Numerical simulation method for magnetic field of linear variable differential transformer

Abstract

The invention discloses a numerical simulation method for a linear variable differential transformer magnetic field, and belongs to the technical field of electromagnetic numerical simulation. Aiming at the problem that the precision is insufficient when the traditional finite element method is used for processing the complex magnetic field problem of the LVDT, numerical simulation is carried out on the magnetic field of the LVDT through the smooth finite element method, structures of a primary coil, a secondary coil and the like included in the LVDT and corresponding physical parameters are considered, and a calculation model based on the smooth finite element method is firstly constructed; and proper current load and boundary conditions are determined, and accurate solution of the model is realized through MATLAB programming. According to the method, the magnetic field distribution of the LVDT under different working conditions can be accurately simulated, the linear relation of the LVDT can be predicted, a powerful basis is provided for optimizing the design of the LVDT and improving the measurement precision and performance of the LVDT, and the method can be widely applied to performance analysis and improvement of the LVDT in the related fields of a displacement measurement system, an electrical control system and the like of the LVDT.

Description

Technical Field

[0001] The invention belongs to the technical field of electromagnetic numerical simulation, and in particular relates to a numerical simulation method for the magnetic field of a linear variable differential transformer. Background Art

[0002] LVDT is widely used in displacement measurement, position feedback control, material property detection and many other aspects, and plays an indispensable role in the modern industrial field. For example, in the aerospace field, it is used to monitor the displacement of aircraft engine blades to ensure stable performance during engine operation; in automobile manufacturing, it can accurately measure the processing accuracy of automobile parts to ensure product quality; in the precision machining industry, it is used to accurately control the position of CNC machine tool tools to improve machining accuracy. Its high precision, high reliability and non-contact measurement characteristics make it popular in occasions with high measurement accuracy requirements.

[0003] With the continuous advancement of science and technology, the requirements for LVDT performance are getting higher and higher. In-depth understanding of its internal magnetic field distribution characteristics is crucial to optimizing LVDT design and improving measurement accuracy. As an important research method, magnetic field numerical simulation can predict the magnetic field behavior of LVDT under different working conditions without conducting actual physical experiments. Through simulation, the impact of changes in magnetic field intensity, direction and distribution on the LVDT output signal can be analyzed, thereby providing a theoretical basis for improving the structural design of LVDT and optimizing coil parameters. Magnetic field numerical simulation can also help researchers conduct in-depth research on the performance changes of LVDT in complex working environments, providing support for improving the stability and reliability of LVDT.

[0004] At present, the methods used for numerical simulation of electromagnetic problems mainly include finite element method (FEM), finite difference method (FDM), finite volume method (FVM) and boundary element method (BEM). However, these traditional methods have certain limitations when applied to LVDT magnetic field simulation. Taking the widely used FEM as an example, when dealing with LVDT magnetic field problems, the calculation accuracy is extremely sensitive to the quality of the unit mesh.

[0005] The internal structure of the LVDT is complex. In order to obtain more accurate simulation results, it is necessary to divide a very fine grid, which will lead to a significant increase in the amount of calculation and a significant increase in the calculation time. Especially when simulating multi-physics field coupling (such as thermal-electromagnetic, force-electromagnetic coupling), the calculation efficiency of FEM is lower and it is difficult to meet the needs of actual engineering. Moreover, the "over-rigidity" nature of the traditional FEM model will lead to a decrease in the accuracy of the numerical solution. When simulating some detailed features of the LVDT magnetic field (such as the change in the magnetic field at the edge of the coil), it may not be able to accurately capture the real physical phenomena. Other methods such as FDM have difficulties in dealing with complex geometric shapes, FVM has relatively low accuracy when dealing with electromagnetic problems, and BEM has a high computational cost when dealing with large-scale problems. They are not suitable for accurate simulation of LVDT magnetic fields.

[0006] Therefore, in response to the above problems, a new numerical simulation method is urgently needed to overcome the shortcomings of these existing methods, meet the needs of accurate simulation of LVDT magnetic fields, and promote the further development of LVDT technology. Summary of the invention

[0007] In view of the problem that the traditional finite element method has insufficient accuracy when dealing with complex magnetic field problems of LVDT, the present invention provides a numerical simulation method for the magnetic field of a linear variable differential transformer.

[0008] In order to achieve the above object, the present invention adopts the following technical solutions:

[0009] A numerical simulation method for a linear variable differential transformer magnetic field, the method comprising the following steps:

[0010] Step 1: Establish a geometric model based on the actual structure and size parameters of the LVDT, and simplify the geometric model; ignore the structural details that have little effect on the magnetic field without affecting the accuracy of the magnetic field simulation;

[0011] Furthermore, the specific operations of step 1 are:

[0012] A geometric model is established and simplified based on the physical model. Various parameters of the geometric model are determined based on the actual parameters of the LVDT. The model includes the primary coil, the secondary coil, and the space area around them, and the relative position relationship and geometric shape and size of each component are clarified.

