基于非结构化网格的三维大地电磁自适应矢量有限元数值模拟
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摘要
大地电磁法(Magnetotellurics,MT)的快速、高精度三维正演是MT三维反演实用化的基础和前提,也是国际研究前沿和热点之一,极具挑战性。有限单元法(Finite Element Method,FEM)以其理论完善、程序通用、适应性强等优点,在MT数值模拟中受到越来越多的应用和重视。论文针对传统FEM中存在的三个问题,即节点型有限元存在伪解、结构化网格对复杂模型的几何离散化误差大、计算精度依赖于对网格的一次性剖分,提出了基于完全非结构化四面体网格的三维大地电磁h-型自适应矢量有限元正演策略,实现了有关的计算流程,以典型模型和国际标准电磁模型的模拟计算,证明了方法的正确和有效,为进一步研究奠定了基础。论文研究分别得到国家863计划项目(2006AA06Z105)和国家自科基金项目(40874072)的支持。论文主要内容如下:
1、研究了完全非结构化的四面体单元剖分及优化方法,提出了四边形控制的局部网格加密策略,不仅减少了对复杂电磁模型剖分的几何离散误差,也可以更精确地对感兴趣的目标区域局部加密,从而提高计算速度和计算精度。
2、从准静态极限下三维MT边值问题出发,导出了基于边的矢量有限元伽辽金有限元公式系统。针对非结构化的四面体单元,采用坐标变换和数值积分方法,实现了MT三维矢量有限单元分析,建立起基于非结构化网格的三维MT矢量有限元计算流程,并对典型模型和国际标准电磁模型进行了数值模拟,结果对比和分析表明,基于非结构化网格的三维MT矢量有限元不仅消除了节点型有限元的伪解,而且具有很高的计算精度和速度,有广阔的应用前景。
3、根据Sobolev函数的向量空间和Hemlholtz空间的分解,推导出基于残差的三维大地电磁矢量有限元后验误差估计公式,为三维大地电磁自适应矢量有限元数值模拟的实现奠定了基础。
4、在完全非结构化四面体单元剖分及优化基础上,结合三维大地电磁矢量有限元后验误差估计公式,提出了基于非结构化网格的三维大地电磁h-型自适应矢量有限元计算策略,保证了对复杂大地电磁模型数值计算的精度和可靠性。
5、实现了基于非结构化网格的三维大地电磁h-型自适应矢量有限元计算流程,对典型模型和国际标准电磁模型进行了数值模拟。结果表明,电磁模型的复杂性总体上不影响方法的收敛性,且可以达到预期的计算精度。因此,h-型自适应矢量有限元可以保证对复杂模型的计算精度和速度,具有很大的应用前景。
The three- dimensional(3-D) magnetotelluric(MT) forward modeling which can be fast solved with high accuracy has become the basis of making the 3-D inversion case practical. Its wide application in geophysical surveying has led it to being mostly focused. The finite element method which has an excellent theory basis and flexibility of modeling complicated problems has been a much popular tool among the numerical techniques. Aiming at the problems which appeared in conventional finite element method, such as the existence of fake solution, the large discretization error encountered when structured mesh was adopted to simulate the complex models and the accuracy being determined by just one time mesh generation and so on, an 3-D h-adaptive MT finite element method which employed the fully unstructured mesh for 3-D MT modeling was introduced. Some standard models were tested which has remarkably validated our algorithm. The work is supported by grants from the National High Technology Research and Development Program of China (863 Program) (2006AA06Z105) and National Natural Science Foundation of China (40874072). The whole framework of our work can be divided into:
1. A systematical study of fully unstructured mesh generation and the corresponding optimal strategy was made. Base on this, a mesh refinement strategy which was controlled by quadrilateral was presented. By using this technique, not only the geometrical discretization errors derived from modeling complicated electromagnetic models were dramatically reduced but also the local refinement could be executed in the area of interest by which both the accuracy and efficiency were mostly improved.
2. Starting from the 3-D MT boundary value problem under the quasi-static limitation, the finite element equation which was derived based on the Galerkin-residual approach was presented. The coordinate transformation and numerical integration was executed on the discretized tetrahedral elements based on which the 3-D MT vector-finite element method was implemented. A whole computation framework for 3-D vector-finite element method with unstructured mesh was given. Based on this some typical models were tested which has demonstrated that our algorithm could distinctively avoid problems caused by the fake solution and both the accuracy and efficiency were enhanced which made our algorithm has a bright future for further application.
3. According to theory of Sobolev vector space and the discretization of Helmholtz space, the error estimate which was suitable for 3-D MT vector-finite element modeling was deduced by which the procedure of adaptive technique was guaranteed.
4. Based on the fully unstructured tetrahedralization and optical strategy, the 3-D magnetotelluric h-adaptive vector-finite element method was presented through combining the error estimate. With this work, the accuracy and creditableness for 3-D MT complicatedly modeling was guaranteed.
