Fast computation model for superconducting pinning Maglev mechanical system

Abstract

Superconducting pinning Maglev, with its passive self-stabilization, low resistance, and noise, holds promise as the next generation of high-speed rail transit. The vehicle dynamic simulation is essential for high-speed operation, and the interaction between the high-temperature superconductor (HTS) and the permanent magnet guideway (PMG), termed the “HTS-PMG Relation,” is fundamental to understanding the system’s dynamics. This paper presents an efficient computation model for the HTS-PMG relation, employing flux penetration characteristics to reduce the dimensionality of the physical field. The model integrates Maxwell’s equations with the Bean E-J constitutive relationship, resulting in reduced-dimensional governing equations and a nonlinear boundary finite difference method (FDM) optimization algorithm. Validated through quasi-static and dynamic experiments, this model accurately simulates the complex hysteresis behaviors of pinning Maglev and significantly enhances computational efficiency compared to traditional finite element methods. The proposed fast HTS-PMG relation computation model enables large-scale dynamic simulations of superconducting Maglev trains and offers a novel approach to studying their dynamic performance and levitation drift.

Introduction

Maglev trains, known for their high speed, quiet operation, and absence of mechanical contact friction, are gradually becoming commercialized and widely adopted^1,2. With the rise of concepts like “Hyperloop” and “Next-Generation Rail Transit,” research institutions in various countries have accelerated the development of new Maglev technologies. The focus of this study, superconducting pinning Maglev, is another type of Maglev that started relatively late. Superconducting pinning Maglev is a type of Maglev system that can achieve self-stabilizing levitation without any control systems^3,4,5,6,7. Its advantages include a simple structure conducive to lightweight design, stable levitation without energy consumption, static levitation capability, no inherent magnetic resistance, low noise, and low energy consumption. These benefits give it tremendous development potential in the future of ultra-high-speed rail transportation^8,9.

In 1986, the discovery of the high-temperature superconductor YBa₂Cu₃O_{7 − x} sparked interest among scientists in applying it to Maglev technology⁵. With continuous technological advancements, superconducting pinning Maglev technology has been widely used in magnetic bearings, flywheel energy storage, and transportation⁹. In 1997, scientists from China and Germany jointly developed a high-temperature superconducting pinning Maglev model car weighing 20 kg, achieving stable levitation with a 7 mm levitation gap¹⁰. In 2000, the Institute of Superconducting Technology at Southwest Jiaotong University successfully developed the world’s first manned high-temperature superconducting pinning Maglev experimental vehicle “Century”, which could carry four people¹¹. In 2004, the Moscow Aviation Institute in Russia developed a corresponding manned high-temperature superconducting pinning Maglev experimental vehicle¹². That same year, the IFW Institute for Solid State Research in Germany successfully developed the high-temperature superconducting pinning Maglev experimental vehicle “SupraTrans I”¹³. Additionally, the AIST National Institute of Advanced Industrial Science and Technology in Japan and the University of L’Aquila in Italy also developed superconducting pinning Maglev model cars¹⁴. In 2014, the Federal University of Rio de Janeiro in Brazil completed the construction of the first 200-meter outdoor high-temperature superconducting pinning Maglev test line and prototype vehicle “Maglev Cobra”¹⁵. This test line, connecting two teaching buildings, features a test vehicle that can accommodate 20 people (Fig. 1)¹⁶. In 2021, a full-scale high-temperature superconducting pinning Maglev engineering prototype vehicle and test line were officially launched at Southwest Jiaotong University (Fig. 2)¹⁷. This prototype vehicle is designed for high-speed operation, with a linear load density of 2.5 t/m, making it the highest load capacity superconducting pinning Maglev vehicle to date.

Fig. 1

Brazilian “Maglev Cobra” Prototype.

Fig. 2

Prototype in Southwest Jiaotong University.

However, after solving the issues of stable levitation and load capacity, the next focus for superconducting pinning Maglev is the vibration problem during high-speed operation. As speed increases, vibration issues during vehicle operation become more pronounced, severely impacting the safety and stability of the vehicle^18,19,20. Therefore, vehicle dynamics research is a crucial step in promoting the engineering application of superconducting pinning Maglev. By using experimental, theoretical, and simulation methods to evaluate the safety and comfort of its operation before actual vehicle testing, potential problems can be identified, significantly reducing trial-and-error costs. Hence, conducting vehicle dynamics research on superconducting pinning Maglev has practical significance and is fundamental for this technology to move towards engineering applications. An accurate dynamic model of superconducting pinning Maglev vehicles requires an essential preliminary study: the “HTS-PMG relation” model.

The “HTS-PMG relation” refers to the interaction between the ground permanent magnet guideway (PMG) and the onboard high-temperature superconductor (HTS). The basic principle involves the interaction between a second-type non-ideal superconducting bulk material and a permanent magnet. Within the superconducting bulk material, induced currents are generated, and the interaction between these induced currents and the magnetic field of the permanent magnet produces electromagnetic forces. When the superconductor transitions into the superconducting state, it traps the magnetic field already inside it, as if the magnetic flux lines are pinned, allowing the superconductor to stably levitate in the magnetic field. This phenomenon is known as the “pinning” effect. With the advancement of computer technology, the study of the HTS-PMG relation in superconducting pinning Maglev has gradually transitioned from theoretical simplification to numerical simulation, especially utilizing the finite element method (FEM) for increasingly convenient and accurate numerical simulations^21,22,23,24. Based on this method, some researchers have conducted simple vibration simulations of superconducting pinning Maglev, but these computations are time-consuming²⁵. In the field of HTS undulators, some researchers have employed inverse analysis to accelerate the FEM simulation of HTS bulks; however, such approaches have not yet been applied to HTS–PMG systems or vehicle dynamics²⁶. From the computation results of levitation and guiding forces, current FEM simulation results align well with experimental test results. However, the common issue with this method is the large computational load and extended time required for computations.

