CN116127699B - Local time step arbitrary high-order precision full-wave solving method based on time domain spectral element method - Google Patents
Local time step arbitrary high-order precision full-wave solving method based on time domain spectral element method Download PDFInfo
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Abstract
The invention discloses a local time step arbitrary high-order precision full-wave solving method based on a time domain spectral element method, which comprises the following steps: carrying out structural modeling according to the model size; according to the model characteristics, carrying out space subdivision refinement on a region with a severe fine structure or unknown quantity change; adopting full conformal mesh subdivision; taking a spectral element method as a solution platform, and adopting a high-order orthogonal basis function to perform Galerkin test on unknown quantity in the wave equation to obtain a wave equation in a semi-discrete format; solving each order partial derivative term of the unknown quantity by adopting any high-order precision format; accelerating the full-wave solving process by adopting a local time step technology; on the premise of ensuring the solving precision, adjusting the sizes of all the subdivision grids to the maximum; on the premise of solving convergence, adjusting the global time step to the maximum; on the premise of solving convergence, the local time step is adjusted to the maximum, and the ratio of the global time step to the local time step is calculated. The invention can improve the calculation efficiency and reduce the calculation cost.
Description
Technical Field
The invention relates to a time domain method for calculating electromagnetism, in particular to a local time step arbitrary high-order precision full-wave solving method based on a time domain spectral element method.
Background
The spectral element method (SETD) combines the advantages of high precision of the spectral method and flexible modeling of the finite element method, and attracts attention of students at home and abroad in the fields of electromagnetic wave propagation, high-power microwave breakdown and multi-physical field simulation. With the increase of the order of the basis function, the calculation error is exponentially reduced, the subdivision size is continuously increased, and the number of unknown quantities is continuously reduced; meanwhile, grid discretization is carried out by adopting a curved hexahedron, so that the modeling flexibility is improved; because the basis function with orthogonality is adopted, the obtained global quality matrix has the characteristics of block diagonal or even diagonal, the inversion speed of the matrix is accelerated, and the solving efficiency of the unknown quantity is further improved.
The spectral element method also develops along with the development of time stepping researches, and the methods such as central difference, dragon-Gregory tower and the like enable the SETD method to have inherent parallelism, and a large matrix equation does not need to be solved at each time step. However, the central differential format has only a second order accuracy in time, which does not match the higher order spatial accuracy of the SETD method [ Kan X, chen R, eng Y, et al Transient analysis ofmicrowave Gunn oscillator using extended spectral element time domain method [ J ]. Radio Science,2011,46 (5): 1-9 ]. The RK time discretization scheme has its own higher order form, whereas the performance in terms of computational efficiency will drop drastically when the order is greater than 4, due to the so-called Butcher barrier [ Enright W H.the Numerical Analysis of Ordinary Differential Equations:Runge-Kutta and General Linear Methods (J.C.Butcher) [ M ]. J.Wiley,2012 ]. Furthermore, the use of explicit time stepping formats is disadvantageous for time stepping, subject to severe constraints of time stability. While center differencing with Explicit format combines the implicit newmark- β Time stepping format [ Xu H, ding D Z, chen R.AHybridExplicit-Implicit Scheme for Spectral-Element Time-Domain Analysis of Multiscale Simulation ] [ J ]. Applied Computational Electromagnetics Society journal,2016,31 (4): 444-449 ], multi-scale modeling can be achieved, but limits the parallelism of SETD. Therefore, the time discretization scheme with high efficiency, flexibility and high-order precision has important significance for the application of the spectral element method in electromagnetic analysis.
Disclosure of Invention
The invention aims to provide a local time step arbitrary high-order precision full-wave solving method based on a time domain spectral element method, which improves the calculating efficiency, solves the problems of smaller global time step and low solving efficiency caused by space and time multiscale, and can shorten the calculating time by times and reduce the calculating cost.
