Perturbation multi-symplectic numerical solution containing disturbance system based on Hamiltonian theory

Abstract

The invention discloses a Hamiltonian theory-based perturbation multi-symplectic numerical solution containing a disturbance system, and belongs to the technical field of numerical calculation of a nonlinear dynamic system. The method comprises the following steps: firstly, establishing a mathematical model containing a disturbance mechanical structure, converting the mathematical model into a Dosin Hamiltonian system, introducing small parameters to unfold the system into a power series form, and decomposing an original system into a series of linear Dosin Hamiltonian equations through a perturbation theory; and carrying out numerical solution on the perturbation equation of each order by adopting a Preissmann Box Dosin difference format, and finally synthesizing a dynamic response solution of the system. According to the method, the influence of small disturbance on long-term dynamic behaviors is effectively captured while the sympathetic structure of the system is maintained, the method has the characteristics of high precision and strong structure maintenance, the method is suitable for dynamic response analysis of disturbance-containing mechanical systems and generalized nonlinear systems, and an efficient and reliable numerical tool is provided for design and optimization of engineering structures such as aircrafts.

Description

Technical Field

[0001] This invention belongs to the field of numerical calculation technology of nonlinear dynamic systems, specifically relating to a perturbation multisymplectic numerical solution method for perturbation-containing systems based on Hamiltonian theory. Background Technology

[0002] With the increasing demands for precision in the analysis of system dynamic characteristics in fields such as aerospace and mechanical engineering, high-fidelity dynamic simulation has become a core component of aircraft design and strength verification. Under ideal conditions, systems can be simulated with high precision using classical numerical methods (such as the Runge-Kutta method). However, small disturbances commonly found in engineering practice—such as continuous aerodynamic micro-excitations from atmospheric turbulence, slow time-varying stiffness parameters due to material fatigue or loose connections, and minor mass and geometric deviations introduced by manufacturing tolerances—transform ideal conservative systems into nonlinear systems with disturbances. Although these small disturbances have small amplitudes, they accumulate and amplify continuously through the nonlinear effects of the system, significantly altering its long-term dynamic response and even inducing critical instability phenomena such as flutter and limit cycle oscillations, posing a serious threat to structural safety.

[0003] Currently, numerical solutions for such perturbation-containing nonlinear systems face two major challenges. First, while conventional numerical algorithms can directly handle nonlinear terms, they struggle to maintain the system's intrinsic physical structure (such as symplectic structures) in long-term simulations, causing the numerical solution to gradually deviate from the true physical trajectory and reducing the reliability of long-term dynamic response predictions. Second, although classical perturbation methods can analytically analyze the effects of small perturbations through asymptotic expansion, their derivation process is complex, and if the resulting perturbation equations are solved using conventional numerical schemes that lack structure preservation properties, they will also face the problems of numerical error accumulation and structural distortion, failing to guarantee that the long-term dynamic characteristics revealed by perturbation theory are faithfully reflected in the numerical solution.

[0004] To ensure structural reliability, the aforementioned uncertainties must be considered in the dynamic analysis. However, traditional Monte Carlo-type probabilistic methods are computationally expensive when dealing with high-dimensional nonlinear systems due to the need for massive samples, making them difficult to apply directly in engineering design. On the other hand, existing perturbation methods often face convergence challenges and complexity bottlenecks when dealing with strongly nonlinear or non-smooth problems.

