场地地震波动模拟中透射边界稳定性问题研究
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摘要
本文对场地地震波动模拟中透射边界的稳定性问题进行了一系列研究。对已有的透射边界稳定措施做了比较分析,并给出了不同措施的适用性。结合粘弹性边界的物理模型提出了一个新的消除透射边界飘移失稳的措施,参考高频滤波的思路探讨了一种新的消除透射边界高频振荡失稳的措施,并用数值试验验证了这两种新的透射边界稳定方法的有效性。将显式有限元-有限差分方法结合透射边界的数值解法的计算结果与解析结果作对比分析,验证这样一套数值解法的计算精度。利用二维模型模拟了自贡土层和山脊地形的场地效应,利用显式有限元-有限差分方法结合透射边界这套数值解法进行数值模拟计算,对比汶川地震的实际强震观测记录来验证这套数值解法对于实际地震波动模拟的有效性。
本文的主要工作如下：
1.将显式有限元-有限差分数值模拟方法与大圆弧假定和Fourier-Bessel级数波函数展开法相结合的解析法作比较分析,不同频率数值解与解析解吻合情况较好,但所对应的网格尺寸要求与频率有关；在满足计算格式稳定性的情况下,计算时间步长无需取的过小就可满足精度需求且能提高计算效率
2.分析了透射边界高频振荡失稳和低频飘移失稳的原因,通过数值试验对已有的稳定性措施进行比较分析,给出了已有稳定措施的适用性：对于完全弹性的问题,采用滤波方法消除高频震荡失稳具有优势；对于粘弹性等本身就存在阻尼的问题,利用阻尼与显式差分格式结合,用差分格式本身的能耗特性来抑制高频失稳具有优势；对于飘移失稳,γ算子方法参数确定没有一个明确的标准,完全依靠经验性试算,使用上应注意参数的取值控制；降阶消飘的方法的关键在于失稳趋势的判定,不存在方法参数的选取,实现上可控性较好。
3.利用在透射边界区附加粘弹性元件的方案,探讨了在波动数值模拟中消除多次透射边界计算失稳的措施,该措施中弹簧和阻尼元件被附加在透射边界区内的单元节点上。数值计算分析表明,该措施是一种处理透射边界计算飘移失稳的有效措施,对透射边界的飘移失稳有较好的抑制作用,但对抑制透射边界的高频振荡失稳该措施没有明显效果。
4.借鉴了透射边界区高频滤波的思路,考虑高频误差振荡首先在人工边界上出现,提出利用人工边界节点之间进行滤波平滑的方法来消除高频震荡失稳的措施,探讨了平滑系数的取值影响。数值计算分析表明,该措施是一种处理透射边界高频振荡失稳的有效措施；并在此措施的基础上,初步提出了一个具有较好稳定性的多向透射公式。
5.利用自贡地形影响强震动观测台阵在汶川地震中获得的强震动记录,以位于山脚下基岩上的台站为参考点采用传统谱比法对场地放大效应进行了初步分析；采用显式有限元—有限差分方法模拟了自贡西山公园山脊场地地形和土层的地震动效应,并进行了数值模拟结果与强震动观测结果的比分析。结果表明：上覆土层对地震动的放大作用相对于地形变化的放大作用更为明显；地形对地震动水平分量的放大效应要明显于对竖直分量的放大效应；采用二维模型对上覆土层的模拟与观测结果较吻合,对于基岩介质部分的模拟在低频范围内也能反映山脊地形对地震动的影响。
最后,笔者对本文进行的研究工作予以了总结,并在此基础上,提出了有待进一步研究解决的问题。
In this paper, stability problems of local transmitting boundary in wave simulation on of site seismic effect is studied. Based on the analysis and comparison of existing stability measures for transmitting boundary, the applicability of each measure is given. A new measure for eliminating the drift instability is proposed based on the combination of the multi-transmitting boundary with the viscous-spring elements, and a new measure for eliminating the high-frequency instability is proposed. Validities of these two suggested measures was tested by numerical experiment. Compared with analytical solution, calculation accuracy of the method which combining the explicit finite element-finite difference method with local transmitting boundary is verified. Ground motion response of the overlaying soil and ridge terrain in Zigong Xishan park was simulated by the explicit finite element-finite difference method. Compared with the observed values of strong motion records in Wenchuan earthquake, Validity of this method to practical wave simulation was tested.
