A ground penetrating radar time domain finite element fast simulation method, system and device

Abstract

The application discloses a ground penetrating radar time domain finite element fast simulation method, system and equipment. The method comprises the following steps: setting geometric parameters and electrical parameters of a two-dimensional complex geoelectric model; constructing a weak form of a two-dimensional electromagnetic wave equation; forming a time domain finite element equation; establishing a linear equation group after central difference time discretization; giving a time step, judging whether the time step meets a CFL stability condition, if yes, directly solving the linear equation group, if not, performing LU decomposition on a coefficient matrix of the linear equation group, and disturbing unstable eigenvalues in the coefficient matrix until absolute values of all eigenvalues of the decomposed coefficient matrix are less than or equal to 1, and generating a reconstruction coefficient matrix; and solving the reconstructed linear equation group, so that the time step breaks through the CFL stability condition. The application can improve the simulation precision of the ground penetrating radar.

Description

Technical Field

[0001] This invention relates to the field of forward modeling of ground penetrating radar, and in particular to a method, system and equipment for rapid simulation of ground penetrating radar using the time-domain finite element method. Background Technology

[0002] Ground penetrating radar (GPR) forward modeling is an important tool for studying the propagation of high-frequency electromagnetic waves in complex media. Through forward modeling, the GPR response characteristics of typical geological bodies can be accurately analyzed, providing crucial references for the interpretation of measured radar data. Currently, the main methods for GPR forward modeling include the time-domain finite difference method, the time-domain pseudospectral method, and the time-domain finite element method. In recent years, the time-domain finite element method has been widely used in GPR forward modeling due to its ability to create unstructured, discrete two-dimensional complex geoelectric models, neglect internal boundary conditions, and standardized programming. To ensure time iteration stability and eliminate numerical dispersion, the time step and grid size must satisfy the Courant–Friedrichs–Lewy (CFL) stability condition when using the time-domain finite element method for GPR forward modeling; that is, the larger the grid size, the longer the time step; and the smaller the grid size, the shorter the time step. However, with the increasing complexity of ground-penetrating radar (GPR) research objects and the continuous improvement of interpretation accuracy, high-precision GPR forward modeling technology for complex, shallow, and small targets has become an urgent need in practical engineering applications. When conducting time-domain finite element forward modeling of GPR for complex, shallow, and small targets, fine meshes are generally used near the target, while sparse meshes are used in other computational regions. However, due to the limitations of CFL stability conditions, fine meshes require short time steps during simulation, leading to reduced computational efficiency, limitations on the algorithm itself, and consequently, low GPR simulation accuracy. Summary of the Invention

[0003] The purpose of this invention is to provide a method, system, and device for rapid simulation of ground-penetrating radar using the time-domain finite element method, in order to solve the problem that in the process of fine mesh generation, the fine mesh requires a short time step for simulation calculation, which leads to reduced computational efficiency and consequently low accuracy of ground-penetrating radar simulation.

[0004] To achieve the above objectives, the present invention provides the following solution:

[0005] A fast time-domain finite element method for ground-penetrating radar includes:

[0006] Define the geometric and electrical parameters of the complex geoelectric model; the geometric parameters include the dimensions of the complex geoelectric model; the electrical parameters include resistivity and relative permittivity.

[0007] The complex geoelectric model is discretized using a Delaunay unstructured grid. A fine grid is used in the target volume region, while a sparse grid is used in the region where the electromagnetic field changes gently. The fine grid has a small spatial step size, while the sparse grid has a large spatial step size.

[0008] Based on the geometric parameters and the electrical parameters, a weak form of the two-dimensional electromagnetic wave equation is constructed according to the Galerkin method;

[0009] Based on the mesh partitioning, the weak form of the two-dimensional electromagnetic wave equation is spatially discretized using the finite element method to construct the finite element equation; the mesh partitioning includes fine mesh partitioning and sparse mesh partitioning.

[0010] The finite element equations are discretized in time using the central difference method to form a system of linear equations after time discretization.

[0011] Given a time step and the number of time steps, determine whether the time step satisfies the CFL stability condition and obtain the first determination result;

[0012] If the first judgment result indicates that the time step satisfies the CFL stability condition, the linear equation system after time discretization is directly solved. Under the time step, the iteration starts from time zero until the number of time steps is reached.

