A fast frequency sweeping method for microwave circuit finite element based on second order arnoldi process

Abstract

The application belongs to the technical field of electromagnetic full-wave simulation, and specifically provides a microwave circuit finite element fast sweeping method based on a second-order Arnoldi process, which is used to overcome the limitations of the prior art in the fast sweeping analysis of non-ideal conductor microwave circuits under non-uniform wave port excitation, and is especially suitable for the wideband S parameter simulation of microwave circuits containing non-ideal conductor losses under non-uniform wave port excitation; in the weak form of the electric field vector Helmholtz equation, the wave port boundary condition, the impedance boundary condition of the non-ideal conductor and the first-order absorbing boundary condition are applied to construct the original finite element system equation; then, the nonlinear term and the fractional order term in the original finite element system equation are creatively mathematically transformed and approximated, the problem is converted into a standard second-order system form, so that the efficient SOAR model order reduction technology is successfully applied to such practical engineering problems, and the calculation precision and solving efficiency are considered.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic full-wave simulation technology, specifically providing a fast frequency sweep method for microwave circuits based on the second-order Arnoldi process using finite element methods. It is particularly suitable for broadband S-parameter simulation of microwave circuits with non-ideal conductor losses under non-uniform wave port excitation. Background Technology

[0002] When numerically modeling semi-open transmission line structures such as microstrip lines and coplanar waveguides using waveport excitation, the electromagnetic field is not completely confined between the signal conductor and the reference ground; some energy diffuses into the surrounding medium and air. To fully capture this open field distribution and avoid introducing false reflections or non-physical coupling between the port edge and the field due to artificial truncation, which would alter the transmission line characteristics, the lateral dimension of the waveport needs to be expanded to a scale sufficient to enclose the microstrip line cross-section and conform to engineering experience criteria in numerical modeling. Although this port setting is physically closer to reality, it results in non-uniform layered media on the port cross-section, making it different from conventional uniform waveguide ports with analytical field distribution, thus constituting a type of non-uniform waveport.

[0003] With the continuous increase in the operating frequency and the continuous reduction in size of microwave circuits, full-wave numerical methods, represented by the finite element method (FEM), have gradually become important analysis and design tools in industry. However, FEM requires global discretization of three-dimensional space, resulting in a huge set of equations and extremely high solution costs. Especially in broadband fine-sweep analysis, the number of frequency points to be calculated increases dramatically, and the inherent point-by-point solution strategy of frequency domain FEM will inevitably bring heavy computation time and memory overhead.

[0004] To alleviate the aforementioned difficulties, a fast frequency sweep method based on Model Reduction (MOR) has been introduced into large-scale finite element systems. For polynomial matrix equations obtained from finite element discretization, commonly used MOR techniques mainly include Asymptotic Waveform Evaluation (AWE) and the Krylov subspace method. Among them, AWE is not stable enough due to ill-conditioned problems in the moment generation process and is usually suitable for low-order approximations; while the Krylov subspace method constructs a more accurate reduced-order model through the Arnoldi process. Zhaojun Bai and Yangfeng Su published their paper "SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue" in the *SIAM Journal on Matrix Analysis and Applications*, Volume 26, Issue 3 (pp. 640-659) in 2005. The paper "Problem" proposes a second-order model reduction technique based on the second-order Arnoldi (SOAR) process. This technique is numerically stable and memory-efficient. Its basic idea is to implicitly linearize the second-order system equations into a first-order system of equations by introducing intermediate variables, and then iteratively generate a Krylov subspace reduction basis using the Arnoldi process. Specifically, the SOAR method requires the system matrix to be in polynomial form (highest order 2) of the sweep variables, and each order matrix to be independent of the sweep variables. However, when there are non-ideal conductor impedance boundaries, half-order terms of the sweep variables will appear in the system equations; when there are non-uniform wave ports, the propagation constant exhibits a nonlinear dependence on the sweep variables. Both of these cases cause the system equations to fail to meet the applicability prerequisites of the SOAR method, thus existing SOAR techniques cannot be directly applied to such engineering scenarios. Therefore, a new method capable of achieving high-precision, high-efficiency, and fast sweep frequency analysis for this type of problem is urgently needed. Summary of the Invention

