CN113627027B - Method and system for simulating electromagnetic field value of non-trivial anisotropic medium - Google Patents
Method and system for simulating electromagnetic field value of non-trivial anisotropic medium Download PDFInfo
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Abstract
The invention provides a numerical simulation method and a numerical simulation system for an electromagnetic field of a non-trivial anisotropic medium, which directly correct a Maxwell electromagnetic field control equation through a current density divergence constraint term to be more perfect, avoid the solution of extra unknown quantity or equation in the solution process and reduce the calculation burden. The method provided by the embodiment of the invention does not need to correct the current density divergence, can avoid the correction failure caused by error correction fields and algorithm restart after correction when the current density divergence correction is carried out on a non-trivial anisotropic medium in the prior art, thereby effectively suppressing the electromagnetic field pseudo solution in low-frequency application, obviously improving the convergence speed and the calculation efficiency of a solver, greatly shortening the solving time, ensuring the accuracy of numerical solution, and having important significance on the problems of electromagnetic response calculation of a complex medium, fine inversion imaging of an underground geological structure and the like.
Description
Technical Field
The invention relates to the technical field of electromagnetic field numerical simulation, in particular to a method and a system for simulating a non-trivial anisotropic medium electromagnetic field numerical value.
Background
Pseudo solutions (spurrious solutions) refer to numerical solutions without physical meaning, are commonly found in electromagnetic field numerical simulation problems, and are generally closely related to adopted numerical methods, control equations and the like. Pseudo solutions are generally considered to result from improper approximation of the rotation operator zero space in maxwell's equations or the eigenvectors in bispin's equations, but are usually caused by multiple factors. Electromagnetic field pseudo-solutions tend to cause non-unique problems in the solution and produce invalid solutions, resulting in slow convergence of numerical algorithms and inaccurate numerical solutions. Therefore, in the face of different problems, various technical approaches have been proposed to try to suppress the spurious solutions, such as interleaving grids, finite element functions with continuous derivatives, solving the magnetic field for the electric field or solving the electric field for the magnetic field, current density divergence correction and current density divergence constraint, etc.
Current density divergence correction is widely applied to low frequency electromagnetic problems, especially to numerical simulations of the geophysical Magnetotelluric Method (MT) and related methods. When the frequency is low, the conductivity term in the double rotation equation is almost negligible; since the gradient space is the null space of the rotation operator, the sum of the gradient of any scalar and the true solution is the solution of the equation at this time, and thus a false solution occurs. By introducing a current density divergence correction technology, in addition to a double-rotation-degree main control equation, a current continuity equation is additionally solved by taking the current redundancy divergence corresponding to the iterative electric field solution of the main control equation as a source, and the iterative electric field solution is corrected by the obtained additional solution, so that the pseudo solution is suppressed. The method has good effect in the low-frequency electromagnetic field simulation of the conventional isotropic medium and the trivial (axial) anisotropic medium, and the convergence efficiency is obviously improved.
However, current density divergence correction techniques fail when dealing with media that are not trivial anisotropic (i.e., other than axial anisotropy), because: the current density divergence correction technology solves an additional correction field through electric potential, electric potential gradient components are distributed along three coordinate axes of a rectangular coordinate system all the time and cannot be used for describing the relation of an electric field corresponding to each element in conductivity tensor, namely the electric potential gradient at the moment is an equivalent value meeting a current density divergence condition; however, when it is used to correct the iterative electric field solution and is multiplied again by the conductivity tensor in the main control equation, errors occur because the conductivity tensor at this point is the original, not the equivalent diagonal conductivity tensor corresponding to the additional correction field. Therefore, it is urgently needed to provide a numerical simulation method for the electromagnetic field of the non-trivial anisotropic medium.
Disclosure of Invention
The invention provides a numerical simulation method and a numerical simulation system for an electromagnetic field of a non-trivial anisotropic medium, which are used for overcoming the defects in the prior art.
The invention provides a numerical simulation method of a non-trivial anisotropic medium electromagnetic field, which comprises the following steps:
determining a spatial topological position constrained by current density divergence in a non-trivial anisotropic medium and a conductivity weighting factor;
determining the current density divergence constraint term based on the spatial topological position and the conductivity weighting factor;
correcting a Maxwell electromagnetic field control equation based on the current density divergence constraint term to obtain a correction equation;
and solving the correction equation to obtain the numerical solution of the electromagnetic field in the non-trivial anisotropic medium.
According to the numerical simulation method of the electromagnetic field of the non-trivial anisotropic medium provided by the invention, the spatial topological position is determined based on the following modes:
constructing a structured hexahedral staggered grid;
for an electromagnetic field in the non-trivial anisotropic medium, defining an electric field vector of the electromagnetic field on edges of the structured hexahedral mesh, defining a magnetic field vector of the electromagnetic field on a face of the structured hexahedral mesh, defining a current density divergence of the electromagnetic field on nodes of the structured hexahedral mesh;
determining a spatial topological location of the current density divergence constraint based on the location of the current density divergence.
According to the numerical simulation method of the electromagnetic field of the non-trivial anisotropic medium provided by the invention, the conductivity weighting factor is determined based on the following mode:
for any node of the structured hexahedral staggered grid, averaging diagonal elements in the grid conductivity tensor around the any node to obtain a conductivity tensor diagonal element corresponding to the any node;
a root mean square of the diagonal elements of the conductivity tensor is computed, and an inverse of the root mean square is used as the conductivity weighting factor.
According to the numerical simulation method of the electromagnetic field of the non-trivial anisotropic medium provided by the invention, the current density divergence constraint term is determined based on the spatial topological position and the conductivity weighting factor, and the method specifically comprises the following steps:
determining an alternative current density divergence constraint term based on the conductivity weighting factor and the spatial topological position;
determining the current density divergence constraint term based on the alternative current density divergence constraint term.
According to the numerical simulation method of the electromagnetic field of the non-trivial anisotropic medium provided by the invention, the current density divergence constraint term is determined based on the alternative current density divergence constraint term, and the method specifically comprises the following steps:
and carrying out regularization processing on the alternative current density divergence constraint term to obtain the current density divergence constraint term.