For the study of the HTS-PMG relation, researchers explore both microscopic electromagnetic mechanisms and macroscopic data obtained from experiments²⁷. Due to the complexity of hysteresis effects in the HTS-PMG relation, many scholars have abandoned the description of this complex phenomenon and instead used damping to replace hysteresis effects, and an elastic model to represent stiffness²⁸. By fitting the measured levitation force data, expressions for the elasticity of levitation and guiding forces can be obtained^29,30,31,32. Some have also used the least squares method to fit measured vibration acceleration data of the levitation system, yielding similar elastic expressions³³. The macroscopic phenomenological elastic models of the HTS-PMG relation obtained through theoretical, quasi-static, and dynamic experimental data fitting are simple expressions that consume almost no computational resources. However, because hysteresis effects are not considered, the calculated results will always return to the origin position, which contradicts the fundamental nature of the HTS-PMG relation in superconducting pinning Maglev. To describe the complex hysteresis behavior of HTS-PMG systems, some researchers have attempted to adopt models analogous to the Bouc–Wen hysteresis model. These approaches can effectively reproduce stable hysteresis loops; however, they remain insufficient for fully capturing the various stochastic hysteresis phenomena that emerge under random vibrations in superconducting magnetic levitation systems (as discussed below)³⁴.

The superconducting pinning Maglev exhibits nonlinear hysteresis phenomena in levitation force, resulting in an indeterminate mapping relationship between levitation force and levitation gap, implying that different levitation forces may exist for the same levitation gap. Its prominent behavioral characteristics include the following points:

• 1. Even if the superconductor returns to its original position after movement, its levitation force cannot be restored due to changes in internal magnetization (see Fig. 3(a)).

• 2. During regular cyclic motion of the superconductor, periodic magnetization phenomena occur, gradually forming a stable hysteresis loop in the levitation force-displacement curve (see Fig. 3(b)).

• 3. When the superconductor moves uniformly in the same direction, non-uniform current densities may exist as magnetic flux penetrates, leading to inflection points in the levitation force-displacement curve (see process 3 to 4 in Fig. 3(c), which the motion is continuous and uniform).

Fig. 3

Illustration of Levitation Force Behavior Characteristics in Pinning Maglev (Experimental Results): (a) Characteristic of Irreversible Levitation Force Recovery; (b) Hysteresis Loop Formed by Cyclical Motion; (c) Characteristics of Inflection Points During Uniform Motion.

The levitation force not only depends on the levitation gap but also on the historical motion orbit of the superconductor in the magnetic field. This leads to an infinite variety of levitation force-displacement curves. Due to the absence of a mapping relationship between the levitation force and displacement, traditional nonlinear elastic models cannot fit the levitation force-displacement curve. Moreover, special properties such as inflection points cannot be described by such as Bouc-We or H-M hysteresis mathematical models^35,36. Additionally, since the motion orbit of the superconductor is infinite, table lookup methods cannot describe the levitation force accurately. Therefore, due to the complexity of pinning Maglev levitation force, traditional mathematical formula fitting methods and look-up table methods cannot accurately describe it. Only by starting from the electromagnetic essence inside the superconductor can this issue be addressed. However, traditional FEM are time-consuming, and considering the hundreds of levitation units in pinning Maglev trains, conducting dynamic simulations with them is not practically feasible. Hence, it is necessary to establish a computational model that can rapidly calculate the levitation force from the internal electromagnetic essence of the superconductor.

A superconducting pinning Maglev train equipped with numerous superconducting levitators (containing superconductors within a Dewar vessel). For example, as illustrated in Fig. 4, one section of the train has 72 superconducting levitators and a train with three sets consists of 216 superconducting levitators. If traditional FEM were used to calculate the HTS-PMG relations for all levitators, the computation would be immense and difficult to achieve. Therefore, current dynamic simulations of superconducting pinning Maglev trains employ elastic models. Although this computational method can reflect the basic frequency characteristics of the train during operation, it cannot accurately obtain displacement information, such as levitation gap, nor can it capture the unique behaviors of superconducting pinning Maglev, such as levitation drift (change of the static levitation gap during operation). Thus, there is a need for an HTS-PMG relation model that can both improve computational efficiency and describe complex hysteresis behaviors, to thoroughly investigate the impact of hysteresis effects on train dynamics. In this paper, a simplified computational model based on the internal electromagnetic characteristics of superconductors is proposed, and an optimization algorithm based on the internal magnetic flux variation during superconductor vibration is proposed. The correctness of the proposed model was validated through quasi-static and dynamic experiments, and its computational efficiency was compared with that of traditional FEM models, confirming the effectiveness of the optimization algorithm. Finally, using the proposed HTS-PMG relation fast computation model, dynamic simulations of a full-scale train consisting of three sets of superconducting pinning Maglev were conducted, achieving joint simulation of all 216 levitators and the train dynamics.

Critical state dimensionality reduction model

This section first analyzes the magnetic penetration and hysteresis phenomena inside the YBa₂Cu₃O_7 − x superconductor using the Bean model (critical state model) and proposes a quasi one-dimensional simplified penetration assumption. Secondly, combining Maxwell’s equations and the Bean critical state E-J relationship model, the electromagnetic governing equations of the superconductor under this assumption system are obtained. Finally, the boundary conditions under non-steady external fields are established. Combined with the governing equations, the distribution of internal current density and magnetization intensity of the superconductor can be determined, and the suspension guidance force can be obtained using the magnetic moment force formula. The model established in this section has the characteristics of low degrees of freedom and small computational volume, providing a foundation for subsequent dynamic simulations.

Trapped magnetic flux and penetration assumption

The fundamental equations of electromagnetism, known as Maxwell’s equations, are the fundamental laws governing electromagnetic fields. Superconducting pinning Maglev also adhere to these electromagnetic laws, with their primary distinction from ordinary conductors being the absence of conventional electrical conductivity. According to Ampere’s circuital law from Maxwell’s equations, the relationship between the current density and magnetization (electromagnetic) inside a superconductor can be derived as follows:

$${\mathbf{J}}=\nabla \times {\mathbf{M}}$$

(1)

Faraday’s law of electromagnetic induction can obtain:

$$\nabla \times {\mathbf{E}}= - \frac{{\partial {\mathbf{B}}}}{{\partial t}}$$

(2)

When the external magnetic field changes, type II superconductors tend to maintain the internal magnetic field constant (known as the pinning effect). However, due to the finite critical current density J_c of the superconductor, the changing external magnetic field begins to penetrate from the outer side of the superconductor. Therefore, an assumption is proposed here: after the external field of the superconductor changes, the changing magnetic field will start to penetrate from the outermost side of the superconductor. Since our PMG provides a uniformly oriented magnetic field in the forward direction, we need not consider the axis in the forward direction. Combining the above assumption, the distribution of the internal current density and magnetic field within the superconductor can be considered as a one-dimensional distribution, defined here as the magnetic flux penetration distance, denoted by r. Thus, the fundamental equations can be analyzed in this abstract dimension of r. The magnetic flux penetration distance r here refers to the distance from the penetration position to the center of the superconductor, and its penetration depth is the distance from the superconductor boundary to the center minus the magnetic flux penetration distance.

