A Nonlinear System Model Order Reduction Method for Material Nonlinearity

Abstract

The present invention relates to a method for reducing the order of a nonlinear system model for material nonlinearity. The method realizes the reduction of the order of the nonlinear system model by establishing a parameter model of the nonlinear material properties at the element level. This method can simplify the process of generating a reduced-order model for material nonlinearity problems involving nonlinear effects such as saturation of ferromagnetic materials and improve the solution efficiency of the reduced-order model.

Description

Technical Field

[0001] The present invention belongs to the field of scientific computing, and particularly relates to a method for reducing the order of a nonlinear system model for material nonlinearity. Background Art

[0002] The computational analysis of electromagnetic fields is a key link in the design of electrical equipment such as motors and transformers. The calculation of electromagnetic fields is based on the Maxwell's equations theoretical system, and the main methods include the analytical method and numerical calculation. For most practical problems, due to their complexity, it is difficult to obtain an analytical solution of the electromagnetic field distribution. With the improvement of computer computing power, numerical calculation has also become the main means of electromagnetic field analysis. The numerical calculation methods of electromagnetic fields mainly include finite difference, finite element, finite volume, boundary element, etc. The basic principle of these methods is to discretize the calculation region into grids, discretize the physical equations to be solved on each grid, and obtain a linear or nonlinear algebraic system. Solving this system can obtain an approximation of the electromagnetic field distribution. The accuracy of the numerical calculation results is closely related to the fineness of the spatial discretization and the accuracy of approximating the physical equations in a single grid. However, increasing the density of the spatial discretization and improving the accuracy of approximating the physical equations by the numerical format will both increase the degrees of freedom for solving the finally obtained algebraic system.

[0003] The electromagnetic analysis of electrical equipment often requires calculating the transient changes of the electromagnetic field under time-varying excitation, that is, the time-varying problem. In addition, during the design process, it is necessary to calculate the influence of different design parameters on the electromagnetic performance of the equipment, that is, the parameterization problem. The calculations of the time-varying problem and the parameter problem require repeated assembly and solution of the aforementioned algebraic system. When the number of degrees of freedom to be solved is large, it often requires a long calculation time. The existing model reduction methods are a class of methods that improve the calculation efficiency by reducing the degrees of freedom to be solved. The main idea behind them is to use the prior information or data of the system to find a set of reduced-order bases in the solution space, and obtain its low-order approximation by projecting the original algebraic system, so as to convert the solution of a large-scale system into a problem of a smaller scale. The projection-based model reduction methods include Asymptotic Waveform Evaluation (AWE), Reduced Basis Method (RBM), Proper Orthogonal Decomposition (POD), Proper Generalized Decomposition (PGD), etc. Among them, the Proper Orthogonal Decomposition (POD) is a commonly used model reduction method and is also the main model reduction method considered in this invention. This method performs singular value decomposition on some pre-obtained snapshot solutions and extracts some of the main left singular vectors as the reduced-order bases of the system. The implementation steps are as follows:

[0004] (1) Obtain a series of snapshot solutions of the original parameterized problem defined in the parameter space;

[0005] (2) Perform singular value decomposition on the snapshot solution matrix to obtain the left singular vectors;

[0006] (3) Use the dominant left singular vectors as the reduced-order bases to reduce the order of the original algebraic system.

[0007] The POD method for linear systems has been relatively mature, but the model reduction of nonlinear systems remains a difficult problem. The main issue involved is that the treatment of nonlinear terms in the reduced-order model involves full-order calculations. For example, when using the Newton-Raphson iteration to solve the nonlinear model reduction problem, the full-order Jacobian matrix needs to be constructed in each iteration, which seriously affects the solution efficiency of the reduced-order system. Currently, common nonlinear model reduction methods include the Trajectory Piecewise-Linear Approach (TPWL) and the Discrete Empirical Interpolation Method (DEIM). The Trajectory Piecewise-Linear Approach is mainly used for the time-domain analysis of nonlinear circuits. Its main idea is to approximate the nonlinear system by establishing piecewise-linear reduced-order bases and reduced-order models on the solution trajectory. The main steps to achieve the model reduction of nonlinear systems by combining this method with POD are as follows:

[0008] (1) Solve the full problem of the nonlinear system under a specific excitation and construct a series of reduced-order bases and linear approximation models: Assume that the solution vector at the initial time t0 is x0. During the time-step iteration, if the solutions obtained up to the nth step all fall within a neighborhood of x0, then these solutions are regarded as a set of slices. Using the POD method, a set of local reduced-order bases and linearized reduced-order models can be obtained. Starting from the (n + 1)th step, take the obtained solution vector x n+1 as the new reference point and repeat the same steps to obtain the next set of reduced-order bases and linearized reduced-order models. Repeat this process until the time iteration ends, and a series of local reduced-order bases and linearized reduced-order models can be obtained.

