DGTD electromagnetic field acquisition method and device based on high-order central difference
The DGTD electromagnetic field acquisition method using high-order central difference solves the problems of low computational accuracy and efficiency in existing technologies, achieving higher computational accuracy and efficiency, and enhancing numerical stability.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- BEIJING UNIV OF TECH
- Filing Date
- 2025-02-14
- Publication Date
- 2026-06-05
AI Technical Summary
Existing technologies using the DGTD method for electromagnetic field acquisition suffer from insufficient accuracy, low computational efficiency, high computational complexity, high memory requirements, high implementation difficulty, poor numerical stability, and challenges in parallel computing.
The DGTD electromagnetic field acquisition method using high-order central difference is adopted. By spatially discretizing the target stratigraphic region of the time-domain discontinuous Galerkin method, a polyhedral mesh is established to obtain the nodal scalar basis functions and edge vector basis functions. Weighted testing and high-order central difference are performed to correct the absorbing boundary conditions, obtain the stability conditions, and iteratively update them.
This improves the computational accuracy and efficiency of the DGTD method, reduces the amount of computation, decreases the number of iterations, and enhances numerical stability.
Smart Images
Figure CN120085378B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of geophysical exploration technology, and in particular to a DGTD electromagnetic field acquisition method and apparatus based on high-order central difference. Background Technology
[0002] Transient electromagnetic logging (TEM) refers to the application of transient electromagnetic methods in well logging, a geophysical exploration method. TEM logs infer the geological structure and rock properties near the wellbore by sending transient electromagnetic fields into the well and observing the secondary eddy field around the wellbore. This method is particularly suitable for detecting geological anomalies such as aquifers, faults, and fractures around the wellbore, providing important data for oil and gas exploration and hydrogeological surveys.
[0003] Electromagnetic field acquisition usually refers to the process of measuring, detecting, or capturing electromagnetic fields to obtain relevant information such as electric field strength, magnetic field strength, frequency and amplitude of electromagnetic waves.
[0004] In researching related technologies, the Discontinuous Galerkin Time-domain method (DGTD) is used to acquire electromagnetic fields. However, when using this method, these technologies typically face challenges related to computational complexity, memory requirements, implementation difficulty, numerical stability, parallel computing, and precision control, resulting in insufficient accuracy and low computational efficiency.
[0005] Therefore, how to improve the accuracy and efficiency of electromagnetic field acquisition using DGTD has become a technical problem that the industry urgently needs to solve. Summary of the Invention
[0006] This invention provides a DGTD electromagnetic field acquisition method and apparatus based on high-order central difference, which solves the defects of insufficient accuracy of calculation results and short time step required by the DGTD method in related technologies, resulting in low computational efficiency, and greatly improves the accuracy and computational efficiency of the DGTD method.
[0007] This invention provides a DGTD electromagnetic field acquisition method based on higher-order central difference, comprising:
[0008] Spatial discretization is performed on the target stratigraphic region of the time-domain discontinuous Galerkin method DGTD electromagnetic field model to obtain a polyhedral mesh for each region.
[0009] The nodal scalar basis functions and edge vector basis functions of the mesh elements are obtained from the polyhedral mesh.
[0010] Weak solutions to the basic equations are obtained by weighted testing of the edge vector basis functions based on the first-order Maxwell curl vector equations; the weak solutions to the basic equations are in the semi-discrete form of the electromagnetic field time-domain stepping algorithm.
[0011] Based on the semi-discrete form, a higher-order central difference is performed to obtain a time-discrete form based on the higher-order central difference; the time-discrete form is the electromagnetic field time-domain step formula; the order in the higher-order central difference is greater than or equal to four.
[0012] Based on the electromagnetic field time-domain step formula, the discrete electromagnetic energy after absorption boundary condition correction is obtained.
[0013] Based on the discrete electromagnetic energy corrected for the absorption boundary conditions, the stability conditions are obtained.
[0014] Based on the stability condition and the electromagnetic field time-domain step formula, the electric field strength and magnetic field strength on each edge of the polyhedral mesh are iteratively updated.
[0015] In some embodiments, before performing a higher-order central difference based on the semi-discrete form to obtain a time-discrete form based on the higher-order central difference, the process includes:
[0016] Based on the weak solution of the fundamental equation, and by expanding the electric and magnetic field strengths according to the edge vector basis function set, a time-discrete matrix equation is established.
