Multi-scale finite element airborne electromagnetic three-dimensional forward modeling method based on deformation octree

Abstract

The present invention relates to the field of geophysical electromagnetic calculations, and specifically to a multi-scale finite element airborne electromagnetic three-dimensional forward modeling method based on a deformable octree. Design a complex model for airborne electromagnetic three-dimensional forward simulation, and use an octree grid to partition a uniform half-space model; calculate the coefficient matrix and right-hand side vector of the fine-scale octree grid to synthesize a large linear equation group; construct a multi-scale basis function inside the coarse-scale octree grid and synthesize an interpolation operator, perform dimensionality reduction processing on the large linear equation group to obtain a multi-scale finite element target equation group; use a direct solver to solve the multi-scale finite element linear equation group to obtain the electric field on the edge of the coarse-scale octree grid; and obtain the magnetic field value at the receiving point based on the electric field on the edge of the coarse-scale octree grid and the interpolation operator. The present invention uses the octree grid to effectively reduce the number of invalid grids, accurately simulate the airborne electromagnetic response of complex models, reduce the number of unknown solutions to the forward equations, and thus improve the computational efficiency of airborne electromagnetic three-dimensional numerical simulation.

Description

Technical Field

[0001] The invention relates to the field of geophysical electromagnetic calculation, and in particular to a multi-scale finite element aeroelectromagnetic three-dimensional forward modeling method based on a deformation octree. Background Art

[0002] The airborne electromagnetic method has many advantages, such as high resolution, no need for ground personnel to approach, and high detection efficiency. The airborne electromagnetic detection area is usually large, generating a large amount of detection data. This will greatly affect the calculation efficiency of 3D airborne electromagnetic inversion and reduce the practicality of 3D airborne electromagnetic inversion. The most time-consuming steps in the 3D airborne electromagnetic inversion calculation process are 3D airborne electromagnetic forward modeling and adjoint forward modeling. Therefore, it is currently necessary to develop an algorithm that can perform fast 3D airborne electromagnetic forward modeling on complex models.

[0003] At present, in the field of three-dimensional airborne electromagnetics, the main methods for realizing three-dimensional forward calculation are finite element method, finite difference method, finite volume method and integral equation method. Compared with other forward simulation methods, finite element method has great advantages in three-dimensional electromagnetic simulation of complex media, but finite element method must solve large linear equations, with low computational efficiency, and cannot be directly applied to three-dimensional fine inversion of large airborne electromagnetic data. The multi-scale finite element method uses two grids of different scales nested in each other to reduce the number of unknown solutions of large linear equations through interpolation operators. It has been widely used in many fields such as fluid mechanics and computational electromagnetics. In the field of geophysical electromagnetic computing technology, multi-scale finite element method is also used to simulate electromagnetic field calculations, but non-deformed hexahedral grids are used for calculation. This regular grid has poor ability to fit terrain and underground ore bodies, and more grids are needed to fit complex models, which increases the number of unknown solutions of large linear equations and affects the efficiency of forward calculation. Summary of the invention

[0004] In order to solve the problems existing in the prior art, the present invention provides a multi-scale finite element aero-electromagnetic three-dimensional forward modeling method based on deformable octree. The scheme includes: designing a complex model of aero-electromagnetic three-dimensional forward simulation, using octree grid to partition the uniform half-space model; calculating the coefficient matrix and right-hand vector of the fine-scale octree grid to synthesize a large linear equation group; constructing a multi-scale basis function inside the coarse-scale octree grid and synthesizing an interpolation operator, reducing the dimension of the large linear equation group to obtain a multi-scale finite element target equation group; using a direct solver to solve the multi-scale finite element linear equation group to obtain the electric field on the edge of the coarse-scale octree grid; according to the electric field on the edge of the coarse-scale octree grid and the interpolation operator, the magnetic field value at the receiving point is obtained. The present invention uses the octree grid to effectively reduce the number of invalid grids, accurately simulate the aero-electromagnetic response of complex models, reduce the number of unknown solutions of the forward equation, and thus improve the computational efficiency of aero-electromagnetic three-dimensional numerical simulation.