The works in this study include:
1. The method which combining the explicit finite element-finite difference method with local transmitting boundary is compared with Fourier-Bessel wave function expansion method combined with big circle assumption. At different frequencies, mesh size requirement of numerical solution corresponding to analytical solution is also different. The precision can be satisfied and the calculation efficiency can be improved by suitable time step without too small step.
2. Based on the Cause of drift instability and high-frequency instability, the applicability of previously used stability measures is discussed by numerical experiment. To the perfect elasticity problem, the filtering method is more suitable to eliminate high-frequency instability; to the situation existing damping itself such as viscoelastic problem, explicit difference scheme with damps is effective to eliminate high-frequency instability by energy consumption of difference scheme. As for drift instability, the method of adding modified operatorγB00 is needed to be carefully used because of the empirical feature and uncertainty of parameter value. We recommend the method of reduced order because of the certainty and controllability of parameter value.
3. The measure was studied for eliminating instability of the transmitting boundary by means of adding viscous-spring elements in the artificial boundary area. In the measure, the springs and dampers are emplaced on the finite element nodes next to the artificial boundary in the artificial boundary area. Validity of the suggested measure is tested by numerical analysis for the topographical site. The results show that the measure is a valid one to solve the drift instability of multi-transmitting boundary, which is effective for eliminating the drift instability but no visible effect on the high-frequency instability.
4. The filtering method using the neighbor node of boundary node is suggested to eliminate high-frequency instability. In this method, the errors of high frequency oscillation in the vertical and parallel direction are all considered. The value of smoothing coefficient is also suggested in this paper. The results show that the measure is a valid one to solve the high-frequency instability of multi-transmitting boundary. According to this filtering method, a preliminary multi-direction transmitting formula is suggested.
5. Site amplification was analyzed for the ridge terrain of Zigong Xishan park using the strong motion record obtained in Wenchuan earthquake by traditional spectral ration method. The reference seismic station of spectral ration is located near the foot of the ridge. Ground motion response of the ridge terrain in Zigong Xishan park was simulated by the explicit finite element-finite difference method. The results of numerical simulation were compared with the observed values of strong motion records, which shows that this numerical simulation model could reflect the ridge terrain basically. Numerical modeling results show these conclusions as follows. Amplification action of the overlaying soil to seismic acceleration is more obvious than that of the topographic change. Horizontal component of the seismic ground motion is amplified by topography variation more evidently than vertical component. The two-dimensional model built in this paper could well simulate the ground motion on overlaying soil and the results coincided well with the observation results. Simulation for bedrock could reflect the ground motion response of the ridge terrain in low frequency range.
At last, the work in this study is summarized, and the problems need to be studied moor deeply are pointed out.
本文的主要工作如下：
1.将显式有限元-有限差分数值模拟方法与大圆弧假定和Fourier-Bessel级数波函数展开法相结合的解析法作比较分析,不同频率数值解与解析解吻合情况较好,但所对应的网格尺寸要求与频率有关；在满足计算格式稳定性的情况下,计算时间步长无需取的过小就可满足精度需求且能提高计算效率
2.分析了透射边界高频振荡失稳和低频飘移失稳的原因,通过数值试验对已有的稳定性措施进行比较分析,给出了已有稳定措施的适用性：对于完全弹性的问题,采用滤波方法消除高频震荡失稳具有优势；对于粘弹性等本身就存在阻尼的问题,利用阻尼与显式差分格式结合,用差分格式本身的能耗特性来抑制高频失稳具有优势；对于飘移失稳,γ算子方法参数确定没有一个明确的标准,完全依靠经验性试算,使用上应注意参数的取值控制；降阶消飘的方法的关键在于失稳趋势的判定,不存在方法参数的选取,实现上可控性较好。
3.利用在透射边界区附加粘弹性元件的方案,探讨了在波动数值模拟中消除多次透射边界计算失稳的措施,该措施中弹簧和阻尼元件被附加在透射边界区内的单元节点上。数值计算分析表明,该措施是一种处理透射边界计算飘移失稳的有效措施,对透射边界的飘移失稳有较好的抑制作用,但对抑制透射边界的高频振荡失稳该措施没有明显效果。
4.借鉴了透射边界区高频滤波的思路,考虑高频误差振荡首先在人工边界上出现,提出利用人工边界节点之间进行滤波平滑的方法来消除高频震荡失稳的措施,探讨了平滑系数的取值影响。数值计算分析表明,该措施是一种处理透射边界高频振荡失稳的有效措施；并在此措施的基础上,初步提出了一个具有较好稳定性的多向透射公式。
5.利用自贡地形影响强震动观测台阵在汶川地震中获得的强震动记录,以位于山脚下基岩上的台站为参考点采用传统谱比法对场地放大效应进行了初步分析；采用显式有限元—有限差分方法模拟了自贡西山公园山脊场地地形和土层的地震动效应,并进行了数值模拟结果与强震动观测结果的比分析。结果表明：上覆土层对地震动的放大作用相对于地形变化的放大作用更为明显；地形对地震动水平分量的放大效应要明显于对竖直分量的放大效应；采用二维模型对上覆土层的模拟与观测结果较吻合,对于基岩介质部分的模拟在低频范围内也能反映山脊地形对地震动的影响。
最后,笔者对本文进行的研究工作予以了总结,并在此基础上,提出了有待进一步研究解决的问题。
