CN117911651A - Three-dimensional structured sub-grid dividing method - Google Patents
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Abstract
The invention discloses a three-dimensional structured sub-grid dividing method, which comprises the following steps: step 1: selecting grids with overlapped thickness in space; step 2: the two grids are offset in space, so that the magnetic field of the coarse grid coincides with the corresponding magnetic field of the fine grid; step 3: using the pulse superposition scheme, the outermost refinement mesh is removed in each time sub-step of the refinement mesh, at the end of the loop the mesh is expanded back to its original size, and the missing field components are calculated using interpolation and extrapolation. An efficient low reflection algorithm is presented for fine resolution analysis of sub-regions of the FDTD modeling. When using cubic spline interpolation, the subgrid method has been widely tested and demonstrated to introduce reflections below 50dB over a wide frequency range. The calculation cost of the fine resolution obtained by the algorithm is moderate and is several orders of magnitude lower than that of the whole problem analysis under the fine resolution.
Description
Technical Field
The invention relates to the technical field of three-dimensional sub-grid division, in particular to a three-dimensional structured sub-grid division method.
Background
In electromagnetic simulation, time domain finite difference (FDTD) techniques have been widely and effectively used to solve various electromagnetic problems. In some problems of electromagnetic simulation, the accuracy of the solution can be greatly improved if finer discretization is used in a specific region of the computation space.
In general, an electric or magnetic field sometimes has a large gradient in only one direction within a limited volume. Such as sharp edges and corners from the vicinity of the current source to conductive and dielectric objects, and small-sized transmission lines connected to antennas radiating into free space with diffusers. If a sufficiently fine grid is used directly in the entire computation space, very large computer resources are required. Therefore, the optimization grid dividing method can greatly improve the simulation speed.
Among the past algorithms, there are three methods of obtaining finer grids in the region, namely, 1) sequential computation; 2) Sub-meshing (progressive meshing) is performed only in space; 3) Grid subdivision (local grid refinement) is performed spatially and temporally.
The earliest sequential calculation method is to use coarse grids to calculate in the whole calculation domain, then use fine grids to recalculate in a limited volume, and the field obtained by coarse grid simulation is used as the boundary value of fine grid calculation. Sub-meshing is only done in space relatively easy to implement and allows hierarchical meshing as well as various meshing steps in different directions. The truncation error of the method is first order, has second order convergence, and can reach second order precision through relatively simple improvement. However, this approach has two limitations. Firstly, the numerical dispersion varies greatly with the grid density, and secondly, the computational efficiency of the method is affected by the need to use the time step corresponding to the smallest grid in the overall computation space.
Two-dimensional sub-grids are implemented in selected sub-fields and spatially and temporally confined to three-dimensional sub-fields, each grid being individually configurable to a time step to achieve high efficiency. Furthermore, the numerical dispersion characteristics vary relatively little among different grids. The linear interpolation is carried out on the electric field, and the space-time average is carried out on the electric field and the magnetic field, so that the resolution of the subareas can be improved by 4 times. However, testing of rectangular waveguide thin metal plates has shown that numerical errors are relatively large due to abrupt changes in the mesh size. This results in relatively large errors in the calculated parameters.
Disclosure of Invention
The invention aims to solve the defects in the prior art and provides a three-dimensional structured sub-grid dividing method.
In order to achieve the above purpose, the present invention adopts the following technical scheme:
a three-dimensional structured sub-meshing method, comprising the steps of:
step 1: selecting grids with overlapped thickness in space;
step 2: the two grids are offset in space, so that the magnetic field of the coarse grid coincides with the corresponding magnetic field of the fine grid;
Step 3: using the pulse superposition scheme, the outermost refinement mesh is removed in each time sub-step of the refinement mesh, at the end of the loop the mesh is expanded back to its original size, and the missing field components are calculated using interpolation and extrapolation.
