A synchronous bidirectional coupling solution method for large deformation of flexible slender beam under uniform pressure based on accurate load stiffness matrix

Abstract

The application discloses a synchronous bidirectional coupling solving method for large deformation of a flexible slender beam under uniform pressure based on an accurate load stiffness matrix and relates to the field of computational mechanics. The method discretizes a beam structure through absolute node coordinate Euler-Bernoulli beam elements, constructs a local curvilinear coordinate system and a uniform pressure vector function, derives an external force vector of the element based on a virtual work principle, obtains an accurate load stiffness matrix through a vector-valued function differential rule, synchronously assembles the load stiffness matrix and an elastic tangent stiffness matrix in a Newton-Raphson iteration step, and realizes synchronous bidirectional coupling solving of load and configuration. The method breaks through the numerical loading limit of a traditional method, avoids error accumulation, improves calculation stability and precision, can accurately analyze large deformation and even post-buckling behavior of the flexible slender beam under uniform pressure, and is suitable for design optimization of related equipment in the fields of ocean engineering and mechanical engineering.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of computational mechanics and nonlinear finite element analysis, and particularly relates to a synchronous bidirectional coupling solution method for large deformation of a flexible slender beam under uniform pressure based on an accurate load stiffness matrix. BACKGROUND

[0002] The static large deformation analysis of a flexible slender beam structure under uniform pressure is a key mechanical problem in the design of marine engineering equipment such as underwater pontoon stiffeners and flexible mechanical arms. The particularity of this problem lies in that the uniform fluid static pressure borne is a typical "non-conservative force", and its direction is not fixed but always perpendicular to the current configuration after deformation, as shown in FIG. 1. This physical nature leads to a strong nonlinear coupling relationship between the load direction and the structural deformation, that is, the load direction is a function of the deformation result, and the deformation result is simultaneously affected by the load direction, forming a complex "load-configuration" synchronous bidirectional feedback system. Figure 1

[0003] In the existing nonlinear finite element analysis practice, this kind of problem is usually simplified and solved by using a "unidirectional sequential asynchronous coupling" method. The core of this method is that in each load increment step (outer loop), the load direction of the current step is fixed based on the approximate configuration obtained in the last increment step, and then the new configuration is solved by Newton-Raphson iteration step (inner loop) under the fixed load. This asynchronous decoupling processing method of "updating the load first and then fixing the solution" has inherent defects, and the fundamental reason is that the additional stiffness contribution due to the change of the load direction with the configuration, that is, the load stiffness matrix, is completely ignored. From the mathematical nature, the load stiffness matrix is the derivative of the node force vector to the node coordinate vector, and its derivation involves complex vector-valued differential operation, so it is often avoided or approximated roughly in most existing technologies.

[0004] Due to the lack of load stiffness matrix, the overall tangent stiffness matrix in the nonlinear solution process cannot fully reflect the real stiffness of the system, thereby causing a series of significant numerical calculation obstacles, which are specifically as follows:

[0005] (1) There is a numerical loading limit: when the external pressure exceeds a certain critical threshold, the numerical instability occurs due to the inaccurate tangent stiffness matrix, and the iterative calculation completely diverges, so that the real large deformation response of the structure under high pressure cannot be obtained.

[0006] (2) The calculation accuracy is restricted by systematic errors: the load direction of the current increment step is severely dependent on the approximate configuration of the last step, and this recursive approximation will introduce systematic errors, which will be accumulated between the increment steps, resulting in a serious deviation of the final calculation result from the true solution.

[0007] ​(3) Calculation efficiency and accuracy are difficult to be considered: a large number of incremental steps are often divided to control the cumulative error, which significantly sacrifices the calculation efficiency; and insufficient number of incremental steps will result in large single-step approximation error, leading to distorted results.