[0013] If the first judgment result indicates that the time step does not meet the CFL stability condition, the coefficient matrix of the time-discrete linear equation system is decomposed, and the unstable eigenvalues ​​in the decomposed coefficient matrix are perturbed using the eigenvalue perturbation method until the absolute value of all eigenvalues ​​of the decomposed coefficient matrix is ​​less than or equal to 1. The coefficient matrix is ​​then reconstructed to generate the linear equation system after the coefficient matrix is ​​reconstructed.

[0014] Solve the linear equation system after reconstructing the coefficient matrix, and iterate from time zero at the time step until the time step number is reached.

[0015] Optionally, the weak form of the two-dimensional electromagnetic wave equation is:

[0016]

[0017] Where Ω is the calculated area of ​​the complex geoelectric model; σ is the conductivity of the medium; μ is the magnetic permeability of the medium; ε is the relative permittivity of the medium; E is the electric field; and S is the emitter wavelet function. For the Laplace operator.

[0018] Optionally, the finite element equation is:

[0019]

[0020] Where E is the vector composed of the electric field values ​​at the three nodes of the triangle; M is the mass matrix; K′ is the damping matrix; K is the stiffness matrix; and S is the source vector.

[0021] Optionally, the time-discrete linear equation system is as follows:

[0022]

[0023] Where Δt is the time step; E t E represents the electric field strength at the current moment. t+Δt E represents the electric field strength at the next time step Δt from the current time t; t-Δt S is the electric field strength at the previous time Δt at the current time t; t Let be the source energy injected at the current time t.

[0024] Optionally, determining whether the time step satisfies the CFL stability condition to obtain a first determination result specifically includes:

[0025] When the formula The condition is established that the time step satisfies the CFL stability condition; where, l min For the minimum spatial step size, v max The maximum speed at which electromagnetic waves propagate in a medium;

[0026] When the formula This is not true; the stated time step does not satisfy the CFL stability condition.

[0027] Optionally, the perturbation of unstable eigenvalues ​​in the decomposed coefficient matrix using the eigenvalue perturbation method specifically includes:

[0028] Using formula Perturb the unstable eigenvalues ​​in the decomposed coefficient matrix; wherein, λ is the eigenvalue after the i-th perturbation; i Let be the i-th eigenvalue; γ is the perturbation coefficient, γ∈[0.5,0.9].

[0029] A rapid time-domain finite element simulation system for ground-penetrating radar includes:

[0030] A geometric and electrical parameter setting module is used to set the geometric and electrical parameters of a complex geoelectric model; the geometric parameters include the dimensions of the complex geoelectric model; the electrical parameters include resistivity and relative permittivity.

[0031] The discretization and mesh generation module is used to discretize complex geoelectric models using Delaunay unstructured meshes. Fine mesh generation is used in the target volume region, while sparse mesh generation is used in regions with gentle electromagnetic field changes. The fine mesh has a small spatial step size, while the sparse mesh has a large spatial step size.

[0032] A weak form construction module for the two-dimensional electromagnetic wave equation is used to construct the weak form of the two-dimensional electromagnetic wave equation based on the geometric parameters and the electrical parameters according to the Galerkin method.

[0033] The finite element equation construction module is used to construct the finite element equation by spatially discretizing the weak form of the two-dimensional electromagnetic wave equation based on the mesh partitioning and using the finite element method; the mesh partitioning includes fine mesh partitioning and sparse mesh partitioning.

[0034] The time discretization module is used to discretize the finite element equations in time using the central difference method, forming a system of linear equations after time discretization.

[0035] The first judgment module is used to determine whether the time step satisfies the CFL stability condition given a time step and the number of time steps, and to obtain the first judgment result.

[0036] The time-discrete linear equations solving module is used to directly solve the time-discrete linear equations if the first judgment result indicates that the time step satisfies the CFL stability condition. Under the time step, iterates from time zero until the number of time steps is reached.

[0037] The decomposition and perturbation module is used to decompose the coefficient matrix of the time-discrete linear equation system if the first judgment result indicates that the time step does not meet the CFL stability condition, and to perturb the unstable eigenvalues ​​in the decomposed coefficient matrix using the eigenvalue perturbation method until the absolute value of all eigenvalues ​​of the decomposed coefficient matrix is ​​less than or equal to 1, and to reconstruct the coefficient matrix to generate the linear equation system after reconstructing the coefficient matrix.

