CN113656750B - Magnetic induction intensity calculation method of strong magnetic medium based on space wave number domain - Google Patents
Magnetic induction intensity calculation method of strong magnetic medium based on space wave number domain Download PDFInfo
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Abstract
The method comprises the steps of dividing an initial prism model containing an abnormal body to obtain a series of nodes, and carrying out magnetic susceptibility assignment on each node, wherein the magnetic susceptibility is a scalar; setting a Gaussian parameter, calculating discrete offset wave number, and calculating the intensity of the background field magnetic field in a spatial domain to obtain a magnetization intensity model; converting an equation which is satisfied by the magnetic potential and the magnetization intensity of the abnormal field in the spatial domain into a one-dimensional ordinary differential equation in the spatial wavenumber domain; and equivalently converting the boundary value problem model satisfied by the abnormal magnetic potential of the space wave number domain into a variation problem model and solving the variation problem model by combining the set boundary condition of the abnormal magnetic potential of the space wave number domain to obtain the abnormal magnetic potential of the space wave number domain and the magnetic field strength of the abnormal magnetic field of the space wave number domain, converting the abnormal magnetic potential of the space wave number domain into the magnetic potential of the abnormal magnetic field of the space domain and the magnetic field strength of the abnormal magnetic field of the space wave number domain, and obtaining the magnetic field strength of the space domain according to the output abnormal magnetic field strength of the space domain. The invention can more accurately carry out magnetic field numerical simulation on the ferromagnetic medium.
Description
Technical Field
The invention belongs to the technical field of strong magnet numerical simulation, and particularly relates to a magnetic induction intensity calculation method of a strong magnetic medium based on a space wavenumber domain.
Background
When the magnetic susceptibility value of the medium is greater than 0.1 SI, it is generally considered to be a ferromagnetic medium, and the self-demagnetizing field in the magnetic field formed by it is not negligible. At present, the magnetic induction intensity of the magnetic field of the ferromagnetic medium is usually calculated by neglecting a self-demagnetizing field, namely, the calculation is approximate as a weak magnetic condition, so that the magnetic induction intensity obtained by numerical simulation has larger deviation with an actual magnetic induction intensity value, and the magnetic measurement data is not favorably processed and explained.
For the calculation of the magnetic field of a ferromagnetic medium, research methods of related scholars are mainly divided into space-domain and frequency-domain methods. Among The spatial domain methods, The literature (volume a. The application of electronic computers to The calculation of effective magnetization [ J ]. geographic magnetization 1963.11(1): 51-58.) proposes to represent The effective magnetization of each volume element inside The magnet by a vector series and use successive approximation to solve, but when The magnetic susceptibility is high, The vector series converges slowly or even diverges, and The calculation efficiency is low. The literature (pursuis, m.b. j., and j.p. fill, a new iterative method for computing the magnetic field at high magnetic susceptibilities: geophisics, 2005.70: 53-62.) calculates the magnetic field of high susceptibility magnetic bodies on an iterative basis by means of a piecewise model defined by spherical voxels of uniform arbitrary diameter, but the high susceptibility calculation results are subject to large errors. In the frequency domain method, a new Gaussian FFT method is proposed based on a shift sampling method in documents (Wu, L.Y., and G.Tian, High-precision Fourier transforming of potential fields: Geophysics, 2014.79, No. 5, G59-G68, doi: 10.1190/geo 2014-0039.1), and is successfully applied to numerical simulation of gravity and magnetism, but is still only applicable to the case of low magnetic susceptibility. The literature (Likun, Daishikun, Chenganui, etc., three-dimensional numerical simulation of magnetic anomaly field integral solution in space and wave number mixed domain, geophysical report 2019.62 (11)) utilizes the characteristic that magnetic potential three-dimensional space domain integral is convolution to perform two-dimensional Fourier transform along the horizontal direction, and converts the three-dimensional integral problem satisfied by the magnetic potential in the space domain into the mutually independent vertical one-dimensional integral problem among different wave numbers, but is only suitable for the condition of low magnetic susceptibility.
Based on the current research situation, it is known that strong magnetism is rarely considered in the current magnetic induction intensity calculation method, and some calculation methods considering strong magnetism have low efficiency and low precision. Therefore, it is imperative to provide a high-efficiency and high-precision magnetic induction intensity calculation method suitable for strong magnets.
Disclosure of Invention
Aiming at the problems that the self-demagnetization effect is less considered in the calculation of the magnetic induction intensity of the strong magnet at present, and the research of the self-demagnetization effect is less considered, the precision is insufficient and the efficiency is low, the invention aims to provide a magnetic induction intensity calculation method of the strong magnetic medium based on the space wave number domain.