[0005] Therefore, seeking a balance between computational resources and engineering accuracy, and developing a numerical method that can efficiently and accurately solve nonlinear systems with disturbances while preserving their intrinsic physical structure, has become an urgent technical need in the field of aircraft dynamics design and analysis. Summary of the Invention

[0006] To address the aforementioned technical problems, this invention provides a perturbation-based multisymplectic numerical solution method for perturbed systems based on Hamiltonian theory. This method is used to solve the dynamic response of nonlinear systems with small perturbations. It utilizes the Hamiltonian system to obtain a Hamiltonian mathematical model of the mechanical structure, fully considering various small perturbation parameters in the structure, and establishes a perturbation expansion model of the perturbed multisymplectic Hamiltonian system. Based on the conventional Preissmann Box multisymplectic difference scheme, after obtaining the expanded multisymplectic Hamiltonian equations using perturbation theory, multiple Hamiltonian equations are numerically solved, ultimately yielding the dynamic response of the perturbed nonlinear system. The obtained results balance the accuracy of theoretical analysis with the structure-preserving characteristics of symplectic algorithms, better reflecting real-world conditions and demonstrating stronger engineering applicability.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] A perturbational multisymplectic numerical solution method for perturbational systems based on Hamiltonian theory, the method comprising:

[0009] Step 1: Establish a simplified mathematical model of the mechanical system or mechanical component, define system parameters, state variables and disturbance terms, and determine the initial conditions and boundary conditions;

[0010] Step 2: Import the simplified mathematical model into the Hamiltonian system, and use the multisymplectic form to represent the system to obtain the multisymplectic Hamiltonian system representation with perturbation;

[0011] Step 3: Define small parameters ,use The solution of the perturbation-containing multi-Sinheim Hamiltonian system is expressed as: The power series form is obtained, and the power series form is substituted back into the perturbated multi-Synthetic Hamiltonian system. A comparison is then made. By taking the coefficients of the same power terms, we obtain a series of multisymplectic Hamiltonian equations, including the nominal equations and perturbation equations of various orders;

[0012] Step 4: Construct the Preissmann Box multisymplectic difference scheme, and numerically solve the series of multisymplectic Hamiltonian equations to obtain the nominal solution and perturbation solutions of each order. The power series form is combined with the perturbation solutions of each order to obtain the numerical solution of the perturbation-containing multi-Synthetic Hamiltonian system, thereby obtaining the structural dynamic response under small perturbations.

[0013] Furthermore, in step 1, the simplified mathematical model of the mechanical system or mechanical component includes a disturbance term, initial conditions, and boundary conditions. The disturbance term originates from continuous aerodynamic micro-excitation, slow time-varying perturbations of structural stiffness and damping parameters, and mass and geometric micro-deviations.

[0014] Furthermore, in step 2, the simplified mathematical model is represented as a differential equation containing two antisymmetric matrices and a state vector; the perturbation term is introduced into the antisymmetric matrix to form the perturbation-containing multi-symmetric Hamiltonian system representation.

[0015] Furthermore, the two antisymmetric matrices correspond to symplectic structures in the spatial and temporal directions, respectively, to maintain the physical conservation properties of the system; the state vector contains the system's generalized coordinates and generalized momentum.

[0016] Furthermore, in step 3, the nominal equation is a linear multisinus Hamiltonian equation obtained when the small parameter is zero, and its solution is the nominal solution; the perturbation equations of each order are linear multisinus Hamiltonian equations obtained by comparing the coefficients of the same power terms of the small parameter.

[0017] Furthermore, let the small parameter If the result is zero, the nominal equation is obtained, which is an undisturbed linear multisinus Hamiltonian system;

[0018] Extract the small parameters The coefficients of the first power term yield the first-order perturbation equation, which is a linear equation for the first-order perturbation solution;

[0019] Extract the small parameters The coefficients of the quadratic term yield the second-order perturbation equation, which is a linear equation for the second-order perturbation solution;

[0020] Similarly, by extracting the coefficients of each power term of the small parameters, a complete sequence of perturbation equations is obtained, where each equation is a linear multisymplectic Hamiltonian equation.

[0021] Furthermore, in step 4, constructing the Preissmann Box multisymplectic difference scheme includes:

[0022] The nominal equation and the perturbation equations of each order are discretized independently.

[0023] The Euler midpoint scheme is used for discretization in both the spatial and temporal directions.

[0024] The discretization scheme for the nominal equation is as follows: on the space-time grid, the midpoint value is approximated by the average of the function values ​​of adjacent grid points, and the partial derivatives are approximated by the central difference.

