CN111898294B - Multi-frequency three-dimensional finite element numerical simulation method for electric dipole source - Google Patents
Multi-frequency three-dimensional finite element numerical simulation method for electric dipole source Download PDFInfo
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Abstract
The invention relates to a multi-frequency three-dimensional finite element numerical simulation method of an electric dipole source, which uses hexahedral units to mesh a research area and provides a prior data preparation for finite element calculation; then, based on a double-rotation equation of an electric field vector, deducing an integral weak solution form of a finite element equation by adopting a variational method, and carrying out unit analysis by bilinear interpolation; then the integral rigidity matrix is assembled, non-zero elements of the rigidity matrix are stored by adopting a sparse storage mode method, a large-scale sparse complex coefficient equation set is formed through a linked list data format, and preprocessing is carried out to form an electric field equation about frequency omega; solving an equation by adopting a model reduced order method based on Krylov subspace projection to obtain an electric field finite element numerical solution. According to the method, a large sparse equation set is transformed from a finite element formula based on an electric field, the electric field is converted into a frequency function, and the multi-frequency controllable source finite element can be rapidly solved through one-time matrix decomposition and multiple times of substitution, so that the calculation efficiency is greatly improved.
Description
Technical Field
The invention relates to a frequency domain controllable source electromagnetic method, which is an active source method for detecting underground mediums with different depths by emitting harmonic waves with a plurality of frequencies, in particular to an electric dipole source multi-frequency three-dimensional finite element numerical simulation method.
Background
The controllable source electromagnetic method overcomes the defect that the signal of the magnetotelluric field is weak and difficult to observe, greatly enhances the anti-interference capability, is widely applied to various aspects such as mineral resources, natural gas, geothermal energy, hydrologic engineering, environmental protection, coal field goaf water enrichment investigation and the like, and becomes an important geophysical exploration technology. The controllable source electromagnetic method usually adopts a grounded electric dipole as a transmitting source, and a series of electromagnetic waves with different frequencies are transmitted to observe mutually orthogonal electromagnetic field components in a remote area, so that the electromagnetic sounding purpose is achieved by using a calculation formula of the Carniya resistivity in the magnetotelluric sounding.
With the progress of science and technology and the development of computer technology, the forward and backward performance of a controllable source electromagnetic method is transited from one-dimensional, 2.5-dimensional to three-dimensional development stage, and because an underground medium is a three-dimensional space body in a strict sense, and the exploration of an electric method has a volume effect, the three-dimensional high-precision high-efficiency forward and backward performance is a main research direction at the present stage. Forward modeling is the basis of inversion, which can be truly put to practical use only by implementing efficient and fast forward algorithms. The three-dimensional forward modeling mainly uses a numerical calculation method, and the three-dimensional numerical simulation method applied to frequency domain electromagnetism at present mainly comprises a finite volume method, a finite difference method, an integral equation method and a finite element method. The finite element method has complete theoretical basis and theoretical system, flexible mesh subdivision, and can simulate any shape target body, and has been widely studied and applied in recent years, and occupies the half-wall Jiangshan of frequency domain electromagnetic method numerical simulation.
Finally, a large sparse complex coefficient equation set is formed by the controllable source electromagnetic finite element forward modeling, and how to quickly solve the equation set becomes a key for improving the calculation speed. For the frequency domain electromagnetic method, frequency point-by-frequency point calculation is usually carried out, and a Pardiso solver based on an LU matrix decomposition method and Open MP is mainly adopted, and a Mumps solver based on a multi-wavefront method is mainly adopted. Or a Krylov subspace iteration method is adopted, and compared with a direct method, the method has small occupied memory and high calculation speed, and becomes a very popular method for solving a large-scale disease state equation set. In addition, the finite element-infinite element combination and the regional decomposition method become a method for reducing the grid and the calculated amount and realizing the fast forward.
Although there are many methods for increasing the calculation speed, for the three-dimensional finite element forward modeling problem of the multi-frequency controllable source electromagnetic method, the calculation amount of each frequency point is still large due to the huge number of grid nodes, for the forward modeling problem of multiple frequencies, the calculation time is approximately equal to the multiple of the frequency, for large-scale inversion, the forward modeling of multiple frequencies needs to be calculated for each iteration, and the calculation efficiency is still a main bottleneck.
