直流电场拟解析近似理论实现技术研究
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摘要
积分方程方法是基于Maxwell方程的积分形式的数值方法,它往往通过求解目标表面或者目标体内的等效源来间接获得场的解。积分方程方法是一种在解决三维电场数值模拟时有很多优点的方法,它可以避免微分方程方法中场值传递过程中的误差积累问题。出于对利用积分方程法快速进行三维模拟的需要,出现了很多的近似方法,如Born近似方法、局部非线性近似方法和拟线性近似方法等。拟解析近似方法是一种解决电磁场散射问题时提出的快速求解积分方程的近似方法,它适用于强散射和大扰动问题,可以避免传统数值方法中遇到的求解大型代数方程组或大型矩阵问题,将其引入直流电场的数值模拟,可以加快三维直流电场模拟的速度。因此,本文研究了利用拟解析近似方法求解三维直流电场的积分方程的实现。
首先本文对孙建国应用拟解析近似方法求解直流电场积分方程的理论公式进行了研究。从直流电场的基本理论出发推导出直流电场满足的积分方程,利用总场与背景场和异常场的关系得到异常电场的积分方程。根据拟线性近似方法的基本思想,推导直流电场的拟线性近似理论公式,对拟线性近似方法进行分析,从而引出直流拟解析近似方法。对直流电场的拟解析近似理论进行研究,在理论公式中的分析中,利用电反射张量的不同假设,将拟解析近似细分为两种近似方法,一种是假设电反射张量为标量,给出了标量拟解析近似的理论公式;另一种假设电反射张量为二阶张量,给出了张量拟解析近似的理论公式。根据拟解析近似理论的初始公式,对其进行改造,引申出迭代拟解析近似,得到了拟解析近似序列,通过拟解析近似序列迭代可求出积分方程n阶拟解析近似解,进而使拟解析近似方法达到更高的精度。
接下来我们考虑在拟解析近似理论实现过程中遇到的几个亟待解决的问题。第一个是磁并矢格林函数的问题。在本课题中用到的格林函数是直流电场磁并矢格林函数,它与前人所用到的格林函数不同,尚未出现其数学表达式,因此,需要对其进行研究。我们从矢量位方程出发,利用位与场的关系,推导出全空间位并矢格林函数与磁并矢格林函数的关系,利用已知的全空间标量位格林函数进行推导,得到全空间磁并矢格林函数。第二个是数值积分方法的选取。在利用拟解析近似方法的过程中,一个无法避免的问题是数值积分问题,如何选取合适的数值积分方法成为拟解析近似方法实现的一个关键。通过对高斯积分的研究,我们发现高斯积分具有很多的优点,尤其是它易于推导得到多重求积公式,因此我们选取高斯积分公式来完成数值积分。通过研究发现,当高斯节点数为5时,它既能够保持很高的精度,又具有较快的计算速度。第三个是磁并矢格林函数积分奇异性问题。为了解决这个问题,我们对格林函数的奇异性处理方法进行了研究,对Yaghjian处理张量格林函数奇异性的方法进行了分析和总结后,提出了拟源并矢方法处理磁并矢格林函数的积分奇异性。这种方法将积分区域分为奇异点小邻域和无奇异性区域两部分,无奇异性区域可直接进行数值积分,对奇异点小邻域积分可以通过数学方法得到拟源并矢,然后将两部分积分相加,这样就在积分区域上消除了磁并矢格林函数的积分奇异性,使得格林函数奇异性问题得到解决。
在解决了拟解析近似方法实现中遇到的难题后,本文对孙建国应用拟解析近似方法求解直流电场积分方程的理论进行了实现。在研究过程中,分别进行了标量和张量拟解析近似方法的实现,对不同形状异常体模型,不同的电导率变化,不同类型的场源进行拟解析近似模拟。
在标量拟解析近似方法实现过程中,首先在均匀场源的情况下选取一系列不同的电导率对比的球体异常体进行计算,求得标量拟解析近似解,利用Born近似方法得到Born近似解。均匀场中的球体异常体模型的异常电场是具有解析解的,因而可以得到异常电场的解析解。然后将标量拟解析近似解、Born近似解分别同解析解进行对比,得到各自的误差。当背景电导率与异常电导率对比很小时,Born近似解和标量拟解析近似解都能较好的逼近解析解,但标量拟解析近似解比Born近似解更接近解析解。当背景电导率与异常电导率对比增大时,Born近似解的误差也逐渐增大,不能再适用于该模型的电场模拟,标量拟解析近似解的误差虽然也增大,但仍能满足误差要求,其解可以用于电场数值模拟。随着背景电导率与异常电导率对比的进一步增大,Born近似解已经完全失去了意义,标量拟解析近似解也超过了允许的误差范围,但我们利用标量拟解析近似迭代方法得到迭代解,一般情况下进行三到五次迭代,得到的结果就能满足误差要求,能够很好的与解析解相拟合。随后本文又对异常体为长方体、椭球体的地电模型进行了数值模拟,得到了长方体异常体和椭球体异常体产生的异常电场的拟解析近似解。接下来选取场源为点源场,我们同样选取了一系列的模型进行了模拟,并将标量拟解析近似解与Born近似解和解析解进行对比,可以看出标量拟解析近似方法在进行大扰动的地电模型模拟时有着很大的优势。同样我们也进行了长方体异常体和椭球体异常体在点源场中产生的异常电场数值模拟,得到了不同背景电导率与异常电导率对比情况下的异常电场。
进行张量拟解析近似方法实现研究时,我们采用与标量拟解析近似实现研究相同的方式。可以得出下面的结论,张量拟解析近似方法能够很好的适应于大扰动地电模型的异常电场的数值模拟,具有很高的精度,能够用于计算如球体、长方体和椭球体等形状的异常体在均匀场和点源场中的响应。
通过对拟解析近似方法的实现研究,可以看出该方法是一种可以用于求解三维直流电场的积分方程的近似方法,它的应用可以推进三维直流电场正反演模拟的进程,实现三维直流电法勘探数据的快速正反演,因此拟解析近似方法具有很好的应用前景。
The integral equation method is numerical method based on the integral form of the Maxwell equations. It usually passes to solve surface of the target or the equivalent source in the body of the target, and acquires the solution of the field. The integral equation method has many advantages when using in resolving the 3D D.C field modeling problems, it can avoid the cumulative error in the transmission the field, which is gained in the differential eguation methods. In order