[0005] The present invention adopts the following technical scheme, which is a multi-scale finite element aero-electromagnetic three-dimensional forward modeling method based on a deformation octree, including:

[0006] Design complex models for airborne electromagnetic 3D forward simulation;

[0007] The uniform half-space model is divided by using an octree grid to obtain a fine-scale octree grid; the fine-scale octree grids are combined to obtain a coarse-scale octree grid;

[0008] Calculating the coefficient matrix and right end vector of the fine-scale octree grid by using coordinate transformation and Gaussian integral;

[0009] According to the position information of the edges of the fine-scale octree grid, the coefficient matrix and the right-hand vector of the fine-scale octree grid are expanded to obtain a large linear equation system of the multi-scale finite element algorithm;

[0010] Solve the local Maxwell equations within the coarse-scale octree grid and impose boundary conditions to obtain the multi-scale basis functions of the coarse-scale octree grid;

[0011] Expanding the multi-scale basis function according to the position information of the edges of the coarse-scale octree grid to obtain an interpolation operator matrix of a multi-scale finite element algorithm;

[0012] Using the interpolation operator matrix to perform dimensionality reduction processing on the large linear equation group to obtain a target equation group of multi-scale finite element;

[0013] A direct solver is used to solve the target equation group to obtain the tangential electric field at the edge of the coarse-scale octree grid; the tangential electric field at the edge of the fine-scale octree grid is obtained according to the interpolation operator; and the magnetic field value at the receiving point is obtained in combination with Faraday's law of electromagnetic induction.

[0014] Furthermore, the complex model includes: the correspondence between the undulating surface, the abnormal ore body, the type of the transceiver, the transmission frequency, the flight altitude, and the octree grid.

[0015] Furthermore, the uniform half-space model is partitioned using an octree grid to obtain a fine-scale octree grid, specifically:

[0016] The uniform half-space model is finely divided using octree grids, and the grid is expanded to obtain multiple fine-scale octree grids after division.

[0017] The coarse-scale octree grid is composed of a set number of fine-scale octree grids, and different coarse-scale octree grids do not overlap each other.

[0018] Furthermore, after obtaining the coarse-scale octree grid, the following steps are also included:

[0019] According to the terrain and underground ore body information near the measuring point, the coarse-scale octree grid and the fine-scale octree grid are deformed respectively. The steps are as follows:

[0020] Acquire terrain and underground ore body information near the transceiver according to the complex model;

[0021] The location of the transceiver is set as the center of the grid, and the node coordinates of the fine-scale octree grid and the coarse-scale octree grid are changed respectively to obtain the local grids corresponding to different transceivers.

[0022] Furthermore, the coefficient matrix and right-hand side vector of the fine-scale octree grid are expanded to obtain a large linear equation system for the multi-scale finite element algorithm, which is:

[0023] Set continuity constraints on the hanging edges, and expand the coefficient matrix of the fine-scale octree grid and the corresponding position of the necklace hanging edge on the right end;

[0024] The large linear equations of the multiscale finite element algorithm are synthesized based on the locations of the edges of the expanded fine-scale octree mesh.

[0025] Furthermore, the continuity constraint condition is specifically:

[0026]

[0027] Among them, E1 is the electric field value on the hanging edge, and E2 and E3 are the electric field values ​​on the edges on both sides of the hanging edge at the interface of grids of different levels.

[0028] Furthermore, the local Maxwell equations are solved within the coarse-scale octree grid and boundary conditions are imposed to obtain the multi-scale basis functions of the coarse-scale octree grid, including:

[0029] Substitute the multi-scale basis function into the electric field double curl equation, set the right side of the double curl equation to zero, and the expression is:

[0030]

[0031] in, represents the multi-scale basis function on the lth edge of the i-th coarse-scale octree grid, and the discretized multi-scale basis function is a vector of dimension nied, where nied is the number of edges of the fine-scale octree grid in the i-th coarse-scale octree grid; ω is the angular frequency, μ0 is the free space magnetic permeability, σ is the electrical conductivity, represents the i-th coarse-scale octree grid;

[0032] The multi-scale basis function on each edge of the coarse-scale octree grid is solved in turn to obtain the multi-scale basis function corresponding to the coarse-scale octree grid.

[0033] Furthermore, the boundary condition is:

[0034]

[0035] Where n represents the normal unit vector pointing outward on the outer surface of the coarse-scale octree mesh, represents the boundary condition on the lth edge of the i-th coarse-scale octree grid. The discretized boundary condition is 1 at the lth edge and decreases linearly to 0 along the outer edge of the coarse-scale octree grid. represents the multi-scale basis function on the lth edge of the i-th coarse-scale octree grid, Represents the outer surface of the i-th coarse-scale octree grid.

[0036] Furthermore, the multi-scale basis function is expanded to obtain an interpolation operator matrix of a multi-scale finite element algorithm, including:

[0037] Expand the corresponding columns of the coarse-scale octree grid hanging edges into two columns, and the value of each column is half of the corresponding column before expansion;

[0038] The interpolation operator matrix of the multi-scale basis function is obtained by combining the numbers of the edges of the expanded coarse-scale octree grid and the edges of the fine-scale octree grid.

[0039] Further, the interpolation operator matrix is ​​used to perform dimension reduction processing on the large linear equation system, including:

[0040] The large linear equation system is: K d E d =B d , the expression for dimensionality reduction is as follows:

[0041] K c =R T K d R

[0042] E d =RE c

[0043] B c =R T B d

[0044] The target equation group K of the multi-scale finite element after dimensionality reduction is obtained c E c =B c .