[0017] In some embodiments, obtaining the time discrete form based on higher-order central difference includes:
[0018] The derivatives of the electric and magnetic fields with respect to time are approximated using the higher-order central difference method. The approximate values are then substituted into the time discrete matrix equation, and electric field intensity terms at multiple different iteration times are introduced to obtain the electromagnetic field time-domain step formula based on the higher-order central difference.
[0019] In some embodiments, obtaining the discrete electromagnetic energy after absorbing boundary condition correction includes:
[0020] By substituting the absorption boundary conditions into the expression for discrete electromagnetic energy and using Green's function to deduce it in reverse, and then performing correction processing, we obtain the discrete electromagnetic energy after the absorption boundary conditions are corrected.
[0021] In some embodiments, obtaining the stability condition includes:
[0022] After performing a vector identity transformation on the discrete electromagnetic energy corrected for the absorption boundary conditions, the inequality is obtained by applying the inequality relations containing the second norm.
[0023] Based on the inequality, we perform identity transformations and simplifications to obtain the stability conditions.
[0024] This invention provides a DGTD electromagnetic field acquisition device based on high-order central difference, comprising:
[0025] The spatial discretization module is used to spatially discretize the target stratigraphic region of the time-domain discontinuous Galerkin method DGTD electromagnetic field model to obtain a polyhedral mesh for each region.
[0026] The function determination module is used to obtain the nodal scalar basis functions and edge vector basis functions of the mesh elements based on the polyhedral mesh.
[0027] The weighted testing module is used to perform weighted testing on the edge vector basis functions based on the first-order Maxwell curl vector equations to obtain weak solutions to the basic equations; the weak solutions to the basic equations are semi-discrete forms of the electromagnetic field time-domain stepping algorithm.
[0028] The difference determination module is used to perform higher-order central difference based on the semi-discrete form to obtain a time-discrete form based on the higher-order central difference; the time-discrete form is an electromagnetic field time-domain step formula; the order in the higher-order central difference is greater than or equal to four.
[0029] The electromagnetic energy determination module is used to obtain the discrete electromagnetic energy after absorption boundary condition correction according to the electromagnetic field time-domain step formula.
[0030] The stability determination module is used to obtain the stability conditions based on the discrete electromagnetic energy after correction of the absorption boundary conditions.
[0031] The field value update module is used to iteratively update the electric field intensity and magnetic field intensity on each edge of the polyhedral mesh according to the stability condition and the electromagnetic field time-domain step formula.
[0032] In some embodiments, the difference determination module is used for:
[0033] Based on the weak solution of the fundamental equation, and by expanding the electric and magnetic field strengths according to the edge vector basis function set, a time-discrete matrix equation is established.
[0034] The present invention provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the DGTD electromagnetic field acquisition method based on higher-order central difference.
[0035] The present invention provides a non-transitory computer-readable storage medium storing a computer program thereon, wherein the computer program, when executed by a processor, implements the DGTD electromagnetic field acquisition method based on higher-order central difference.
[0036] This invention provides a computer program product, including a computer program that, when executed by a processor, implements the DGTD electromagnetic field acquisition method based on higher-order central difference.
[0037] This invention provides a DGTD electromagnetic field acquisition method and apparatus based on higher-order central difference. Starting from the establishment of the electromagnetic field time-domain step formula, it applies a higher-order central difference method when it is necessary to approximate the time derivative with central difference, thereby reducing the magnitude of the introduced error and improving the stability conditions. This reduces the number of iterations required to calculate the same total time, thereby reducing the computational load and greatly improving the computational efficiency and accuracy of the DGTD method. Attached Figure Description
[0038] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with the invention and, together with the description, serve to explain the principles of the invention.
[0039] To more clearly illustrate the technical solutions in this invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.
[0040] Figure 1 This is a flowchart illustrating the DGTD electromagnetic field acquisition method based on higher-order central difference provided by the present invention.
[0041] Figure 2 This is a schematic diagram of the structure of the DGTD electromagnetic field acquisition device based on high-order central difference provided by the present invention.
[0042] Figure 3 This is a schematic diagram of the structure of the electronic device provided by the present invention. Detailed Implementation
[0043] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.