According to the numerical simulation method for the electromagnetic field of the non-trivial anisotropic medium provided by the invention, the correction equation is solved to obtain the numerical solution of the electromagnetic field in the non-trivial anisotropic medium, and the method specifically comprises the following steps:
based on the structured hexahedral staggered grid, performing discrete processing on the correction equation and a calculation domain corresponding to the correction equation by adopting a finite volume method to obtain a first discrete result corresponding to the correction equation and a second discrete result corresponding to the calculation domain;
determining a boundary condition for the second discrete result and determining a sparse system of linear equations for the electromagnetic field based on the first discrete result and the boundary condition;
and solving the sparse linear equation set based on a Krylov subspace solver to obtain the electromagnetic field numerical solution.
According to the numerical simulation method of the electromagnetic field of the non-trivial anisotropic medium provided by the invention, the sparse linear equation set is solved based on a Krylov subspace solver to obtain the numerical solution of the electromagnetic field, and the method specifically comprises the following steps:
and iteratively solving the sparse linear equation set by adopting a quasi-minimum residual method based on a Krylov subspace solver to obtain the electromagnetic field numerical solution.
The invention also provides a non-trivial anisotropic medium electromagnetic field numerical simulation system, comprising:
a first determination module for determining a current density divergence constrained spatial topological position in a non-trivial anisotropic medium and a conductivity weighting factor;
a second determination module to determine the current density divergence constraint term based on the spatial topological position and the conductivity weighting factor;
the correction module is used for correcting a Maxwell electromagnetic field control equation based on the current density divergence constraint term to obtain a correction equation;
and the solving module is used for solving the correction equation to obtain the electromagnetic field numerical solution in the non-trivial anisotropic medium.
The invention also provides an electronic device, which comprises a memory, a processor and a computer program stored on the memory and capable of running on the processor, wherein the processor executes the computer program to realize the steps of the non-trivial anisotropic medium electromagnetic field numerical simulation method.
The present invention also provides a non-transitory computer readable storage medium having stored thereon a computer program which, when executed by a processor, implements the steps of the non-trivial anisotropic medium electromagnetic field numerical simulation method as described in any of the above.
According to the numerical simulation method and system for the electromagnetic field of the non-trivial anisotropic medium, provided by the invention, the Maxwell electromagnetic field control equation is directly corrected through the current density divergence constraint term, so that the Maxwell electromagnetic field control equation is more perfect, the solution of extra unknown quantity or equation in the solution process is avoided, and the calculation burden is reduced. The method provided by the embodiment of the invention does not need to correct the current density divergence, can avoid the correction failure caused by error correction fields and algorithm restart after correction when the current density divergence correction is carried out on a non-trivial anisotropic medium in the prior art, thereby effectively suppressing the electromagnetic field pseudo solution in low-frequency application, obviously improving the convergence speed and the calculation efficiency of a solver, greatly shortening the solving time, ensuring the accuracy of numerical solution, and having important significance on the problems of electromagnetic response calculation of a complex medium, fine inversion imaging of an underground geological structure and the like. In addition, the method provided by the embodiment of the invention is relatively simpler and more direct to realize, and the practicability is improved.
Drawings
In order to more clearly illustrate the technical solutions of the present invention or the prior art, the drawings needed to be used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without creative efforts.
FIG. 1 is a flow chart of a numerical simulation method of electromagnetic field of non-trivial anisotropic medium provided by the present invention;
FIG. 2 is a cut-away view of a finite volume three-dimensional grid of a numerical simulation model of a non-trivial anisotropic medium electromagnetic field provided by the present invention;
FIG. 3 is a schematic diagram of the spatial topological location of the current density divergence constraint provided by the present invention;
FIG. 4 is a schematic diagram of electrical parameters of a one-dimensional layered model of different types of anisotropic media provided by the present invention;
FIG. 5 is a schematic diagram of a three-dimensional COMMEM 3D-2 model of a heterogeneous anisotropic medium provided by the present invention;
FIG. 6 is a comparison graph of the efficiency test results of the half-space model, the one-dimensional layer model and the three-dimensional COMMEM 3D-2 model of any anisotropic medium provided by the present invention;
FIG. 7 is a comparison graph of the accuracy test of the one-dimensional layered model response of any anisotropic medium provided by the present invention;
FIG. 8 is a schematic diagram of an asymmetric dike model of any anisotropic medium provided by the present invention;
FIG. 9 is a comparison graph of the accuracy test of the asymmetric dike model response of any anisotropic medium provided by the present invention;
FIG. 10 is a comparison graph of the accuracy test of the response of the three-dimensional COMMEM 3D-2 model of any anisotropic medium provided by the present invention;
FIG. 11 is a schematic structural diagram of a numerical simulation system for electromagnetic field of non-trivial anisotropic medium provided by the present invention;
fig. 12 is a schematic structural diagram of an electronic device provided in the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings, and it is obvious that the described embodiments are some, but not all embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Since numerical simulation of electromagnetic fields of non-trivial anisotropic media is crucial to electromagnetic response calculation of complex media and detailed inversion imaging of subsurface geological formations, it is necessary to develop new technical means to deal with such problems. Therefore, the embodiment of the invention provides a numerical simulation method of the electromagnetic field of the non-trivial anisotropic medium.
Fig. 1 is a schematic flow chart of a method for numerically simulating an electromagnetic field of a non-trivial anisotropic medium according to an embodiment of the present invention, as shown in fig. 1, the method includes:
s1, determining a spatial topological position and a conductivity weighting factor of current density divergence constraint in a non-trivial anisotropic medium;
s2, determining the current density divergence constraint term based on the space topological position and the conductivity weighting factor;
s3, correcting a Maxwell electromagnetic field control equation based on the current density divergence constraint term to obtain a correction equation;
and S4, solving the correction equation to obtain a numerical solution of the electromagnetic field in the non-trivial anisotropic medium.
Specifically, in the method for simulating the electromagnetic field value of the nontrivial anisotropic medium provided in the embodiment of the present invention, the execution subject is a server, the server may be a local server or a cloud server, and the local server may be a computer, which is not specifically limited in the embodiment of the present invention. In the embodiment of the invention, the electromagnetic field numerical simulation method for the non-trivial anisotropic medium is aimed at, the non-trivial anisotropic medium can be a medium with non-trivial anisotropy, and the non-trivial anisotropy can comprise horizontal anisotropy, dip anisotropy, any anisotropy and the like, wherein any anisotropy refers to that an electrical main axis and a coordinate axis are not coincident. Non-trivial anisotropic media have differences in their physical or mechanical properties, such as absorbance, refractive index, conductivity, and tensile strength, measured along different directions.