The technical scheme for realizing the purpose of the invention is as follows: a local time step arbitrary high-order precision full-wave solving method based on a time domain spectral element method comprises the following six steps:
step 1, carrying out structural modeling on the geometric dimension of a target according to the full-wave solving requirement;
step 2, preliminarily setting the maximum subdivision grid size according to one tenth of medium wavelength, adopting finer subdivision grid size for the region with more intense fine structure and field change according to model size characteristics and frequency characteristics, and carrying out full conformal subdivision on the whole model;
step 3, taking a spectral element method as a solution platform, and adopting a high-order orthogonal basis function to perform Galerkin test on unknown quantity in the wave equation to obtain a semi-discrete format, namely a wave equation in a space discrete format;
step 4, adopting any high-order precision to replace a direct differential format for unknown quantity solving, and sequentially obtaining any order partial derivative;
step 5, adopting a local time step technology, adopting a larger time step for a linear region which does not need to be thinned in space according to a stability condition, adopting a smaller time step for a region (such as a nonlinear region) with more intense time change of a space thinned region and a field, and obtaining a field value of the smaller time step through field interpolation calculation of the larger time step;
and 6, optimizing the subdivision size and the multi-stage time steps, continuously adjusting the mesh subdivision size of the solving target, determining the maximum subdivision size on the premise of meeting the precision requirement, adjusting the time steps, and determining the multi-stage time steps on the premise of meeting the solving convergence to obtain the maximum acceleration efficiency.
Compared with the prior art, the invention has the remarkable advantages that:
(1) According to the method, an arbitrary high-order precision method (ADER) is adopted, an arbitrary order expansion unknown quantity of Taylor expansion can be selected, and when the expansion order is not larger than the order of a base function, the calculation error is exponentially reduced along with the increase of the expansion order;
(2) The method adopts a local time stepping method (LTS), and when the multi-scale phenomenon is obvious, particularly when the unknown quantity occupied by the fine structure is small, obvious acceleration effect can be obtained;
(3) On the premise of meeting the stability condition, the time step size of the method can be increased along with the increase of ADER order.
(4) The spectrum element method is used as a platform, global conformal subdivision is adopted, and the method has a good accelerating effect on an electromagnetic target with small electric size, so that the diagonal characteristic of a mass matrix block can be maintained, the calculation speed can be increased, less unknown quantity can be maintained, the calculation memory is reduced, and the calculation cost is reduced.
Drawings
FIG. 1 is a schematic diagram of a coarse grid and fine grid coupling solution process incorporating a local time stepping technique.
FIG. 2 is a graph of simulation result error contrast using Taylor formulas of different orders to develop unknowns.
FIG. 3 is a graph showing the comparison of simulation result errors by adopting different ratios between coarse grids and fine grids.
Fig. 4 is a schematic diagram of a waveguide coupler having a multi-scale structure.
Fig. 5 is a graph of electric field strength at field probe 1 for a waveguide directional coupler using LTS-ADER-SETD and ADER-SETD methods.
Fig. 6 is a graph of electric field strength contrast at field probe 2 for a waveguide directional coupler using the LTS-ADER-SETD method and the ADER-SETD method.
Fig. 7 is a graph of electric field strength contrast at field probe 3 for a waveguide directional coupler using LTS-ADER-SETD and ADER-SETD methods.
Fig. 8 is a graph of electric field strength at field probe 4 for a waveguide directional coupler using the LTS-ADER-SETD method and the ADER-SETD method.
Fig. 9 is a graph comparing the S parameters of waveguide directional couplers using the LTS-ADER-SETD method and the ADER-SETD method with those found in the references.
