Multi-stage time stepping lossy model numerical simulation method based on wave equation

Abstract

The invention discloses a multistage time stepping lossy model numerical simulation method based on a wave equation. The method comprises the following steps: modeling a model with an ultrathin lossy structure; grid division is carried out on the model, and spatial discretization is carried out on a solution domain; deriving a second-order wave equation containing a lossy medium, introducing a model boundary condition, carrying out a Galerkin test, and converting the second-order wave equation into a lossy global matrix partial differential equation in combination with spatial discretization; all unknown quantities of the partial differential equation are written as a Taylor expansion, the partial differential equation is substituted into the Taylor expansion, and order reduction processing of high-order partial derivative terms is carried out; solving an unknown quantity in the taylor expansion after the order reduction processing by adopting a multi-stage stepping scheme; giving a multi-stage time stepping stability condition; and the speed-up ratio is obtained through theoretical analysis, and test verification is carried out. According to the method, not only can a flexible spatial discretization method be adopted, but also high-precision and stable numerical solution can be realized, particularly, the method has relatively high calculation efficiency for an ultrathin lossy structure, and the calculation cost is obviously reduced.

Description

Technical Field

[0001] The invention relates to a transient high-efficiency simulation method for a lossy model, in particular to a multi-level time-stepping lossy model numerical simulation method based on a wave equation. Background Art

[0002] The wave equation can be derived from Maxwell's equations. However, the wave equation only needs to store the unknown quantities of the electric field, while Maxwell's equations need to store the unknown quantities of both the electric field and the magnetic field, which will result in greater memory overhead. At the same time, [C.-Y.Tian, Y.Shi, KM Shum, and CH Chan, “Wave equationbased discontinuous Galerkin time domain method for co-simulation of electromagnetics-circuitsystems,” IEEE Trans.Antennas Propag., vol.68, no.4, pp.3026–3036, Apr.2020.] and [C.-Y.Tian, Y.Shi, and CH Chan, “Interior penalty discontinuous Galerkin time-domain method based on wave equation for 3-D electromagnetic modeling,” IEEE Trans.Antennas Propag., vol.65, no.12, pp.7174–7184, Dec.2017.] both show that, under the same numerical analysis algorithm, the wave equation has higher simulation efficiency than the Maxwell equations.

[0003] High-conductivity lossy materials are widely used due to their low internal resistance, high current-carrying capacity, significant skin effect, and rapid electromagnetic wave attenuation. For example, coating the exterior of certain equipment with conductive composite materials can absorb radar waves and reduce target detection signals. However, these coatings are typically very thin, forming cross-scale features with the surrounding structure. This poses challenges for transient full-wave simulations. Global time stepping is limited by these lossy structures, and more efficient simulation strategies are urgently needed. Summary of the Invention

[0004] The purpose of the present invention is to provide a multi-level time-stepping lossy model numerical simulation method based on the wave equation. This method can not only describe the model infinitely accurately by increasing the expansion order of the Taylor expansion, but also significantly improve the computational efficiency and reduce the computational cost for lossy models, especially high conductivity structures containing ultra-thin structures.

[0005] The technical solutions for achieving the purpose of the present invention are:

[0006] A multi-level time-stepping lossy model numerical simulation method based on wave equations, comprising the steps of:

[0007] Step 1: Model the lossy model with ultra-thin structure, perform mesh division, spatially discretize the solution domain, and derive mesh information and node information;

[0008] Step 2: Derive the second-order wave equation containing a lossy medium, introduce the model boundary conditions, and perform a Galerkin test to transform the frequency-domain second-order wave equation into a lossy global matrix partial differential equation by combining spatial discretization.

[0009] Step 3: Write all unknowns of the lossy global matrix partial differential equation as Taylor expansion, substitute the lossy global matrix partial differential equation into the Taylor expansion, and perform order reduction of high-order partial derivatives;

[0010] Step 4: Use a multi-stepping scheme to solve the unknowns in the reduced-order Taylor expansion. The unknowns in the ultra-thin lossy structure are solved using a small time step, while the unknowns in other areas are solved using a large time step, to obtain the numerical simulation value of the lossy model.

[0011] Step 5: List the multi-level time step stability conditions and perform critical multi-level time step tests;

[0012] In step 6, the numerical discrete results of the lossy structure are obtained based on step 4, the speedup ratio is obtained through theoretical analysis, and a simulation resource overhead test is performed based on the time consumption of step 5. The speedup ratio of the overhead test is compared with the speedup ratio of the theoretical analysis. If the difference between the speedup ratio of the overhead test and the speedup ratio of the theoretical analysis does not meet the requirements, the numerical simulation needs to be repeated.

