CN115795231B - Space wave number mixed domain three-dimensional strong magnetic field iteration method and system - Google Patents
Space wave number mixed domain three-dimensional strong magnetic field iteration method and system Download PDFInfo
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Abstract
The invention relates to the technical field of magnetic prospecting, and discloses a space wave number mixed domain three-dimensional strong magnetic field iteration method and a system, wherein the method comprises the following steps: constructing a three-dimensional target model of a target area containing an abnormal body; solving the magnetization intensity of the target area according to the relation between the magnetic susceptibility and the magnetization intensity; determining a three-dimensional poisson equation based on magnetization intensity, transforming the three-dimensional poisson equation to a wave number domain through arbitrarily sampled two-dimensional Fourier forward transformation, solving the wave number domain abnormal field magnetic field intensity of a target area, and performing arbitrarily sampled two-dimensional Fourier reverse transformation on the wave number domain abnormal field magnetic field intensity to obtain the space domain abnormal field magnetic field intensity of the target area; the method comprises the steps of carrying out iterative convergence judgment on the field intensity of the abnormal field in the spatial domain, and solving the magnetic induction intensity in the spatial domain of the target area according to the relation between the field intensity of the abnormal field in the spatial domain and the magnetic induction intensity in the spatial domain when the iterative convergence condition is met.
Description
Technical Field
The invention relates to the technical field of magnetic prospecting, in particular to a space wave number mixed domain three-dimensional strong magnetic field iteration method and system.
Background
Magnetic prospecting is one of the important geophysical prospecting means, and when the magnetic susceptibility of a medium is greater than 0.1SI during the magnetic prospecting process, the medium is generally considered to be a ferromagnetic medium, and the self-demagnetizing field formed by the medium is not negligible. The self-demagnetizing field of the target is usually ignored in calculating the magnetic induction intensity of the magnetic field of the ferromagnetic medium, namely, the magnetic induction intensity is approximately calculated under the weak magnetic condition, so that the value of the magnetic induction intensity obtained by numerical simulation has larger deviation from the value of the actual magnetic induction intensity. The magnetic susceptibility of strong magnetite is basically larger than 0.1SI, which is a strong magnetic condition, if the self-demagnetizing effect is ignored, the magnetic measurement data interpretation will be wrong. In the existing strong magnetic field calculation numerical simulation, the numerical simulation method using Fourier transformation basically only can handle the situation of uniform sampling, and when the space domain model is complex or the distribution of the wave number domain spectrum is uneven, the accuracy of the numerical simulation can be affected to a certain extent, so that the accuracy of the numerical simulation is reduced. Therefore, the existing strong magnetic field simulation calculation method has the problem of lower precision.
Disclosure of Invention
The invention provides a three-dimensional strong magnetic field iteration method and a three-dimensional strong magnetic field iteration system for a spatial wave number mixed domain, which are used for solving the problem of low precision of the existing strong magnetic field simulation calculation method.
In order to achieve the above object, the present invention is realized by the following technical scheme:
in a first aspect, the present invention provides a method for iterating a three-dimensional strong magnetic field in a spatial wave number mixed domain, including:
constructing a three-dimensional target model of a target area containing an abnormal body, splitting the three-dimensional target model to obtain a series of nodes, and carrying out susceptibility assignment on each node according to susceptibility distribution data to obtain the susceptibility of each node;
solving the magnetization intensity of the target area according to the relation between the magnetic susceptibility and the magnetization intensity;
determining a three-dimensional poisson equation based on the magnetization intensity, transforming the three-dimensional poisson equation into a wave number domain through arbitrarily sampled two-dimensional Fourier positive transformation, and solving to obtain abnormal field magnetic positions of the wave number domain of a target area;
solving the wave number domain abnormal field magnetic field intensity of the target area according to the relation between the wave number domain abnormal field magnetic field position and the wave number domain abnormal field magnetic field intensity, and performing two-dimensional Fourier inverse transformation of arbitrary sampling on the wave number domain abnormal field magnetic field intensity to obtain the space domain abnormal field magnetic field intensity of the target area;
and carrying out iterative convergence judgment on the magnetic field intensity of the abnormal field in the spatial domain, solving the magnetic induction intensity in the spatial domain of the target area according to the relation between the magnetic field intensity of the abnormal field in the spatial domain and the magnetic induction intensity in the spatial domain when the iterative convergence condition is met, and recalculating the magnetization intensity in the target area when the iterative convergence condition is not met.
Optionally, the performing iterative convergence judgment on the field strength of the abnormal field in the spatial domain includes:
carrying out iterative computation on the magnetic field intensity of the space domain through a tight operator to obtain the total magnetic field intensity of the space domain, wherein the tight operator is shown in the following formula:
where j represents the number of iterations, H j+1 Representing the spatial domain total field strength of the current iteration,representing the field strength of the background field of the spatial domain of the current iteration, < >>Represents the field intensity of the abnormal field in the spatial domain of the iteration, H j The total field intensity of the spatial domain of the previous iteration is represented, and χ represents the magnetic susceptibility;
and carrying out iterative convergence judgment on the calculated spatial domain total field magnetic field intensity, wherein the judgment condition of iterative convergence is shown as follows:
|H j+1 -H j |/H j+1 ＜10 -4 ；
wherein ,Hj+1 Representing the total field intensity of the spatial domain of the iteration, H j Representing the spatial domain total field magnetic field strength of the last iteration.