Detailed Description
The method provided by the invention is a local time step arbitrary high-order precision full-wave solving method based on a time domain spectral element method (the method is abbreviated as an LTS-ADER-SETD method in the patent, and any high-order precision full-wave solving method without the local time step is abbreviated as an ADER-SETD method), the method introduces a local time step technology and any high-order precision explicit solving method into the time domain spectral element method, and because a plurality of electromagnetic targets comprise time and space multi-scale problems, the solving area can be divided by the local time step method, grids with different sizes are adopted for conformal subdivision, and unknown quantity is subjected to Taylor expansion and reduction by any high-order precision method, so that simulation precision can be improved. The method specifically comprises the following steps:
firstly, carrying out structural modeling on the full-wave solution based on the geometric dimension of a target according to the requirement of the full-wave solution;
secondly, preliminarily setting the maximum subdivision grid size according to one tenth of medium wavelength, adopting finer subdivision grid size for a region with more intense change of a fine structure or a field along time (the change rate of an electric field is five times that of other regions in the region compared with other regions) according to model size characteristics and frequency characteristics, and carrying out full conformal subdivision on the whole model;
thirdly, taking a spectral element method as a solution platform, and adopting a high-order orthogonal basis function to perform Galerkin test on unknown quantity in the wave equation to obtain a semi-discrete format, namely a wave equation in a space discrete format;
fourthly, adopting any high-order precision to replace a direct differential format for unknown quantity solving, and sequentially obtaining any order partial derivatives;
fifthly, adopting a local time step technology, adopting a larger time step for a linear region which does not need to be thinned in space according to a stability condition, adopting a smaller time step for a region (such as a nonlinear region) with more intense time change of a fine structure and a field, and obtaining a field value of the smaller time step through field interpolation calculation of the larger time step, wherein the larger and smaller time steps are relatively speaking, and represent that the linear region step length which needs to be thinned is larger than the region step length of the spatial thinning region and the region step length with more intense time change of the field;
and step six, optimizing the subdivision size and the multi-stage time step, continuously adjusting the mesh subdivision size of the solving target, determining the maximum subdivision size on the premise of meeting the precision requirement, adjusting the time step, and determining the multi-stage time step on the premise of meeting the solving convergence to obtain the maximum acceleration efficiency.
In the first step, structural modeling is carried out on the microwave device based on the geometric dimension of the target according to the full-wave solving requirement, the three-dimensional structural shape of the microwave device is drawn in 3D drawing software according to the structural dimension of the microwave device, then the model is segmented into different areas according to conformal requirements, and for a complex structure, operations of setting material numbers of certain areas, setting parameters of the structure and the like are carried out for the convenience of algorithm processing;
in the second step, the maximum subdivision grid size is preliminarily set according to one tenth of medium wavelength, and finer subdivision grid size is adopted for the area with more intense fine structure or field change according to model size characteristics and frequency characteristics, and full conformal subdivision is carried out on the whole model, specifically: the method is provided for conformal grid, and globally adopts conformal grid subdivision. Adopting finer subdivision size for the area with severe change of the fine structure and the field along with time, and carrying out grid subdivision on the area, wherein the area is called a fine grid area; for a linear region where space does not need to be refined, after meshing, the region is called a coarse mesh region.
In the third step, a spectral element method is taken as a solution platform, and a Galerkin test is carried out on the unknown quantity in the wave equation by adopting a high-order orthogonal basis function to obtain a wave equation in a semi-discrete format, namely a space discrete format, specifically comprising the following steps: wave motion in a lossy medium
Whereas in an lossless medium, the conductivity is 0, which can be written as
Wherein sigma, epsilon and mu are respectively conductivity (S/m) and dielectric constant (F/m) 2 ) Permeability (H/m), vacuum dielectric constant ε 0 ＝8.85×10 -12 F/m 2 Vacuum permeability mu 0 ＝1.2566×10 -6 H/m, E is electric field intensity (V/m), the unknown quantity is spatially discretized, the Galerkin test is carried out on each discretized unknown quantity by adopting a high-order orthogonal basis function, the spectrum precision can be improved through polynomial order, the calculation error can be reduced exponentially, the orthogonal basis function enables the variation to solve partial differential equation, the quality matrix is diagonal or even diagonal, and the matrix inversion speed is greatly accelerated.