[0013] Compared with the prior art, the present invention has the following significant advantages:

[0014] (1) The method of the present invention adopts multi-level time stepping technology, which can solve the time multi-scale problem caused by the time stability conditions of lossy models and lossless models, and significantly improve the computational efficiency.

[0015] (2) For the low-order terms in the high-order expansion terms, the low-order expansion can be used directly by utilizing the stacking relationship; and the high-order partial derivatives can be reduced to the same order as the low-order ones, thus avoiding the complex high-order derivation and unnecessary storage resource consumption.

[0016] (3) Compared with the global unified time stepping technique, no new unknowns are generated and memory consumption is almost not increased.

[0017] (4) The present invention combines the relationship between the local coordinate system and the reference coordinate system of PML at any rotation angle to derive a lossy model boundary processing method of PML+ABC at any angle, which can be more flexibly applied to the simulation of irregular lossy models. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 Schematic diagram of a high-power RF absorber model with an ultra-thin lossy structure.

[0019] Figure 2 Schematic diagram of the electric field intensity at the field probe of a high-power RF absorber (comparison between global unified time stepping and multi-level time stepping methods).

[0020] Figure 3 Schematic diagram of the reflection coefficient of a high-power RF absorber (comparison between global unified time stepping and multi-level time stepping methods). DETAILED DESCRIPTION

[0021] This embodiment provides a multi-level time-stepping lossy model numerical simulation method based on the wave equation, comprising the following steps:

[0022] First, the lossy model is constructed. Generally, due to the high conductivity and low current skin depth of lossy structures, only a very thin lossy structure is required to achieve functionality. The ultra-thin structure, the dielectric substrate, and the air form a multi-scale structure. To facilitate flexible meshing, the more flexible tetrahedron can be used to model local regions, which can then be uniformly converted to hexahedrons for spatial discretization of the solution domain. The mesh numbers, the corresponding material numbers within the mesh, the corresponding node numbers within the mesh, and the spatial coordinates of the nodes are exported to complete the model construction.

[0023] Secondly, the wave equation of the lossy model can be written as:

[0024]

[0025] Where μ is the magnetic permeability, ε is the dielectric constant, σ is the conductivity, t is the time, and E is the unknown electric field. According to the expression of thermal power density P in the electrothermal relationship d =σ|E| 2, we can see that this model is lossy, and the electric field energy is converted into thermal power dissipation. To ensure good absorption effect, PML (perfectly matched layer) is used for absorption at the boundary, and ABC (absorption boundary condition) is used to truncate the calculation threshold. The frequency domain wave equation containing PML can be written as:

[0026]

[0027] in is the matching matrix of the corner vertex area, ξ takes values of x, y, z, σ ξ is the conductivity of the distribution in different directions, ω is the angular frequency, and ε0 is the dielectric constant of vacuum. If the reference coordinate system is used to set the PML, then according to the mapping relationship between the reference coordinate system (u, v, w) and the global coordinate system (x, y, z), it can be seen that:

[0028]

[0029]

[0030] So if the PML parameter setting adopts the reference coordinate system, then the vertex matrix [Λ] xyz It is necessary to express the relationship of the reference angle matrix [Λ] as follows:

[0031] [Λ] xyz =Q T [Λ]Q (6)

[0032] Then the frequency domain wave equation in the reference coordinate system with PML can be written as:

[0033]

[0034] Expanding the unknown electric field in the above equation and performing the Galerkin test yields:

[0035]

[0036] The subscripts ei and ej represent the numbers of the unknown quantities, N ej is the expanded basis function, N ei is the test basis function. In addition, the solution domain outside the PML is truncated using ABC, which can be expressed as:

[0037]

[0038] Substituting the ABC boundary into the above expansion equation, we can obtain:

[0039]

[0040] Will Substitute into the above formula and multiply both sides by Sx S y S z ,have to:

[0041]

[0042] in, Then we can get:

[0043]

[0044] And because σ x , σ y , σ z The three and σ cannot be non-zero at the same time and belong to different regions, so the last term is simplified directly:

[0045]

[0046]

[0047] in

[0048]

[0049] Multiply both sides by the magnetic permeability and transfer to the time domain:

[0050]

[0051] This gives a compact form of the second-order wave matrix equation:

[0052]

[0053] in

[0054]

[0055] [C] ij =[T] ij =με∫N ei ·N ej dv (25)