Step S1 is performed first, determining the spatial topological position of the current density divergence constraint in the non-trivial anisotropic medium and the conductivity weighting factor. The current density divergence constraint refers to a constraint condition on the current density divergence in the non-trivial anisotropic medium, which is introduced in the embodiment of the invention to avoid the problem of pseudo solution in the numerical simulation process.
The current density divergence constraint can be determined by an electromagnetic field model in the non-trivial anisotropic medium, i.e. a numerical simulation model of the electromagnetic field of the non-trivial anisotropic medium, which can be a structured hexahedral mesh grid as shown in fig. 2. The current density divergence constraints may include node constraints, edge constraints, and the like. The nodes are nodes in the structured hexahedron staggered grid, and the edges are edges in the structured hexahedron staggered grid.
In fig. 2, a three-dimensional rectangular coordinate system is established with the right direction as the positive direction of the x axis, the downward direction as the positive direction of the z axis, and the left direction as the positive direction of the y axis, the coordinates of the center node of the structured hexahedron staggered grid are (i, j, k), the coordinates of the left lower node are (i-1, j +1, k + 1), the coordinates of the right lower node are (i +1, j +1, k + 1), and the coordinates of the center node of the right lower edge are (i +1, j, k + 1). And each edge in the structured hexahedron staggered grid has a corresponding electric field vector and a direction of the electric field vector.
The current density divergence refers to the divergence of the current density, and takes the value of a scalar. The current density, which is the product of the conductivity tensor and the electric field vector, can be expressed as:
J＝σE
where J denotes the current density, σ denotes the conductivity tensor, and E denotes the electric field vector.
The spatial topological position of the current density divergence constraint in the non-trivial anisotropic medium can be characterized by a suitable node or edge in the structured hexahedral staggered grid, which is not specifically limited in the embodiments of the present invention.
The conductivity of a non-trivial anisotropic medium is the tensor σ, and the conductivity tensor has off-diagonal elements, i.e., it needs to be characterized by at least 4 electrical parameters. Since the constraint object in an embodiment of the invention may be the current density divergence defined at the nodes, the conductivity weighting factor λ needs to be estimated from the multi-element conductivity tensor in the grid around the nodes. Wherein the conductivity weighting factor λ takes the value of a scalar.
Then, step S2 is performed, and according to the spatial topological position and the conductivity weighting factor, the current density divergence constraint term can be determined. The current density divergence constraint term is used to modify maxwell's electromagnetic field control equations and is therefore introduced as an additional term.
And then, executing step S3, and correcting a Maxwell electromagnetic field control equation (which can be simply referred to as a Maxwell equation) according to the current density divergence constraint term to obtain a correction equation.
In accordance with geophysical magnetotelluric Method (MT), maxwell's electromagnetic field control equations in the absence of external field sources, irrespective of displacement current, with e -iωt The specific form expressed by a second order partial differential equation of the frequency domain electric field is as follows:
where ω is the angular frequency and μ is the magnetic permeability, which may take the value of vacuum magnetic permeability.
When the maxwell electromagnetic field control equation is corrected according to the current density divergence constraint term, the current density divergence constraint term can be introduced into the maxwell electromagnetic field control equation to obtain a correction equation shown as the following, and the correction equation can suppress a pseudo solution.
Where M is a current density divergence constraint term.
And finally, executing the step S4, solving the correction equation to obtain the electromagnetic field numerical solution in the non-trivial anisotropic medium. When the correction equation is solved, a proper numerical method, a discrete correction equation and a calculation domain thereof can be selected, then a linear equation set is constructed by processing boundary conditions or field sources, and finally an electromagnetic field numerical solution in the non-trivial anisotropic medium can be obtained through the linear equation set.
According to the numerical simulation method of the electromagnetic field of the non-trivial anisotropic medium provided by the embodiment of the invention, the Maxwell electromagnetic field control equation is directly corrected through the current density divergence constraint term, so that the Maxwell electromagnetic field control equation is more perfect, the solution of extra unknown quantity or equation in the solution process is avoided, and the calculation load is reduced. The method provided by the embodiment of the invention does not need to correct the current density divergence, can avoid the correction failure caused by error correction fields and algorithm restart after correction when the current density divergence correction is carried out on a non-trivial anisotropic medium in the prior art, thereby effectively suppressing the electromagnetic field pseudo solution in low-frequency application, obviously improving the convergence speed and the calculation efficiency of a solver, greatly shortening the solving time, ensuring the accuracy of numerical solution, and having important significance on the problems of electromagnetic response calculation of a complex medium, fine inversion imaging of an underground geological structure and the like. In addition, the method provided by the embodiment of the invention is relatively simpler and more direct to realize, and the practicability is improved.
On the basis of the above embodiment, in the non-trivial anisotropic medium electromagnetic field numerical simulation method provided in the embodiment of the present invention, the spatial topological position is determined based on the following manner:
constructing a structured hexahedral staggered grid;
for an electromagnetic field in the non-trivial anisotropic medium, defining an electric field vector of the electromagnetic field on edges of the structured hexahedral mesh, defining a magnetic field vector of the electromagnetic field on a face of the structured hexahedral mesh, defining a current density divergence of the electromagnetic field on nodes of the structured hexahedral mesh;
determining a spatial topological location of the current density divergence constraint based on the location of the current density divergence.
Specifically, in the embodiment of the present invention, when determining the spatial topological position, the structured hexahedral staggered mesh shown in fig. 2 may be constructed first. Then, for the electromagnetic field in the non-trivial anisotropic medium, defining the electric field vector E of the electromagnetic field on the edges of the structured hexahedral staggered grid, defining the magnetic field vector H of the electromagnetic field on the surface of the structured hexahedral staggered grid, and defining the current density divergence of the electromagnetic field on the nodes of the structured hexahedral staggered grid. And finally, according to the position of the current density divergence, namely the node of the structured hexahedral staggered grid, the spatial topological position of the current density divergence constraint can be determined. In the embodiment of the invention, the space topological position of the node or the edge as the current density divergence constraint can be determined according to the advantage analysis. For example, the current density divergence constraint may be directly applied to the node, or the current density divergence may be averaged to the edge before applying the constraint, which is not specifically limited in the embodiment of the present invention.