Wherein N is ei Testing basis functions for physical domains, N ej And (3) expanding a basis function for a physical domain, wherein e is an electric field coefficient, v is a volume, s is an area, and residual integration is 0 on the whole solving domain according to the continuous Galerkin test requirement. The above compact format can be written as being consistent in terms of divergence and rotation by considering the closed-domain boundary condition PEC, i.e. perfect electrical boundary
Wherein T is a mass matrix, S is a stiffness matrix, ts is a damping matrix, and the specific expression is
[T] ij ＝ε∫(J -1 Φ i )·(J -1 Φ j )|J|dξdηdζ (1.6)
[T S ] ij ＝σ∫(J -1 Φ i )·(J -1 Φ j )|J|dξdηdζ (1.7)
Wherein J is Jacobian matrix required by mapping relation, phi i 、Φ j Respectively the basis functions N ei 、N ej Mapping in a reference domain, i, j values are 0,1,..t, t is the order of the basis function; whereas the open domain boundary condition PML + ABC, i.e. the perfect matching layer and the first order impedance absorption boundary condition, is considered.
In the fourth step, the unknown quantity solving adopts any high-order precision to replace a direct differential format, and any order partial derivative term is obtained in sequence. In particular, for the closed domain boundary condition PEC,
i.e.
Wherein the method comprises the steps ofp is defined as the first time derivative of the electric field, I is a unit array, and the unknowns u can be developed from the taylor series as follows:
because of M -1 L is a time-independent spatial matrix, so
So (1.12) can be written approximately as
Any higher order precision format is used as an explicit time stepping format, and then the time stability condition is determined, wherein the stability criterion is that
Z is a growth factor, and lambda is a matrix M -1 Complex eigenvalues of L. The absolute stability region is determined jointly by the real and imaginary parts of λΔt, and will increase with increasing summation order. The use of a higher order ADER time stepping format allows for a larger time stepping to be used to ensure convergence of the solution process. However, since the process of solving complex eigenvalues is time consuming, it is also possible to combine the sampling theorem, give an empirically based stability criterion,
wherein C is the Courant constant, l min Is the minimum subdivision size, c is the speed of light, f max To solve for the maximum of the frequency range.
In the fifth step, the solving area is decomposed into two subdomains omega according to the thickness of the subdivision grid c And omega f Coarse and fine grids are relative concepts when directly satisfying one tenth of a waveThe long or larger mesh is a coarse mesh, and the coarse mesh is generally several times the fine mesh in size, Δt c And Deltat f For two subfields of different time steps, there is the following identity relationship:
Δt f :Δt c ＝1:N LTS (1.16)
in equation (1.13), the unknown quantity update requires the value of the unknown quantity at the previous moment and the time derivative of each step, and the ADER format is evolved to the following matrix to represent the coupling between different areas, and the coupling process diagram is shown in figure 1
For fine areas, small time steps are used for calculation, while for neighboring coarse grid areas, small time step values are interpolated, for coarse grid areas, large time steps are used for solving, in particular,
first, assume that in a fine-meshed area, the time n delta t is an integer large time step c Electric field value u of (n is a positive integer) f (nΔt c ) Is known, the value of the i-th electric field derivative at the last small time step in a large time stepElectric field value u at integer large time step moments of coarse grid region c (nΔt c ) At the beginning of the iteration, 0, the value will be updated continuously after the source is added. According to the above assumption or updated value, first, the i-order electric field derivative value of the fine grid region at the time of the whole number of time steps is obtained>
Then the electric field value u of the next big time step of the coarse grid area is obtained c [(n+1)Δt c ]The coupling matrix proposed above will be used here,
then the electric field derivative of the coarse grid region time at the moment of large time steps can be reduced to be in any high-order precision time stepping format
The formula 1.20 can be written as,
next, t=nΔt is calculated c +mΔt f I-order electric field derivative value of coarse grid region at moment
Fine mesh region at t=nΔt c +mΔt f Can be updated as
Fine mesh region at t=nΔt c +(N LTS -1)Δt f The i-order electric field derivative value of (2) can be updated as
Thus, the updating of the initial assumption value is completed, and the iteration solution is continuously carried out to obtain the value at the moment of the large time step of all the simulations.