[0045] The beneficial effects of the present invention are as follows: according to the technical means proposed in the present invention, the three-dimensional airborne electromagnetic calculation space is discretized by using deformable octree grids, the number of invalid grids is reduced, and the fitting of the terrain and the underground ore body is achieved with fewer grids; multi-scale finite elements are used to solve the electric field double curl equation, and the electric field is approximated by using two sets of coarse and fine grids of different scales, which greatly reduces the number of unknown solutions of large linear equations in finite element forward modeling, and further reduces the amount of calculation required for forward modeling in terms of both the number of grids and the dimension of the linear equations, thereby improving the calculation efficiency of airborne electromagnetic three-dimensional fast forward modeling, and achieving fast and high-precision forward simulation calculation of electromagnetic responses of complex models. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0047] Figure 1 It is a schematic diagram of the process flow of a multi-scale finite element airborne electromagnetic three-dimensional forward modeling method based on a deformation octree according to an embodiment of the present invention;

[0048] Figure 2 A schematic diagram of a coarse-scale and fine-scale octree grid according to an embodiment of the present invention;

[0049] Figure 3 A schematic diagram of a process of generating a local grid according to an embodiment of the present invention;

[0050] Figure 4 A schematic diagram of an octree grid according to an embodiment of the present invention;

[0051] Figure 5 A schematic diagram of algorithm accuracy verification according to an embodiment of the present invention;

[0052] Figure 6 A schematic diagram showing a comparison of calculation time between a multi-scale finite element and a traditional finite element according to an embodiment of the present invention;

[0053] Figure 7 It is a schematic diagram of a comparative test of resistivity change between a multi-scale finite element and a traditional finite element according to an embodiment of the present invention. DETAILED DESCRIPTION

[0054] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0055] The schematic diagram of the process flow of the multi-scale finite element airborne electromagnetic three-dimensional forward modeling method based on deformation octree in an embodiment of the present invention is as follows: Figure 1 As shown, including:

[0056] Design complex models for airborne electromagnetic 3D forward simulation;

[0057] In a specific embodiment of the present invention, the complex model includes information on undulating surface, underground ore body information, transceiver type, transceiver distance, transmission frequency, flight altitude and other parameters used for airborne electromagnetic three-dimensional forward modeling.

[0058] The uniform half-space model is divided by using an octree grid to obtain a fine-scale octree grid; based on the fine-scale octree grid, a coarse-scale octree grid in the coarse-scale octree grid is obtained by combining;

[0059] The embodiment of the present invention utilizes the local grid technology to avoid fine subdivision of the entire measurement area. Before generating the local grid, the uniform half-space model needs to be finely subdivided. The densely gridded area of ​​the uniform half-space model does not need to be too large. Since the Dirichlet boundary condition is used as the constraint condition of the electric field double curl equation to be solved, the grid should be expanded outside the densely gridded area.

[0060] Assume that the fine-scale grid is Ω d , N d is the number of fine-scale octree grid cells, Represents the i-th fine-scale grid, and then a larger-scale octree grid is nested on the fine-scale grid as the coarse-scale octree grid in the multi-scale finite element. Assume that the coarse-scale grid is Ω c , The coarse-scale octree grid can be represented as Where N c is the number of coarse-scale octree grid cells, is the i-th coarse-scale octree grid unit; Figure 2 As shown, it is a schematic diagram of a fine-scale octree grid in a coarse-scale octree grid. The thicker line segments are the edges of the coarse-scale octree grid, and different fine-scale octree grids represent fine-scale octree grids with different electrical parameters.

[0061] It should be noted that the coarse-scale and fine-scale octree grids here are both octree grids, and it is not required that the fine-scale octree grids within the same coarse-scale octree grid have the same level. When dividing the calculation area, the relationship between the coarse-scale and fine-scale octree grids can be adjusted as needed to improve the grid quality and reduce the calculation time.

[0062] In a specific embodiment of the present invention, after generating two sets of coarse and fine grids, in order to generate local grids at different transceiver devices, the embodiment of the present invention adopts a method of deforming the grids and assigning different electrical parameters so that the two sets of grids can accurately simulate the undulating terrain and the underground ore body; Figure 3 As shown, a schematic diagram of a local grid is shown, taking a fine-scale grid as an example. In an embodiment of the present invention, the two sets of grids are deformed according to the undulating terrain information around the measuring point and the underground ore body information, so that the grids fit the terrain and the ore body better, and different conductivities are given to grids at different positions, so as to achieve the goal of generating a local grid. The advantage of this method is that there is no need to finely divide the entire airborne electromagnetic survey area. When calculating the airborne electromagnetic response at a certain position, only the grid needs to be changed to realize the three-dimensional forward simulation calculation of the airborne electromagnetic, which greatly reduces the number of grids and greatly improves the calculation efficiency of the three-dimensional airborne electromagnetic forward calculation.