[0056]

[0057] According to the mapping space (ξ, η, ζ) basis function (Φ i , Φ i ) Divergence compatibility and curl compatibility, resulting in a degenerate matrix:

[0058]

[0059] [C] ij =[T] ij =με∫(J -1 Φ i)·(J -1 Φ j )|J|dξdηdζ (30)

[0060]

[0061] Then, the compact format of the second-order wave matrix equation obtained in (21) is transformed into a system of equations. The first-order partial derivatives of the unknown quantities in the system of equations can be written as:

[0062]

[0063]

[0064] To facilitate the demonstration of the deduction process, all unknown quantities are restored to the transient format. After the first-order Taylor expansion, the first-order partial derivatives of each unknown quantity are substituted:

[0065]

[0066] After the second-order Taylor expansion, substituting it into the first-order Taylor expansion, we can get:

[0067]

[0068] By analogy, any order of Taylor expansion can be performed on the unknown quantity, depending on the required accuracy. For lower-order terms within higher-order expansions, the stacking relationship allows the lower-order expansion to be used directly. High-order partial derivatives can also be reduced to lower-order terms, avoiding the complex high-order derivatives and unnecessary storage resource consumption.

[0069] Then, the unknowns in the Taylor expansion after order reduction are solved using a multi-stepping scheme, and the unknowns in the ultra-thin lossy structure are solved using a small time step Δt s Solve, and the unknowns in the lossless region use a large time step Δt lSolve, the time step ratio of the lossless region and the lossy region is N. When solving the coupled unknowns in the lossy and lossless regions, the vectors are divided according to different time steps. When solving the lossless region, the partial derivatives of each order in the lossy region are still used; when solving the lossy region, the partial derivatives of each order in the lossless region are also applied. Since the unknowns are solved using different time steps, the vectors solved in small time steps are time interpolated according to the relationship between time steps to ensure the consistency of the global time solution. The interpolation stepping process is as follows: the unknowns after the first hour time step in the lossy region are obtained by the previous large time step and the partial derivatives of the unknowns in all regions; the unknowns after the mth (2≤m≤N, and is an integer) hour time step are obtained by the unknowns after the m-1th hour time step and the partial derivatives of the unknowns in all regions. Due to the overlapping relationship in Taylor expansions of different orders and the coupling relationship between partial derivatives and unknown quantities, when solving higher-order expansions, it is only necessary to store the results of the polynomials in the previous-order expansion.

[0070] The multi-level time stepping stability condition is not only related to the size of the mesh size in different regions, but also to the material parameters of each region (such as dielectric constant, magnetic permeability and electrical conductivity).

[0071] For the long-term stability of the zero-loss region, the time step is mainly affected by the fluctuation term and needs to satisfy the following empirical formula:

[0072]

[0073] For long-term stability in the lossy region, the time step is mainly affected by the lossy term and needs to satisfy the following empirical formula:

[0074]

[0075] Among them, C is an empirical constant, which is related to the subdivision type and spatial dimension, l min is the minimum grid size of the region, c0 is the speed of light in vacuum, μ r To solve the magnetic permeability of the region, ε r To solve for the dielectric constant of the region. It can be seen that for lossy regions with very large conductivity (≥10^4S / m), the choice of time step will be completely dominated by the conductivity.

[0076] The initial time step ratio N0 can also be selected by the above formula:

[0077]

[0078] In addition, due to the influence of the subdivision type, numerical discretization method, spatial dimension, etc., Δt l , Δt sThe selection of the actual time step ratio N should select a critical value after multiple tests.

[0079] Finally, since the unknown quantity and the machine solution time have a nearly linear growth relationship, the theoretical speedup ratio of the proposed multi-level time stepping method can be written as:

[0080]

[0081] Where T 全局 , T 局部 , DOFs 全局 , DOFs 有耗 , DOFs 无耗 The computation time for global unified time stepping, the computation time for multi-level time stepping, the number of unknown quantities to be solved globally, the number of unknown quantities to be solved in the lossy region, and the number of unknown quantities to be solved in the lossless region are respectively represented. Because the dimensions of the computational matrices are relatively small, excessive computational memory consumption is avoided. The proposed method and acceleration effect can be verified through lossy model simulations.

