Synchronous bidirectional coupling solving method for large deformation of flexible slender beam under uniformly distributed pressure based on precise load stiffness matrix

Abstract

The invention discloses a synchronous bidirectional coupling solving method for large deformation of a flexible slender beam under uniformly distributed pressure based on an accurate load stiffness matrix, and relates to the field of computational mechanics. According to the method, a local curve coordinate system and a uniformly distributed pressure vector function are constructed through an absolute node coordinate Euler-Bernoulli beam unit discrete beam structure, a unit external force vector is derived based on a virtual work principle, and a precise load stiffness matrix is obtained through a vector value function differential rule. The load stiffness matrix and the elastic tangent stiffness matrix are synchronously assembled in the Newton-Rafson iteration step, and synchronous bidirectional coupling solution of the load and the configuration is achieved. According to the method, the numerical value loading limit of a traditional method is broken through, error accumulation is avoided, the calculation stability and precision are improved, the large deformation and even post-buckling behavior of the flexible slender beam under the uniformly distributed pressure can be accurately analyzed, and the method is suitable for design optimization of related equipment in the fields of ocean engineering, mechanical engineering and the like.

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 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: (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.

[0005] (2) The calculation accuracy is restricted by systematic errors: the load direction of the current increment step depends heavily on the approximate configuration of the last step, and this recursive approximation will introduce systematic errors, which will accumulate between the increment steps, resulting in a serious deviation of the final calculation result from the true solution.

[0006] ​(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 too large single-step approximation error, leading to distorted results.

[0007] 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

[0008] 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, so as 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.

[0009] 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: S1, establishing a finite element model of a flexible slender beam: using an absolute node coordinate Euler-Bernoulli beam element to discretize the flexible slender beam in space, the current configuration of the Euler-Bernoulli beam element being uniquely determined by a node coordinate vector and an element shape function ; S2, constructing a local curve coordinate system of the center line of the Euler-Bernoulli beam element: 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 center line of the Euler-Bernoulli beam element is constructed, and a normal vector is obtained; S3, constructing an elastic force vector of the beam element and its tangent stiffness matrix: based on the local curve coordinate system constructed in step S2, an elastic force vector of the absolute node coordinate beam element and its corresponding tangent stiffness matrix are constructed through the continuous medium mechanics theory; S4, constructing a uniform pressure vector function: based on the normal vector , a uniform pressure vector function perpendicular to the current configuration of the center line of the Euler-Bernoulli beam element is constructed; 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 integrated along the center line of the element, and the element shape function is combined to obtain the element external force vector acting on the nodes of the Euler-Bernoulli beam element. ; S6, deriving load stiffness matrix: deriving the load stiffness matrix of the unit external force vector with respect to the node coordinate vector ; 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 simultaneously ; 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 simultaneously ; 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 simultaneously ; 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 simultaneously ; S8, synchronous bidirectional coupling solution: based on the overall tangent stiffness matrix ; S8, synchronous bidirectional coupling solution: based on the overall tangent stiffness matrix ; S8, synchronous bidirectional coupling solution: based on the overall tangent stiffness matrix ;

[0010] 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, which covers 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, unit external force vector derivation, load stiffness matrix derivation, overall tangent stiffness matrix construction and synchronous bidirectional coupling solution, forming a complete and closed technical system. The core technical feature is to integrate the accurate derivation of the load stiffness matrix into each Newton-Raphson iteration step, so as to realize the synchronous interaction of load and configuration. The present application completely abandons the defects of 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 traditional method by completely including the additional stiffness contribution of load direction change with configuration; at the same time, the load related parameters are accurately calculated based on the current configuration in each iteration step, which avoids the error accumulation caused by recursive approximation, significantly improves the solution accuracy, and does not need to divide a large number of incremental steps, which balances the calculation efficiency while ensuring the accuracy, and provides reliable support for accurate mechanical analysis of flexible slender beam structure.

[0011] Further, in step S1, the absolute node coordinate Euler-Bernoulli beam element is a three-dimensional absolute node coordinate Euler-Bernoulli beam element, and when dealing with plane problems, perpendicular to X-YPlane.

[0012] The application expands the adaptability of the method to complex spatial deformation through the setting of three-dimensional units, and realizes flexible coverage of three-dimensional and plane analysis scenes through 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 three-dimensional units guarantee the accuracy of spatial deformation description, and the limitation of plane scenes simplifies the calculation process, realizing the balance between precision and efficiency in different scenes.

[0013] In some embodiments of the application, each absolute nodal coordinate Euler-Bernoulli beam element in step S1 comprises 2 nodes, each node having 7 degrees of freedom, the first to third degrees of freedom representing spatial position coordinates, the fourth degree of freedom representing a torsion angle around the beam center line, and the fifth to seventh degrees of freedom representing gradient coordinates of the node.