[0005] The purpose of this invention is to provide a fast frequency sweep method for microwave circuits based on the second-order Arnoldi process (SOAR) using finite element methods, to overcome the limitations of existing technologies in fast frequency sweep analysis of non-ideal conductor microwave circuits under non-uniform wave port excitation. This invention transforms the problem into a standard second-order system form by performing reasonable mathematical transformations and approximations on the nonlinear and fractional terms in the system equations, thus successfully applying the efficient SOAR model order reduction technique to such practical engineering problems, achieving a balance between computational accuracy and solution efficiency.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] A fast frequency sweep method for microwave circuits based on the second-order Arnoldi process using finite element analysis is characterized by the following steps:

[0008] Step 1. In the weak form of the electric field vector Helmholtz equation, apply wave port boundary conditions, impedance boundary conditions of non-ideal conductors, and first-order absorbing boundary conditions, and use Whitney basis functions to discretize the field quantities. Combined with Galerkin matching, construct the original finite element system equations.

[0009] Step 2. Approximate the propagation constant introduced by the waveport boundary conditions in the original finite element system equations by approximating the nonlinear relation matrix terms in the original finite element system equations as frequency sweep variables. With constant The product form is used to reconstruct the finite element system equations;

[0010] Step 3. For the frequency sweep variables introduced by the impedance boundary conditions of the non-ideal conductor in the finite element system equations reconstructed in Step 2. The 1 / 2 square root term, at the central expansion point Perform a Taylor series expansion at the point, truncate it to the second order, and reconstruct the second-order finite element system equations;

[0011] Step 4. Expand at the center point In step 3, the second-order finite element system equations are linearized and transformed into an equivalent first-order linear system form applicable to the second-order Arnoldi process, and the system matrix and input vector are extracted.

[0012] Step 5. Using the system matrix and input vector as input, construct the Krylov subspace iteratively using a second-order Arnoldi process to generate a set of orthogonal reduced-order bases;

[0013] Step 6. Using the Krylov subspace reduction basis, perform a projection transformation on each coefficient matrix in the reconstructed finite element system equation in Step 2, mapping it from the high-dimensional finite element space to the low-dimensional Krylov subspace to obtain the reduced-order system equation.

[0014] Step 7. Transfer the sweep frequency variable Substituting into the reduced-order system equations, we obtain the compressed solution;

[0015] Step 8. Map the compressed solution from the low-dimensional space back to the original high-dimensional finite element space, restore the approximate solution of the original finite element system equations, and calculate the S-parameters of each port.

[0016] Furthermore, in step 1, the original finite element system equations are expressed as:

[0017] ,

[0018] in, This is the curl-curl stiffness matrix. It is a purely imaginary unit. For free space wavenumber, Contribution matrix to the absorbing boundary Let be the propagation constant. Contribute matrix to waveport, Contribute a matrix to the impedance boundary. For the quality matrix, Let the vector be the unknown variable to be solved. This is the activation vector.

[0019] Furthermore, the specific steps of step 2 are as follows:

[0020] Define sweep frequency variable Define the propagation constant at a non-uniform wave port. With free space wavenumber Relationship: , As an intermediate variable;

[0021] Then the nonlinear relation matrix terms The approximate term is expressed as:

[0022] ,

[0023] in, It is a constant. ;

[0024] Substituting the approximation terms into the original finite element system equations, we obtain the reconstructed finite element system equations:

[0025] ,

[0026] constant Merging , After that, it was converted to , Then and The coefficients of the two items are combined into The reconstructed finite element system equations are simplified to:

[0027] .

[0028] Furthermore, the specific process of step 3 is as follows:

[0029] frequency sweep variable 1 / 2 square root term At the central unfolding point Expanded to:

[0030] ,

[0031] make , The offset relative to the center expansion point is substituted into the finite element system equations reconstructed in step 2. The Taylor series is then truncated to second order and rearranged into... If the polynomial is given, then the equations of the second-order finite element system are expressed as:

[0032] ,

[0033] Abbreviated as:

[0034] ,

[0035] in, .

[0036] Furthermore, the specific process of step 4 is as follows:

[0037] The second-order finite element system equations are transformed into equivalent first-order linear equations applicable to the second-order Arnoldi process, expressed as:

[0038] ,

[0039] in, for The unit diagonal matrix, The total number of degrees of freedom of the finite element system. This represents an intermediate variable in the linearization process; To linearize the system matrix, ; Input vector, .