[0082] Example

[0083] The implementation effect is demonstrated by taking a high-power RF absorbing device that integrates a 0.1um thick new ceramic material on a high thermal conductivity SiC substrate 11 as an example. Figure 1 As shown, the RF absorber also includes a conductive microstrip line 13 and a metal via 10. The ultra-thin lossy structure 12 formed by the new ceramic material has a melting point greater than 3400°C, which ensures excellent thermal reliability when absorbing high-power microwaves. The new ceramic material also has an ultra-high conductivity of 350,000 S / m and a relative dielectric constant of 4.2. The relative dielectric constant of the SiC substrate is 9.7. The dimensional parameters of the high-power RF absorber are shown in Table 1. During the numerical simulation, a PML absorption layer with a thickness of 0.3 mm was set in the x, y, and z directions of the model, and the outermost layer used an ABC truncation of the calculation domain. A modulated Gaussian source with a center frequency of 15 GHz was injected into Port A, and the total simulation time was 200 ps. Compared with the global unified time-stepping method, which uses a maximum time step of 0.167 fs, the multi-level time-stepping technique employed by this invention allows the time step in the lossless region to be 100 times larger than the global unified time step, using a time step of 16.7 fs. In the lossy region, the time step of 0.167 fs is still used. Due to the high conductivity of the new ceramic material, the electromagnetic wave skin depth is relatively small, resulting in rapid attenuation, requiring an extremely small time step to ensure the convergence of this method. Figure 2 The transient time-varying electric field obtained by the field probe Probe using two methods is: Figure 3The reflection coefficient of port PortA simulated for the two methods shows obvious return loss below 22 GHz. Figure 2 and Figure 3 Transient simulations demonstrate a high degree of consistency between the two methods, verifying that the proposed method does not degrade simulation accuracy compared to the global time-stepping approach. Furthermore, as shown in Table 2, the multi-level time-stepping scheme reduces computational time by over 90% while maintaining the same level of memory consumption. This further highlights the superior computational efficiency of the multi-level time-stepping approach for lossy numerical simulations based on the wave equation.

[0084] Table 1 Structural dimensions of high-power RF absorber devices

[0085]

[0086] Table 2 Computational resource consumption during the simulation of high-power RF absorber devices

[0087]

[0088] The present invention proposes a multi-level time-stepping lossy model numerical simulation method based on the wave equation. In the ultra-thin structure area with extremely high conductivity, a sufficiently small time step is used to ensure the stability of the lossy structure, while the time step can be amplified in the lossless area according to the stability relationship, which can achieve higher computational efficiency. At the same time, by using Taylor expansions of different orders with a stacking relationship, high-order precision numerical simulation can be easily achieved. Compared with the global unified time-stepping technology, no new unknowns are generated and memory consumption is almost not increased. The present invention can not only adopt a flexible spatial discretization method, but also achieve high-precision and stable numerical solutions, especially for ultra-thin lossy structures, it has high computational efficiency and significantly reduces computational costs.

[0089] Obviously, those skilled in the art may make various changes and modifications to the embodiments of the present invention without departing from the spirit and scope of the embodiments of the present invention. Thus, if such changes and modifications of the embodiments of the present invention fall within the scope of the claims and their equivalents, the present invention is intended to include such changes and modifications.

Claims

1. A multi-level time-stepping lossy model numerical simulation method based on wave equation, characterized in that: Including steps: Step 1: Model the lossy model with ultra-thin structure, perform mesh division, spatially discretize the solution domain, and derive mesh information and node information; Step 2: Derive the second-order wave equation containing a lossy medium, introduce the model boundary conditions, and perform a Galerkin test to transform the frequency-domain second-order wave equation into a lossy global matrix partial differential equation by combining spatial discretization. Step 3: Write all unknowns of the lossy global matrix partial differential equation as Taylor expansion, substitute the lossy global matrix partial differential equation into the Taylor expansion, and perform order reduction of high-order partial derivatives; Step 4: Use a multi-stepping scheme to solve the unknowns in the reduced-order Taylor expansion. The unknowns in the ultra-thin lossy structure are solved using a small time step, while the unknowns in other areas are solved using a large time step, to obtain the numerical simulation value of the lossy model. Step 5: List the multi-level time step stability conditions and perform critical multi-level time step tests; In step 6, based on the number of unknown quantities of the lossy model obtained in step 4, the theoretical speedup ratio is calculated, and the test speedup ratio is obtained based on the test in step 5. If the difference between the test speedup ratio and the theoretical speedup ratio does not meet the requirements, the numerical simulation needs to be repeated.

2. The multi-level time-stepping lossy model numerical simulation method based on the wave equation according to claim 1 is characterized in that: The lossy model is modeled using tetrahedrons, and then uniformly converted into hexahedrons for spatial discretization of the solution domain. The grid number, the number of the material in the corresponding grid, the node number in the corresponding grid, and the node space coordinates are exported to complete the model construction.