[0025] The discretization scheme for the first-order perturbation equation is as follows: based on the discretization scheme of the nominal equation, the right-hand side terms are replaced with first-order perturbation terms, while maintaining the same discrete operators;

[0026] The same construction method is used for the discrete schemes of higher-order perturbation equations to maintain the consistency of mathematical form among the discrete schemes of each order.

[0027] Furthermore, the numerical solution of the system is composed of a linear superposition of the nominal solution, the first-order perturbation solution, the second-order perturbation solution, and so on up to the Nth-order perturbation solution. In a second aspect, the present invention provides an electronic device, comprising: one or more processors; and a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned perturbation multisymplectic numerical solution method for perturbed systems based on Hamiltonian theory.

[0028] Thirdly, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enable the processor to implement the aforementioned perturbation multisymplectic numerical solution method for perturbed systems based on Hamiltonian theory.

[0029] The beneficial effects of this invention are as follows:

[0030] High precision and structural integrity are equally important: By combining perturbation theory with multisymplectic geometric algorithms, the process of decomposing the nonlinear perturbation system into a series of linear multisymplectic Hamiltonian equations ensures both high precision in numerical calculations and strict preservation of the inherent physical properties of the original system, such as its symplectic structure. This effectively avoids the structural drift problem that occurs in long-term simulations with traditional algorithms.

[0031] Accurate prediction of long-term dynamic behavior: This method can accurately capture the impact of small disturbances on the long-term dynamic behavior of the system, and is more reliable in predicting critical instability phenomena such as flutter and limit cycle oscillations, providing a powerful tool for stability analysis of mechanical systems with disturbances.

[0032] Strong engineering applicability: Compared with the traditional Monte Carlo method, this invention significantly reduces the computational cost; compared with the classical perturbation method, it has better convergence and applicability, providing a general numerical framework for the dynamic response analysis of aircraft wing components and a class of generalized nonlinear systems.

[0033] It provides greater optimization space for engineering design: Through efficient and reliable numerical prediction, it provides more sufficient analytical basis and greater optimization room for the design of high-performance aircraft, which has important engineering application value. Attached Figure Description

[0034] Figure 1 This is a flowchart of a perturbation multisymplectic numerical solution method for perturbation-containing systems based on Hamiltonian theory according to the present invention.

[0035] Figure 2 This is a simplified schematic diagram of the mechanical component structure used in this invention;

[0036] Figure 3 This is a schematic diagram of the nominal solution of the present invention using a simplified mathematical model;

[0037] Figure 4 This invention uses a simplified mathematical model of heat conduction process. Temperature distribution diagram at time;

[0038] Figure 5 This invention uses a simplified mathematical model of heat conduction process. Temperature distribution diagram at time;

[0039] Figure 6 This invention uses a simplified mathematical model of heat conduction process. The relative error diagram between the solutions of various methods and the analytical solution of the system. Detailed Implementation

[0040] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0041] like Figure 1 As shown, this invention proposes a perturbational multisymplectic numerical solution method for perturbational systems based on Hamiltonian theory, comprising the following steps:

[0042] Step 1: Based on the characteristics of mechanical structures under general working conditions, establish a simplified mathematical model of the mechanical system or mechanical component, define the corresponding system parameters, state variables and disturbance terms, and determine the initial conditions and boundary conditions under the working conditions.

[0043] In real-world operating environments, the dynamic behavior of an aircraft wing is influenced by a variety of small disturbances. These disturbances are dispersed across the aerodynamic environment, structural properties, and flight conditions, and may include: continuous random aerodynamic excitations from atmospheric turbulence; slow time-varying perturbations of structural stiffness and damping parameters due to batch differences in material properties or slight loosening of connectors; and instantaneous micro-fluctuations in state parameters such as angle of attack and Mach number caused by flight control system responses or gusts.