[0063] Calculating the coefficient matrix and right end vector of the fine-scale octree grid by using coordinate transformation and Gaussian integral;

[0064] For the vector finite element method, the deformed octree grid does not have an analytical expression for the coefficient matrix, so it is necessary to introduce coordinate transformation and Gaussian integral to obtain the coefficient matrix and right-end vector of the fine-scale octree grid, and transform the deformed fine-scale grid into a regular hexahedral grid through coordinate transformation, and then use Gaussian integral to obtain the coefficient matrix and right-end vector of the fine-scale octree grid. Specifically: the embodiment of the present invention uses the finite element method to obtain the coefficient matrix and right-end vector of each fine-scale octree grid unit, starting from the electric field double curl equation:

[0065]

[0066] Where E s For a field, E p is the secondary field, ω is the angular frequency, μ0 is the free space magnetic permeability, σ represents the electrical conductivity, σ a =σ-σ0, σ0 is the background conductivity, σ a is the abnormal conductivity; in order to ensure the uniqueness of the solution, this embodiment adopts the following Dirichlet boundary condition:

[0067] E s =0∈Γ d

[0068] When using the finite element method for forward simulation, the calculation area needs to be divided and the Galerkin method is applied in each unit to discretize the electric field to the edges of the grid to obtain the integral form of the double curl equation:

[0069]

[0070] Where N is the vector finite element basis function, represents the i-th fine-scale octree grid, κ 2 =iωμ0σ, Using coordinate transformation and Gaussian integral, we get the linear equations of the fine-scale octree grid:

[0071]

[0072] in is the cell matrix of the i-th fine-scale octree grid, with a matrix dimension of 12×12. and They are the tangential electric field and the right end vector at the edge of the fine-scale octree grid, and their dimensions are both 1×12.

[0073] The coefficient matrix and right-hand side vector of the fine-scale octree grid are expanded according to the position information of the edges of the fine-scale octree grid, and a large linear equation system of the multi-scale finite element algorithm is obtained;

[0074] The octree grid realizes fine division of the calculation area by evenly dividing the hexahedral grid into eight parts. Each division will increase the level of the grid. When the levels of adjacent grids are different, the problem of hanging edges of the grid needs to be solved. In order to ensure the continuity of the electric field, the embodiment of the present invention imposes the following constraint conditions on the hanging edges of the octree grid:

[0075]

[0076] Among them, E1 is the electric field value on the hanging edge, E2 and E3 are the electric field values ​​on the edges on both sides of the hanging edge at the interface of grids of different levels. Since the above constraints are added, when expanding the coefficient matrix of the grid, it is only necessary to change the row and column where the hanging edge is located, and change it to two rows with each row value being half of the original, and the corresponding columns are processed in the same way. The corresponding edges of the expanded rows and columns are changed to the edges on both sides of the hanging edge. After expanding all grid coefficient matrices, the coefficient matrices and the values ​​in the right-hand vector are placed in the corresponding positions to synthesize a large linear equation system.

[0077] In a specific embodiment of the present invention, Figure 4FIG. 1 is a schematic diagram of an octree mesh with different adjacent mesh levels. When performing octree meshing, the embodiment of the present invention sets the level difference of adjacent meshes to be no greater than 1. When the adjacent mesh levels are different, hanging nodes and hanging edges will be generated. For the vector finite element method, if no constraints are imposed on the hanging edges, the continuity of the electric field on the outer surface of the mesh will be destroyed. Therefore, the embodiment of the present invention imposes the following constraints on the hanging edges:

[0078]

[0079] E5=E14

[0080] E9=E16

[0081] Among them, E7 and E10 are Figure 4 The electric field on the corresponding edges of the medium level 2 grid, edges 7 and 10 are hanging edges on the interface, E5 and E9 are Figure 4 The electric field on the corresponding edges of the level 2 grid, edge 5 and edge 9 are the extended edges on the interface; E13, E14, E15 and E16 correspond to the tangential electric field at the edges of the level 1 grid on the interface. After the constraints are applied, the electric field on the hanging edge and the extended edge on the interface is replaced by the electric field of the level 1 grid on the interface. Therefore, the electric field in the level 2 grid is discretized into the following format:

[0082]

[0083] Where E is Figure 4 Electric field in the medium level 2 grid, N j is a vector function on the jth edge. Since the format of the vector function has changed, the embodiment of the present invention needs to expand the coefficient matrix and the right end vector of the fine-scale octree grid. Suppose the coefficient matrix generated by the grid is:

[0084]

[0085] According to the change of the vector function, the matrix is ​​expanded to:

[0086] After the right-hand side vector is also expanded in the same way, all coefficient matrices can be combined with the right-hand side vector to form a large linear equation system.