[0040] Furthermore, in step 6, the reduced-order system equations are expressed as:

[0041] ,

[0042] in, , It is a reduced-order basis.

[0043] Furthermore, in step 8, the approximate solution to the original finite element system equations is: , For compression, It is a reduced-order basis.

[0044] Based on the above technical solution, the beneficial effect of the present invention is that it provides a fast frequency sweep method for microwave circuits based on the second-order Arnoldi process using finite element methods, which has the following advantages:

[0045] First, the efficient matrix filling strategy: This invention uses a hash table-based DOK storage format to organize matrix blocks, which significantly reduces the time overhead of element positioning during the matrix filling process.

[0046] Second, the invention employs an efficient model simplification mechanism by introducing intermediate variables. Constructing an approximate constant term Furthermore, by combining Taylor series expansion to handle the impedance boundary conditions of non-ideal conductors, these two approximation methods significantly reduce the complexity of the finite element system model while maintaining a certain level of accuracy.

[0047] Third, port modeling that is close to actual engineering: By introducing non-uniform wave ports, the physical characteristics of this invention are closer to the real engineering environment, thereby ensuring the accuracy and effectiveness of this invention in practical applications.

[0048] Fourth, the optimized linear system solution strategy: This invention adopts a strategy of one-time decomposition and repeated solution of the system matrix. In each linear solution process, the system matrix is ​​not decomposed or explicitly inverted repeatedly, which greatly improves the calculation speed and efficiency. Attached Figure Description

[0049] Figure 1 This is a flowchart illustrating the fast frequency sweep method for microwave circuits based on the second-order Arnoldi process using finite element methods in this invention.

[0050] Figure 2 This is a 3D structural schematic diagram of a third-order microstrip interdigital filter in an embodiment of the present invention.

[0051] Figure 3 This is a 2D planar dimension diagram of the third-order microstrip interdigitated filter in an embodiment of the present invention.

[0052] Figure 4 This is a comparison chart of the S-parameter calculation results of the microwave circuit finite element fast frequency sweep method based on the second-order Arnoldi process in this embodiment of the invention and the reference results of the commercial simulation software HFSS. Detailed Implementation

[0053] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0054] This embodiment provides a fast frequency sweep method for microwave circuits based on the second-order Arnoldi process using finite element methods. The process is as follows: Figure 1 As shown, the specific steps include:

[0055] Step 1. In the weak form of the electric field vector Helmholtz equation, apply wave port boundary conditions, impedance boundary conditions of non-ideal conductors, and first-order absorbing boundary conditions, and use Whitney basis functions to discretize the field quantities. Combined with Galerkin matching, construct the original finite element system equations.

[0056] The electric field vector Helmholtz equation is discretized using the Galerkin weighted residual method based on the vector finite element method. Its corresponding weak form is:

[0057] ,

[0058] in, This is the test function (weight function). Let be the electric field vector to be determined. For the finite element solution domain, For free space wavenumber, These are the relative permeability and relative permittivity of the corresponding tetrahedrons, respectively. The unit vector of the outer boundary phase; For boundaries, including waveport boundaries Impedance boundary of non-ideal conductor and absorption boundary ,Right now ; For inner product, For the outer product, Curl;

[0059] Apply the following boundary conditions sequentially to the weak-form area integral:

[0060] Waveport Boundary Conditions (WPBC):

[0061] ,

[0062] Impedance boundary conditions (IBCs) are used to characterize the surfaces of non-ideal conductors.

[0063] ,

[0064] First-order absorbing boundary conditions (ABC):

[0065] ,

[0066] in, Let be the propagation constant. To excite the electric field, It is the magnetic field vector. It is a purely imaginary unit; For the surface impedance of a non-ideal conductor, The electrical conductivity of the metal, These are frequency and permeability, respectively. Indicates and Wave impedance corresponding to adjacent tetrahedrons at the boundary Free-space wave impedance;

[0067] If the boundary contains an ideal electrical conductor boundary PEC boundary conditions can also be applied. ;

[0068] Using Whitney basis functions for electric field Discretize the system and combine the Galerkin matching method to construct the original finite element system equations:

[0069] ,

[0070] in, This is the curl-curl stiffness matrix. Contribution matrix to the absorbing boundary Contribute matrix to waveport, Contribute a matrix to the impedance boundary. For the quality matrix, For the activation vector, Let the vector be the unknown variable to be solved.