[0044] To avoid the model becoming overly complex and difficult to identify due to multiple disturbance source coupling, when establishing a simplified mathematical model, priority should be given to independent disturbance sources with clear physical mechanisms that play a dominant role in the system's dynamic response. Simultaneously, based on the essence of the dynamic system, more fundamental physical quantities directly reflected in the system's energy function (Hamiltonian) should be selected as the disturbance carriers. Based on the above criteria, define the corresponding parameter variables and disturbance terms, and determine the initial and boundary conditions under the operating conditions.

[0045] Step 2: Import the simplified mathematical model into the Hamiltonian system, and use the multisymplectic form to represent the system, thereby obtaining the multisymplectic Hamiltonian system representation with perturbation.

[0046] Define a two-dimensional multi-symmetric Hamiltonian system, and assume... It is a finite-dimensional phase space. for The function on, , If the two matrices are antisymmetric, then equation (1) is called the Dossing Hamiltonian equation:

[0047] (1)

[0048] It is an antisymmetric matrix; It is a state vector; It is a smooth function; Representation function about The gradient; Let be two independent variables in the base space, for example, for hyperbolic equations. For elliptic equations, .

[0049] A desirable characteristic of multisymmetric structures is their strict concept of local conservation and the ability to derive a conservation law containing a differential binary form, namely the multisymmetric conservation law. The multisymmetric conservation law is boundary-independent, therefore it is a local conservation law. Antisymmetric matrix , Corresponding to the 2-form of the symptotic system:

[0050] (2)

[0051] in, yes The inner product on the top, this symplectic form is also the symplectic structure of the multisymplectic Hamiltonian system. Multisymplectic systems have different symplectic structures in different directions. Corresponding to Direction, symptotic structure Corresponding to Direction. For example, the Dossin Hamiltonian equation (1) possesses the differential form of the Dossin conservation law:

[0052] (3)

[0053] in, The sign is for the outer product; the perturbation term may exist in the antisymmetric matrix. , Using the above process, a simplified mathematical model and the multi-Synthetic Hamiltonian representation of the perturbation term are obtained.

[0054] Step 3: Define small parameters ,use The solution of the perturbation-containing multi-Sinheim Hamiltonian system is expressed as: The power series form is obtained, and the power series form is substituted back into the perturbated multi-Synthetic Hamiltonian system. A comparison is then made. By finding the coefficients of the same power terms, we obtain a series of multisymplectic Hamiltonian equations, including the nominal equations and perturbation equations of various orders.

[0055] Define small parameters ,when When the perturbation-containing multi-Sympholm system is transformed into a linear equation, this linear equation is called the nominal equation of the perturbation-containing multi-Sympholm system. Applying the perturbation method to the perturbation-containing multi-Sympholm system, we assume that a small perturbation exists in system (1):

[0056] (4)

[0057] in, For the disturbance term, use small parameters The solution to the perturbation-containing multi-Synthetic Hamiltonian system is expressed as:

[0058] (5)

[0059] in, This is called the nominal solution, which is the solution to the nominal equation. It is a positive integer. Called of The perturbation solution. If for The analytic function, substituting the solution (5) back into the Dossin Hamiltonian equation (1), and... Expanding this into a Taylor series yields:

[0060] (6)

[0061] Among them, operators , ;

[0062] Since this equation applies to any Both are true; compare both sides of the equation. With coefficients of the same power, a series of multisymplectic Hamiltonian equations can be obtained:

[0063] (7)

[0064] The nominal solution can be obtained from the first equation of the system of equations (7). Accordingly, the first equation is called the nominal equation. Substituting into the second equation of the system of equations (7), we can obtain the solution. First-order perturbation solution Accordingly, the second equation is called the first-order perturbation equation, and then the process is repeated to derive the equation in sequence. The value of is used to finally obtain the solution to the original equation. ,

[0065] .