[0087] It should be noted that each row and column of the unexpanded coefficient matrix corresponds to the corresponding edges of the fine-scale octree grid, but since the hanging edges and extended edges are replaced by other edges on the interface, the corresponding rows and columns also represent the edge numbers after replacement. By synthesizing according to the edge numbers, the large linear equations of the finite element method can be obtained.

[0088] Solve the local Maxwell equations within the coarse-scale octree grid and impose boundary conditions to obtain the multi-scale basis functions of the coarse-scale octree grid;

[0089] The local double-curl equations are solved for each coarse-scale octree grid to obtain the multi-scale basis function. The local double-curl equations are as follows:

[0090]

[0091] in is the multi-scale basis function on the lth edge of the i-th coarse-scale octree grid, and the discretized multi-scale basis function is a vector of dimension nied, where nied is the number of fine-scale octree mesh edges in the i-th coarse-scale octree mesh.

[0092] Since the right-hand side of the double curl equation is 0, in order to make the solution of the multi-scale basis function unique and ensure the continuity of the electric field between adjacent coarse-scale octree grids, the embodiment of the present invention imposes the following boundary conditions on the multi-scale basis function:

[0093]

[0094] Where n represents the normal unit vector pointing outward on the outer surface of the coarse-scale octree mesh, represents the constraint condition on the lth edge of the i-th coarse-scale octree grid. The discretized constraint condition has a value of 1 in the lth tangential direction and decreases linearly to 0 along the outer surface of the coarse-scale octree grid. Represents the outer surface of the i-th coarse-scale octree grid; after solving the multi-scale basis functions on 12 edges in combination with the boundary conditions, the multi-scale basis function R of the i-th coarse-scale octree grid can be obtained. i ,

[0095] Expanding the multi-scale basis function according to the position information of the edges of the coarse-scale octree grid in the octree grid to obtain an interpolation operator matrix of the multi-scale finite element algorithm;

[0096] After obtaining the multi-scale basis function of each coarse-scale octree grid, the embodiment of the present invention processes the multi-scale basis function according to the octree information of the grid; the coarse-scale octree grid and the fine-scale octree grid in this embodiment are both octree grids, so there are also situations where the levels of adjacent grids are different in the coarse-scale octree grid, and the multi-scale basis function is essentially an interpolation operator from the field value on the edge of the coarse-scale octree grid to the field value on the edge of the fine-scale octree grid, and when the levels of adjacent grids are different, the edge of the coarse-scale octree grid may be a hanging edge or an extended edge, and in this case, it is necessary to interpolate the multi-scale basis function R iIt needs to be expanded before it can be used to synthesize a multi-scale interpolation operator matrix.

[0097] The embodiment of the present invention still uses Figure 4 For example, if the grid of level 1 and the grid of level 2 are coarse-scale octree grids in the multi-scale finite element method, and the same constraints are imposed on the electric field on the edges, the following formula is available for the grid of level 2:

[0098]

[0099] in is the tangential electric field value at the edge of the fine-scale octree grid in the coarse-scale octree grid. The multi-scale basis function is expanded according to the above formula, and the following formula is obtained:

[0100]

[0101] to:

[0102]

[0103] In the multi-scale finite element calculation process, the coarse-scale octree grid is also an octree grid. When the levels of adjacent grids are different, the hanging edge problem needs to be solved. In order to ensure the continuity of the electric field, the electric field on the coarse-scale octree grid is subject to the same boundary conditions as the fine-scale octree grid. Therefore, the multi-scale basis function needs to be expanded. The dimension of the multi-scale basis function of the i-th coarse-scale octree grid is nied×12. The column corresponding to the hanging edge is expanded into two columns, and the value of each column is half of the original value to complete the expansion. After the multi-scale basis function of all coarse-scale octree grids is expanded, the multi-scale basis function interpolation operator R can be obtained according to the number of the coarse-scale octree grid edges and the fine-scale octree grid edges in the grid. The dimension of R is nded×nced. nded is the number of fine-scale octree grid edges, and nced is the number of coarse-scale octree grid edges.

[0104] It should be noted that the difference from synthesizing the overall matrix is ​​that when synthesizing the interpolation operator, when multiple coefficients are placed in the same position, the coefficients are not added but the average of the coefficients is calculated to obtain the correct interpolation operator coefficients.

[0105] Using the interpolation operator matrix to perform dimensionality reduction processing on the large linear equation group to obtain a target equation group of multi-scale finite element;

[0106] In the embodiment of the present invention, the large linear equation system K d E d =B d Perform dimensionality reduction processing, the specific formula is as follows:

[0107] Kc =R T K d R

[0108] E d =RE c

[0109] B c =R T B d

[0110] After dimensionality reduction, the multi-scale finite element linear equation system K is obtained. c E c =B c , the number of unknown solutions of the system of equations is nced, and the number of unknown solutions of the large linear system of equations is nded. After the above processing, the number of unknown solutions of the linear system of equations is greatly reduced.