Step six, firstly dissecting according to the uniform size of the large grid, and dissecting according to the maximum size of the limitation if some structures limit the size of the large grid; other unrestricted portions may be as coarse as possible when the CFL conditions are met, where the maximum size is tried (so that the S parameter is unchanged, correct) because the time steps are limited by the large grid minimum size. Then, the subdivision is determined by the fine structure according to the subdivision size of the small mesh.
Then a larger LTS (local time step) is set, say 10 times, and then the time step is increased (larger) until the LTS diverges, and the LTS is changed by a multiple of the LTS by adopting the diverged LTS, and the LTS still diverges, so that the limited large grid is illustrated. A time step is determined.
Finally, a smaller LTS, e.g., 5 times, is set, and then the LTS multiple is reduced (smaller) until it diverges. LTS is determined.
Examples
Taking a rectangular resonant cavity with a size of 1.0m x 0.5m x 1.5m as an example, the high accuracy of the proposed ADER-SETD method and its local time stepping scheme was verified. Here, the cavity is excited by the TE101 mode and the entire computational domain is divided into 6 cubic units. First, a total time step number of 800 steps, a step size of 83.39ps, was selected and the convergence error of the global time step ADER-SETD method under a series of fixed-order spatial basis functions (ns=3, 4,5 and 6) was studied. As shown in fig. 2, for the global time step ADER-SETD method, the error decreases exponentially with increasing integral order over time. When the time integral order satisfies the criterion that Nt is not less than ns+1, the precision level is affected by the spatial basis function order.
Then, a rectangular resonant cavity with the size of 1.0mx0.5mx1.5m is still selected, the accuracy of the proposed LTS-ADER-SETD method is verified, the time increment of 2 cells in 6 cells (subdivision units) is smaller, and the time increment of other elements is the same as that of the global time stepping ADER-SETD method. According to the stability criterion of the high-order space basis function, the time step increment of the ADER-SETD method of global time stepping is 16.68ps, and 4000 steps of time are obtained. To secure the calculation accuracy, as described above, the order Nt of the time integration is set to ns+1. When nlts=10, 100 or 1000, the time step increment of the LTS-ADER-SETD method is small. As shown in fig. 3, the convergence error of LTS-ADER-SETD at the increasing order of the spatial basis function illustrates that the proposed LTS-ADER-SETD method inherits the spectral accuracy of the ADER-SETD method with a global time step, even with a very large time step ratio.
To demonstrate the efficiency of the LTS-ADER-SETD method, the example model employed herein was a five-branch E-face 3dB waveguide directional coupler operating in the terahertz band provided in the literature [ Dai B, zhang B, fanY. Design of a 600GHz Broadband 3dB Hybrid Coupler[C ]//2020 13th UK-Europe-China Workshop on Millimetre-Waves and Terahertz Technologies (UCMMT) 2020 ]. The coupler is a commonly used microwave device for power synthesis and distribution, and particularly has a design with strict requirements on coupling degree. The classical five-branch structure not only can ensure the wider bandwidth of terahertz wave bands, but also can reduce the processing difficulty caused by excessive branches. The coupler is of a symmetrical structure, wherein the long side a=0.381 mm, the short side b= 0.1905mm of the waveguide port, the height k=0.061 mm of the discontinuous structure at the upper end and the lower end of the waveguide, the width h=0.075 mm, the heights of the branches at the two sides are c=0.124 mm, the width e=0.05 mm, the heights of the three branches in the middle are d=0.114 mm, the width f=0.075 mm, the gap lengths at the two sides at the left side and the right side are the same as the width of the discontinuous structure, and the gap lengths g=0.05 mm at the middle. In the simulation of LTS-ADER-SETD, the waveguide coupler Z-direction uses PML as the absorption boundary and ABC truncated computational domain, as shown in fig. 4, four field probes are placed at four ports.