[0014] 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 comprises 2 nodes, each node having 7 degrees of freedom, and the degrees of freedom cover spatial position coordinates, gradient coordinates and torsion angle around the beam center line. This technology adapts to modeling requirements of different levels 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. For simple beams, 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 due to missing degrees of freedom, and providing comprehensive and accurate deformation information support for subsequent mechanical parameter derivation and coupled solution.

[0015] Further, in step S2, the local curve coordinate system is composed of tangent vector , normal vector and binormal vector . Based on the theory of spatial curve differential geometry, the local reference system closely matched with the current configuration of the beam element center line is constructed by three orthogonal vectors, which clearly defines the constituent elements of the coordinate system. The rigidity and accuracy of the local curve coordinate system are ensured, and the three orthogonal vectors can accurately reflect the spatial geometric pose 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 binormal vector provide reliable geometric references for the derivation of mechanical parameters such as elastic force vector, so that all subsequent mechanical calculations are based on the local coordinate system matched with the current configuration of the beam, significantly improving the accuracy and rationality of the mechanical calculation.

[0016] Further, in step S4, the uniform pressure vector function is where p is a pressure scalar value.

[0017] Further, in step S5, the unit external force vector The specific calculation process is: through the virtual work principle, using the shape function matrix The uniform pressure vector function is integrated along the unit length , that is .

[0018] Further, in step S6, the derivation process of the load stiffness matrix uses the differential rule of vector value function, specifically, the derivative of the normal vector with respect to the node coordinate vector .

[0019] Preferably, the load stiffness matrix is an asymmetric matrix. It can accurately reflect the mechanical nature of non-conservative forces. The structure of the asymmetric matrix avoids the distortion of stiffness characteristics caused by the approximate processing of symmetric matrices in traditional methods, so that the overall tangent stiffness matrix can fully and truly reflect the mechanical characteristics of the system. Based on the asymmetric matrix, the iterative solution can obtain a more accurate search direction, effectively avoiding the iteration divergence problem caused by matrix approximation, and greatly improving the stability and precision of the nonlinear solving process.

[0020] Further, in steps S7 and S8, each iteration step of the Newton-Raphson iteration method includes the recalculation, assembly and solution of the load stiffness matrix . The present application guarantees the realization of synchronous bidirectional coupling from the process, and the dynamic update of the load stiffness matrix in each iteration step makes the overall tangent stiffness matrix reflect the elastic stiffness and load additional stiffness in the current configuration in real time, avoiding the iteration error caused by the lag of stiffness matrix update in the traditional method. The real-time interaction between the load and the configuration in the iteration process is ensured, which significantly improves the stability and convergence speed of the iterative solution, so that the method can stably handle large deformation and even post-buckling complex nonlinear problems.

[0021] The application also comprises the application of the flexible slender beam large deformation synchronous bidirectional coupling solving method based on the precise load stiffness matrix under uniform pressure, including the application of the large deformation and post-buckling behavior analysis 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 fields of marine engineering, mechanical engineering and the like, which is difficult to accurately solve by traditional methods. The method can provide reliable calculation results for such analysis through accurate coupling solving, 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.

[0022] Compared with the prior art, the application has the following beneficial effects: (1) The application eliminates the numerical loading limit: the method of the application 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 the complex mechanical behavior of the structure such as large deformation and even post-buckling is no longer restricted by the numerical stability.

[0023] (2) The solving 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.

[0024] (3) The calculation stability is fundamentally improved: since the load stiffness matrix accurately derived at each iteration step is synchronized into the overall tangent stiffness matrix, 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

[0025] Figure 1 A schematic view of a flexible slender beam under the action of uniform pressure.

[0026] Figure 2 A general flowchart of the synchronous bidirectional coupling solving method of the embodiment of the application.

[0027] Figure 3 A flowchart of the unidirectional sequential asynchronous coupling solving method.

[0028] Figure 4 A schematic view of the absolute node Euler-Bernoulli beam element model used in the embodiment of the application.

[0029] Figure 5 This is the dimensionless displacement-load factor balance path of the traditional method.

[0030] Figure 6 This is the dimensionless displacement-load factor balance path in an embodiment of the present invention.

[0031] Figure 7 This is a comparison of the accuracy of the X-direction displacement results obtained by the traditional method and the present invention.

[0032] Figure 8 This is a comparison of the accuracy of the Y-direction displacement results obtained by the traditional method and the present invention. Detailed Implementation

[0033] The accompanying drawings illustrate the technical solutions of the embodiments of the present invention in more detail. Throughout the drawings, the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions. The described embodiments are some, but not all, embodiments of the present invention. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention. The embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0034] It should be noted that if the embodiments of this application involve directional indicators (such as up, down, left, right, front, back, etc.), the directional indicators are only used to explain the relative positional relationship and movement of the components in a certain specific posture (as shown in the figure). If the specific posture changes, the directional indicators will also change accordingly.

