The Lumped mass Chebyshev spectral element method for seismic response analysis of horizontally layered soil sites
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摘要: 提出了一种用于水平成层场地地震反应分析的时域高阶显式算法. 首先，将覆盖土层和基岩划分为若干个切比雪夫谱单元，在模型底部设置多次透射人工边界；其次，以切比雪夫正交多项式构建高阶单元位移模式，通过高斯−洛巴托积分严格导出对角形式的切比雪夫谱单元集中质量矩阵，结合中心差分时域逐步积分格式，建立了高效的集中质量切比雪夫谱元波动模拟方法；最后，利用日本Kik-net强震台网提供的不同类型场地上获得的实际地震观测记录检验了本文方法的有效性. 该方法避免了传统切比雪夫谱元法由于具有一致质量矩阵形式而造成的计算效率不高的问题。数值结果表明，本文方法能够较好地预测Ⅰ1，Ⅱ和Ⅳ类场地在较弱地震和中等强度地震作用下的地面运动特征，每个波长内仅需布置少量单元即可取得较高精度的计算结果。Abstract: A time-domain high-order explicit method for the seismic response analysis of horizontally layered soil sites is proposed. The upper soil and bedrock are discretized by several Chebyshev spectral elements. The multi-transmitting artificial boundary is set at the bottom of the model. The Chebyshev orthogonal polynomials are employed for establishing high-order element displacement field. By means of the Gauss-Lobatto quadrature, the lumped mass matrix, which has diagonal form, for the Chebyshev spectral element is rigorously derived. Combined with the central difference time-stepping scheme, an efficient lumped mass Chebyshev spectral element method for simulating wave motion is constructed. Earthquake records obtained from different kinds of sites provided by the Kik-net strong earthquake network are used to examine the validity of the proposed method. This method overcomes the shortage in efficiency of conventional Chebyshev spectral element method resulted from having consistent form ofmass matrix. Numerical results show that the proposed method can give reasonable prediction on the ground motions of Ⅰ 1, Ⅱ and Ⅳ type sites under weak or moderate earthquakes, and good accuracy can be achieved only by deploying a small number of elements per wavelength.
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图 1 水平成层场地切比雪夫谱元模型
Figure 1. Chebyshev spectral element model of horizontal layered soil site
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图 2 切比雪夫谱单元的一阶MTF插值方案
Figure 2. Interpolation scheme for first-order MTF in Chebyshev spectral element
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图 3 均匀半空间模型在雷克波入射下的地表和底部位移反应
Figure 3. Displacement responses of ground surface and bottom of a homogeneous half-space model under Ricker wave incidence
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图 4 四个Kik-net台站的土层剪切波速图
Figure 4. Shear wave velocity profiles at the four Kik-net stations
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图 5 不同强度地震动作用下四个台站地表加速度反应时程
（a） 较弱地震动；（b） 中等强度地震动；（c） 较强地震动
Figure 5. Ground acceleration response histories of four stations under ground motions with different intensities
（a） Weak ground motions；（b） Moderate ground motions； （c） Strong ground motions
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图 6 不同强度地震动作用下四个台站的地表加速度反应谱
（a） 较弱地震动；（b） 中强地震动；（c） 较强地震动
Figure 6. Ground acceleration response spectra of four stations under ground motions with different intensities
（a） Weak ground motions；（b） Moderate ground motion； （c） Strong ground motions
表 1 GLC节点上的高斯−洛巴托积分权系数
Table 1 Gauss-Lobatto quadrature weights based on GLC points
谱单元阶次 GLC节点坐标 积分权系数 1 ±1 1 2 ±1, 0 0.333 3, 1.333 3 3 ±1, ±0.5 0.111 1, 0.888 9 4 ±1, ±0.707 1, 0 0.066 7, 0.533 3, 0.8 5 ±1, ±0.809 0, ±0.309 0 0.04, 0.360 7, 0.599 3 
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表 2 选用Kik-net台站的基本信息
Table 2 Basic information of selected Kik-net stations
台站名称 覆盖层厚度/m 等效剪切波速/（m·s−1） 场地类别 TCGH08 2 170 Ⅰ1 MYGH10 34 330 Ⅱ KMMH14 88 200 Ⅲ IKRH02 ＞108 112 Ⅳ
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表 3 不同强度地震动作用下各台站计算PGA与实测PGA对比
Table 3 Comparison of computed PGA and recorded PGA for the stations under ground motions different intensities
台站名称 较弱地震动 中等强度地震动 较强地震动 计算PGA/g 实测PGA/g 计算PGA/g 实测PGA/g 计算PGA/g 实测PGA/g TCGH08 0.090 0.065 0.142 0.145 0.226 0.204 MYGH10 0.066 0.067 0.154 0.156 0.240 0.235 KMMH14 0.121 0.083 0.227 0.144 0.369 0.228 IKRH02 0.082 0.084 0.102 0.104 0.329 0.227
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表 4 不同强度地震动作用下各台站PGA放大倍数
Table 4 Amplification factors of PGA for the stations under ground motions different intensities
台站名称 较弱地震动 中等强度地震动 较强地震动 井下实测
峰值/g计算放大
倍数实测放大
倍数井下实测
峰值/g计算放大
倍数实测放大
倍数井下实测
峰值/g计算放大
倍数实测放大
倍数TCGH08 0.015 5.96 4.31 0.037 3.80 3.89 0.028 8.21 7.40 MYGH10 0.016 4.14 4.22 0.042 3.67 3.71 0.065 3.69 3.62 KMMH14 0.016 7.36 5.07 0.031 7.36 4.69 0.035 10.48 6.48 IKRH02 0.018 4.54 4.63 0.023 4.45 4.55 0.069 4.75 3.28
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