[0013] Step 2: Discretize the problem domain, divide the tetrahedral mesh using adaptive mesh technology, adjust the mesh density according to the complexity of different regions in the geometric model and initialize the material parameters, the physical parameters include the current density, conductivity, magnetic permeability of the coil and the magnetic permeability of the surrounding medium;

[0014] Furthermore, the specific operations of step 2 are:

[0015] Step 2.1: Use adaptive meshing technology to divide the tetrahedral mesh, and adjust the mesh density according to the complexity of different areas of the geometric model; use denser meshes in areas where the magnetic field changes drastically, such as coil windings, and use sparser meshes in other areas where the magnetic field changes slowly;

[0016] Step 2.2: Initialize material parameters, wherein the physical parameters include the current density, electrical conductivity, magnetic permeability of the coil and the magnetic permeability of the surrounding medium;

[0017] Step 2.3: The expression of primary coil input current is as follows:

[0018] I(t)=I m sin(2πf·t)

[0019] In the formula, I m represents the amplitude of the input current; f represents the frequency of the input current; t represents time.

[0020] Step 3: Based on the Maxwell equations, interpolate the domain variables, and reconstruct the smooth domain based on edges, nodes, and faces based on the smooth finite element method. Apply the gradient smoothing technique to recalculate the smooth domain interpolation function and apply the generalized smooth Galerkin weak form to establish the discrete system equations.

[0021] Furthermore, the specific operations of step 3 are:

[0022] Step 3.1: According to Maxwell's equations, the governing equations of the eddy current field are expressed as follows:

[0023]

[0024] In the formula, H is the magnetic field intensity, J is the current density, E is the electric field intensity, and B is the magnetic flux density; represents the partial differential operator; t represents time;

[0025] Step 3.2: The constitutive relationship of electromagnetic performance is as follows:

[0026] B=μH

[0027] J=σE

[0028] Where μ is the magnetic permeability and σ is the electrical conductivity. In order to reduce the scale of calculation, the vector magnetic potential A is introduced as an unknown function in the control equation, which is expressed as follows:

[0029]

[0030] The spin-free field is expressed as the gradient of a scalar function, further expressed as:

[0031]

[0032] In the formula, represents the gradient of the electric scalar potential;

[0033] Step 3.3: Consider the sinusoidal steady-state eddy current field as a special case of the general transient eddy current field. The governing equation of the boundary value problem of the sinusoidal steady-state eddy current field is as follows:

[0034]

[0035] In the formula, v is the magnetic reluctance, ω is the angular frequency of the sinusoidal change, is the coil current density, φ is the electric scalar potential, the dots above the field variables indicate that they are complex values; j represents the imaginary unit; Ω represents the problem domain; Ω1 represents the eddy current region in the problem domain;

[0036] Step 3.4: In the tetrahedral element, the vector magnetic potential and electric scalar potential are expressed as linear interpolation functions at each node as follows:

[0037]

[0038] in, is the shape function of node i, A i and φ i are the vector magnetic potential and electric scalar potential of node i; represents the vector magnetic potential of unit e; represents the electric scalar potential of unit e;

[0039] Step 3.5: The magnetic flux density of the linear tetrahedral element is expressed as follows:

[0040]

[0041] In the formula, Respectively represent the vector magnetic potential components of the ith node along the x, y, and z directions; Represent the unit vectors along the x, y, and z directions respectively;

[0042] Step 3.6: The control equation is discretized into the following form:

[0043]

[0044] Step 3.7: In order to make the governing equations have a symmetric coefficient matrix, introduce the following function:

[0045]

[0046] Step 3.8: Substitute it into the magnetic flux density formula of the linear tetrahedron element to obtain the following governing equation:

[0047]

[0048] Step 3.9: The vector magnetic potential contains components in the X, Y, and Z directions. The above control equations can be simplified into the following algebraic equations:

[0049]

[0050] Where K is the overall coefficient matrix, is the solution vector, F is the right-hand side vector, which is provided by the linear tetrahedral unit k e , and f e The contribution of linear tetrahedral elements is expressed as follows:

[0051]

[0052] In the formula, are the vector magnetic potential and scalar potential of the ith node of the eth unit, is the source current vector of the i-th node of the e-th unit; T represents the transpose of the vector and matrix;

[0053] Step 3.10: The magnetic flux density and current density of the e-th unit are as follows:

[0054]

[0055] In the formula, Respectively represent the vector magnetic potential components of the ith node along the x, y, and z directions; They represent the magnetic flux density components of unit e along the x, y, and z directions respectively; represents the source current density of unit e; represents the current density of unit e; They respectively represent the partial differential of the shape function (shape function coefficient) of the i-th node.

[0056] Step 3.11: Based on the tetrahedral element mesh, the problem domain is further subdivided into smooth subdomains without overlap and gap. The smooth subdomains are constructed by connecting the body centers, face centers, edge centers and nodes of the elements.