The total simulation time is 90ps, the time step used in ADER-SETD is 9fs, while the LTS-ADER-SETD method used can use a time step of 54fs in the coarse grid area and a time step of 9fs only in the fine grid area. In the simulation process, a modulated Gaussian excitation source with a center frequency of 600GHz is adopted, and fig. 5-8 show time domain waveforms of electric fields at four observation points of the structure, so that the electric field values calculated by adopting two methods are consistent, and the accuracy of using local time steps is illustrated. FIG. 9 is a simulated comparison of the S parameters using ADER-SETD, LTS-ADER-SETD as set forth in this patent with the reference, and it can be seen that the use of ADER-SETD and LTS-ADER-SETD is consistent with the results of the above references. Table 1 shows that the use of the simulation resources by the global unified time step ADER-SETD is compared with the use of the simulation resources by the local time step (LTS-ADER-SETD), and it can be seen that the simulation time is saved by nearly 80% under the condition that the occupied memory is almost unchanged.
Table 1 comparison of resource occupancy
Claims (7)
1. A local time step arbitrary high-order precision full-wave solving method based on a time domain spectral element method is characterized by comprising the following steps:
step 1, carrying out structural modeling on a target based on the geometric dimension of the target according to the full-wave solving requirement;
step 2, setting an initial value of the maximum subdivision grid size, adopting finer subdivision sizes for the area with severe change of a fine structure and a field along with time according to the size characteristics and the frequency characteristics of the model, and carrying out full conformal subdivision on the whole model;
step 3, taking a spectral element method as a solution platform, and adopting a high-order orthogonal basis function to perform Galerkin test on unknown quantity in the wave equation to obtain a semi-discrete format, namely a wave equation in a space discrete format;
step 4, adopting any high-order precision format to replace a direct differential format for unknown quantity solving, and sequentially obtaining any order partial guide item; the method comprises the following steps: for closed domain boundary conditions PEC:
i.e.
Wherein T is a mass matrix, S is a stiffness matrix, T s In order to provide a damping matrix,p is the first time derivative of the electric field, e is the electric field coefficient, I is the unit array, and the unknown u is developed by the Taylor series:
because of M -1 L is a spatial matrix that does not depend on time variation, so:
so the equation (1.5) is approximately written as:
step 5, adopting a local time step technology, adopting a larger time step for a linear region which does not need to be thinned in space according to a stability condition, adopting a smaller time step for a region with a fine structure and a field which is violent in change along with time, and obtaining a field value of the smaller time step through field interpolation calculation of the larger time step; the method specifically comprises the following steps:
decomposing the solving area into fine grid areas omega according to the thickness of the split grid c And coarse grid region Ω f The coarse mesh and the fine mesh are opposite, and the mesh meeting the tenth wavelength size or larger is the coarse mesh, deltat c And Deltat f For two regions of different time steps, the two have the following identity relationship:
Δt f :Δt c ＝1:N LTS (1.7)
in equation (1.6), the updating of the unknown quantity is requiredThe unknown value and its time derivative are changed from any high-order precision format to fine lattice region omega c And coarse grid region Ω f The coupling matrix between the two is:
first, assume that in a fine-meshed area, the time n delta t is an integer large time step c Electric field value u of (2) f Is known, n is a positive integer whose i-th electric field derivative has a value at the last small time step in a large time stepElectric field value u at integer large time step moments of coarse grid region c 0 at the beginning of the iteration, and after the source is added, the value is updated continuously; according to the above assumption or updated value, first, the i-order electric field derivative value of the fine grid region at the time of integer large time step is obtained>
Then the electric field value u of the next big time step of the coarse grid area is obtained c [(n+1)Δt c ]
Based on the coupling matrix, then the electric field derivative of the coarse grid region time at the moment of large time steps is reduced by any high-order precision time stepping format as follows:
(1.11) the formula is:
then t=nΔt is calculated c +mΔt f I-order electric field derivative value of coarse grid region at moment
Fine mesh region at t=nΔt c +mΔt f The updating is as follows:
fine mesh region at t=nΔt c +(N LTS -1)Δt f The i-order electric field derivative value of (2) is updated as:
and continuously carrying out iterative solution based on the steps to obtain the value at the moment of the large time step of all the simulations.