to make the 3D modeling more quickly by The integral equation method, some Approximation method was turned up, such as Born method、Extended Born approximation and Quasi-linear Method and so on. Quasi-analytical Method was put forward while resolving the electromagnetic problems. It can be used in the large pertubation problems, and can avoid to solving the large matrix problems, so it was introduced in the numeracal modeling of the D.C. electric field.
First of all, this paper studies the formulas which appeared in Sun.J' paper. We derived the integral equations of the D.C. electric field from the basic theory of electric field, using the relation of the normal field and the anomalous field, the integral equation can be derived. Based on the idea of the Quasi-linear method, and analysis the formulas of the Quasi-linear method, the Quasi-analytical method can be put forwared. In this method we get two kinds of apporximation methods by different assumptions. They are called the Scalar Quasi-analytical method and the tensor Quasi-analytical method. According to the Quasi-analytical formulas get the iterative Quasi-analytical equation. Using the iterative Quasi-analytical equation we can make the result more accurately.
Next, we have to consider some problems meeting in the process of realizing the Quasi-analytical method. The first is problem of the magnetic dyadic Green's function. The Green's function used in this paper is different from the Green's function before, and it has not explicit expression. So we must study it. We can derive the relationship of the two kinds Green'function from the relation between the potential and the field, and we know the expression of the potentil dyadic Green's function, so the explicit expression can be presented. The second problem is the numeracal integral method. On the study of the Gauss integral method, we find that it has many advantages; especially the method can be extended to multi-dimensions numeracal integral. After experimentding with new method of Gauss integral method, we find that when the number of the nodes is 5, the method has higher precision and higher speed. The third problem is the singularity of the magnetic dyadic Green's function. to make out the problem, we studied the treatment method of the tensor Green's function, which was put forwarded by Yaghjian, and we get the quasi source dyadic method to solve the singularity. This method split the integral area into two parts, which are the small neighbourhood area of the singularity and the non singularity area. We calculate the integral on the neighbourhood area of the singularity to get the quasi source dyadic by methods of mathematics, and calulate the integral on the non singularity area directly. Finally integral on the whole area can be got by adding the two parts integral, and we resoved the problem.
