A method for determining a stiffness matrix null space by an electromagnetic field finite element solver

Abstract

The present application relates to a kind of electromagnetic field finite element solver determination stiffness matrix zero space method, comprising the following steps: loop through all nodes;For each node, loop through all connected edges;For each node, find a edge connected with this node, the reciprocal of the length of the edge is taken as the coefficient, and the zero-order vector basis function of interpolation point in the edge is multiplied, and the product obtained is taken as the weighted vector;For each node, find all edges connected with this node and calculate the weighted vector, according to the principle that inflow is positive and outflow is negative or according to the principle that inflow is negative and outflow is positive, take their algebraic sum, which is equal to the gradient of auxiliary potential scalar basis function of the node;Repeat the above operation for all nodes, a matrix equation with the number of rows as the number of nodes and the number of columns as the number of edges can be obtained, transpose the coefficient matrix, and each column is the vector of the null space of the corresponding stiffness matrix.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of electromagnetic simulation, more particularly, to a method for determining the null space of a stiffness matrix of an electromagnetic field finite element solver. BACKGROUND

[0002] In the computation of electromagnetic vector finite element method, continuous electromagnetic field is discretely characterized by finite vector basis functions. The second-order wave equation of electric field obtained from Maxwell's equations is discretely obtained by using the finite element method to obtain the system matrix, which is composed of the sum of the stiffness matrix and the mass matrix. The mass matrix is obtained by the inner product of the linearly independent vector basis functions. The stiffness matrix is obtained by the inner product of the vector basis functions under the action of the curl operator. Since the originally independent basis functions are no longer independent under the action of the curl operator, the stiffness matrix has singularity, and its null space has non-zero vectors.

[0003] With the decrease of frequency, the contribution of the stiffness matrix gradually dominates, and the system matrix will tend to be singular, which is the low frequency breakdown problem. It is very important to quickly and accurately obtain the null space of the stiffness matrix for other problems, such as spurious mode problem, direct current parameter solving and iterative solver acceleration, etc. The null space of the stiffness matrix can be obtained by solving the corresponding eigenvalue problem, but the calculation cost is huge.

[0004] In addition to solving the eigenvalue problem, when the null space of the stiffness matrix is obtained in practice, the tree-cotree splitting method is often used. This method regards the finite element mesh as an undirected graph, and divides the edges in the mesh into two groups by using tree-cotree splitting: edges belonging to the tree and edges belonging to the cotree. The basis functions belonging to the null space of the stiffness matrix can be found by the edges belonging to the tree. Then the corresponding basis functions found are replaced by divergence-free basis functions composed of the tag-curl at the nodes of the mesh, so as to eliminate the null space. But this method has limited application scope, and it is difficult to apply to high-order interpolatory vector bases.

[0005] The preceding description is to provide general background information and does not necessarily constitute prior art. SUMMARY

[0006] The application aims to provide a method for determining the zero space of a stiffness matrix by an electromagnetic field finite element solver, which can be used to obtain the zero space of the stiffness matrix in a system matrix of an electromagnetic vector finite element method without solving an eigenvalue problem or using tree-conjugate tree decomposition, and can be expanded to high-order basis functions.

[0007] The application provides a method for determining the zero space of a stiffness matrix by an electromagnetic field finite element solver, which comprises the following steps:

[0008] S1: loop through all nodes;

[0009] S2: for each node, loop through all connected edges;

[0010] S3: for each node, find an edge connected to the node, multiply the inverse of the length of the edge by the zero-order vector basis function of the interpolation point on the edge to obtain a weighted vector;

[0011] S4: for each node, find all edges connected to the node and calculate the weighted vectors, and take the algebraic sum of the weighted vectors according to the principle that the inflow is positive and the outflow is negative or the principle that the inflow is negative and the outflow is positive, that is, equal to the gradient of the auxiliary potential scalar basis function of the node;

[0012] S5: perform steps S3 and S4 on all nodes to obtain a matrix equation with the number of rows equal to the number of nodes and the number of columns equal to the number of edges, and transpose the coefficient matrix, and each column is a vector of the zero space of the corresponding stiffness matrix.

[0013] Further, the step S4 is suitable for selecting high-order interpolation vector basis functions as the basis functions for representing the electromagnetic field according to the matrix equation obtained by the finite element method.