[0009] (2) Through weighted averaging, use the obtained reduced-order bases and models for the time-domain simulation of arbitrary excitations to achieve the reduction of the system.

[0010] For general parameterized problems, since the solutions corresponding to different parameters do not have the time-step dependence of time-varying systems, this method has not been seen in the literature applied to the model reduction of nonlinear systems with general parameterized problems.

[0011] Another more widely used nonlinear model reduction method is the Discrete Empirical Interpolation Method (DEIM). This method combines spatial interpolation and projection to approximate nonlinear terms. Its main implementation steps are as follows:

[0012] (1) Obtain a series of slice solutions of the original parameterized problem defined in the parameter space;

[0013] (2) Perform singular value decomposition on the slice solution matrix to obtain the left singular vectors, and take the dominant left singular vectors as the reduced-order bases;

[0014] (3) Based on the reduced-order basis obtained in step (2), a set of key interpolation points are obtained by using the discrete empirical interpolation algorithm;

[0015] (4) The nonlinear system is reduced in order by using the reduced-order basis obtained in (2) and the key interpolation points obtained in (3).

[0016] In the implementation process of this algorithm, some details are described as follows: In step (1), due to the more complex dependence relationship between the solution and parameters in the nonlinear system, more slice solutions are often required than in the linear system to obtain a reduced-order basis with higher approximation accuracy; the selection of the key interpolation points in step (3) depends on the reduced-order basis; in step (4), by introducing the key interpolation points, although the full-order calculations in the nonlinear term and Jacobian matrix evaluation can be avoided, the reduced-order system is still nonlinear. Summary of the Invention

[0017] To solve the above technical problems, the present invention proposes a method for reducing the order of a nonlinear system model for material nonlinearity, which is a POD model reduction method for calculating material nonlinearity problems in low-frequency electromagnetic fields, to solve the problem of low reduction efficiency for this type of nonlinear problems at present. Although as described in the previous part, the POD method based on the discrete empirical interpolation technique can reduce the order of a nonlinear system, the implementation process is complex. It is necessary to implement the discrete empirical interpolation method, obtain the required interpolation points through iteration, and in each iteration, a linear system needs to be constructed and solved. For the reduced-order low-order nonlinear system obtained, iterative methods such as Newton-Raphson and fixed-point still need to be used for solution, and the reduction efficiency is limited.

[0018] Due to the magnetic saturation effect involved in magnetic materials, material nonlinearity is one of the most common types of nonlinear problems in the analysis of low-frequency electromagnetic fields in electrical equipment such as motors and transformers. The present invention proposes a method for reducing the order of a nonlinear system model based on the unit-level material parameter model for material nonlinearity problems, that is, while obtaining the slice solutions for constructing the reduced-order basis, the unit material parameters corresponding to the slice solutions are stored, and these data are used to establish a unit-level material parameter model. When solving the reduced-order problem, the approximate construction of the nonlinear system matrix can be based on the unit-level material parameter model. This method is easy to implement, and the required unit material data can be obtained synchronously while obtaining the slice solutions, without the need to implement additional calculation programs. In the process of solving the reduced-order system, through the value of the unit-level material parameter model, a linear reduced-order system can be directly obtained, without the need to call the nonlinear solution process again, which can improve the calculation efficiency of the reduced-order model.

[0019] To achieve the above object, the technical solution adopted by the present invention is:

[0020] A method for model order reduction of a nonlinear system for material nonlinearity, comprising the following steps:

[0021] Consider the equation with the magnetic vector potential a as the system variable:

[0022]

[0023] where b is the magnetic induction intensity, satisfying where is the curl operator, j0 represents the current density, which is the excitation source of the system; μ is the magnetic permeability, which depends on the magnetic induction intensity under the magnetic saturation effect of the magnetic material and is described by the hysteresis loop; λ represents the relevant design or operating parameters, Λ is the parameter space, and since the electromagnetic field distribution depends on the parameter λ, the magnetic vector potential and the magnetic induction intensity in Equation (1) are denoted as functions of λ, a(λ) and b(λ);