[0057] Step 3.12: Magnetic flux density at node k and the electric scalar potential gradient Use the following formula:

[0058]

[0059] In the formula, Smooth subdomain The borders of Represent the unit vectors along the x, y, and z directions respectively; They represent the partial differential of the shape function (shape function coefficient) of the kth node respectively; n s represents the number of smooth subdomains associated with the kth node; Respectively represent the vector magnetic potential components of the kth node along the x, y, and z directions;

[0060] Smooth subdomain The volume is expressed in the following form:

[0061]

[0062] In the formula, represents the number of units associated with the kth node; represents the volume of the jth element associated with the kth node;

[0063] Step 3.13: After the gradient smoothing operation, the formula is as follows:

[0064]

[0065] Where n r For the border The unit external normal vector on ; They represent the partial differential of the shape function (shape function coefficient) of the kth node after the smooth domain is re-divided and before the smooth domain is not divided; N k represents the shape function of the kth node; r can represent the three directions of x, y, and z respectively;

[0066] Furthermore, the discretization model using the smooth finite element method in step 3 is an edge-based smooth finite element model ES-FEM and a surface-based smooth finite element model FS-FEM. In ES-FEM, the calculation accuracy is improved by processing the area related to the edge, and in FS-FEM, the simulation effect is improved by processing the area related to the surface.

[0067] Step 4: Apply current load and determine eddy current field boundary conditions; the magnetic field is an eddy current problem domain with boundaries, the problem domain consists of an eddy current region with non-zero conductivity and a non-eddy region containing a given source current, the boundary conditions include the external boundary conditions of the LVDT and the boundary conditions of the internal interface, the external boundary conditions are set according to the actual working environment of the LVDT, and since the domain variables are continuous at the interface between the two media, the internal boundary conditions will be automatically satisfied;

[0068] Furthermore, the specific operations of step 4 are:

[0069] Step 4.1: Determine the composition and boundary of the eddy current problem domain, and set the normal component of the given magnetic flux density and the tangential component of the magnetic field intensity on the boundary.

[0070] Step 4.2: The boundary conditions are expressed as follows:

[0071]

[0072] Where n is the unit normal vector on the boundary; A represents the vector magnetic potential, Γ B and Γ H represents the outer boundary of the problem domain; Γ 12 Represents the internal interface of the problem domain; and They represent the vector magnetic potential of the eddy current zone and the non-eddy current zone on the interface, respectively; v1 and v2 represent the magnetic resistivity of the eddy current zone and the non-eddy current zone on the interface, respectively; n 12 represents the internal interface Γ 12 The unit normal vector of .

[0073] Step 5: Solve the discrete linear equations to obtain the numerical simulation results of the LVDT magnetic field;

[0074] Furthermore, the step 5 utilizes MATLAB programming to assemble and solve discrete linear equations to obtain numerical simulation results of the LVDT magnetic field.

[0075] In the step of solving the discretized model using the smooth finite element method, parallel computing technology is used to improve the solution efficiency, and the computing tasks are distributed to multiple computing cores or processors for simultaneous computing.

[0076] In the step of solving the discretized model using the smooth finite element method, error analysis and verification are performed on the simulation results, and the accuracy of the simulation is evaluated by comparing with experimental data.

[0077] In the step of solving the discretized model using the smooth finite element method, data mining and analysis are performed on the simulation results to extract valuable information for LVDT design and performance optimization, such as the magnetic field distribution law and linear relationship.

[0078] The method further includes a step of post-processing the numerical simulation results in step 5, wherein the post-processing includes visually displaying the magnetic field distribution of the LVDT, calculating the average value, maximum value or minimum value of the vector magnetic potential, magnetic flux density, and current density in a specific area, and predicting the linear relationship of the LVDT.

[0079] Furthermore, the visualization of the magnetic field distribution of the LVDT uses a cloud diagram or a vector diagram to display the distribution of the magnetic field intensity and direction in the LVDT and its surrounding space.

[0080] Compared with the prior art, the present invention has the following advantages:

[0081] (1) Simplify the processing and improve accuracy: Under the same mesh unit type conditions, S-FEM can directly use FEM data for subsequent processing, greatly reducing the workload of preprocessing. Compared with FEM, S-FEM only needs a small modification to achieve higher accuracy.

[0082] (2) Superior performance of FS-FEM: The best approximate solution provided by FS-FEM is very close to the reference result and outperforms other methods in terms of computational accuracy. Specifically, compared with FEM, the computational errors of FS-FEM in magnetic vector potential, magnetic flux density, and current density are reduced by 52%, 5%, and 12%, respectively.

