| 有限元三角网格中波动的频散分析及数值仿真 |
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| 引用本文： | 王圣柱,周建科,张奎华,王艺豪.有限元三角网格中波动的频散分析及数值仿真[J].地震学报,2015,37(3):493-507. |
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| 作者姓名： | 王圣柱 周建科 张奎华 王艺豪 |
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| 作者单位： | 1.中国山东青岛 266580 中国石油大学(华东)地球科学与技术学院 |
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| 基金项目： | 国家“十二五”重点科技攻关项目(2011ZX05002-002)与中石化科技重大攻关项目(P13020)联合资助. |
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| 摘 要： | 波动问题有限元离散后会引起数值误差， 数值频散的本质就是数值误差传播引起的非物理解. 数值频散不仅没有实际意义， 而且还会影响对真实波动现象的认识. 为厘清有限元三角网格中波动数值频散的影响因素， 本文推导了集中质量矩阵和一致质量矩阵的频散函数， 同时给出了组合质量矩阵的频散函数， 并对不同质量矩阵的数值频散进行了对比研究. 理论分析和数值计算结果表明: 有限元三角网格中波动的数值频散受网格布局、 波传播方向、 单元网格纵横比以及质量矩阵的影响； 一致质量矩阵的数值频散比集中质量矩阵更易受到波传播方向的影响； 不合理的三角网格单元会对数值相速度(数值频散)产生不良影响； 正三角网格中波动的数值频散几乎不受波传播方向的影响； 一致质量矩阵与集中质量矩阵的线性组合能够有效地压制数值频散.
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| 关 键 词： | 有限元法 三角网格 波动方程 数值频散 组合质量矩阵 数值仿真 |
| 收稿时间： | 2014-05-25 |
Dispersion analysis and numerical simulation of wave motion in finite element algorithm with triangular meshes |
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| Institution: | 1.School of Geosciences， China University of Petroleum， Shandong Qingdao 266580， China2.Western New Prospect Research Institute， Shengli Oilfield Company， SINOPEC， Shandong Dongying 257061， China3.School of Geodesy and Geomatics， Wuhan University， Wuhan 430079， China |
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| Abstract: | The finite element discretization of wave motion can cause numerical error. The numerical dispersion is in essence the non-physical solution caused by the propagation of numerical error. Not only has the numerical dispersion no practical meaning, but also it can affect the understanding of real fluctuations. In order to clarify the influence factors of numerical dispersion in finite element algorithm with triangular meshes, the dispersion functions of lumped mass matrix and consistent mass matrix are derived respectively, and the dispersion function of combined mass matrix is also given. And then the numerical dispersion of different mass matrices are compared. The results of theoretical analysis and numerical simulation indicate: ① The numerical dispersion in finite element algorithm with triangular meshes depend on the mesh layout, direction of wave propagation, the ratio of vertical length to horizontal length and the mass matrix; ② The numerical dispersion of consistent mass marix is more easily affected by wave propagation direction than that of lumped mass matrix; ③ The irrational triangular meshes have a bad effect on numerical phase velocity (numerical dispersion); ④ Equilateral triangular meshes can provide superior results regardless of the propagation direction; ⑤ The linear combination of lumped mass matrix and consistent mass matrix can effectively suppress numerical dispersion. |
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