Local time stepping method based on arbitrarily high-order finite-difference time-domain method

Abstract

The present invention discloses a local time stepping method based on an arbitrary high-order finite-difference time-domain method, which uses an arbitrary high-order finite-difference time-domain method to implement iteration of electromagnetic fields using different time steps. The present invention can effectively solve the problem of low efficiency of the traditional finite-difference time-domain method when analyzing multi-scale problems. The present invention is simple to program and has strong practical engineering application value.

Description

Technical Field

[0001] The present invention belongs to the numerical calculation technology of finite-difference time-domain method, in particular to a local time stepping method based on an arbitrary high-order finite-difference time-domain method. Background Art

[0002] Multi-scale problems appear in many practical electronic engineering fields, such as electromagnetic interference of complex systems, electromagnetic compatibility and communication channel analysis. Due to the existence of fine structures, the traditional finite-difference time-domain method needs to use a small-size mesh to simulate the physical model. At the same time, due to the limitation of stability conditions, the time iteration step is too small, resulting in serious time-consuming system simulation. Summary of the invention

[0003] The object of the present invention is to provide a local time stepping technique based on an arbitrary high-order finite-difference time-domain method.

[0004] The technical solution to achieve the purpose of the present invention is: a local time stepping method based on an arbitrary high-order finite-difference time-domain method, the steps are as follows:

[0005] Step 1: Establish the geometric model of the target and discretize the model using non-uniform Yee grid;

[0006] Step 2: Use Maxwell equations to describe the relationship between the electric field and magnetic field on the Yee grid. In Maxwell equations, the partial differential of the electromagnetic field with respect to space is replaced by the central difference, and the partial differential of the electromagnetic field with respect to time is expanded according to the high-order Taylor formula.

[0007] Step 3: Calculate the different time steps for each Yee grid unit to meet the stability condition, and arrange them in order from small to large as Δt i ,i=1,2,…,N,N is the number of different time steps, where Δt i+1 =N i Δt i , N i is a natural number, let the time step Δt i The corresponding Yee grid is a fine grid, and the time step is Δt i+1 The corresponding Yee grid is a coarse grid. At the beginning, i=1;

[0008] Step 4: Iteratively solve the electromagnetic field in the fine grid. The electromagnetic field iteration N of the fine grid i After time steps, the time step length Δt i+1 The electromagnetic field of the corresponding Yee grid starts to iterate. If there is no coarse grid in the adjacent position of the fine grid iteration during the iteration process, the fine grid electromagnetic field is iterated according to the Taylor formula expansion; if the coarse grid electromagnetic field is used in the fine grid iteration process, it is obtained by interpolation of the known coarse grid electromagnetic field information through the Taylor formula;

[0009] Step 5: When the electromagnetic field of the coarse grid has completed a time step Δt i+1 , return to step 4 until the electromagnetic field of the fine grid has been iterated N times. i N i+1 time steps, the electromagnetic field of the coarse grid is iterated N times. i+1 time step, let i = i + 1, and set the next time step Δt i+1 The corresponding Yee grid is used as a coarse grid, and the previous coarse grid becomes a fine grid. Return to step 4 to iteratively solve the electromagnetic field on the coarse and fine grids until the electromagnetic fields of all grids are iterated, and the electromagnetic field value information at any time in the space is obtained.

[0010] Preferably, in the Yee grid unit, the electromagnetic field satisfies the passive differential form of the Maxwell curl equation, specifically:

[0011]

[0012]

[0013] Where ε and μ represent the dielectric constant and magnetic permeability respectively, E and H represent the electric field strength and magnetic field strength respectively;

[0014] Expanding, arranging and simplifying the curl of Maxwell's curl equation, we get

[0015]

[0016] in E x , E y and E z are the three components of the electric field in the rectangular coordinate system x, y and z, H x , H y and H z They are the three components of the magnetic field in the rectangular coordinate system x, y and z;