[0038] The module for solving the linear equations after reconstructing the coefficient matrix is ​​used to solve the linear equations after reconstructing the coefficient matrix. Under the time step, it iterates from time zero until the number of time steps is reached.

[0039] An electronic device includes a memory and a processor, the memory storing a computer program, and the processor running the computer program to cause the electronic device to perform the fast simulation method of ground penetrating radar time domain finite element method as described above.

[0040] A computer-readable storage medium storing a computer program that, when executed by a processor, implements the fast simulation method for ground-penetrating radar in the time domain using finite element method as described above.

[0041] According to specific embodiments provided by the present invention, the present invention discloses the following technical effects: The present invention provides a method, system and device for rapid simulation of ground-penetrating radar using time-domain finite element method. Under the condition of breaking the stability condition of CFL at a given time step, the weak form of the electromagnetic wave equation is discretized by the finite element method and the central difference method to generate a linear equation system after time discretization. The eigenvalues ​​of the coefficient matrix of the electric field intensity at the current time in the linear equation system after time discretization are artificially perturbed to keep the absolute value of all eigenvalues ​​of the coefficient matrix less than or equal to 1, thereby breaking the limitation of the CFL stability condition, and thus enabling rapid and high-precision simulation of GPR using a large time step. Attached Figure Description

[0042] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0043] Figure 1 This is a flowchart of the fast simulation method for ground-penetrating radar using the time-domain finite element method provided by the present invention;

[0044] Figure 2 This is a flowchart of the ground-penetrating radar time-domain finite element rapid simulation method provided in Embodiment 2 of the present invention. Detailed Implementation

[0045] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. 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 are within the scope of protection of the present invention.

[0046] The purpose of this invention is to provide a method, system, and equipment for rapid simulation of ground penetrating radar using the time-domain finite element method, which can improve the simulation accuracy of ground penetrating radar.

[0047] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0048] Example 1

[0049] Figure 1 The flowchart of the ground-penetrating radar time-domain finite element fast simulation method provided by the present invention is as follows: Figure 1 As shown, a fast time-domain finite element simulation method for ground-penetrating radar includes:

[0050] Step 101: Set the geometric parameters and electrical parameters of the complex geoelectric model; the geometric parameters include the dimensions of the complex geoelectric model; the electrical parameters include resistivity and relative permittivity.

[0051] In practical applications, the size, resistivity σ, and relative permittivity ε of the two-dimensional complex geoelectric model are set; the position and movement step of the transmitting and receiving antennas, the transmit sub-wave function S, and the center frequency f0 are determined.

[0052] Step 102: Discretize the complex geoelectric model using a Delaunay unstructured grid. Use a fine grid in the target area and a sparse grid in areas with gradual electromagnetic field changes. The fine grid has a small spatial step size, while the sparse grid has a large spatial step size. The target area refers to the vicinity of the target body, and the areas with gradual electromagnetic field changes refer to other areas with gradual changes in high-frequency electromagnetic fields.

[0053] Step 103: Based on the geometric parameters and the electrical parameters, construct the weak form of the two-dimensional electromagnetic wave equation according to the Galerkin method.

[0054] In practical applications, the weak form of the two-dimensional electromagnetic wave equation is:

[0055]

[0056] Where Ω is the calculated area of ​​the complex geoelectric model; σ is the electrical conductivity of the medium; μ is the magnetic permeability of the medium; ε is the relative permittivity of the medium; E is the electric field; S is the emitted wavelet function; ▽ 2 It is the Laplace operator.

[0057] Step 104: Based on the mesh partitioning, the weak form of the two-dimensional electromagnetic wave equation is spatially discretized using the finite element method to construct the finite element equation; the mesh partitioning includes fine mesh partitioning and sparse mesh partitioning.

[0058] In practical applications, the finite element equation is:

[0059]

[0060] Where E is the vector composed of the electric field values ​​at the three nodes of the triangle; M is the mass matrix; K′ is the damping matrix; K is the stiffness matrix; and S is the source vector.

[0061] Step 105: Discretize the finite element equations in time using the central difference method to form a set of linear equations after time discretization.