With the above assumptions and definitions, Eq. (1) is transformed into the new dimensional form:

$$J(r,t)=\frac{{\partial M(r,t)}}{{\partial r}}$$

(3)

And Eq. (2) is transformed into:

$$\frac{{\partial E(r,t)}}{{\partial r}}=\frac{{\partial B(r,t)}}{{\partial t}}$$

(4)

Then, we still need the E-J constitutive relationship equation to relate the electric field to the current density. We use the Bean model here because of its simplicity and its ability to effectively reflect the hysteresis characteristics during the superconductor’s motion. The general form of the Bean model is expressed as³⁷:

$${\mathbf{J}}=\left\{ \begin{gathered} 0{\text{ }}{\mathbf{E}}=0 \hfill \\ {J_{\text{c}}}{\mathbf{n}}{\text{ }}{\mathbf{E}} \ne 0 \hfill \\ \end{gathered} \right.$$

(5)

where n is a unit vector in the direction of the electric field vector.

From Eq. (4), the general solution for E can be obtained, but not the particular solution, which requires additional constraints such as Ampere’s law. However, based on the characteristics of magnetic flux penetration in the superconductor, the absence of current in the non-penetrated part implies zero electric field, and each unidirectional movement inevitably results in the distribution of electric fields in the same direction. Therefore, the direction of the electric field gradient is consistent with that of the electric field. Combining Eq. (4) Eq. (5) can be transformed into the constitutive relation for E-J in the new dimension:

$$J(r,t)={J_{\text{c}}}{\text{sign}}(E(r,t))$$

(6)

Combining Eq. (4) and Eq. (6) yields the relationship between current and magnetic field:

$$J= - {J_{\text{c}}}{\text{sign}}(\partial B/\partial t)$$

(7)

Combining Eq. (3) and Eq. (7), we obtain its electromagnetic governing equation:

$${J_{\text{c}}}{\text{sign}}\left( {\frac{{\partial B}}{{\partial t}}} \right)+\frac{{\partial M}}{{\partial r}}=0$$

(8)

Equation (8) simplifies the originally complex system of three-dimensional electromagnetic governing equations into a one-dimensional equation, significantly streamlining the solving process. At this stage, the electromagnetic fundamental equations in terms of magnetic flux penetration depth dimension are essentially determined. Next, we consider the external field variations and boundary conditions to solve this equation.

Boundary conditions under unsteady external magnetic fields

The spatial distribution of the magnetic field on the PMG is complex and highly nonlinear, making it difficult to describe directly. Here, first assume that the superconductor is in a time-varying uniform field, which corresponds to different positions above the PMG at different times, while the positional distribution of the PMG’s magnetic field is nonlinear.

First, let R be the boundary of the superconductor in dimension r. The aforementioned external field assumption can be described as:

$$\forall {B_{{\text{ex}}}}(r,t)={B_{{\text{ex}}}}(0,t)$$

(9)

Where B_ex is the external field of the PMG at this moment. The magnetic field inside the superconductor can be described as:

$${\mathbf{B}}={{\mathbf{B}}_{{\text{ex}}}}+{\mu _0}{\mathbf{M}}$$

(10)

When the external magnetic field changes, the changing magnetic field first penetrates from point R (i.e., the outer surface), while the magnetic field at the center remains unchanged (i.e., flux pinning). This boundary condition is described as:

$$B(0,t)=B(0,0)$$

(11)

$$B(R,t)={B_{{\text{ex}}}}(0,t)$$

(12)

Integrating the current density over the r-dimension yields:

$$M(0,t)=\int_{R}^{0} {J(r,t){\text{d}}r=(B(0,t)} - {B_{{\text{ex}}}}(0,t))/{\mu _0}$$

(13)

Unlike ideal superconductors, when a type-II non-ideal superconductor undergoes phase transition, it does not expel any magnetic field, and there is no current inside. Therefore, the initial condition of field cooling for the superconductor is described as:

$$J(r,0)=0$$

(14)

In this way, the electromagnetic governing equation for the superconductor under an unsteady external field, the boundary magnetic field conditions, and the initial current density distribution conditions of the superconductor are obtained. When an external field is given, the internal current distribution and magnetization strength of the superconductor can be determined.

Derivation of levitation forces and description of time-varying magnetic fields

For the levitation force acting on the superconductor, it is generally calculated using the Lorentz force, as follows:

$${\mathbf{F}}=\int {{\mathbf{J}} \times {\mathbf{B}}} {\text{d}}v$$

(15)

However, calculating in this manner involves an additional integral computation, while the previous operations already performed integrals over J with respect to r and t to derive the current density distribution. Integrating the vector product of J and B over the spatial volume again would significantly increase the computational load. Additionally, converting the abstract r-dimension back into the regular spatial dimension would complicate the process, contradicting the purpose of model simplification and fast computation. Since the calculation process also yields the magnetic field distribution within the superconductor, thereby providing its magnetization distribution, directly using the magnetic moment levitation force expression is simpler and requires less computation. The magnetic moment levitation force expression is as follows:

$${F_z}=\int {M\frac{{\partial {B_{{\text{ex}}}}}}{{\partial z}}{\text{d}}r}$$

(16)

It can also be written in terms of current as follows:

$${F_z}=\int {\int {J\frac{{\partial {B_{{\text{ex}}}}}}{{\partial z}}{{\text{d}}^{\text{2}}}r} }$$

(17)