Finally, after solving the problems of the Quasi-analytical method, this paper studies the realizing the Quasi-analytical method using to solving the integral equation of the D.C.electic field. In the process, we realize the scalar and tensor Quasi-analytical method make use of the different shape of the anomalous body, and different conductance rate, different type of the field source.
In the process of realizing the scalar Quasi-analytical method, first of all, we must confirm that whether this method is feasible or not, and whether this method has high precision or not. At the aim of this, we calculate the anomalous electric field caused by anomalous sphere models which has different conductance rate to the surrounding medium, which is caused by the uniform field and the point source field, because anomalous sphere has analytical solution. Through the numerical analysis of the scalar quasi-analytical result and the analytical solution about the anomalous, also we get the Born approximation result, and analysis its precision, we find that the scalar quasi-analytical method has high precision whether the model with large perturbation. When the perturbation is very large, the iterative scalar Quasi-analytical solution is also accuracy. So we validate the feasible of the the scalar Quasi-analytical method, and make sure that the scalar Quasi-analytical method is excelled than Born approximation. Next we prove this method having better adaptability. So we choose the anomalous ellipsoid and anomalous cube located in the the uniform field and the point source field, and calculate the anomalous field using the scalar Quasi-analytical method and the Born appoximation method. After these researches, we can sum up that the scalar Quasi-analytical method is excellent method for solving the 3D D.C. electric field integral equations.
When we carry on realzing of the tensor Quasi-analytical method, we use the same way as the scalar Quasi-analytical method. We get the following conclusions: This method can be applied to large perturbation problems with high precision; it can be used to modeling anomalous body in the uniform field and point source field, the anomalous body can take on sphere, ellipsoid, tube and so on.
Through the studies of realizing of the Quasi-analytical method, we can find than this method can be used to solve the integral equations of the 3D D.C. electric field, its application can promote the fast forward and inverse modeling of the D.C. electic field, so it has bright foreground.
首先本文对孙建国应用拟解析近似方法求解直流电场积分方程的理论公式进行了研究。从直流电场的基本理论出发推导出直流电场满足的积分方程,利用总场与背景场和异常场的关系得到异常电场的积分方程。根据拟线性近似方法的基本思想,推导直流电场的拟线性近似理论公式,对拟线性近似方法进行分析,从而引出直流拟解析近似方法。对直流电场的拟解析近似理论进行研究,在理论公式中的分析中,利用电反射张量的不同假设,将拟解析近似细分为两种近似方法,一种是假设电反射张量为标量,给出了标量拟解析近似的理论公式;另一种假设电反射张量为二阶张量,给出了张量拟解析近似的理论公式。根据拟解析近似理论的初始公式,对其进行改造,引申出迭代拟解析近似,得到了拟解析近似序列,通过拟解析近似序列迭代可求出积分方程n阶拟解析近似解,进而使拟解析近似方法达到更高的精度。