[0083] (3) Accurate prediction of LVDT differential voltage: In the calculation of differential voltage for different ranges and input voltage frequencies, the FS-FEM results are in good agreement with the experimental data. The minimum error observed is only 4.8%, which is 13.8% lower than that of FEM, making FS-FEM a reliable tool for predicting the linear relationship of LVDT under actual working conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] Figure 1 A physical picture of the LVDT used in the experiment of the present invention;

[0085] Figure 2 The physical picture and size diagram of the secondary coil of the LVDT used in the experiment of the present invention;

[0086] Figure 3 The numerical model diagram of the LVDT used in the experiment of the present invention;

[0087] Figure 4 It is the solution area of ​​the eddy current field problem set in the present invention;

[0088] Figure 5 Schematic diagram of the smooth domain of ES-FEM and FS-FEM in the present invention;

[0089] Figure 6 A numerical simulation flow chart of the smooth finite element method used in the present invention;

[0090] Figure 7 The vector magnetic potential distribution diagram of the LVDT along the ab side after using the present invention;

[0091] Figure 8 A comparison diagram of the average error of the vector magnetic potential of the LVDT along the ab side after using the present invention;

[0092] Fig. 9 A comparison diagram of the surface vector magnetic potential results of the LVDT after using the present invention;

[0093] Fig.10 The magnetic flux density distribution diagram along the ab side of the LVDT after using the present invention;

[0094] Fig.11 This is a comparison diagram of the average error of the magnetic flux density of the LVDT along the ab side after using the present invention;

[0095] Fig.12 A comparison diagram of the LVDT surface magnetic flux density results after using the present invention;

[0096] Fig.13 The current density distribution diagram of the LVDT along the ab side after using the present invention;

[0097] Fig.14 This is a comparison diagram of the average error of the current density of the LVDT along the ab side after using the present invention;

[0098] Fig.15 A comparison diagram of the LVDT surface current density results after using the present invention;

[0099] Fig.16 The comparison chart of simulation results and experimental results of LVDT differential voltage at 10KHz;

[0100] Fig.17 This is a comparison chart of the average error between the simulation results and the experimental results of the LVDT differential voltage at 10KHz;

[0101] Fig.18 The comparison chart of simulation results and experimental results of LVDT differential voltage at 30KHz;

[0102] Fig.19 This is a comparison chart of the average error between the simulation results and the experimental results of the LVDT differential voltage at 30KHz;

[0103] Fig. 20 The comparison chart of simulation results and experimental results of LVDT differential voltage at 50KHz;

[0104] Fig.21 The figure shows the average error comparison between the simulation results and the experimental results of the LVDT differential voltage at 50KHz. DETAILED DESCRIPTION

[0105] In order to gain a deeper understanding of the present invention, we will provide a comprehensive and detailed description of the present invention. However, the present invention has multiple implementations and is not limited to the specific examples listed herein. The presentation of these examples is intended to deepen the comprehensive understanding of the disclosure of the present invention.

[0106] Attached Figure 1The physical picture of the non-contact linear variable differential transformer (LVDT) used in the experiment of the present invention is composed of a main coil and two secondary coils symmetrically distributed relative to the main coil, wherein the two secondary coils are connected in series in reverse. When the main coil undergoes differential displacement, the secondary coil generates a sinusoidal signal whose amplitude is proportional to the displacement of the geometric center. Therefore, the LVDT can be used to measure the relative displacement in the seismic attenuation system (SAS).

[0107] Attached Figure 2 Taking the LVDT structure shown in the figure as an example, the enlarged view of the secondary coil of the LVDT used in the experiment of the present invention, the geometric dimensions and relative positions of the three coils of the LVDT are shown in the figure, and its material parameters and working parameters are obtained at the same time. The specific parameters are as follows:

[0108] D1(mm)=60, D2(mm)=56, D3(mm)=30, D4(mm)=26, L1(mm)=13.5, L2(mm)=18, L3(mm)=300, magnetic permeability of coil or air (H / m)=4π×10 -7 , coil conductivity (S / m) = 5.998×10 7 .

[0109] Primary coil input current formula:

[0110] I(t)=I m sin(2πf·t)

[0111] Among them, I m is the amplitude of the input current, and f is the frequency of the input current.

[0112] Mesh density: FEM / S-FEM: 12420 nodes, 70645 elements, 37260 degrees of freedom, reference solutions: 85709, 497461, 257127.

[0113] Attached Figure 3 This is the numerical model of the LVDT used in the experiment of the present invention. The left figure is a schematic diagram of the position of the reference point on the LVDT, and the right figure is a grid model of the LVDT.

[0114] Attached Figure 4 It is the solution area of ​​the eddy current field problem set in the present invention, which clearly indicates that the problem area is divided into two parts (eddy current area and non-eddy current area), and the boundary conditions of the normal component and the tangential component of the specified field variable are respectively given.