Thus, knowing the temporal values of the external field B_ex at time t allows the calculation of the corresponding levitation force on the superconductor. However, during the motion process, what is most easily obtained is the position of the superconductor relative to the PMG, and calculating the levitation force also requires solving the spatial derivatives of B_ex. So, it is necessary to obtain the spatial distribution of B_ex. Assuming the cryogenic position as the origin, z represents the vertical position with downward direction as positive, and y represents the horizontal position with leftward direction as positive. In reference²⁹, it is calculated that the vertical magnetic field above the PMG exhibits an exponential distribution in the vertical direction, while the horizontal magnetic field exhibits an approximately proportional distribution in the horizontal direction. Thus, the distribution of B_ex in these two directions is represented as:

$${B_{{\text{exz}}}}(z,y)={\alpha _1}{e^{{\beta _1}\cdot z}}+{\eta _1}{y^{{\varphi _1}}}+{\gamma _1}$$

(18)

$${B_{{\text{exy}}}}(z,y)={\alpha _2}y({\eta _2}{e^{{\beta _1}\cdot z}}+{\gamma _2})$$

(19)

Where B_exz is the vertical external field, B_exy is the horizontal external field; α₁, α₂, β₁, φ₁, γ₁, γ₂ are parameters to be determined, which can be obtained by fitting experimental values using the least squares method. Thus, Eq. (4) is expressed as:

$$E(r)=\frac{{\partial B_{\text{ex}}}}{{\partial z}} \cdot \frac{{\partial z}}{{\partial t}}$$

(20)

Equation (7) is expressed as:

$$J= - {J_{\text{c}}}(\frac{{\partial {B_{{\text{ex}}}}}}{{\partial z}} \cdot \frac{{\partial z}}{{\partial t}})$$

(21)

Equation (8) is expressed as:

$${J_{\text{c}}}{\text{sign}}(\frac{{\partial B}}{{\partial z}} \cdot \frac{{\partial z}}{{\partial t}})+\frac{{\partial M}}{{\partial r}}=0$$

(22)

Henceforth, the essential requirements transition from the temporal variability of the external field to solely capturing the motion of the superconductor. Consequently, the model founded upon the r dimension becomes applicable for dynamic computations.

Nonlinear boundary FDM optimization algorithm

This Section primarily focuses on the specific numerical solution methods for the dimensionality reduction model of pinning Maglev force established in Sect. 2, employing primarily FDM methods. Firstly, the FDM formulation of the governing equations and boundary conditions is established. Secondly, an analysis of the errors induced by the FDM is conducted. Finally, a nonlinear boundary optimization algorithm tailored for the characteristics of superconducting flux penetration is introduced.

FDM formulation of governing equations and boundary conditions

FDM is a numerical technique used to solve partial differential equations. Its core idea involves discretizing the domain with a finite grid, and then transforming the partial differential equations to a system of difference equations. Here, the grid divides the superconductor into $M'$a segments in the r dimension and $N'$b segments in the time dimension, with grid sizes in each dimension given by:

$$\Delta r=R/M'$$

(23)

$$\Delta t=T/N'$$

(24)

where Δr is the grid size in the r-dimension, Δt is the grid size in the time t-dimension, and T is the total computation time. Thus, r and t are represented as:

$$r=m\Delta r$$

(25)

$$t=n\Delta t$$

(26)

where m denotes the m-th grid, and n denotes the n-th grid.

Write the governing Eq. (8) in the form of a FDM equation:

$${J_{\text{c}}}{\text{sign}}\left( {\frac{{B(m\Delta r,(n+1)\Delta t) - B(m\Delta r,n\Delta t)}}{{\Delta t}}} \right)+\frac{{M\left( {(m+1)\Delta r,n\Delta t} \right) - M\left( {m\Delta r,n\Delta t} \right)}}{{\Delta r}}=0$$

(27)

The current distribution can be expressed as Eq. (7):

$$J= - {J_{\text{c}}}{\text{sign}}\left( {\frac{{B(m\Delta r,(n+1)\Delta t) - B(m\Delta r,n\Delta t)}}{{\Delta t}}} \right)$$

(28)

The boundary condition Eq. (11) can be written in differential form:

$$B(0,n\Delta t)=B(0,\Delta t)$$

(29)

The boundary condition Eq. (12) can be written as:

$$B(M'\Delta r,n\Delta t)={B_{{\text{ex}}}}(\Delta r,n\Delta t)$$

(30)

Combining the spatial distribution of the magnetic field from Eq. (18) with Eq. (19) yields the relationship between the external field and position.

Therefore, the fundamental electromagnetic governing equations and boundary condition equations for the superconductor have both been transformed into discrete differential forms. Solving the partial differential governing equations has also been converted into solving a system of discrete equations.

Next, express the levitation force integral Eq. (16) in differential form as well:

$${F_z}(n\Delta t)=\sum\limits_{{m=1}}^{M} {M(m\Delta r,n\Delta t)} \frac{{\partial {B_{{\text{ex}}}}}}{{\partial z}}=\sum\limits_{{m=1}}^{M} {\sum\limits_{{m=1}}^{M} {J(m\Delta r,n\Delta t)} \frac{{\partial {B_{{\text{ex}}}}}}{{\partial z}}}$$

(31)

Thus, the numerical solution method for the reduced-dimensional computation model of pinning magnetic levitation force has been fully explained. However, two issues remain unresolved: (1) The error issue introduced by the FDM, specifically the errors generated by the finite grid density. While increasing grid density can reduce errors, they persist to some extent. (2) The problem of grid partitioning in the r-dimension, where too few partitions can affect computational accuracy, while too many can significantly increase computational workload and reduce speed. These issues will be addressed in the following two sections.

Nonlinear boundary optimization algorithm

The subsequent sections of this paper will rely on the fast computation model of levitation force established in this paper. Given that the computational load of vehicle dynamics itself is substantial, and considering that an actual engineering train will have hundreds of superconducting levitators, the computational load for each step of levitation force calculation will be at least hundreds of times that of a single levitator. Coupled with the extensive time span of dynamic simulations, the total computation time for the entire large-scale model will significantly increase. Therefore, this section establishes a nonlinear boundary condition based on the magnetic flux penetration characteristics during the operation of the superconducting pinning Maglev.

