A Multipole Algorithm Configuration Method

Abstract

This invention provides a method for extracting and configuring resistance and inductance (RL) parameters based on the Fast Multipole Algorithm (FMM). This includes proposing a novel basis function, using centroid-center basis functions to decompose surface currents, configuring an accurate impedance model, proposing a novel method for calling the Fast Multipole Model (FMM), and preconditions. The centroid-center basis function introduced in this invention can transform traditional edge-based vector bit integration operations into surface-based scalar bit integration operations. The new integration method is easier to operate and faster. The accurate impedance model enables this invention to perform wideband simulations from low to high frequencies. Correspondingly, the FMM algorithm only requires surface elements instead of edges. The Fast Multipole Algorithm is also easier to configure.

Description

Technical Field

[0001] This invention relates to the field of algorithm optimization for parasitic parameter extraction, and more particularly to a method for RL parameter extraction and configuration based on the multipole algorithm. Background Technology

[0002] Existing algorithms for solving impedance parameters based on area integral equations require calculating the current distribution on the conductor surface. To discretize the surface current, current techniques employ Relative-Warranty (RWG) basis functions. RWG basis functions are established on the edges of each surface element. The vector potential integral calculation of RWG basis functions is relatively complex, and when used in conjunction with the Fast Multipole Algorithm, the RWG current needs to be decomposed into elements in the x, y, and z directions for invocation. This is computationally and configurationally time-consuming and difficult. Edge-based preconditioning matrix filling is also relatively time-consuming. Summary of the Invention

[0003] In view of the above problems, the present invention is proposed to provide a multipole algorithm configuration method that overcomes or at least partially solves the above problems.

[0004] According to one aspect of the present invention, a fast multipole algorithm configuration method is provided, the configuration method comprising:

[0005] The conductor surface is divided into N rectangular / triangular hybrid elements. p The number of edges is N e The number of loop currents is N l ;

[0006] Establish a sparse mapping matrix M from loop current to edge;

[0007] Establish sparse mapping matrices A1, A2, A3 from edges to surface elements;

[0008] The conductor surface current is discretized using the centroid-midpoint basis function proposed for the first time in this paper;

[0009] The sparse mapping matrix is ​​applied to map the loop current onto the surface element;

[0010] Establish the precondition matrix;

[0011] Accelerating matrix-vector multiplication based on the element configuration FMM algorithm;

[0012] Post-processing yields the RL parameters.

[0013] Optionally, the sparse mapping matrix M has a dimension of N. l ×N e The dimensions of A1, A2, and A3 are all N. e ×N p .

[0014] Optionally, the non-zero elements of the matrix M are 1 or -1. If the current direction defined on the edge is the same as the loop current direction, it is 1; otherwise, it is -1.

[0015] Optionally, the non-zero elements in each column of the sparse mapping matrices A1, A2, A3 correspond to the numbers of the face element edges, and the values ​​of the matrices are the x, y, and z components of the difference between the coordinates of the midpoint of the corresponding edge and the midpoint of the face element.

[0016] Optionally, the method for obtaining the sparse mapping matrices A1, A2, A3 specifically includes:

[0017] Obtain surface element S i and S j All current vectors ρ ik ,ρ jl ;

[0018] k,l are surface elements S i and S j The index above;

[0019] If ρ ik If the direction of the current is the same as the direction of the current on the side, then A1(k,i)=ρ il (x),A2(k,i)=ρ ik (y),A3(k,i)=ρ ik (z), otherwise, A1(k,i)=-ρ ik (x),A2(k,i)=-ρ ik (y),A3(k,i)=-ρ ik (z). Here, x, y, z represent the x-, y-, and z-components of the current vector;

[0020] If ρ jl The direction is the same as the direction of the side current, A1(l,j)=ρ j1 (x),A2(l,j)=ρ j1 (y),A3(l,j)=ρ jl (z), otherwise, A1(l,j)=-ρ jl (x),A2(l,j)=-ρ jl (y),A3(l,j)=-ρ jl (z).

[0021] Optionally, based on mapping the loop current onto a surface element, the formula is as follows:

[0022] The linear equations for solving the RL parameters through loop current analysis are further transformed into...

[0023]

[0024] Where P represents a scalar bit integral matrix based on surface elements;

[0025] M(A1PI p1 +A2PI p2 +A3PI p3 ) = V l ;

[0026] in

[0027] This invention provides a method for extracting and configuring RL parameters based on the multipole algorithm. It includes converting the vector bit integral operation of the RWG basis functions into the scalar bit integral calculation of the impulse function. Since the impulse function is built on surface elements, the new integral calculation method is easier to operate and faster. Correspondingly, using the fast multipole algorithm only requires surface elements instead of edges, making the fast multipole algorithm easier to configure.

