Sickness-preserving finite element solution method for dynamic response of structure under Hamiltonian system

Abstract

The invention discloses a pungent-keeping finite element solving method for dynamic response of a structure under a Hamiltonian system. According to the method, a finite element method is applied to a Hamiltonian equation, a Pocking finite element method is deduced, and it is proved that the Pocking finite element method is of a strict Pocking structure. And on the other hand, programmed solution of a high-order iteration format is realized by utilizing mathematical calculation software. In addition, the flexibility of the pungency-preserving finite element method to the linear Hamiltonian system is also discussed. A plurality of numerical examples verify the excellent performance of the Pocking finite element method in the structural dynamic response problem, and prove the capability of keeping energy conservation and excellent stability of the Pocking finite element method in the long-term dynamic simulation process, and compared with a traditional numerical method, the Pocking finite element method has remarkable advantages.

Description

Technical Field

[0001] The present invention belongs to the field of dynamic response analysis of beam and wing structures, and in particular relates to a symplectic finite element solution method for structural dynamic response under a Hamiltonian system. Background Art

[0002] Beams and wing structures are widely used in aerospace and civil engineering, and accurate prediction of their dynamic response is crucial for structural safety and stability. However, beams and wing structures often exhibit complex dynamic characteristics, such as multimodal coupling, nonlinear vibration, and high-frequency vibration response, which makes traditional numerical analysis methods face significant challenges in terms of accuracy and stability. Summary of the Invention

[0003] To overcome these technical challenges, the present invention provides a symplectic finite element method for solving the dynamic response of structures in Hamiltonian systems. By accurately calculating and analyzing the dynamic behavior of these structures, the symplectic finite element method provides a powerful theoretical tool for studying the stability, vibration response, and bifurcation behavior of complex engineering structures in nonlinear dynamic environments. This method demonstrates exceptional accuracy and conservation properties when dealing with structural systems with complex boundary conditions and high-dimensional degrees of freedom, making it suitable for demanding engineering design and optimization applications.

[0004] The symplectic finite element method (SFM), as an efficient numerical analysis method, has been widely used in the field of structural dynamics. In particular, the SFM offers significant advantages in dynamic response analysis of complex structures such as beams and wings, with its high accuracy over long calculation times and excellent symplectic properties (energy conservation).

[0005] The introduction of the symplectic finite element method (FEM) provides a new approach to addressing these issues. By introducing a symplectic structure, the FEM maintains the energy conservation of the system during long-term calculations, avoiding numerical dissipation and spurious vibrations, thereby significantly improving the accuracy of the numerical solution. Especially when it comes to large-scale structures and long-term response analysis, the FEM offers greater efficiency and accuracy than traditional FEM and differential methods. Furthermore, the FEM can adapt to the nonlinear characteristics of complex materials and structures, such as geometric and material nonlinearities, making it widely applicable in practical engineering applications.

[0006] In summary, the symplectic finite element method has important theoretical significance and engineering application value in the dynamic response analysis of beams and wing structures, and can provide strong technical support for the design, optimization and safety assessment of related structures.

[0007] In order to achieve the above object, the present invention adopts the following technical solutions:

[0008] A symplectic finite element method for solving the dynamic response of a structure under a Hamiltonian system is used for dynamic response analysis under free vibration conditions, including the following steps:

[0009] Step 1: Find the weak solution of the Hamiltonian equation in the integral sense based on the finite element method, and construct the symplectic finite element algorithm by solving the weak solution;

[0010] Step 2: Use mainstream commercial mathematical operation software to construct iterative formats of different-order symplectic finite element methods;

[0011] Step 3: Based on the Pade symplectic iteration scheme and the accuracy analysis of the finite element method at the unit nodes, the symplectic finite element method is proved to be able to maintain the symplectic structure.

[0012] Step 4: Import the structural dynamics equations under free vibration into the Hamiltonian system by introducing dual variables;

[0013] Step 5: Use the iterative format of the symplectic finite element method to process the Hamiltonian equation and analyze the accuracy of the dynamic response calculation structure and its symplectic properties.