[0115] Attached Figure 6This is a numerical simulation flow chart of the magnetic field of a linear variable differential transformer (LVDT) based on the smooth finite element method of the present invention. First, the model is imported, the problem domain is discretized, the material parameters are initialized, and the domain variables are interpolated. The following are performed simultaneously: SFEM: 1) reconstructing the smooth subdomain, applying the gradient smoothing operation, calculating the smooth domain interpolation function, and applying the generalized smooth Galerkin weak form; FEM: 2) constructing the interpolation function and applying the Galerkin weak form; then applying the set loads and boundary conditions to solve the discrete equations.

[0116] Therefore, the numerical simulation method of the magnetic field of a linear variable differential transformer disclosed in the present invention specifically includes the following steps:

[0117] Step 1: Establish a geometric model based on the actual structure and size parameters of the LVDT, and simplify the geometric model; ignore the structural details that have little effect on the magnetic field without affecting the accuracy of the magnetic field simulation;

[0118] Furthermore, the specific operations of step 1 are:

[0119] A geometric model is established and simplified based on the physical model. Various parameters of the geometric model are determined based on the actual parameters of the LVDT. The model includes the primary coil, the secondary coil, and the space area around them, and the relative position relationship and geometric shape and size of each component are clarified.

[0120] Step 2: Discretize the problem domain, divide the tetrahedral mesh using adaptive mesh technology, adjust the mesh density according to the complexity of different regions in the geometric model and initialize the material parameters, the physical parameters include the current density, conductivity, magnetic permeability of the coil and the magnetic permeability of the surrounding medium;

[0121] Furthermore, the specific operations of step 2 are:

[0122] Step 2.1: Set the mesh density: FEM / S-FEM: 12420 nodes, 70645 elements, 37260 degrees of freedom, reference solution: 85709, 497461, 257127, use adaptive meshing technology to divide the tetrahedral mesh, and adjust the mesh density according to the complexity of different areas of the geometric model; use denser meshes in areas where the magnetic field changes drastically, such as coil windings, and use sparser meshes in other areas where the magnetic field changes slowly;

[0123] Step 2.2: Initialize material parameters, wherein the physical parameters include the current density, electrical conductivity, magnetic permeability of the coil and the magnetic permeability of the surrounding medium;

[0124] Step 2.3: The expression of primary coil input current is as follows:

[0125] I(t)=I m sin(2πf·t)

[0126] In the formula, I m represents the amplitude of the input current; f represents the frequency of the input current; t represents time.

[0127] Step 3: Based on the Maxwell equations, interpolate the domain variables, and reconstruct the smooth domain based on edges, nodes, and faces based on the smooth finite element method. Apply the gradient smoothing technique to recalculate the smooth domain interpolation function and apply the generalized smooth Galerkin weak form to establish the discrete system equations.

[0128] Furthermore, the specific operations of step 3 are:

[0129] Step 3.1: According to Maxwell's equations, the governing equations of the eddy current field are expressed as follows:

[0130]

[0131] In the formula, H is the magnetic field intensity, J is the current density, E is the electric field intensity, and B is the magnetic flux density; represents the partial differential operator; t represents time;

[0132] Step 3.2: The constitutive relationship of electromagnetic performance is as follows:

[0133] B=μH

[0134] J=σE

[0135] Where μ is the magnetic permeability and σ is the electrical conductivity. In order to reduce the scale of calculation, the vector magnetic potential A is introduced as an unknown function in the control equation, which is expressed as follows:

[0136]

[0137] The spin-free field is expressed as the gradient of a scalar function, further expressed as:

[0138]

[0139] In the formula, represents the gradient of the electric scalar potential;

[0140] Step 3.3: Consider the sinusoidal steady-state eddy current field as a special case of the general transient eddy current field. The governing equation of the boundary value problem of the sinusoidal steady-state eddy current field is as follows:

[0141]

[0142] In the formula, v is the magnetic reluctance, ω is the angular frequency of the sinusoidal change, is the coil current density, φ is the electric scalar potential, the dots above the field variables indicate that they are complex values; j represents the imaginary unit; Ω represents the problem domain; Ω1 represents the eddy current region in the problem domain;

[0143] Step 3.4: In the tetrahedral element, the vector magnetic potential and electric scalar potential are expressed as linear interpolation functions at each node as follows:

[0144]

[0145] in, is the shape function of node i, A i and φ i are the vector magnetic potential and electric scalar potential of node i; represents the vector magnetic potential of unit e; represents the electric scalar potential of unit e;

[0146] Step 3.5: The magnetic flux density of the linear tetrahedral element is expressed as follows:

[0147]

[0148] In the formula, Respectively represent the vector magnetic potential components of the ith node along the x, y, and z directions; Represent the unit vectors along the x, y, and z directions respectively;

[0149] Step 3.6: The control equation is discretized into the following form:

[0150]

[0151]

[0152] Step 3.7: In order to make the governing equations have a symmetric coefficient matrix, introduce the following function:

[0153]

[0154] Step 3.8: Substitute it into the magnetic flux density formula of the linear tetrahedron element to obtain the following governing equation:

[0155]