The applicant further researches that the key problem of the three-dimensional finite element simulation method of the frequency domain electromagnetic method is to realize the rapid calculation of multiple frequency points, the existing electromagnetic controllable source finite element method mainly carries out cyclic calculation through frequencies, the calculation time of each frequency is almost the same, if the number of frequency points is large, the calculation time is many times that of a single frequency point, and the calculation efficiency is very low.
Disclosure of Invention
The main bottleneck of finite element fast forward modeling of the current three-dimensional terrestrial controllable source electromagnetic method is the technical problem of solving a multi-frequency large-scale sparse equation set.
In order to achieve the above task, the present invention adopts the following technical solutions:
the method for simulating the multi-frequency three-dimensional finite element numerical value of the electric dipole source is characterized by comprising the following steps of:
firstly, meshing a research area by adopting a hexahedral unit meshing method, and providing early-stage data preparation for finite element calculation;
secondly, deriving an integral weak solution form of a finite element equation by adopting a variational method based on a double-rotation equation of an electric field vector, and carrying out unit analysis by adopting a hexahedral bilinear interpolation method;
where E is the electric field vector, k is the wave number at the quasi-static limit, ω is the angular frequency, μ is the permeability,
is the source current density.
Thirdly, assembling the whole rigidity matrix, storing non-zero elements of the rigidity matrix by adopting a sparse storage mode method, and forming a large-scale sparse complex coefficient equation set through a linked list data format:
(K 1e +iωK 2e )·E e ＝-iωb e (2)
wherein K is 1e And K 2e A cell stiffness matrix of 8×8, E e Is the electric field vector of the unit node, b e Is the cell load.
Fourth, preprocessing a large sparse matrix to form an electric field equation about frequency omega as follows:
E(ω)＝g ω (A)X (3)
wherein g ω (A)＝-iω(A+iωI) -1 ，A＝M -1 C，X＝M -1 b；
And fifthly, solving an equation (3) by adopting a model order reduction method based on Krylov subspace projection to obtain an electric field finite element numerical solution.
According to the multi-frequency three-dimensional finite element numerical simulation method of the electric dipole source, the large sparse equation set is transformed from the finite element formula based on the electric field, the electric field is converted into a function of frequency, and the multi-frequency controllable source finite element can be rapidly solved through once matrix decomposition and multiple times of substitution, so that the calculation efficiency is greatly improved. Compared with the prior art, the technical innovation brought is that:
1. by preprocessing a finite element matrix equation, expressing an electric field as a function related to omega, introducing a transfer function, and constructing an effective approximation of the matrix by adopting a Krylov subspace method, the quick solution of a large sparse equation set is realized.
2. The adopted model order reduction method is irrelevant to the emission frequency of the source, so that the solution of a plurality of frequency points can be realized by only one model order reduction and multiple times of back generation, the speed is quite high, the speed is improved by more than ten times compared with the speed of the traditional finite element method, the multi-frequency forward computation of the three-dimensional controllable source electromagnetic method can be realized on a common computer, and a foundation is provided for inversion.
Drawings
Fig. 1 is a hexahedral unit, wherein (a) is a parent unit; (b) drawing a subunit;
FIG. 2 is a model reduced order algorithm diagram;
FIG. 3 is a graph of electric field horizontal components of a two-layer model reduced order solution, 3DFEM versus analytical solution, where (a) is the electric field horizontal component; (b) graph is relative error;
FIG. 4 is a graph of the contrast of the Carnikov resistivity of a two-layer model reduced solution, 3DFEM and analytical solution, where (a) is the Carnikov resistivity; (b) graph is relative error;
FIG. 5 is a diagram of a low-resistance body model, wherein (a) is a cross-sectional view and (b) is a plan view;
FIG. 6 is a plan view of multi-frequency Carnitia apparent resistivity.
The invention is described in further detail below with reference to the drawings and examples.
Detailed Description
The embodiment provides a multi-frequency three-dimensional finite element numerical simulation method of an electric dipole source, which comprises the following specific implementation steps:
firstly, mesh subdivision is carried out on a research area by adopting a hexahedral unit subdivision method, nodes and units are integrally numbered according to the sequence from top to bottom, from back to front and from left to right to form global node numbers and unit numbers, unit local node numbers are formed on each unit according to the anticlockwise direction, a resistivity value is given to each unit, and preparation of early-stage data is provided for finite element calculation.