[0044] It should be noted that the terms "first," "second," etc., used in this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion; for example, a process, method, system, product, or device that comprises a series of steps, units, or modules is not necessarily limited to those explicitly listed, but may include other steps, units, or modules not explicitly listed or inherent to such processes, methods, products, or devices.
[0045] The Discontinuous Galerkin Time-domain (DGTD) method is a special type of time-domain finite element method. It not only possesses the advantages of traditional time-domain finite element methods, such as the ability to use unstructured meshes to reduce shape approximation errors, but also allows for direct analysis and solution of broadband structures and transient problems. Furthermore, due to the introduction of numerical flux techniques, the DGTD method avoids the calculation of the global system matrix, enabling element-by-element solutions and offering greater flexibility.
[0046] However, since the time-domain step formula for electromagnetic fields requires approximation of the time derivative using a central difference, errors are inevitably introduced. Furthermore, the second-order central difference is typically used, leading to further errors in the calculation results. Secondly, the stability condition is also related to the order of the central difference; a larger order results in a larger time step, thus reducing the total number of iterations. Therefore, increasing the order of the central difference to improve computational efficiency and accuracy is feasible.
[0047] Figure 1 This is a flowchart illustrating the DGTD electromagnetic field acquisition method based on higher-order central difference provided by the present invention, as shown below. Figure 1 As shown, the method includes:
[0048] Step 110: Spatial discretization is performed on the target formation region (the target formation includes the oil layer region, the upper and lower surrounding rock formation regions, and the wellbore region, etc.) of the time-domain discontinuous Galerkin method DGTD electromagnetic field model to obtain the polyhedral mesh of each region.
[0049] Step 120: Obtain the node scalar basis functions of the mesh elements based on the polyhedron mesh, and then obtain the edge vector basis functions;
[0050] Step 130: Based on the first-order Maxwell curl vector equations, a weighted test is performed on the edge vector basis functions to obtain a weak solution to the basic equations; the weak solution to the basic equations is a semi-discrete form of the electromagnetic field time-domain stepping algorithm.
[0051] Step 140: Based on the semi-discrete form, perform higher-order central difference to obtain the time-discrete form based on the higher-order central difference; the time-discrete form is the electromagnetic field time-domain step formula.
[0052] Step 150: Based on the electromagnetic field time-domain step formula, obtain the discrete electromagnetic energy after the absorption boundary condition correction.
[0053] Step 160: Based on the discrete electromagnetic energy corrected by the absorption boundary conditions, the stability conditions are obtained.
[0054] Step 170: Based on the stability condition and the electromagnetic field time-domain step formula, the electric field intensity and magnetic field intensity on each edge of the polyhedral mesh are iteratively updated.
[0055] The following sections will explain each step in detail.
[0056] In step 110, the target stratigraphic region of the time-domain discontinuous Galerkin method DGTD electromagnetic field model is spatially discretized to obtain a polyhedral mesh for each region.
[0057] It should be noted that the target stratigraphic region can be determined based on the actual area to be calculated, and no further limitations are imposed here. Furthermore, when spatially discretizing the target stratigraphic region, since the electromagnetic field is generally an irregular three-dimensional space, polyhedral meshes such as tetrahedrons and hexahedrons can be used for spatial discretization. For example, a three-dimensional tetrahedral mesh can be used to freely partition the target stratigraphic region. The specific method can be determined based on the actual selected target stratigraphic region and the actual electromagnetic field space, and no further limitations are imposed here.
[0058] In this embodiment of the invention, the target stratigraphic region of the DGTD electromagnetic field model is spatially discretized, that is, the target stratigraphic region is... Can be divided into Each polyhedral mesh is composed of [number] polyhedral meshes, and each polyhedral mesh is... This means, that is:
[0059]
[0060] in, yes The approximate computational domain is composed of multiple discretized polyhedral meshes.
[0061] Step 120: Obtain the node scalar basis functions and edge vector basis functions of the mesh elements based on the mesh partitioning.
[0062] In this example, tetrahedral mesh elements are selected. Each node is defined, resulting in the node scalar basis functions of the mesh elements, expressed as:
[0063]
[0064] in, Indicates the first The set of node scalar basis functions corresponding to each tetrahedral mesh, where These represent the node scalar basis functions of the four nodes of the tetrahedral mesh.