Optionally, the relationship between the magnetic field intensity of the abnormal field in the spatial domain and the magnetic induction intensity in the spatial domain is expressed as:
B a ＝μH a ；
wherein ,Ba For the magnetic induction intensity of the space domain, H a For the field intensity of the abnormal field in the spatial domain, mu is the absolute permeability of the medium, the unit is H/m, and the relation between mu and the magnetic susceptibility satisfies the following expression:
μ＝μ 0 (1+χ)；
wherein ,μ0 For permeability in vacuum, mu 0 ＝4π×10 -7 H/m, χ represents magnetic susceptibility.
Optionally, the solving the magnetization of the target area according to the relation between the magnetic susceptibility and the magnetization comprises:
determining the field intensity of a background field of a space domain of a target area according to the earth main magnetic field model;
determining the field intensity of the abnormal field in the spatial domain of the target area according to the field intensity generated by the abnormal body;
taking the sum of the background field magnetic field intensity of the space domain and the abnormal field magnetic field intensity of the space domain as the total field magnetic field intensity of the space domain, and solving to obtain the magnetization intensity of the target area according to the relation between the magnetic susceptibility and the magnetization intensity;
the relationship between susceptibility and magnetization is shown as follows:
M＝χH＝χ(H 0 +H a )；
wherein M represents magnetization intensity, χ represents magnetic susceptibility, H 0 Represents the field strength of background field in the space domain, H a The abnormal field magnetic field intensity in the space domain is represented, and H represents the total field magnetic field intensity in the space domain.
Optionally, the recalculating the magnetization of the target region includes:
bringing the field intensity of the abnormal field in the space domain into a relation between the magnetic susceptibility and the magnetic intensity, and recalculating the magnetic intensity of the target area;
after bringing the field intensity of the abnormal field in the spatial domain into the relation between the magnetic susceptibility and the magnetization intensity, the relation is shown as follows:
M＝χH＝χ(H 0 +H a )；
wherein M represents magnetization intensity, χ represents magnetic susceptibility, H 0 Represents the field strength of background field in the space domain, H a The abnormal field magnetic field intensity in the space domain is represented, and H represents the total field magnetic field intensity in the space domain.
Optionally, the three-dimensional poisson equation is:
wherein ,Ua Represents the abnormal field magnetic potential of the magnetic field in the space domain, M represents the magnetization intensity, wherein, i, j and k are unit vectors in the x, y and z directions respectively;
the above-mentioned expansion is:
wherein ,Mx 、M y 、M z The components of the magnetization M in the x, y, z directions,is the sign of the partial derivative.
Optionally, the arbitrary sampled two-dimensional fourier positive transform formula is:
wherein ,kx The wave number, k, representing the x-direction y Representing wavenumbers in the y-direction, F (x, y) representing a spatial domain function, F (k) x ，k y ) Representing a wavenumber spectrum;
the arbitrary sampling two-dimensional Fourier forward transformation formula is obtained through two one-dimensional Fourier forward transformations, and the two one-dimensional Fourier forward transformations are respectively;
performing one-dimensional Fourier forward transformation in the x direction on f (x, y), wherein the transformation formula is as follows:
wherein x and y represent two mutually perpendicular directions; k (k) x The wavenumbers in the x-direction, F (x, y) represent the spatial domain function, F x (k x Y) is a wave number spectrum obtained by performing one-dimensional Fourier transform on f (x, y) in the x direction;
for F x (k x Y) performing one-dimensional Fourier forward transformation in the y direction, wherein the transformation formula is as follows:
wherein ,ky Represents the wavenumber in the y-direction, F (k) x ，k y ) To a wavenumber spectrum after a two-dimensional fourier transform of f (x, y).
Optionally, the transforming the three-dimensional poisson equation to the wave number domain through any sampled two-dimensional fourier positive transformation, and solving to obtain the abnormal field magnetic bits of the wave number domain of the target area, including:
performing horizontal two-dimensional Fourier forward transformation on the three-dimensional poisson equation to obtain a one-dimensional ordinary differential equation, wherein the one-dimensional ordinary differential equation is shown in the following formula:
wherein ,representing abnormal field magnetic bits in wavenumber domain, +.>Representing the magnetization in the wavenumber domain->X-component, y-component and z-component, k x 、k y Wave numbers in x and y directions are respectively expressed, +.>For partial derivative sign, i is an imaginary number;
under the Cartesian coordinate system, taking the vertical downward direction of the Z axis as the forward direction, and calculating the upper boundary Z of the horizontal ground in the area min Taking the abnormal body far enough away from the ground as a lower boundary Z max The upper and lower boundary conditions thereof satisfy:
upper boundary:
the lower boundary:
wherein ,
and (3) combining a one-dimensional ordinary differential equation with an upper boundary and a lower boundary to obtain:
and obtaining a variation problem equivalent to the boundary value problem by using a variation method:
performing unit subdivision along the z direction, and obtaining the abnormal field magnetic potential of the wave number domain of the target region by adopting a quadratic interpolation function in each unit
Optionally, the relationship between the abnormal field magnetic level in the wave number domain and the abnormal field magnetic field strength in the wave number domain is:
wherein i is an imaginary number;
the arbitrary sampled two-dimensional Fourier inverse transform expression is:
wherein ,kx 、k y Representing wavenumber, F (x, y) is a spatial domain function, F (k) x ，k y ) Representing the wavenumber spectrum.