Secondly, a variation principle is adopted for an electric field vector formula, and a divergence condition is applied to obtain the following form:
thirdly, a node finite element method is adopted, a hexahedral unit is split into a whole area, the hexahedral unit is converted into a mother unit through iso-parametric transformation, the mother unit is a cube with a side length of 1, 8 corner points are taken as nodes, the numbers and coordinates of the mother unit and the child unit are shown in a figure 1 (such as marks 1,2,3,4,5,6,7 and 8 of a figure and a figure b in the figure 1), wherein (a) is the mother unit, (b) is the child unit, and the coordinate relationship between the mother unit and the child unit is as follows:
the following form function is constructed:
ξ i ,η i ,ζ i i (i=1, …, 8) is the node in the parent cell
By the form function constructed as above, the magnitude of the electric field component at each node in the arbitrary unit e can be expressed as:
if the region is divided into n units, the discretized variational equation is as follows:
the extremum problem of the variational equation is calculated, which corresponds to the difference between the multiple functions F (E x ,E y ,E z ) Taking the extremum, through cell analysis, the equation for a cell can be finally expressed as shown in equation (6):
(K 1e +iωK 2e )·E e ＝-iωb e (6)
wherein, the liquid crystal display device comprises a liquid crystal display device,
Fourth, according to the total sequence number of the unit nodes, the unit coefficient matrix K is obtained 1e And K 2e The elements are respectively arranged at the crossing positions of the corresponding rows and columns of the overall rigidity matrix C and M to be expanded into a large sparse matrix of 3Nd multiplied by 3Nd e And b e Expanding into vector matrixes E and b of 3Nd multiplied by 1;
(C+iωM)·E＝-iωb (7)
and finally converting the extreme value solving problem of the discretization variation equation into a solving equation set, wherein the equation set is a large-scale sparse complex coefficient equation set.
Fifth, construct transfer function to let a=m -1 C，X＝M -1 b, then formula (7) becomes formula (8) below:
(A+iωI)·E＝-iωX (8)
where I is a unit array, then E can be seen as a function of ω, as shown in the following equation:
E(ω)＝-iω(A+iωI) -1 X＝-iω(C+iωM) -1 b (9)
constructing a transfer function of a frequency domain:
E(ω)＝g ω (A)X (10)
wherein g ω (A)＝-iω(A+iωI) -1
The key to solving E (ω) is to solve its transfer function g ω (A)。
Sixth, the Krylov subspace method can be used to construct an effective approximation r of such matrices m (A) Such that:
E≈r m (A)·X (11)
wherein, the liquid crystal display device comprises a liquid crystal display device,
q m (z)＝(z-ξ 1 )(z-ξ 2 )...(z-ξ m ),ξ i (i=1,., m) is m poles, p m Is an m-order non-zero polynomial.
Thus, solve r m (A) EtcIs used for solving A in m-dimensional Krylov subspace Q m+1 ＝q m (A) -1 κ m+1 Projection onto (a, X).
An M orthogonal Arnoldi algorithm, namely M orthogonal operation is adopted for the Krylov subspace, so that a standard column orthogonal vector V is obtained m+1 The final expression is as follows:
E＝||b|| M V m+1 g ω (A m+1 )e 1 (12)
wherein, the liquid crystal display device comprises a liquid crystal display device,
A m+1 is A to V m+1 In Krylov subspace Q m+1 From equation (12), it can be seen that the matrix A constructed by the model reduced order algorithm m+1 The dimension of the model is far smaller than the original dimension, so that the inversion can be directly performed, and the matrix before and after the model reduction is irrelevant to the frequency of the source, so that the operation of a plurality of frequency points can be realized only by one model reduction, and the efficiency of the model reduction is dependent on the solution of an equation generated by a plurality of different poles.
FIG. 2 shows the main steps of the model reduction algorithm, where L p Under Cholesky decomposition trigonometric array for the p-th iteration, v j+1 Is an orthogonal basis vector, w is an intermediate vector.
And seventh, numerical derivation, namely solving spatial derivatives of the electric field in different directions. For a cell, the derivative calculation on its node can be expressed as:
the present embodiment uses a structured grid so that the derivative of a node j is the average of the derivatives of the n cells associated with it.
