Transformer winding vibration analysis method based on fine integration method

Abstract

The invention relates to a transformer winding vibration analysis method based on a fine integration method, and belongs to the field of power equipment. Aiming at the problem that an existing method is difficult to give consideration to high-frequency transient response capture precision and calculation efficiency, a multi-degree-of-freedom vibration equation is constructed based on an equivalent mass-spring-damping model, after the multi-degree-of-freedom vibration equation is converted into a state-space equation, recursive decomposition is performed on an index matrix through a fine integration method, and linear processing of electromagnetic force excitation is combined, so that the high-frequency transient response capture precision and the calculation efficiency are obtained. And high-efficiency and high-precision solving of the axial vibration of the winding is realized. According to the method, the characterization capability of high-frequency nonlinear vibration characteristics is effectively improved, numerical oscillation and accumulative errors are suppressed, and reliable technical support is provided for state monitoring and fault early warning of the transformer.

Description

Technical Field

[0001] The invention belongs to the field of electric power equipment and relates to a transformer winding vibration analysis method based on a precise integration method. Background Art

[0002] With the rapid development of ultra-high voltage transmission and renewable energy grid-connected technologies, power transformers face a more complex electromagnetic and mechanical coupling excitation environment during operation. Under fault conditions, high-frequency transient disturbances can induce high-frequency vibrations in the windings. Traditional vibration models, ignoring the time-varying nature of electromagnetic forces and the nonlinearity of structural damping, struggle to accurately capture microsecond-level transient responses. Furthermore, online fault monitoring based on vibration signals places higher demands on simulation accuracy and real-time performance. Existing methods generally struggle to balance computational accuracy and efficiency.

[0003] Existing research primarily focuses on modeling, experimental verification, and numerical simulation techniques for transformer winding vibration. For example, the single-degree-of-freedom equivalent model for axial vibration proposed by Wang Hongfang et al., while capable of qualitatively analyzing the effect of preload on vibration characteristics, cannot characterize multi-degree-of-freedom coupling and local nonlinear effects. The three-dimensional finite element model established by Wang Fenghua's team reveals the influence of preload on natural frequency by introducing nonlinear material constitutive relations, but the calculations are too time-consuming to meet real-time monitoring requirements. The electromagnetic and mechanical coupling field model proposed by Federico Barrero achieves real-time interactive modeling of electromagnetic force and vibration, but is prone to numerical oscillation and truncation errors under high-frequency excitation. In experimental research, some researchers have verified the linear relationship between vibration amplitude and the square of voltage and current, and developed a fault diagnosis method based on the fundamental frequency component, but its sensitivity to high-order harmonics and nonlinear vibration characteristics is insufficient. Other researchers have inverted the preload state based on the vibration transfer function, but this method relies on linear assumptions and has limited adaptability to actual operating conditions. In numerical simulation, the implicit difference method has low computational efficiency, and the explicit method is fast but has large high-frequency response errors. In addition, existing technologies often use linear approximations when dealing with nonlinear and multi-field coupling problems, resulting in significant cumulative errors. The differences in the time scales of the physical fields further aggravate numerical instability, seriously restricting the realization of online high-precision simulation.

[0004] In summary, existing technologies face significant bottlenecks in multi-degree-of-freedom coupling and nonlinear characterization, capturing high-frequency transient responses, and synergistically improving real-time monitoring accuracy and efficiency. The precise integration method, with its precise exponential matrix decomposition and the structural fidelity of the symplectic algorithm, provides a new technical approach to addressing these issues, possessing significant theoretical value and promising engineering applications. Summary of the Invention

[0005] In view of this, an object of the present invention is to provide a transformer winding vibration analysis method based on a precise integration method.