In a second aspect, an embodiment of the present application provides a spatial wave number hybrid domain three-dimensional strong magnetic field iteration system, including a memory, a processor, and a computer program stored on the memory and executable on the processor, the processor implementing the steps of any of the methods of the first aspect described above when the computer program is executed.
The beneficial effects are that:
according to the space wave number mixed domain three-dimensional strong magnetic field iteration method, the problem of the three-dimensional strong magnetic field is reduced to one dimension through arbitrary sampling two-dimensional Fourier forward transformation, the differential equation is solved by means of shape function secondary interpolation, then the solution value is transformed back to the space domain through arbitrary sampling two-dimensional Fourier inverse transformation, the solution value is solved, iteration convergence is carried out on the solution value, finally an optimal solution meeting iteration convergence conditions is obtained, the space domain magnetic induction intensity of a target area is calculated by using the optimal solution, calculation accuracy and calculation efficiency are improved, algorithm parallelism is good, and occupied memory is small.
In addition, the method calculates Fourier transform coefficients in advance, flexibly sets sampling intervals according to the distribution of fields and spectrums, and properly sparsifies and encrypts sampling points according to requirements, and can obtain Fourier oscillation operators e-i in an integration interval kx The method has the advantages that the accuracy and the efficiency are both considered, the truncation effect does not exist, and the boundary problem can be perfectly solved by applying the Fourier transformation method to the partial differential equation solution, and the calculation efficiency is higher.
Drawings
FIG. 1 is a flow chart of a method for iterating a three-dimensional strong magnetic field in a spatial wave number mixed domain according to a preferred embodiment of the present invention;
FIG. 2 is a schematic diagram of a three-dimensional object model of a three-dimensional strong magnetic field iteration method of a spatial wave number mixed domain according to a preferred embodiment of the present invention;
FIG. 3 is one of the split schematic diagrams of the spatial wave number mixed domain three-dimensional strong magnetic field iteration method according to the preferred embodiment of the present invention;
FIG. 4 is a second schematic diagram of the method for iterating the three-dimensional strong magnetic field of the spatial wave number mixed domain according to the preferred embodiment of the present invention;
FIG. 5 is a third schematic diagram of the method for iterating the three-dimensional strong magnetic field of the spatial wave number mixed domain according to the preferred embodiment of the present invention;
FIG. 6 is a schematic diagram of a unit node structure of a three-dimensional strong magnetic field iteration method of a spatial wave number mixed domain according to a preferred embodiment of the present invention;
FIG. 7 is a schematic diagram of boundary conditions of a spatial wavenumber hybrid domain three-dimensional strong magnetic field iterative method according to a preferred embodiment of the invention;
fig. 8 is a graph showing the comparison between the results and the analysis of the spatial wave number mixed domain three-dimensional strong magnetic field iteration method according to the preferred embodiment of the present invention.
Detailed Description
The following description of the present invention will be made clearly and fully, and it is apparent that the embodiments described are only some, but not all, of the embodiments of the present invention. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Unless defined otherwise, technical or scientific terms used herein should be given the ordinary meaning as understood by one of ordinary skill in the art to which this invention belongs. The terms "first," "second," and the like, as used herein, do not denote any order, quantity, or importance, but rather are used to distinguish one element from another. Likewise, the terms "a" or "an" and the like do not denote a limitation of quantity, but rather denote the presence of at least one. The terms "connected" or "connected," and the like, are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. "upper", "lower", "left", "right", etc. are used merely to indicate a relative positional relationship, which changes accordingly when the absolute position of the object to be described changes.
It should be appreciated that an arbitrarily sampled spatial wavenumber domain three-dimensional magnetic field numerical simulation method of the present application may be applied to magnetic prospecting, such as subsurface vein prospecting, petroleum, natural gas prospecting, geological structure inference, and the like, by way of example only, and not limitation.
Example 1, please refer to fig. 1:
the embodiment of the application provides a three-dimensional strong magnetic field iteration method of a spatial wave number mixed domain, which comprises the following steps:
constructing a three-dimensional target model of a target area containing an abnormal body, splitting the three-dimensional target model to obtain a series of nodes, and carrying out susceptibility assignment on each node according to susceptibility distribution data to obtain the susceptibility of each node;
solving the magnetization intensity of the target area according to the relation between the magnetic susceptibility and the magnetization intensity;
determining a three-dimensional poisson equation based on the magnetization intensity, transforming the three-dimensional poisson equation into a wave number domain through arbitrarily sampled two-dimensional Fourier positive transformation, and solving to obtain abnormal field magnetic positions of the wave number domain of a target area;
solving the wave number domain abnormal field magnetic field intensity of the target area according to the relation between the wave number domain abnormal field magnetic field position and the wave number domain abnormal field magnetic field intensity, and performing two-dimensional Fourier inverse transformation of arbitrary sampling on the wave number domain abnormal field magnetic field intensity to obtain the space domain abnormal field magnetic field intensity of the target area;
and carrying out iterative convergence judgment on the magnetic field intensity of the abnormal field in the spatial domain, solving the magnetic induction intensity in the spatial domain of the target area according to the relation between the magnetic field intensity of the abnormal field in the spatial domain and the magnetic induction intensity in the spatial domain when the iterative convergence condition is met, and recalculating the magnetization intensity in the target area when the iterative convergence condition is not met.
