有限元—无限元耦合法在三维直流电和电磁数值模拟中的应用
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摘要
目前在三维直流电阻率法有限元正演数值模拟中存在以下两个问题：第一个问题是,三维地电模型的快速建立及属性添加还存在困难。目前国内多数学者仍然依靠在文本文件中手工输入网格信息的方式进行模型建立及前处理,此方法耗时巨大、容易出错、且无法建立复杂的三维模型。第二个问题是,传统的混合边界条件虽然可以得到较高精度的解,但是由于系统矩阵与电源位置相关,对于电源位置多次改变的装置,如偶极-偶极装置,每改变一次电源位置就需重新形成系数矩阵及解方程组,以此为基础的反演计算将十分耗时。目前广泛采用的解决办法是将混合边界条件换为Dirichlet或Neumann边界条件,即在半无限边界上强制电位为零或其法相导数为零,或假定混合边界条件中的距离项为常数,以上方案要求有限元离散区域必须取得非常大以减小边界条件造成的误差,这必将增加节点数和计算量。
在三维可控源电磁法有限元正演模拟中,数据量大和解方程耗时是主要问题。由于电磁法研究的问题尺度较大,目标体覆盖区域一般为几百米至几千米,造成有限元网格数量众多。Dirichlet外边界条件是目前易于实现且效果较好的外边界条件,但其要求外边界远离场源和目标体,一般范围都要在数万米,这无疑在我们不关心的区域增加了许多有限元网格。考虑到电磁法数值模拟中每个有限元结点包含多个自由度,且待求解的未知数均为复数,使得最终形成的系统矩阵规模庞大,方程组求解困难。
在本文中,我们通过对通用三维前后处理软件GiD进行简单的二次开发,实现了可视化的快速建立复杂三维地电模型,并且实现了前处理、计算和后处理的整合；通过引入无限单元与有限元相结合,形成了有限元-无限元偶和算法,替代了传统的人工边界条件,解决了截断边界条件误差大、节点多的问题。对GiD的二次开发仅需要用简单的脚本语言编写用户自定义“问题类型”,即可在图形化界面中建立并输出可用于已有有限元计算程序的初始模型。Astley波包映射无限元被用来将电位分布或电磁场分布延伸到无限元处,并使其在无限远处衰减为零。我们还提出了一种全新的无限元形函数,它保持了系数矩阵的稀疏对称性,并且与多种常见无限元形函数的对比显示在精度和时间消耗上均占优势。最后,在三维直流电法和三维电磁法正演模拟中，通过若干不同的模型计算验证了本文提出的有限元-无限元耦合算法的正确性和实用性。
本文提出的三维模型的建立方法快速、方便,且具备通用性；提出的有限元-无限元耦合算法可以有效减少有限元网格剖分区域的范围,节省结点数,加快计算速度。在三维直流电法数值模拟中,在仅包含测区的计算范围内即可得到与混合边界条件相当的计算精度；在三维可控源电磁法数值模拟中,本方法可以将各个方向的边界范围均缩小到几千米,计算结果与数万米边界范围下施加Dirichlet边界条件的传统有限元法相差无几。
Currently, there are two problems that baffle the 3D direct current (DC) resistivity finite element forward modeling. The first is how to build complex 3D geoelectrical model efficiently and effectively. Most domestic scholars are still build 3D models within a.txt file and by manual input, which is time consuming, error prone and not applicable for complex ones. The second is, though by enforcing mixed boundary conditions we can get relatively high accuracy in an acceptable discretization domain, because the global system matrix is affected by the locations of sources, once the locations have been changed the matrix has to be formed over again, which would make the forward modeling for some survey configurations, such as dipole-dipole array, very time consuming and worthless for inversions. The solutions commonly used for solving the problems mentioned above include taking replacement of the mixed boundary conditions by Dirichlet or Neumann boundary conditions, that is to say to force potentials or there derivatives to be zero on the subsurface boundaries, or assuming the distance between the source and boundary in the mixed boundary conditions to be a constant. All the schemes represented above need the domain of calculation and discretization to be set so large that the effect of the truncated boundaries can be eliminated to some extent, while what is inevitable is that the number of grid nodes must be increasing and so is the computational complexity.
The same problems also arise from the 3D controlled source electromagnetic (CSEM) finite element modeling. Due to the large scale of the survey domain, which usually extends from hundreds to thousands meters, the numbers of finite element grids are also huge. Application of Dirichlet boundary conditions has been proved to be an effective and handy method for truncating the infinite domain, but the boundaries must be far away from the source and anomalous bodies, usually tens of thousands meters, thereby bringing so many finite elements in the domain that we are not interested in. Considering in electromagnetic problems, the number of degrees of freedoms for each node is typically more than one. Moreover, the unknowns are all complex numbers. As a result, the global matrices will occupy so many memories and bring difficulties for solving.
In this study, fast and visualized pre and post processing for finite element method was achieved by customized development based on the universal 3D modeling software named GiD, while Infinite elements were introduced to form the finite-infinite element coupling method, which can be used to substitute the artificial boundary conditions and reduce the number of finite element nodes. For the customized development on GiD, we only need to program the so-called problem types with simple script language, then models could be build in the graphic interface and output in the format fitting our calculation program. After the calculation process, by using the GiDpost library, the files used for post-processing in GiD could be outputted easily. As to our finite-infinite element coupling method, Astley mapped wave envelope infinite elements were employed to continue the electrical fields to infinity. Meanwhile, a new type of infinite element shape functions was proposed and proved to be the optimal one in both accuracy and time consumption by comparing with several other shape functions during the simulations for 3D DC problems. Finally, the availability and superiority of this coupling algorithm were confirmed by several numerical tests both in 3D DC modeling and 3D CSEM modeling.