[0017] The partial differential of the z component of the magnetic field with respect to space is right Spatial position in rectangular coordinate system The central difference is used for expansion:

[0018]

[0019] The first row of Q in formula (3) multiplied by the first term of U is expanded to:

[0020]

[0021] At the same time, the partial differential of the electromagnetic field with respect to space is expanded using central differences;

[0022] According to the M-order Taylor formula expansion, the electric field and magnetic field values ​​at the next moment are obtained:

[0023]

[0024] Where t represents the time, Δt represents the time step used for iteration. According to formula (3), the high-order derivatives of the electromagnetic field with respect to time in formula (5) are transformed into multiple derivatives with respect to space, and the iterative formula of any high-order (M-order) time-domain finite-difference method is obtained:

[0025]

[0026] Preferably, the specific method for calculating different time steps for each Yee grid unit to meet the stability condition is:

[0027] The Δt of the Yee grid unit is selected to meet the stability condition and consider the time-harmonic field condition, that is,

[0028] f(x,y,z,t)=f0(x,y,z)e jωt (7)

[0029] Where ω represents the angular frequency, f0(x,y,z) represents the field value at the coordinate (x,y,z), and formula (7) is the solution of the following first-order differential equation:

[0030]

[0031] Formula (5) is simplified according to formula (8) and becomes:

[0032]

[0033] Simplifying formula (9), we get:

[0034]

[0035] Where q represents the numerical growth factor. Numerical stability requires that when time t tends to infinity and Δt is small enough, the growth factor needs to satisfy |q|≤1. Let a=ωΔt, and the set of solutions satisfying |q|≤1 is:

[0036]

[0037] From Maxwell's equations, it can be deduced that any right-angle component of the electromagnetic field satisfies the homogeneous wave equation:

[0038]

[0039] Where c represents the propagation speed of light in the medium, considering the solution of plane wave, that is:

[0040]

[0041] Where f1 is a constant and the central difference approximation is used, the second-order derivative in formula (12) is approximately:

[0042]

[0043] Where Δx, Δy and Δz represent the lengths of the Yee grid unit on the x, y and z axes respectively. Substituting formula (13) into formula (14), we get:

[0044]

[0045] The discretized form of the wave equation is:

[0046]

[0047] The relationship between the wave vector and the frequency in the discretized plane wave equation (13) can be rewritten as:

[0048]

[0049] For any k x , k y , k z The sufficient condition for both to hold is

[0050]

[0051] The maximum Δt that satisfies the stability condition is the time step of the Yee grid cell.

[0052] Preferably, the electromagnetic field in the fine grid is expanded using the M (M≥1) order Taylor formula:

[0053]

[0054] Among them U f are the electric and magnetic fields in the fine grid, Δt f is the time step of electromagnetic field iteration in fine grid, n is a positive integer, 1≤n≤N;

[0055] (a) When the electromagnetic field of a fine grid that is not adjacent to the coarse grid is iterated, the above formula is transformed into:

[0056]

[0057] When the electromagnetic field in the fine grid is iterated, the electromagnetic field information in the adjacent coarse grid is needed, and the second term on the right side of the above formula is expanded:

[0058]

[0059] The first term on the right side of the above formula is the normal update part of the fine grid field value, and the second term is the part that needs to be transferred or interpolated between the coarse and fine grids. Expressed as a Taylor formula expansion, the solution is obtained using the electromagnetic field information of the coarse grid at the previous moment:

[0060]

[0061] Thus, the electromagnetic field iteration formula of the fine grid adjacent to the coarse and fine grid is obtained:

[0062]

[0063] When the iteration reaches t+N i Δt f At this moment, it means that the iteration of fine grid field value in a single time step has been completed;

[0064] (b) Field value iteration of coarse grid:

[0065]

[0066] When the coarse grid field value is iterated to the adjacent coarse and fine grids, the fine grid field value information is needed. This information has been obtained in the fine grid electromagnetic field iteration and is directly used for iteration.