In order to achieve the technical purpose, the technical scheme provided by the invention is as follows:
in one aspect, the present invention provides a method for calculating magnetic induction intensity of a ferromagnetic medium based on a spatial wavenumber domain, including:
acquiring a three-dimensional target area containing an abnormal body, and establishing an initial prism model containing the three-dimensional target area;
for the initial prism model edgex、y、zThe directions are respectively divided at equal intervals inx、y、zObtaining a series of nodes in the direction; assigning a magnetic susceptibility to each node based on the magnetic susceptibility distribution data, the magnetic susceptibility being a scalar quantity, usingχRepresenting, at nodes of the anomaly part, each node being assigned according to a susceptibility value of the anomaly; the magnetic susceptibility of the nodes of the abnormal part is 0;
according to prism models and givenxDirection andygaussian parameter in direction, calculatingxAndydirection dispersion offset wavenumber;
calculating the strength of the main earth magnetic field at each node according to the main earth magnetic field model, and taking the strength of the main earth magnetic field as the strength of a spatial domain background field;
obtaining a calculation model of the magnetization intensity according to the intensity of the spatial domain background field magnetic field and the intensity of the spatial domain abnormal field magnetic field;
converting a three-dimensional Laplace equation which meets the magnetic potential and the magnetization intensity of the abnormal field in the spatial domain into a one-dimensional ordinary differential equation in the spatial wave number domain by utilizing two-dimensional Fourier transform;
based on a space wave number domain one-dimensional ordinary differential equation and in combination with a set boundary condition which needs to be met by the magnetic potential of the abnormal field of the space wave number domain, converting an edge value problem model which is met by the magnetic potential of the abnormal field of the space wave number domain into an equivalent variational problem model;
obtaining the magnetic potential of the space wave number domain abnormal field by solving a variational problem model;
obtaining the magnetic field intensity of the spatial wave number domain abnormal field based on the magnetic potential of the spatial wave number domain abnormal field, and converting the magnetic potential of the spatial wave number domain abnormal field and the magnetic field intensity of the spatial wave number domain abnormal field into the magnetic potential of the spatial wave number domain abnormal field and the magnetic field intensity of the spatial wave number domain abnormal field through inverse Fourier transform;
the total magnetic field intensity of the space domain is the sum of the magnetic field intensity of the abnormal field of the space domain and the magnetic field intensity of the background field of the space domain, and the total magnetic field intensity of the space domain is tightened by using a tightening operator to obtain the total magnetic field intensity of the space domain after tightening;
judging whether iterative convergence conditions are met or not, and if yes, outputting a space domain abnormal field magnetic potential and a space domain abnormal field magnetic field strength corresponding to the tightened space domain total magnetic field strength obtained through current calculation;
and solving according to the output spatial domain abnormal field magnetic field intensity to obtain spatial domain magnetic induction intensity.
In another aspect, the present invention provides a magnetic induction calculation apparatus for a ferromagnetic medium based on a spatial wave number domain, including:
the system comprises a first module, a second module and a third module, wherein the first module is used for acquiring a three-dimensional target area containing an abnormal body and establishing an initial prism model containing the three-dimensional target area;
a second module for aligning the initial prism model edgex、y、zThe directions are respectively divided at equal intervals inx、y、zObtaining a series of nodes in the direction; assigning a magnetic susceptibility to each node based on the magnetic susceptibility distribution data, the magnetic susceptibility being a scalar quantity, usingχRepresenting, at nodes of the anomaly part, each node being assigned according to a susceptibility value of the anomaly; the magnetic susceptibility of the nodes of the abnormal part is 0;
a third module for determining the shape of the prism model and the givenxDirection andygaussian parameter in direction, calculatingxAndydirection dispersion offset wavenumber;
the fourth module is used for calculating the strength of the main earth magnetic field at each node according to the main earth magnetic field model and taking the strength of the main earth magnetic field as the strength of a spatial domain background field;
the fifth module is used for obtaining a calculation model of the magnetization intensity according to the ambient field magnetic field intensity of the spatial domain and the abnormal field magnetic field intensity of the spatial domain;
the sixth module is used for converting a three-dimensional Laplace equation which meets the magnetic potential and the magnetization intensity of the abnormal field in the spatial domain into a one-dimensional ordinary differential equation in the spatial wavenumber domain by utilizing two-dimensional Fourier transform;
the seventh module is used for converting an edge value problem model which is satisfied by the magnetic potential of the abnormal field in the space wave number domain into an equivalent variational problem model based on a one-dimensional ordinary differential equation in the space wave number domain and by combining the set boundary condition which is required to be satisfied by the magnetic potential of the abnormal field in the space wave number domain;
the eighth module is used for obtaining the magnetic potential of the space wave number domain abnormal field by solving the variational problem model;
the ninth module is used for solving the magnetic field intensity of the spatial wave number domain abnormal field based on the magnetic potential of the spatial wave number domain abnormal field, and converting the magnetic potential of the spatial wave number domain abnormal field and the magnetic field intensity of the spatial wave number domain abnormal field into the magnetic potential of the spatial wave number domain abnormal field and the magnetic field intensity of the spatial wave number domain abnormal field through inverse Fourier transform;
a tenth module, configured to apply a tightening operator to tighten the total magnetic field strength of the space domain to obtain a tightened total magnetic field strength of the space domain, where the total magnetic field strength of the space domain is a sum of an abnormal magnetic field strength of the space domain and a background magnetic field strength of the space domain;
an eleventh module, configured to determine whether an iterative convergence condition is currently satisfied, and if so, output a spatial domain abnormal field magnetic potential and a spatial domain abnormal field magnetic field strength corresponding to the tightened spatial domain total magnetic field strength obtained through current calculation;
and the twelfth module is used for solving according to the output space domain abnormal field magnetic field intensity to obtain the space domain magnetic induction intensity.