[0062] In practical applications, the time-discrete linear equation system is as follows:

[0063]

[0064] Where Δt is the time step; E t E represents the electric field strength at the current moment. t+Δt E represents the electric field strength at the next time step Δt from the current time t; t-Δt S is the electric field strength at the previous time Δt at the current time t; t Let be the source energy injected at the current time t.

[0065] Step 106: Given a time step and the number of time steps, determine whether the time step satisfies the CFL stability condition and obtain the first judgment result; if yes, proceed to step 107; if no, proceed to step 108.

[0066] In practical applications, step 106 specifically includes: when the formula The condition is established that the time step satisfies the CFL stability condition; where, l min For the minimum spatial step size, v max The maximum speed at which electromagnetic waves propagate in a medium; when the formula This is not true; the stated time step does not satisfy the CFL stability condition.

[0067] Step 107: Directly solve the time-discrete linear equation system. At the time step, iterate from time zero until the number of time steps is reached.

[0068] Step 108: Decompose the coefficient matrix of the time-discrete linear equation system, and perturb the unstable eigenvalues ​​in the decomposed coefficient matrix using the eigenvalue perturbation method until the absolute value of all eigenvalues ​​of the decomposed coefficient matrix is ​​less than or equal to 1. Reconstruct the coefficient matrix to generate the linear equation system after reconstructing the coefficient matrix. In practical applications, the coefficient matrix of the time-discrete linear equation system is decomposed using LU decomposition.

[0069] In practical applications, the method of perturbing unstable eigenvalues ​​in the decomposed coefficient matrix using eigenvalue perturbation specifically includes: using the formula Perturb the unstable eigenvalues ​​in the decomposed coefficient matrix; wherein, λ is the eigenvalue after the i-th perturbation; iγ is the i-th eigenvalue; γ is the perturbation coefficient, γ∈[0.5,0.9].

[0070] Step 109: Solve the linear equation system after reconstructing the coefficient matrix. At the time step, iterate from time zero until the time step number is reached.

[0071] Example 2

[0072] Figure 2 The flowchart of the ground-penetrating radar time-domain finite element rapid simulation method provided in Embodiment 2 of the present invention is as follows: Figure 2 As shown.

[0073] Ground-penetrating radar (GPR) simulation calculations are performed using a fast time-domain finite element method that overcomes the stability condition limitations of the ground-penetrating radar (CFL). The specific steps include:

[0074] Step 1: Set the size, resistivity σ, and relative permittivity ε of the two-dimensional complex geoelectric model; determine the position and movement step of the transmitting and receiving antennas, the transmit sub-wave function S, and the center frequency f0.

[0075] Step 2: Geometric discretization of the complex geoelectric model. The complex geoelectric model is discretized using a Delaunay unstructured mesh (triangular mesh). A fine mesh (small spatial step size) is used near the target body, while a sparse mesh (large spatial step size) is used in other regions where the high-frequency electromagnetic field changes gradually, thus finely discretizing the complex geoelectric model.

[0076] Step 3: Derive the weak form of the two-dimensional electromagnetic wave equation satisfied by ground-penetrating radar. The two-dimensional electromagnetic wave equation is derived using the Galerkin method. The weak form is:

[0077]

[0078] Where E is the electric field, μ, ε, and σ are the magnetic permeability, dielectric constant, and conductivity of the medium, respectively; S is the wavelet function of the emission source, and Ω is the calculated area of ​​the complex geoelectric model.

[0079] Step 4: Applying the finite element method to discretize equation (1), the finite element equation formed by equation (1) can be derived as follows:

[0080]

[0081] Where E represents the vector composed of the electric field values ​​at the three nodes of the triangle, M, K′, and K represent the formed mass matrix, damping matrix, and stiffness matrix, respectively; and S represents the source vector.

[0082] Step 5: Apply the second-order central difference method to discretize equation (1) over time. The time-discrete equation is:

[0083]

[0084] Where Δt represents the time step, E t E represents the electric field strength at the current time t. t+Δt E represents the electric field strength at the next time step Δt from the current time. t-Δt S represents the electric field strength at the previous time Δt at the current time; t This represents the source energy injected at the current moment.