[0065] The edge vector basis functions can be obtained from the node scalar basis functions of the two nodes of the edge. The edge vector basis functions can be expressed as:
[0066]
[0067] In the formula Number the edges of the tetrahedral mesh. and edge Two nodes, Let be the length of the edge. From the above relationship, we can obtain the edge vector basis function set, expressed as:
[0068]
[0069] in, Indicates the first The edge vector basis function set corresponding to each tetrahedral mesh, where These represent the edge vector basis functions of the six edges of the tetrahedral mesh.
[0070] Step 130: Based on the first-order Maxwell curl vector equations, the vector basis functions are weighted and tested. By adding numerical flux and the vector basis functions, weak solutions to the basic equations are obtained. Then, the electric field strength and magnetic field strength are combined according to the edge vector basis function set to obtain the semi-discrete form of the electromagnetic field time-domain step algorithm.
[0071] First, using the first-order Maxwell curl vector equation as the fundamental equation, its form in a non-uniform, lossy medium is as follows:
[0072]
[0073] In the formula, The dielectric constant of the medium. The magnetic permeability of the medium, The conductivity of the medium, For electric field strength, The magnetic field strength, For time.
[0074] Secondly, based on the first-order Maxwell's curl vector equations, a weighted test of the vector basis functions is performed, yielding:
[0075]
[0076] In the formula, Indicates the first The dielectric constant of a tetrahedral grid cell. Indicates the first The permeability of each tetrahedral grid cell Indicates the first The conductivity of each tetrahedral grid cell. Indicates the first The electric field intensity of each tetrahedral grid cell. Indicates the first The magnetic field strength of each tetrahedral grid cell. Indicates the first The first of the edge vector basis functions of the tetrahedral mesh element Each edge vector basis function.
[0077] Then, considering the units in the DGTD method and its adjacent units The electromagnetic field components at the interface are discontinuous, which contradicts the actual continuity of electromagnetic field components. Therefore, a numerical flux needs to be introduced, generally in the form of:
[0078]
[0079] In the formula, Represents tetrahedral mesh elements The tangential component of the electric field intensity, Represents tetrahedral mesh elements The tangential component of the magnetic field strength, Represents tetrahedral mesh elements electric field strength, Represents tetrahedral mesh elements magnetic field strength, Represents tetrahedral mesh elements electric field strength, Represents tetrahedral mesh elements magnetic field strength, and The coefficient of the mutation term, and is the coefficient of the dissipative mutation term.
[0080] After adding the numerical flux, we get:
[0081]
[0082] By applying vector identity transformations and the divergence theorem, a weak solution to the DGTD can be obtained, expressed as:
[0083]
[0084] Finally, based on the obtained weak solution, and by expanding the electric field strength and magnetic field strength according to the edge vector basis function set, the semi-discrete form of the electromagnetic field time-domain step algorithm is obtained, so as to establish the time discrete matrix equation.
[0085] The electric and magnetic field strengths are expanded using the edge vector basis functions and expressed as follows:
[0086]
[0087] In the formula, Indicates the first The first element in the edge vector basis function set of the tetrahedral mesh element. Each edge vector basis function This indicates the number of basis functions within the edge vector basis function set. Indicates the first The coefficients of the electric field intensity of each tetrahedral mesh element after expansion using the edge vector basis functions. Indicates the first The coefficients of the magnetic field strength of a tetrahedral grid cell after expansion using the edge vector basis functions.
[0088] Based on the above expansion method, the weak solution of the DGTD can be obtained in the semi-discrete form of the electromagnetic field time-domain stepping algorithm, expressed as:
[0089]
[0090] The semi-discrete form of the DGTD equation described above can be expressed in the form of a time-discrete matrix equation as follows:
[0091]
[0092] In the formula, , and The mass matrix of the mesh elements is represented as:
[0093]
[0094] In the formula, The rigidity matrix representing the mesh element is expressed as:
[0095]
[0096] In the formula, The surface flux matrix of the mesh element is represented as:
[0097]
[0098] In the formula, The excitation term representing the mesh element is expressed as:
[0099]
[0100] In the formula, This represents the current density.
