
In this paper, explicit finite element-finite difference method is studied systematically, including theoretical analysis, algorithm improvement, and practical application. The effect of the numerical simulation method on wave propagation and the calculation stability of the method combination of local artificial boundary conditions are studied quantitatively through theoretical derivation. Improvements about simulating source problems and scattering problems are presented, and these approaches are approved by the numerical simulation. A finite element and explicit finite element-finite difference method coupled method is presented, and using this method to simulate process of vibrator working. In the earthquake using the data obtained, the site effect is analyzed with the Wenchuan earthquake data. Main tasks are as follows:
1. Based on the recursion formula of the integration method which can be used to solve dynamic equation of structural system, the transfer function of wave is presented. For one-dimensional uniform discrete model, the transfer function of numerical wave is put forward and numerical experiments are provided to demonstrate the correctness of the result. The effect of the explicit integration formula on cut-off frequency, frequency dispersion and numerical dissipation of wave propagation in finite element meshes is studied. The present paper aims to provide academic reference for the explicit integration formula application in infinite domain wave problem.
2.Considering the numerical simulation of wave with explicit finite element-finite deference method and Local Transmitting boundary, the circulation coefficient is derived and calculated, the stability condition of numerical simulation is presented and tested by the numerical experiment; Based on the circulation coefficient, the effect of the explicit integration depressing and eliminating the high-frequency induced by local transmitting boundary is studied quantitatively; And how to use this effect to calculate steadily is discussed, that is how to take the time-discrete step and size of the space model values to ensure stability of calculation and decrease calculation amount to the full.
3. In this chapter, improvements about simulating source problems and scattering problems using explicit finite element-finite difference method combined Local Transmitting boundary are presented respectively. For wave source problems,the artificial wave velocity is suggested to be different value, which can be calculated through correlation processing of data. Numerical experiments show that solving wave source problems with this approach can improve numerical accuracy without increasing the risk of calculation instability. For wave scattering problems, a improvement about wave inputting is presented, that is incident wave in the wave field separating use the discrete numerical solution instead of using the analytical solution, numerical experiments approve the measure can eliminate the drift instability.
4. This chapter gives an explicit finite element and finite element-finite difference method coupling method. The calculation model is divided into two regions, the general structure and the surrounding area is simulated with finite element method, and the other part of the soil using explicit finite element finite difference method, the data of the two region communicate in the overlap region, This method is approved by numerical experiments representing the wave source problem and the scattering problem. Proposed coupling method used for establishing numerical model to simulate the vibrator working process, the effect of various factors, including soil density and stiffness, surface layer thickness, soft, hard interlayer, and the radius of the bottom plate of the vibrator on energy is studied, providing references for how to input energy to earth.
5. Based on the strong motion accelerations recorded by the topographical effect observation array in Xishan park, Zigong city from the mainshock of Wenchuan earthquake,2008, the site effects of the ridge in the observation array site are studied through Horizontal-to-Vertical Spectral Ratio (HVSR) method, and the reliability of results is discussed. The strong motion accelerations of the observation array site are also simulated by the two-dimensional site model and analysis method, which are based on the explicit finite element-finite deference method and local transmitting boundary. This study indicates that the results of the numerical simulation and the observation data are strong consistent, and the simple two-dimensional site model is suitable for simulating the topographical effects on ground motions.
引文
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