基于电场双旋度方程的三维可控源电磁法有限单元法数值模拟
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摘要
大地电磁法由于天然场源信号的微弱性和随机性,在野外记录和数据采集方面需要花费巨大的努力。可控源电磁法由于人工源的加入恰恰可以解决大地电磁法的这个缺点,因此在矿产普查、油气勘探等方面得到了广泛的应用。当前对于可控源电磁法的研究主要集中于一维层状地质体的近似模拟或沿走向无限延伸地质体的二维模拟,但严格来讲,地球物理电磁场问题都应该在三维空间中进行讨论。三维条件下的可控源数值模拟不仅仅是一维、二维情况下的简单的扩展,许多一维、二维条件下不曾出现的问题在三维情况下都出现了。随着计算机硬件技术的不断发展,三维可控源电磁法正演逐渐变的可行。因此,本文就三维可控源电磁法数值模拟进行了深入的研究。
从可控源电磁法的基本原理出发,推导了基于电场双旋度方程的边值问题,利用广义变分原理,把边值问题转换为变分问题,从而得到了基于电场双旋度方程的积分弱解形式,为其后的有限元计算奠定了理论基础。在准静态近似条件下,分别推导了水平电偶极子在空中和大地的远区电场闭合表达式,并以此作为有限元计算中的外边界条件,解决了边界条件加载的困难。把应用于地震模拟中的伪delta函数引入到可控源电磁法中的有限元模拟中,并把其扩展到三维情况下,用源周围的小块区域代替偶极子源,从而消除了源点的奇异性,提高了方程组的稳定性。针对三维可控源电磁法有限元模拟中形成的刚度系数矩阵巨大而且稀疏的特点,采用了全稀疏按行压缩存储的方法,极大的节省了存储空间,为在个人电脑上实现几万甚至几十万自由度的计算提供了保证。对于大型、稀疏复系数方程组的求解来说,由于其系数矩阵的条件数非常大,导致了方程组的严重病态,为了保证迭代法的快速收敛,本文采用不完全LDL~T预处理技术,降低了矩阵的条件数,加快了收敛速度,为迭代法高效求解大型方程组奠定基础。在预处理技术做保证的条件下,引入Krylov子空间迭代法,计算表明,该迭代方法结合预处理技术后收敛速度非常快,是高效快速求解电磁有限元模拟中形成的大型复系数方程组的最佳选择。
由于源的加入,使可控源电磁法理论和数值模拟非常复杂,本文的三维有限元数值模拟基于电场的双旋度方程,避免了对电偶极子源求旋度,减少了程序编制的复杂性;同时,引入散度条件,保证电源点以外研究区域中电场散度为零,从而避免了伪解的出现,使有限元计算在理论上更加完备。
Due to the weakness and randomness of natural source in magnetotelluric method, a great deal of effort is spent in the field records and data acquisition. This problem can be solved by controlled source electromagnetic method due to artificial sources, and so it is widely used in nineral prospecting, oil and gas exploration, and so on. However, the currently controlled source electromagnetic research is limited to one-dimensional layered modeling or two-dimensional simulation for geologic body of unlimited extension along strike. Strictly speaking, the geophysical electromagnetic field problems should be discussed in three dimensional space. Under the condition of three-dimension, numerical simulation of controlled source is not a simple expansion of one dimension, two-dimension cases. But it also encounters many problems which do not appear in one-dimension and two-dimension cases. With the development of computer hardware technology, three dimensional controlled source electromagnetic forward modeling become available. So, in this paper, three dimensional controlled source electromagnetic problems are studied in depth.
From the basic principles of controlled source electromagnetic, boundary value problem based on double electric field curl equation is derived, and then transformed into variational problem by generalized variational principle from which weak solutions form is obtained based on double curl equation of the electric field. In quasi-static conditions, the closed expressions far from source in the air and the earth is derived for electric dipole source, and by using it as the outer boundary conditions of the finite element method, difficulties in loading the boundary conditions is solved. The pseudo delta function in seismic method is used to simulate the electric dipole source in finite element simulation, In this paper, the author extend it into three-dimensional case from which singularity from source is eliminated, and also the the stability of the equation is remarkably. For sparse and large stiffness matrix in controlled source electromagnetic three-dimensional finite element simulation, full sparse compressed row storage method is used. And through it a great deal of storage space is saved by which large-scale modeling is becoming available in personal computers for hundreds of thousands of degrees of freedoms. Because of a huge condition number, large-scale, sparse complex coefficient equations is badly in-conditioned. In order to ensure the rapid convergence of the method, incomplete precondition LDLT technology is used. By using it, condition number of matrix is reduced from which the, convergence rate is accelerated.Under the guarantee of preconditions, Krylov subspace iteration method is introduced. The results show that Krylov subspace iteration combined with precondition, convergence is very quickly, and it is fast and efficient for solving electromagnetic finite element simulation of large-scale complex coefficient equations.