[0101] Step 140: Based on the semi-discrete form, perform higher-order central difference to obtain the time-discrete form based on higher-order central difference, that is, the electromagnetic field time-domain step formula based on higher-order central difference.
[0102] The method provided by this invention is applicable to higher-order central difference finite element methods of order four or greater. In this embodiment, a fourth-order central difference finite element method is used as an example for illustration.
[0103] With integer time steps, the electromagnetic field is sampled alternately along the time axis, and the electric field in Sampling, magnetic field in Sampling, among which Let be the time to iterate to the nth step. Simultaneously, using the fourth-order central difference to approximate the time derivative, we obtain:
[0104]
[0105]
[0106] Substituting the approximate result with respect to the time derivative into the time discrete matrix equation of DGTD, and then through backward approximation and identity transformation, the electromagnetic field time-domain stepping formula based on the fourth-order central difference is obtained, expressed as:
[0107]
[0108] In the formula, , and The mass matrix of the mesh elements is represented as:
[0109]
[0110] In the formula, The rigidity matrix representing the mesh element is expressed as:
[0111]
[0112] In the formula, The surface flux matrix of the mesh element is represented as:
[0113]
[0114] In the formula, The excitation term representing the mesh element is expressed as:
[0115]
[0116] Step 150: Based on the electromagnetic field time-domain step formula of the higher-order central difference, obtain the discrete electromagnetic energy after the absorption boundary condition correction.
[0117] In this embodiment, before the introduction of the corrected discrete electromagnetic energy, the recursive equation for the discrete electromagnetic energy is the discrete electromagnetic energy obtained through a fourth-order central difference, expressed as:
[0118]
[0119] In the formula, Represented as two adjacent subdivided mesh elements and boundary surface, Representing units respectively and The electric field strength and magnetic field strength, express The normal vector of the boundary surface. It is represented as an arbitrarily connected finite element group. It is represented as discrete electromagnetic energy within an arbitrarily connected finite element group.
[0120] The aforementioned progressive equation for discrete electromagnetic energy is inversely derived using Green's function, and then the metallic boundary conditions, absorbing boundary conditions, and... , Substituting this into the equation, we obtain the corrected discrete electromagnetic energy, expressed as:
[0121]
[0122] In the formula, Represented as two adjacent subdivided mesh elements and boundary surface, express The normal vector of the boundary surface. Represented as units The electric field strength and magnetic field strength, It is represented as an arbitrarily connected finite element group. It is represented as discrete electromagnetic energy within an arbitrarily connected finite element group. It is represented as the corrected discrete electromagnetic energy within an arbitrarily connected finite element group. Represented as a subdivided mesh element The propagation speed of internal electromagnetic waves, Represented as a subdivided mesh element The permeability within, Represented as a subdivided mesh element The dielectric constant within.
[0123] Step 160: Based on the discrete electromagnetic energy corrected by the absorption boundary conditions, the corrected discrete electromagnetic energy is transformed by vector identity and then inequality relations containing the second norm are applied to obtain inequalities containing time steps; based on the inequalities, identity transformation and simplification are performed to obtain the stability conditions.
[0124] Specifically, in order for the recursive equation of the corrected discrete electromagnetic energy to be stable, it is necessary to Time step under stability conditions Below is and The positive definite quadratic form.
[0125] There are two constants in any finite element. and The following relationships exist:
[0126]
[0127] The speed of light in the subdivided grid cells is Set two dimensionless constants and The following relationships exist:
[0128]
[0129] In the formula and Determined by the geometry of the finite element and the form of the fundamental field, curl represents curl. and These are the symmetric positive definite permittivity and permeability, respectively.