In another aspect, the present invention provides a computer device comprising a memory and a processor, the memory storing a computer program, the processor implementing the following steps when executing the computer program:
acquiring a three-dimensional target area containing an abnormal body, and establishing an initial prism model containing the three-dimensional target area;
for the initial prism model edgex、y、zThe directions are respectively divided at equal intervals inx、y、zObtaining a series of nodes in the direction; assigning a magnetic susceptibility to each node based on the magnetic susceptibility distribution data, the magnetic susceptibility being a scalar quantity, usingχRepresenting, at nodes of the anomaly part, each node being assigned according to a susceptibility value of the anomaly; the magnetic susceptibility of the nodes of the abnormal part is 0;
according to prism models and givenxDirection andygaussian parameter in direction, calculatingxAndydirection dispersion offset wavenumber;
calculating the strength of the main earth magnetic field at each node according to the main earth magnetic field model, and taking the strength of the main earth magnetic field as the strength of a spatial domain background field;
obtaining a calculation model of the magnetization intensity according to the intensity of the spatial domain background field magnetic field and the intensity of the spatial domain abnormal field magnetic field;
converting a three-dimensional Laplace equation which meets the magnetic potential and the magnetization intensity of the abnormal field in the spatial domain into a one-dimensional ordinary differential equation in the spatial wave number domain by utilizing two-dimensional Fourier transform;
based on a space wave number domain one-dimensional ordinary differential equation and in combination with a set boundary condition which needs to be met by the magnetic potential of the abnormal field of the space wave number domain, converting an edge value problem model which is met by the magnetic potential of the abnormal field of the space wave number domain into an equivalent variational problem model;
obtaining the magnetic potential of the space wave number domain abnormal field by solving a variational problem model;
obtaining the magnetic field intensity of the spatial wave number domain abnormal field based on the magnetic potential of the spatial wave number domain abnormal field, and converting the magnetic potential of the spatial wave number domain abnormal field and the magnetic field intensity of the spatial wave number domain abnormal field into the magnetic potential of the spatial wave number domain abnormal field and the magnetic field intensity of the spatial wave number domain abnormal field through inverse Fourier transform;
the total magnetic field intensity of the space domain is the sum of the magnetic field intensity of the abnormal field of the space domain and the magnetic field intensity of the background field of the space domain, and the total magnetic field intensity of the space domain is tightened by using a tightening operator to obtain the total magnetic field intensity of the space domain after tightening;
judging whether iterative convergence conditions are met or not, and if yes, outputting a space domain abnormal field magnetic potential and a space domain abnormal field magnetic field strength corresponding to the tightened space domain total magnetic field strength obtained through current calculation;
and solving according to the output spatial domain abnormal field magnetic field intensity to obtain spatial domain magnetic induction intensity.
In yet another aspect, the present invention also provides a computer readable storage medium having a computer program stored thereon, the computer program when executed by a processor implementing the steps of:
acquiring a three-dimensional target area containing an abnormal body, and establishing an initial prism model containing the three-dimensional target area;
for the initial prism model edgex、y、zThe directions are respectively divided at equal intervals inx、y、zObtaining a series of nodes in the direction; assigning a magnetic susceptibility to each node based on the magnetic susceptibility distribution data, the magnetic susceptibility being a scalar quantity, usingχRepresenting, at nodes of the anomaly part, each node being assigned according to a susceptibility value of the anomaly; the magnetic susceptibility of the nodes of the abnormal part is 0;
according to prism models and givenxDirection andygaussian parameter in direction, calculatingxAndydirection dispersion offset wavenumber;
calculating the strength of the main earth magnetic field at each node according to the main earth magnetic field model, and taking the strength of the main earth magnetic field as the strength of a spatial domain background field;
obtaining a calculation model of the magnetization intensity according to the intensity of the spatial domain background field magnetic field and the intensity of the spatial domain abnormal field magnetic field;
converting a three-dimensional Laplace equation which meets the magnetic potential and the magnetization intensity of the abnormal field in the spatial domain into a one-dimensional ordinary differential equation in the spatial wave number domain by utilizing two-dimensional Fourier transform;
based on a space wave number domain one-dimensional ordinary differential equation and in combination with a set boundary condition which needs to be met by the magnetic potential of the abnormal field of the space wave number domain, converting an edge value problem model which is met by the magnetic potential of the abnormal field of the space wave number domain into an equivalent variational problem model;
obtaining the magnetic potential of the space wave number domain abnormal field by solving a variational problem model;
obtaining the magnetic field intensity of the spatial wave number domain abnormal field based on the magnetic potential of the spatial wave number domain abnormal field, and converting the magnetic potential of the spatial wave number domain abnormal field and the magnetic field intensity of the spatial wave number domain abnormal field into the magnetic potential of the spatial wave number domain abnormal field and the magnetic field intensity of the spatial wave number domain abnormal field through inverse Fourier transform;
the total magnetic field intensity of the space domain is the sum of the magnetic field intensity of the abnormal field of the space domain and the magnetic field intensity of the background field of the space domain, and the total magnetic field intensity of the space domain is tightened by using a tightening operator to obtain the total magnetic field intensity of the space domain after tightening;
judging whether iterative convergence conditions are met or not, and if yes, outputting a space domain abnormal field magnetic potential and a space domain abnormal field magnetic field strength corresponding to the tightened space domain total magnetic field strength obtained through current calculation;
and solving according to the output spatial domain abnormal field magnetic field intensity to obtain spatial domain magnetic induction intensity.