[0085] Step 6: Multiply both sides of equation (3) by After sorting, we can obtain:

[0086]

[0087] Equation (4) is the time-domain iterative scheme discretized using the central difference method. Equation (4) can be simplified as follows:

[0088] E t+Δt =AE t +BE t-Δt +CS t (5)

[0089] Where A represents E t The coefficient matrix, B represents E t-△t The coefficient matrix, C represents S t The coefficient matrix.

[0090] To ensure that the time iteration of equation (5) satisfies the CFL stability condition, the eigenvalues ​​of the coefficient matrix A must satisfy λ. i |≤1, where λ i Let be the i-th eigenvalue of the coefficient matrix A.

[0091] The parameters of the complex geoelectric model include the size of the complex geoelectric model, resistivity σ and relative permittivity ε, the position and movement step size of the transmitting and receiving antennas, the transmit wavelet function S and the center frequency f0. Based on the parameters of the complex geoelectric model, in order to stably calculate the propagation of electromagnetic waves in the medium, the distance traveled by the electromagnetic wave in time Δt must be less than the minimum spacing between adjacent grids, that is:

[0092]

[0093] Among them, l min v is the minimum spatial step size for fine mesh subdivision in step 2. max It is the maximum speed at which electromagnetic waves propagate in a medium.

[0094] When the selected time step Δt is outside the stability condition, the coefficient matrix A contains unstable eigenvalues ​​and no longer satisfies formula (5).

[0095] Step 7: Perform eigenvalue decomposition on the coefficient matrix A to obtain:

[0096] A=UΛU -1 (7)

[0097] Matrix U contains the eigenvectors of coefficient matrix A, and Λ is a diagonal matrix that is a vector composed of all the eigenvalues ​​of coefficient matrix A.

[0098] Step 8: Perturb the unstable eigenvalues ​​of coefficient matrix A so that all eigenvalues ​​of coefficient matrix A satisfy |λ i If |≤1, the perturbation formula is:

[0099]

[0100] in, This indicates that under the different triangular regions formed by the subdivision in step 2, E in the wave equation t Different eigenvalues ​​of the coefficient matrix A, where γ represents the perturbation coefficient, typically between 0.5 and 0.9.

[0101] Step 9: Reconstruct the coefficient matrix A.

[0102]

[0103] Reconstructed matrix All eigenvalues ​​satisfy |λ i |≤1.

[0104] Step 10: Reconstruct the coefficient matrix after eigenvalue perturbation The time-discrete solution can be expressed as:

[0105]

[0106] Note: Formula (5) only represents the iterative change of the electric field in the time domain at time t. Formula (10) shows that the electric field iterates from zero until the time step number n is reached.

[0107] This invention addresses the problem that existing ground-penetrating radar (GPR) forward modeling methods using the finite element method (FEA) are constrained by the CFL stability condition in the selection of time and spatial steps. This makes it difficult to balance computational accuracy and efficiency; large time steps exceed the stability condition, resulting in low accuracy, while small time steps, although satisfying the CFL stability condition, suffer from low computational efficiency. Mathematically, the CFL stability condition essentially requires that the absolute values ​​of the eigenvalues ​​of the coefficient matrix on the right-hand side are all less than or equal to 1. This invention, even with a given time step exceeding the CFL stability condition, artificially perturbs the eigenvalues ​​of the left-hand coefficient matrix in the large sparse linear equations discretized by the finite element method and central difference method, ensuring that the absolute values ​​of all eigenvalues ​​are less than or equal to 1. This overcomes the CFL stability condition, enabling fast and high-precision GPR simulations using large time steps.

[0108] Example 3

[0109] In order to implement the method corresponding to Embodiment 1 above and achieve the corresponding functions and technical effects, a ground-penetrating radar time-domain finite element rapid simulation system is provided below.

[0110] A rapid time-domain finite element simulation system for ground-penetrating radar includes:

[0111] A geometric and electrical parameter setting module is used to set the geometric and electrical parameters of a complex geoelectric model; the geometric parameters include the dimensions of the complex geoelectric model; the electrical parameters include resistivity and relative permittivity.

[0112] The discretization and mesh generation module is used to discretize complex geoelectric models using Delaunay unstructured meshes. Fine mesh generation is used in the target volume region, while sparse mesh generation is used in regions with gentle electromagnetic field changes. The fine mesh has a small spatial step size, while the sparse mesh has a large spatial step size.