[0130] Substituting the boundary conditions into the modified discrete electromagnetic energy expression above, we obtain an expression containing the above inequality, expressed as:
[0131]
[0132] Using the two sets of inequalities mentioned above, we can obtain the inequalities for each term in the modified discrete electromagnetic energy expression, as follows:
[0133]
[0134]
[0135]
[0136]
[0137] Sum the results of each inequality Substituting these expressions into the above formula for the corrected discrete electromagnetic energy, we obtain the inequality relation regarding the corrected discrete electromagnetic energy, which is expressed as:
[0138]
[0139] use To abbreviate the above expression, therefore, in the abbreviation... equal:
[0140]
[0141] To ensure the stability of the recurrence equation, the matrix must be a positive definite quadratic form, therefore the above... The coefficients within the parentheses in the expression are all greater than zero, which is represented as:
[0142]
[0143] The first expression in the above relation can be further transformed to obtain the following relation:
[0144]
[0145] Since the transformations for each expression are consistent, they can be transformed similarly and then combined to obtain the following result including the time step. Relationship:
[0146]
[0147] After further transformation, the time step that meets the stability condition can be obtained:
[0148]
[0149] In the formula, Formula is a unit Surface area For unit volume, For unit The speed of light in , , , , These four are units. and unit The permeability and permittivity, and Is with unit A dimensionless constant related to the geometry and the order of the basis functions.
[0150] Step 170: Iteratively update the electromagnetic field time-domain stepping formula based on the stability condition and the higher-order central difference.
[0151] In this embodiment, tetrahedral mesh elements are used as the basic elements, first-order edge vector basis functions are used, and electromagnetic field time-domain step formulas based on fourth-order central difference are used for iteration.
[0152] The electromagnetic field time-domain stepping formula based on the fourth-order central difference is expressed as:
[0153]
[0154] The stability conditions for the tetrahedral mesh element, the first-order edge vector basis function, and the electromagnetic field time-domain step formula iteration based on the fourth-order central difference are expressed as:
[0155]
[0156] In the formula, Formula is a unit Surface area For unit volume, For unit The speed of light in , , , , These four are units. and unit The permeability and permittivity, and Is with unit A dimensionless constant related to the geometry and the order of the basis functions.
[0157] The following relationship exists:
[0158]
[0159] In the formula For unit and unit The area of the intersection surface, which is related to the unit The maximum value of the ratio of surface areas is generally less than or equal to Therefore, the above expression can be simplified to:
[0160]
[0161] The following relationship exists:
[0162]
[0163] In the formula This represents the order of the basis functions used in the recursive formula. Since we are studying first-order basis functions, here... ,Right now:
[0164]
[0165] The above and Substituting the relational expression into the initial expression of the stability condition, we obtain the stability condition under this precondition:
[0166]
[0167] The electromagnetic field time-domain step formula, updated with the above time step, yields the electric and magnetic field intensities on each edge of the target mesh at different iteration times.
[0168] In summary, this embodiment of the invention starts with the establishment of the electromagnetic field time-domain step formula. When it is necessary to approximate the time derivative with central difference, a higher-order central difference method is used, which reduces the magnitude of the introduced error. At the same time, the stability condition is improved, the number of iterations required to calculate the same total time is reduced, the amount of computation is reduced, and the computational efficiency and accuracy of the DGTD method are greatly improved.
[0169] The DGTD electromagnetic field acquisition device based on higher-order central difference provided by the present invention will be described below. The DGTD electromagnetic field acquisition device based on higher-order central difference described below can be referred to in correspondence with the DGTD electromagnetic field acquisition method based on higher-order central difference described above.
[0170] Figure 2 This is a schematic diagram of the structure of the DGTD electromagnetic field acquisition device based on high-order central difference provided by the present invention, as shown below. Figure 2 As shown, the device includes:
[0171] Spatial discretization module 210 is used to spatially discretize the target stratigraphic region of the time-domain discontinuous Galerkin method DGTD electromagnetic field model to obtain a polyhedral mesh for each region.
[0172] The function determination module 220 is used to obtain the nodal scalar basis functions and edge vector basis functions of the mesh elements based on the polyhedral mesh.
[0173] The weighted test module 230 is used to perform weighted tests on the edge vector basis functions based on the first-order Maxwell curl vector equations to obtain weak solutions to the basic equations; the weak solutions to the basic equations are semi-discrete forms of the electromagnetic field time-domain stepping algorithm.
[0174] The difference determination module 240 is used to perform higher-order central difference based on the semi-discrete form to obtain the time-discrete form based on the higher-order central difference; the time-discrete form is the electromagnetic field time-domain step formula; the order in the higher-order central difference is greater than or equal to four.
[0175] The electromagnetic energy determination module 250 is used to obtain the discrete electromagnetic energy after the absorption boundary condition correction according to the electromagnetic field time-domain step formula.