[0014] Beneficial effects:

[0015] The present invention establishes a symmetric finite element method dynamic model for beam and wing structures, which can efficiently and accurately analyze the dynamic response of the structure based on the consideration of complex dynamic characteristics. By combining the symmetric structure and high-order approximation technology, the symmetric finite element method can maintain energy conservation during long-term calculations, avoiding numerical dissipation and the generation of spurious vibrations. Compared with traditional numerical methods such as the Runge-Kutta method, the symmetric finite element method shows higher accuracy and stability when dealing with complex phenomena such as multi-modal coupling, nonlinear vibration and high-frequency response. Specifically, the symmetric finite element method can accurately capture the response characteristics of beam and wing structures in a variety of dynamic environments, including large deformation, material nonlinearity and the dynamic behavior of multi-degree-of-freedom systems. The application of the symmetric finite element method significantly improves the calculation accuracy of the structural response, especially in long-term simulations, it can avoid the accumulation of errors, and accurately predict key characteristics such as the resonant frequency, modal coupling and amplitude response of the structure while maintaining the conservation of system energy. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 It is a symplectic finite element iterative format flow chart of the symplectic finite element solution method for the dynamic response of a structure under a Hamiltonian system of the present invention;

[0017] Figure 2 is the finite element mesh diagram of the wing structure;

[0018] Figure 3 It is the analysis diagram of the 3 dynamic responses of the wing structure endpoint nodes using different methods;

[0019] Figure 4(a) is a diagram showing the long-term dynamic response calculation results of the wing structure endpoint node 3 using the RK2 method;

[0020] Figure 4(b) is a diagram showing the long-term dynamic response calculation results of the wing structure endpoint node 3 under different methods;

[0021] Figure 5(a) shows the long-term calculation results of the Hamiltonian function of the wing structure under different methods;

[0022] Figure 5(b) shows the long-term calculation results of the Hamiltonian function of the wing structure under different methods;

[0023] Figure 6 It is an analysis flow chart of the symplectic finite element solution method for the dynamic response of a structure under a Hamiltonian system according to the present invention. DETAILED DESCRIPTION

[0024] In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other. Figure 6 As shown, the present invention proposes a symplectic finite element solution method for structural dynamic response under a Hamiltonian system, comprising the following steps:

[0025] Step (1) Find the weak solution of the Hamiltonian equation in the integral sense based on the finite element method, and construct the symplectic finite element algorithm by solving the weak solution.

[0026] Consider a linear Hamiltonian system, namely the Hamiltonian function is a variable The quadratic form of At this point, the equation can be written as formula (1):

[0027] (1)

[0028] in, represents the time variable, is a symmetric matrix, For a symplectic matrix satisfying , from this we can see It is an infinitesimal symplectic matrix.

[0029] Given interval ,in Indicates the time interval, Indicates the Time nodes, in the above time interval The continuous finite element method is used for the above formula, and its m-order finite element satisfies formula (2):

[0030] (2)

[0031] in, For variables of The specific formula is shown in formula (3):

[0032] (3)

[0033] The above formula indicates that in a given interval Select m time nodes Construct the Lagrangian difference function and use it as of Finite element approximation .in, represents the ith m-th order Lagrangian basis function, Indicates that in the interval The i-th m-th order Lagrangian basis function The coefficient of , j represents the j-th time node.

[0034] In formula (2) is a weighting function, and its specific form is shown in formula (4):

[0035] (4)

[0036] in, Indicated by The weight function is composed of polynomials of order.

