The mass property model and its implementation in the time-domain spectral element method
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摘要: 研究了构建时域谱单元质量特性模型的数学机制，针对时域切比雪夫谱单元和勒让德谱单元建立了一种直接导出谱单元一致质量矩阵和集中质量矩阵的统一数学方法，对比分析两种谱单元质量特性模型的特征，并从物理角度探讨了谱单元质量特性模型的合理性。研究表明，数值积分点与谱单元节点选取是否一致是决定时域谱单元形成一致质量模型或集中质量模型的根本原因，当采用高斯-勒让德积分计算谱单元质量模型时将导出一致质量矩阵，而采用高斯-洛巴托积分则导出集中质量矩阵。而集中质量模型更具有物理合理性，两种谱单元质量特性模型优劣相当，均可取得很好的动力问题分析结果。Abstract: The mathematical mechanism of constructing mass property model for the time-domain spectral elements is studied in this paper. A unified mathematical method for directly deriving consistent and lumped mass matrix is established for the time-domain Chebyshev and Legendre spectral elements. The characteristics of two mass property models of the spectral elements are analyzed through comparison. Meanwhile, the rationality of mass property model of spectral element is discussed from physical perspective. This study reveals that the formation of consistent or lumped mass matrix in time-domain spectral elements depends on whether the quadrature points are coincident with the element nodes or not. To be specific, the Gauss-Legendre quadrature results in consistent mass matrix for spectral elements, and the Gauss-Lobatto quadrature leads to lumped mass matrix. The lumped mass matrix is more reasonable in physics. The two mass property models of spectral elements have comparable performance and they can both achieve good results for dynamic problems.
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图 2 一维集中质量切比雪夫谱单元（左）和勒让德谱单元（右）质量分布特征
（a） 3节点单元；（b） 4节点单元；（c） 5节点单元
Figure 2. Mass distribution characteristics of 1-D lumped mass Chebyshev （left） and Legendre （right） spectral elements
（a） 3-node element；（b） 4-node element；（c） 5-node element
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图 3 二维集中质量切比雪夫（左）和勒让德（右）谱单元质量分布特征
（a） 9节点单元；（b） 16节点单元；（c） 25节点单元
Figure 3. Mass distribution characteristics of 2-D lumped mass Chebyshev （left） and Legendre （right） spectral elements
（a） 9-node element；（b） 16-node element；（c） 25-node element
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图 4 一维5节点切比雪夫谱单元一致质量模型数学图示
Figure 4. Mathematical representation for consistent mass model of 1-D 5-node Chebyshev element
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图 5 一维5节点切比雪夫谱单元集中质量模型数学图示
Figure 5. Mathematical representation for lumped mass model of 1-D 5-node Chebyshev element
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图 6 不同谱单元数量下简支梁的前6阶固有频率计算误差
Figure 6. Error of first six natural frequencies with different number of elements
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图 9 离散化和积分过程引起的质量特性误差
Figure 9. Error in mass property caused by discretization and quadrature
表 1 GLL节点坐标和积分权系数
Table 1 Abscissas of GLL points and quadrature weights
谱单元阶次 GLL节点坐标 积分权系数 1 （−1，1） 1，1 2 （−1，0，1） 0.333 333，1.333 333，0.333 333 3 （−1，−0.447 214，0.447 214，1） 0.166 667，0.833 333，0.833 333，0.166 667 4 （−1，−0.654 654，0，0.654 654，1） 0.1，0.544 444，0.711 111，0.544 444，0.1 5 （−1，−0.765 055，−0.285 232，
0.285 232，0.765 055，1）0.066 667，0.378 475，0.554 858，
0.554 858，0.378 475，0.066 667
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表 2 GLC节点坐标和高斯-洛巴托积分权系数
Table 2 Abscissas of GLC points and Gauss-Lobatto quadrature weights
谱单元阶次 GLC节点坐标 积分权系数 1 （−1，1） 1，1 2 （−1，0，1） 0.333 333，1.333 333，0.333 333 3 （−1，−0.5，0.5，1） 0.111 111，0.888 889，0.888 889，0.111 111 4 （−1，−0.707 107，0，0.707 107，1） 0.066 667，0.533 333，0.8，0.533 333，0.066 667 5 （−1，−0.809 017，−0.309 017，
0.309 017，0.809 017，1）0.04，0.360 743，0.599 257，
0.599 257，0.360 743，0.04
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表 3 简支梁前6阶固有频率计算结果
Table 3 First six natural frequencies of simply supported beam
谱单元 单元阶次 固有频率 /Hz 1阶 2阶 3阶 4阶 5阶 6阶 切比雪夫
（集中质量）4 26.093 6 103.808 9 230.472 6 408.857 7 633.689 4 1 092.290 1 5 26.093 6 103.794 3 231.362 4 406.506 8 626.318 0 876.153 0 6 26.093 6 103.793 9 231.425 2 406.355 0 625.330 3 884.533 9 切比雪夫
（一致质量）4 26.093 7 103.818 0 231.485 4 410.983 5 647.442 7 1 025.303 2 5 26.093 6 103.794 3 231.453 3 406.625 7 627.393 8 887.768 5 6 26.093 6 103.793 9 231.418 6 406.356 5 625.379 6 886.020 8 勒让德
（集中质量）4 26.093 7 103.823 4 231.410 2 411.380 2 646.890 0 1 054.997 5 5 26.093 6 103.794 4 231.465 2 406.682 7 627.654 9 884.530 9 6 26.093 6 103.793 9 231.418 6 406.359 7 625.409 9 886.581 6 解析解 26.093 5 103.791 8 231.395 7 406.225 1 624.820 5 883.181 3
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