[0014] The method for determining the zero space of a stiffness matrix by an electromagnetic field finite element solver provided by the application can be used to obtain the zero space of the stiffness matrix in a system matrix of an electromagnetic vector finite element method without solving an eigenvalue problem or using tree-conjugate tree decomposition, and can be expanded to high-order basis functions. BRIEF DESCRIPTION OF DRAWINGS

[0015] Figure 1 The flowchart of the method for determining the zero space of a stiffness matrix by an electromagnetic field finite element solver provided by the embodiment of the application is shown.

[0016] Figure 2 The triangular mesh diagram provided by the embodiment of the application is shown.

[0017] Figure 3 The triangular parameter diagram provided by the embodiment of the application is shown. DETAILED DESCRIPTION

[0018] The specific embodiments of the present application are described in further detail below with reference to the drawings and examples. The following examples are intended to illustrate the present application and are not intended to limit the scope of the present application.

[0019] The terms "first", "second", "third", "fourth" and the like in the description and claims of the present application are used for distinguishing between similar objects and are not necessarily used to describe a particular sequential or chronological order.

[0020] Example 1

[0021] The method for determining the stiffness matrix null space of the electromagnetic field finite element solver of the present application is derived from:

[0022] Original problem description: A boundary value problem defined in a region Ω can be written as Lφ = f, where L is a self-adjoin operator, φ is an unknown function, and f is a known function. Electromagnetic fields in the physical world are governed by Maxwell's equations. By simplifying Maxwell's equations and removing electromagnetic coupling, the Helmholtz equation for a pure electric field (or magnetic field) can be obtained. In the frequency domain, given a frequency, the Helmholtz equation operator L, the unknown electric (or magnetic) field φ, and the known system excitation f all conform to this boundary value problem, so the unknown electromagnetic field can be obtained by solving this boundary value problem. The following problem description takes the electric field as an example:

[0023] The Helmholtz equation for the electric field is as follows formula (1):

[0024]

[0025] where E is the unknown electric field and J is the known excitation current. The boundary of the boundary value problem can be divided into an outer boundary and an inner boundary. The surface of the region Ω is the outer boundary, which is generally specified as an artificial boundary condition, commonly used such as the first-order radiation boundary condition as follows formula (2):

[0026]

[0027] where

[0028] The inner boundary is generally an ideal conductor and a conductor with finite electrical conductivity. The tangential electric field component on the surface of the ideal conductor is 0, i.e.

[0029]

[0030] The conductor with finite electrical conductivity can be described by the following impedance boundary condition:

[0031]

[0032] Space Discretization: Physical quantities in the real world are continuous, but computers can only deal with discrete problems, so we need to discretize the original continuous problem. The region Ω of the electric field distribution to be solved is meshed with tetrahedrons, so that Ω becomes a collection of all tetrahedral elements, and a set of known sub-domain basis functions where each basis function is defined in a certain sub-domain of the total region Ω (i.e. in a specific mesh element, such as within a triangle or a tetrahedron), the unknown electric field to be solved can be represented by the sum of the products of known basis functions and unknown coefficients x i , that is,

[0033] According to the Rayleigh-Ritz method or the Galerkin method, the matrix equation of the system is obtained:

[0034] AX = B

[0035] Here X is the vector of unknown coefficients x i , A is the coefficient matrix, and the i-row j-column element of A is where is the stiffness matrix part, is the finite conductivity conductor part, is the first-order absorbing boundary condition part, is the material matrix part. B is the known excitation. By solving the system equation, the unknown coefficient X is obtained, and finally the electric field to be solved is obtained.

[0036] Figure 1 The flowchart of the method for determining the stiffness matrix null space of the electromagnetic field finite element solver provided by the embodiment of the application is shown in FIG. 1. Please refer to Figure 1 The method for determining the stiffness matrix null space of the electromagnetic field finite element solver provided by the embodiment of the application comprises the following steps:

[0037] S1: loop through all nodes;

[0038] S2: for each node, loop through all connected edges;

[0039] S3: for each node, find an edge connected to the node, take the reciprocal of the length of the edge as the coefficient, and multiply the zero-order vector basis function of the interpolation point on the edge to obtain the product as the weighted vector;

[0040] S4: For each node, find all the edges connected to this node and calculate the weighted vector, take their algebraic sum according to the principle of inflow positive and outflow negative or inflow negative and outflow positive, which is equal to the gradient of the auxiliary scalar potential basis function of the node;

[0041] Further, the step S4 is suitable for selecting high-order interpolation vector basis functions as the basis functions representing the electromagnetic field, and obtaining the matrix equation according to the finite element method.

[0042] S5: Perform the steps S3 and S4 on all nodes, and a matrix equation with the number of rows equal to the number of nodes and the number of columns equal to the number of edges can be obtained. Transpose the coefficient matrix, and each column is a vector in the null space of the corresponding stiffness matrix.