接下来我们考虑在拟解析近似理论实现过程中遇到的几个亟待解决的问题。第一个是磁并矢格林函数的问题。在本课题中用到的格林函数是直流电场磁并矢格林函数,它与前人所用到的格林函数不同,尚未出现其数学表达式,因此,需要对其进行研究。我们从矢量位方程出发,利用位与场的关系,推导出全空间位并矢格林函数与磁并矢格林函数的关系,利用已知的全空间标量位格林函数进行推导,得到全空间磁并矢格林函数。第二个是数值积分方法的选取。在利用拟解析近似方法的过程中,一个无法避免的问题是数值积分问题,如何选取合适的数值积分方法成为拟解析近似方法实现的一个关键。通过对高斯积分的研究,我们发现高斯积分具有很多的优点,尤其是它易于推导得到多重求积公式,因此我们选取高斯积分公式来完成数值积分。通过研究发现,当高斯节点数为5时,它既能够保持很高的精度,又具有较快的计算速度。第三个是磁并矢格林函数积分奇异性问题。为了解决这个问题,我们对格林函数的奇异性处理方法进行了研究,对Yaghjian处理张量格林函数奇异性的方法进行了分析和总结后,提出了拟源并矢方法处理磁并矢格林函数的积分奇异性。这种方法将积分区域分为奇异点小邻域和无奇异性区域两部分,无奇异性区域可直接进行数值积分,对奇异点小邻域积分可以通过数学方法得到拟源并矢,然后将两部分积分相加,这样就在积分区域上消除了磁并矢格林函数的积分奇异性,使得格林函数奇异性问题得到解决。
在解决了拟解析近似方法实现中遇到的难题后,本文对孙建国应用拟解析近似方法求解直流电场积分方程的理论进行了实现。在研究过程中,分别进行了标量和张量拟解析近似方法的实现,对不同形状异常体模型,不同的电导率变化,不同类型的场源进行拟解析近似模拟。
在标量拟解析近似方法实现过程中,首先在均匀场源的情况下选取一系列不同的电导率对比的球体异常体进行计算,求得标量拟解析近似解,利用Born近似方法得到Born近似解。均匀场中的球体异常体模型的异常电场是具有解析解的,因而可以得到异常电场的解析解。然后将标量拟解析近似解、Born近似解分别同解析解进行对比,得到各自的误差。当背景电导率与异常电导率对比很小时,Born近似解和标量拟解析近似解都能较好的逼近解析解,但标量拟解析近似解比Born近似解更接近解析解。当背景电导率与异常电导率对比增大时,Born近似解的误差也逐渐增大,不能再适用于该模型的电场模拟,标量拟解析近似解的误差虽然也增大,但仍能满足误差要求,其解可以用于电场数值模拟。随着背景电导率与异常电导率对比的进一步增大,Born近似解已经完全失去了意义,标量拟解析近似解也超过了允许的误差范围,但我们利用标量拟解析近似迭代方法得到迭代解,一般情况下进行三到五次迭代,得到的结果就能满足误差要求,能够很好的与解析解相拟合。随后本文又对异常体为长方体、椭球体的地电模型进行了数值模拟,得到了长方体异常体和椭球体异常体产生的异常电场的拟解析近似解。接下来选取场源为点源场,我们同样选取了一系列的模型进行了模拟,并将标量拟解析近似解与Born近似解和解析解进行对比,可以看出标量拟解析近似方法在进行大扰动的地电模型模拟时有着很大的优势。同样我们也进行了长方体异常体和椭球体异常体在点源场中产生的异常电场数值模拟,得到了不同背景电导率与异常电导率对比情况下的异常电场。
进行张量拟解析近似方法实现研究时,我们采用与标量拟解析近似实现研究相同的方式。可以得出下面的结论,张量拟解析近似方法能够很好的适应于大扰动地电模型的异常电场的数值模拟,具有很高的精度,能够用于计算如球体、长方体和椭球体等形状的异常体在均匀场和点源场中的响应。
通过对拟解析近似方法的实现研究,可以看出该方法是一种可以用于求解三维直流电场的积分方程的近似方法,它的应用可以推进三维直流电场正反演模拟的进程,实现三维直流电法勘探数据的快速正反演,因此拟解析近似方法具有很好的应用前景。
The integral equation method is numerical method based on the integral form of the Maxwell equations. It usually passes to solve surface of the target or the equivalent source in the body of the target, and acquires the solution of the field. The integral equation method has many advantages when using in resolving the 3D D.C field modeling problems, it can avoid the cumulative error in the transmission the field, which is gained in the differential eguation methods. In order to make the 3D modeling more quickly by The integral equation method, some Approximation method was turned up, such as Born method、Extended Born approximation and Quasi-linear Method and so on. Quasi-analytical Method was put forward while resolving the electromagnetic problems. It can be used in the large pertubation problems, and can avoid to solving the large matrix problems, so it was introduced in the numeracal modeling of the D.C. electric field.