[0006] In order to achieve the above object, the present invention provides the following technical solutions:

[0007] A transformer winding vibration analysis method based on Precise Integration Method (PIM) includes the following steps:

[0008] Step 1: Establish a mathematical model of transformer winding axial vibration: Based on the equivalent mass-spring-damper model, establish the second-order differential equation of winding axial vibration, which is expressed as:

[0009]

[0010] Where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, x(t) is the displacement vector, and F(t) is the electromagnetic force vector;

[0011] Step 2: Construct the state-space equation: Convert the second-order differential equation into a first-order state-space equation:

[0012]

[0013] Among them, the state vector The system matrix H is:

[0014]

[0015] Excitation vector f(t)=[0,M -1 F(t)] T ;

[0016] Step 3: Apply the precise integration method to solve: divide the time step τ into m=2 N subintervals, calculate the exponential matrix T = e of the system matrix H Ht =(e HΔt ) m , through the recursive formula Solve for the state vector;

[0017] Step 4: Extract vibration response characteristics: Extract displacement, velocity, and acceleration parameters from the state vector z(t) to analyze the winding vibration characteristics.

[0018] Furthermore, in step 1, the mass matrix M is composed of the mass of the coil, the mass of the pressure plate and the mass of the support plate, and the stiffness matrix K includes the linear stiffness coefficient a and the nonlinear hardening coefficient b of the pad, wherein the pad stress-strain relationship satisfies σ=aε+bε 3 .

[0019] Furthermore, the exponential matrix T in step 3 is Hτ Calculated by Taylor series expansion, the truncation order is 4, and the expansion is:

[0020] e H t≈I+Ht+(Ht) 2 / 2+(Ht) 3 / 6+(Ht) 4 / twenty four

[0021] Where I is the identity matrix.

[0022] Furthermore, in step 3, the electromagnetic force F(t) is linearly approximated within the subinterval:

[0023]

[0024] The corresponding integral term analytical expression is:

[0025]

[0026] Furthermore, the value range of the time step τ in step 3 is 0.001s≤τ≤0.01s, and the sub-interval division parameter N≥10.

[0027] Furthermore, the damping coefficient of the damping matrix C in step 1 is set to 0.01% of the parallel spring stiffness.

[0028] Furthermore, the vibration characteristic analysis in step 4 includes time domain waveform comparison, frequency response spectrum analysis, and calculation error comparison with the Newmark-β method and the Runge-Kutta method.

[0029] A transformer winding vibration analysis system, used to implement the method, comprising:

[0030] a model building module configured to establish an equivalent mass-spring-damper model according to winding structure parameters and generate a state space equation;

[0031] an electromagnetic force calculation module, connected to the model building module, configured to obtain electromagnetic force distribution data of the winding coil through finite element analysis and input the electromagnetic force vector F(t) into the model building module;

[0032] a fine integration solver connected to the model building module, configured to receive the state space equation and the system matrix H, and perform matrix exponential calculation and state vector recursive solution;

[0033] a feature extraction module, connected to the precise integration solver, and configured to extract displacement, velocity, and acceleration parameters from the solution results;

[0034] Among them, the output end of the electromagnetic force calculation module is connected to the excitation term input end of the model construction module, the output end of the model construction module is connected to the input end of the fine integration solver, and the output end of the fine integration solver is connected to the input end of the feature extraction module.

[0035] Furthermore, the electromagnetic force calculation module generates the electromagnetic force vector F(t) in the following manner:

[0036] A two-dimensional axisymmetric finite element model of the core-winding system was constructed. The leakage magnetic field distribution was solved based on the time-varying load current, and the axial electromagnetic force on each coil was calculated using the Lorentz force formula.

[0037] The time-varying load current is provided by a Matlab / Simulink simulation circuit, and the electromagnetic force calculation module inputs F(t) into the excitation item of the model construction module through a data interface.

[0038] Furthermore, the refined integral solver includes a parallel computing unit configured to split the recursive calculation task of the exponential matrix T into multi-threaded tasks and assign them to the multi-core processor for execution;

[0039] The parallel computing unit is further configured to: set the time step τ to m=2 N The subintervals are divided into multiple calculation batches, each batch independently calculates the subinterval matrix index and synchronizes the intermediate calculation results through the cache.

[0040] The beneficial effects of the present invention are:

[0041] (1) Through the fine decomposition of the exponential matrix and the recursive solution strategy, the contradiction between the calculation accuracy and efficiency of the traditional numerical method is broken, the cumulative error of the high-frequency transient response solution is significantly reduced, and the calculation time is greatly reduced to meet the real-time requirements of online monitoring.