[0111] The target equations are solved by a direct solver to obtain the tangential electric field at the edge of the coarse-scale octree grid; the tangential electric field at the edge of the fine-scale octree grid is obtained according to the interpolation operator; and the magnetic field value at the receiving point is obtained in combination with Faraday's law of electromagnetic induction.

[0112] The embodiment of the present invention adopts the pardiso direct solver to obtain the multi-scale finite element linear equation group, obtains the tangential electric field at the edge of the coarse-scale octree grid, and then uses the interpolation operator to obtain the tangential electric field at the edge of the fine-scale octree grid; the magnetic field value at the transceiver is obtained according to Faraday's law of electromagnetic induction, and completes the multi-scale finite element aero-electromagnetic three-dimensional fast forward modeling based on the deformable octree.

[0113] In another specific embodiment of the present invention, Figure 5 A schematic diagram of algorithm accuracy verification is shown in the figure to verify that this method can obtain correct three-dimensional airborne electromagnetic forward modeling results, specifically:

[0114] The calculation accuracy of the present invention is verified by calculating the uniform half-space model. The height of the transceiver is set to 30m from the ground surface, the transceiver is a horizontal coplanar transceiver, the transceiver distance is 10m, the transmission frequency is between 100Hz and 100000Hz, and the air conductivity is 10 -7 S / m, underground conductivity is 10 -2 S / m; this model has a one-dimensional semi-analytical solution, where Figure 5 (a) Relative error of the finite element forward modeling results using fine-scale octree grids; Figure 5 (b) is the relative error of the results of multi-scale finite element forward modeling; Figure 5(c) is the relative error of the finite element forward calculation results using the coarse-scale octree grid; the above relative errors are obtained by comparing with the one-dimensional semi-analytical solution; it can be seen from the figure that the finite element calculation results obtained using the fine-scale octree grid have a smaller error than the multi-scale finite element calculation results, and the maximum error does not exceed 5%, while the finite element calculation results obtained using the coarse-scale octree grid have a larger relative error, with the maximum error being around 14%. It can be seen that the calculation error of the multi-scale finite element algorithm is similar to the calculation error of the fine-scale octree grid finite element algorithm, and has higher calculation accuracy.

[0115] Further Figure 6 The table shows a comparison of the calculation time of a multi-scale finite element and a traditional finite element given in this embodiment. This embodiment uses the same computing resources on the same computer to perform three forward modeling calculations, namely, fine-scale octree grid finite element forward modeling, multi-scale finite element forward modeling, and coarse-scale octree grid finite element forward modeling. Both sets of grids are used for the above-mentioned accuracy verification. Figure 6 Shown Figure 5 The time required for the three forward calculations is as follows. According to the information in the figure, the number of fine-scale octree grids is 216208, and the number of coarse-scale octree grids is 19900. The corresponding number of grid edges, that is, the number of unknown solutions of the finite element method, are 700104 and 70220, respectively. The multi-scale finite element method uses both coarse and fine-scale octree grids for calculation, but the number of unknown solutions of the linear equations finally solved is 70220, the number of edges of the coarse-scale octree grid.

[0116] According to the information in the table, when the three methods are on the same computer and use the same computing resources, the time required for the coarse-scale octree mesh finite element method to calculate the forward response of a measuring point at one frequency is 15.21s, and the time required for the multi-scale finite element algorithm and the fine-scale octree mesh finite element algorithm are 152.53s and 1195.45s respectively; among them, the multi-scale finite element algorithm needs to calculate the multi-scale basis functions of each coarse-scale octree mesh, so the longest equation assembly time is 144.61s, and the equation solution time is similar to the coarse-scale octree mesh finite element method time of 7.92s. The forward modeling time of the multi-scale finite element is 7.83 times that of the fine-scale octree mesh finite element, and the multi-scale finite element algorithm is relatively slow compared to the coarse-scale octree mesh, but according to Figure 5 As shown in the figure, the calculation accuracy of the multi-scale finite element algorithm is higher than that of the coarse-scale octree grid finite element algorithm. Under the condition of meeting the calculation accuracy, it successfully reduces the number of unknown solutions of the linear equations of the finite element algorithm and reduces the calculation time, thus achieving the purpose of three-dimensional fast aeroelectromagnetic forward modeling.