[0067] Compared with the prior art, the present invention has the following significant advantages: (1) The present invention can implement iteration of non-uniform Yee grid electric and magnetic fields with different time steps, thus reducing the time required for calculation. (2) The present invention can obtain higher accuracy than the traditional time domain difference method. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 It's Yee Grid.

[0069] Figure 2 It is a schematic diagram of iterations of different time steps in the calculation area.

[0070] Figure 3 is a schematic diagram of the metal target model.

[0071] Figure 4 It is a comparison diagram of the time domain waveforms of the observation points. DETAILED DESCRIPTION

[0072] The present invention is further described in detail below in conjunction with the accompanying drawings.

[0073] The present invention is a local time stepping technique based on an arbitrary high-order finite-difference time-domain method, the steps of which are as follows:

[0074] Step 1: Establish the geometric model of the target and discretize the model using non-uniform Yee grid.

[0075] Step 2: Use Maxwell's equations to describe the relationship between the electric field and magnetic field on the Yee grid. In Maxwell's equations, the partial differential of the electromagnetic field with respect to space is replaced by the central difference, and the partial differential of the electromagnetic field with respect to time is expanded according to the high-order Taylor formula.

[0076] Maxwell's curl equation in known passive differential form:

[0077]

[0078]

[0079] Where ε and μ represent the dielectric constant and magnetic permeability respectively, E and H represent the electric field strength and magnetic field strength respectively;

[0080] Expanding, arranging and simplifying the curl of Maxwell's curl equation, we get

[0081]

[0082] in E x , E y and E z are the three components of the electric field in the rectangular coordinate system x, y and z, H x , H y and H z They are the three components of the magnetic field in the rectangular coordinate system x, y and z;

[0083] The partial differential of the z component of the magnetic field with respect to space is right Spatial position in rectangular coordinate system The central difference is used for expansion:

[0084]

[0085] The first row of Q in formula (3) multiplied by the first term of U is expanded to:

[0086]

[0087] At the same time, the partial differential of the electromagnetic field with respect to space is expanded using central differences;

[0088] According to the M-order Taylor formula expansion, the electric field and magnetic field values ​​at the next moment are obtained:

[0089]

[0090] Where t represents the time, Δt represents the time step used for iteration. According to formula (3), the high-order derivatives of the electromagnetic field with respect to time in formula (5) are transformed into multiple derivatives with respect to space, and the iterative formula of any high-order (M-order) time-domain finite-difference method is obtained:

[0091]

[0092] Step 3: Calculate the different time steps for each Yee grid unit to meet the stability condition, and arrange them in order from small to large as Δt i ,i=1,2,…,N,N is the number of different time steps, where Δt i+1 =N i Δt i , N i is a natural number, let the time step Δt i The corresponding Yee grid is a fine grid, and the time step is Δt i+1 The corresponding Yee grid is a coarse grid. At the beginning, i=1;

[0093] The Δt of the Yee grid unit is selected to meet the stability condition and consider the time-harmonic field condition, that is,

[0094] f(x,y,z,t)=f0(x,y,z)e jωt (7)

[0095] Where ω represents the angular frequency, f0(x,y,z) represents the field value at the coordinate (x,y,z), and formula (7) is the solution of the following first-order differential equation:

[0096]

[0097] Formula (5) is simplified according to formula (8) and becomes:

[0098]

[0099] Simplifying formula (9), we get:

[0100]

[0101] Where q represents the numerical growth factor. Numerical stability requires that when time t tends to infinity and Δt is small enough, the growth factor needs to satisfy |q|≤1. Let a=ωΔt, and the set of solutions satisfying |q|≤1 is:

[0102]

[0103] From Maxwell's equations, it can be deduced that any right-angle component of the electromagnetic field satisfies the homogeneous wave equation:

[0104]

[0105] Where c represents the propagation speed of light in the medium, considering the solution of plane wave, that is:

[0106]

[0107] Where f1 is a constant. Using the central difference approximation, the second-order derivative in formula (12) is approximately:

[0108]

[0109] Where Δx, Δy and Δz represent the lengths of the Yee grid unit on the x, y and z axes respectively. Substituting formula (13) into formula (14), we get:

[0110]

[0111] The difference approximation of the other two second-order derivatives in formula (12) also has a similar form, so the discrete form of the wave equation is

[0112]

[0113] The relationship between the wave vector and the frequency in the discretized plane wave equation (13) can be rewritten as:

[0114]

[0115] For any k x , k y , k z The sufficient condition for both to hold is

[0116]

[0117] The maximum Δt that satisfies the stability condition is the time step of the Yee grid cell.