[0042] (2) Based on the multi-degree-of-freedom modeling of the equivalent mass-spring-damper model, the spatial coupling effect of the winding structure and the nonlinear stiffness characteristics of the pad are fully restored. Combined with the refined loading of the electromagnetic force time-varying excitation, the microsecond-level transient vibration details are effectively captured, and the fault feature identification capability is improved.

[0043] (3) The inherent fidelity of the symplectic algorithm structure of the fine integration method is utilized to avoid the numerical oscillation problem of the traditional explicit method under high-frequency excitation, ensure energy conservation and phase accuracy in long-term simulation, and provide a reliable data basis for winding state evaluation.

[0044] (4) Through the collaborative solution of electromagnetic-mechanical coupling fields and the intelligent extraction of vibration response characteristics, it can adapt to the vibration analysis needs of transformers with different voltage levels and winding structures, and provide universal technical support for design optimization, fault warning and health management.

[0045] Other advantages, objects, and features of the present invention will be described in part in the following description and, in part, will be apparent to those skilled in the art upon examination of the following description or may be learned from practice of the present invention. The objects and other advantages of the present invention may be realized and obtained through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be described in detail below with reference to the accompanying drawings, in which:

[0047] Figure 1 Schematic diagram of the solution process of the winding axial vibration precise integration method

[0048] Figure 2 Simulink circuit schematic diagram for power transformer

[0049] Figure 3 The current-time curves of the high-voltage and low-voltage side windings are shown in Figure 2. (a) is the current waveform of the high-voltage side winding; (b) is the current waveform of the low-voltage side winding.

[0050] Figure 4 The curve of the axial electromotive force on the high-voltage side conductor changing with time;

[0051] Figure 5 is the axial displacement-time curve of each mass block;

[0052] Figure 6 is the axial velocity-time curve of each mass block;

[0053] Figure 7 Comparison of displacement solutions between PIM and ode15s;

[0054] Figure 8 Comparison of the speed solution results of PIM and ode15s. DETAILED DESCRIPTION

[0055] The following describes the embodiments of the present invention by means of specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments are only schematic illustrations of the basic concept of the present invention, and the following embodiments and features in the embodiments can be combined with each other without conflict.

[0056] Among them, the accompanying drawings are only for illustrative purposes and represent only schematic diagrams rather than actual pictures, and should not be understood as limiting the present invention. In order to better illustrate the embodiments of the present invention, some parts of the accompanying drawings may be omitted, enlarged or reduced, and do not represent the dimensions of actual products. For those skilled in the art, it is understandable that some well-known structures and their descriptions may be omitted in the accompanying drawings.

[0057] The same or similar numbers in the drawings of the embodiments of the present invention correspond to the same or similar parts; in the description of the present invention, it should be understood that if there are terms such as "upper", "lower", "left", "right", "front", "back", etc. indicating directions or positional relationships, they are based on the directions or positional relationships shown in the drawings. They are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific direction, be constructed and operate in a specific direction. Therefore, the terms describing the positional relationship in the drawings are only used for illustrative purposes and cannot be understood as limiting the present invention. For ordinary technicians in this field, the specific meanings of the above terms can be understood according to specific circumstances.

[0058] The present invention aims to propose a precise integration method for transformer winding axial vibration analysis, which is characterized by the following steps:

[0059] 1) Establishing a mathematical model of axial vibration

[0060] Based on the actual structural parameters of the transformer winding, a mathematical model of the winding's axial vibration is established using the equivalent mass-spring-damper model. Under non-short-circuit conditions, this model considers the tangent modulus of the pad under the preload force to be a constant value and describes the winding's axial vibration behavior under electromagnetic excitation. The equation of motion is:

[0061]

[0062] Among them, M, C, and K are mass, damping, and stiffness matrices, respectively; x(t) is the displacement vector; and F(t) is the electromagnetic force vector.

[0063] 2) Constructing state space equations

[0064] The above second-order differential equation is converted into a first-order state space form to facilitate numerical solution:

[0065]

[0066] in, is the state vector, and H is the system matrix:

[0067]

[0068] f(t)=[0,M -1 F(t)]T is the excitation vector.