Compared with the prior art, the invention has the advantages that:
1. the invention can more accurately carry out magnetic field numerical simulation on the ferromagnetic medium by considering the self-demagnetization effect.
2. For the condition that the right end item of the model has unknown parameters, the iterative method is used for solving, and a solving idea is provided for similar problems.
3. Further, in order to reduce the truncation effect of the fourier transform, an edge expansion process is adopted.
4. And a tightening operator is added, so that the algorithm can be stably converged.
5. The invention reduces the three-dimensional problem into one dimension through Fourier transformation, improves the calculation precision and the calculation efficiency, and has good algorithm parallelism.
6. The method has great significance for estimating the reserves of the iron ore deposit and inverting and explaining the actually measured data.
Drawings
FIG. 1 is a flow chart in one embodiment of the present invention;
FIG. 2 is a schematic representation of an initial prism model in an embodiment of the present invention;
FIG. 3 is a schematic representation of the initial prismatic model of FIG. 2, subdivided;
FIG. 4 is a schematic diagram of a target region and an anomaly in an embodiment of the invention;
FIG. 5 is a diagram illustrating a numerical solution, an analytical solution, and an absolute error of an isotropic sphere model according to an embodiment of the present invention; wherein (a), (b) and (c) each representsB ax The numerical solution of (a) is,B ax solution of (2)Analysis and solution ofB ax The absolute error of the numerical solution and the analytic solution of (a); (d) respectively, e and f representB ay The numerical solution of (a) is,B ay an analytical solution ofB ay The absolute error of the numerical solution and the analytic solution of (a); (g) (h), (i) respectively representB az The numerical solution of (a) is,B az an analytical solution ofB az The absolute error of the numerical solution and the analytic solution of (a);
fig. 6 is an internal structural diagram of a computer device in an embodiment of the present invention.
Detailed Description
For the purpose of promoting a clear understanding of the objects, aspects and advantages of the embodiments of the invention, reference will now be made to the drawings and detailed description, wherein there are shown in the drawings and described in detail, various modifications of the embodiments described herein, and other embodiments of the invention will be apparent to those skilled in the art. The exemplary embodiments of the present invention and the description thereof are provided to explain the present invention and not to limit the present invention.
Referring to fig. 1, the method for calculating magnetic induction intensity of a ferromagnetic medium based on a spatial wavenumber domain provided by the invention comprises the following steps:
(S1) acquiring a three-dimensional target region including an anomaly, and creating an initial prism model including the three-dimensional target region, as shown in fig. 2, wherein the initial prism model is a cuboid or a cube, and the anomaly may be in any shape. The anomaly in fig. 2 is a cuboid, and the anomaly is located inside the initial prism model.
(S2) for the initial prism model edgex、y、zThe directions are respectively divided at equal intervals inx、y、zA series of nodes are obtained in the direction as shown in fig. 3.
Then, the magnetic susceptibility is assigned to each node according to the magnetic susceptibility distribution data, the magnetic susceptibility is scalar, and the magnetic susceptibility is used for assigning the magnetic susceptibility to each nodeχExpressed in SI.
It is understood that a person skilled in the art can select a suitable assignment manner according to the existing method. In one embodiment of the invention, for the nodes of the anomaly portion, each node is assigned according to the susceptibility value of the anomaly; the magnetic susceptibility of the nodes of the non-anomalous body part is 0;
(S3) according to the prism model and the givenxDirection andygaussian parameter in direction, calculatingxAndydirection dispersion offset wavenumber;
(S4) calculating the main earth magnetic field strength at each node as the spatial domain background field strength according to the main earth magnetic field model;
(S5) obtaining a calculation model of the magnetization intensity according to the spatial domain background field magnetic field intensity and the spatial domain abnormal field magnetic field intensity;
(S6) converting a three-dimensional Laplace equation which meets the magnetic potential and the magnetization intensity of the abnormal field in the spatial domain into a one-dimensional ordinary differential equation in the spatial wavenumber domain by utilizing two-dimensional Fourier transform;
(S7) converting the boundary value problem model satisfied by the abnormal magnetic potential of the space wave number domain into an equivalent variational problem model based on a one-dimensional ordinary differential equation of the space wave number domain and by combining the set boundary condition to be satisfied by the abnormal magnetic potential of the space wave number domain;
(S8) obtaining the magnetic potential of the space wave number domain abnormal field by solving a variational problem model;
(S9) obtaining the magnetic field intensity of the space wave number domain abnormal field based on the magnetic potential of the space wave number domain abnormal field, and converting the magnetic potential of the space wave number domain abnormal field and the magnetic field intensity of the space wave number domain abnormal field into the magnetic potential of the space wave number domain abnormal field and the magnetic field intensity of the space wave number domain abnormal field through inverse Fourier transform;
(S10) the total magnetic field strength of the space domain is the sum of the magnetic field strength of the abnormal field of the space domain and the magnetic field strength of the background field of the space domain, and the total magnetic field strength of the space domain is tightened by a tightening operator to obtain the total magnetic field strength of the space domain after tightening;
(S11) judging whether the iterative convergence condition is met or not, if not, returning (S8), and if so, outputting the space domain abnormal field magnetic potential and the space domain abnormal field magnetic field strength corresponding to the tightened space domain total magnetic field strength obtained by current calculation;
and (S12) solving according to the output spatial domain abnormal field magnetic field intensity to obtain spatial domain magnetic induction intensity.