[0037] Therefore, according to formula (2) to formula (4), the following equations can be obtained:

[0038] (5)

[0039] The first Expanding the equation we can get:

[0040] (6)

[0041] According to the above expansion, the equations in equation (6) can be written in matrix form:

[0042] (7)

[0043] Among them, the matrix Representation and Matrix The unit matrix of the same dimension. After transposing the above equation, we can get

[0044] (8)

[0045] At this time, according to Cramer's law, we can directly derive the matrix equation in formula (8): and The relationship between:

[0046] (9)

[0047] in, Representing variables At the time node The value of .

[0048] Step (2) uses mathematical operation software to realize the construction of different order symplectic finite element method iterative formats.

[0049] Given in step (1) and The relationship between contains complex calculations such as differentials, integrals, and determinants, so it is impossible to manually calculate high-order finite element expressions. For this reason, the present invention proposes to calculate arbitrary-order symplectic finite element expressions through programming with the help of mathematical tools.

[0050] The calculation program of the first-order symplectic finite element formula is written by referring to the derivation process of formula (2) to formula (9) in step (1). First, select interpolation points and calculate the interpolation basis function in formula (3). Then select the weighting function of the form (4) , through differential and integral operations, we get the inner product coefficient matrix and give the matrix expression in the form of formula (6). Finally, according to Carmer's rule, we derive and The relationship between is formula (9), which can be obtained by calculating the determinant. The specific program flow chart is as follows: Figure 1 shown.

[0051] In the specific calculation, the standard interval is selected as Then we calculated it using the program. Order and Order symplectic finite element expression.

[0052] when The augmented matrix of the equation calculated by the program is:

[0053] (10)

[0054] in, represents the calculation step size, Representation and infinitesimal symplectic matrix The unit matrix of the same order. Further calculation of the determinant can obtain the third-order symplectic finite element expression:

[0055] (11)

[0056] The augmented matrix of the equation obtained by running the program when m=4 is:

[0057] (12)

[0058] Calculating the determinant yields the 4th-order symplectic finite element expression:

[0059] (13)

[0060] like Figure 2 Shown is the finite element mesh diagram of the wing structure.

[0061] Step (3) proves the preservation of symplectic structure by the symplectic finite element method based on the Pade symplectic iteration format and the accuracy analysis of the finite element method at the unit nodes.

[0062] The derivation has been obtained and The relationship between:

[0063] (14)

[0064] Expand the above determinant, and the determinants corresponding to the above numerators and denominators are all about the matrix The highest order does not exceed The matrix polynomial of is written as:

[0065] (15)

[0066] in, and It is an infinitesimal symplectic matrix of Next, we prove that Equation (15) is a 2m-order Pade approximation. In the proof, we first introduce the superconvergence of the continuous finite element and the definition of the Pade approximation formula, and then use this to show that Equation (15) satisfies the definition of the Pade approximation.

[0067] For a linear Hamiltonian system with constant coefficients, >1st order continuous finite element approximation at time nodes There is a highest order superconvergence on .

[0068] (16)

[0069] in, express At the time node The value at express The value at the initial moment, express of Finite element approximation At the time node The value at is the Hamilton coefficient matrix The model, Represents the time interval between any two time nodes, constant and It doesn’t matter. In particular, if we take t=0 as the starting point, then at the first node There is a higher order superconvergence estimate at .

[0070] (17)

[0071] Pade approximation is a kind of rational fraction approximation. Let It can be expressed as a power series:

[0072] (18)

[0073] in, represents the polynomial variable, express of Power, express The constant coefficient of . At this time, if there is:

[0074] (19)

[0075] in, is a polynomial of order up to l, The highest order is polynomial of order, Indicates about variables of A high-order small quantity. Then called of Pade approximation.

[0076] If you assume is the initial value, then according to the equation satisfied by the linear Hamilton system, in the small interval I_(j+1) The exact solution can be expressed as:

[0077] (20)

[0078] The analysis shows that the estimation accuracy of formula (17) matches the Pade approximation accuracy. According to formula (15) and formula (20), we can know that:

[0079] (twenty one)

[0080] It can be found that the form of formula (21) satisfies the definition of (m,m)-order Pade approximation. Therefore, it can be proved that formula (21) is a 2m-order Pade approximation.