[0113] A weak form construction module for the two-dimensional electromagnetic wave equation is used to construct the weak form of the two-dimensional electromagnetic wave equation based on the geometric parameters and the electrical parameters according to the Galerkin method.

[0114] The finite element equation construction module is used to construct the finite element equation by spatially discretizing the weak form of the two-dimensional electromagnetic wave equation based on the mesh partitioning and using the finite element method; the mesh partitioning includes fine mesh partitioning and sparse mesh partitioning.

[0115] The time discretization module is used to discretize the finite element equations in time using the central difference method, forming a system of linear equations after time discretization.

[0116] The first judgment module is used to determine whether the time step satisfies the CFL stability condition given a time step and the number of time steps, and to obtain the first judgment result.

[0117] The time-discrete linear equations solving module is used to directly solve the time-discrete linear equations if the first judgment result indicates that the time step satisfies the CFL stability condition. Under the time step, iterates from time zero until the number of time steps is reached.

[0118] The decomposition and perturbation module is used to decompose the coefficient matrix of the time-discrete linear equation system if the first judgment result indicates that the time step does not meet the CFL stability condition, and to perturb the unstable eigenvalues ​​in the decomposed coefficient matrix using the eigenvalue perturbation method until the absolute value of all eigenvalues ​​of the decomposed coefficient matrix is ​​less than or equal to 1, and to reconstruct the coefficient matrix to generate the linear equation system after reconstructing the coefficient matrix.

[0119] The module for solving the linear equations after reconstructing the coefficient matrix is ​​used to solve the linear equations after reconstructing the coefficient matrix. Under the time step, it iterates from time zero until the number of time steps is reached.

[0120] Example 4

[0121] This invention provides an electronic device including a memory and a processor. The memory stores a computer program, and the processor runs the computer program to enable the electronic device to perform the ground-penetrating radar time-domain finite element fast simulation method provided in Embodiment 1.

[0122] In practical applications, the aforementioned electronic devices can be servers.

[0123] In practical applications, electronic devices include: at least one processor, memory, bus, and communication interface.

[0124] The processor, communication interface, and memory communicate with each other via a communication bus.

[0125] A communication interface is used to communicate with other devices.

[0126] The processor is used to execute programs, specifically the methods described in the above embodiments.

[0127] Specifically, the program may include program code, which includes computer operation instructions.

[0128] The processor may be a central processing unit (CPU), an application-specific integrated circuit (ASIC), or one or more integrated circuits configured to implement embodiments of the present invention. The electronic device may include one or more processors of the same type, such as one or more CPUs; or it may include processors of different types, such as one or more CPUs and one or more ASICs.

[0129] Memory is used to store programs. Memory may include high-speed RAM, and may also include non-volatile memory, such as at least one disk drive.

[0130] Based on the description of the above embodiments, this application provides a storage medium storing computer program instructions thereon, which can be executed by a processor to implement the methods described in any embodiment.

[0131] The ground-penetrating radar time-domain finite element rapid simulation system provided in this application exists in various forms, including but not limited to:

[0132] (1) Mobile communication devices: These devices are characterized by their mobile communication capabilities and primarily aim to provide voice and data communication. These terminals include: smartphones (e.g., iPhones), multimedia phones, feature phones, and low-end phones, etc.

[0133] (2) Ultra-mobile personal computer devices: These devices fall under the category of personal computers, possessing computing and processing capabilities, and generally also have mobile internet access capabilities. These terminals include PDAs, MIDs, and UMPCs, such as the iPad.

[0134] (3) Portable entertainment devices: These devices can display and play multimedia content. This category includes: audio and video players (such as iPods), handheld game consoles, e-books, as well as smart toys and portable car navigation devices.

[0135] (4) Other electronic devices with data interaction functions.

[0136] Specific embodiments of the subject matter have now been described. Other embodiments are within the scope of the appended claims. In some cases, the actions recited in the claims can be performed in a different order and still achieve the desired result. Furthermore, the processes depicted in the drawings do not necessarily require a specific or sequential order to achieve the desired result. In some embodiments, multitasking and parallel processing can be advantageous.