It is understood that, in the step (S3) of the present invention, the gaussian parameter may be set by referring to the method for setting the gaussian parameter among the existing methods in the art, and the setting may be performedxAndyand calculating direction discrete offset wave number.
In step (S3) of an embodiment of the present invention, axNumber of Gaussian points of directionN x Interval [ -1,1 [ ]]Upper gauss pointt a Gaussian coefficient ofA a Wherein, in the step (A),a=1,2,...,N x (ii) a Given ayNumber of Gaussian points of directionN y Interval [ -1,1 [ ]]Upper gauss pointt b Gaussian coefficient ofA b Wherein, in the step (A),b=1,2,...,N y 。
in step (S3) of an embodiment of the present invention,xandythe direction dispersion offset wavenumber is calculated by the following method:
in the formula (I), the compound is shown in the specification,
wherein:k x to representxThe number of shifted wave-numbers of the directions,
to representxThe number of direction base waves is the number of direction base waves,NN x representing the initial prism model edgexThe number of nodes obtained by direction equal-interval subdivision,
representing the initial prism model edgexUnit interval length adopted in the direction equal interval division;k y to representyThe number of shifted wave-numbers of the directions,
to representyThe number of direction base waves is the number of direction base waves,NN y representing the initial prism model edgeyThe number of nodes obtained by direction equal-interval subdivision,
representing the initial prism model edgeyThe unit interval length adopted when the direction is equally spaced.
In the step (S4) of an embodiment of the present invention, the method includes:
calculating the intensity of the main earth magnetic field at each node according to the model IGRF of the main earth magnetic field, and taking the intensity as the intensity H of the background field magnetic field in the space domain at each node0It is the background field in numerical simulation, i.e. the magnetic field when there is no anomaly, and the unit is A/m. Magnetic field intensity H of spatial domain background field0The components of the three directions are respectively represented as H 0x 、H y0、H z0：
Wherein:
represents H0The L2 norm of (a),αis the magnetic tilt angle of the target area,βis the target area declination.
In step (S5) of an embodiment of the present invention, the calculation model of the magnetization is:
wherein HaThe magnetic field intensity generated by the abnormal body at each node, namely the magnetic field intensity of the abnormal field in the space domain is represented, the abnormal field in numerical simulation, namely the magnetic field generated by the abnormal magnetic susceptibility is represented in the unit of A/m, and the three components are respectively H ax 、H ay 、H az . The total magnetic field intensity H of the space domain at each node is the ambient field magnetic field intensity H of the space domain0Magnetic field intensity H corresponding to spatial domain abnormal fieldaAnd (4) summing.
In step (S6) of one embodiment of the present invention, the magnetic potential U of the spatial domain abnormal field a Three-dimensional Laplace equation satisfied by magnetization M
In the formula (I), the compound is shown in the specification,
is a Laplace operator, is unfolded into
，
Showing the intensity of magnetization
And (5) solving divergence. The above equation (7) is developed as follows:
the two-dimensional Fourier transform of equation (7) is performed, and in order to reduce the truncation effect of the Fourier transform, an edge-expanding process, that is, an edge-expanding process is adoptedx，yThe directions are extended for a certain distance on the basis of the target area to avoid the influence of the truncation effect on the field value calculation.
Obtaining one-dimensional ordinary differential equation of space wave number domain through two-dimensional Fourier transform, and reservingzThe direction is the spatial domain:
wherein
Representing the magnetic potential of the anomalous field in the spatial wavenumber domain,
、
、
in order to be the wave-number domain magnetization,k x 、k y are respectively asx、yOffset wavenumber of directions. Equation (8) which is the magnetic potential of the anomalous field in the space wavenumber domain
A one-dimensional ordinary differential equation satisfied.
In step (S7), in order to obtain a definite solution of the control equation, appropriate boundary conditions are given. Taking the Z axis vertically downwards as the forward direction and taking the horizontal ground as the upper boundary Z in the target area under a Cartesian coordinate systemminTaking a sufficient distance from the underground to the abnormal body as a lower boundary ZmaxAnd the upper and lower boundary conditions meet:
an upper boundary:
（9）
lower bound:
（10）
wherein the content of the first and second substances,
。
and (3) simultaneous equations (8), (9) and (10) to obtain an edge value problem model satisfied by the abnormal magnetic field magnetic potential in the space wavenumber domain:
converting an edge value problem model satisfied by the magnetic potential of the space wave number domain abnormal field into an equivalent variational problem model by using a variational method:
in a Cartesian coordinate system, unit subdivision is carried out along the z direction, and a quadratic interpolation function, namely a space wave number domain abnormal field magnetic potential, is adopted in each unit
Changing twice within a cell.
Magnetic potential of space wave number domain abnormal field
The right term of the satisfied one-dimensional ordinary differential equation comprises a background field and an abnormal field, and the abnormal field is unknown, so that an iterative solution is adopted, and a solution thought is provided for similar problems (namely, under the condition that the right term of the model has unknown parameters, an iterative method is proposed for solving).