5. The 3-D magnetotelluric h-adaptive vector-finite element algorithm with unstructured mesh was implemented. The numerical results have shown that the complicity of electromagnetic models generally will not effect the convergence with satisfactory accuracy being obtained. Thus, 3-D h-adaptive vector-finite element has remarkably guaranteed the accuracy and efficiency by which a broad application in the future can be predicated.
1、研究了完全非结构化的四面体单元剖分及优化方法,提出了四边形控制的局部网格加密策略,不仅减少了对复杂电磁模型剖分的几何离散误差,也可以更精确地对感兴趣的目标区域局部加密,从而提高计算速度和计算精度。
2、从准静态极限下三维MT边值问题出发,导出了基于边的矢量有限元伽辽金有限元公式系统。针对非结构化的四面体单元,采用坐标变换和数值积分方法,实现了MT三维矢量有限单元分析,建立起基于非结构化网格的三维MT矢量有限元计算流程,并对典型模型和国际标准电磁模型进行了数值模拟,结果对比和分析表明,基于非结构化网格的三维MT矢量有限元不仅消除了节点型有限元的伪解,而且具有很高的计算精度和速度,有广阔的应用前景。
3、根据Sobolev函数的向量空间和Hemlholtz空间的分解,推导出基于残差的三维大地电磁矢量有限元后验误差估计公式,为三维大地电磁自适应矢量有限元数值模拟的实现奠定了基础。
4、在完全非结构化四面体单元剖分及优化基础上,结合三维大地电磁矢量有限元后验误差估计公式,提出了基于非结构化网格的三维大地电磁h-型自适应矢量有限元计算策略,保证了对复杂大地电磁模型数值计算的精度和可靠性。
5、实现了基于非结构化网格的三维大地电磁h-型自适应矢量有限元计算流程,对典型模型和国际标准电磁模型进行了数值模拟。结果表明,电磁模型的复杂性总体上不影响方法的收敛性,且可以达到预期的计算精度。因此,h-型自适应矢量有限元可以保证对复杂模型的计算精度和速度,具有很大的应用前景。
The three- dimensional(3-D) magnetotelluric(MT) forward modeling which can be fast solved with high accuracy has become the basis of making the 3-D inversion case practical. Its wide application in geophysical surveying has led it to being mostly focused. The finite element method which has an excellent theory basis and flexibility of modeling complicated problems has been a much popular tool among the numerical techniques. Aiming at the problems which appeared in conventional finite element method, such as the existence of fake solution, the large discretization error encountered when structured mesh was adopted to simulate the complex models and the accuracy being determined by just one time mesh generation and so on, an 3-D h-adaptive MT finite element method which employed the fully unstructured mesh for 3-D MT modeling was introduced. Some standard models were tested which has remarkably validated our algorithm. The work is supported by grants from the National High Technology Research and Development Program of China (863 Program) (2006AA06Z105) and National Natural Science Foundation of China (40874072). The whole framework of our work can be divided into:
1. A systematical study of fully unstructured mesh generation and the corresponding optimal strategy was made. Base on this, a mesh refinement strategy which was controlled by quadrilateral was presented. By using this technique, not only the geometrical discretization errors derived from modeling complicated electromagnetic models were dramatically reduced but also the local refinement could be executed in the area of interest by which both the accuracy and efficiency were mostly improved.
2. Starting from the 3-D MT boundary value problem under the quasi-static limitation, the finite element equation which was derived based on the Galerkin-residual approach was presented. The coordinate transformation and numerical integration was executed on the discretized tetrahedral elements based on which the 3-D MT vector-finite element method was implemented. A whole computation framework for 3-D vector-finite element method with unstructured mesh was given. Based on this some typical models were tested which has demonstrated that our algorithm could distinctively avoid problems caused by the fake solution and both the accuracy and efficiency were enhanced which made our algorithm has a bright future for further application.
3. According to theory of Sobolev vector space and the discretization of Helmholtz space, the error estimate which was suitable for 3-D MT vector-finite element modeling was deduced by which the procedure of adaptive technique was guaranteed.
4. Based on the fully unstructured tetrahedralization and optical strategy, the 3-D magnetotelluric h-adaptive vector-finite element method was presented through combining the error estimate. With this work, the accuracy and creditableness for 3-D MT complicatedly modeling was guaranteed.
5. The 3-D magnetotelluric h-adaptive vector-finite element algorithm with unstructured mesh was implemented. The numerical results have shown that the complicity of electromagnetic models generally will not effect the convergence with satisfactory accuracy being obtained. Thus, 3-D h-adaptive vector-finite element has remarkably guaranteed the accuracy and efficiency by which a broad application in the future can be predicated.
引文
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