Posterior analysis method for three-dimensional initial geometric defects of aircraft skin panel

Abstract

The invention provides a posterior analysis method for three-dimensional initial geometric defects of an aircraft skin panel, which comprises the following steps of: establishing a three-dimensional finite element model of a skin panel structure based on a solid shell unit, and constructing a proxy equilibrium equation of nonlinear mechanical response analysis of the structure by adopting the three-dimensional initial geometric defects of the skin panel structure; the invention provides a posteriori analysis method for three-dimensional initial geometric defects of an aircraft skin panel. According to the method, only a proxy balance equation needs to be established for a perfect structure without defects at a time, for different initial geometric defects, only additional items caused by the defects in the proxy balance equation need to be modified, and then the minimum-scale proxy balance equation is recalculated.

Description

Technical Field

[0001] This invention belongs to the field of structural mechanics modeling and analysis technology, specifically relating to a method for post-hoc analysis of three-dimensional initial geometric defects in aircraft skin panels. Background Technology

[0002] Initial geometric defects are inevitably introduced into aircraft skin panels during manufacturing and assembly, especially during assembly under multi-factor coupling constraints. These defects can easily lead to deviations from the ideal configuration, significantly reducing the structure's ultimate load-bearing capacity. Therefore, the impact of initial geometric defects must be considered when analyzing the load-bearing performance of aircraft skin panels. A classic approach to incorporating initial geometric defects into the calculation of the structure's nonlinear mechanical response is to establish a finite element model of the structure with defects and perform nonlinear analysis using conventional finite element methods to obtain the load-displacement response curve. The load value corresponding to the extreme point of this curve represents the structure's ultimate load-bearing capacity. However, this approach not only involves a large computational burden for a single defect analysis but also requires rebuilding the finite element model for different initial geometric defects and iteratively solving large-scale nonlinear equilibrium equations. This approach is essentially a defect-prior method, resulting in high computational costs and low analysis efficiency.

[0003] The applicant's prior invention patent (a fast reanalysis method for the load-bearing response of thin-walled structures with geometric defects, patent number ZL201911010489.X) and paper (Computers & Mathematics with Applications, 2020, 79(12): 3429-3446.) propose a fast reanalysis method for the load-bearing response of thin-walled structures with geometric defects, which can significantly improve computational efficiency. However, through further research and practical application, the applicant found the following three shortcomings in the proposed solution: 1) The above work all uses two-dimensional plate and shell elements, which cannot simulate the stress and deformation of aircraft skin panels along the thickness direction, nor can it consider the three-dimensional initial geometric defects of the structure; 2) The above work is based on the simplified assumption of decoupling geometric defects from structural deformation, and approximately converts the geometric defect influence terms to the right end (i.e., the load end) of the reduced-order structural model, which differs from the actual situation and introduces additional errors; 3) The above work provides two calculation methods for the defect influence terms for initial geometric defects of different sizes, which require designers to make a judgment (whether the defect size exceeds 20% of the wall thickness) before selecting the appropriate method. However, the defect shape may be extremely complex, and inaccurate judgment will affect the accuracy of the analysis. Furthermore, this approach is not suitable for panels with variable thickness. Summary of the Invention

[0004] To overcome the low efficiency of existing technologies in reanalyzing three-dimensional initial geometric defects, this invention establishes a three-dimensional finite element model of a skin panel structure based on solid shell elements. Using the three-dimensional initial geometric defects of the skin panel structure, a surrogate equilibrium equation for structural nonlinear mechanical response analysis is constructed, and a posterior analysis method for three-dimensional initial geometric defects in aircraft skin panels is proposed. This method only requires establishing the surrogate equilibrium equation once for a perfectly defect-free structure. For different initial geometric defects, only the additional terms caused by the defects in the surrogate equilibrium equation need to be modified, and then the surrogate equilibrium equation on a very small scale needs to be recalculated.

[0005] The technical solution of this invention is as follows:

[0006] A method for post-hoc analysis of three-dimensional initial geometric defects in aircraft skin panels includes the following steps:

[0007] Step 1: Obtain the initial three-dimensional geometric defects of the aircraft skin panels and construct the defect displacement field for structural finite element analysis. ;

[0008] Step 2: Construct hybrid solid shell elements. Use hybrid solid shell elements to construct a three-dimensional finite element model of the defect-free skin panel and establish the nonlinear equilibrium equations of the skin panel structure.

[0009] Step 3: Construct a surrogate equilibrium equation for the nonlinear equilibrium equation of the skin panel structure, and embed the influence of defects into the surrogate equilibrium equation through a posterior method to obtain the surrogate equilibrium equation for the skin panel with initial geometric defects. By solving the surrogate equilibrium equation for the skin panel with initial geometric defects, the nonlinear response curve of the skin panel with initial geometric defects is obtained, thus realizing the posterior analysis of the three-dimensional initial geometric defects of the aircraft skin panel.

[0010] In a further preferred embodiment, the surrogate equilibrium equations of the skin panel with initial geometric defects include first-order, second-order, and third-order surrogate expansions of the defect-free skin panel with respect to the generalized displacement, as well as additional surrogate expansions of the generalized displacement after the introduction of defects.

[0011] In a further optimized approach, for skin panels with different initial geometric defects, during posterior analysis, step 1 is first repeated to establish the defect displacement field of the structure. Then, in steps 2 and 3, we only need to solve the additional surrogate expansion terms of the generalized displacement after introducing the new initial geometric defects, and combine them with the first, second and third order surrogate expansion terms of the generalized displacement of the defect-free skin panel that have been calculated, to obtain and solve the surrogate equilibrium equation of the skin panel with the new initial geometric defects, and then obtain the nonlinear response curve of the skin panel with the new initial geometric defects.