[0081] Next, we prove that the Pade approximation formula in formula (21) is unique and satisfies the symplectic property. Assume There are two different Pade approximations:

[0082] (twenty two)

[0083] in, Indicates that it is different from The highest order of a polynomial is l, Indicates that it is different from The highest order is According to the precision analysis formula (19) of the Pade format, we know that:

[0084] (twenty three)

[0085] Riding on both sides Then we can get:

[0086] (twenty four)

[0087] Since formula (23) is a polynomial with an order not exceeding (l+m), in order for formula (23) to hold true, only:

[0088] (25)

[0089] And because and It is not 0, so we have:

[0090] (26)

[0091] This proves that any form The Pade approximation of is unique.

[0092] if is an infinitesimal symplectic matrix, then for sufficiently small ,when Pade diagonal approximation It is a Xin formation.

[0093] set up .make is an even-order polynomial, is an odd-order polynomial. To prove It is a symplectic matrix. We only need to prove that the following equation holds:

[0094] (27)

[0095] Because B is a symplectic matrix, and is an even-order polynomial, It is an odd-order polynomial, so according to the basic properties of the symplectic matrix:

[0096] (28)

[0097] The superscript T represents the transpose of the matrix. From this we can deduce:

[0098] (29)

[0099] Comparative analysis shows that the right side of the above formula is obviously equal, so the left side is also equal, so the Pade approximation formula in formula (21) is unique, and the approximation formula satisfies the symplectic property. This proves that the derivation and The relationship between them satisfies uniqueness and has the corresponding symplectic properties.

[0100] From the above proof, we can know that the m-order symplectic finite element derived and The relationship between is the (m,m)-order Pade approximation formula, and the formula is unique. From this, we can give its general expression:

[0101] (30)

[0102] From the above proof, we know that no matter how the weighting function is selected, the final determinant expression should be unique. Therefore, we use the program to verify whether the final structure is unique when different weighting functions are selected. Here we take the m=4th order symplectic finite element expression as an example. When the weighting function is When , the inner product coefficient matrix can be obtained according to the differential and integral operations:

[0103] (31)

[0104] The augmented matrix of the equation is now: (32)

[0105] Calculating the determinant yields the 4th-order symplectic finite element expression:

[0106] (33)

[0107] When the weighting function is changed to When the inner product coefficient matrix is:

[0108] (34)

[0109] The augmented matrix is:

[0110] (35)

[0111] The final 4th-order symplectic finite element expression is:

[0112] (36)

[0113] Verification found that when different weighting functions are selected, the inner product results of the Lagrange basis function and the weighting function change, but the final augmented matrix and determinant calculation results remain unchanged.

[0114] Step (4) imports the structural dynamics equations under free vibration into the Hamiltonian system by introducing dual variables.

[0115] The free vibration equation can usually be expressed as:

[0116] (37)

[0117] in, is the mass matrix, is the stiffness matrix, is the displacement variable at the structural node. Introducing the dual variable After that, the above equation of motion can be reformulated as a Hamiltonian system in matrix form:

[0118] (38)

[0119] make:

[0120] (39)

[0121] Then equation (30) can be expressed as:

[0122] (40)

[0123] in, is the displacement variable of the structural node and dual variables The variables that constitute it together. Due to the constant coefficient moment Meet the conditions , is an infinitesimal symplectic matrix. Equation (32) is a linear Hamiltonian equation. It should be noted that different engineering structures will have different mass and stiffness matrices. The specific process of constructing these matrices can refer to the spatial finite element method, where the global mass / stiffness matrix is ​​composed of the unit mass / stiffness matrices. After introducing the free vibration equation into the linear Hamiltonian system, the symplectic finite element method can be used to predict the response of the structure at different time points. First, the order of the symplectic finite element method iteration scheme is selected. and iteration step size Then, the infinitesimal symplectic matrix obtained from equation (31) is Substitute into the final iterative formula. For example, when and When , the iterative formula can be expressed as:

[0124] (41)

[0125] in, is an infinitesimal symplectic matrix composed of the mass matrix and the stiffness matrix, is with The identity matrix of corresponding dimensions.