[0071] The matrix blocks of the finite element system equations are stored using a sparse matrix storage format based on hash tables. Specifically, the row and column indices of the non-zero elements of the matrix are used as the key, and the values ​​of the non-zero elements are used as the value.

[0072] Step 2. Address the propagation constant introduced by the waveport boundary conditions in the finite element system equations. An approximation is made by defining the propagation constant under non-uniform wave port excitation. With free space wavenumber The mathematical relationship approximates the nonlinear relationship matrix terms of the swept frequency variables as swept frequency variables. With constant The product form ;

[0073] Set the sweep start frequency FREQ_START, end frequency FREQ_END, sweep point count FREQ_NUM, and segment count SEG_NUM. Use equal-interval sampling to acquire all frequency points within the entire frequency band, then distribute these frequency points to each sub-band as evenly as possible. Let the frequency points within a certain frequency band be... ,in, The center frequency of this sub-band, the sweep frequency variable (Right now Free space wavenumber) Let the vacuum permeability and vacuum permittivity be respectively, then the range of sweep frequency variables corresponding to this sub-frequency band is: ,in, This is the central expansion point of this sub-band. The following will elaborate on the subsequent process in detail regarding the range of frequency sweep variables.

[0074] Define the relationship between the propagation constant and the wavenumber in a non-uniform port: ,Will Represented as:

[0075] ,

[0076] Among them, the frequency sweep variable ,constant ;

[0077] In this embodiment, Off-center expansion point The rate of change does not exceed 1% over a relatively large range (preferably, this rate of change threshold is set to 1%), and therefore can be approximated as a constant; this approximation of a constant is one of the core innovations of this invention, and its definition is derived from the introduced intermediate variable. Export;

[0078] Will Substituting the approximate expression into the original finite element system equations, we can write:

[0079] ,

[0080] constant Merging , After that, it was converted to , Then and The coefficients of the two items are combined into The equation simplifies to

[0081] ,

[0082] Step 3. Address the frequency sweep variables introduced by the impedance boundary conditions of the non-ideal conductor in the reconstructed finite element system equations. The 1 / 2 square root term, at the selected central expansion point We perform a Taylor series expansion at the point, truncate it to the second order, and reconstruct a new second-order finite element system equation;

[0083] After approximation in step 2, the frequency sweep variable appears in the equation. 1 / 2 order matrix terms This invention uses Taylor series to... At the point of expansion Expand here:

[0084] ,

[0085] Combining the above formula, let Substituting into the finite element equation, This represents the offset relative to the center expansion point; then the Taylor series is truncated to second order and rearranged into... The polynomial is then expressed as:

[0086] ,

[0087] The equation can be simplified as follows:

[0088]

[0089] This second-order equation will change the original equation at a single-sweep frequency point. Calculation and expansion points By connecting these points, we can obtain the frequency sweep points. Irrelevant, only related to the point of expansion Related matrix items , , and activation vector :

[0090] ;

[0091] Step 4. Expand at the center point In step 3, the reconstructed finite element system equations are linearized, transforming them into an equivalent first-order linear system form suitable for a second-order Arnoldi process, and the system matrix is ​​extracted. With input vector ;

[0092] Obtain matrix items unrelated to the frequency sweep point from step 3. , , , The second-order reconstructed finite element system equations This is transformed into an equivalent first-order linear equation form applicable to second-order Arnoldi processes:

[0093] ,

[0094] in, for The unit diagonal matrix, The total number of degrees of freedom of the finite element system; This represents the intermediate variables introduced for linearization, and the linearized system matrix. Input vector ;

[0095] Step 5. Using the system matrix and input vector as input, construct the Krylov subspace iteratively using a second-order Arnoldi process to generate a set of orthogonal reduced-order bases. ;

[0096] A second-order Arnoldi process is used to construct a Krylov subspace for the above first-order linear system, generating a reduced-order basis:

[0097] ,

[0098] in, The pre-defined price reduction base order (i.e., the Krylov subspace dimension) ), These are the orthonormal basis vectors generated successively through a second-order Arnoldi process;

[0099] Step 6. Using the Krylov subspace reduced basis, perform a projection transformation on each coefficient matrix in the reconstructed finite element system equation in Step 2, mapping it from the high-dimensional finite element space to the low-dimensional Krylov subspace, to obtain a reduced-order system equation with a scale much smaller than the original system.