Combining equation (6) and equation (7), it can be seen that the magnetization M is determined by the background field H0And an anomalous field H a Sum and product of magnetic susceptibility, and H = H0+H a And an abnormal field H a Unknown, so the first iteration assumes H a And (3) the sum of the abnormal field and the background field in the first iteration is replaced by the background field, so that the one-dimensional partial differential equation is changed into the one-dimensional ordinary differential equation to be solved to obtain a first abnormal field, and then the sum of the abnormal field and the background field is used as a right-end term total field to be solved for the next time.
And the problem solved each time is a variational problem, the variational problem is subjected to unit analysis and total synthesis item by item to obtain a five-diagonal equation consisting of all nodes, the quick and efficient solution can be realized by adopting a catch-up method, and the magnetic potential of the spatial wave number domain abnormal field at each node is obtained
。
In step (S9) of one embodiment of the present invention, the magnetic potential is based on the anomalous field in the space wave number domain
And obtaining the magnetic field intensity of the abnormal field in the space wave number domain by the following formula:
wherein the content of the first and second substances,iis the number of the imaginary numbers,
presentation pairzThe differential is obtained by the differential analysis,
representing magnetic potential pairs of space wavenumber domain abnormal fieldszAnd (6) carrying out differentiation.
The magnetic potential U of the abnormal field in the spatial wave number domain and the magnetic field strength of the abnormal field in the spatial wave number domain can be obtained through Fourier inverse transformation a And spatial domain anomalous field strength H a 。
In the step (S10) of an embodiment of the present invention, a tightening operator is applied to the total magnetic field strength of the space domain to obtain the total magnetic field strength of the space domain after tightening, where the total magnetic field strength of the space domain is the sum of the magnetic field strength of the abnormal field of the space domain and the magnetic field strength of the background field of the space domain.
Wherein the content of the first and second substances,jrepresenting the number of iterations, H j+1Is as followsjThe total magnetic field strength of the spatial domain calculated in +1 iteration,
are respectively the firstjAmbient field strength in the spatial domain and anomalous field strength in the spatial domain, H, in +1 iteration j Is shown asjAnd calculating the total magnetic field intensity of the space domain in the secondary iteration.
It is understood that the preset iteration termination condition refers to a preset model calculation constraint condition for constraining the whole model to converge in the performance calculation process, so that the model can output a result meeting the condition. In the present invention, the iteration termination condition in step (S11) may be set as:
when the above iteration convergence condition is satisfied, the iteration is stopped, wherein
Is shown asjThe total magnetic field strength of the tightened space domain obtained by calculation in the secondary iteration,
is shown asjAnd (5) calculating the total magnetic field intensity of the tightened space domain in +1 iteration.
Of course, in practical applications, a person skilled in the art may set other iteration termination conditions based on the prior art, the conventional technical means in the field, or the common general knowledge, and is not limited to the iteration termination conditions set in the above preferred embodiments of the present application.
Finally, in step (S12), the magnetic induction B is induced by the abnormal field a (unit is T) and the magnetic field intensity H of the spatial domain abnormal field a From the relationship of (1), the magnetic induction B can be obtained a And further to obtain B a Three components ofB ax ，B ay ，B az 。
Wherein the content of the first and second substances,μis the absolute permeability of the ferromagnetic medium, in units of H/m.
Absolute magnetic permeabilityμAndχthe relationship between them is shown in equation (17),μ 0in order to obtain a magnetic permeability in a vacuum,
。
the accuracy and the efficiency of the method for calculating the magnetic induction intensity of the strong magnetic medium based on the space wave number domain are tested.
The test computer is configured to i5-4590, the main frequency is 3.30GHz, and the memory is 12 GB.
The target area is a three-dimensional prismatic structure, the size of which is 1000m × 1000m × 1000m,x，ythe direction of each side expanding is 3000 m. The anomaly in the target region is an isotropic sphere with a radius of 200m, the sphere center being at the model center (500 m, 500m, 500 m), which is schematically shown in fig. 4.
The anomalous body is an isotropic sphere, and the sphere magnetic susceptibility of the anomalous body is 2 SI. The background magnetic field intensity of the model region is 50000nT, the magnetic dip angle of the target region is 45 degrees, and the magnetic declination angle of the target region is 9 degrees.x，y，z The number of direction nodes is 101,101,101. Under the condition of single thread, edge expanding FFT is adopted, iteration is needed for 7 times when the convergence condition is achieved, the time is 56s, the occupied memory is 1.38GB, the occupied memory is low, and the efficiency is high. B thereof ax ，B ay ，B az The absolute error of the analytic solution is different from the field value by more than two orders of magnitude, and the numerical calculation requirement is met. FIG. 5 is a model calculationB ax ，B ay ，B az The numerical solution, analytical solution and absolute error of the components are shown in FIG. 5, wherein (a), (b) and (c) representB ax The numerical solution of (a) is,B ax an analytical solution ofB ax The absolute error of the numerical solution and the analytic solution of (a); in FIG. 5, (d), (e) and (f) representB ay The numerical solution of (a) is,B ay an analytical solution ofB ay The absolute error of the numerical solution and the analytic solution of (a); in FIG. 5, (g), (h) and (i) representB az The numerical solution of (a) is,B az an analytical solution ofB az And resolving the absolute error of the solution.