[0126] Example:

[0127] To better understand the characteristics of this invention and its applicability to practical engineering, the inventors first analyzed the governing dynamic equations of a two-dimensional linear Hamiltonian system and employed the symplectic finite element method to solve the structural displacement and overall energy change. Subsequently, the proposed method of introducing dual variables into the Hamiltonian system and mathematical software calculations using complex symplectic finite element iterative formulas were combined to solve the endpoint displacement response of the wing structure under free vibration.

[0128] The first to third order symplectic finite element method, central symplectic difference scheme (SCD), second order Runge-Kutta (RK2) and fourth order Runge-Kutta (RK4) are used to solve the examples. Figure 3 The simulation results in show that the RK2 method deviates significantly from the actual response over a long period of time. Although the SCD calculation results do not diverge, the phase difference gradually accumulates over time, causing a large deviation from the analytical solution. The results obtained by the RK4 method can represent the actual system response to a certain extent, but compared with the first-order symplectic finite element method, the second-order symplectic finite element method, and the third-order symplectic finite element method, the error between the RK4 results and the analytical solution is relatively large. In addition, by comparing Figure 3The three simulation curves for the first-order, second-order, and third-order symplectic finite element methods (SFMs) show that as the order of the SFM increases, the numerical solution gradually approaches the exact solution, indicating a gradual improvement in the accuracy of the method. Furthermore, by comparing the curves for the second-order and third-order symplectic finite element methods, it is noted that when 𝑚 is greater than or equal to 2, the impact of the order on fitting accuracy decreases. Therefore, the third-order symplectic finite element method is generally suitable for system simulation. The present invention also performs long-term calculations of the structural response. Figures 4(a) and 4(b) show the long-term continuous calculation results of different methods. The results in Figures 4(a) and 4(b) further demonstrate that the SFM remains stable during the continuous calculation process, while traditional methods such as RK2 and RK4 exhibit numerical dissipation over time.

[0129] In addition, for Hamiltonian systems, the Hamiltonian function has the dimension of energy, indicating the conservation of the system energy. Figure 5 (a) and Figure 5 (b) show the Hamiltonian functions of linear systems obtained by the symplectic finite element method, SCD, RK2, and RK4 methods. H By analyzing Figure 4 (a) and Figure 4 (b), it can be seen that the symplectic finite element method and SCD maintain the H The value is -27.25. However, although the fourth-order Runge-Kutta method has high accuracy, its Hamiltonian function H The Hamiltonian function of the second-order Runge-Kutta method H This phenomenon shows that the proposed method performs well in terms of conservation compared with the second-order and fourth-order Runge-Kutta methods.

[0130] The above are only specific steps of the present invention and do not constitute any limitation to the scope of protection of the present invention; it can be extended to other fields of structural dynamics analysis, and any technical solutions formed by equivalent transformation or equivalent replacement fall within the scope of protection of the present invention.

[0131] Some parts of the present invention are well known to those skilled in the art and are not described in detail.

Claims

1. A symplectic finite element method for solving the dynamic response of a structure under a Hamiltonian system, used for dynamic response analysis under free vibration conditions, characterized in that: The steps include: Step 1: Find the weak solution of the Hamiltonian equation in the integral sense based on the finite element method, and construct the symplectic finite element algorithm by solving the weak solution; Step 2: Realize the construction of iterative formats of symplectic finite element method of different orders; Step 3: Based on the Pade symplectic iteration scheme and the accuracy analysis of the finite element method at the unit nodes, the symplectic finite element method is proved to be able to maintain the symplectic structure. Step 4: Import the structural dynamics equations under free vibration into the Hamiltonian system by introducing dual variables; Step 5: Use the iterative format of the symplectic finite element method to process the Hamiltonian equation and analyze the accuracy of the dynamic response calculation structure and its symplectic properties.