In this paper, stability problems of local transmitting boundary in wave simulation on of site seismic effect is studied. Based on the analysis and comparison of existing stability measures for transmitting boundary, the applicability of each measure is given. A new measure for eliminating the drift instability is proposed based on the combination of the multi-transmitting boundary with the viscous-spring elements, and a new measure for eliminating the high-frequency instability is proposed. Validities of these two suggested measures was tested by numerical experiment. Compared with analytical solution, calculation accuracy of the method which combining the explicit finite element-finite difference method with local transmitting boundary is verified. Ground motion response of the overlaying soil and ridge terrain in Zigong Xishan park was simulated by the explicit finite element-finite difference method. Compared with the observed values of strong motion records in Wenchuan earthquake, Validity of this method to practical wave simulation was tested.
The works in this study include:
1. The method which combining the explicit finite element-finite difference method with local transmitting boundary is compared with Fourier-Bessel wave function expansion method combined with big circle assumption. At different frequencies, mesh size requirement of numerical solution corresponding to analytical solution is also different. The precision can be satisfied and the calculation efficiency can be improved by suitable time step without too small step.
2. Based on the Cause of drift instability and high-frequency instability, the applicability of previously used stability measures is discussed by numerical experiment. To the perfect elasticity problem, the filtering method is more suitable to eliminate high-frequency instability; to the situation existing damping itself such as viscoelastic problem, explicit difference scheme with damps is effective to eliminate high-frequency instability by energy consumption of difference scheme. As for drift instability, the method of adding modified operatorγB00 is needed to be carefully used because of the empirical feature and uncertainty of parameter value. We recommend the method of reduced order because of the certainty and controllability of parameter value.
3. The measure was studied for eliminating instability of the transmitting boundary by means of adding viscous-spring elements in the artificial boundary area. In the measure, the springs and dampers are emplaced on the finite element nodes next to the artificial boundary in the artificial boundary area. Validity of the suggested measure is tested by numerical analysis for the topographical site. The results show that the measure is a valid one to solve the drift instability of multi-transmitting boundary, which is effective for eliminating the drift instability but no visible effect on the high-frequency instability.
4. The filtering method using the neighbor node of boundary node is suggested to eliminate high-frequency instability. In this method, the errors of high frequency oscillation in the vertical and parallel direction are all considered. The value of smoothing coefficient is also suggested in this paper. The results show that the measure is a valid one to solve the high-frequency instability of multi-transmitting boundary. According to this filtering method, a preliminary multi-direction transmitting formula is suggested.
5. Site amplification was analyzed for the ridge terrain of Zigong Xishan park using the strong motion record obtained in Wenchuan earthquake by traditional spectral ration method. The reference seismic station of spectral ration is located near the foot of the ridge. Ground motion response of the ridge terrain in Zigong Xishan park was simulated by the explicit finite element-finite difference method. The results of numerical simulation were compared with the observed values of strong motion records, which shows that this numerical simulation model could reflect the ridge terrain basically. Numerical modeling results show these conclusions as follows. Amplification action of the overlaying soil to seismic acceleration is more obvious than that of the topographic change. Horizontal component of the seismic ground motion is amplified by topography variation more evidently than vertical component. The two-dimensional model built in this paper could well simulate the ground motion on overlaying soil and the results coincided well with the observation results. Simulation for bedrock could reflect the ground motion response of the ridge terrain in low frequency range.
At last, the work in this study is summarized, and the problems need to be studied moor deeply are pointed out.
引文
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