[0156] Step 3.9: The vector magnetic potential contains components in the X, Y, and Z directions. The above control equations can be simplified into the following algebraic equations:

[0157]

[0158] Where K is the overall coefficient matrix, is the solution vector, F is the right-hand side vector, which is provided by the linear tetrahedral unit k e , and f e The contribution of linear tetrahedral elements is expressed as follows:

[0159]

[0160] In the formula, are the vector magnetic potential and scalar potential of the ith node of the eth unit, is the source current vector of the i-th node of the e-th unit; T represents the transpose of the vector and matrix;

[0161] Step 3.10: The magnetic flux density and current density of the e-th unit are as follows:

[0162]

[0163] In the formula, Respectively represent the vector magnetic potential components of the ith node along the x, y, and z directions; They represent the magnetic flux density components of unit e along the x, y, and z directions respectively; represents the source current density of unit e; represents the current density of unit e; They respectively represent the partial differential of the shape function (shape function coefficient) of the i-th node.

[0164] Step 3.11: Based on the tetrahedral element mesh, the problem domain is further subdivided into smooth subdomains without overlap and gap. The smooth subdomains are constructed by connecting the body centers, face centers, edge centers and nodes of the elements.

[0165] Step 3.12: Magnetic flux density at node k and the electric scalar potential gradient Use the following formula:

[0166]

[0167] In the formula, Smooth subdomain The borders of Represent the unit vectors along the x, y, and z directions respectively; They represent the partial differential of the shape function (shape function coefficient) of the kth node respectively; n s represents the number of smooth subdomains associated with the kth node; Respectively represent the vector magnetic potential components of the kth node along the x, y, and z directions;

[0168] Smooth subdomain The volume is expressed in the following form:

[0169]

[0170] In the formula, represents the number of units associated with the kth node; represents the volume of the jth element associated with the kth node;

[0171] Step 3.13: After the gradient smoothing operation, the formula is as follows:

[0172]

[0173] Where n r For the border The unit external normal vector on ; They represent the partial differential of the shape function (shape function coefficient) of the kth node after the smooth domain is re-divided and before the smooth domain is not divided; N k represents the shape function of the kth node; r can represent the three directions of x, y, and z respectively;

[0174] Step 4: Apply current load and determine eddy current field boundary conditions; the magnetic field is an eddy current problem domain with boundaries, the problem domain consists of an eddy current region with non-zero conductivity and a non-eddy region containing a given source current, the boundary conditions include the external boundary conditions of the LVDT and the boundary conditions of the internal interface, the external boundary conditions are set according to the actual working environment of the LVDT, and since the domain variables are continuous at the interface between the two media, the internal boundary conditions will be automatically satisfied.

[0175] Furthermore, the specific operations of step 4 are:

[0176] Step 4.1: Determine the composition and boundary of the eddy current problem domain, and set the normal component of the given magnetic flux density and the tangential component of the magnetic field intensity on the boundary.

[0177] Step 4.2: The boundary conditions are expressed as follows:

[0178]

[0179]

[0180] Where n is the unit normal vector on the boundary; A represents the vector magnetic potential, Γ B and Γ H represents the outer boundary of the problem domain; Γ 12 Represents the internal interface of the problem domain; and They represent the vector magnetic potential of the eddy current zone and the non-eddy current zone on the interface, respectively; v1 and v2 represent the magnetic resistivity of the eddy current zone and the non-eddy current zone on the interface, respectively; n 12 represents the internal interface Γ12 The unit normal vector of .

[0181] Step 5: Solve the discrete linear equations to obtain the numerical simulation results of the LVDT magnetic field;

[0182] Furthermore, the step 5 utilizes MATLAB programming to assemble and solve discrete linear equations to obtain numerical simulation results of the LVDT magnetic field.

[0183] In the step of solving the discretized model using the smooth finite element method, parallel computing technology is used to improve the solution efficiency, and the computing tasks are distributed to multiple computing cores or processors for simultaneous computing.

[0184] In the step of solving the discretized model using the smooth finite element method, error analysis and verification are performed on the simulation results, and the accuracy of the simulation is evaluated by comparing with experimental data.

[0185] In the step of solving the discretized model using the smooth finite element method, data mining and analysis are performed on the simulation results to extract valuable information for LVDT design and performance optimization, such as the magnetic field distribution law and linear relationship.

[0186] Step 6: Follow the above steps to obtain the numerical solutions required for this experiment; use the obtained numerical solutions for post-processing and plotting, and compare them with the reference solutions and experimental results to verify the accuracy and effectiveness of the method.

[0187] In summary, the present invention has the following advantages:

[0188] (1) Simplify the processing and improve accuracy: Under the same mesh unit type conditions, S-FEM can directly use FEM data for subsequent processing, greatly reducing the workload of preprocessing. Compared with FEM, S-FEM only needs a small modification to achieve higher accuracy.