Firstly, the magnetic flux penetration in the superconductor starts from the outer surface in the r dimension. For vibrations, the displacement is not large; it is merely repetitive motion, so the magnetic flux will repeatedly penetrate near the R surface. Secondly, the central part of the superconductor typically does not experience magnetic flux penetration most of the time, unless there is a large-scale motion like unloading after field cooling. However, this model is mainly used for vibration calculations, so it only needs to consider the magnetic flux penetration near the R surface. Therefore, it is only necessary to compute the magnetic flux penetration areas within the superconductor for each time step, while the remaining areas inherit the values from the previous step. This method of variable boundary conditions is also known as the nonlinear boundary method.

The nonlinear boundary optimization algorithm proposed in this section follows these steps:

• 1. Assume the magnetic flux penetration depth for the current time step is 1 ∆r.

• 2. Substitute the boundary condition at the outermost boundary to solve the electromagnetic equation within this penetration depth.

• 3. Compare the magnetic field at the deepest part of the computed region with the magnetic field on the outer side of the computed region.

• 4. If they match, it indicates that the calculation boundary meets the conditions, and the computed result is adopted; if they do not match, it indicates that the calculation depth needs to be increased, and steps (1) to (4) are repeated.

The logic of the nonlinear boundary optimization algorithm in this paper is illustrated in Fig. 4. The figure shows the computation domain at time t (the n^th time step), where h represents the number of grids in the computation domain of the new optimization algorithm (illustratively, in practice, it constantly changes). It can be intuitively seen from the figure that the area of the computation domain is significantly reduced, which will greatly reduce the computation load for this time step. Additionally, the boundary conditions have also changed, from the original single boundary R to two boundaries, R and R’.

Fig. 4

Schematic diagram of the nonlinear boundary optimization algorithm (gray represents the original computation domain, blue represents the computation domain using the new algorithm. This diagram is for illustrative purposes only and does not represent the actual number of grids).

The FDM forms of these two boundaries are:

$$R=M'\Delta r$$

(32)

$$R'=(M' - h)\Delta r$$

(33)

where

$$h=1,2,3, \cdots$$

(34)

At this point, the current distribution obtained in the n-th time step is given by:

$$J(r,n\Delta t)=[{J_{1n}},{\text{ }}{J_{2n}}, \cdots {\text{,}}{J_{{j_{\text{R}}}n}}{\text{]}}$$

(35)

And the current distribution obtained in the (n + 1)-th time step is given by:

$$J(r,(n+1)\Delta t)=[{J_{1n+1}},{\text{ }}{J_{2n+1}}, \cdots {\text{,}}{J_{{j_{\text{R}}}n+1}}{\text{]}}$$

(36)

Combining with Eq. (7) the relationship between the current density distribution at the n-th time step and the (n + 1)-th time step can be determined as:

$$\left\{ \begin{gathered} {J_{n+1}}={J_n}{\text{ }}m \leqslant M' - h \hfill \\ {J_{n+1}}={J_{\text{c}}}{\text{sign(}}\frac{{\partial B}}{{\partial t}}){\text{ }}m>M' - h \hfill \\ \end{gathered} \right.$$

(37)

The magnetic field at the R′ boundary of the superconductor at the (n + 1)-th time step can be obtained from the n-th step, which means the magnetic field boundary condition at the (n + 1)-th step is:

$$\left\{ \begin{gathered} {B_{n+1}}={B_n} \hfill \\ {B_{n+1}}={B_{{\text{ex}}}}(m,(n+1)\Delta t)+{M_{n+1}} \hfill \\ \end{gathered} \right.{\text{ }}\begin{array}{*{20}{c}} {m \leqslant M' - h} \\ {m>M' - h} \end{array}$$

(38)

where, it is only necessary to know:

$$\begin{gathered} M(R,{t_{n+1}})=\int\limits_{0}^{R} {J{\text{d}}r=\sum\limits_{{m=0}}^{{M'}} {J(m\Delta r,{\text{ (}}n+1)\Delta } } t)\Delta r \hfill \\ =\sum\limits_{{m=0}}^{{M' - h - 1}} {J(m\Delta r,{\text{ }}n\Delta } t)\Delta r+\sum\limits_{{m=M' - h}}^{{M'}} {J(m\Delta r,{\text{ (}}n+1)\Delta } t)\Delta r \hfill \\ \end{gathered}$$

(39)

$$M(R',{t_{n+1}})=\int\limits_{0}^{{R'}} {J{\text{d}}r} =\sum\limits_{{n=0}}^{{M' - h - 1}} {J(m\Delta r,{\text{ }}} (n+1)\Delta t)$$

(40)

To further reduce the computational load, let the number of grid points in the calculation domain for the n-th time step be h_n, and for the (n + 1)-th time step be h_n+1. It is only necessary to compute the second term of Eq. (39), i.e.:

$$M(R,{t_{n+1}})=M(R',{t_n})+\sum\limits_{{m=M' - h}}^{{M'}} {J(m\Delta r,{\text{ (}}n+1)\Delta } t)\Delta r$$

(41)

$$M(R',{t_{n+1}})=M(R',{t_n})$$

(42)

At this point, the construction of the nonlinear boundary optimization algorithm model based on the FDM is complete. The nonlinear boundary optimization algorithm, established in this section based on the flux penetration characteristics inside the superconductor, will significantly reduce the computational load in the subsequent dynamic simulation calculations. This makes it feasible to consider flux penetration and nonlinear hysteresis characteristics in vehicle dynamics simulations. The next necessary step is to verify the accuracy of this model and whether there is a significant improvement in computation speed.

Model computation efficiency and validation

This subsection primarily aims to validate the correctness and efficiency of the fast computation model of the nonlinear hysteresis HTS-PMG relation with superconducting pinning Maglev established in this paper. Firstly, the levitation force-displacement curve of the superconductors on the PMG was measured, and a limited dataset was used to fit the undetermined parameters of the external field in the model. Secondly, the calculation values of the new model were used to verify the inflection point characteristics that are difficult to obtain with traditional hysteresis models. Finally, dynamic experiments and FEM vibration simulations were conducted, using experimental data and well-established simulation methods to validate the new model’s correctness and efficiency under transient conditions.