2. The symplectic finite element method for solving the structural dynamic response under a Hamiltonian system according to claim 1, characterized in that: The first step comprises: Within a given time interval, the Hamiltonian equations are discretized using the continuous finite element method; a high-order finite element approximation formula is constructed, and a system of equations is established through Lagrange interpolation functions and weighting functions; the system of equations is written in matrix form, and the relationship between the variables at the time nodes is derived using Cramer's rule, thus completing the construction of the symplectic finite element algorithm.

3. The symplectic finite element method for solving the structural dynamic response under a Hamiltonian system according to claim 1, characterized in that: The relationship between variables at time nodes derived using Cramer's law is: （9） in, For variables of The finite element approximation formula is Representing variables At the time node The value of Indicated by The weight function composed of polynomials of order; matrix Representation and Matrix Identity matrices of the same dimensions; is an infinitesimal symplectic matrix; represents the i-th m-order Lagrangian basis function; t represents time.

4. The symplectic finite element method for solving the dynamic response of a structure under a Hamiltonian system according to claim 1, characterized in that: The second step includes: Select interpolation points in a given interval and calculate the interpolation basis function; select the weighting function and obtain the inner product coefficient matrix through differentiation and integration operations.

5. The symplectic finite element method for solving the structural dynamic response under a Hamiltonian system according to claim 4, characterized in that: The second step also includes: The matrix expression is given, and the relationship between the variables is derived according to Cramer's rule; the symplectic finite element expressions of different orders are obtained by calculating the determinant.

6. The symplectic finite element method for solving the structural dynamic response under a Hamiltonian system according to claim 3, characterized in that: The third step includes: The relationship between the variables is derived and expressed as a matrix polynomial. It is shown that the relationship satisfies the definition of Pade approximation, that is, it can approximate the solution of the Hamiltonian system with high accuracy at a given order. It is proved that the Pade approximation formula is unique and satisfies the symplectic property, that is, it can maintain the energy conservation and symplectic structure of the system in numerical calculations.

7. The symplectic finite element method for solving the structural dynamic response under a Hamiltonian system according to claim 6, characterized in that: The third step also includes: further confirming the uniqueness and symplectic structure preservation of the symplectic finite element method through verification with different weighting functions, thereby proving the stability and accuracy of the method in long-term calculations.

8. The symplectic finite element method for solving the structural dynamic response under a Hamiltonian system according to claim 7, characterized in that: The Pade approximation of order 2m is: （21） in, and It is an infinitesimal symplectic matrix of Matrix polynomial of order, Represents the calculation step size, C is a constant, and O represents a high-order infinitesimal.

9. The symplectic finite element method for solving the structural dynamic response under a Hamiltonian system according to claim 1, characterized in that: The fourth step includes: The free vibration equation is reformulated as a Hamiltonian system in matrix form; the dual variables are defined, and the displacement variables and the dual variables are combined into an extended variable vector; an infinitesimal symplectic matrix is ​​constructed so that the dynamic equations satisfy the standard form of the Hamiltonian system; through the above transformation, the structural dynamics equations can be applied to the solution framework of the symplectic finite element method, thus providing a basis for subsequent numerical analysis.

10. The symplectic finite element method for solving the structural dynamic response under a Hamiltonian system according to claim 1, characterized in that: The fifth step includes: The derived symplectic finite element iterative formula is applied to the Hamiltonian system to perform numerical calculations of the structural dynamic response; by selecting an appropriate iteration order and time step, combined with the dynamic parameters of the specific structure, the response of the structure at different time points is calculated; the accuracy of the calculation results is analyzed to verify the ability of the symplectic finite element method to maintain energy conservation and avoid numerical dissipation in long-term simulations; and the superiority of the symplectic finite element method in complex dynamic problems is evaluated.

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