Then the magnetic field is calculated as:
finally, the Carnitia apparent resistivity is obtained as shown in equation (18):
and (3) constructing a two-layer model, placing the origin of coordinates in the center of an electrical source, placing the coordinate axes of the two layers of models in a right-handed spiral manner, placing the coordinate axes of the two layers of models in a Z-axis downward positive manner, placing the electrical source along an X-axis, transmitting current 1A, wherein the number of frequency points is 12, the lowest frequency is 2Hz, the highest frequency is 4096Hz, and the coordinates of measuring points are (-1500, -2400,0). The resistivity of the first stratum is 500 omega-m, and the thickness is 200 m The resistivity of the second layer is 50Ω·m, and a single repeated pole is selected
The research area is divided into grids of 51 multiplied by 47 multiplied by 54, the grid division boundary is +/-20 ten thousand meters, so that the boundary field value is ensured to be zero, and the grids are encrypted at the measuring points and the field sources, so that the singularities of the field sources are reduced, and the errors brought by equidistant interpolation are reduced.
FIG. 3 shows a two-layer model reduced order solution, a comparison graph of electric field horizontal components of a 3DFEM and an analytic solution, wherein (a) the graph is a comparison graph of electric field horizontal components, the fitting degree is good, and (b) the graph is a relative error graph, 3DFEM refers to a three-dimensional finite element direct solution, compared with a theoretical value, the error of high frequency and low frequency is large, but the relative error of most points is smaller than 5%, and the average relative error between the model reduced order solution and the analytic solution is 1.72%.
FIG. 4 shows a graph of the contrast of the Carnikov resistivity of a two-layer model reduced solution, 3DFEM and analytical solution, where (a) is the Carnikov resistivity; the graph (b) is the relative error. The abscissa is frequency, in Hz, and the ordinate is apparent resistivity, in Ω m. The apparatus resistivity refers to apparent resistivity, the Relative error refers to analytical solution, analytic solution refers to model reduction solution, and the 3DFEM refers to three-dimensional finite element direct solution.
Fig. 5 shows a low-resistance body model diagram, wherein (a) is a sectional view and (b) is a plan view. Wherein A and B are positions of electric dipole sources, electric dipole moment is equal to Idy, background resistivity is equal to 100 Omegam, left high-resistance body resistivity is equal to 1000 Omegam, x coordinates are respectively from 4500m to 4800m, y coordinates are respectively from 2400m to 3000m, right low-resistance body resistivity is equal to 10 Omegam, x coordinates are respectively from 6800m to 7100m, and y coordinates are respectively from 2400m to 3000m. The burial depths of two abnormal bodies are 100 m-300 m, air refers to an air area, and Surface refers to the ground.
FIG. 6 shows a plane view of the resistivity of the multi-frequency Carnitia view, the resistivity ranges from 40Ω m to 320 Ω m, the ordinate marks the detection frequency of each plane view, the horizontal coordinates refer to the measurement line and the measurement point respectively, the measurement line ranges from 1800m to 4800m, and the measurement point ranges from 4000m to 8400m.
The error between the 3DFEM and the theoretical value is 2.4%, the relative error between the model reduced solution and the analytical solution is 2.1%, and the error is increased, mainly because the numerical value derivation is needed for solving the magnetic field, numerical value calculation error is introduced, and the relative error of the Carniya resistivity is increased. It is worth mentioning that the model order reduction method is stable in calculation in the high frequency band, and the error is far smaller than 3DFEM. The comparison of the operational efficiency of this model is shown in table 1 below.
Table 1: calculation rate comparison table
Under the same calculation condition, the operation time of the 3DFEM is more than 10 times of the model reduced order solution. It can be seen that the result of the model reduced order solution has consistency with the result of the 3DFEM solution, and even the calculation accuracy is higher than that of the 3DFEM solution, mainly because the numerical calculation error brought by the model reduced order solution is smaller than that brought by the 3DFEM solution directly solving the large sparse matrix. From an error perspective, it is feasible to solve a large sparse matrix in a model reduced order within the error tolerance. From the aspect of operation rate, the model reduction only needs to solve an equation containing a large sparse matrix once, and a required orthogonal array can be constructed through multiple matrix backlashes, so that the solution of a plurality of frequency points becomes quick, and when 3DFEM is used for direct solution, each frequency point needs to solve an equation containing a large sparse matrix once, so that the model reduction algorithm is far inferior in the aspect of operation rate. The 3DFEM solution based on the frequency point parallelism only distributes each frequency point to different threads for processing, in the CSAMT terrestrial three-dimensional forward solving, more frequency points need to be calculated, the 3DFEM parallelism can only increase the operation rate by increasing the number of threads, and the influence of the number of the model reduced intermediate frequency points on the operation time is not dominant, so that the model reduction is applicable to the CSAMT terrestrial three-dimensional forward.