In the above embodiment, by establishing a three-dimensional target model of a target area containing isomers, splitting the three-dimensional target model to obtain a series of nodes, assigning each node, calculating the magnetization intensity of the target area according to the assignment result, calculating the wave number domain abnormal field magnetic potential of the target area through arbitrary sampling two-dimensional fourier positive transformation, calculating the space domain abnormal field magnetic field intensity of the target area through arbitrary sampling two-dimensional fourier inverse transformation, performing iterative convergence judgment on the space domain abnormal field magnetic field intensity, and when the space domain abnormal field magnetic field intensity meets the iterative convergence condition, obtaining the value of the space domain magnetic induction intensity of the target area through the relation between the space domain abnormal field magnetic field intensity and the space domain magnetic induction intensity, thereby completing calculation; and when the field intensity of the abnormal field in the space domain does not meet the iterative convergence condition, returning to the magnetization intensity calculation step, calculating the magnetization intensity again, and calculating the field intensity of the abnormal field in the space domain by the value of the magnetization intensity obtained by the recalculation until the field intensity of the abnormal field in the space domain meets the iterative convergence condition.
The three-dimensional target model is split to obtain a series of nodes, and the method comprises any one of the following modes:
mode one: the three directions of the space domain x, y and z are evenly split, wherein the x, y and z are three directions which are respectively vertical;
mode two: non-uniform subdivision is adopted for a preset first area, and encryption is carried out; wherein the first region satisfies the following formula:
ρ i for the residual density ρ of the corresponding node of the first region j The residual density of the jth node around the corresponding node of the first area is given, and n is the number of the nodes around the first area; omega is weight, and the value range is (0, 1).
Mode three: sparse sampling is carried out on a preset second area, wherein the second area meets the following formula:
ρ i′ for the remaining density of the corresponding node of the second region, ρ j′ The remaining density of the jth node around the corresponding node of the second area is given, and n is the number of the nodes around the first area; omega is weight, and the value range is (0, 1).
Example 2, please refer to fig. 2-7:
the invention provides a space wave number mixed domain three-dimensional strong magnetic field iteration method which comprises the following steps:
step one: model building
And (5) performing geological modeling on the calculated area of the numerical simulation. The size of the whole calculation region is firstly determined, and then the distribution of abnormal bodies, which can be any complex condition and any complex shape, are determined, wherein the abnormal bodies are required to be in the calculation region. A schematic diagram of a simple model is shown in FIG. 2, in which the anomaly is a sphere.
Step two: model subdivision
Modeling in a spatial domain:
after the model is built, the model is split, and the sampling points in the x, y and z directions are Nx, ny and Nz respectively. One of the advantages of the invention is that the model subdivision is arbitrary in the x, y and z directions, the non-uniform subdivision can be adopted at the place where the abnormal body of the model changes fast for encryption, and the sparse sampling can be carried out at the place where the abnormal body changes slow or where the abnormal body does not change. It is also possible that all three directions are uniformly sampled, as shown in fig. 4. For the model shown in fig. 2, for a better fit to the sphere, the subdivision and sampling in the horizontal direction may be as shown in fig. 3, as well as the non-uniform subdivision in the z-direction.
Determining wavenumber k from spatial domain subdivision x ，k y The cut-off frequency (k) x ，k y Maximum positive and minimum negative of (a) and k x ，k y Is a sampling mode of the system.
The cut-off spectrum is related to the minimum subdivision interval in the corresponding direction of the spatial domain, and the minimum subdivision interval in the x direction is set as Deltax min The minimum subdivision interval in the y direction is delta y min The corresponding cut-off frequency is:
sampling within the cut-off frequency can ensure that all spectral information is sampled. After determining the cut-off frequency, the number of samples is determined again, assuming k x ，k y The number of samples of (a) is Nkx and Nky, respectively.
Can select uniform sampling, namely k x ，k y The arrangement intervals are the same; the method can also select even sampling in the logarithmic domain, and the logarithmic domain sampling is suitable for selecting wave numbers of magnetic numerical simulation.
When sampling logarithmic interval, the wave number is set to be in the range of [ -k max ,k max ]The number of the wave number domain sampling points is 2M+1, the sampling is carried out on the digital domain sampling at equal intervals, and the sampling interval is
wherein ,kmin Is a decimal, typically 10 -6 ～10 -3 。
The wave number is arranged at [ -k max ,0]Upper is
Wavenumbers are arranged in [0, k ] max ]Upper is
k x ，k y The logarithmic domain sampling of (a) can be performed by using the sampling methods of the formula (1) and the formula (2), thereby giving the arrangement of the spatial domains x, y, z and the wavenumber domains kx, ky.
It is also possible to sample the segmented uniform subdivision in the wavenumber domain, as shown in fig. 5.
Step three: model parameter susceptibility assignment
The susceptibility assignment is made to the nodes in fig. 3 or fig. 4. The abnormal body part is assigned to each corresponding node according to the magnetic susceptibility of the abnormal body, the magnetic susceptibility on the node without the abnormal part is 0, the magnetic susceptibility is represented by χ, the scalar is represented by SI, and the unit is SI.