[0117] Figure 7The present invention is a schematic diagram of a multi-scale finite element and a traditional finite element test with resistivity changes in an embodiment of the present invention; in order to demonstrate that the multi-scale finite element method can capture resistivity changes in coarse-scale octree grids, the present invention further designs a comparative experiment with resistivity changes in coarse-scale octree grids. A real three-dimensional airborne electromagnetic forward simulation model usually involves complex terrain and multiple underground ore bodies. Underground resistivity changes may be frequent. If these resistivity change areas are finely divided, a large number of grids will be generated to affect the calculation efficiency. However, if the forward model is divided using a larger-scale grid, these changes will be ignored. However, these resistivity changes may have a huge impact on the calculation results of the forward simulation. In the face of this problem, the multi-scale finite element method solves the local electric field double curl equation, and the multi-scale basis function can capture the resistivity changes inside the coarse-scale octree grid. Therefore, the multi-scale finite element method can use a fine-scale octree grid to finely divide the underground area, and use the coarse-scale octree grid to interpolate it to reduce the amount of calculation required for the forward calculation. This embodiment gives the following experimental examples for verification:

[0118] First, this embodiment finely divides the half-space model and sets the air conductivity to 10 -7 S / m, underground conductivity is 10 -2 S / m, and randomly insert grids with a conductivity of 0.009S / m into the underground grid. Figure 7 (a) is a schematic diagram of a fine-scale octree grid, after which a larger coarse-scale octree grid is nested on the basis of the fine-scale octree grid; Figure 7 (b) is a schematic diagram of a coarse-scale octree grid, and the conductivity of each coarse-scale octree grid is the arithmetic mean of the conductivity of the internal fine-scale octree grid; then, this embodiment performs a three-dimensional airborne electromagnetic forward calculation on the above model, and the transceiver is located at the center of the grid and 30m above the ground. The transceiver is an HCP transceiver and a horizontal coplanar transceiver. The transceiver distance is 10m, and the transmission frequency is between 100Hz and 100000Hz; since there is no analytical solution for the forward modeling result of the model, the embodiment of the present invention uses the calculation result of the fine-scale octree grid finite element as the standard solution, and uses the multi-scale finite element algorithm and the coarse-scale octree grid finite element algorithm to calculate the model to obtain the forward calculation result, and compares it with the fine-scale octree grid finite element calculation result, showing the difference between the calculation results of the two methods and the fine-scale octree grid finite element calculation results, and comparing the two methods' ability to capture conductivity changes; as shown in FIG. Figure 7(c) shows the relative error between the coarse-scale octree grid finite element and the multi-scale finite element calculation results relative to the fine-scale octree grid finite element calculation results. According to the information in the figure, the use of coarse-scale octree grid for forward simulation of the above model will ignore the details of the resistivity change, resulting in a large difference between the calculation results and the fine-scale octree grid finite element calculation results. The calculation error increases with the emission frequency and the maximum error is about 15%. The multi-scale finite element method uses multi-scale basis functions to capture the detailed resistivity changes, which improves the occurrence of this situation to a certain extent. The maximum error is about 5%, and the calculation results are more similar to the fine-scale octree grid finite element calculation results.

[0119] In summary, according to the technical means provided by the present invention, the multi-scale finite element method is combined with the deformation octree for fast forward simulation calculation of three-dimensional airborne electromagnetics. According to the characteristics of the deformation octree grid, fewer grids are used to achieve accurate fitting of complex terrain and underground ore bodies. The multi-scale finite element method is used to greatly reduce the number of unknown solutions of the large linear equations generated by the forward equation. Under the premise of ensuring the calculation accuracy, the computational efficiency of the three-dimensional airborne electromagnetic forward simulation is improved in terms of the number of grids and the order of the equation, and the three-dimensional airborne electromagnetic fast forward simulation of complex models is realized.

[0120] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A multi-scale finite element aero-electromagnetic three-dimensional forward modeling method based on deformation octree, characterized in that: include: Design complex models for airborne electromagnetic 3D forward simulation; Octree grid is used to divide the uniform half-space model and obtain a fine-scale octree grid; Combining the fine-scale octree grids to obtain a coarse-scale octree grid; Calculating the coefficient matrix and right end vector of the fine-scale octree grid by using coordinate transformation and Gaussian integral; The coefficient matrix and right-hand side vector of the fine-scale octree grid are expanded according to the position information of the edges of the fine-scale octree grid, and a large linear equation system of the multi-scale finite element algorithm is obtained; Solve the local Maxwell equations within the coarse-scale octree grid and impose boundary conditions to obtain the multi-scale basis functions of the coarse-scale octree grid; Expanding the multi-scale basis function according to the position information of the edges of the coarse-scale octree grid to obtain an interpolation operator matrix of the multi-scale finite element algorithm; Using the interpolation operator matrix to perform dimensionality reduction processing on the large linear equation group to obtain a target equation group of multi-scale finite element; The target equations are solved by a direct solver to obtain the tangential electric field at the edges of the coarse-scale octree grid; Obtaining the tangential electric field at the edge of the fine-scale octree grid according to the interpolation operator; Combined with Faraday's law of electromagnetic induction, the magnetic field value at the receiving point is obtained.