Validation of levitation and guidance forces under Quasi-Static conditions

This subsection uses the SCML-01 measurement device from Southwest Jiaotong University to measure the levitation and guidance forces of four 3-seed YBa₂Cu₃O_{7 − x} high-temperature superconductors. The dimensions of the superconductor and the PMG are shown in Fig. 5. The SCML-01 device can simultaneously measure the levitation and guidance forces during movement in two directions. Since the generation of levitation and guidance forces in superconductors requires physical displacement, and this device enables movement of the superconductor at velocities below 1 mm/s, the measured results are therefore referred to as “quasi-static”¹⁸. The PMG chosen is a Halbach array track with dimensions of 30 + 20 + 30 + 20 + 30 mm, and the superconductors are 64 × 32 × 13 mm³. The superconductor is pre-installed in a cryogenic container filled with liquid nitrogen to immerse the superconductor. After the bottom surface of the superconductor (also the seed surface) is fully cooled at the field-cooling position (30 mm optimal field-cooling height was chosen for this experiment¹⁸, the pressing experiment begins.

Fig. 5

Experimental equipment and materials: (a) levitation and guidance force measurement device; (b) pmg (left) and superconductors (right).

Two sets of data were measured. In the first set, the superconductor was moved 20 mm downward from the field-cooled position, then returned to the field-cooled position, repeating this cycle three times. In the second set, the superconductor was moved 20 mm downward from the field-cooled position to the working height, followed by three cycles of lateral movement with a maximum lateral displacement of 10 mm. Both experiments simultaneously recorded the position information of the superconductor, levitation force, and guidance force. However, in the first experiment, since there was no lateral displacement, the guidance force remained zero and thus does not need to be considered.

After obtaining the relevant experimental data, the vertical and lateral movements in the new model were set to be consistent with the experiments (i.e., the movement process and speed). To ensure consistent motion, the collected displacement time-domain signals were directly used as input signals for the computation model. The undetermined parameters in Eqs. (18) and (19) were obtained through least squares fitting, and then the experimental data were compared with the data calculated using the new model. Various grid number were tested, and free vibration simulations were conducted (with a time step of 1 ms and a duration of 10 s). The variance root σ (using the data calculated with a mesh count of 10000 as the standard.) and simulation time t (recorded using MATLAB’s timing tool tic) were documented, resulting in Table 1. After repeated simulations, it was determined that setting the grid number in the r-dimension to 120 provided the optimal balance between calculation accuracy and efficiency.

Table 1 Error and calculation time.

Figure 6(a) shows the curve of levitation force versus vertical displacement obtained from the first set of experiments and the curve calculated using the new model. The new model fits the experimental data well, accurately capturing the irreversibility of the levitation force and the hysteresis loop formed by cyclic motion as mentioned in the introduction of this paper. Figure 6(b) shows the curve of guidance force versus lateral displacement obtained from the second set of experiments and the curve calculated using the new model. The new model fits the experimental data well, accurately capturing the irreversibility and hysteresis loop of the guidance force. This demonstrates that the model accurately describes the most important dynamic relationships: the levitation force-vertical displacement relationship and the guidance force-lateral displacement relationship. The results of the proposed method are also compared with FEM at FCH = 20 mm and 40 mm, and a similarly high fitting accuracy is observed.

Fig. 6

The measured and calculated curves of levitation force and guidance force (field-cooling height of 30 mm): (a) Levitation force curve for vertical motion; (b) Guidance force curve for lateral motion.

Finally, to verify whether this model can compute the unique and challenging-to-describe ‘inflection point’ characteristic of pinning Maglev, a third set of measurement experiments was conducted in this section. The third set of measurement experiments involved introducing some small-cycle motions into the first set’s typical large-cycle motion, resulting in multiple ‘Inflection Points.’ Fig. 7 presents the measured and calculated curves of levitation force and vertical displacement for the third set of experiments. Since the ‘inflection points’ cause only a slight change in the curve’s trajectory, an enlargement is applied here for better observation. From the figure, it can be observed that the calculated values still closely match the experimental measurements, indicating that the new model is also capable of describing the unique property of pinning Maglev levitation force.

Fig. 7

The enlarged plot of the measured and calculated values of the ‘inflection point’ characteristic of the levitation force.

Verification of the correctness and computational efficiency under free vibration

Due to factors such as air friction and liquid nitrogen oscillation present in small-scale dynamic experiments, using a levitation system composed of several superconductors, as in Sect. 4.1, would result in significant damping. Additionally, the hysteresis effect of pinning Maglev also consumes energy. Therefore, small-scale vibration experiments are difficult to accurately demonstrate the full impact of the levitation force hysteresis effect, as they are mixed with many other damping factors. Consequently, it is challenging to validate the applicability of the new model in dynamic scenarios using experimental measurements in the time domain.

Hence, in this section, we employ FEM simulation, a well-established and reliable method for superconducting pinning Maglev, to validate the dynamic levitation and guidance force of the new model. The FEM model is built using the COMSOL software, employing dynamic meshing. The PMG utilizes a residual flux density magnetization model, along with a model for the superconductor’s E-J characteristics. For specific modeling methods, refer to reference³⁸, and the simulation model is illustrated in Fig. 8.

Fig. 8

FEM simulation model schematic (magnetic flux density).

In order to simulate its free vibration, a constraint equation was added to the superconductor in the FEM simulation. This constraint equation represents the motion differential equation of the levitation system itself:

$$m\ddot {z}+mg={F_z}$$

(43)

which F_z is the levitation force obtained from the simulation multiplied by the length of the block in the longitudinal direction, m is the mass, and g is the gravitational acceleration. The solver for Eq. (43) also employs the forward difference method. This allows for the FEM simulation of free vibration in the vertical direction with a single degree of freedom.

In this simulation, the four superconducting blocks and a field cooling height of 30 mm used in Sect. 4.1 are applied again, with the mass set to 10 kg. The time-domain results of the simulated levitation force are shown in Fig. 9(a). The time-domain signals of free vibration calculated by the FEM and the new model are quite consistent, indicating that the energy loss calculated by both methods is similar, thus verifying the accuracy of the energy loss computation by the new model. After interpolating and unifying the sample number of this set of time-domain signals, a Fourier transform is performed to obtain its frequency-domain signal, as shown in Fig. 9(b). The frequency-domain signal of the new model’s free vibration results is very close to that calculated by the FEM, verifying the accuracy of the new model’s stiffness computation.

Fig. 9

Simulation results of levitation force in free vibration using fem and the new proposed model: (a) time-domain simulation results of levitation force; (b) amplitude-frequency characteristics of levitation force.