[0024] Adopt a numerical method to discretize Equation (1) to obtain the following system matrix:

[0025] K λ (b)a = j0 (2)

[0026] where K λ (b) is the system matrix corresponding to the relevant design or operating parameters λ; for the finite element method, the system matrix K λ (b) is assembled from the element matrices of each discrete element; denote the element matrix of the tetrahedral element as K ele , since the magnetic vector potential a is discretized using edge elements, the dimension of K ele is 6×6, and each element of the element matrix K ele is obtained by the following integral:

[0027]

[0028] where w i , w j are the interpolation functions corresponding to the i-th and j-th edges respectively; in the nonlinear iterative solution process, for the elements in the nonlinear material region, the reluctivity ν(b) = 1 / μ(b) depends on the magnetic induction intensity of the current iteration step and is given by the hysteresis loop;

[0029] Use the proper orthogonal decomposition method to perform model order reduction on Equation (2).

[0030] Furthermore, the use of the proper orthogonal decomposition method to perform model order reduction on Equation (2) includes:

[0031] Give a set of parameter sample sets in the parameter space Λ where n sis the number of parameter samples. Solve Equation (2) of the system matrix corresponding to this set of parameter samples to obtain a set of slice solutions and form a matrix Suppose the degree of freedom of Equation (2) of the system matrix is N, then the dimension of matrix S is N×n s ; Use a non-linear solution algorithm to solve Equation (2); During this solution process, at the same time, obtain a set of data corresponding to the permeability of each unit in the non-linear material region for the parameter sample set That is Store this set of data to construct a unit-level material parameter model; After obtaining the matrix S of the slice solution, perform singular value decomposition on it, as shown in Equation (4):

[0032] S = ΦΣΨ T (4)

[0033] where Φ and Ψ T are the left and right singular matrices of matrix S respectively, and the dimensions are N×N and n s ×n s . The dimension is N×n s , where is a diagonal matrix with the diagonal elements being the singular values of matrix S, denoted as are the singular values of S;

[0034] According to a set threshold ∈, select the smallest k value that satisfies the condition to truncate Φ, and obtain a matrix Φ with dimension N×k k = [φ1, φ2, …, φ k , where φ i is the left singular vector corresponding to the singular value σ i , that is, obtain a set of reduced-order bases {φ1, φ2, …, φ k} of the solution space; Let a = Φ k a r , and obtain a reduced-order system with the number of degrees of freedom being k:

[0035]

[0036] where K λ (b) is a non-linear system matrix

[0037] Furthermore, the approximate construction method of the non-linear system matrix K λ (b) in the reduced-order model includes:

[0038] Using the values of the permeability of each unit stored during the slice solution generation process at different parameters, a material parameter model at the unit level is established by the response surface method, denoted as μ ele (λ), and through the material parameter model, a linear system matrix is obtained for approximating the original non-linear system matrix K λ (b), and a corresponding reduced-order model is obtained:

[0039]

[0040] Denote the unit matrix corresponding to the linear system as The unit material parameters of

[0041]

[0042] are given by the unit-level material model, that is: ele (λ) = 1 / μ ele (λ) is the magnetic resistivity on the unit when the parameter value is λ.

[0043] Beneficial effects:

[0044] The present invention provides a model reduction method for material non-linearity problems in low-frequency electromagnetic fields. Compared with the existing non-linear model reduction methods in the literature, the implementation process is simpler, the obtained reduced-order system is linear, and there is no need for non-linear iterative processes, with higher reduction efficiency. The present invention can be combined with traditional numerical calculation methods and applied to the efficient optimization design of electrical equipment and digital twin technology. Brief description of the drawings

[0045] Figure 1 It is a flow chart of the non-linear system model reduction method for material non-linearity of the present invention. Detailed implementation manners

[0046] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0047] As Figure 1 described, the non-linear system model reduction method for material non-linearity of the present invention will be illustrated by taking the model reduction problem of a static magnetic field system involving magnetic saturation as an example, which specifically includes the following steps:

[0048] Consider the following equation with the magnetic vector potential a as the system variable:

[0049]

[0050] where b is the magnetic induction intensity, satisfying where is the curl operator, j0 represents the current density, which is the excitation source of the system. μ is the magnetic permeability, which depends on the magnetic induction intensity under the magnetic saturation effect of the magnetic material. This dependence is generally described by the hysteresis loop (B-H curve). Due to the introduction of this dependence, Equation (1) is a nonlinear problem. In the process of designing electrical equipment, it is often necessary to calculate the distribution of the electromagnetic field under different design and operating parameters, that is, to solve the parametric problem. In Equation (1), λ is used to represent the relevant design or operating parameters, and the parameter space involved is denoted as Λ. Since the electromagnetic field distribution depends on the parameter λ, the magnetic vector potential and magnetic induction intensity in Equation (1) are denoted as functions of λ, a(λ) and b(λ).