3. The multi-level time-stepping lossy model numerical simulation method based on the wave equation according to claim 1 is characterized in that: PML is used for absorption at the model boundary. The second-order wave equation in the frequency domain including PML is: Where μ is the magnetic permeability, ε is the dielectric constant, σ is the conductivity, and E is the unknown electric field. is the matching matrix, ξ takes values of x, y, z, σ ξ is the conductivity of the distribution in different directions, ω is the angular frequency, is the density operator, and x, y, and z represent the coordinates of the global coordinate system.

4. The multi-level time-stepping lossy model numerical simulation method based on the wave equation according to claim 3 is characterized in that: The parameter setting of PML adopts the reference coordinate system, and the second-order wave equation in the frequency domain is: Among them, the mapping matrix between the reference coordinate system and the global coordinate system is QQ T =I, u, v, w are the coordinates of the reference coordinate system.

5. The multi-level time-stepping lossy model numerical simulation method based on the wave equation according to claim 4 is characterized in that: The second-order wave equation in the frequency domain is transformed into a lossy global matrix partial differential equation by combining spatial discretization: in, e is an unknown quantity. Combining the divergence compatibility and curl compatibility between the mapping space (ξ,η,ζ) and the original space (u,v,w), the coefficient matrices in the equation are specifically: [C] ij =with∫(J -1 F i )·(J -1 F j )|J|dξdηdζ Where the subscripts i and j represent the numbers of the basis functions Φ, J is the Jacobian matrix, and n is the normal vector.

6. According to the multi-level time-stepping lossy model numerical simulation method based on the wave equation in claim 5, in step 3, all unknowns of the lossy global matrix partial differential equation are written as Taylor expansions, and the lossy global matrix partial differential equation is substituted into the Taylor expansion to perform order reduction processing of high-order partial derivatives, and the specific process is: The lossy global matrix partial differential equation is transformed into a system of equations, and the first-order partial derivatives of the unknown quantities in the system of equations are written as: Restore all unknowns to the transient format, expand them by the first order Taylor, and substitute the first order partial derivatives of each unknown: After the second-order Taylor expansion, substitute it into the first-order Taylor expansion and we get: By analogy, according to the required accuracy, perform Taylor expansion of any order on the unknown quantity. For the low-order terms in the high-order expansion terms, use the stacking relationship to directly use the low-order expansion; the high-order partial derivatives are reduced to low-order processing.

7. The multi-level time-stepping lossy model numerical simulation method based on the wave equation according to claim 1 is characterized in that: In step 4, when the unknown quantity in the ultra-thin lossy structure is solved by a small time step, and the unknown quantity in other regions is solved by a large time step, the specific stepping process is as follows: the unknown quantity in the ultra-thin lossy structure is solved by a small time step Δt s Solve, and the unknowns in the lossless region use a large time step Δt l To solve, the time step ratio of the lossless area to the lossy area is N. When solving the lossless area, the partial derivatives of each order in the lossy area must be used; when solving the lossy area, the partial derivatives of each order in the lossless area must be used. The interpolation step process is: the unknown quantity after the first hour time step in the lossy area is obtained by the previous large time step and the partial derivatives of the unknown quantities in all areas; the unknown quantity after the mth hour time step is obtained by the unknown quantity after the m-1th hour time step and the partial derivatives of the unknown quantities in all areas, 2≤m≤N, and it is an integer.

8. The multi-level time-stepping lossy model numerical simulation method based on the wave equation according to claim 4 is characterized in that: The multi-level time step stability conditions in step 5 include: For the lossless region, the time step satisfies: For lossy regions, the time step satisfies: Among them, C is an empirical constant, which is related to the subdivision type and spatial dimension, l min is the minimum grid size of the region, c0 is the speed of light in vacuum, μ r To solve the magnetic permeability of the region, ε r To solve the dielectric constant of the region; the initial time step ratio N0:

9. The multi-level time-stepping lossy model numerical simulation method based on the wave equation according to claim 4 is characterized in that: The theoretical speedup ratio in step 6 is: Where T 全局 , T 局部 , DOFs 全局 , DOFs 有耗 , DOFs 无耗 They are respectively the computing time consumed by the global unified time step, the computing time consumed by the multi-level time step, the number of all unknown quantities to be solved globally, the number of unknown quantities to be solved in the consumption area, and the number of unknown quantities to be solved in the non-consumption area.

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