[0066] Step 4: Construct a special Preissmann Box multisymplectic difference scheme to numerically solve the series of multisymplectic Hamiltonian equations, obtaining the nominal solution and the multi-order perturbation solution respectively. Use the nominal solution and the multi-order perturbation solution to obtain the numerical solution of the perturbation-containing multisymplectic Hamiltonian system, thereby obtaining the structural dynamic response under small perturbations.

[0067] First, a special Preissmann Box multisymplectic difference scheme is constructed, and then... , for At grid points The discrete representation of the dossinic equation system (1) is given by:

[0068] (8)

[0069] The symplectic structure corresponding to the discrete scheme (8) is also a discrete 2-form:

[0070] (9)

[0071] in, The discrete variational equation that satisfies the discrete scheme (8) is:

[0072] (10)

[0073] For the system of dossin equations (1) in The Euler midpoint method is used to discretize the multisymplectic equations in all directions, resulting in the Preissmann Box scheme, which is a special multisymplectic difference scheme.

[0074] (11)

[0075] in, They are respectively Step size in direction:

[0076] ,

[0077] After obtaining the Box format, the perturbation equation system (7) is constructed using a difference scheme. First, the nominal equations are constructed using a difference scheme:

[0078] (12)

[0079] in, They are respectively Step size in direction,

[0080] ;

[0081] The nominal solution can be obtained using the difference scheme (12). Next, solve the first-order perturbation equation and construct the difference scheme:

[0082] (13)

[0083] in, They are respectively Step size in direction,

[0084]

[0085] The first-order perturbation solution can be obtained using the difference scheme (13). The first-order solution of the method of this invention can be obtained using the nominal solution and the first-order perturbation solution.

[0086] (14)

[0087] Continue solving the second-order perturbation equations and construct the difference scheme:

[0088] (15)

[0089] in, They are respectively Step size in direction,

[0090]

[0091] The second-order perturbation solution can be obtained using the difference scheme (15). The second-order solution of the method of this invention can be obtained using the nominal solution, the first-order perturbation solution, and the second-order perturbation solution.

[0092] (16)

[0093] By continuing to solve the problem using the above method, the solution of the method of this invention of arbitrary order can be obtained, and the solution of the original equation can be expressed accordingly, thus obtaining the structural dynamic response under small perturbations.

[0094] Step 5: By combining the dynamic response of the same structure under undisturbed conditions, compare the analytical accuracy and structural retention characteristics of the results obtained by the present invention, and prove that the results obtained by the present invention are more consistent with the real situation and have stronger engineering applicability.

[0095] Based on step 4 order solution Generally, the accuracy of a second-order solution is sufficient. This solution is then compared with some known analytical solutions. Based on all the above results and processes, a Hamiltonian dossin perturbation numerical solution method that conforms to the real-world situation is established.

[0096] Example:

[0097] To better understand the characteristics of this invention and its applicability to engineering practice, this invention first establishes a simplified model of a component structure of an aircraft wing, a certain saturated porous elastic rod, and imports the simplified mathematical model and disturbance terms into a Hamiltonian system. The system is then represented using a multisymplectic form. The solution of the original system is expressed as By substituting the power series form of the equations back into the original multisymplectic system for comparison, a series of multisymplectic Hamiltonian equations are obtained. A special Preissmann Box multisymplectic difference scheme is constructed to obtain the multi-order perturbation solutions of the original equations. Finally, these solutions are compared with known analytical solutions to establish a Hamiltonian multisymplectic perturbation numerical solution method that conforms to the real situation.

[0098] A simplified structural model of a component of an aircraft wing is shown below. Figure 2 As shown, this is a saturated porous elastic rod with a length of [missing information]. The cross-sectional height is Based on the assumptions of the continuum mechanics model, the displacement of a material particle can be denoted as:

[0099] (17)

[0100] in, Representing the mobile phase and solid phase respectively. Indicator rod at The position of direction, Indicates time, For the initial region of the porous elastic rod, consider a local thermal equilibrium state, meaning that thermal equilibrium is achieved at every spatial point, and the fluid and solid phases have the same temperature. In the absence of other heat sources within the porous elastic rod, its local thermal equilibrium heat conduction equation can be written as follows:

[0101] (18)

[0102] in, The thermal conductivity coefficient of the fluid-solid two-phase system is... The density of the fluid-solid two phases, To represent the specific heat capacity of a fluid-solid two-phase system, some dimensionless quantities and constants are introduced:

[0103]

[0104] At the same time We can obtain the dimensionless local thermal equilibrium heat conduction equation for a saturated porous elastic rod:

[0105] (19)

[0106] This yields a simplified mathematical model, which is then imported into the Hamiltonian system and represented using multisymplectic forms. Regular variables are then introduced. (19) can be transformed into the standard dosine form:

[0107] (20)

[0108] in,

[0109]

[0110] Take dimensionless parameters The dimensionless boundary conditions and initial conditions for the above equations are given.

[0111] (twenty one)

[0112] The analytical solution to this equation, i.e. the nominal solution to the perturbation system, is:

[0113] (twenty two)

[0114] Image as Figure 3 As shown, it can be seen that as time progresses, the porous elastic rod gradually tends towards local thermal equilibrium, which is approximately [value missing]. The time is reached. Now, let's give the perturbation quantity in this example, which exists in a dimensionless quantity. middle,

[0115] (twenty four)

[0116] dimensionless quantities in the nominal system Disturbance term Select spatial step size Time step small parameters Numerical simulations were performed using the first-order and second-order methods of this invention. At that time, the temperature distribution during the heat conduction process of the porous elastic rod is as follows: Figure 4 As shown. Clearly, the impact of small disturbances on the system cannot be ignored. The second-order method of this invention has higher accuracy than all other methods, with a maximum error of only 1.4%, which fully demonstrates the accuracy of the method of this invention.

[0117] Figure 5 for The temperature distribution during heat conduction in a porous elastic rod varies with time, and the impact of disturbances on the system becomes increasingly significant. However, the second-order method of this invention can consistently simulate the actual situation well, demonstrating the effectiveness of the proposed method. Figure 6 Given Error graphs between the solutions obtained by various methods and the exact solutions show that neither the conventional Preissmann Box scheme nor the traditional Runge-Kutta method can match the accuracy of the method proposed in this invention. Overall, the method proposed in this invention completes the required work with high quality and efficiency, providing a highly engineering-applicable general numerical tool for the dynamic response analysis of perturbed aircraft wings and even a class of generalized nonlinear Hamiltonian systems, and offering greater optimization and analysis space for high-performance aircraft design.

[0118] In a second aspect, the present invention provides an electronic device, comprising: one or more processors; and a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned perturbation multisymplectic numerical solution method for perturbation-containing systems based on Hamiltonian theory.

[0119] Thirdly, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enable the processor to implement the aforementioned perturbation multisymplectic numerical solution method for perturbed systems based on Hamiltonian theory.

[0120] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A perturbational multisymplectic numerical solution method for perturbed systems based on Hamiltonian theory, characterized in that, The method includes: Step 1: Establish a simplified mathematical model of the mechanical system or mechanical component, define system parameters, state variables and disturbance terms, and determine the initial conditions and boundary conditions; Step 2: Import the simplified mathematical model into the Hamiltonian system, and use the multisymplectic form to represent the system to obtain the multisymplectic Hamiltonian system representation with perturbation; Step 3: Define small parameters ,use The solution of the perturbation-containing multi-Sinheim Hamiltonian system is expressed as: The power series form is obtained, and the power series form is substituted back into the perturbated multi-Synthetic Hamiltonian system. A comparison is then made. By taking the coefficients of the same power terms, we obtain a series of multisymplectic Hamiltonian equations, including the nominal equations and perturbation equations of various orders; Step 4: Construct the Preissmann Box multisymplectic difference scheme, and numerically solve the series of multisymplectic Hamiltonian equations to obtain the nominal solution and perturbation solutions of each order. The power series form is combined with the perturbation solutions of each order to obtain the numerical solution of the perturbation-containing multi-Synthetic Hamiltonian system, thereby obtaining the structural dynamic response under small perturbations.