[0008] Therefore, developing a synchronous bidirectional coupling solving method capable of accurately considering the load-configuration coupling effect and effectively introducing the load stiffness matrix has important engineering application value and theoretical significance for breaking the existing loading limit, improving the calculation accuracy and efficiency, and accurately predicting the large deformation and post-buckling behavior of flexible structures. SUMMARY

[0009] The present application aims to overcome at least one of the above-mentioned defects of the prior art, and provides a synchronous bidirectional coupling solving method for large deformation of a flexible slender beam under uniform pressure based on an accurate load stiffness matrix, to realize synchronous interaction analysis and solving of load and configuration, thereby breaking the loading limit caused by numerical instability and greatly improving the stability and accuracy of calculation.

[0010] The present application protects a synchronous bidirectional coupling solving method for large deformation of a flexible slender beam under uniform pressure based on an accurate load stiffness matrix, comprising the following steps:

[0011] S1, establishing a finite element model of a flexible slender beam: using absolute node coordinate Euler-Bernoulli beam elements to discretize the flexible slender beam in space, the current configuration of the Euler-Bernoulli beam element being uniquely determined by the node coordinate vector and the element shape function ;

[0012] S2, constructing a local curve coordinate system of the Euler-Bernoulli beam element centerline: based on the spatial curve differential geometry theory and the finite element model of step S1, a local curve coordinate system of any point on the Euler-Bernoulli beam element centerline is constructed, obtaining the normal vector ;

[0013] S3, constructing the elastic force vector and its tangent stiffness matrix of the beam element: based on the local curve coordinate system established in step S2, the elastic force vector and its corresponding tangent stiffness matrix of the absolute node coordinate beam element are constructed through the continuous medium mechanics theory;

[0014] S4, constructing the uniform pressure vector function: based on the normal vector , the uniform pressure vector function perpendicular to the current configuration of the Euler-Bernoulli beam element centerline is constructed;

[0015] S5, deriving the uniform pressure vector and calculating the element external force vector: based on the virtual work principle, the uniform pressure vector function is differentiated to obtain the uniform pressure vector, and the element external force vector is calculated. Integrating along the element centerline, combining the element shape function , the element external force vector acting on the node of the Euler-Bernoulli beam element is obtained ;

[0016] S6, deriving the load stiffness matrix: taking the derivative of the element external force vector with respect to the node coordinate vector , the load stiffness matrix corresponding to the uniform pressure is obtained , wherein ;

[0017] S7, constructing the overall tangent stiffness matrix: in each Newton-Raphson iteration step of the nonlinear finite element solution, the load stiffness matrix is calculated synchronously with the elastic force vector of the absolute node coordinate Euler-Bernoulli beam element , the tangent stiffness matrix corresponding to the elastic force vector is obtained , and the overall tangent stiffness matrix of the system is assembled , wherein ;

[0018] S8, synchronous bidirectional coupling solution: based on the overall tangent stiffness matrix and the residual force vector , the Newton-Raphson iteration method is used to solve the increment of the node coordinate vector , and the node coordinate vector is updated , realizing the synchronous bidirectional coupling solution of load and configuration until the convergence criterion is met.

[0019] The present application protects the core process of the synchronous bidirectional coupling solution method of flexible slender beam large deformation under uniform pressure based on accurate load stiffness matrix, covering eight key steps of finite element model establishment, local curve coordinate system construction, elastic force vector and tangent stiffness matrix construction, uniform pressure vector function construction, element external force vector derivation, load stiffness matrix derivation, overall tangent stiffness matrix construction and synchronous bidirectional coupling solution, forming a complete and closed-loop technical system. The core technical feature is to integrate the accurate derivation of the load stiffness matrix into each Newton-Raphson iteration step, realizing the synchronous interaction of load and configuration. The present application completely abandons the defects of the traditional one-way sequential asynchronous coupling method which fixes the load direction first and then solves the configuration, and breaks through the numerical loading limit of the traditional method by completely including the additional stiffness contribution of the load direction changing with the configuration. At the same time, the load related parameters are accurately calculated based on the current configuration in each iteration step, avoiding the error accumulation caused by recursive approximation, significantly improving the solution accuracy, and without the need to divide a large number of increment steps, ensuring the accuracy while considering the calculation efficiency, providing reliable support for the accurate mechanical analysis of flexible slender beam structure.