In the embodiment of the invention, the spatial topological position of current density divergence constraint is determined through the structured hexahedral staggered grid, so that the spatial topological position is more visual.
On the basis of the above embodiments, the electromagnetic field numerical simulation method for the non-trivial anisotropic medium provided in the embodiments of the present invention, the conductivity weighting factor is determined based on the following manner:
for any node of the structured hexahedron staggered grid, averaging diagonal elements in grid conductivity tensor around the any node to obtain conductivity tensor diagonal elements corresponding to the any node;
a root mean square of a diagonal element in the conductivity tensor is computed, and an inverse of the root mean square is used as the conductivity weighting factor.
Specifically, in the embodiment of the present invention, when determining the conductivity weighting factor, since the conductivity of the nontrivial anisotropic medium has the non-diagonal elements, that is, at least the conductivity of the nontrivial anisotropic medium needs to be characterized by 4 electrical parameters, for any node of the structured hexahedron-interleaved grid, that is, any node, the volume of the diagonal elements in the grid conductivity tensor around the any node may be averaged to the any node, so as to obtain the conductivity tensor diagonal element corresponding to the any node, which may be represented as the conductivity tensor diagonal element corresponding to the any node
Then, the Root Mean Square (RMS) of the diagonal elements of the conductivity tensor can be calculated by a Root Mean Square (RMS) method, and the inverse of the calculated Root Mean Square is used as the conductivity weighting factor. Namely, the method comprises the following steps:
and the lambda is a conductivity weighting factor, and each node of the structured hexahedral staggered grid corresponds to one conductivity weighting factor.
In the embodiment of the invention, the diagonal elements in the grid conductivity tensor around the nodes in the structured hexahedron staggered grid are averaged to the nodes to obtain the conductivity tensor diagonal elements corresponding to the nodes, and the conductivity weighting factor is obtained by calculating the root mean square of the conductivity tensor diagonal elements, so that the obtained conductivity weighting factor is more accurate.
On the basis of the foregoing embodiment, the method for numerically simulating an electromagnetic field of a non-trivial anisotropic medium provided in the embodiment of the present invention, wherein the determining the current density divergence constraint term based on the spatial topological position and the conductivity weighting factor specifically includes:
determining an alternative current density divergence constraint term based on the conductivity weighting factor and the spatial topological position;
determining the current density divergence constraint term based on the alternative current density divergence constraint term.
Specifically, in the embodiment of the present invention, when determining the current density divergence constraint term, the candidate current density divergence constraint term at the spatial topological position may be determined according to the conductivity weighting factor. Namely, the following steps are included:
then, according to the alternative current density divergence constraint term, the current density divergence constraint term can be determined. In the embodiment of the present invention, the alternative current density divergence constraint term may be directly used as the current density divergence constraint term, or the current density divergence constraint term may be obtained by processing the alternative current density divergence constraint term, which is not specifically limited in the embodiment of the present invention.
In the embodiment of the invention, as the alternative current density divergence constraint term is equal to 0, the maxwell electromagnetic field control equation is not influenced when the alternative current density divergence constraint term is used as the final current density divergence constraint term to correct the maxwell electromagnetic field control equation, the problem of false solution can be avoided, the electromagnetic field numerical simulation efficiency of the non-trivial anisotropic medium can be improved, new problems can not be introduced, and the reliability is improved.
On the basis of the foregoing embodiment, the non-trivial anisotropic medium electromagnetic field numerical simulation method provided in the embodiment of the present invention, determining the current density divergence constraint term based on the candidate current density divergence constraint term specifically includes:
and carrying out regularization processing on the alternative current density divergence constraint term to obtain the current density divergence constraint term.
Specifically, in the embodiment of the present invention, in order to balance the constraint term and the double rotation term in the original maxwell electromagnetic field control equation and make the sum of the two approximate to the vector Laplacian term, so as to compress the trivial null space, when the current density divergence constraint term is determined according to the alternative current density divergence constraint term, the alternative current density divergence constraint term may be subjected to regularization, that is, the alternative current density divergence constraint term is converted into:
i.e. to obtain the current density divergence constraint term.
Accordingly, there are:
the correction equation can be expressed as:
in the correction equation, the first two terms to the left of the equal sign compress the non-trivial null space of the solution, thereby suppressing the spurious solution.
As shown in fig. 3, which is a schematic diagram of topological positions of current density divergence constraints, part (a) of fig. 3 represents node constraints, and part (b) represents edge constraints. In FIG. 3, the node constraint term is expressed as
And the expression of the edge constraint term is->
The two implement functions similarly. />
On the basis of the foregoing embodiment, the method for numerically simulating an electromagnetic field of a non-trivial anisotropic medium provided in the embodiment of the present invention, which is implemented by solving the correction equation to obtain a numerical solution of the electromagnetic field in the non-trivial anisotropic medium, specifically includes:
based on the structured hexahedral staggered grid, performing discrete processing on the correction equation and a calculation domain corresponding to the correction equation by adopting a finite volume method to obtain a first discrete result corresponding to the correction equation and a second discrete result corresponding to the calculation domain;
determining a boundary condition for the second discrete result and determining a sparse system of linear equations for the electromagnetic field based on the first discrete result and the boundary condition;
and solving the sparse linear equation set based on a Krylov subspace solver to obtain the electromagnetic field numerical solution.
Specifically, in the embodiment of the present invention, when the correction equation is solved to obtain the electromagnetic field numerical solution in the nontrivial anisotropic medium, the correction equation and the calculation domain corresponding to the correction equation may be discretized by using a finite volume method according to the structured hexahedron staggered grid shown in fig. 2. The finite volume method emphasizes that discrete equations are constructed from a physical point of view, each discrete equation is a representation of conservation of certain physical quantity on finite volume, the physical concept of the derivation process is clear, the coefficient of the discrete equation has certain physical significance, and the discrete equation can be guaranteed to have conservation characteristics.