2. The multi-scale finite element aero-electromagnetic three-dimensional forward modeling method based on deformation octree according to claim 1 is characterized by: The complex model includes: undulating surface, abnormal ore body, type of transceiver, transmission frequency, flight altitude, and the corresponding relationship between octree grids.

3. The multi-scale finite element aero-electromagnetic three-dimensional forward modeling method based on deformation octree according to claim 1 is characterized by: The uniform half-space model is divided by using the octree grid to obtain a fine-scale octree grid, which is: The uniform half-space model is finely divided using octree grids, and the grid is expanded to obtain multiple fine-scale octree grids after division. The coarse-scale octree grid is composed of a set number of fine-scale octree grids, and different coarse-scale octree grids do not overlap each other.

4. The multi-scale finite element aero-electromagnetic three-dimensional forward modeling method based on deformation octree according to claim 1 is characterized by: After obtaining the coarse-scale octree grid, it also includes: According to the terrain and underground ore body information near the measuring point, the coarse-scale octree grid and the fine-scale octree grid are deformed respectively. The steps are as follows: Acquire terrain and underground ore body information near the transceiver according to the complex model; The location of the transceiver is set as the center of the grid, and the node coordinates of the fine-scale octree grid and the coarse-scale octree grid are changed respectively to obtain the local grids corresponding to different transceivers.

5. The multi-scale finite element aero-electromagnetic three-dimensional forward modeling method based on deformation octree according to claim 1, characterized in that: The coefficient matrix and right-hand side vector of the fine-scale octree grid are expanded to obtain the large linear equation system of the multi-scale finite element algorithm, which is: Set continuity constraints on the hanging edges, and expand the coefficient matrix of the fine-scale octree grid and the corresponding position of the necklace hanging edge on the right end; The large linear equations of the multiscale finite element algorithm are synthesized based on the locations of the edges of the expanded fine-scale octree mesh.

6. The multi-scale finite element aero-electromagnetic three-dimensional forward modeling method based on deformation octree according to claim 5 is characterized by: The continuity constraint conditions are specifically: Among them, E1 is the electric field value on the hanging edge, and E2 and E3 are the electric field values ​​on the edges on both sides of the hanging edge at the interface of grids of different levels.

7. The multi-scale finite element airborne electromagnetic three-dimensional forward modeling method based on deformation octree according to claim 1 is characterized by: Solving the local Maxwell equations within the coarse-scale octree grid and applying boundary conditions, we obtain the multi-scale basis functions of the coarse-scale octree grid, including: Substitute the multi-scale basis function into the electric field double curl equation, set the right side of the double curl equation to zero, and the expression is: in, represents the multi-scale basis function on the lth edge of the i-th coarse-scale octree grid, and the discretized multi-scale basis function is a vector of dimension nied, where nied is the number of edges of the fine-scale octree grid in the i-th coarse-scale octree grid; ω is the angular frequency, μ0 is the free space magnetic permeability, σ is the electrical conductivity, represents the i-th coarse-scale octree grid; The multi-scale basis function on each edge of the coarse-scale octree grid is solved in turn to obtain the multi-scale basis function corresponding to the coarse-scale octree grid.

8. The multi-scale finite element aero-electromagnetic three-dimensional forward modeling method based on deformation octree according to claim 1 is characterized by: The boundary conditions are: Where n represents the normal unit vector pointing outward on the outer surface of the coarse-scale octree mesh, represents the boundary condition on the lth edge of the i-th coarse-scale octree grid. The discretized boundary condition is 1 at the lth edge and decreases linearly to 0 along the outer edge of the coarse-scale octree grid. represents the multi-scale basis function on the lth edge of the i-th coarse-scale octree grid, Represents the outer surface of the i-th coarse-scale octree grid.

9. The multi-scale finite element airborne electromagnetic three-dimensional forward modeling method based on deformation octree according to claim 1, characterized in that: The multi-scale basis function is expanded to obtain an interpolation operator matrix of a multi-scale finite element algorithm, including: Expand the corresponding columns of the coarse-scale octree grid hanging edges into two columns, and the value of each column is half of the corresponding column before expansion; The interpolation operator matrix of the multi-scale basis function is obtained by combining the numbers of the edges of the expanded coarse-scale octree grid and the edges of the fine-scale octree grid.

10. The multi-scale finite element aero-electromagnetic three-dimensional forward modeling method based on deformation octree according to claim 1, characterized in that: The interpolation operator matrix is ​​used to perform dimension reduction processing on the large linear equation system, including: The large linear equation system is: K d E d =B d , the expression for dimensionality reduction is as follows: K c =R T K d R AND d =RE c B c =R T B d The target equation group K of the multi-scale finite element after dimensionality reduction is obtained c E c =B c .

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