From Fig. 9(a), it can also be observed that when the levitation body undergoes free vibration from the field-cooling height, the initial amplitude decays rapidly, but as time progresses, the decay rate decreases. This indicates that the smaller the amplitude, the smaller the AC losses inside the superconductor. Additionally, Fig. 9(b) shows that the main frequency band of the free vibration is relatively broad, indicating that the equivalent stiffness changes as the amplitude decreases. Table 2 shows the grid count and computational time for simulating 10 s of free vibration using both the FEM and the proposed model. Under both methods, the processor load approaches full utilization (AMD R7 3700X); however, a significant difference is observed in memory usage, which varies over time—the values reported in the table represent typical peak usage. It can be observed that the computational time of the proposed model is several orders of magnitude lower than that of the FEM, making it feasible to apply the proposed model to full-train simulation calculations.

Table 2 Comparison table of computational time.

Dynamic experiments and validation

Although subsection 4.2 mentioned that conducting vibration experiments with small models would result in some energy loss due to external interference, and that the hysteresis characteristic of the pinning Maglev levitation force consumes relatively little energy, making it difficult to validate the time-domain signal of the new model’s vibration due to interference potentially overshadowing the hysteresis effect itself, the pinning Maglev system has a relatively large equivalent stiffness despite its small equivalent damping. Therefore, its natural frequency, a key parameter, would not be significantly affected by external factors³⁹. Hence, this subsection conducts an experiment to measure the natural frequency of a small pinning Maglev model.

The vibration experiment used the same four superconductors as in the quasi-static levitation force experiment in subsection 4.1, with the same 30 mm levitation height and PMG. The four pieces were installed at the bottom of an insulated container. The bottom of the insulated container is 3 mm thick, so raising the container by 27 mm achieves a 30 mm field-cooled height. The insulated container used in this experiment is shown in Fig. 10(a), with an accelerometer mounted at the center of the upper surface of the container. To avoid significant pitching motion, the two rows of superconductor pieces were spaced a certain distance apart in the longitudinal direction, as shown in Fig. 10(b).

Fig. 10

Photos of the vibration experiment levitation unit: (a) Insulated container and sensor location; (b) Inside the insulated container and the position of the superconducting blocks.

After field cooling, the cooling plate was removed to allow the insulated container to levitate freely. Subsequently, the levitation system was set into free vibration using the impact method (hitting the center of the insulated container vertically with a rubber hammer). The acquired vibration acceleration time-domain signal was converted into the frequency domain, and the frequency with the maximum amplitude was taken to obtain its natural frequency. In the experiment, it was found that when the total mass of the levitation system was 9.2 kg (including the mass of liquid nitrogen and counterweights), the natural frequency of the levitation system was around 7.9 Hz. Due to the aforementioned external damping, the vibration decay was faster than in the simulation, so only 8 s of time-domain signals were used, and the frequency coordinates after Fourier transform could only be accurate to one decimal place. After repeated tapping and loading/unloading, the natural frequency did not change significantly, indicating that the equivalent vertical levitation stiffness did not change much after being disturbed. To generate more data for verification, four different total masses of the levitating bodies (by adjusting the counterweights installed on the insulated container) were set: 4.2 kg, 9.2 kg, 14.2 kg, and 19.2 kg. Since it is easier to change system parameters in numerical simulations, the new model in this section was used to perform free vibration simulations for masses ranging from 2 kg to 20 kg, increasing the mass by 1 kg each time.

Figure 11 displays the natural frequencies of the single-degree-of-freedom levitation system under different masses. As shown in the figure, the difference between the natural frequencies calculated using the new model and those measured in experiments is small, validating the correctness of the model in dynamic simulations. Moreover, it can be observed that the natural frequency of the levitation system initially decreases with increasing mass. However, as the mass continues to increase (approximately after 8 kg), the natural frequency tends to stabilize, indicating that adding additional weight does not significantly alter its natural frequency. This is because with greater loading, the levitation height decreases, resulting in an increase in the equivalent levitation stiffness. Since the natural frequency equals the square root of the mass divided by the equivalent stiffness, the change is relatively small.

Fig. 11

The natural frequency of the single-degree-of-freedom levitation system under different masses.

Vehicle dynamics simulation of full-scale train and full-quantity levitators

To verify whether the fast computation model of the HTS-PMG relation proposed in this paper can achieve the electromechanical-coupled dynamic simulation of the superconducting pinning Maglev train with full-size trains and full-quantity levitators, a dynamic model of the full-size train is constructed here, and co-simulation is performed with the fast computation model proposed in this paper.

Full-size superconducting pinning Maglev train dynamic model

Based on mature and reliable multibody dynamics theory, there are currently several commercial multibody dynamics simulation software. Therefore, the vehicle dynamics model section of this paper was completed using UM (Universal Mechanism) multibody dynamics simulation software. At the same time, input-output interfaces were established in the model and connected to the HTS-PMG relation fast computation model proposed in this paper (in the MATLAB environment).

Based on the design of the superconducting pinning Maglev train at Southwest Jiaotong University (Fig. 4), a vehicle dynamics model was constructed. Figure 12 illustrates the schematic diagram of the pinning Maglev vehicle dynamics model. For specific modeling details of the vehicle dynamics model, please refer to the literature¹⁷. Such forces act on the levitators and the PMG, which can be customized to include curves and track irregularities. Levitation forces act in the vertical direction between the levitators and the PMG, while guidance forces act in the lateral direction. The train consists of 3 cars, 18 levitation frames, and 216 superconducting levitators, with a Maglev force element interface set at the bottom of each levitator. Each force element interface is connected to a HTS-PMG relation fast computation model. The entire train dynamics model consists of 21 rigid bodies, each with 6 degrees of freedom, totaling 126 degrees of freedom. Thus, the entire joint simulation model includes 1 mechanical dynamics module of vehicle and 216 electromagnetic modules of levitator.

Fig. 12

Superconducting pinning Maglev train dynamics simulation model.

The simulated track irregularity data is based on measurements conducted on the experimental line at Southwest Jiaotong University as described in⁴⁰. The simulated field cooling height is set to 40 mm, resulting in an initial levitation height (the height at which the train statically levitates after the field cooling and mechanical support release) of approximately 15 mm (obtained from the simulation). The simulation speed is set to 600 km/h, which is also the target speed for the prototype vehicle at Southwest Jiaotong University.