[0176] The stability determination module 260 is used to obtain the stability conditions based on the discrete electromagnetic energy after correction of the absorption boundary conditions.
[0177] The field value update module 270 is used to iteratively update the electric field intensity and magnetic field intensity on each edge of the polyhedral mesh according to the stability condition and the electromagnetic field time-domain step formula.
[0178] In summary, this embodiment of the invention starts with the establishment of the electromagnetic field time-domain step formula. When it is necessary to approximate the time derivative with central difference, a higher-order central difference method is used, which reduces the magnitude of the introduced error. At the same time, the stability condition is improved, the number of iterations required to calculate the same total time is reduced, the amount of computation is reduced, and the computational efficiency and accuracy of the DGTD method are greatly improved.
[0179] Figure 3 This is a schematic diagram of the structure of the electronic device provided by the present invention, such as... Figure 3 As shown, the electronic device may include a processor 310, a communications interface 320, a memory 330, and a communications bus 340, wherein the processor 310, the communications interface 320, and the memory 330 communicate with each other via the communications bus 340. The processor 310 can call logical commands stored in the memory 330 to execute the methods described in the above embodiments, for example:
[0180] The target stratigraphic region of the time-discontinuous Galerkin method DGTD electromagnetic field model is spatially discretized to obtain a polyhedral mesh for each region. Based on the polyhedral mesh, the nodal scalar basis functions and edge vector basis functions of the mesh elements are obtained. Weak solutions to the fundamental equations are obtained by weighted testing of the edge vector basis functions based on the first-order Maxwell curl vector equations. The weak solutions to the fundamental equations are in a semi-discrete form of the electromagnetic field time-domain step algorithm. Based on the semi-discrete form, a higher-order central difference is performed to obtain a time-discrete form based on the higher-order central difference. The time-discrete form is the electromagnetic field time-domain step formula. Based on the electromagnetic field time-domain step formula, the discrete electromagnetic energy after correction for absorbing boundary conditions is obtained. Based on the discrete electromagnetic energy after correction for absorbing boundary conditions, the stability condition is obtained. Based on the stability condition and the electromagnetic field time-domain step formula, the electric and magnetic field intensities on each edge of the polyhedral mesh are iteratively updated.
[0181] Furthermore, when the logical commands in the aforementioned memory can be implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this invention, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several commands to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.
[0182] The processor in the electronic device provided in this embodiment of the invention can call logical instructions in the memory to implement the above method. Its specific implementation method is the same as the aforementioned method implementation method and can achieve the same beneficial effects, which will not be repeated here.
[0183] This invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, is implemented to perform the methods provided in the above embodiments.
[0184] The specific implementation method is the same as the aforementioned method implementation method and can achieve the same beneficial effects, so it will not be repeated here.
[0185] This invention provides a computer program product, including a computer program that, when executed by a processor, implements the method described above.
[0186] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs. Those skilled in the art can understand and implement this without any creative effort.
[0187] Through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, and of course, it can also be implemented by hardware. Based on this understanding, the above technical solutions, in essence or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.
[0188] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.
Claims
1. A DGTD electromagnetic field acquisition method based on higher-order central difference, characterized in that, include: Spatial discretization is performed on the target stratigraphic region of the time-domain discontinuous Galerkin method DGTD electromagnetic field model to obtain a polyhedral mesh for each region. The nodal scalar basis functions and edge vector basis functions of the mesh elements are obtained from the polyhedral mesh. Weak solutions to the basic equations are obtained by weighted testing of the edge vector basis functions based on the first-order Maxwell curl vector equations; the weak solutions to the basic equations are in the semi-discrete form of the electromagnetic field time-domain stepping algorithm. Based on the semi-discrete form, a higher-order central difference is performed to obtain a time-discrete form based on the higher-order central difference; the time-discrete form is the electromagnetic field time-domain step formula; the order in the higher-order central difference is greater than or equal to four. Based on the electromagnetic field time-domain step formula, the discrete electromagnetic energy after absorption boundary condition correction is obtained. Based on the discrete electromagnetic energy corrected for the absorption boundary conditions, the stability conditions are obtained. The time step that satisfies the stability condition is expressed as: ； In the formula, For time step, For unit Surface area For unit volume, For unit The speed of light in , , , , These four are units. and unit The permeability and permittivity, and Is with unit Dimensionless constants related to the geometry and order of the basis functions; The following relationship exists: ； For unit and unit The area of the intersection surface; With unit The ratio of surface areas The maximum value is less than or equal to , ; The following relationship exists: , Let be the order of the basis functions; Based on the stability condition and the electromagnetic field time-domain step formula, the electric field strength and magnetic field strength on each edge of the polyhedral mesh are iteratively updated. Before performing higher-order central difference based on the semi-discrete form to obtain the time-discrete form based on the higher-order central difference, the process includes: Based on the weak solution of the fundamental equation, and by expanding the electric and magnetic field strengths according to the edge vector basis function set, a time-discrete matrix equation is established.