And 6, repeating the steps 2-5, optimizing the subdivision size and the multi-stage time steps, continuously adjusting the mesh subdivision size of the solving target, determining the maximum subdivision size on the premise of meeting the precision requirement, adjusting the time step, and determining the multi-stage time step on the premise of meeting the solving convergence to obtain the maximum acceleration efficiency.
2. The method for solving the full wave of any high-order precision of local time step based on the time domain spectral element method according to claim 1, wherein in the step 1, the structural modeling of the target based on the geometric dimension of the target according to the full wave solving requirement is specifically as follows: firstly, drawing a three-dimensional structure shape model of the microwave device in 3D drawing software according to the structure size of the microwave device, then cutting the model into different areas according to conformal requirements, and finally performing conformal subdivision.
3. The method for solving the full wave of any high-order precision of the local time step based on the time domain spectral element method according to claim 1, wherein the initial value of the maximum subdivision grid size is one tenth of the medium wavelength.
4. The method for solving the full wave of any high-order precision in local time steps based on the time domain spectral element method according to claim 1, wherein the change rate of an electric field in a region with intense field change along with time is five times or more than that of an electric field in other regions.
5. The method for solving the full wave of the arbitrary high-order precision of the local time step based on the time domain spectral element method according to claim 1, wherein in the step 3, the unknown quantity in the wave equation is subjected to a galkin test by adopting a high-order orthogonal basis function to obtain the wave equation in a semi-discrete format, namely a space discrete format, specifically comprising the following steps:
the wave equation in the lossy medium was determined as:
whereas in the unconstrained medium, the conductivity is 0, the wave equation in the unconstrained medium is:
wherein sigma, epsilon and mu are respectively conductivity, dielectric constant and magnetic conductivity, E is electric field intensity, unknown quantity is spatially discretized, and Galerkin test is carried out on each discrete unknown quantity by adopting a high-order orthogonal basis function, wherein:
wherein N is ei Testing basis functions for physical domains, N ej For the physical domain expansion basis function, e is an electric field coefficient, v is a volume, s is an area, and according to the continuous Galerkin test requirement, the residual integral is 0 on the whole solving domain, so the above formula (1.18) is formed by considering the closed domain boundary condition PEC, namely a perfect electric boundary, according to the divergence compatibility and the rotation compatibility, the wave equation of the space discrete format is formed by the following steps:
wherein T is a mass matrix, S is a stiffness matrix, T s The damping matrix comprises the following specific expression:
[T] ij ＝ε∫(J -1 Φ i )·(J -1 Φ j )|J|dξdηdζ (1.21)
[T S ] ij ＝σ∫(J -1 Φ i )·(J -1 Φ j )|J|dξdηdζ (1.22)
wherein J is a Jacobian matrix required by a mapping relation, phi i 、Φ j Respectively the basis functions N ei 、N ej Mapping in a reference domain, i, j values are 0,1,..t, t is the order of the basis function; consider the open domain boundary condition pml+abc, i.e., the perfect matching layer and the first order impedance absorption boundary condition.
6. The method for solving the full wave of the arbitrary high-order precision of the local time step based on the time domain spectral element method according to claim 1, wherein the arbitrary high-order precision format is an explicit time step format, and the time stability condition is required to be determined, and the stability condition is as follows:
wherein Z is a growth factor and lambda is a matrix M -1 Complex eigenvalue of L, deltat is
Wherein C is the Courant constant, l min Is the minimum subdivision size, c is the speed of light, f max To solve for the maximum of the frequency range.
7. The method for solving the full wave of any high-order precision of local time step based on the time domain spectral element method according to claim 1, wherein the step 6 is specifically:
firstly, dividing a coarse mesh region according to the dividing size of the coarse mesh, and then dividing a fine mesh region according to the dividing size of the fine mesh;
setting a time step larger than the original time step for a linear region which is not required to be thinned in space, and then increasing the time step multiple until the linear region diverges to determine the time step of the linear region;
for the region where the fine structure and the field change drastically with time, a time step smaller than the current time step is set, and then the time step multiple is reduced until divergence, and the time step is determined.
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