[0069] 3) Apply the precise integration method to solve

[0070] The precise integration method is used to perform high-precision numerical solutions to the state space equations. The specific steps are as follows:

[0071] a. Calculate the matrix exponential

[0072] Divide the time step τ into m=2 N subintervals, each subinterval Δt = τ / m. The matrix index is calculated by the recursive formula:

[0073] T=e Ht =(e HΔt ) m

[0074] Subinterval matrix index e HΔt Taylor series expansion is used for efficient calculation to ensure numerical stability.

[0075] b. Recursively solve the state vector

[0076] The state vector for the next time step is calculated using the following recursive formula:

[0077]

[0078] For linear time-invariant systems, f(s) is approximately a constant or linear function within the subinterval, simplifying the integral calculation. The integral uses a high-order quadrature method to ensure accuracy.

[0079] 4) Extract vibration response features

[0080] The vibration response characteristic parameters of the winding, such as displacement, velocity and acceleration, are extracted from the solved state vector, and its frequency response characteristics and time domain behavior are analyzed to provide a basis for winding state assessment and fault diagnosis.

[0081] 5) Verification and application

[0082] By comparing this method with traditional numerical integration methods (such as the Newmark-β method and the Runge-Kutta method), the advantages of this method in terms of computational accuracy and efficiency are verified. This method is then applied to the vibration analysis of actual transformer windings to evaluate its feasibility and effectiveness in engineering practice.

[0083] In summary, the present invention provides a high-precision and high-efficiency method for analyzing the axial vibration of transformer windings. The method is applicable to linear steady-state systems under non-short-circuit conditions, can effectively capture the dynamic response characteristics of the windings, and provide technical support for the design optimization and operation and maintenance of transformers.

[0084] In this example, a three-phase, double-winding, oil-immersed power transformer, SFZ-40000 / 110, with a rated capacity of 40MVA and a rated voltage of 110 / 10.5kV, using copper conductors for the windings and high-density fiberboard for the insulation spacers, is analyzed for winding axial vibration under high-voltage power supply and low-voltage load operation. This process involves the use of a precise integration method to improve the accuracy of the analysis. The specific implementation process is as follows: Figure 1 shown.

[0085] First, the winding is regarded as a multi-degree-of-freedom system consisting of n coils, a top pressure plate, and a bottom support plate, and a multi-degree-of-freedom mass-spring-damper system physical model is constructed.

[0086] The mechanical properties of the pad are nonlinear, and its stress-strain relationship is described in polynomial form:

[0087] σ=aε+bε 3

[0088] Among them, the linear stiffness coefficient a = 16.2 MPa, and the nonlinear hardening coefficient b = 584 MPa.

[0089] The spacer has an initial strain, introduced by pre-compression during the manufacturing process. Referring to transformer design specifications, the spacer prestress at normal operating point should be 3 MPa. Taking a 110 kV transformer winding as an example, the stress change caused by electromagnetic force is typically in the kPa range. When the compressive force is sufficiently high, the electromagnetic force has almost no effect on the compressive force distribution. At this point, the spacer's Young's modulus is approximately 4.1904e7 Pa.

[0090] During the winding's forced vibration, the pad releases oil under pressure, or absorbs oil during rebound. This is the primary factor contributing to oil damping in dynamic vibration. To simulate the system's energy dissipation, viscous damping is introduced, with the damping coefficient set to 0.01% of the parallel spring stiffness.

[0091] The clamping stiffness between the top platen and the bottom support plate and the supporting stiffness from the bottom support plate to the ground are much higher than the stiffness of the pads and are defined as 100 times their stiffness under prestrain conditions to characterize the stiffness effect of the core, clamps, and supporting structure.

[0092] Because high-voltage and low-voltage windings share similarities in physical structure and operating principles, the analysis process and methods are identical. To simplify the analysis, the high-voltage winding is used as an example to effectively verify the universality and applicability of the calculation method while avoiding duplication of work. The subsequent analysis will focus on the characteristics of the high-voltage winding, whose basic parameters are shown in Table 1 below. However, the conclusions and methods obtained are also applicable to the low-voltage winding.