[0118] Step 4: Iteratively solve the electromagnetic field in the fine grid. The electromagnetic field iteration N of the fine grid i After time steps, the time step length Δt i+1 The electromagnetic field of the corresponding Yee grid begins to iterate, and the electromagnetic field iteration on each grid will use the electromagnetic field on the adjacent grid. It can be judged from the position information that if there is no coarse grid adjacent to the fine grid iteration during the iteration process, the fine grid electromagnetic field will be iterated according to the Taylor formula expansion. If the coarse grid electromagnetic field is used in the fine grid iteration process, it can be obtained by interpolating the coarse grid electromagnetic field information at the previous moment through the Taylor formula;

[0119] The electromagnetic field in the fine grid is expanded using the M-order Taylor formula:

[0120]

[0121] Among them U f are the electric and magnetic fields in the fine grid, n is a positive integer, 1≤n≤N.

[0122] When the grid (i, j+1, k) is a coarse grid and the grid (i, j, k) is a fine grid, before iterating the electromagnetic field on the grid (i, j+1, k), it is necessary to know The four surrounding magnetic fields (such as Figure 1 blue box), The Δt of the grid is different from that of the grid (i, j+1, k), so it is necessary to calculate it by interpolation. The operation is shown in (a). When grid (i, j+1, k) and grid (i, j, k) are fine grids, the operation is shown in (b).

[0123] (a) When iterating the electromagnetic field of a fine grid that is not adjacent to the coarse grid, formula (19) can be transformed into:

[0124]

[0125] When the electromagnetic field in the fine grid is iterated, the electromagnetic field information in the adjacent coarse grid is needed. The second term on the right side of formula (19) is expanded according to formula (3):

[0126]

[0127] The first term on the right side of the above formula is the normal update part of the fine grid field value, and the second term is the part that needs to be transferred or interpolated between the coarse and fine grids. As an unknown quantity, this term can be expressed as the expanded form of Taylor's formula and solved using the electromagnetic field information of the coarse grid at the previous moment:

[0128]

[0129] From this, we can get the electromagnetic field iteration formula of the fine grid adjacent to the coarse and fine grid:

[0130]

[0131] When the iteration reaches t+N i Δt f moment, indicating that a single time step (Δt c ) within the fine grid field value iteration.

[0132] (b), coarse grid (time step is Δt c )’s field value iteration:

[0133]

[0134] When the coarse grid field value iterates to the adjacent coarse and fine grids, the fine grid field value information is needed. This information has been obtained in the fine grid electromagnetic field iteration and can be used directly for iteration.

[0135] Step 5: The above is a local time step. When the electromagnetic field of the fine grid is iterated N times, i N i+1 time steps, the electromagnetic field of the coarse grid is iterated N times. i+1 time steps, let i = i + 1, and set the next time step Δt i+1 The corresponding Yee grid is used as a coarse grid, and the previous coarse grid becomes a fine grid. By analogy, the field value iterations in all non-uniform Yee grids can be completed until the total time of all field value iterations meets the requirements.

[0136] Data post-processing, extracting relevant physical parameters based on the calculated field values.