[0137] The systems, devices, modules, or units described in the above embodiments can be implemented by computer chips or entities, or by products with certain functions. A typical implementation device is a computer. Specifically, a computer can be, for example, a personal computer, laptop computer, cellular phone, camera phone, smartphone, personal digital assistant, media player, navigation device, email device, game console, tablet computer, wearable device, or any combination of these devices.

[0138] For ease of description, the above apparatus is described by dividing it into various functional units. Of course, in implementing this application, the functions of each unit can be implemented in one or more software and / or hardware components. Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0139] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create a machine for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0140] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0141] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0142] In a typical configuration, a computing device includes one or more processors (CPU), input / output interfaces, network interfaces, and memory.

[0143] Memory may include non-persistent storage in computer-readable media, such as random access memory (RAM) and / or non-volatile memory, such as read-only memory (ROM) or flash RAM. Memory is an example of computer-readable media.

[0144] Computer-readable media, including both permanent and non-permanent, removable and non-removable media, can store information using any method or technology. Information can be computer-readable instructions, data structures, program modules, or other data. Examples of computer storage media include, but are not limited to, phase-change memory (PRAM), static random access memory (SRAM), dynamic random access memory (DRAM), other types of random access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technologies, and CD-ROM.

[0145] Digital multifunction optical disc (DVD) or other optical storage, magnetic cassette tape, magnetic tape, disk storage or other magnetic storage devices

[0146] Or any other non-transmission medium that can be used to store information that can be accessed by a computing device. As defined herein, computer-readable media does not include transient media, such as modulated data signals and carrier waves.

[0147] It should also be noted that the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0148] This application can be described in the general context of computer-executable instructions that are executed by a computer, such as program modules. Generally, program modules include routines, programs, objects, components, data structures, etc., that perform specific transactions or implement specific abstract data types. This application can also be practiced in distributed computing environments where transactions are performed by remote processing devices connected via a communication network. In distributed computing environments, program modules can reside in local and remote computer storage media, including storage devices.

[0149] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the systems disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the descriptions are relatively simple; relevant parts can be referred to the method section.

[0150] This document uses specific examples to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. Furthermore, those skilled in the art will recognize that, based on the ideas of the present invention, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A fast time-domain finite element simulation method for ground-penetrating radar, characterized in that, include: Define the geometric and electrical parameters of the complex geoelectric model; the geometric parameters include the dimensions of the complex geoelectric model. The electrical parameters include resistivity and relative permittivity; The complex geoelectric model is discretized using a Delaunay unstructured grid. A fine grid is used in the target volume region, while a sparse grid is used in the region where the electromagnetic field changes gently. The fine grid has a small spatial step size, while the sparse grid has a large spatial step size. Based on the geometric parameters and the electrical parameters, a weak form of the two-dimensional electromagnetic wave equation is constructed according to the Galerkin method; Based on network partitioning, the weak form of the two-dimensional electromagnetic wave equation is spatially discretized using the finite element method to construct the finite element equation. The mesh partitioning includes fine mesh partitioning and sparse mesh partitioning; The finite element equations are discretized in time using the central difference method to form a system of linear equations after time discretization. Given a time step and the number of time steps, determine whether the time step satisfies the CFL stability condition and obtain the first determination result; If the first judgment result indicates that the time step satisfies the CFL stability condition, the linear equation system after time discretization is directly solved. Under the time step, the iteration starts from time zero until the number of time steps is reached. If the first judgment result indicates that the time step does not meet the CFL stability condition, the coefficient matrix of the time-discrete linear equation system is decomposed, and the unstable eigenvalues ​​in the decomposed coefficient matrix are perturbed using the eigenvalue perturbation method until the absolute value of all eigenvalues ​​of the decomposed coefficient matrix is ​​less than or equal to 1. The coefficient matrix is ​​then reconstructed to generate the linear equation system after the coefficient matrix is ​​reconstructed. The method of perturbing unstable eigenvalues ​​in the decomposed coefficient matrix using eigenvalue perturbation specifically includes: Using formula Perturb the unstable eigenvalues ​​in the decomposed coefficient matrix; wherein, Let be the eigenvalue after the i-th perturbation; Let i be the i-th eigenvalue; The disturbance coefficient is... ; Solve the linear equation system after reconstructing the coefficient matrix, and iterate from time zero at the time step until the time step number is reached.