On the basis of considering the self-demagnetization effect, a three-dimensional problem is reduced to one dimension by adopting standard edge expanding FFT (fast Fourier transform), only the z direction is reserved, a one-dimensional finite element method is applied, and shape function quadratic interpolation is adopted in an element, so that iterative solution is carried out on a differential equation, and a tightening operator is applied to ensure the convergence of the algorithm, thereby greatly improving the calculation precision and the calculation efficiency, and the algorithm has good parallelism and occupies less memory. The invention can be used in the inversion algorithm of magnetic measurement data, and improves the accuracy and efficiency of inversion.
In one embodiment, a computer device is provided, which may be a server, and its internal structure diagram may be as shown in fig. 6. The computer device includes a processor, a memory, a network interface, and a database connected by a system bus. Wherein the processor of the computer device is configured to provide computing and control capabilities. The memory of the computer device comprises a nonvolatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, a computer program, and a database. The internal memory provides an environment for the operation of an operating system and computer programs in the non-volatile storage medium. The database of the computer device is used to store sample data. The network interface of the computer device is used for communicating with an external terminal through a network connection. The computer program is executed by a processor to realize the above-mentioned method for calculating magnetic induction of a ferromagnetic medium based on a spatial wavenumber domain.
Those skilled in the art will appreciate that the architecture shown in fig. 6 is merely a block diagram of some of the structures associated with the disclosed aspects and is not intended to limit the computing devices to which the disclosed aspects apply, as particular computing devices may include more or less components than those shown, or may combine certain components, or have a different arrangement of components.
In one embodiment, a computer device is provided, which includes a memory and a processor, the memory stores a computer program, and the processor implements the steps of the magnetic induction intensity calculation method based on the strong magnetic medium in the spatial wave number domain in the above embodiments when executing the computer program.
In one embodiment, a computer-readable storage medium is provided, on which a computer program is stored, which when executed by a processor implements the steps of the method for calculating magnetic induction of a strong magnetic medium based on the spatial wavenumber domain in the above-described embodiments.
It will be understood by those skilled in the art that all or part of the processes of the methods of the embodiments described above can be implemented by hardware instructions of a computer program, which can be stored in a non-volatile computer-readable storage medium, and when executed, can include the processes of the embodiments of the methods described above. Any reference to memory, storage, database, or other medium used in the embodiments provided herein may include non-volatile and/or volatile memory, among others. Non-volatile memory can include read-only memory (ROM), Programmable ROM (PROM), Electrically Programmable ROM (EPROM), Electrically Erasable Programmable ROM (EEPROM), or flash memory. Volatile memory can include Random Access Memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in a variety of forms such as Static RAM (SRAM), Dynamic RAM (DRAM), Synchronous DRAM (SDRAM), Double Data Rate SDRAM (DDRSDRAM), Enhanced SDRAM (ESDRAM), Synchronous Link DRAM (SLDRAM), Rambus Direct RAM (RDRAM), direct bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM).
The technical features of the above embodiments can be arbitrarily combined, and for the sake of brevity, all possible combinations of the technical features in the above embodiments are not described, but should be considered as the scope of the present specification as long as there is no contradiction between the combinations of the technical features.
The above-mentioned embodiments only express several embodiments of the present application, and the description thereof is more specific and detailed, but not construed as limiting the scope of the invention. It should be noted that, for a person skilled in the art, several variations and modifications can be made without departing from the concept of the present application, which falls within the scope of protection of the present application. Therefore, the protection scope of the present patent shall be subject to the appended claims.
Claims (10)
1. The method for calculating the magnetic induction intensity of the strong magnetic medium based on the space wave number domain is characterized by comprising the following steps of:
acquiring a three-dimensional target area containing an abnormal body, and establishing an initial prism model containing the three-dimensional target area;
for the initial prism model edgex、y、zThe directions are respectively divided at equal intervals inx、y、zObtaining a series of nodes in the direction; assigning a magnetic susceptibility to each node based on the magnetic susceptibility distribution data, the magnetic susceptibility being a scalar quantity, using
Representing, at nodes of the anomaly part, each node being assigned according to a susceptibility value of the anomaly; the magnetic susceptibility of the nodes of the abnormal part is 0;
according to prism models and givenxDirection andygaussian parameter in direction, calculatingxAndydirection dispersion offset wavenumber;
calculating the strength of the main earth magnetic field at each node according to the main earth magnetic field model, and taking the strength of the main earth magnetic field as the strength of a spatial domain background field;
obtaining a calculation model of the magnetization intensity according to the intensity of the spatial domain background field magnetic field and the intensity of the spatial domain abnormal field magnetic field;
converting a three-dimensional Laplace equation which meets the magnetic potential and the magnetization intensity of the abnormal field in the spatial domain into a one-dimensional ordinary differential equation in the spatial wave number domain by utilizing two-dimensional Fourier transform;
based on a space wave number domain one-dimensional ordinary differential equation and in combination with a set boundary condition which needs to be met by the magnetic potential of the abnormal field of the space wave number domain, converting an edge value problem model which is met by the magnetic potential of the abnormal field of the space wave number domain into an equivalent variational problem model;
obtaining the magnetic potential of the space wave number domain abnormal field by solving a variational problem model;
obtaining the magnetic field intensity of the spatial wave number domain abnormal field based on the magnetic potential of the spatial wave number domain abnormal field, and converting the magnetic potential of the spatial wave number domain abnormal field and the magnetic field intensity of the spatial wave number domain abnormal field into the magnetic potential of the spatial wave number domain abnormal field and the magnetic field intensity of the spatial wave number domain abnormal field through inverse Fourier transform;
the total magnetic field intensity of the space domain is the sum of the magnetic field intensity of the abnormal field of the space domain and the magnetic field intensity of the background field of the space domain, and the total magnetic field intensity of the space domain is tightened by using a tightening operator to obtain the total magnetic field intensity of the space domain after tightening;
judging whether iterative convergence conditions are met or not, and if yes, outputting a space domain abnormal field magnetic potential and a space domain abnormal field magnetic field strength corresponding to the tightened space domain total magnetic field strength obtained through current calculation;
and solving according to the output spatial domain abnormal field magnetic field intensity to obtain spatial domain magnetic induction intensity.