[0051] The numerical methods such as finite difference and finite element can be used to discretize Equation (1) to obtain the following system matrix:

[0052] K λ (b)a = j0 (2)

[0053] where K λ (b) is the system matrix corresponding to the relevant parameter λ. For the finite element method, the system matrix K λ (b) is assembled from the element matrices of each discrete element. For a tetrahedral element, its element matrix is denoted as K ele , since the magnetic vector potential a is generally discretized by edge elements, the dimension of K ele is 6×6, and the elements of the element matrix K ele are obtained by the following integrals:

[0054]

[0055] where w i , w j are the interpolation functions corresponding to the i-th and j-th edges respectively; in the process of nonlinear iterative solution, for the elements in the nonlinear material region, the material coefficient ν(b) = 1 / μ(n) depends on the magnetic induction intensity of the current iteration step, which is given by the B-H curve, and dΩ is the volume differential of the calculation region Ω.

[0056] Next, the POD method is used to reduce the order of the model of Equation (2). First, a set of parameter sample sets are given in the parameter space Λ where n sis the number of parameter samples. Solve the system matrix formula (2) corresponding to this set of parameter samples to obtain a set of slice solutions and form a matrix Suppose the degree of freedom of the system matrix formula (2) is N, then the dimension of the matrix S is N×n s . Since the system matrix formula (2) is a nonlinear system, nonlinear solution algorithms such as Newton-Raphson need to be used for solution. During this solution process, a set of data corresponding to the permeability of each unit in the nonlinear material region for the parameter sample set can be obtained simultaneously , that is In the method proposed by the present invention, this data needs to be stored to construct the unit-level material parameter model. After obtaining the slice solution matrix, it can be subjected to singular value decomposition in the same manner as the linear system, as shown in formula (4):

[0057] S = ΦΣΨ T (4)

[0058] where Φ and Ψ T are the left and right singular matrices of the matrix S respectively, and the dimensions are N×N and n s ×n s . with the dimension of N×n s , where is a diagonal matrix with the diagonal elements being the singular values of the matrix S, denoted as are the singular values of S. According to a set threshold ∈, select the smallest k value that satisfies the condition to truncate Φ, and obtain a matrix Φ with the dimension of N×k k = [φ1, φ2, …, φ k , where φ i is the left singular vector corresponding to the singular value σ i , that is, a set of reduced-order bases {φ1, φ2, …, φ k} of the solution space is obtained. Let a = Φ k a r , and a reduced-order system with the number of degrees of freedom being k is obtained:

[0059]

[0060] For the nonlinear system, the construction method of the nonlinear system matrix K λ (b) needs to be specially considered. According to During the nonlinear solution process, the solution a obtained from the reduced-order system needs to be rRestore to the full order, and then evaluate the nonlinear terms in the system matrix, including the Jacobian matrix of the system, which involves a large amount of computation. Therefore, to ensure the reduction efficiency, it is necessary to reduce the computation by approximating the nonlinear terms in the system matrix.

[0061] In view of the material nonlinearity problem, the present invention proposes a nonlinear system matrix approximation method based on the element-level material model. By using the values of the permeability of each element stored during the generation of the slice solution under different parameters, a material parameter model at the element level is established by using methods such as response surface, denoted as μ ele (λ), that is, the response model. Through this response model, an approximate linear system matrix is obtained and the corresponding reduced-order model:

[0062]

[0063] Denote the element matrix corresponding to the linear system matrix as Different from Equation (3), the element material parameters are given by the element-level material model, that is:

[0064]

[0065] The above nonlinear model reduction method can be summarized into the following steps Figure 1 :

[0066] As Figure 1 shown, the present invention can be decomposed into three steps: data generation, data modeling, and constructing the model reduction. Figure 1 The relevant steps in the left box in are the conventional steps of the traditional POD model reduction, and the steps in the right box are the main technical points of the model reduction for the material nonlinearity problem proposed in the present invention. The main feature is that when constructing the reduced-order model, the element-level material data and model corresponding to the slice solution are combined to improve the efficiency of generating and solving the reduced-order model.