[0020] Further, in the step S1, the absolute nodal coordinate Euler-Bernoulli beam element is a three-dimensional absolute nodal coordinate Euler-Bernoulli beam element, when a plane problem is processed, perpendicular to X-Y a plane.

[0021] The application expands the adaptability of the method to complex spatial deformation through the setting of the three-dimensional element, and realizes the flexible coverage of three-dimensional and plane analysis scenes through the targeted processing of plane problems. The universality and engineering applicability of the method are greatly improved, which can not only cope with the complex bending and torsional coupling deformation analysis of three-dimensional structures such as flexible mechanical arms in ocean engineering, but also efficiently complete the mechanical response calculation of simple plane beam structures through the simplified processing of plane problems. The characteristics of the three-dimensional element guarantee the accuracy of the spatial deformation description, and the limitation of the plane scene simplifies the calculation process, realizing the balance between precision and efficiency in different scenes.

[0022] In some embodiments of the application, in the step S1, each of the absolute nodal coordinate Euler-Bernoulli beam elements contains 2 nodes, each node has 7 degrees of freedom, the first to third degrees of freedom represent spatial position coordinates, the fourth degree of freedom represents a torsion angle around the center line of the beam, and the fifth to seventh degrees of freedom represent gradient coordinates of the node.

[0023] The number of nodes and the degrees of freedom of the absolute nodal coordinate Euler-Bernoulli beam element are limited, which clearly indicates that the element contains 2 nodes, each node has 7 degrees of freedom, and the degrees of freedom cover spatial position coordinates, gradient coordinates and torsion angle around the center line of the beam. This technology adapts to modeling requirements of different degrees of refinement through flexible node number configuration, and the comprehensive degree of freedom design ensures complete capture of beam structure deformation information. Therefore, the flexibility and adaptability of the method are enhanced, and for simple beam structures, fewer nodes can be used for simplified modeling to improve calculation efficiency; for complex structures, the number of nodes can be increased to improve modeling accuracy, and the comprehensive degree of freedom design can completely capture the axial, bending and torsional deformation coupling effects of the beam, avoiding incomplete deformation description caused by missing degrees of freedom, and providing comprehensive and accurate deformation information support for subsequent mechanical parameter derivation and coupling solution.

[0024] Further, in the step S2, the local curve coordinate system is composed of a tangent vector , a normal vector and a binormal vector The application is based on the spatial curve differential geometry theory, and a local reference system closely matched with the current configuration of the beam element center line is constructed by three orthogonal vectors, and the components of the coordinate system are clearly defined. The rigidity and accuracy of the local curve coordinate system are ensured, and the three orthogonal vectors can accurately reflect the spatial geometric posture of the beam center line, the normal vector provides a direct direction basis for the construction of the uniform pressure vector function, and the tangent vector and the binormal vector provide a reliable geometric reference for the derivation of the elastic force vector and other mechanical parameters, so that all subsequent mechanical calculations are based on the local coordinate system matched with the current configuration of the beam, and the accuracy and rationality of the mechanical calculation are significantly improved.

[0025] Further, in step S4, the uniform pressure vector function is , wherein p is a pressure scalar value.

[0026] Further, in step S5, the element external force vector The specific calculation process is as follows: by the virtual work principle, the shape function matrix is used to integrate the uniform pressure vector function along the element length , that is, .

[0027] Further, in step S6, the derivation process of the load stiffness matrix uses the differential rule of the vector value function, and specifically, the derivative of the normal vector with respect to the node coordinate vector is calculated.

[0028] Preferably, the load stiffness matrix is an asymmetric matrix. The mechanical nature of the non-conservative force can be accurately reflected, the structure of the asymmetric matrix avoids the distortion of the stiffness characteristics caused by the approximate processing of the symmetric matrix in the traditional method, so that the overall tangent stiffness matrix can completely and truly reflect the mechanical characteristics of the system, the iterative solution based on the asymmetric matrix can obtain a more accurate search direction, effectively avoids the iteration divergence problem caused by the matrix approximation, and greatly improves the stability and precision of the nonlinear solution process.