Overall, by the pre and post processing method proposed in this paper, the 3D models can be built efficiently and applicable to any calculation program, while the finite-infinite element coupling method could derive solutions with high accuracy in a comparatively small descretization domain, which helps to reduce the number of degrees of freedoms and speed up the computation.
在三维可控源电磁法有限元正演模拟中,数据量大和解方程耗时是主要问题。由于电磁法研究的问题尺度较大,目标体覆盖区域一般为几百米至几千米,造成有限元网格数量众多。Dirichlet外边界条件是目前易于实现且效果较好的外边界条件,但其要求外边界远离场源和目标体,一般范围都要在数万米,这无疑在我们不关心的区域增加了许多有限元网格。考虑到电磁法数值模拟中每个有限元结点包含多个自由度,且待求解的未知数均为复数,使得最终形成的系统矩阵规模庞大,方程组求解困难。
在本文中,我们通过对通用三维前后处理软件GiD进行简单的二次开发,实现了可视化的快速建立复杂三维地电模型,并且实现了前处理、计算和后处理的整合；通过引入无限单元与有限元相结合,形成了有限元-无限元偶和算法,替代了传统的人工边界条件,解决了截断边界条件误差大、节点多的问题。对GiD的二次开发仅需要用简单的脚本语言编写用户自定义“问题类型”,即可在图形化界面中建立并输出可用于已有有限元计算程序的初始模型。Astley波包映射无限元被用来将电位分布或电磁场分布延伸到无限元处,并使其在无限远处衰减为零。我们还提出了一种全新的无限元形函数,它保持了系数矩阵的稀疏对称性,并且与多种常见无限元形函数的对比显示在精度和时间消耗上均占优势。最后,在三维直流电法和三维电磁法正演模拟中，通过若干不同的模型计算验证了本文提出的有限元-无限元耦合算法的正确性和实用性。
本文提出的三维模型的建立方法快速、方便,且具备通用性；提出的有限元-无限元耦合算法可以有效减少有限元网格剖分区域的范围,节省结点数,加快计算速度。在三维直流电法数值模拟中,在仅包含测区的计算范围内即可得到与混合边界条件相当的计算精度；在三维可控源电磁法数值模拟中,本方法可以将各个方向的边界范围均缩小到几千米,计算结果与数万米边界范围下施加Dirichlet边界条件的传统有限元法相差无几。
Currently, there are two problems that baffle the 3D direct current (DC) resistivity finite element forward modeling. The first is how to build complex 3D geoelectrical model efficiently and effectively. Most domestic scholars are still build 3D models within a.txt file and by manual input, which is time consuming, error prone and not applicable for complex ones. The second is, though by enforcing mixed boundary conditions we can get relatively high accuracy in an acceptable discretization domain, because the global system matrix is affected by the locations of sources, once the locations have been changed the matrix has to be formed over again, which would make the forward modeling for some survey configurations, such as dipole-dipole array, very time consuming and worthless for inversions. The solutions commonly used for solving the problems mentioned above include taking replacement of the mixed boundary conditions by Dirichlet or Neumann boundary conditions, that is to say to force potentials or there derivatives to be zero on the subsurface boundaries, or assuming the distance between the source and boundary in the mixed boundary conditions to be a constant. All the schemes represented above need the domain of calculation and discretization to be set so large that the effect of the truncated boundaries can be eliminated to some extent, while what is inevitable is that the number of grid nodes must be increasing and so is the computational complexity.
The same problems also arise from the 3D controlled source electromagnetic (CSEM) finite element modeling. Due to the large scale of the survey domain, which usually extends from hundreds to thousands meters, the numbers of finite element grids are also huge. Application of Dirichlet boundary conditions has been proved to be an effective and handy method for truncating the infinite domain, but the boundaries must be far away from the source and anomalous bodies, usually tens of thousands meters, thereby bringing so many finite elements in the domain that we are not interested in. Considering in electromagnetic problems, the number of degrees of freedoms for each node is typically more than one. Moreover, the unknowns are all complex numbers. As a result, the global matrices will occupy so many memories and bring difficulties for solving.
In this study, fast and visualized pre and post processing for finite element method was achieved by customized development based on the universal 3D modeling software named GiD, while Infinite elements were introduced to form the finite-infinite element coupling method, which can be used to substitute the artificial boundary conditions and reduce the number of finite element nodes. For the customized development on GiD, we only need to program the so-called problem types with simple script language, then models could be build in the graphic interface and output in the format fitting our calculation program. After the calculation process, by using the GiDpost library, the files used for post-processing in GiD could be outputted easily. As to our finite-infinite element coupling method, Astley mapped wave envelope infinite elements were employed to continue the electrical fields to infinity. Meanwhile, a new type of infinite element shape functions was proposed and proved to be the optimal one in both accuracy and time consumption by comparing with several other shape functions during the simulations for 3D DC problems. Finally, the availability and superiority of this coupling algorithm were confirmed by several numerical tests both in 3D DC modeling and 3D CSEM modeling.
Overall, by the pre and post processing method proposed in this paper, the 3D models can be built efficiently and applicable to any calculation program, while the finite-infinite element coupling method could derive solutions with high accuracy in a comparatively small descretization domain, which helps to reduce the number of degrees of freedoms and speed up the computation.
引文
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