2. The method for rapid simulation of ground-penetrating radar using the time-domain finite element method according to claim 1, characterized in that, The weak form of the two-dimensional electromagnetic wave equation is: in, The calculated area for a complex geoelectric model; The conductivity of the medium; ρ is the magnetic permeability of the medium; is the relative permittivity of the medium; E For electric field; For the emission subwave function; For the Laplace operator.

3. The method for rapid simulation of ground-penetrating radar using the time-domain finite element method according to claim 1, characterized in that, The finite element equation is: in, It is a vector composed of the electric field values ​​at the three nodes of the triangle; This is the quality matrix; Here is the damping matrix; Here is the stiffness matrix; This is the source vector.

4. The method for rapid simulation of ground-penetrating radar using the time-domain finite element method according to claim 3, characterized in that, The time-discrete linear equation system is as follows: in, For time step; Let be the electric field strength at the current moment; Indicates the next time step at the current time t. The electric field strength at any given moment; For the previous time t at the current time The electric field strength at any given moment; Let be the source energy injected at the current time t.

5. The method for rapid simulation of ground-penetrating radar using the time-domain finite element method according to claim 4, characterized in that, The step of determining whether the time step satisfies the CFL stability condition to obtain a first determination result specifically includes: When the formula The condition is established that the time step satisfies the CFL stability condition; wherein, l min For the minimum spatial step size, v max The maximum speed at which electromagnetic waves propagate in a medium; When the formula This is not true; the stated time step does not satisfy the CFL stability condition.

6. A rapid simulation system for ground-penetrating radar using the time-domain finite element method, characterized in that, The ground-penetrating radar time-domain finite element rapid simulation system executes the ground-penetrating radar time-domain finite element rapid simulation method according to any one of claims 1-5, and the ground-penetrating radar time-domain finite element rapid simulation system includes: A geometric and electrical parameter setting module is used to set the geometric and electrical parameters of a complex geoelectric model; the geometric parameters include the dimensions of the complex geoelectric model; the electrical parameters include resistivity and relative permittivity. The discretization and mesh generation module is used to discretize complex geoelectric models using Delaunay unstructured meshes. Fine mesh generation is used in the target volume region, while sparse mesh generation is used in regions with gentle electromagnetic field changes. The fine mesh has a small spatial step size, while the sparse mesh has a large spatial step size. A weak form construction module for the two-dimensional electromagnetic wave equation is used to construct the weak form of the two-dimensional electromagnetic wave equation based on the geometric parameters and the electrical parameters according to the Galerkin method. The finite element equation construction module is used to construct the finite element equation by spatially discretizing the weak form of the two-dimensional electromagnetic wave equation based on the mesh partitioning and using the finite element method; the mesh partitioning includes fine mesh partitioning and sparse mesh partitioning. The time discretization module is used to discretize the finite element equations in time using the central difference method, forming a system of linear equations after time discretization. The first judgment module is used to determine whether the time step satisfies the CFL stability condition given a time step and the number of time steps, and to obtain the first judgment result. The time-discrete linear equations solving module is used to directly solve the time-discrete linear equations if the first judgment result indicates that the time step satisfies the CFL stability condition. Under the time step, iterates from time zero until the number of time steps is reached. The decomposition and perturbation module is used to decompose the coefficient matrix of the time-discrete linear equation system if the first judgment result indicates that the time step does not meet the CFL stability condition, and to perturb the unstable eigenvalues ​​in the decomposed coefficient matrix using the eigenvalue perturbation method until the absolute value of all eigenvalues ​​of the decomposed coefficient matrix is ​​less than or equal to 1, and to reconstruct the coefficient matrix to generate the linear equation system after reconstructing the coefficient matrix. The module for solving the linear equations after reconstructing the coefficient matrix is ​​used to solve the linear equations after reconstructing the coefficient matrix. Under the time step, it iterates from time zero until the number of time steps is reached.

7. An electronic device, characterized in that, The device includes a memory and a processor, the memory being used to store a computer program, and the processor running the computer program to enable the electronic device to perform the ground-penetrating radar time-domain finite element rapid simulation method as described in any one of claims 1-5.

8. A computer-readable storage medium, characterized in that, It stores a computer program that, when executed by a processor, implements the ground-penetrating radar time-domain finite element rapid simulation method as described in any one of claims 1-5.

Images