[0093] Table 1 Basic parameters of SFZ-40000 / 110 transformer high voltage winding

[0094] parameter Numerical Geometric height (mm) 1220 Total number of segments 74 Coil radial size (mm) 90 Coil inner radius (mm) 443 Number of gears / pad width (mm) 20 / 50 Coil preload force (kN) 270 Preload pressure (MPa) 3 Mass of conductor per column (kg) 1472 Platen mass (kg) 121.353 Pallet mass (kg) 145.624

[0095] In order to accurately obtain the electromagnetic force under load operation, a two-dimensional axisymmetric model of the core-winding is first established with the coil as the basic unit. The time-varying load current is obtained by solving the simulation circuit built on the Matlab / Simulink platform, such as Figure 2 , as the excitation source applied to the high and low voltage windings, the curve of its amplitude changing with time is as follows Figure 3 The leakage magnetic field distribution is solved by the finite element method, and then the axial electromagnetic force on each coil on the high-voltage side is calculated and plotted on Figure 4 .

[0096] Based on the electromagnetic force data calculated through the finite element analysis above, a multi-degree-of-freedom mass-spring-damper system was constructed for dynamic analysis. The electromagnetic force, acting as an external force, acts on the system together with the preload, gravity, pad spring force, and damping force.

[0097] Set the time step τ = 0.001s, divided into m = 2 20 = 1048576 subintervals, and the subinterval step size Δt = τ / m is a very small time segment. The Taylor expansion truncation order is selected as 4, and the matrix index can be calculated by the following formula:

[0098]

[0099] The electromagnetic force F(t) is approximately a linear function in the subinterval and can be expressed as:

[0100]

[0101] Then the analytical format of the second term on the right side of the recursive formula (Duhamel integral term) is:

[0102]

[0103] The initial displacement and velocity are set to zero, and the displacement and velocity of the high-voltage wire disc are extracted from the solved state variables and the waveforms are drawn, as shown in the figure. Figure 5 、 Figure 6 shown.

[0104] To verify the accuracy and efficiency of this method, we compared it with the ode15s solver method in Matlab software. The calculation conditions were set to be the same. Taking the first coil as an example, we compared the calculation results of PIM and ode15s solver, and plotted the displacement and velocity time history responses on Figure 7 、 Figure 8As can be seen from the comparison chart, the time history curves of the two methods are highly consistent during the main response phase. Furthermore, the physical quantities calculated for all masses at all time points using the two ordinary differential equation solving methods show the following maximum deviations: 2.54e-4mm for displacement calculations and 2.72e-4m / s for velocity calculations.

[0105] The computational time of the PIM method and the ode15s solver were measured and compared on a 12th Gen Intel(R) Core(TM) i5-12490F processor. The results showed that the PIM method completed the calculation in just 2.65 seconds, while the ode15s solver took 5.29 seconds. This comparison, based on the same hardware platform, fully demonstrates the PIM method's superior computational efficiency and effectively validates the effectiveness of the proposed method in a real-world environment.

[0106] The above results show that the application of PIM precise integration method in the analysis of transformer winding axial vibration meets the actual engineering requirements and its calculation speed is faster than the traditional time difference method.

[0107] In summary, by conducting comparative experiments with the ode15s solver in Matlab software, key data such as displacement and velocity time-history response are visually compared and quantitatively analyzed under the premise of ensuring consistency of calculation conditions (such as model parameters, initial conditions, error tolerance, and hardware operating platform, etc.), and this embodiment provides an efficient and accurate method to solve the axial vibration of the transformer winding under non-fault conditions. This method can not only provide accurate vibration data support for the design optimization of the transformer, help improve the stability and reliability of the transformer structure, but also quickly and accurately obtain vibration response characteristics in the transformer operation status monitoring, so as to timely detect potential vibration anomalies, provide a strong basis for preventive maintenance and fault warning of the equipment, and effectively ensure the safe and stable operation of the transformer.

[0108] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not limiting. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solutions, which should all be included in the scope of the claims of the present invention.