[0028] The above description is merely an overview of the technical solution of the present invention. In order to better understand the technical means of the present invention and to implement it in accordance with the contents of the specification, and in order to make the above and other objects, features and advantages of the present invention more apparent and understandable, specific embodiments of the present invention are described below. Attached Figure Description

[0029] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the following description of the embodiments will be briefly introduced. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0030] Figure 1 A flowchart of an RL parameter extraction and configuration method based on the multipole algorithm provided in an embodiment of the present invention;

[0031] Figure 2 This is a schematic diagram of the arrow indicator of the sparse mapping matrix provided in an embodiment of the present invention;

[0032] Figure 3 This is a schematic diagram of the algorithm for generating these three mapping matrices provided in an embodiment of the present invention. Detailed Implementation

[0033] Exemplary embodiments of the present disclosure will now be described in more detail with reference to the accompanying drawings. While exemplary embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the disclosure to those skilled in the art.

[0034] The terms "comprising" and "having," and any variations thereof, in the specification, embodiments, claims, and drawings of this invention are intended to cover non-exclusive inclusion, such as including a series of steps or units.

[0035] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0036] like Figure 1 As shown, the surface is divided into rectangular / triangular hybrid elements, and the number of elements is N. p The number of edges is N e The number of loop currents is N l .

[0037] Construct a sparse mapping matrix M from loop currents to edges. The dimension of matrix M is N. l ×N e The non-zero elements of matrix M are 1 or -1. If the current direction defined on the edge is the same as the current direction of the loop, it is 1; otherwise, it is -1.

[0038] Construct sparse mapping matrices A1, A2, A3 from edges to surface elements, where the dimensions of A1, A2, and A3 are all N. e ×N p The specific methods for obtaining the sparse mapping matrices A1, A2, and A3 include: obtaining the surface element S. i and S j All current vectors ρ ik and ρ jl k,l are surface elements S i and S j The index above, if ρ ik If the direction of the current is the same as the direction of the current on the side, then A1(k,i)=ρ ik (x),A2(k,i)=ρ ik (y),A3(k,i)=ρ ik (z), otherwise, A1(k,i)=-ρ ik (x),A2(k,i)=-ρ ik (y),A3(k,i)=-ρ ik (z); where x, y, z represent the x-, y-, and z-components of the current vector; if ρ jl The direction is the same as the direction of the side current.

[0039] A1(l,j)=ρ jl (x),A2(l,j)=ρ jl (y),A3(l,j)=ρ jl (z), otherwise, A1(l,j)=-ρ jl (x),A2(l,j)=-ρjl (y),A3(l,j)=-ρ jl (z).

[0040] The loop current is mapped onto surface elements. The linear equations for solving the RL parameters from the loop current analysis are further transformed into...

[0041]

[0042] Where P represents a scalar bit integral matrix based on surface elements;

[0043] M(A1PI p1 +A2PI p2 +A3PI p3 ) = V l ;

[0044] in

[0045] Through the above transformation, the matrix-vector multiplication PI is accelerated using only the Fast Multipole Algorithm. p1 ,PI p2 ,PI p3 Taking triangulation as an example, the number of edges is 1.5 times the number of face elements. This operation reduces the number of operations in the Fast Multipole algorithm by 1.5 times. Furthermore, the Fast Multipole algorithm itself is designed for impulse functions, and the new formula directly uses the Fast Multipole algorithm without any vector decomposition. In addition, the direct solution complexity of the Fast Multipole algorithm is reduced, and the integral calculation is even faster after being converted to scalar bitwise integration.

[0046] Scalar bit integrals are 9-16 times faster than vector bit integrals (vector bit integrals require searching the edges of two facets separately; if the facets are triangular, the number of edge combinations is 9; if the facets are rectangular, the number of edge combinations is 16). The number of edges is generally 1.5-2 times the number of facets.

[0047] like Figure 2 As shown, the circular arrow represents the loop current l. i The direction is indicated by the straight arrow, which shows the direction from edge e1 to e5. Loop l i It contains these five edges, and

[0048] The directions of e2, e4, and e5 are related to l. i If the directions are the same, then

[0049] M(l i [e2,e4,e5])=1,M(l i ,[e1,e3])=-1.