[0029] Further, in steps S7 and S8, each iteration step of the Newton-Raphson iteration method includes the calculation of the load stiffness matrix The re-calculation, assembly and solution of the tangent stiffness matrix. The application guarantees the realization of synchronous bidirectional coupling from the process, and the dynamic updating of the load stiffness matrix in each iteration step, so that the overall tangent stiffness matrix can reflect the elastic stiffness and load additional stiffness under the current configuration in real time, avoiding the iteration error caused by the updating of the stiffness matrix lagging behind the configuration change in the traditional method, ensuring the real-time interaction of the load and the configuration in the iteration process, and significantly improving the stability and convergence speed of the iteration solution, so that the method can stably handle large deformation and even post-buckling and other complex nonlinear problems.

[0030] The application also includes the application of the flexible slender beam large deformation synchronous bidirectional coupling solving method based on the accurate load stiffness matrix under uniform pressure, including the application of analyzing the large deformation and even post-buckling behavior of the flexible slender beam under the action of uniform pressure. The application clearly defines the core application field and targeted use of the method, highlights the engineering value of the method, and the large deformation and post-buckling behavior analysis of the flexible slender beam is a key technical problem in the field of ocean engineering, mechanical engineering and the like, which is difficult to accurately solve by the traditional method. The method can provide reliable calculation results for such analysis through accurate coupling solution, provide strong support for the design, optimization and safety evaluation of flexible structures, help engineers accurately predict the mechanical response of the structure under extreme working conditions, reduce the engineering risk caused by structural failure, and has important engineering practical significance.

[0031] Compared with the prior art, the application has the following beneficial effects:

[0032] (1) The numerical loading limit is eliminated: the method can stably solve a maximum uniform pressure working condition far exceeding the critical value of the traditional method, which fundamentally breaks through the numerical loading upper limit due to the neglect of the load stiffness matrix, so that the accurate simulation of complex mechanical behaviors such as large deformation and even post-buckling of the structure is no longer restricted by numerical stability.

[0033] (2) The solution accuracy is essentially improved: based on strict mathematical derivation, the method accurately calculates the load direction based on the current configuration at each iteration step, which fundamentally avoids the error accumulation caused by recursive approximation in the traditional "unidirectional sequential asynchronous coupling" method, thereby significantly improving the accuracy of the large deformation solution.

[0034] (3) The calculation stability is fundamentally improved: since the load stiffness matrix derived accurately is incorporated into the overall tangent stiffness matrix at each iteration step, the stiffness contribution of non-conservative forces is fully considered, so that the numerical stability of the nonlinear Newton-Raphson iteration process is fundamentally improved, and the divergence problem caused by the inaccuracy of the tangent stiffness matrix in the traditional method is effectively avoided. BRIEF DESCRIPTION OF DRAWINGS

[0035] Figure 1 It is a schematic diagram of the flexible slender beam under the action of uniform pressure.

[0036] Figure 2 The overall flow chart of the synchronous bidirectional coupling solution method of the embodiment of the present application.

[0037] Figure 3 The flow chart of the unidirectional sequential asynchronous coupling solution method.

[0038] Figure 4 The schematic diagram of the absolute node Euler-Bernoulli beam element model used in the embodiment of the present application.

[0039] Figure 5 The dimensionless displacement-load factor balance path of the traditional method.

[0040] Figure 6 The dimensionless displacement-load factor balance path of the embodiment of the present application.

[0041] Figure 7 The accuracy comparison of the X-direction displacement results obtained by the traditional method and the present application.

[0042] Figure 8 The accuracy comparison of the Y-direction displacement results obtained by the traditional method and the present application. DETAILED DESCRIPTION

[0043] The accompanying drawings of the embodiments, the technical solutions in the embodiments of the present application are described in more detail. In the drawings, the same or similar notations represent the same or similar elements or elements with the same or similar functions throughout. The described embodiments are part of the embodiments of the present application, not all of the embodiments. The embodiments described below by referring to the drawings are exemplary, and are intended to explain the present application, and cannot be understood as a limitation of the present application. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor are within the scope of protection of the present application. The embodiments of the present application are described in detail below in conjunction with the drawings.