[0137] In order to verify the correctness and effectiveness of the present invention, the electromagnetic scattering characteristics of a metal model are analyzed below. Example: The calculation model is as follows Figure 2 As shown, the CFDTD calculation area grid number is 64*204*850, the total field boundary area is set at the 23rd and 42nd grids in the x direction, the 23rd and 182nd grids in the y direction, and the 235th and 635th grids in the z direction. The incident angle θ = 90°. The incident pulse center frequency is 150GHz, the pulse width is 2.4ns, Δx=Δy=10·Δz=0.0002m, the iteration time step is 6000 steps, the grid where the observation point is located is (17, 17, 285), in the CFDTD calculation area, the number of PML to extrapolation grids in the x and y directions is 6, the number of extrapolation to total field grids is 6, the number of total field to scatterer distance grids is 5, and the number of PML layers is 10. In the z direction, the number of PML to extrapolation grids is 60, the number of extrapolation to total field grids is 60, the number of total field to scatterer distance grids is 50, and the number of PML layers is 100. CFDTD takes 21484s. The local time stepping technology based on arbitrary high-order finite-difference time-domain method (LTS-ADER-FDTD) calculation area grid is 64*204*94, set at the 23rd and 42nd grids in the x direction, the 23rd and 182nd grids in the y direction, and the 23rd and 72nd grids in the z direction. The 33rd to 37th rows of grids and the 58th to 62nd rows of grids in the z direction are all fine grids with a size of Δx=Δy=10·Δz=0.0002m. The other rows of grids are all coarse grids with a size of Δx=Δy=Δz=0.0002m. The coarse grid iteration time step is 1000 steps. The observation point is located at the grid (17, 17, 28). LTS takes 2281s.

[0138] Table 1 Comparison of time consumption of each algorithm

[0139] algorithm time consuming CFDTD (Uniform Grid) 21484s ADER-FDTD (non-uniform grid) 7264s LTS-ADER-FDTD (non-uniform grid) 2281s

Claims

1. A local time stepping method based on an arbitrary high-order finite-difference time-domain method, characterized in that: Here are the steps: Step 1: Establish the geometric model of the target and discretize the model using non-uniform Yee grid; Step 2: Use Maxwell equations to describe the relationship between the electric field and magnetic field on the Yee grid. In Maxwell equations, the partial differential of the electromagnetic field with respect to space is replaced by the central difference, and the partial differential of the electromagnetic field with respect to time is expanded according to the high-order Taylor formula. Step 3: Calculate the different time steps for each Yee grid unit to meet the stability condition, and arrange them in order from small to large as Δt i ,i=1,2,…,N,N is the number of different time steps, where Δt i+1 =N i Δt i , N i is a natural number, let the time step Δt i The corresponding Yee grid is a fine grid, and the time step is Δt i+1 The corresponding Yee grid is a coarse grid. At the beginning, i=1; Step 4: Iteratively solve the electromagnetic field in the fine grid. The electromagnetic field iteration N of the fine grid i After time steps, the time step length Δt i+1 The electromagnetic field of the corresponding Yee grid starts to iterate. If there is no coarse grid in the adjacent position of the fine grid iteration during the iteration process, the fine grid electromagnetic field is iterated according to the Taylor formula expansion; if the coarse grid electromagnetic field is used in the fine grid iteration process, it is obtained by interpolation of the known coarse grid electromagnetic field information through the Taylor formula; Step 5: When the electromagnetic field of the coarse grid has completed a time step Δt i+1 , return to step 4 until the electromagnetic field of the fine grid has been iterated N times. i N i+1 time steps, the electromagnetic field of the coarse grid is iterated N times. i+1 time step, let i = i + 1, and set the next time step Δt i+1 The corresponding Yee grid is used as a coarse grid, and the previous coarse grid becomes a fine grid. Return to step 4 to iteratively solve the electromagnetic field on the coarse and fine grids until the electromagnetic fields of all grids are iterated, and the electromagnetic field value information at any time in the space is obtained.