Step four: calculating magnetization M corresponding to the node
Calculating the earth main magnetic field intensity H at each node according to the earth main magnetic field model IGRF 0 The unit of the background field in the numerical simulation, namely the magnetic field without abnormality is A/m, and the components in three directions are respectively expressed as H 0x 、H 0y 、H 0z And this main magnetic field value is taken as the magnetic field initial value. The strength of the magnetic field generated by the abnormal body at the node is H a Is an abnormal field in numerical simulation, namely a magnetic field generated by abnormal magnetic susceptibility, the unit is A/m, and three components are H respectively 0x 、H 0y 、H 0z . The total field H is the sum of the background field and the anomaly field.
The three components of the background field are calculated by the following formula, where H 0 The I represents the background field H 0 And (2) is the L2 norm of the study area, alpha is the study area dip angle, and beta is the study area declination angle.
H 0x ＝||H 0 ||·cos(α)·cos(β)；
H 0y ＝||H 0 ||·cos(α)·sin(β)；
H 0z ＝||H 0 ||·sin(α)；
Thus, H of each node is obtained 0 The magnetization M is then calculated by the formula:
M＝χH＝χ(H 0 +H a )；
step five: obtaining wave number domain magnetic bits through arbitrary sampling two-dimensional Fourier transformOne-dimensional ordinary differential equation
Space domain magnetic field magnetic potential U a And the magnetization M satisfies the equation as follows:
the above equation is subjected to a two-dimensional fourier transform.
The principle of the two-dimensional fourier transform of arbitrary samples here is as follows:
the two-dimensional fourier transform formula is:
in the formula kx 、k y Representing wavenumber, F (x, y) is a spatial domain function, F (k) x ,k y ) Representing the wavenumber spectrum.
The two-dimensional transformation is completed through two one-dimensional Fourier forward transformation, and the principle of the one-dimensional Fourier forward transformation is described first.
The one-dimensional fourier positive transforms can be expressed as:
wherein kx Representing wavenumber, F (x) is a spatial domain function, F (k) x ) Is wave number spectrum.
The forward transform integral in the above equation is discretized to obtain:
wherein N represents the number of units, e j Represents the j-th cell, where i is an imaginary number.
F (x) is interpolated by a quadratic function. When the quadratic interpolation shape function fitting is adopted in the units, three node coordinates in any unit are respectively set as x 1 、x 2 、x 3 ，x 2 Is the midpoint, satisfy x 1 +x 3 ＝2x 2 The intra-cell nodes are shown in fig. 6.
Each node has a value f (x 1 )、f(x 2 )、f(x 3 ) F (x) is expressed by using a quadratic function:
f(x)＝N 1 f(x 1 )+N 2 f(x 2 )+N 3 f(x 3 )；
wherein ,N1 、N 2 、N 3 Representing the function of the quadratic interpolation, respectively,
the above formula (3) can be written as:
order theFor intra-cell Fourier transform node coefficients, W 1 、W 2 、W 3 And respectively representing the Fourier transform coefficients corresponding to each node, wherein the above formula is abbreviated as:
when the wave number is k x When the value is not 0, N is 1 、N 2 、N 3 Substitution into W 1 、W 2 、W 3 In (2), the intra-cell fourier transform node coefficients can be obtained:
W 1 、W 2 、W 3 the integral kernel functions all includeIt is at [ x ] 1 ,x 3 ]The upper unit integral analysis solution is:
thus, k can be obtained x W when it is not 0 1 、W 2 、W 3 The semi-analytical solution is:
when the wave number is k x When the number of the organic light emitting diode is 0,W 1 0 、W 2 0 、W 3 0 the Fourier transform coefficients when the wave number is 0 are respectively represented, and the Fourier transform node coefficients under the zero wave number can be obtained by simple integration, and the Fourier transform node coefficients are:
and accumulating the analysis expressions of different units to obtain a final one-dimensional Fourier positive transformation result. It is easy to know that when the space domain and frequency domain subdivision is unchanged, the Fourier transform node coefficient W 1 、W 2 、W 3 and W1 0 、W 2 0 、W 3 0 The Fourier transform coefficients are calculated and stored in advance, so that repeated calculation can be reduced, the algorithm efficiency is improved, and the method is one of the advantages of the algorithm.
The two-dimensional fourier transform is performed after one-dimensional fourier transform is performed on x:
after that for F (k) x Y) performing a one-dimensional fourier transform in the y-direction:
the principle of the two one-dimensional fourier transforms is exactly the same as the process, and therefore will not be described in detail.
The spatial wave number mixed domain one-dimensional ordinary differential equation is obtained through Fourier transformation of arbitrary sampling, and the z direction is reserved as a spatial domain:
the above is the abnormal field magnetic position of wave number domainIn the one-dimensional ordinary differential equation satisfied, wherein +.>Representing abnormal field magnetic bits in wavenumber domain, +.>Representing the magnetization in the wavenumber domain->X-component, y-component and z-component, k x 、k y The wave numbers in the x and y directions are shown. The vertical direction is kept in a space domain, so that the vertical direction can be arbitrarily split.