[0044] It should be noted that if the present application embodiments involve directional indications (such as up, down, left, right, front, back, etc.), the directional indications are only used to explain the relative position relationship, motion condition, etc. between the components in a certain posture (as shown in the drawings), if the certain posture changes, the directional indications also change accordingly.

[0045] In addition, if the description of "first", "second" and the like is involved in the embodiments of the present application, the description of "first", "second" and the like is only for the purpose of description, and cannot be understood as indicating or implying the relative importance of the indicated technical features or implicitly indicating the number of the indicated technical features. Therefore, the features limited by "first", "second" can be explicitly or implicitly included at least one of the features. In addition, the technical solutions of various embodiments can be combined with each other, but it must be based on the realization of ordinary skilled in the art, when the combination of technical solutions appears contradictory or cannot be realized, it should be considered that the combination of technical solutions does not exist, also not within the protection scope required by the present application.

[0046] Embodiments

[0047] 1. Establish the finite element model of flexible slender beam:

[0048] As shown in Figure 1 , the elastic modulus of cantilever beam ; cross-sectional area ; section moment of inertia . Length . The full length of the cantilever beam is subjected to uniform pressure P. Point A is the fixed point, and point B is the free point.

[0049] The cantilever beam is discretized by using 10 absolute node coordinate Euler-Bernoulli beam elements, each element has 2 nodes, and each node has 7 degrees of freedom (3 spatial position coordinates, 3 gradient coordinates and 1 torsion angle around the center line of the beam). The current configuration of the element is uniquely determined by the node coordinate vector and the element shape function interpolation.

[0050] 2. Key vector and matrix derivation

[0051] 2.1 Building local curve coordinate system of Euler-Bernoulli beam element center line and building elastic force vector of beam element and its tangent stiffness matrix

[0052] Based on the theory of spatial curve differential geometry, the local curve coordinate system of the center line of the beam element at any point is constructed, as shown in Figure 4 . The normal vector of the point is obtained . Based on this coordinate system, the elastic force vector of the absolute node coordinate beam element and its corresponding tangent stiffness matrix are derived by the theory of continuous medium mechanics.

[0053] 2.2 Construction of uniform pressure vector function and derivation of uniform pressure vector and calculation of element external force vector

[0054] The uniform pressure acting on the beam element is a vector, and its direction is always perpendicular to the current configuration of the beam center line. According to this, the pressure vector function can be expressed as .

[0055] The element external force vector equivalent to the unit node is obtained by the following integration:

[0056]

[0057] where, is the shape function matrix of the element, is the length of the beam element. The integration is realized by Gauss numerical integration.

[0058] 2.3 Derivation of the exact asymmetric load stiffness matrix

[0059] The load stiffness matrix is the derivative of the element external force vector with respect to the node coordinate vector :

[0060]

[0061] The calculation of this derivative involves the derivation of the normal vector with respect to . The final is an asymmetric matrix that exactly describes the stiffness effect of the pressure direction change with the configuration.

[0062] 3. Simultaneous bidirectional coupling solution

[0063] Referring to the complete flow shown in Figure 2 , the implementation process includes an outer loop (load control) and an inner loop (Newton iteration).

[0064] 3.1 Outer loop (load control): starting from the initial load ( ), the load factor ( ) is updated step by step until the target load is reached. Among them, the relationship between is .

[0065] 3.2 Inner loop (Newton iteration): at each load level, iteration is performed until convergence:

[0066] (1) Simultaneously calculate the force and the residual: according to the current configuration and the load level , calculate the elastic force , , and assemble the residual vector .

[0067] (2) Simultaneously calculate and assemble the stiffness matrix: simultaneously calculate the tangent stiffness matrix and load stiffness matrix , and assemble the global tangent stiffness matrix .

[0068] (3) Solution and update: solve the linear equations get the increment , and update the node coordinate vector: .