First of all, this paper studies the formulas which appeared in Sun.J' paper. We derived the integral equations of the D.C. electric field from the basic theory of electric field, using the relation of the normal field and the anomalous field, the integral equation can be derived. Based on the idea of the Quasi-linear method, and analysis the formulas of the Quasi-linear method, the Quasi-analytical method can be put forwared. In this method we get two kinds of apporximation methods by different assumptions. They are called the Scalar Quasi-analytical method and the tensor Quasi-analytical method. According to the Quasi-analytical formulas get the iterative Quasi-analytical equation. Using the iterative Quasi-analytical equation we can make the result more accurately.
Next, we have to consider some problems meeting in the process of realizing the Quasi-analytical method. The first is problem of the magnetic dyadic Green's function. The Green's function used in this paper is different from the Green's function before, and it has not explicit expression. So we must study it. We can derive the relationship of the two kinds Green'function from the relation between the potential and the field, and we know the expression of the potentil dyadic Green's function, so the explicit expression can be presented. The second problem is the numeracal integral method. On the study of the Gauss integral method, we find that it has many advantages; especially the method can be extended to multi-dimensions numeracal integral. After experimentding with new method of Gauss integral method, we find that when the number of the nodes is 5, the method has higher precision and higher speed. The third problem is the singularity of the magnetic dyadic Green's function. to make out the problem, we studied the treatment method of the tensor Green's function, which was put forwarded by Yaghjian, and we get the quasi source dyadic method to solve the singularity. This method split the integral area into two parts, which are the small neighbourhood area of the singularity and the non singularity area. We calculate the integral on the neighbourhood area of the singularity to get the quasi source dyadic by methods of mathematics, and calulate the integral on the non singularity area directly. Finally integral on the whole area can be got by adding the two parts integral, and we resoved the problem.
Finally, after solving the problems of the Quasi-analytical method, this paper studies the realizing the Quasi-analytical method using to solving the integral equation of the D.C.electic field. In the process, we realize the scalar and tensor Quasi-analytical method make use of the different shape of the anomalous body, and different conductance rate, different type of the field source.
In the process of realizing the scalar Quasi-analytical method, first of all, we must confirm that whether this method is feasible or not, and whether this method has high precision or not. At the aim of this, we calculate the anomalous electric field caused by anomalous sphere models which has different conductance rate to the surrounding medium, which is caused by the uniform field and the point source field, because anomalous sphere has analytical solution. Through the numerical analysis of the scalar quasi-analytical result and the analytical solution about the anomalous, also we get the Born approximation result, and analysis its precision, we find that the scalar quasi-analytical method has high precision whether the model with large perturbation. When the perturbation is very large, the iterative scalar Quasi-analytical solution is also accuracy. So we validate the feasible of the the scalar Quasi-analytical method, and make sure that the scalar Quasi-analytical method is excelled than Born approximation. Next we prove this method having better adaptability. So we choose the anomalous ellipsoid and anomalous cube located in the the uniform field and the point source field, and calculate the anomalous field using the scalar Quasi-analytical method and the Born appoximation method. After these researches, we can sum up that the scalar Quasi-analytical method is excellent method for solving the 3D D.C. electric field integral equations.
When we carry on realzing of the tensor Quasi-analytical method, we use the same way as the scalar Quasi-analytical method. We get the following conclusions: This method can be applied to large perturbation problems with high precision; it can be used to modeling anomalous body in the uniform field and point source field, the anomalous body can take on sphere, ellipsoid, tube and so on.
Through the studies of realizing of the Quasi-analytical method, we can find than this method can be used to solve the integral equations of the 3D D.C. electric field, its application can promote the fast forward and inverse modeling of the D.C. electic field, so it has bright foreground.
引文
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