[0100] After the Krylov subspace reduced basis calculation is completed, the finite element system equations are... After reducing the order, we get:

[0101] ,

[0102] The reduced-order matrices and reduced-order excitation vectors are calculated through the following projection:

[0103] ;

[0104] Step 7. Transfer the sweep frequency variable Substituting the equations into the reduced-order system equations, we can directly solve the low-dimensional matrix equations to obtain the compressed solution. ;

[0105] The dimension of the above order reduction system is The matrix size is extremely small, and it can be solved quickly using a dense matrix solver to obtain a compressed solution. ;

[0106] Step 8. Decompress Mapping back from the low-dimensional space to the original high-dimensional finite element space to restore an approximate solution to the original system equations: ,use Calculate the S-parameters for each port;

[0107] Compressed Mapping back from the low-dimensional space to the original high-dimensional finite element space:

[0108] ,

[0109] The S-parameters are calculated by performing a line integral over the electric field at the port location:

[0110] ,

[0111] in, and These represent the amplitudes of the incident wave and the reflected wave, respectively. Indicates in Under port excitation, S-parameters measured at the port Indicates the first The voltage at the port, Indicates the excitation port excitation incident voltage, , It is the voltage integration line on the port. It is the unit direction vector of the integral line.

[0112] To verify the effectiveness, accuracy, and computational efficiency of the fast frequency sweeping method proposed in this invention, the following numerical calculation examples are given.

[0113] like Figure 2 The image shown is composed of gold, copper, and tungsten. , , A 3D structural diagram of a third-order microstrip interdigital filter constructed from three metals with finite conductivity, with a GND ground plane thickness of 0.01. The three grounding vias are made of the same material as the metal microstrip they are connected to, and the thickness of the inner wall of the vias is 1. Substrate material model Its thickness is 0.1 The size of the air box surrounding the entire solid model is Other sizes, such as Figure 3 The details show that non-uniform wave port excitation is used, and the cutoff boundary consists of three types: WPBC boundary, IBC boundary, and ABC boundary.

[0114] The entire model, after discretization, has a tetrahedral mesh of 219,619 elements and 269,578 unknowns. The order of the reduced Krylov subspace basis is set. The frequency is set to 20, the sweep start frequency FREQ_START is 17 GHz, the end frequency FREQ_END is 27 GHz, and the number of fast sweep frequency points is set to 401.

[0115] Set the number of segments to SEG_NUM, and take the center frequency of each segment as the expansion point for model order reduction calculation.

[0116] The fast frequency sweep algorithm of this invention is used to calculate the above-mentioned model constructed from a non-ideal conductor. The number of segments, SEG_NUM, is set to 1, and the S-parameters are used as the metric. The results are compared with the reference results obtained using the simulation software HFSS. Figure 4 As shown, from Figure 4 It is evident that the method of this invention matches the reference results well, and the precision and accuracy are guaranteed to a certain extent.

[0117] The fast frequency sweeping method of this invention significantly reduces the computation time and greatly improves computational efficiency while maintaining comparable accuracy and precision to the traditional point-by-point frequency sweeping method. The table below shows a comparison of the computation time consumption of the fast frequency sweeping method of this invention when using the point-by-point frequency sweeping method and when the number of segments SEG_NUM is 1 and 3, respectively. It is evident that for sweeping 401 frequency points, the point-by-point frequency sweeping method requires 10049.06 s, while the fast frequency sweeping method of this invention with SEG_NUM 1 requires only 62.54 s, representing a 99.38% improvement in computational efficiency, approximately 160.7 times; with SEG_NUM 3, it requires only 96.72 s, representing a 99.04% improvement in computational efficiency, approximately 103.9 times. This clearly demonstrates the high computational efficiency of this invention.

[0118] The above description is merely a specific embodiment of the present invention. Any feature disclosed in this specification may be replaced by other equivalent or similar features unless otherwise specified. All disclosed features, or steps in all methods or processes, may be combined in any way except for mutually exclusive features and / or steps.