[0043] It should be noted that since the electric potential at a node can be obtained by integrating a certain vector representing the electric field along a route passing through the interpolation points, there is a linear transformation relationship between the vector basis functions of the function space constituting the solution of the original electromagnetic field functional problem and the gradient of the scalar potential basis function defined on the nodes. The null space of the stiffness matrix can be processed by Helmholtz decomposition through this relationship, thereby avoiding solving the eigenvalue problem. The specific reason is explained as follows:

[0044] We define the scalar function φ defined on Δ as The p+1 order Lagrange polynomial basis expansion on the function space, that is,

[0045]

[0046] And according to the basic mathematical theory, since the gradient of the scalar function φ is a divergence-free vector function, the divergence-free vector function can be expanded by the p-order rotationally conservative vector interpolation polynomial basis. Selecting high-order interpolation vector basis functions (such as GWP vector interpolation basis functions), that is,

[0047]

[0048] where e k is a constant coefficient, is the kth p-order GWP basis function. In Figure 2 , φ1, φ2, … φ m represent the values of the scalar potential at the m interpolation points in the FEM grid respectively. If we take φ1 at vertex 1 in the attached Figure 2 as the reference point (i.e. φ1=0), then for the scalar potential φ u at any point u, it can be calculated according to the following formula:

[0049]

[0050] Figure 2 The integral line can be selected as a straight line without loss of generality, but the formula is established for any curve.

[0051] Substitute the expansion of the foregoing , the following formula can be obtained:

[0052]

[0053] The formula clearly reveals that there is a linear mapping between the scalar degrees of freedom [phi1, phi2, …, phi m ] and the vector degrees of freedom [e1, e2, …, e k ].

[0054] Specifically, for the first-order or higher-order interpolation vector basis function, the corresponding homogeneous form of the basis function will be used in the above steps, and after the null space of the corresponding homogeneous form basis function is obtained, the linear relationship between the homogeneous form and the general form is converted.

[0055] For example, in the triangular mesh shown in Figure 3 , the following equation is obtained at node 6:

[0056]

[0057] Writing such an equation for each point can obtain a linear equation group. The coefficients 1 / l mn items are arranged in the order of the basis function N i , and the coefficient matrix of the equation group is extracted, and after transposition, the result is obtained. The null space vector corresponding to node 6 is as follows:

[0058] [0, 0, 1 / l 16 , 0, 1 / l 26 , 0, 1 / l 36 , 0, 1 / l 46 , 1 / l 56 ] T

[0059] It is noted that

[0060]

[0061] The rows of the matrix are linearly related, and one is randomly discarded, and the remaining one is the null space of the stiffness matrix.

[0062] Based on the description above, the present application has the following advantages:

[0063] The method for determining the zero space of the stiffness matrix of the electromagnetic field finite element solver is used to obtain the zero space of the stiffness matrix in the system matrix of the electromagnetic vector finite element method, and the zero space of the stiffness matrix in the system matrix of the electromagnetic vector finite element method is obtained without solving the eigenvalue problem or using tree-conjugate tree decomposition, and can be expanded to high-order basis functions.

[0064] The above merely illustrates the specific embodiments of the present application, but the protection scope of the present application is not limited thereto, any person skilled in the art can easily think of the changes or replacements within the technical range disclosed by the present application, which should be covered within the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims

1. A method for determining the null space of a stiffness matrix using an electromagnetic field finite element solver, comprising the following steps: S1: Loop through all nodes; S2: For each node, loop through all connected edges; S3: For each node, find an edge connected to this node, use the inverse of the length of the edge as the coefficient, and multiply it by the zero-order vector basis function of the interpolation point on the edge, and use the product as the weighted vector; S4: For each node, find all edges connected to this node and calculate the weighted vector. According to the principle that inflow is positive and outflow is negative, or according to the principle that inflow is negative and outflow is positive, take their algebraic sum, which is equal to the gradient of the auxiliary potential scalar basis function of the node; S5: Perform steps S3 and S4 on all nodes to obtain a matrix equation with the number of rows being the number of nodes and the number of columns being the number of edges. Transpose this coefficient matrix, and each column is the vector of the null space of the corresponding stiffness matrix.

2. The method for determining the stiffness matrix null space using an electromagnetic field finite element solver according to claim 1, wherein: The step S4 is suitable for selecting a high-order interpolation vector basis function as a basis function for characterizing the electromagnetic field, according to the matrix equation obtained by the finite element method.

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