Claims

1. A transformer winding vibration analysis method based on the Precise Integration Method (PIM), characterized by: The following steps are involved: Step 1: Establish a mathematical model of transformer winding axial vibration: Based on the equivalent mass-spring-damper model, establish the second-order differential equation of winding axial vibration, which is expressed as: Where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, x(t) is the displacement vector, and F(t) is the electromagnetic force vector; Step 2: Construct the state-space equation: Convert the second-order differential equation into a first-order state-space equation: Among them, the state vector The system matrix H is: Excitation vector f(t)=[0,M -1 F(t)] T ; Step 3: Apply the precise integration method to solve: divide the time step τ into m=2 N subintervals, calculate the exponential matrix T = e of the system matrix H Ht =(e HΔt ) m , through the recursive formula Solve for the state vector; Step 4: Extract vibration response characteristics: Extract displacement, velocity, and acceleration parameters from the state vector z(t) to analyze the winding vibration characteristics.

2. The transformer winding vibration analysis method based on the precise integration method according to claim 1 is characterized in that: In step 1, the mass matrix M is composed of the mass of the coil, the mass of the pressure plate and the mass of the support plate, and the stiffness matrix K includes the linear stiffness coefficient a and the nonlinear hardening coefficient b of the pad, where the pad stress-strain relationship satisfies σ=aε+bε 3 .

3. The transformer winding vibration analysis method based on the precise integration method according to claim 1 is characterized in that: In step 3, the exponential matrix T=e Hτ Calculated by Taylor series expansion, the truncation order is 4, and the expansion is: to Ht ≈I+Ht+(Ht) 2 / 2+(Ht) 3 / 6+(Ht) 4 / 24 Where I is the identity matrix.

4. The transformer winding vibration analysis method based on the precise integration method according to claim 1 is characterized in that: In step 3, the electromagnetic force F(t) is linearly approximated within the subinterval: The corresponding integral term analytical expression is:

5. The transformer winding vibration analysis method based on the precise integration method according to claim 1 is characterized in that: The value range of the time step τ in step 3 is 0.001s≤τ≤0.01s, and the sub-interval division parameter N≥10.

6. The transformer winding vibration analysis method based on the precise integration method according to claim 1 is characterized in that: The damping coefficient of the damping matrix C in step 1 is set to 0.01% of the parallel spring stiffness.

7. The transformer winding vibration analysis method based on the precise integration method according to claim 1 is characterized in that: The vibration characteristic analysis in step 4 includes time domain waveform comparison, frequency response spectrum analysis, and calculation error comparison with the Newmark-β method and the Runge-Kutta method.

8. A transformer winding vibration analysis system, used to implement the method according to any one of claims 1 to 7, characterized in that: include: a model building module configured to establish an equivalent mass-spring-damper model according to winding structure parameters and generate a state space equation; an electromagnetic force calculation module, connected to the model building module, configured to obtain electromagnetic force distribution data of the winding coil through finite element analysis and input the electromagnetic force vector F(t) into the model building module; a fine integration solver connected to the model building module, configured to receive the state space equation and the system matrix H, and perform matrix exponential calculation and state vector recursive solution; a feature extraction module, connected to the precise integration solver, and configured to extract displacement, velocity, and acceleration parameters from the solution results; Among them, the output end of the electromagnetic force calculation module is connected to the excitation term input end of the model construction module, the output end of the model construction module is connected to the input end of the fine integration solver, and the output end of the fine integration solver is connected to the input end of the feature extraction module.

9. The transformer winding vibration analysis system according to claim 8, characterized in that: The electromagnetic force calculation module generates the electromagnetic force vector F(t) in the following manner: A two-dimensional axisymmetric finite element model of the core-winding system was constructed. The leakage magnetic field distribution was solved based on the time-varying load current, and the axial electromagnetic force on each coil was calculated using the Lorentz force formula. The time-varying load current is provided by a Matlab / Simulink simulation circuit, and the electromagnetic force calculation module inputs F(t) into the excitation item of the model construction module through a data interface.

10. The transformer winding vibration analysis system according to claim 8, characterized in that: The precise integral solver includes a parallel computing unit configured to split the recursive calculation task of the exponential matrix T into multi-threaded tasks and distribute them to the multi-core processor for execution; The parallel computing unit is further configured to: set the time step τ to m=2 N The subintervals are divided into multiple calculation batches, each batch independently calculates the subinterval matrix index and synchronizes the intermediate calculation results through the cache.

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