2. The DGTD electromagnetic field acquisition method based on higher-order central difference as described in claim 1, characterized in that, The obtained time discrete form based on higher-order central difference includes: The derivatives of the electric and magnetic fields with respect to time are approximated using the higher-order central difference method. The approximate values are then substituted into the time discrete matrix equation, and electric field intensity terms at multiple different iteration times are introduced to obtain the electromagnetic field time-domain step formula based on the higher-order central difference.
3. The DGTD electromagnetic field acquisition method based on high-order central difference as described in claim 1, characterized in that, The obtained discrete electromagnetic energy after absorption boundary condition correction includes: By substituting the absorption boundary conditions into the expression for discrete electromagnetic energy and using Green's function to deduce it in reverse, and then performing correction processing, we obtain the discrete electromagnetic energy after the absorption boundary conditions are corrected.
4. The DGTD electromagnetic field acquisition method based on high-order central difference as described in claim 1, characterized in that, The stability conditions obtained include: After performing a vector identity transformation on the discrete electromagnetic energy corrected for the absorption boundary conditions, the inequality is obtained by applying the inequality relations containing the second norm. Based on the inequality, we perform identity transformations and simplifications to obtain the stability conditions.
5. A DGTD electromagnetic field acquisition device based on high-order central difference, characterized in that, include: The spatial discretization module is used to spatially discretize the target stratigraphic region of the time-domain discontinuous Galerkin method DGTD electromagnetic field model to obtain a polyhedral mesh for each region. The function determination module is used to obtain the nodal scalar basis functions and edge vector basis functions of the mesh elements based on the polyhedral mesh. The weighted testing module is used to perform weighted testing on the edge vector basis functions based on the first-order Maxwell curl vector equations to obtain weak solutions to the basic equations; the weak solutions to the basic equations are semi-discrete forms of the electromagnetic field time-domain stepping algorithm. The difference determination module is used to perform higher-order central difference based on the semi-discrete form to obtain a time-discrete form based on the higher-order central difference; the time-discrete form is an electromagnetic field time-domain step formula; the order in the higher-order central difference is greater than or equal to four. The electromagnetic energy determination module is used to obtain the discrete electromagnetic energy after absorption boundary condition correction according to the electromagnetic field time-domain step formula. The stability determination module is used to obtain the stability conditions based on the discrete electromagnetic energy after correction of the absorption boundary conditions. The time step that satisfies the stability condition is expressed as: ； In the formula, For time step, For unit Surface area For unit volume, For unit The speed of light in , , , , These four are units. and unit The permeability and permittivity, and Is with unit Dimensionless constants related to the geometry and order of the basis functions; The following relationship exists: ； For unit and unit The area of the intersection surface; With unit The ratio of surface areas The maximum value is less than or equal to , ; In the formula The following relationship exists: , Let be the order of the basis functions; The field value update module is used to iteratively update the electric field intensity and magnetic field intensity on each edge of the polyhedral mesh according to the stability condition and the electromagnetic field time-domain step formula. The difference determination module is used for: Based on the weak solution of the fundamental equation, and by expanding the electric and magnetic field strengths according to the edge vector basis function set, a time-discrete matrix equation is established.
6. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the DGTD electromagnetic field acquisition method based on higher-order central difference as described in any one of claims 1 to 4.
7. A non-transitory computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the DGTD electromagnetic field acquisition method based on higher-order central difference as described in any one of claims 1 to 4.
8. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by the processor, it implements the DGTD electromagnetic field acquisition method based on higher-order central difference as described in any one of claims 1 to 4.