Vehicle dynamics simulation results

Due to the numerous superconducting levitators on the pinning Maglev train, they have been assigned specific numbers. The first digit represents the carriage number (1, 2, 3 from front to back), the second digit represents the levitation frame number for each carriage (1 to 6 from front to back), and the third digit represents the levitator number on each frame (from front to back, left side: 1–6; right side: 7–12). The vertical gap and lateral offset of the levitators are critical indicators of operational safety as they directly reflect the risk of track impact and derailment. Given the large number of levitators, observations were focused on the vertical gap and lateral offset signals of three specific units: 1-1-1, 2–3-6, and 3–6-12, corresponding to positions near the front, middle, and rear of the train, respectively. Figure 13 illustrates the vibration characteristics of their levitation gap and lateral offset. From Fig. 13(a), it can be observed that the levitation gap decreases significantly after the vehicle is in motion and eventually stabilizes. Track irregularity is introduced starting from 15 s, with removal at 30 s. After the vibration ceases, the levitation gap is observed to decrease from 15 mm to 10–11 mm, indicating a noticeable vertical drift during operation at the ultra-high speed of 600 km/h. In Fig. 13(b), some lateral drift of the levitators is also observed, but the magnitude is negligible.

Fig. 13

Temporal domain plots of selected levitation units: (a) levitation unit vertical gap; (b) levitation unit lateral offset.

The vibration characteristics of three carriages during the train operation are depicted in Fig. 14. It can be observed that the vibration responses of the three carriages are similar, with significantly reduced amplitudes compared to the levitators. This indicates the excellent performance of the air springs in isolating vibrations caused by irregularities in the PMG. Additionally, the downward drift in the vertical stable positions of the carriages matches that of the levitators, both decreasing by approximately 5 mm. However, there is a discrepancy in the lateral stable positions of the carriages compared to the levitators, showing an offset of about 0.3 mm. By observing the stable levitation gaps of all levitation units, it is evident that the vertical stable levitation gaps on the left and right sides of the train are not consistent. This results in an overall rotation around the x-axis of the train, leading to lateral displacement of the carriages. Nevertheless, the lateral displacement of the carriages remains within a very small range.

Fig. 14

The time-domain plots of the cars’ vibrations are shown in: (a) for vertical vibrations and (b) for lateral vibrations.

Table 3 shows the Sperling index during the train’s operation, which reflects the comfort level experienced by passengers¹⁹. Due to the dual isolation provided by levitation forces and air springs, most high-frequency vibrations are not transmitted to the train compartments, resulting in a relatively high level of passenger comfort. Specifically, the front car exhibits the lowest comfort level, while the middle car experiences the least discomfort.

Table 3 Sperling index of the train.

This simulation involved the mechanical vibration of 21 rigid bodies of a three-car train and the magnetic flux variation inside the 216 levitators, making it the most extensive, realistic, and comprehensive electromechanical coupled simulation of a full-size superconducting pinning Maglev train to date. The computational efficiency of this simulation is comparable to that of simulating a single levitator using FEM. With a time step set to 1 ms and 120 reduced-dimensional grids for the superconductors within each levitator, simulating 40 s of operation on a civilian-grade 8-core CPU (AMD R7 3700X) took only about 6 h. In contrast, simulating the vibration of just one levitator for the same duration using finite element methods would require approximately 2 h and consume more memory.

Limitation analysis

The proposed fast computation model is intended for application in the simulation of full-scale superconducting pinning maglev trains, where hundreds of HTS–PMG interactions must be efficiently computed within a complex system. Therefore, substantial model simplification and computational resource optimization constitute the primary objectives of this study. The simulation results presented in this section demonstrate that the proposed fast computation model successfully achieves these objectives.

However, to attain this efficiency, the model adopts a critical-state E–J constitutive relation that neglects relaxation effects arising from current density redistribution, and the governing equations do not account for thermal effects induced by AC losses within the superconductor. Additionally, the modeled motion is limited to small-stroke vibrations. As a result, the proposed fast computation model is primarily suited for vibration simulations of large-scale systems—such as superconducting pinning maglev trains—and is not applicable to large-displacement motions in small systems or to high-fidelity simulations that emphasize internal HTS bulk behavior.

Conclusion

The main focus of this study is to develop a HTS-PMG relation model that can serve vehicle dynamics computations. Utilizing the magnetic flux penetration characteristics of superconductors, a reduced-dimensional HTS-PMG relation model based on the critical current density (J_c) of the superconductor is proposed in this paper. Additionally, a nonlinear boundary optimization algorithm based on the magnetic flux penetration characteristics during superconductor vibration is introduced, forming a fast computation model for the HTS-PMG relation. The correctness of the proposed model is verified through quasi-static and dynamic experiments, and its computational efficiency is compared with traditional FEM. Finally, based on the proposed fast computation model for the HTS-PMG relation, dynamic simulations of a full-size superconducting pinning Maglev train are conducted, with synchronous simulation of all 216 levitators. The main research conclusions are as follows:

• 1. The computational results of the proposed fast computation model are in good agreement with quasi-static and dynamic experimental results, accurately simulating various magnetic hysteresis characteristics of pinning Maglev.

• 2. The proposed fast computation model significantly reduces computation time while maintaining the same accuracy as conventional FEM simulations.

• 3. The proposed fast computation model can simulate not only the vibration response of superconducting pinning Maglev systems but also the phenomenon of suspension drift during vibration.

• 4. The proposed fast computation model enables dynamic and magnetic flux penetration simulations of full-size trains with all levitators.

• 5. The proposed fast computation model is able to simulate the vibration levitation drift of a full-size train during operation, which is difficult to achieve using traditional finite element methods.

• 6. The proposed fast computation model is suitable for dynamic simulation of large and complex systems such as maglev trains, but not suitable for detailed simulation analysis of a small number of HTS bulks.

In summary, this study provides an efficient tool for researching the vibration and levitation performance evolution of superconducting pinning Maglev trains. The dynamic simulation method for train dynamics and levitation flux proposed in this paper offers a more realistic and efficient means for studying the dynamic performance and levitation drift of superconducting pinning Maglev trains.