In summary, in the electric dipole source multi-frequency three-dimensional finite element numerical simulation method provided by the embodiment, a Krylov subspace projection algorithm based on model reduction is introduced, a model reduction form of a finite element stiffness matrix is deduced, and a frequency domain transfer function is constructed; constructing a matrix which is far smaller than the dimension of the finite element stiffness matrix by adopting a standard orthogonal vector sequence, wherein the matrix is irrelevant to frequency, and can realize the rapid solution of a multi-frequency-point finite element equation by one-time model reduction; the pseudo delta function is introduced, so that the singularity of source points is eliminated, the method is applicable to three-dimensional finite element numerical simulation of a complex background model, and a foundation is laid for multi-source solving; the method is characterized in that a layered medium model analysis solution is used as a standard, and compared with a finite element algorithm (3 DFEM) based on a Pardiso direct solver, the calculation time of the model reduction method is less than 1/10 of that of the model reduction method, the average relative error is 1.72%, and the high-efficiency three-dimensional finite element numerical solution is realized under the condition that the precision requirement is met.
The present invention is not limited to the specific details of the above preferred embodiments, and various simple modifications can be made to the technical solution of the present invention within the scope of the technical concept of the present invention, and all the simple modifications belong to the protection scope of the present invention.
In addition, the specific features described in the above embodiments may be combined in any suitable manner, and in order to avoid unnecessary repetition, various possible combinations are not described separately.
In addition, any combination of the various embodiments of the present embodiment may be performed, so long as the concept of the technical solution of the present invention is not violated, and the present invention should also be considered as the protection scope of the present invention.
Claims (1)
1. The method for simulating the multi-frequency three-dimensional finite element numerical value of the electric dipole source is characterized by comprising the following steps of:
firstly, meshing a research area by adopting a hexahedral unit meshing method, and providing early-stage data preparation for finite element calculation;
secondly, deriving an integral weak solution form of a finite element equation by adopting a variational method based on a double-rotation equation of an electric field vector, and carrying out unit analysis by adopting a hexahedral bilinear interpolation method;
where E is the electric field vector, k is the wave number at the quasi-static limit, ω is the angular frequency, μ is the permeability,
is the source current density;
thirdly, assembling the whole rigidity matrix, storing non-zero elements of the rigidity matrix by adopting a sparse storage mode method, and forming a large-scale sparse complex coefficient equation set through a linked list data format:
(K 1e +iωK 2e )·E e ＝-iωb e (2)
wherein K is 1e And K 2e A cell stiffness matrix of 8×8, E e Is the electric field vector of the unit node, b e Is the cell load;
fourth, preprocessing a large sparse matrix to form an electric field equation about frequency omega as follows:
E(ω)＝g ω (A)X (3)
wherein g ω (A)＝-iω(A+iωI) -1 ，A＝M -1 C，X＝M -1 b；
And fifthly, solving an equation (3) by adopting a model order reduction method based on Krylov subspace projection to obtain an electric field finite element numerical solution.
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| WO2007102973A2 (en) * | 2006-03-08 | 2007-09-13 | Exxonmobil Upstream Research Company | Efficient computation method for electromagnetic modeling |
| CN108509693A (en) * | 2018-03-13 | 2018-09-07 | 中南大学 | Three-dimensional frequency domain controllable source method for numerical simulation |
| CN110058315A (en) * | 2019-05-29 | 2019-07-26 | 中南大学 | A kind of three dimensional anisotropic radio frequency magnetotelluric self-adapting finite element forward modeling method |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| WO2007102973A2 (en) * | 2006-03-08 | 2007-09-13 | Exxonmobil Upstream Research Company | Efficient computation method for electromagnetic modeling |
| CN108509693A (en) * | 2018-03-13 | 2018-09-07 | 中南大学 | Three-dimensional frequency domain controllable source method for numerical simulation |
| CN110058315A (en) * | 2019-05-29 | 2019-07-26 | 中南大学 | A kind of three dimensional anisotropic radio frequency magnetotelluric self-adapting finite element forward modeling method |
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