[0189] (2) Excellent performance of FS-FEM: Figure 7-Figure 8 , Figure 10-11 , Figure 13-14 It can be seen from the comparison of numerical results and average errors that the best approximate solution provided by FS-FEM is very close to the reference result and outperforms other methods in terms of calculation accuracy. Specifically, compared with FEM, the calculation errors of FS-FEM in magnetic vector potential, magnetic flux density and current density are reduced by 52%, 5% and 12%, respectively.

[0190] (3) Accurate prediction of LVDT differential voltage: Figure 16-Figure 21The numerical results and average error comparison show that the FS-FEM results are in good agreement with the experimental data in the differential voltage calculations for different ranges and input voltage frequencies. The minimum error observed is only 4.8%, which is 13.8% lower than that of FEM, making FS-FEM a reliable tool for predicting the linear relationship of LVDTs under actual operating conditions.

[0191] The contents not described in detail in the specification of the present invention belong to the prior art known to the professional and technical personnel in the field. Although the illustrative specific embodiments of the present invention are described above to facilitate the understanding of the present invention by the technical personnel in the field, it should be clear that the present invention is not limited to the scope of the specific embodiments. For the ordinary technical personnel in the field, as long as various changes are within the spirit and scope of the present invention defined and determined by the attached claims, these changes are obvious, and all inventions and creations using the concept of the present invention are protected.

Claims

1. A numerical simulation method for the magnetic field of a linear variable differential transformer, characterized in that: The method comprises the following steps: Step 1: Establish a geometric model based on the actual structure and size parameters of the LVDT, and simplify the geometric model; Step 2: Discretize the problem domain, divide the tetrahedral mesh using adaptive mesh technology, adjust the mesh density according to the complexity of different regions in the geometric model and initialize the material parameters, the physical parameters include the current density, conductivity, magnetic permeability of the coil and the magnetic permeability of the surrounding medium; Step 3: Based on the Maxwell equations, interpolate the domain variables, and reconstruct the smooth domain based on edges, nodes, and faces based on the smooth finite element method. Apply the gradient smoothing technique to recalculate the smooth domain interpolation function and apply the generalized smooth Galerkin weak form to establish the discrete system equations. Step 4: Apply current load and determine eddy current field boundary conditions; the magnetic field is an eddy current problem domain with boundaries, the problem domain consists of an eddy current region with non-zero conductivity and a non-eddy region containing a given source current, the boundary conditions include the external boundary conditions of the LVDT and the boundary conditions of the internal interface, the external boundary conditions are set according to the actual working environment of the LVDT, and since the domain variables are continuous at the interface between the two media, the internal boundary conditions will be automatically satisfied; Step 5: Solve the discrete linear equations to obtain the numerical simulation results of the LVDT magnetic field.

2. The method for numerical simulation of magnetic field of a linear variable differential transformer according to claim 1, characterized in that: The specific operations of step 1 are: A geometric model is established and simplified based on the physical model. Various parameters of the geometric model are determined based on the actual parameters of the LVDT. The model includes the primary coil, the secondary coil and the spatial area around them, and clarifies the relative position relationship and geometric shape and size of each component.

3. The method for numerical simulation of magnetic field of a linear variable differential transformer according to claim 2, characterized in that: The specific operations of step 2 are: Step 2.1: Use adaptive meshing technology to divide the tetrahedral mesh and adjust the mesh density according to the complexity of different areas of the geometric model; Step 2.2: Initialize material parameters, wherein the physical parameters include the current density, electrical conductivity, magnetic permeability of the coil and the magnetic permeability of the surrounding medium; Step 2.3: The expression of primary coil input current is as follows: I(t)=I m sin(2πf·t) In the formula, I m represents the amplitude of the input current; f represents the frequency of the input current; t represents time.