And a first discrete result, namely a plurality of discrete equations, corresponding to the correction equation obtained through discrete processing, and a second discrete result, namely a calculation domain corresponding to each discrete equation, corresponding to the calculation domain.
Then, the boundary condition of the computation domain corresponding to each discrete equation can be determined, and considering the passive case, the boundary condition can be a first kind of boundary condition, i.e., dirichlet boundary condition. The field value on the boundary is determined by the electromagnetic response field value of the boundary one-dimensional model, and the right-end term of the equation can be formed by sorting, so that a sparse linear equation set of the electromagnetic field can be constructed according to each discrete equation and the boundary condition of the calculation domain corresponding to each discrete equation. The sparse linear equation set comprises a plurality of discrete equations, so that the sparse linear equation set is a large sparse linear equation set.
And finally, solving the obtained sparse linear equation set through a Krylov subspace solver to obtain a final electromagnetic field numerical solution, wherein the electromagnetic field numerical solution is a result of numerical simulation of the electromagnetic field of the non-trivial anisotropic medium. The Krylov subspace solver has the main idea that a residual vector is constructed for each iteration step recursion, namely the residual vector of the nth step is obtained by multiplying a certain polynomial of a coefficient matrix a by a first residual vector. However, it should be noted that the selection of the iterative polynomial should make the constructed residual vectors orthogonal to each other in a certain inner product sense, so as to ensure minimal residual error and achieve the purpose of fast convergence.
In the embodiment of the invention, when the correction equation is solved, the finite volume method, the discrete correction equation and the calculation domain corresponding to the correction equation are used, the boundary condition of the calculation domain corresponding to each discrete equation is determined, the sparse linear equation set of the electromagnetic field is constructed through the discrete equation and the boundary condition, and the Krylov subspace solver is used for solving the sparse linear equation set, so that the solving speed is greatly improved, and the numerical simulation efficiency of the electromagnetic field of the non-trivial anisotropic medium is further improved.
On the basis of the foregoing embodiment, the method for numerically simulating an electromagnetic field of a nontrivial anisotropic medium provided in the embodiment of the present invention solves the sparse linear equation set based on a Krylov subspace solver to obtain a numerical solution of the electromagnetic field, and specifically includes:
and iteratively solving the sparse linear equation set by adopting a quasi-minimum residual error method based on a Krylov subspace solver to obtain the electromagnetic field numerical solution.
Specifically, in the embodiment of the present invention, a quasi-minimum residual method (QMR) may be used to iteratively solve the sparse linear equation set through a Krylov subspace solver, so as to obtain an electromagnetic field numerical solution, which may further improve the solving speed and the efficiency of the electromagnetic field numerical simulation of the non-trivial anisotropic medium.
In the following, the application of the field of geoelectromagnetic Method (MT) numerical simulation in geophysics is taken as an example, and the simulation is realized by a three-dimensional numerical algorithm and in a three-dimensional space. The first kind of boundary condition is that a thicker air layer is applied to the top of the model, is set to be low in conductivity and is synchronously split and solved with the lower earth model; in the accuracy test, for MT, the electromagnetic field obtained by numerical simulation needs to be converted into impedance, and then apparent resistivity needs to be calculated.
1) And (3) testing efficiency:
the embodiment of the invention provides three different types of non-trivial anisotropic media and a trivial anisotropic medium for a half-space model, a one-dimensional layered model and a three-dimensional model, so as to comprehensively test the effect of the electromagnetic field numerical simulation method of the non-trivial anisotropic medium provided by the embodiment of the invention. Wherein, the three different kinds of non-trivial anisotropic media include any anisotropic media, horizontal anisotropic media and inclined anisotropic media, and the trivial anisotropic media include axial anisotropic media. Table 1 shows electrical parameters of different anisotropic half-space models, in the test process, spatial dimensions of a computational domain along X, Y and Z axis directions are set to be 192km,192km and 330km respectively, and the computational domain is divided into 26 × 26 × 59 hexahedral meshes.
TABLE 1 semi-space model Electrical parameters of different kinds of Anisotropic media
FIG. 4 shows electrical parameters of one-dimensional layered models of different kinds of anisotropic media, where LM1 represents any anisotropy, LM2 represents horizontal anisotropy, LM3 represents dip anisotropy, LM4 represents axial anisotropy, and the spatial dimensions and grid profile of the model computational domain are the same as those of the above-mentioned half-space model. FIG. 5 shows the three-dimensional COMMEM 3D-2 model electrical parameters for different types of anisotropic media. In fig. 5, (a) is a top view, the abscissa is the east-west direction, the ordinate is the north-south direction, only block1 (block 1), block2 (block 2), and block3 (block 3) can be seen in (a), and Profile is a position where the ordinate takes a value of 0. (b) The section (a) is a cross-sectional view of the section along the Profile, with the abscissa being the east-west direction and the ordinate being the depth direction. (b) Block1 (block 1), block2 (block 2), block3 (block 3) and block4 (block 4) can be seen in the section.
The electrical parameters for each block in fig. 5 are as follows:
block 1.1S/m (Isotropic Medium)
block3 0.01/1/0.01S/m，[30°/0°/0°]
Block 4S/m (isotropic medium)
For any anisotropic media, all angles in brackets are retained; for horizontally anisotropic media: strike holding block3 brackets; for an anisotropic media: reserving a Dip angle of block 2; for axially anisotropic media: all angles in brackets are discarded. COMMIMI 3D-2 is a standard model of a checking numerical algorithm provided by an international electromagnetic induction research community, an original model is provided for an isotropic algorithm, and isotropic parameters in the original model are modified into anisotropic parameters in the embodiment of the invention; the spatial dimensions are 125.626km,125.626km and 145km respectively, and are subdivided into 46 × 46 × 40 hexahedral meshes.
Through the practical test of the algorithm, the calculation efficiency of all models of different types of anisotropic media is obviously improved, the time consumption is obviously shortened, and the electromagnetic field numerical simulation method for the non-trivial anisotropic media in the embodiment of the invention is shown to be not only specific to all types of non-trivial anisotropic media, but also specific to trivial anisotropic media, and the trivial anisotropic media can be regarded as a special case of the non-trivial anisotropic media. But to avoid redundancy, only the most general case of non-trivial anisotropy to be solved in embodiments of the present invention, i.e. the case of numerically simulated efficiency of electromagnetic fields for any anisotropic medium, is shown here.