Claims

1. A fast frequency sweep method for microwave circuits based on the second-order Arnoldi process using finite element analysis, characterized in that, Includes the following steps: Step 1. In the weak form of the electric field vector Helmholtz equation, apply wave port boundary conditions, impedance boundary conditions of non-ideal conductors, and first-order absorbing boundary conditions, and use Whitney basis functions to discretize the field quantities. Combined with Galerkin matching, construct the original finite element system equations. Step 2. Approximate the propagation constant introduced by the waveport boundary conditions in the original finite element system equations by approximating the nonlinear relation matrix terms in the original finite element system equations as frequency sweep variables. With constant The product form is used to reconstruct the finite element system equations; Step 3. For the frequency sweep variables introduced by the impedance boundary conditions of the non-ideal conductor in the finite element system equations reconstructed in Step 2. The 1 / 2 square root term, at the central expansion point Perform a Taylor series expansion at the point, truncate it to the second order, and reconstruct the second-order finite element system equations; Step 4. Expand at the center point In step 3, the second-order finite element system equations are linearized and transformed into an equivalent first-order linear system form applicable to the second-order Arnoldi process, and the system matrix and input vector are extracted. Step 5. Using the system matrix and input vector as input, construct the Krylov subspace iteratively using a second-order Arnoldi process to generate a set of orthogonal reduced-order bases; Step 6. Using the Krylov subspace reduction basis, perform a projection transformation on each coefficient matrix in the reconstructed finite element system equation in Step 2, mapping it from the high-dimensional finite element space to the low-dimensional Krylov subspace to obtain the reduced-order system equation. Step 7. Transfer the sweep frequency variable Substituting into the reduced-order system equations, we obtain the compressed solution; Step 8. Map the compressed solution from the low-dimensional space back to the original high-dimensional finite element space, restore the approximate solution of the original finite element system equations, and calculate the S-parameters of each port.

2. The microwave circuit finite element fast frequency sweep method based on the second-order Arnoldi process according to claim 1, characterized in that, In step 1, the original finite element system equations are expressed as: ， in, This is the curl-curl stiffness matrix. It is a purely imaginary unit. For free space wavenumber, Contribution matrix to the absorbing boundary Let be the propagation constant. Contribute matrix to waveport, Contribute a matrix to the impedance boundary. For the quality matrix, Let the vector be the unknown variable to be solved. This is the activation vector.

3. The microwave circuit finite element fast frequency sweep method based on the second-order Arnoldi process according to claim 2, characterized in that, The specific steps of step 2 are as follows: Define sweep frequency variable Define the propagation constant at a non-uniform wave port. With free space wavenumber Relationship: , As an intermediate variable; Then the nonlinear relation matrix terms The approximate term is expressed as: ， in, It is a constant. ; Substituting the approximation terms into the original finite element system equations, we obtain the reconstructed finite element system equations: ， constant Merging , After that, it was converted to , Then and The coefficients of the two items are combined into The reconstructed finite element system equations are simplified to: 。 4. The microwave circuit finite element fast frequency sweep method based on the second-order Arnoldi process according to claim 3, characterized in that, The specific process of step 3 is as follows: frequency sweep variable 1 / 2 square root term At the central unfolding point Expanded to: ， make , The offset relative to the center expansion point is substituted into the finite element system equations reconstructed in step 2. The Taylor series is then truncated to second order and rearranged into... If the polynomial is given, then the equation of the second-order finite element system is expressed as: ， Abbreviated as: ， in, .

5. The microwave circuit finite element fast frequency sweep method based on the second-order Arnoldi process according to claim 4, characterized in that, The specific process of step 4 is as follows: The second-order finite element system equations are transformed into equivalent first-order linear equations applicable to the second-order Arnoldi process, expressed as: ， in, for The unit diagonal matrix, The total number of degrees of freedom of the finite element system. This represents an intermediate variable in the linearization process; To linearize the system matrix, ; Input vector, .

6. The microwave circuit finite element fast frequency sweep method based on the second-order Arnoldi process according to claim 3, characterized in that, In step 6, the reduced-order system equations are expressed as: ， in, , It is a reduced-order basis.

7. The microwave circuit finite element fast frequency sweep method based on the second-order Arnoldi process according to claim 1, characterized in that, In step 8, the approximate solution to the original finite element system equations is: , For compression, It is a reduced-order basis.

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