4. The method for numerical simulation of magnetic field of a linear variable differential transformer according to claim 3, characterized in that: The specific operations of step 3 are: Step 3.1: According to Maxwell's equations, the governing equations of the eddy current field are expressed as follows: In the formula, H is the magnetic field intensity, J is the current density, E is the electric field intensity, and B is the magnetic flux density; represents the partial differential operator; t represents time; Step 3.2: The constitutive relationship of electromagnetic performance is as follows: B=μH J=σE Where μ is the magnetic permeability and σ is the electrical conductivity. To reduce the scale of calculation, the vector magnetic potential A is introduced as an unknown function in the control equation, which is expressed as follows: The spin-free field is expressed as the gradient of a scalar function, further expressed as: In the formula, represents the gradient of the electric scalar potential; Step 3.3: Consider the sinusoidal steady-state eddy current field as a special case of the general transient eddy current field. The governing equation of the boundary value problem of the sinusoidal steady-state eddy current field is as follows: In the formula, v is the magnetic reluctance, ω is the angular frequency of the sinusoidal change, is the coil current density, φ is the electric scalar potential, the dots above the field variables indicate that they are complex values; j represents the imaginary unit; Ω represents the problem domain; Ω1 represents the eddy current region in the problem domain; Step 3.4: In the tetrahedral element, the vector magnetic potential and electric scalar potential are expressed as linear interpolation functions at each node as follows: in, is the shape function of node i, A i and φ i are the vector magnetic potential and electric scalar potential of node i; represents the vector magnetic potential of unit e; represents the electric scalar potential of unit e; Step 3.5: The magnetic flux density of the linear tetrahedral element is expressed as follows: In the formula, Respectively represent the vector magnetic potential components of the ith node along the x, y, and z directions; Represent the unit vectors along the x, y, and z directions respectively; Step 3.6: The control equation is discretized into the following form: Step 3.7: In order to make the governing equations have a symmetric coefficient matrix, introduce the following function: Step 3.8: Substitute it into the magnetic flux density formula of the linear tetrahedron element to obtain the following governing equation: Step 3.9: The vector magnetic potential contains components in the X, Y, and Z directions. The above control equations are simplified into the following algebraic equations: Where K is the overall coefficient matrix, is the solution vector, F is the right-hand side vector, which is provided by the linear tetrahedral unit k e , and f e The contribution of linear tetrahedral elements is expressed as follows: In the formula, are the vector magnetic potential and scalar potential of the ith node of the eth unit, is the source current vector of the i-th node of the e-th unit; T represents the transpose of the vector and matrix; Step 3.10: The magnetic flux density and current density of the e-th unit are as follows: In the formula, Respectively represent the vector magnetic potential components of the ith node along the x, y, and z directions; They represent the magnetic flux density components of unit e along the x, y, and z directions respectively; represents the source current density of unit e; represents the current density of unit e; They represent the partial differential of the shape function of the i-th node respectively; Step 3.11: Based on the tetrahedral element mesh, the problem domain is further subdivided into smooth subdomains without overlap and gap. The smooth subdomains are constructed by connecting the body centers, face centers, edge centers and nodes of the elements. Step 3.12: Magnetic flux density at node k and the electric scalar potential gradient Use the following formula: In the formula, Smooth subdomain The borders of Represent the unit vectors along the x, y, and z directions respectively; They represent the partial differential of the shape function of the kth node; n s represents the number of smooth subdomains associated with the kth node; Respectively represent the vector magnetic potential components of the kth node along the x, y, and z directions; Smooth subdomain The volume is expressed in the following form: In the formula, represents the number of units associated with the kth node; represents the volume of the jth element associated with the kth node; Step 3.13: After the gradient smoothing operation, the formula is as follows: Where n r For the border The unit external normal vector on ; They represent the partial differential of the shape function of the kth node after the smooth domain is re-divided and before the smooth domain is not divided; N k represents the shape function of the kth node; r represents the x, y, and z directions respectively.

5. The method for numerical simulation of magnetic field of a linear variable differential transformer according to claim 4, characterized in that: The specific operations of step 4 are: Step 4.1: Determine the composition and boundary of the eddy current problem domain, and set the normal component of the given magnetic flux density and the tangential component of the magnetic field intensity on the boundary; Step 4.2: The boundary conditions are expressed as follows: Where n is the unit normal vector on the boundary; A represents the vector magnetic potential, Γ B and Γ H represents the outer boundary of the problem domain; Γ 12 Represents the internal interface of the problem domain; and They represent the vector magnetic potential of the eddy current zone and the non-eddy current zone on the interface, respectively; v1 and v2 represent the magnetic resistivity of the eddy current zone and the non-eddy current zone on the interface, respectively; n 12 represents the internal interface Γ 12 The unit normal vector of .

6. The method for numerical simulation of magnetic field of a linear variable differential transformer according to claim 5, characterized in that: The specific operations of step 5 are: Using MATLAB programming, the discrete linear equations are assembled and solved to obtain the numerical simulation results of the LVDT magnetic field.

7. The method for numerical simulation of magnetic field of a linear variable differential transformer according to claim 6, characterized in that: The discretization models using the smooth finite element method in step 3 are the edge-based smooth finite element model ES-FEM and the surface-based smooth finite element model FS-FEM. In ES-FEM, the calculation accuracy is improved by processing the edge-related areas, and in FS-FEM, the simulation effect is improved by processing the surface-related areas.

8. The method for numerical simulation of magnetic field of a linear variable differential transformer according to claim 7, characterized in that: The method further includes a step of post-processing the numerical simulation results in step 5, wherein the post-processing includes visually displaying the magnetic field distribution of the LVDT, calculating the average value, maximum value or minimum value of the vector magnetic potential, magnetic flux density, and current density in a specific area, and predicting the linear relationship of the LVDT.

9. The method for numerical simulation of magnetic field of a linear variable differential transformer according to claim 8, characterized in that: The visualization displays the magnetic field distribution of the LVDT, and uses cloud diagrams or vector diagrams to show the distribution of the magnitude and direction of the magnetic field in the LVDT and the space around it.

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