The variation of the normalized relative residual error of a half-space model, a one-dimensional layered model and a three-dimensional COMMEM 3D-2 model of any anisotropic medium along with the iteration number is shown in the set of FIG. 6, and it is marked that the test period is 500 seconds and the polarization mode is XY mode when the time is used. The method comprises the steps of obtaining a non-trivial anisotropic medium electromagnetic field numerical simulation method, obtaining a current density divergence correction method, and obtaining a current density divergence correction value by using a non-trivial anisotropic medium electromagnetic field numerical simulation method. In fig. 6, (a) corresponds to a half-space model, (b) corresponds to a one-dimensional layer model, and (c) corresponds to a three-dimensional commmi 3D-2 model.
Fig. 6 illustrates that the electromagnetic field numerical simulation method for the non-trivial anisotropic medium provided in the embodiment of the present invention can suppress the pseudo-solution well, and significantly improve the electromagnetic field numerical simulation convergence efficiency of the non-trivial anisotropic medium.
2) And (3) testing the accuracy:
the embodiment of the invention carries out accuracy test aiming at the condition of the one-dimensional layered model of any anisotropic medium. The model layering and model parameters of the one-dimensional layered model are shown in table 2.
TABLE 2 one-dimensional layer model Electrical parameters for any Anisotropic Medium
The model calculation domain has spatial dimension of 402km,402km and 530km, and is divided into 30 multiplied by 59 hexahedral grids, the test period is 1-10000 seconds, and the response calculation measuring point is positioned in the center of the calculation domain. Fig. 7 is a comparison between the electromagnetic response calculated by the numerical simulation method for the electromagnetic field of the non-trivial anisotropic medium provided in the embodiment of the present invention and the electromagnetic response calculated by the existing analytic method, where the electromagnetic response calculated by the numerical simulation method for the electromagnetic field of the non-trivial anisotropic medium provided in the embodiment of the present invention corresponds to the label of the symbol and the electromagnetic response calculated by the existing analytic method corresponds to the label of the dashed line. In fig. 7, the left part is a graph of apparent resistivity versus Period (Period), and the right part is a graph of Impedance phase versus Period (Period).
As can be seen from fig. 7, comparing the xy, yx, xx, yy four modes, the electromagnetic responses are all well matched, and it is verified that the electromagnetic field numerical simulation method for the non-trivial anisotropic medium provided in the embodiment of the present invention can accurately calculate the electromagnetic response of the one-dimensional layer model of any anisotropic medium.
3) And (3) testing the accuracy:
according to the embodiment of the invention, the accuracy test is carried out aiming at the condition of the asymmetric dike model of any anisotropic medium. The asymmetric dike model is shown in fig. 8. The asymmetric dike model comprises three parts (a), (b) and (c), and the electrical parameters of the three parts are shown in table 3:
TABLE 3 asymmetric dike model electrical parameters for any anisotropic media
The model calculation domain has the spatial dimension of 60km,60km and 148km, and is divided into 70 multiplied by 44 hexahedral grids, the test period is 10 seconds, the section is located in the middle of the X axis and extends from the center to the right end along the Y axis. Fig. 9 shows a comparison of the electromagnetic response calculated by the non-trivial anisotropic medium electromagnetic field numerical simulation method provided in the embodiment of the present invention, which corresponds to the label of the symbol, and the electromagnetic response calculated by the existing analytic method corresponds to the label of the dashed line, with the electromagnetic response calculated by the two-dimensional conventional method.
It can be seen from fig. 9 that the electromagnetic responses of the given modes can also be well matched, and it is verified that the electromagnetic response of the asymmetric dike model of any more complex anisotropic medium can be accurately calculated by the non-trivial anisotropic medium electromagnetic field numerical simulation method provided in the embodiment of the present invention. In addition, the method for simulating the electromagnetic field value of the non-trivial anisotropic medium provided by the embodiment of the invention is also suitable for the application case with not very low frequency.
4) And (3) testing the accuracy:
the embodiment of the invention carries out accuracy test on the condition of the three-dimensional COMMEMI 3D-2 model of any anisotropic medium. The three-dimensional COMMEM 3D-2 model is schematically shown in FIG. 5, and all angles in square brackets of each block are preserved for any anisotropic medium. The spatial dimension and the grid section of the model calculation domain are consistent with the relevant description of the model in the embodiment (namely 125.626km,145km and 46 multiplied by 40), the test period is 1-10000 seconds, and the coordinate of the test point for calculating response is (62500, 0).
FIG. 10 is a graph showing the comparison of the electromagnetic response calculated by the numerical simulation method for the electromagnetic field of the non-trivial anisotropic medium provided in the embodiment of the present invention, which corresponds to the notation of the symbol, and the electromagnetic response calculated by the numerical simulation method for the electromagnetic field of the non-trivial anisotropic medium provided in the embodiment of the present invention, which corresponds to the notation of the dashed line. As can be seen from FIG. 10, the responses of the four modes can be matched with each other well, and the electromagnetic response of the complex three-dimensional COMMEM 3D-2 model of any anisotropic medium can be accurately calculated by the non-trivial anisotropic medium electromagnetic field numerical simulation method provided by the embodiment of the invention.
In summary, the embodiment in the efficiency test and the embodiment in the accuracy test strongly explain the effectiveness of the non-trivial anisotropic medium electromagnetic field numerical simulation method provided in the embodiment of the invention in suppressing the pseudo solution in the non-trivial anisotropic medium electromagnetic field numerical simulation, avoid the defects of the existing current density divergence correction method, improve the convergence efficiency of the algorithm, ensure the accuracy of the solution, have general applicability, provide a new effective means for the non-trivial anisotropic medium electromagnetic field high-efficiency numerical simulation, and provide calculation guarantee for the problems of electromagnetic response calculation of complex media, fine inversion imaging of underground geological structures and the like.