Further, in step 3, the interpolation scheme employs one or more of tri-linear interpolation, cubic spline, or conformal spline for achieving smooth transitions between coarse and fine meshes.
Compared with the prior art, the invention has the beneficial effects that:
the method can analyze the FDTD modeled subareas with fine resolution;
the grid density is increased by two times (recursion is possible), and the fine grid is spatially shifted relative to the coarse grid, smooth transition between the coarse and fine grids can be realized by interpolation and extrapolation, and reflection below 50dB is introduced in a wide frequency range;
The method can obviously improve the calculation precision without increasing calculation load, and the calculation cost of the obtained fine resolution is moderate and is several orders of magnitude lower than the calculation cost of the whole problem analysis under the fine resolution.
Drawings
The accompanying drawings are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate the invention and together with the embodiments of the invention, serve to explain the invention.
FIG. 1 is a flow chart of a three-dimensional structured sub-meshing method according to the present invention;
FIG. 2 is an overlapping grid employed in an embodiment of the present invention; FIG. 3 is a schematic diagram of an arrangement of grids in three-dimensional space according to an embodiment of the present invention;
Fig. 4 is a schematic diagram of instantaneous transfer of information by interpolation in an embodiment of the invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments.
Referring to fig. 1, a three-dimensional structured sub-meshing method includes the steps of:
step 1: selecting grids with overlapped thickness in space;
step 2: the two grids are offset in space, so that the magnetic field of the coarse grid coincides with the corresponding magnetic field of the fine grid;
Step 3: using the pulse superposition scheme, the outermost refinement mesh is removed in each time sub-step of the refinement mesh, at the end of the loop the mesh is expanded back to its original size, and the missing field components are calculated using interpolation and extrapolation.
In this embodiment, in step 3, the interpolation scheme employs one or more of tri-linear interpolation, cubic spline, or conformal spline for achieving a smooth transition between coarse and fine meshes.
For a better understanding of the technical solution of the present invention, the following description is further given with reference to the accompanying drawings and the specific embodiments.
Step 1: this step selects the refinement factor of the two grids and the overlap of the two grids, as shown in fig. 2 and 3, the fine and coarse grids being offset in all three directions.
In fig. 2, the coarse and fine grids are offset in the z-direction, and the gray areas represent overlapping pulsed grid areas;
Step 2: the two grids are spatially offset such that the magnetic field of the coarse grid coincides with the corresponding magnetic field of the dense grid (see "cross" in fig. 2).
This grid arrangement is chosen after a number of numerical experiments on other topologies. The criterion used is the minimum reflection from the fine grid. Another grid offset would coincide the electric fields.
Step 3: the outermost refinement mesh is removed in each time sub-step of the refinement mesh, at the end of the loop the mesh expands back to its original size and the missing field components are calculated using interpolation and extrapolation, the gray areas in fig. 2 being transition (impulse) areas.
As a further description of the inventive arrangements, the following are specified: (uppercase letters indicate the field amount of the coarse grid, lowercase letters indicate the field amount of the fine grid)
(1)Time t＝n。
A) E n is obtained from E n in the area of grid overlap using two-dimensional and three-dimensional interpolation methods.
(2)Time t＝n+1/4。
A) H (n+1/4) in the sub-grid was obtained using FDTD.
(3)Time t＝n+1/2。
A) Folding the fine mesh (removing the outermost layer) and obtaining e (n+1/2) using FDTD;
b) H (n+1/2) was obtained using FDTD in the coarse grid.
(4)Time t＝n+3/4。
A) Folding the fine mesh and obtaining h (n+3/4) using FDTD;
b) Immediately insert H (n+1/4) and H (n+3/4), then spatially optimize H (n+1/2);
c) The grid is extended using temporal extrapolation and spatial interpolation to obtain the missing h (n+3/4) values.
(5)Time t＝n+1。
A) E (n+1) and E (n+1) were obtained using FDTD.