2. The perturbational multisymplectic numerical solution method for perturbation-containing systems based on Hamiltonian theory according to claim 1, characterized in that, In step 1, the simplified mathematical model of the mechanical system or mechanical component includes a disturbance term, initial conditions, and boundary conditions. The disturbance term comes from continuous aerodynamic micro-excitation, slow time-varying perturbations of structural stiffness and damping parameters, and mass and geometric micro-deviations.

3. The perturbational multisymplectic numerical solution method for perturbed systems based on Hamiltonian theory according to claim 1, characterized in that, In step 2, the simplified mathematical model is represented as a differential equation containing two antisymmetric matrices and a state vector; the perturbation term is introduced into the antisymmetric matrix to form the perturbation-containing multi-symmetric Hamiltonian system representation.

4. The perturbational multisymplectic numerical solution method for perturbed systems based on Hamiltonian theory according to claim 3, characterized in that, The two antisymmetric matrices correspond to the symplectic structures in the spatial and temporal directions, respectively, and are used to maintain the physical conservation properties of the system; the state vector contains the system's generalized coordinates and generalized momentum.

5. The perturbational multisymplectic numerical solution method for perturbed systems based on Hamiltonian theory according to claim 1, characterized in that, In step 3, the nominal equation is a linear multisinus Hamiltonian equation obtained when the small parameter is zero, and its solution is the nominal solution; the perturbation equations of each order are linear multisinus Hamiltonian equations obtained by comparing the coefficients of the same power terms of the small parameter.

6. The perturbational multisymplectic numerical solution method for perturbed systems based on Hamiltonian theory according to claim 5, characterized in that, Let the small parameter If the result is zero, the nominal equation is obtained, which is an undisturbed linear multisinus Hamiltonian system; Extract the small parameters The coefficients of the first power term yield the first-order perturbation equation, which is a linear equation for the first-order perturbation solution; Extract the small parameters The coefficients of the quadratic term yield the second-order perturbation equation, which is a linear equation for the second-order perturbation solution; Similarly, by extracting the coefficients of each power term of the small parameters, a complete sequence of perturbation equations is obtained, where each equation is a linear multisymplectic Hamiltonian equation.

7. The perturbational multisymplectic numerical solution method for perturbed systems based on Hamiltonian theory according to claim 1, characterized in that, In step 4, constructing the Preissmann Box multisymplectic difference scheme includes: The nominal equation and the perturbation equations of each order are discretized independently. The Euler midpoint scheme is used for discretization in both the spatial and temporal directions. The discretization scheme for the nominal equation is as follows: on the space-time grid, the midpoint value is approximated by the average of the function values ​​of adjacent grid points, and the partial derivatives are approximated by the central difference. The discretization scheme for the first-order perturbation equation is as follows: based on the discretization scheme of the nominal equation, the right-hand side terms are replaced with first-order perturbation terms, while maintaining the same discrete operators; The same construction method is used for the discrete schemes of higher-order perturbation equations to maintain the consistency of mathematical form among the discrete schemes of each order.

8. The perturbational multisymplectic numerical solution method for perturbed systems based on Hamiltonian theory according to claim 7, characterized in that, The numerical solution of the system is formed by the linear superposition of the nominal solution, the first-order perturbation solution, the second-order perturbation solution, and so on up to the Nth-order perturbation solution.

9. An electronic device, characterized in that, include: One or more processors; Memory, used to store one or more programs; When one or more programs are executed by the one or more processors, the one or more processors implement the perturbation multisymplectic numerical solution method for perturbation-containing systems based on Hamiltonian theory as described in any one of claims 1-8.

10. A computer-readable storage medium, characterized in that, It stores executable instructions that, when executed by a processor, enable the processor to implement the perturbation multisymplectic numerical solution method for perturbation-containing systems based on Hamiltonian theory as described in any one of claims 1-8.

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