Step six: applying one-dimensional function method to wave number domain magnetic positionPerforming iterative solution
In order to obtain the definite solution of the control equation (4), a proper boundary condition is required to be given, the schematic diagram of the boundary condition is shown in fig. 7, the vertical downward direction of the Z axis is taken as the forward direction under the Cartesian coordinate system, the horizontal ground of the calculation area is taken as the upper boundary Zmin, and the ground far enough away from the abnormal body is taken as the lower boundary Zmax. The upper and lower boundary conditions thereof satisfy:
upper boundary:the lower boundary: />
wherein ,
the boundary value problem of simultaneous spatial wave number mixed domain magnetic potential satisfaction:
and obtaining a variation problem equivalent to the boundary value problem by using a variation method:
in the Cartesian coordinate system shown in FIG. 7, the cells are split along the z-direction, and a quadratic interpolation function, i.e., magnetic bits, is used in each cellAnd changes twice within the cell.
The right end term of the normal differential equation (5) is satisfied and contains a background field and an abnormal field, and the abnormal field is unknown, so that iterative solution is adopted. The combination equation m=χh=χ (H 0 +H a ) And equation->It can be seen that the magnetization M is determined by the background field H 0 And an abnormal field H a The product of the sum and the magnetic susceptibility is determined, and H=H 0 +H a While the abnormal field H a Unknown, therefore, the first iteration hypothesis H a 0, the sum of the initial anomaly field and the background field is replaced by the background field, thereby dividing the one-dimensional partial differentialThe process is changed into a one-dimensional ordinary differential equation solution, a first abnormal field is obtained, and then the sum of the obtained abnormal field and background field is used as the total field of the right-end term to carry out the next solution.
Step seven: after each iteration, magnetic bits based on wave number domainObtaining the abnormal field magnetic field intensity of wave number domain +.>
Wave number domain magnetic potentialAbnormal magnetic bit in wave number domain>The following relationship is satisfied: />
Where i is an imaginary number.
Step eight: method for obtaining abnormal field intensity H of spatial domain by using arbitrary sampling two-dimensional Fourier inverse transformation method a
The application of arbitrary sampling two-dimensional Fourier inverse transformation is also a great innovation of the patent, and the arbitrary sampling can be ensured when the numerical simulation of the magnetic field of the patent is inversely transformed back to the spatial domain, thereby improving the precision and the efficiency.
The two-dimensional arbitrary sampling Fourier inverse transformation formula is:
in the formula kx 、k y Representing wavenumber, F (x, y) is a spatial domain function, F (k) x ,k y ) Representing the wavenumber spectrum. The reverse transformation formula is identical to the forward transformation formula in form,the principle is the same and will not be described again.
Step nine: h after the magnetic field intensity of the space domain is obtained a Iterative computation using tight operators
The tight operator is shown as follows:
where j represents the number of iterations. H j+1 For the total field of the current iteration,respectively a background field and an abnormal field of the iteration, H j Is the total field of the last iteration. Thus, the total field magnetic field strength of the spatial domain is obtained after calculation by the tight operator.
Step ten: judging iteration convergence condition
When the following formula is satisfied,
|H j+1 -H j |/H j+1 ＜10 -4 ；
the iteration stops. If not, returning to the step six. H j For the total field magnetic field intensity of the space domain obtained by the last calculation, H j+1 The total magnetic field intensity obtained by the calculation is obtained.
After the iteration stop condition is met, outputting the space domain magnetic field magnetic bit U obtained by current solving a And its corresponding H a ，U a Namely, the abnormal magnetic field magnetic position, H a Abnormal field strength.
Step eleven: solving the magnetic induction intensity B of the space domain a Ending the numerical simulation
From abnormal field magnetic induction intensity B a (in T) and the field strength H of the abnormal field a Can be used to determine the magnetic induction B a Further obtain B a Is defined by three components B ax ，B ay ，B az 。
B a ＝μH a ；
Wherein mu is the absolute permeability of the medium and the unit is H/m. The relationship of μ to χ satisfies the following equation:
μ＝μ 0 (1+χ)；
wherein μ0 For permeability in vacuum, mu 0 ＝4π×10 -7 H/m。
Example 3, please refer to fig. 8:
the accuracy and efficiency of the spatial wave number mixed domain three-dimensional strong magnetic field iteration method provided by the invention are checked.
The test computer is configured as i7-11800, the main frequency is 2.30GHz, the memory is 32GB, and the computer configuration is only used as an example and is not limited.
Designing a sphere model, wherein the background magnetic field strength is 50000nT, the magnetic inclination angle is 45 degrees, and the magnetic deflection angle is 5.9 degrees. Model calculation size is 500m×500m, range: x direction-250 m, y direction-250 m, z direction 0-500 m. The abnormal sphere model is centered at (0 m,250 m), the sphere radius is 100m, the sphere magnetic susceptibility is 1SI, and the model schematic diagram is shown in figure 1. The horizontal direction was split unevenly, and as shown in fig. 2, the horizontal direction was split with a minimum interval of 1m and a maximum interval of 32m, and the sampling interval was gradually increased from 1m to 32m outside the abnormal body. The z direction adopts an equally spaced split mode. And the number of the nodes in the three directions is 101. The sampling range of the wave number domain is-0.25, the sampling modes of the wave number domain are 101, the sampling modes are all sectional uniform sampling, the sampling modes are shown in fig. 4, the sampling modes are divided into three sections according to the spectral energy distribution, 31 nodes are respectively taken by 0-0.04 for example, 11 nodes are respectively taken by 0.04-0.1, 11 nodes are respectively taken by 0.1-0.25, the negative wave number and the positive wave number are symmetrical, and the total number is 101. After 6 iterations, the ground field value B ax 、B ay 、B az The relative root mean square error is 0.19%, 0.24% and 0.25% respectively compared with the strong magnetic sphere analytical solution. As shown in fig. 7, 0.8GB of memory is occupied, taking 3.28s.