[0069] Iterate until the residual norm satisfies.

[0070] This process realizes the synchronous bidirectional coupling solution of load and configuration by considering the load stiffness matrix in each iteration step, the calculation is extremely stable, and there is no numerical loading limit.

[0071] Effect verification

[0072] To quantitatively verify the superiority of the present application, the same flexible slender beam model is solved by the traditional unidirectional sequential asynchronous coupling method and the synchronous bidirectional coupling method of the present application respectively. The comparison of the calculation results is shown in Figures 5-8 .

[0073] Figures 5-6 (Non-dimensional displacement-load factor balance path comparison) shows that the method of the present application has significant advantages in calculation stability and loading capacity. The traditional method has a loading limit of about 0.1, and the relative error with the analytical solution is about 10 % when the load factor is less than 0.1. When the load factor exceeds this limit, the solution fails and no convergent solution can be obtained. While the method of the present application successfully breaks through this limit and remains stable convergence in the entire test range (0 ), and obtains a complete structural balance path, proving its ability to handle large deformation problems.

[0074] Figures 7-8 The comparison of the solution accuracy further reveals the fundamental improvement of the present application in calculation accuracy and robustness. The relative error of the traditional method with the analytical solution fluctuates dramatically (the order of magnitude is as high as 1) with the calculation process, and only when the increment step number falls into a certain narrow window, the relative error with the analytical solution can be less than 1%, showing extreme sensitivity to calculation parameters. In contrast, the relative error of the method of the present application with the analytical solution is always stable at a very low level (10 -7 order of magnitude), and is not sensitive to the increment step number, fundamentally overcoming the error accumulation problem caused by recursive approximation in the traditional method, and realizing the order of magnitude improvement in accuracy.

[0075] In summary, combined with Figure 3 and Figures 5-8 , it is shown that the present application realizes the essential transcendence of the traditional method in calculation stability, loading limit and solution accuracy by introducing the load stiffness matrix and the synchronous bidirectional coupling solution strategy.

[0076] The above embodiments are only used to illustrate the technical solutions of the present application but not to limit the present application. Although the present application is explained in detail with reference to the above preferred embodiments, those skilled in the art should understand that the technical solutions of the present application can be modified or equivalently replaced without departing from the spirit and scope of the present application. Those skilled in the art can also make other changes and the like in the design of the present application within the spirit of the present application, as long as the technical effects of the present application are not deviated. The changes made according to the spirit of the present application should be included in the scope of the present application.

Claims

1. A synchronous two-way coupled solution method for large deformation of flexible slender beams under uniformly distributed pressure based on an accurate load stiffness matrix, characterized in that, Includes the following steps: S1. Establish the finite element model of the flexible slender beam: The flexible slender beam is spatially discretized using Euler-Bernoulli beam elements with absolute nodal coordinates. The current configuration of the Euler-Bernoulli beam elements is determined by the nodal coordinate vectors. and unit shape function The interpolation is uniquely determined, where, The coordinates are the unit arc length coordinates; S2. Constructing the local curvilinear coordinate system of the Euler-Bernoulli beam element centerline: Based on the theory of spatial curve differential geometry and combined with the finite element model described in step S1, construct the local curvilinear coordinate system at any point on the centerline of the Euler-Bernoulli beam element to obtain the normal vector. ; S3: Constructing the elastic force vector and tangent stiffness matrix of the beam element: Based on the local curvilinear coordinate system established in step S2, the elastic force vector of the absolute nodal coordinate beam element is constructed using the theory of continuum mechanics. and its corresponding tangent stiffness matrix ; S4: Construction of the uniformly distributed pressure vector function: based on the normal vector Construct a uniform pressure vector function perpendicular to the centerline of the Euler-Bernoulli beam element in the current configuration; S5. Derive the uniformly distributed pressure vector and calculate the element external force vector: Based on the principle of virtual work, derive the uniformly distributed pressure vector function... Integrating along the element centerline, combined with the element shape function The element external force vectors acting on the nodes of the Euler-Bernoulli beam element are obtained. ; S6. Derive the load stiffness matrix: For the external force vector of the element... Regarding the node coordinate vector By taking the derivative, we obtain the load stiffness matrix corresponding to the uniformly distributed pressure. ,in , The symbol for partial derivative calculations; S7. Construct the overall tangent stiffness matrix: In each Newton-Raphson iteration step of the nonlinear finite element solution, the load stiffness matrix is ​​calculated synchronously. The elastic force vector of the Euler-Bernoulli beam element with absolute nodal coordinates The corresponding tangent stiffness matrix And assembled into the overall tangent stiffness matrix of the system. ,in ; S8. Synchronous bidirectional coupling solution: Based on the global tangent stiffness matrix and residual force vector The node coordinate vector is solved using the Newton-Raphson iterative method. Increment and update the node coordinate vector. This enables synchronous bidirectional coupling of load and configuration for solution until the convergence criterion is met.