[0035] Furthermore, if the embodiments of this application involve descriptions such as "first" or "second," these descriptions are for descriptive purposes only and should not be construed as indicating or implying their relative importance or implicitly specifying the number of technical features indicated. Therefore, features defined with "first" or "second" may explicitly or implicitly include at least one of those features. Additionally, the technical solutions of various embodiments can be combined with each other, but this must be based on the ability of those skilled in the art to implement them. If the combination of technical solutions is contradictory or impossible to implement, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection claimed in this application.

[0036] Example

[0037] 1. Establish a finite element model of the flexible slender beam: like Figure 1 As shown, the elastic modulus of the cantilever beam ; cross-sectional area ; cross-sectional moment of inertia . length The cantilever beam is subjected to a uniform pressure P along its full length. Point A is the fixed point and point B is the free point.

[0038] The cantilever beam is discretized by 10 absolute nodal coordinate Euler-Bernoulli beam elements, each of which has 2 nodes and 7 degrees of freedom (3 spatial position coordinates, 3 gradient coordinates and 1 torsion angle about the beam centerline) at each node. The current configuration of the element is uniquely determined by the nodal coordinate vector and the element shape function interpolation.

[0039] 2. Key vector and matrix derivation 2.1 Construction of the local curvilinear coordinate system of the Euler-Bernoulli beam element centerline and construction of the elastic force vector of the beam element and its tangent stiffness matrix Based on the theory of spatial curve differential geometry, the local curvilinear coordinate system of the centerline 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 nodal coordinate beam element and its corresponding tangent stiffness matrix are derived by the theory of continuum mechanics.

[0040] 2.2 Construction of the uniform pressure vector function and derivation of the uniform pressure vector and calculation of the element external force vector The uniform pressure acting on the beam element is a vector whose direction is always perpendicular to the current configuration of the beam centerline. Accordingly, the pressure vector function can be expressed as .

[0041] Through the virtual work principle, the element external force vector equivalent to the nodes of the element is obtained by the following integral: where is the shape function matrix of the element, is the length of the beam element. The integral is realized by Gauss numerical integration.

[0042] 2.3 Derivation of the exact asymmetric load stiffness matrix The load stiffness matrix is the derivative of the element external force vector with respect to the nodal coordinate vector : The calculation of this derivative involves the derivation of the normal vector with respect to . The final obtained The stiffness effect of the change of the pressure direction with the configuration is accurately described for the asymmetric matrix.

[0043] 3. Synchronous bidirectional coupling solution Referring to Figure 2 the complete flow shown, the implementation process includes an outer loop (load control) and an inner loop (Newton iteration).

[0044] 3.1 Outer loop (load control): starting from an initial load (P0) , the load factor (P) is updated step by step until the target load P is reached . Among them, the relationship between P and P is . .

[0045] 3.2 Inner loop (Newton iteration): at each load level, iteration is performed until convergence: (1) Synchronous calculation of force and residual: according to the current configuration and load level , the elastic force , is calculated, and the residual vector is assembled.

[0046] (2) Synchronous calculation and assembly of stiffness matrix: the tangent stiffness matrix corresponding to the beam element elastic force vector and the load stiffness matrix are calculated synchronously, and the overall tangent stiffness matrix is assembled.

[0047] (3) Solution and update: solve the linear equation to get the increment , and update the node coordinate vector: .

[0048] Iterate until the residual norm satisfies.

[0049] This process realizes synchronous bidirectional coupling solution of load and configuration by synchronously considering the load stiffness matrix at each iteration step, the calculation is extremely stable, and there is no numerical loading limit.

[0050] Effect verification To quantitatively verify the superiority of the present application, the same flexible slender beam model is solved by the traditional one-way 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 .

[0051] Figures 5-6(Dimensionless displacement-load factor equilibrium path comparison) shows that the method of the present application has a significant advantage in calculating stability and loading capacity. The traditional method has a loading limit of about , and beyond this limit, the failure is solved, and the convergent solution cannot be obtained. While the method of the present application successfully breaks through this limit, it remains stable and convergent within the entire test range ( ), and obtains a complete structural equilibrium path, proving its ability to handle large deformation problems.

[0052] 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 and the analytical solution fluctuates dramatically with the calculation process (the order of magnitude is as high as 1), and only when the incremental step number falls within a narrow window, the relative error with the analytical solution can be less than 1%, showing extreme sensitivity to the calculation parameters. In contrast, the relative error of the method of the present application and the analytical solution is always stable at a very low level (10 -7 order of magnitude), and is not sensitive to the incremental step number, fundamentally overcoming the error accumulation problem caused by the recursive approximation of the traditional method, and achieving an order of magnitude improvement in accuracy.