Wherein (a) in FIG. 8 is B ax Numerical solution, (B) is B ax Analytical solution (c) is B ax Absolute error of numerical solution and analytic solution; (d) Is B ay Numerical solution, (e) is B ay Analytical solution (f) is B ay Absolute error of numerical solution and analytic solution; (g) Is B az Numerical solution, (h) is B az Analytical solution (i) is B az The absolute error of the numerical solution and the analytical solution.
The embodiment of the application also provides a three-dimensional strong magnetic field iteration system of the spatial wave number mixed domain, which comprises a memory, a processor and a computer program stored on the memory and capable of running on the processor, wherein the processor realizes the steps of any one of the three-dimensional strong magnetic field iteration methods of the spatial wave number mixed domain when executing the computer program.
The spatial wave number mixed domain three-dimensional strong magnetic field iteration system can realize all the embodiments of the spatial wave number mixed domain three-dimensional strong magnetic field iteration method, can achieve the same beneficial effects, and is not described in detail herein.
The foregoing describes in detail preferred embodiments of the present invention. It should be understood that numerous modifications and variations can be made in accordance with the concepts of the invention by one of ordinary skill in the art without undue burden. Therefore, all technical solutions which can be obtained by logic analysis, reasoning or limited experiments based on the prior art by the person skilled in the art according to the inventive concept shall be within the scope of protection defined by the claims.
Claims (9)
1. The method for iterating the three-dimensional strong magnetic field of the spatial wave number mixed domain is characterized by comprising the following steps of:
constructing a three-dimensional target model of a target area containing an abnormal body, splitting the three-dimensional target model to obtain a series of nodes, and carrying out susceptibility assignment on each node according to susceptibility distribution data to obtain the susceptibility of each node;
solving the magnetization intensity of the target area according to the relation between the magnetic susceptibility and the magnetization intensity;
determining a three-dimensional poisson equation based on the magnetization intensity, transforming the three-dimensional poisson equation into a wave number domain through arbitrarily sampled two-dimensional Fourier positive transformation, and solving to obtain abnormal field magnetic positions of the wave number domain of a target area;
solving the wave number domain abnormal field magnetic field intensity of the target area according to the relation between the wave number domain abnormal field magnetic field position and the wave number domain abnormal field magnetic field intensity, and performing two-dimensional Fourier inverse transformation of arbitrary sampling on the wave number domain abnormal field magnetic field intensity to obtain the space domain abnormal field magnetic field intensity of the target area;
carrying out iterative convergence judgment on the magnetic field intensity of the abnormal field in the space domain, solving the magnetic induction intensity in the space domain of the target area according to the relation between the magnetic field intensity of the abnormal field in the space domain and the magnetic induction intensity in the space domain when the iterative convergence condition is met, and recalculating the magnetization intensity in the target area when the iterative convergence condition is not met;
the iterative convergence judgment on the field intensity of the abnormal field in the spatial domain comprises the following steps:
carrying out iterative computation on the magnetic field intensity of the space domain through a tight operator to obtain the total magnetic field intensity of the space domain, wherein the tight operator is shown in the following formula:
where j represents the number of iterations, H j+1 Representing the spatial domain total field strength of the current iteration,representing the field strength of the background field of the spatial domain of the current iteration, < >>Represents the field intensity of the abnormal field in the spatial domain of the iteration, H j The total field intensity of the spatial domain of the previous iteration is represented, and χ represents the magnetic susceptibility;
and carrying out iterative convergence judgment on the calculated spatial domain total field magnetic field intensity, wherein the judgment condition of iterative convergence is shown as follows:
|H j+1 -H j |/H j+1 <10 -4 ；
wherein ,Hj+1 Representing the total field intensity of the spatial domain of the iteration, H j Representing the spatial domain total field magnetic field strength of the last iteration.
2. The spatial wave number mixed domain three-dimensional strong magnetic field iteration method according to claim 1, wherein the relationship between the spatial domain abnormal field magnetic field intensity and the spatial domain magnetic induction intensity is expressed as:
B a ＝μH a ；
wherein ,Ba For the magnetic induction intensity of the space domain, H a For the field intensity of the abnormal field in the spatial domain, mu is the absolute permeability of the medium, the unit is H/m, and the relation between mu and the magnetic susceptibility satisfies the following expression:
μ＝μ 0 (1+χ)；
wherein ,μ0 For permeability in vacuum, mu 0 ＝4π×10 -7 H/m, χ represents magnetic susceptibility.