As shown in fig. 11, on the basis of the above embodiments, in an embodiment of the present invention, there is provided a non-trivial anisotropic medium electromagnetic field numerical simulation system, including:
a first determination module 111 for determining a current density divergence constrained spatial topological position in a non-trivial anisotropic medium and a conductivity weighting factor;
a second determination module 112 for determining the current density divergence constraint term based on the spatial topological position and the conductivity weighting factor;
the correction module 113 is configured to correct a maxwell electromagnetic field control equation based on the current density divergence constraint term to obtain a correction equation;
and the solving module 114 is used for solving the correction equation to obtain the electromagnetic field numerical solution in the non-trivial anisotropic medium.
On the basis of the foregoing embodiment, in the non-trivial anisotropic medium electromagnetic field numerical simulation system provided in the embodiment of the present invention, the first determining module is specifically configured to:
constructing a structured hexahedral staggered grid;
for an electromagnetic field in the non-trivial anisotropic medium, defining an electric field vector of the electromagnetic field on edges of the structured hexahedral mesh, defining a magnetic field vector of the electromagnetic field on a face of the structured hexahedral mesh, defining a current density divergence of the electromagnetic field on nodes of the structured hexahedral mesh;
determining a spatial topological location of the current density divergence constraint based on the location of the current density divergence.
On the basis of the foregoing embodiment, in the non-trivial anisotropic medium electromagnetic field numerical simulation system provided in the embodiment of the present invention, the first determining module is further specifically configured to:
for any node of the structured hexahedral staggered grid, averaging diagonal elements in the grid conductivity tensor around the any node to obtain a conductivity tensor diagonal element corresponding to the any node;
a root mean square of the diagonal elements of the conductivity tensor is computed, and an inverse of the root mean square is used as the conductivity weighting factor.
On the basis of the foregoing embodiment, in the non-trivial anisotropic medium electromagnetic field numerical simulation system provided in the embodiment of the present invention, the second determining module is specifically configured to:
determining an alternative current density divergence constraint term based on the conductivity weighting factor and the spatial topological position;
determining the current density divergence constraint term based on the alternative current density divergence constraint term.
On the basis of the foregoing embodiment, in the non-trivial anisotropic medium electromagnetic field numerical simulation system provided in the embodiment of the present invention, the second determining module is further specifically configured to:
and carrying out regularization processing on the alternative current density divergence constraint term to obtain the current density divergence constraint term.
On the basis of the foregoing embodiment, in the non-trivial anisotropic medium electromagnetic field numerical simulation system provided in the embodiment of the present invention, the solving module is specifically configured to:
based on the structured hexahedron staggered grid, performing discrete processing on the correction equation and a calculation domain corresponding to the correction equation by adopting a finite volume method to obtain a first discrete result corresponding to the correction equation and a second discrete result corresponding to the calculation domain;
determining a boundary condition for the second discrete result and determining a sparse system of linear equations for the electromagnetic field based on the first discrete result and the boundary condition;
and solving the sparse linear equation set based on a Krylov subspace solver to obtain the electromagnetic field numerical solution.
On the basis of the foregoing embodiment, the solving module of the non-trivial anisotropic medium electromagnetic field numerical simulation system provided in the embodiment of the present invention is further specifically configured to:
and iteratively solving the sparse linear equation set by adopting a quasi-minimum residual error method based on a Krylov subspace solver to obtain the electromagnetic field numerical solution.
Specifically, the functions of the modules in the non-trivial anisotropic medium electromagnetic field numerical simulation system provided in the embodiment of the present invention correspond to the operation flows of the steps in the foregoing method embodiments one to one, and the achieved effects are also consistent, for which specific reference is made to the foregoing embodiments, which are not described again in the embodiment of the present invention.
Fig. 12 illustrates a physical structure diagram of an electronic device, which may include, as shown in fig. 12: a Processor (Processor) 1210, a communication Interface (Communications Interface) 1220, a Memory (Memory) 1230, and a communication bus 1240, wherein the Processor 1210, the communication Interface 1220, and the Memory 1230 communicate with each other via the communication bus 1240. Processor 1210 may invoke logic instructions in memory 1230 to perform the non-trivial anisotropic medium electromagnetic field numerical simulation methods provided by the various embodiments described above, the methods comprising: determining a spatial topological position constrained by current density divergence in a non-trivial anisotropic medium and a conductivity weighting factor; determining the current density divergence constraint term based on the spatial topological position and the conductivity weighting factor; correcting a Maxwell electromagnetic field control equation based on the current density divergence constraint term to obtain a correction equation; and solving the correction equation to obtain the numerical solution of the electromagnetic field in the non-trivial anisotropic medium.
In addition, the logic instructions in the memory 1230 may be implemented in software functional units and stored in a computer readable storage medium when the logic instructions are sold or used as a stand-alone product. Based on such understanding, the technical solution of the present invention may be embodied in the form of a software product, which is stored in a storage medium and includes instructions for causing a computer device (which may be a personal computer, a server, or a network device) to execute all or part of the steps of the method according to the embodiments of the present invention. And the aforementioned storage medium includes: a U-disk, a removable hard disk, a Read-Only Memory (ROM), a Random Access Memory (RAM), a magnetic disk or an optical disk, and other various media capable of storing program codes.
In another aspect, the present invention also provides a computer program product comprising a computer program stored on a non-transitory computer readable storage medium, the computer program comprising program instructions, which when executed by a computer, enable the computer to perform the method for numerical simulation of electromagnetic field of non-trivial anisotropic medium provided by the above embodiments, the method comprising: determining a spatial topological position constrained by current density divergence in a non-trivial anisotropic medium and a conductivity weighting factor; determining the current density divergence constraint term based on the spatial topological position and the conductivity weighting factor; correcting a Maxwell electromagnetic field control equation based on the current density divergence constraint term to obtain a correction equation; and solving the correction equation to obtain the electromagnetic field numerical solution in the nontrivial anisotropic medium.
In yet another aspect, the present invention further provides a non-transitory computer-readable storage medium having stored thereon a computer program, which when executed by a processor, is implemented to perform the method for numerical simulation of electromagnetic fields of a non-trivial anisotropic medium provided by the above embodiments, the method comprising: determining a spatial topological position constrained by current density divergence in a non-trivial anisotropic medium and a conductivity weighting factor; determining the current density divergence constraint term based on the spatial topological position and the conductivity weighting factor; correcting a Maxwell electromagnetic field control equation based on the current density divergence constraint term to obtain a correction equation; and solving the correction equation to obtain the numerical solution of the electromagnetic field in the non-trivial anisotropic medium.