Since the electric sources makes controlled source electromagnetic theory and numerical simulation very complex, The electric field double curl equation in three-dimensional finite element numerical simulation is adoped to avoids curling the electric dipole source by which the complexity of programming is reduced; Meanwhile, the introduction of divergence conditions ensures divergence to be zero for the electric field outside the source and avoid a spurious solution and make electromagnetic finite element method more complete in theory.
从可控源电磁法的基本原理出发,推导了基于电场双旋度方程的边值问题,利用广义变分原理,把边值问题转换为变分问题,从而得到了基于电场双旋度方程的积分弱解形式,为其后的有限元计算奠定了理论基础。在准静态近似条件下,分别推导了水平电偶极子在空中和大地的远区电场闭合表达式,并以此作为有限元计算中的外边界条件,解决了边界条件加载的困难。把应用于地震模拟中的伪delta函数引入到可控源电磁法中的有限元模拟中,并把其扩展到三维情况下,用源周围的小块区域代替偶极子源,从而消除了源点的奇异性,提高了方程组的稳定性。针对三维可控源电磁法有限元模拟中形成的刚度系数矩阵巨大而且稀疏的特点,采用了全稀疏按行压缩存储的方法,极大的节省了存储空间,为在个人电脑上实现几万甚至几十万自由度的计算提供了保证。对于大型、稀疏复系数方程组的求解来说,由于其系数矩阵的条件数非常大,导致了方程组的严重病态,为了保证迭代法的快速收敛,本文采用不完全LDL~T预处理技术,降低了矩阵的条件数,加快了收敛速度,为迭代法高效求解大型方程组奠定基础。在预处理技术做保证的条件下,引入Krylov子空间迭代法,计算表明,该迭代方法结合预处理技术后收敛速度非常快,是高效快速求解电磁有限元模拟中形成的大型复系数方程组的最佳选择。
由于源的加入,使可控源电磁法理论和数值模拟非常复杂,本文的三维有限元数值模拟基于电场的双旋度方程,避免了对电偶极子源求旋度,减少了程序编制的复杂性;同时,引入散度条件,保证电源点以外研究区域中电场散度为零,从而避免了伪解的出现,使有限元计算在理论上更加完备。
Due to the weakness and randomness of natural source in magnetotelluric method, a great deal of effort is spent in the field records and data acquisition. This problem can be solved by controlled source electromagnetic method due to artificial sources, and so it is widely used in nineral prospecting, oil and gas exploration, and so on. However, the currently controlled source electromagnetic research is limited to one-dimensional layered modeling or two-dimensional simulation for geologic body of unlimited extension along strike. Strictly speaking, the geophysical electromagnetic field problems should be discussed in three dimensional space. Under the condition of three-dimension, numerical simulation of controlled source is not a simple expansion of one dimension, two-dimension cases. But it also encounters many problems which do not appear in one-dimension and two-dimension cases. With the development of computer hardware technology, three dimensional controlled source electromagnetic forward modeling become available. So, in this paper, three dimensional controlled source electromagnetic problems are studied in depth.
From the basic principles of controlled source electromagnetic, boundary value problem based on double electric field curl equation is derived, and then transformed into variational problem by generalized variational principle from which weak solutions form is obtained based on double curl equation of the electric field. In quasi-static conditions, the closed expressions far from source in the air and the earth is derived for electric dipole source, and by using it as the outer boundary conditions of the finite element method, difficulties in loading the boundary conditions is solved. The pseudo delta function in seismic method is used to simulate the electric dipole source in finite element simulation, In this paper, the author extend it into three-dimensional case from which singularity from source is eliminated, and also the the stability of the equation is remarkably. For sparse and large stiffness matrix in controlled source electromagnetic three-dimensional finite element simulation, full sparse compressed row storage method is used. And through it a great deal of storage space is saved by which large-scale modeling is becoming available in personal computers for hundreds of thousands of degrees of freedoms. Because of a huge condition number, large-scale, sparse complex coefficient equations is badly in-conditioned. In order to ensure the rapid convergence of the method, incomplete precondition LDLT technology is used. By using it, condition number of matrix is reduced from which the, convergence rate is accelerated.Under the guarantee of preconditions, Krylov subspace iteration method is introduced. The results show that Krylov subspace iteration combined with precondition, convergence is very quickly, and it is fast and efficient for solving electromagnetic finite element simulation of large-scale complex coefficient equations.
Since the electric sources makes controlled source electromagnetic theory and numerical simulation very complex, The electric field double curl equation in three-dimensional finite element numerical simulation is adoped to avoids curling the electric dipole source by which the complexity of programming is reduced; Meanwhile, the introduction of divergence conditions ensures divergence to be zero for the electric field outside the source and avoid a spurious solution and make electromagnetic finite element method more complete in theory.
引文
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