[0050] This invention proposes a novel CM basis function to discretize the surface current of a conductor into local currents on surface elements, such as... Figure 3 As shown, the basis function connects the centroid of the face element to the midpoint of the edge. The algorithm demonstrates how to generate these three mapping matrices;

[0051] The existing linear equation system for solving the RL parameters using loop current analysis is as follows:

[0052] Z s I l +Z l I l =V l

[0053] Z here l I represents the system matrix calculated by integrating the loop current. l V represents the loop current vector. l Indicates voltage excitation, Z s This represents the surface impedance. It can be converted to... using the M matrix.

[0054] Z s I l +MZ e M T I l =V l

[0055] The Z of the present invention e This represents the system matrix calculated using the potential integral of the current vector on the edge.

[0056] Traditional method Z e The integral calculations during matrix filling are extremely time-consuming, and configuring the FMM based on the original formula requires decomposing the edge currents and calling the FMM algorithm separately, making the configuration complex. Both the direct solution path and precondition filling of the FMM require direct vector bit integration calculations, which are also extremely time-consuming.

[0057] Z s I l This is sparse matrix-vector multiplication, which doesn't require FMM acceleration. Here, we only need to use FMM to accelerate the second part. To configure the fast FMM algorithm, the second part of the above equation is again transformed into...

[0058] M(A1PI p1 +A2PI p2 +A3PI p3 ) = V l ;

[0059] in

[0060] FMM can be directly applied to PI. p1 ,PI p2,PI p3 Note that the FMM algorithm is specifically applied to this type of matrix-vector multiplication. Currently known literature mentions FMM algorithms for calculating RL parameters, but these cannot be directly applied. The FMM algorithm is a mature and classic algorithm, and will not be elaborated upon in this invention. This invention's configuration using the FMM algorithm offers the following advantages.

[0061] (1) Direct call, no need for internal decomposition of side current.

[0062] (2) Processing surface elements saves 1.5-2 times the computation time compared to existing technologies.

[0063] (3) The direct path of FMM involves integral calculation. This invention converts vector bit integrals into scalar bit integrals, which can save 9-16 times the calculation time.

[0064] This invention establishes a preconditioning matrix that only requires the diagonal elements of the P matrix, compared to the traditional edge-based preconditioning matrix (Z). e (Diagonal), which can also save 9-16 times the calculation time.

[0065] Beneficial effects: This invention proposes a method to transform the vector bit integral operation of RWG basis functions into the scalar bit integral calculation of impulse functions. Since the impulse functions are built on surface elements, this new integration method is easier to operate and faster to compute. Correspondingly, the Fast Multipole Algorithm (FMA) only needs to be based on surface elements instead of edges, making the FMA easier to configure.

[0066] The above specific embodiments further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above are merely specific embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for extracting and configuring RL parameters based on the fast multipole algorithm, characterized in that, The configuration method includes: The conductor surface is divided into rectangular and / or triangular mixed elements, the number of which is... The number of edges is The number of loop currents is ; Establish a sparse mapping matrix M from loop current to edge; Establish a sparse mapping matrix from edge to surface element. The conductor surface current is discretized using centroid-midpoint basis functions, including: Get face element and All current vectors and ; k,l are surface elements and The index above; if If the direction of the current is the same as the direction of the current on the side, then , , Otherwise, , , ; Where x, y, z represent the x-, y-, and z-components of the current vector; if The direction is the same as the direction of the side current. , ,otherwise, , ; The sparse mapping matrix is ​​applied to map the loop current onto the surface element; Establish the precondition matrix; Accelerating matrix-vector multiplication based on the element configuration FMM algorithm; Post-processing yields the RL parameters.

2. The RL parameter extraction and configuration method based on the fast multipole algorithm according to claim 1, characterized in that, The dimension of the sparse mapping matrix M is , All dimensions are .

3. The RL parameter extraction and configuration method based on the fast multipole algorithm according to claim 1, characterized in that, The non-zero elements of the matrix M are 1 or -1. If the current direction defined on the edge is the same as the loop current direction, it is 1; otherwise, it is -1.

4. The RL parameter extraction and configuration method based on the fast multipole algorithm according to claim 1, characterized in that, The sparse mapping matrix The non-zero elements in each column correspond to the numbers of the face element edges, and the values ​​of the matrix are the x, y, and z components of the difference between the coordinates of the midpoint of the corresponding edge and the midpoint of the face element.

5. The RL parameter extraction and configuration method based on the fast multipole algorithm according to claim 1, characterized in that, Based on mapping the loop current onto a surface element, the formula is as follows: The linear equations for solving the RL parameters through loop current analysis are further transformed into... ； Where P represents a scalar bit integral matrix based on surface elements; ； in , , .

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