2. The local time stepping method based on arbitrary high-order finite-difference time-domain method according to claim 1, characterized in that: In the Yee grid unit, the electromagnetic field satisfies the Maxwell curl equation in passive differential form, specifically: Where ε and μ represent the dielectric constant and magnetic permeability respectively, E and H represent the electric field strength and magnetic field strength respectively; Expanding, arranging and simplifying the curl of Maxwell's curl equation, we get in E x , E y and E z are the three components of the electric field in the rectangular coordinate system x, y and z, H x , H y and H z They are the three components of the magnetic field in the rectangular coordinate system x, y and z; The partial differential of the z component of the magnetic field with respect to space is right Spatial position in rectangular coordinate system The central difference is used for expansion: The first row of Q in formula (3) multiplied by the first term of U is expanded to: At the same time, the partial differential of the electromagnetic field with respect to space is expanded using central differences; According to the M-order Taylor formula expansion, the electric field and magnetic field values ​​at the next moment are obtained: Where t represents the time, Δt represents the time step used for iteration. According to formula (3), the high-order derivatives of the electromagnetic field with respect to time in formula (5) are transformed into multiple derivatives with respect to space, and the iterative formula of any high-order (M-order) time-domain finite-difference method is obtained:

3. The local time stepping method based on arbitrary high-order finite-difference time-domain method according to claim 2, characterized in that: The specific method for calculating the different time steps for each Yee grid unit to meet the stability condition is: The Δt of the Yee grid unit is selected to meet the stability condition and consider the time-harmonic field condition, that is, f(x,y,z,t)=f0(x,y,z)e jωt (7) Where ω represents the angular frequency, f0(x,y,z) represents the field value at the coordinate (x,y,z), and formula (7) is the solution of the following first-order differential equation: Formula (5) is simplified according to formula (8) and becomes: Simplifying formula (9), we get: Where q represents the numerical growth factor. Numerical stability requires that when time t tends to infinity and Δt is small enough, the growth factor needs to satisfy |q|≤1. Let a=ωΔt, and the set of solutions satisfying |q|≤1 is: From Maxwell's equations, it can be deduced that any right-angle component of the electromagnetic field satisfies the homogeneous wave equation: Where c represents the propagation speed of light in the medium, considering the solution of plane wave, that is: Where f1 is a constant and the central difference approximation is used, the second-order derivative in formula (12) is approximately: Where Δx, Δy and Δz represent the lengths of the Yee grid unit on the x, y and z axes respectively. Substituting formula (13) into formula (14), we get: The discretized form of the wave equation is: The relationship between the wave vector and the frequency in the discretized plane wave equation (13) can be rewritten as: For any k x , k y , k z The sufficient condition for both to hold is The maximum Δt that satisfies the stability condition is the time step of the Yee grid cell.

4. The local time stepping method based on arbitrary high-order finite-difference time-domain method according to claim 1, characterized in that: The electromagnetic field in the fine grid is expanded using the M (M ≥ 1) order Taylor formula: Among them U f are the electric and magnetic fields in the fine grid, n is a positive integer, 1≤n≤N, Δt f is the time step for electromagnetic field iteration in the fine grid; (a) When the electromagnetic field of a fine grid that is not adjacent to the coarse grid is iterated, the above formula is transformed into: When the electromagnetic field in the fine grid is iterated, the electromagnetic field information in the adjacent coarse grid is needed, and the second term on the right side of the above formula is expanded: Among them U c is the electric field and magnetic field in the coarse grid. The first term on the right side of the above formula is the normal update part of the fine grid field value, and the second term is the part that needs to be transferred or interpolated between the coarse and fine grids. Expressed as a Taylor formula expansion, the solution is obtained using the electromagnetic field information of the coarse grid at the previous moment: Thus, the electromagnetic field iteration formula of the fine grid adjacent to the coarse and fine grid is obtained: When the iteration reaches t+N i Δt f At this moment, it means that the iteration of fine grid field value in a single time step has been completed; (b) Field value iteration of coarse grid: When the coarse grid field value iterates to the adjacent coarse and fine grids, the fine grid field value information is needed. This information has been obtained in the fine grid electromagnetic field iteration and is directly used for iteration.

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