(6) And (5) circulating.
In this embodiment, interpolation plays an important role in the algorithm of the present invention,
Three different schemes are used, namely tri-linear interpolation, cubic spline and conformal spline.
Tri-linear interpolation is the easiest to implement and, to date, the fastest.
The extension of cubic splines to three dimensions is expected to improve the algorithm because it guarantees not only the continuity of the domain, but also the continuity of the domain derivative. However, such algorithms sometimes introduce local oscillations, which may manifest themselves in algorithm degradation at higher frequencies. Thus, another spline algorithm, a conformal spline, is also considered, which preserves the convexity of the interpolation function.
The inherent interpolation scheme is an instantaneous transmission of information in the context of time domain finite difference algorithms, considering waves propagating along an axis, as shown in fig. 4, linear interpolation produces non-zero electric fields between points and at some locations where the actual electric field is zero.
Thus, interpolation may compromise the stability of the FDTD scheme, typically obtaining a stable solution without increasing the stability, but may not be in some cases.
For tri-linear interpolation and two spline formats, if the maximum value calculated by FDTD using the current stability criterion is less than 0.77 and 0.92, respectively, then FDTD remains stable;
In the calculation, 0.7 and 0.9dt max were set for the linear interpolation and the spline interpolation, respectively, and no instability was observed in these values.
The foregoing is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art, who is within the scope of the present invention, should make equivalent substitutions or modifications according to the technical scheme of the present invention and the inventive concept thereof, and should be covered by the scope of the present invention.
Claims (2)
1. A method of three-dimensional structured sub-meshing comprising the steps of:
step 1: selecting grids with overlapped thickness in space;
step 2: the two grids are offset in space, so that the magnetic field of the coarse grid coincides with the corresponding magnetic field of the fine grid;
step 3: the outermost refinement mesh is removed in each time sub-step of the refinement mesh, at the end of the loop the mesh is expanded back to its original size and the missing field components are calculated using interpolation and extrapolation.
2. The three-dimensional structured sub-meshing method according to claim 1, wherein in step 3, the interpolation scheme employs one or more of tri-linear interpolation, cubic spline, or conformal spline for achieving smooth transition between coarse and fine meshes.
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Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US6453275B1 (en) * | 1998-06-19 | 2002-09-17 | Interuniversitair Micro-Elektronica Centrum (Imec Vzw) | Method for locally refining a mesh |
| CN111062156A (en) * | 2019-11-26 | 2020-04-24 | 北京安怀信科技股份有限公司 | Mesh division method and device based on mesh convergence factor |
| CN111737835A (en) * | 2020-06-28 | 2020-10-02 | 大连理工大学 | Three-period minimum curved surface-based three-dimensional porous heat dissipation structure design and optimization method |
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Patent Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US6453275B1 (en) * | 1998-06-19 | 2002-09-17 | Interuniversitair Micro-Elektronica Centrum (Imec Vzw) | Method for locally refining a mesh |
| CN111062156A (en) * | 2019-11-26 | 2020-04-24 | 北京安怀信科技股份有限公司 | Mesh division method and device based on mesh convergence factor |
| CN111737835A (en) * | 2020-06-28 | 2020-10-02 | 大连理工大学 | Three-period minimum curved surface-based three-dimensional porous heat dissipation structure design and optimization method |
Non-Patent Citations (2)
| Title |
|---|
| XIHAO LI: "Stable FDTD simulations with subgridding at the time step of the coarse grid", 《2015 IEEE MTT-S INTERNATIONAL CONFERENCE ON NUMERICAL ELECTROMAGNETIC AND MULTIPHYSICS MODELING AND OPTIMIZATION (NEMO)》, 25 February 2016 (2016-02-25) * |
| 吴小倩: "基于FDTD的由粗到细网格逐步逼近计算电磁场", 《科技通报》, 31 May 2006 (2006-05-31) * |
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