[0053] In summary, combined with Figure 3 and Figures 5-8 , it shows that the present application, by introducing the load stiffness matrix and the synchronous bidirectional coupling solving strategy, has fundamentally surpassed the traditional method in terms of calculation stability, loading limit and solution accuracy.

[0054] The above embodiments are only used to illustrate the technical solutions of the present application and not to limit it. Although the present application has been described in detail with reference to the above preferred embodiments, those skilled in the art should understand that modifications or equivalent replacements of the technical solutions of the present application should not deviate from the spirit and scope of the present application. Those skilled in the art can make other changes within the spirit of the present application and use them in the design of the present application, as long as they do not deviate from the technical effects of the present application. These changes made in accordance with the spirit of the present application should be included in the scope of protection claimed by the present application.

Claims

1. A method for solving large deformation of flexible slender beam under uniform pressure based on synchronous bidirectional coupling of precise load stiffness matrix, characterized in that, The method comprises the following steps: S1, establishing a finite element model of a flexible slender beam: discretizing the flexible slender beam in space by using absolute node coordinate Euler-Bernoulli beam elements, the current configuration of the Euler-Bernoulli beam element being uniquely determined by the node coordinate vector and the element shape function interpolated, wherein, is an element arc length coordinate; S2, constructing a local curve coordinate system of a center line of an Euler-Bernoulli beam element: based on spatial curve differential geometry theory, in combination with the finite element model described in step S1, a local curve coordinate system at any point of the center line of the Euler-Bernoulli beam element is constructed, and a normal vector is obtained ; S3: constructing the elastic force vector of the beam unit and its tangent stiffness matrix: based on the local curvilinear coordinate system established in step S2, the elastic force vector of the absolute node coordinate beam unit is constructed by the continuous medium mechanics theory and the tangent stiffness matrix corresponding thereto ; S4: Construction of the uniform pressure vector function: based on the normal vector , a uniform pressure vector function is constructed which is perpendicular to the current configuration of the center line of the Euler-Bernoulli beam element; S5, deriving uniform pressure vector and calculating element external force vector: based on the virtual work principle, the uniform pressure vector function Integrating along the element center line, combining the element shape function , the element external force vector acting on the Euler-Bernoulli beam element node ; S6, Deriving the load stiffness matrix: Differentiating the element force vector with respect to the node coordinate vector is the partial derivative symbol;​​​ 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 simultaneously the elastic force vector of the absolute node coordinate euler-bernoulli beam element the corresponding tangent stiffness matrix and assembled into the overall tangent stiffness matrix of the system wherein ; S8, synchronously bidirectionally coupling solving: based on the whole tangent stiffness matrix and residual force vector , solving the node coordinate vector increment by Newton-Raphson iteration method , and updating the node coordinate vector , realizing the synchronous bidirectionally coupling solving of load and configuration until meeting the convergence criterion.

2. The method of claim 1, wherein, In the step S1, the absolute nodal coordinate Euler-Bernoulli beam element is a three-dimensional absolute nodal coordinate Euler-Bernoulli beam element.

3. The method of claim 1, wherein, In the step S1, each absolute nodal coordinate Euler-Bernoulli beam element comprises two nodes, each node has seven degrees of freedom, the first to third degrees of freedom represent spatial position coordinates, the fourth degree of freedom represents a torsion angle around a beam center line, and the fifth to seventh degrees of freedom represent gradient coordinates of the node.

4. The method of claim 1, wherein, In step S2, the local curvilinear coordinate system comprises a tangent vector , a normal vector and a binormal vector , which, in the case of a planar problem, are perpendicular to the X-Y plane.

5. The method of claim 1, wherein, In step S4, the uniform pressure vector function is where p is the pressure scalar value.

6. The method of claim 1, wherein, In step S5, the unit external force vector is calculated as follows: by the principle of virtual work, using the shape function matrix to integrate the uniform pressure vector function along the unit length , i.e. where T is the matrix transpose symbol and d is the differential symbol.

7. The method of claim 1, wherein, In step S6, the load stiffness matrix is derived using the differentiation rule for vector-valued functions, specifically the derivation of the normal vector with respect to the node coordinate vector .

8. The synchronous bidirectional coupling solution of large deformation of flexible slender beam under uniform pressure based on precise load stiffness matrix according to any one of claims 1-7, characterized in that, The load stiffness matrix is a non-symmetric matrix.

9. The method of claim 1, wherein, In steps S7 and S8, each iteration step of the Newton-Raphson iteration comprises a recalculation, assembly and solution of the load stiffness matrix K.

10. The application of the method for solving the large deformation of the flexible slender beam under the uniform pressure by the synchronous bidirectional coupling based on the precise load stiffness matrix according to any one of claims 1-7 or 9, characterized in that, The method comprises solving large deformation and even post-buckling behavior of a flexible slender beam under the action of uniform pressure.

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