3. The method of claim 1, wherein solving the magnetization of the target region according to the relationship between the magnetic susceptibility and the magnetization comprises:
determining the field intensity of a background field of a space domain of a target area according to the earth main magnetic field model;
determining the field intensity of the abnormal field in the spatial domain of the target area according to the field intensity generated by the abnormal body;
taking the sum of the background field magnetic field intensity of the space domain and the abnormal field magnetic field intensity of the space domain as the total field magnetic field intensity of the space domain, and solving to obtain the magnetization intensity of the target area according to the relation between the magnetic susceptibility and the magnetization intensity;
the relationship between susceptibility and magnetization is shown as follows:
M＝χH＝χ(H 0 +H a )；
wherein M represents magnetization intensity, χ represents magnetic susceptibility, H 0 Represents the field strength of background field in the space domain, H a The abnormal field magnetic field intensity in the space domain is represented, and H represents the total field magnetic field intensity in the space domain.
4. The method of iterative spatial wavenumber hybrid domain three-dimensional strong magnetic field according to claim 1, wherein the recalculating the magnetization of the target region comprises:
bringing the field intensity of the abnormal field in the space domain into a relation between the magnetic susceptibility and the magnetic intensity, and recalculating the magnetic intensity of the target area;
after bringing the field intensity of the abnormal field in the spatial domain into the relation between the magnetic susceptibility and the magnetization intensity, the relation is shown as follows:
M＝χH＝χ(H 0 +H a )；
wherein M represents magnetization intensity, χ represents magnetic susceptibility, H 0 Represents the field strength of background field in the space domain, H a The abnormal field magnetic field intensity in the space domain is represented, and H represents the total field magnetic field intensity in the space domain.
5. The method of claim 1, wherein the three-dimensional poisson equation is:
wherein ,Ua Represents the abnormal field magnetic potential of the magnetic field in the space domain, M represents the magnetization intensity, wherein, i, j and k are unit vectors in the x, y and z directions respectively;
the above-mentioned expansion is:
wherein ,Mx 、M y 、M z The components of the magnetization M in the x, y and z directions are respectively, the x, y and z are three mutually perpendicular directions,is the sign of the partial derivative.
6. The method according to claim 1, wherein the arbitrary sampled two-dimensional fourier transform formula is:
wherein ,kx The wave number, k, representing the x-direction y Representing wavenumbers in the y-direction, F (x, y) representing a spatial domain function, F (k) x ,k y ) Representing a wavenumber spectrum;
the arbitrary sampling two-dimensional Fourier forward transformation formula is obtained through two one-dimensional Fourier forward transformations, and the two one-dimensional Fourier forward transformations are respectively;
performing one-dimensional Fourier forward transformation in the x direction on f (x, y), wherein the transformation formula is as follows:
wherein x and y represent two mutually perpendicular directions; k (k) x The wavenumbers in the x-direction, F (x, y) represent the spatial domain function, F x (k x Y) is a wave number spectrum obtained by performing one-dimensional Fourier transform on f (x, y) in the x direction;
for F x (k x Y) performing one-dimensional Fourier forward transformation in the y direction, wherein the transformation formula is as follows:
wherein ,ky Represents the wavenumber in the y-direction, F (k) x ,k y ) To a wavenumber spectrum after a two-dimensional fourier transform of f (x, y).
7. The method for iterating the three-dimensional strong magnetic field in the spatial wave number mixed domain according to claim 1, wherein the transforming the three-dimensional poisson equation to the wave number domain through the arbitrary sampled two-dimensional fourier transform, solving to obtain the abnormal field magnetic bits in the wave number domain of the target region, comprises:
performing horizontal two-dimensional Fourier forward transformation on the three-dimensional poisson equation to obtain a one-dimensional ordinary differential equation, wherein the one-dimensional ordinary differential equation is shown in the following formula:
wherein ,representing abnormal field magnetic bits in wavenumber domain, +.>Representing the magnetization in the wavenumber domain->X-component, y-component and z-component, k x 、k y Wave numbers in x and y directions are respectively expressed, +.>For partial derivative sign, i is an imaginary number;
under the Cartesian coordinate system, taking the vertical downward direction of the Z axis as the forward direction, and calculating the upper boundary Z of the horizontal ground in the area min Taking the abnormal body far enough away from the ground as a lower boundary Z max The upper and lower boundary conditions thereof satisfy:
upper boundary:
the lower boundary:
wherein ,
and (3) combining a one-dimensional ordinary differential equation with an upper boundary and a lower boundary to obtain:
and obtaining a variation problem equivalent to the boundary value problem by using a variation method:
performing unit subdivision along the z direction, and obtaining the abnormal field magnetic potential of the wave number domain of the target region by adopting a quadratic interpolation function in each unitWherein δF represents the variation of F.
8. The iterative method for three-dimensional strong magnetic field in spatial wave number mixed domain according to claim 1, wherein the relationship between abnormal field magnetic potential in wave number domain and abnormal field magnetic field strength in wave number domain is:
wherein i is an imaginary number,is the abnormal field magnetic position in wave number domain，/>The abnormal field magnetic field intensity in the wave number domain;
the arbitrary sampled two-dimensional Fourier inverse transform expression is:
wherein ,kx 、k y Representing wavenumber, F (x, y) is a spatial domain function, F (k) x ,k y ) Representing the wavenumber spectrum.
9. A spatial wavenumber hybrid domain three-dimensional strong magnetic field iteration system comprising a memory, a processor and a computer program stored on the memory and executable on the processor, characterized in that the processor implements the steps of the method of any of the preceding claims 1 to 8 when executing the computer program.
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