2. The synchronous two-way coupled solution method for large deformation of flexible slender beams under uniformly distributed pressure based on an accurate load stiffness matrix, as described in claim 1, is characterized in that... In step S1, the absolute nodal coordinate Euler-Bernoulli beam element is a three-dimensional absolute nodal coordinate Euler-Bernoulli beam element.

3. The synchronous two-way coupled solution method for large deformation of flexible slender beams under uniformly distributed pressure based on an accurate load stiffness matrix, as described in claim 1, is characterized in that... In step S1, each absolute node coordinate Euler-Bernoulli beam element contains 2 nodes, each node has 7 degrees of freedom, the 1st to 3rd degrees of freedom represent the spatial position coordinates, the 4th degree of freedom represents the torsion angle around the beam centerline, and the 5th to 7th degrees of freedom represent the gradient coordinates of the node.

4. The synchronous two-way coupled solution method for large deformation of flexible slender beams under uniformly distributed pressure based on an accurate load stiffness matrix, as described in claim 1, is characterized in that... In step S2, the local curve coordinate system includes tangent vectors. Normal vector and binormal vector When dealing with planar problems, Perpendicular to the XY plane.

5. The synchronous two-way coupled solution method for large deformation of flexible slender beams under uniformly distributed pressure based on an accurate load stiffness matrix, as described in claim 1, is characterized in that... In step S4, the uniformly distributed pressure vector function is: , where p is the pressure scalar value.

6. The synchronous two-way coupled solution method for large deformation of flexible slender beams under uniformly distributed pressure based on an accurate load stiffness matrix, as described in claim 1, is characterized in that... In step S5, the unit external force vector The specific calculation process is as follows: using the principle of virtual work, and utilizing the shape function matrix... For the uniformly distributed pressure vector function Along the element length Integrating, that is , where T is the matrix transpose symbol and d is the differential symbol.

7. The synchronous two-way coupled solution method for large deformation of flexible slender beams under uniformly distributed pressure based on an accurate load stiffness matrix, as described in claim 1, is characterized in that... In step S6, the load stiffness matrix The derivation process employs the differential rule of vector-valued functions, specifically, it involves applying the normal vector... Regarding node coordinate vectors Differentiate it.

8. The synchronous two-way coupled solution method for large deformation of flexible slender beams under uniformly distributed pressure based on an accurate load stiffness matrix, as described in any one of claims 1 to 7, is characterized in that... The load stiffness matrix It is an asymmetric matrix.

9. The synchronous two-way coupled solution method for large deformation of flexible slender beams under uniformly distributed pressure based on an accurate load stiffness matrix, as described in claim 1, is characterized in that... In steps S7 and S8, each iteration of the Newton-Raphson iterative method includes adjusting the load stiffness matrix. The recalculation, assembly, and solution.

10. An application of the synchronous two-way coupled solution method for large deformation of a flexible slender beam under uniformly distributed pressure based on an accurate load stiffness matrix as described in any one of claims 1 to 7 or 9, characterized in that, This includes solving the problem of large deformation and even post-buckling behavior of flexible slender beams under uniformly distributed pressure.

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