[0067] It is mentioned in the present invention that the response surface method can be used to construct the element-level material parameter model, and different data modeling methods such as polynomial fitting, Kriging method, and neural network can also be used to obtain the response model.

[0068] Although the present invention is directed to the material nonlinearity problem of low-frequency electromagnetic fields, it can also be applied to the reduction of the material nonlinearity problem in the numerical analysis of other physical fields.

[0069] It is easy for those skilled in the art to understand that the above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A method for reducing the order of a nonlinear system model for material nonlinearity, characterized in that, Including the following steps: Consider the equation with the magnetic vector potential a as the system variable: where b is the magnetic induction intensity, satisfying where is the curl operator, j0 represents the current density and is the excitation source of the system; μ is the magnetic permeability, which depends on the magnetic induction intensity under the magnetic saturation effect of the magnetic material and is described by the hysteresis loop; λ represents the relevant design or operating parameters, Λ is the parameter space, and since the electromagnetic field distribution depends on the parameter λ, the magnetic vector potential and magnetic induction intensity in Equation (1) are denoted as functions of λ, a(λ) and b(λ); Adopt a numerical method to discretize Equation (1) to obtain the following system matrix: K λ (b) a = j0 (2) where K λ (b) is the system matrix corresponding to the relevant design or operating parameter λ; for the finite element method, the system matrix K λ (b) is assembled from the element matrices of each discrete element; denote the element matrix of the tetrahedral element as K ele , since the magnetic vector potential a is discretized using edge elements, the dimension of K ele is 6×6, and the elements of the element matrix K ele are obtained by the following integrals: where, w i , w j are the interpolation functions corresponding to the i-th and j-th edges respectively; in the process of non-linear iterative solution, for the elements in the non-linear material region, the magnetic resistivity v(b) = 1 / μ(b) depends on the magnetic induction intensity of the current iteration step and is given by the hysteresis loop; Use the proper orthogonal decomposition method to perform model reduction on Equation (2).

2. The method for reducing the order of a nonlinear system model for material nonlinearity according to claim 1, characterized in that, The use of the proper orthogonal decomposition method to perform model reduction on Equation (2) includes: Give a set of parameter sample sets in the parameter space Λ where n s is the number of parameter samples. Solve equation (2) of the system matrix corresponding to this set of parameter samples to obtain a set of slice solutions and form a matrix Suppose the degree of freedom of equation (2) of the system matrix is N, then the dimension of matrix S is N×n s ; Use a nonlinear solution algorithm to solve equation (2); During this solution process, at the same time, obtain a set of data corresponding to the permeability of each unit in the nonlinear material region for the parameter sample set That is Store this set of data to construct a unit-level material parameter model; After obtaining the matrix S of the slice solution, perform singular value decomposition on it, as shown in equation (4): S = ΦΣΨ T (4) Among them, Φ and Ψ T are the left and right singular matrices of matrix S, with dimensions N×N and n s ×n s . with dimension N×n s , where is a diagonal matrix with the diagonal elements being the singular values of matrix S, denoted as are the singular values of S; According to the set threshold ∈, select the one that satisfies the condition Truncate Φ with the smallest k value to obtain a matrix Φ of dimension N×k k =[φ1, φ2, …, φ k , where φ i is the left singular vector corresponding to the singular value σ i , that is, a set of reduced-order bases {φ1, φ2, …, φ k} of the solution space is obtained; let a = Φ k a r , and a reduced-order system with k degrees of freedom is obtained: Among them, K λ (b) is a non-linear system matrix.

3. The method for reducing the order of a nonlinear system model for material nonlinearity according to claim 2, characterized in that, Nonlinear system matrix K in the reduced-order model λ The approximate construction method of (b) includes: Using the values of the permeability of each unit stored during the slice solution generation process under different parameters, a material parameter model at the unit level is established by the response surface method, denoted as μ ele (λ), and through the material parameter model, a linear system matrix is obtained for approximating the original nonlinear system matrix K λ (b), and a corresponding reduced-order model is obtained: Denote the element matrix corresponding to the linear system as The element material parameters of are given by the element-level material model, i.e.: where dΩ is the volume differential of the computational domain Ω, and v ele (λ) = 1 / μ ele (λ) is the magnetic resistivity of the element when the parameter takes the value of λ.

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