The above-described embodiments of the apparatus are merely illustrative, and the units described as separate parts may or may not be physically separate, and parts displayed as units may or may not be physical units, may be located in one place, or may be distributed on a plurality of network units. Some or all of the modules may be selected according to actual needs to achieve the purpose of the solution of the present embodiment. One of ordinary skill in the art can understand and implement it without inventive effort.
Through the above description of the embodiments, those skilled in the art will clearly understand that each embodiment can be implemented by software plus a necessary general hardware platform, and certainly can also be implemented by hardware. With this understanding in mind, the above-described technical solutions may be embodied in the form of a software product, which can be stored in a computer-readable storage medium such as ROM/RAM, magnetic disk, optical disk, etc., and includes instructions for causing a computer device (which may be a personal computer, a server, or a network device, etc.) to execute the methods described in the embodiments or some parts of the embodiments.
Finally, it should be noted that: the above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it should be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.
Claims (8)
1. A numerical simulation method of an electromagnetic field of a non-trivial anisotropic medium is characterized by comprising the following steps:
determining a spatial topological position constrained by current density divergence in a non-trivial anisotropic medium and a conductivity weighting factor;
determining the current density divergence constraint term based on the spatial topological position and the conductivity weighting factor;
correcting a Maxwell electromagnetic field control equation based on the current density divergence constraint term to obtain a correction equation;
solving the correction equation to obtain an electromagnetic field numerical solution in the nontrivial anisotropic medium;
wherein the spatial topological position is determined based on:
constructing a structured hexahedral staggered grid;
for an electromagnetic field in the non-trivial anisotropic medium, defining an electric field vector of the electromagnetic field on edges of the structured hexahedral interleaved grid, defining a magnetic field vector of the electromagnetic field on faces of the structured hexahedral interleaved grid, defining a current density divergence of the electromagnetic field on nodes of the structured hexahedral interleaved grid;
determining a spatial topological position of the current density divergence constraint based on the position of the current density divergence;
solving the correction equation to obtain a numerical solution of the electromagnetic field in the nontrivial anisotropic medium, specifically comprising:
based on the structured hexahedral staggered grid, performing discrete processing on the correction equation and a calculation domain corresponding to the correction equation by adopting a finite volume method to obtain a first discrete result corresponding to the correction equation and a second discrete result corresponding to the calculation domain;
determining a boundary condition for the second discrete result and determining a sparse system of linear equations for the electromagnetic field based on the first discrete result and the boundary condition;
and solving the sparse linear equation set based on a Krylov subspace solver to obtain the electromagnetic field numerical solution.
2. A non-trivial anisotropic medium electromagnetic field numerical simulation method according to claim 1, wherein the conductivity weighting factor is determined based on:
for any node of the structured hexahedral staggered grid, averaging diagonal elements in the grid conductivity tensor around the any node to obtain a conductivity tensor diagonal element corresponding to the any node;
a root mean square of the diagonal elements of the conductivity tensor is computed, and an inverse of the root mean square is used as the conductivity weighting factor.
3. The method of numerical simulation of an electromagnetic field of a non-trivial anisotropic medium as set forth in claim 1, wherein said determining said current density divergence constraint term based on said spatial topological position and said conductivity weighting factor specifically comprises:
determining an alternative current density divergence constraint term based on the conductivity weighting factor and the spatial topological position;
determining the current density divergence constraint term based on the alternative current density divergence constraint term.
4. A non-trivial anisotropic medium electromagnetic field numerical simulation method according to claim 3, wherein said determining the current density divergence constraint term based on the candidate current density divergence constraint term, comprises in particular:
and carrying out regularization processing on the alternative current density divergence constraint term to obtain the current density divergence constraint term.
5. The non-trivial anisotropic medium electromagnetic field numerical simulation method of claim 1, wherein the solving the sparse linear equation set based on Krylov subspace solver to obtain the electromagnetic field numerical solution, specifically comprises:
and iteratively solving the sparse linear equation set by adopting a quasi-minimum residual error method based on a Krylov subspace solver to obtain the electromagnetic field numerical solution.
6. A numerical simulation system for electromagnetic fields of non-trivial anisotropic media, comprising:
a first determination module for determining a current density divergence constrained spatial topological position in a non-trivial anisotropic medium and a conductivity weighting factor;
a second determination module to determine the current density divergence constraint term based on the spatial topological position and the conductivity weighting factor;
the correction module is used for correcting a Maxwell electromagnetic field control equation based on the current density divergence constraint term to obtain a correction equation;
the solving module is used for solving the correction equation to obtain an electromagnetic field numerical solution in the non-trivial anisotropic medium;
wherein the spatial topological position is determined based on:
constructing a structured hexahedral staggered grid;
for an electromagnetic field in the non-trivial anisotropic medium, defining an electric field vector of the electromagnetic field on edges of the structured hexahedral interleaved grid, defining a magnetic field vector of the electromagnetic field on faces of the structured hexahedral interleaved grid, defining a current density divergence of the electromagnetic field on nodes of the structured hexahedral interleaved grid;
determining a spatial topological position of the current density divergence constraint based on the position of the current density divergence;
solving the correction equation to obtain a numerical solution of the electromagnetic field in the nontrivial anisotropic medium, specifically comprising:
based on the structured hexahedral staggered grid, performing discrete processing on the correction equation and a calculation domain corresponding to the correction equation by adopting a finite volume method to obtain a first discrete result corresponding to the correction equation and a second discrete result corresponding to the calculation domain;
determining a boundary condition for the second discrete result and determining a sparse system of linear equations for the electromagnetic field based on the first discrete result and the boundary condition;
and solving the sparse linear equation set based on a Krylov subspace solver to obtain the electromagnetic field numerical solution.
7. An electronic device comprising a memory, a processor and a computer program stored on said memory and executable on said processor, characterized in that said processor, when executing said program, implements the steps of the method for numerical simulation of electromagnetic fields of a non-trivial anisotropic medium as claimed in any one of claims 1 to 5.
8. A non-transitory computer readable storage medium having stored thereon a computer program, wherein the computer program when executed by a processor implements the steps of the method for numerical simulation of electromagnetic fields of a non-trivial anisotropic medium as claimed in any one of claims 1 to 5.
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