2. The method according to claim 1, wherein the gaussian parameter comprises:
xnumber of Gaussian points of directionN x Interval [ -1,1 [ ]]Upper gauss pointt a Gaussian coefficient ofA a Wherein, in the step (A),a=1,2,...,N x ；
ynumber of Gaussian points of directionN y Interval [ -1,1 [ ]]Upper gauss pointt b Gaussian coefficient ofA b Wherein, in the step (A),b=1,2,...,N y 。
3. the method of calculating the magnetic induction of a strong magnetic medium based on the spatial wave number domain according to claim 2,xandythe direction dispersion offset wavenumber is calculated by the following method:
in the formula (I), the compound is shown in the specification,
wherein:k x to representxThe number of shifted wave-numbers of the directions,
to representxThe number of direction base waves is the number of direction base waves,NN x representing the initial prism model edgexThe number of nodes obtained by direction equal-interval subdivision,
representing the initial prism model edgexUnit interval length adopted in the direction equal interval division;k y to representyThe number of shifted wave-numbers of the directions,
to representyThe number of direction base waves is the number of direction base waves,NN y representing the initial prism model edgeyThe number of nodes obtained by direction equal-interval subdivision,
representing the initial prism model edgeyThe unit interval length adopted when the direction is equally spaced.
4. The method for calculating the magnetic induction of a strong magnetic medium based on the spatial wavenumber domain according to claim 1, 2 or 3, wherein the calculation model of the magnetization is:
wherein H a Representing the magnetic field strength produced by the anomaly at each node, i.e. its spatial domain anomalous field strength, H0And H represents the spatial domain background field magnetic field strength at each node, and H represents the spatial domain total magnetic field strength at each node.
5. The method according to claim 4, wherein the method is applied to the magnetic induction of the spatial-wavenumber-domain-based ferromagnetic medium for calculating the magnetic potential of the anomalous field in the spatial domainU a And the magnetization M satisfies the three-dimensional Laplace equation
Performing two-dimensional Fourier transform to obtain a space wave number domain one-dimensional ordinary differential equation:
whereiniIs the number of the imaginary numbers,
in order to be the laplacian operator,
showing the intensity of magnetization
Solving divergence;
representing the magnetic potential of the anomalous field in the spatial wavenumber domain,
、
、
are respectively asx、y、zThe wave-number domain magnetization of the direction,k x 、k y are respectively asx、yThe number of shifted wave-numbers of the directions,
。
6. the method for calculating the magnetic induction intensity of the ferromagnetic medium based on the spatial wavenumber domain according to claim 5, wherein the boundary conditions that the magnetic potential of the abnormal field in the spatial wavenumber domain needs to satisfy are as follows:
an upper boundary:
lower bound:
。
7. the method for calculating the magnetic induction intensity of the ferromagnetic medium based on the spatial wavenumber domain according to claim 6, wherein the boundary conditions that the magnetic potential of the abnormal field in the spatial wavenumber domain needs to satisfy and the one-dimensional ordinary differential equation in the spatial wavenumber domain jointly form an edge value problem model that the magnetic potential of the abnormal field in the spatial wavenumber domain satisfies; and converting the boundary value problem model satisfied by the magnetic potential of the abnormal field in the space wavenumber domain into an equivalent variational problem model by using a variational method.
8. The magnetometer based on the strong magnetic medium in the space wave number domain as claimed in claim 7The calculation method is characterized in that a variational problem model is solved by a shape function method to obtain the magnetic potential of the space wave number domain abnormal field
。
9. The method according to claim 8, wherein the tightening is performed by using a tightening operator to obtain the total magnetic field strength of the tightened spatial domain, and the method comprises the following steps:
wherein the content of the first and second substances,jrepresenting the number of iterations, H j+1Is as followsjThe total magnetic field strength of the spatial domain calculated in +1 iteration,
are respectively the firstjAmbient field strength in the spatial domain and anomalous field strength in the spatial domain, H, in +1 iteration j Is shown asjAnd calculating the total magnetic field intensity of the space domain in the secondary iteration.
10. The method for calculating the magnetic induction of a strong magnetic medium based on the spatial wavenumber domain according to claim 1, 2, 3, 5, 6, 7, 8 or 9, wherein the iteration termination condition is set as:
wherein
Is shown asjThe total magnetic field strength of the tightened space domain obtained by calculation in the secondary iteration,
is shown asjAnd (5) calculating the total magnetic field intensity of the tightened space domain in +1 iteration.
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