[0012] Beneficial effects

[0013] This invention employs hybrid solid shell elements to construct a three-dimensional finite element model of a skinned panel structure. Compared to traditional two-dimensional plate and shell elements, each node of a solid shell element has only three translational degrees of freedom, eliminating the need to handle non-vector rotational degrees of freedom and simplifying the calculation process. Furthermore, solid shell elements can directly apply three-dimensional material constitutive relations without degenerating into a planar form, thus avoiding the introduction of additional assumptions. Most importantly, three-dimensional initial geometric defects of the structure can be directly introduced into the three-dimensional finite element model. When establishing the surrogate equilibrium equations for a defective structure, this invention directly introduces the action term of the initial geometric defects into the left-hand side (deformation term) of the surrogate equilibrium equations, coupling it with structural deformation, which is more consistent with reality. Moreover, this invention is applicable to initial geometric defects of different sizes when conducting post-hoc recalculation of defects, eliminating the need for manual classification and judgment of defect sizes before calculation.

[0014] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0015] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0016] Figure 1 This is a schematic diagram of the steps of the present invention;

[0017] Figure 2 Schematic diagram of solid shell unit and schematic diagram of three-dimensional finite element model of skin panel;

[0018] Figure 3 A skin panel (curved panel) that bears a uniform compressive load in the plane.

[0019] Figure 4 This is a schematic diagram of the initial geometric defect shape of the skin panel;

[0020] Figure 5 The nonlinear mechanical response curves of the skin panel are obtained by conventional methods and the method of this patent based on three different initial geometric defects. Detailed Implementation

[0021] The embodiments of the present invention are described in detail below. These embodiments are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0022] This embodiment is derived from a common structural type of aircraft skin panels, namely curved panel structure, such as... Figure 3As shown. Two straight edges of the curved panel constrain displacement in the v and w directions, one curved edge constrains displacement in the w direction and applies a uniform in-plane compressive load, and the other curved edge is fixed (constraining displacement in the u, v, and w directions). The straight edge length of the skin panel is L = 150 mm, the curved edge radius is R = 80 mm, the central angle is θ = 90°, and the panel thickness is t = 1.68 mm.

[0023] This invention provides a method for post-hoc analysis of three-dimensional initial geometric defects in aircraft skin panels, with reference to... Figure 1 It includes the following steps:

[0024] Step 1: Numerical simulation of the initial three-dimensional geometric defects.

[0025] The initial three-dimensional geometric defects of aircraft skin panels can be obtained in various ways, such as directly selecting the buckling deformation modes of the structure or using measured geometric defects of the structure. Regardless of the method, it is necessary to extract detailed geometric parameter information of the defects, interpolate and discretize the geometric parameters of the defects according to the nodal coordinates of the elements, and transform them into nodal displacements of the three-dimensional finite element model of the skin panel structure, thereby forming the defect displacement field for structural finite element analysis. .

[0026] In this embodiment, the first-order buckling mode shape of the curved wall panel structure is adopted, such as... Figure 4 As shown, the initial geometric defects are simulated. Detailed geometric parameters of the defects are extracted and transformed into nodal displacements in the three-dimensional finite element model of the skin panel structure, thus forming the defect displacement field of the structure. The partial data of the defect displacement field are shown in Table 1.

[0027] Table 1. Partial data of the defect displacement field

[0028]

[0029] Step 2: Hybrid solid shell elements are constructed using the hybrid stress formula and the assumed natural strain method (ANS method) to overcome the element self-locking problem of solid shell elements in finite element analysis of thin-walled structures (wall thickness much smaller than the in-plane span of the wall panel), ensuring computational accuracy. Then, a three-dimensional finite element model of a perfect, defect-free skinned wall panel is constructed using hybrid solid shell elements, such as... Figure 2 As shown, the nonlinear equilibrium equations for the skin panel structure are established. The specific process is as follows:

[0030] Stress formula for hybrid solid shell unit:

[0031] (1)

[0032] in For element stress vectors, Here is the stress interpolation matrix. These are stress parameters.

[0033] The ANS method was used to calculate the transverse (thickness direction) of the hybrid solid shell unit. Figure 2 The three strain components of )

[0034] (2)

[0035] in, The transverse normal strain at a point in the element. and The transverse shear strain at a point in the element. , , and The element shape function that includes the coordinates of a point within the element. and These are two coordinates in the unit natural coordinate system. Represents coordinates in the natural coordinate system of the unit The transverse normal strain at the location is given, where the origin of the element's natural coordinate system is established at the geometric center of the element, and the coordinates in the element's natural coordinate system are expressed as follows: .

[0036] The three transverse strain components (2) of the element are combined with the in-plane strain components to form the element strain vector E, and the defect displacement field is also combined with the strain vector E. By introducing the element strain expression in a "stress-free" manner, we can obtain:

[0037] (3)

[0038] in, The geometric matrix of the linear strain of the element is obtained from the three transverse strain components and the in-plane strain components. The specific method used is conventional in this field. The geometric matrix of the nonlinear strain of the element is obtained from the three transverse strain components and the in-plane strain components. The specific method used is conventional in this field. For the nodal displacement field of the element, An additional matrix is ​​added to the element strain caused by the introduction of defects.

[0039] Based on the stress (1) and strain (3) calculation formulas of the element, the Hellinger-Reissner energy functional of the three-dimensional hybrid solid shell element is established as follows:

[0040] (4)

[0041] The Hellinger-Reissner energy functional expression is a standard formula in this field; where, Unit energy, For unit volume, For the element constitutive matrix, The structural load factor is... For reference external load, , , All are intermediate matrices

[0042] (5)

[0043] in, For the element constitutive matrix, The structural load factor is... For reference external load.

[0044] Applying the variational principle to the energy functional (4) yields the nonlinear equilibrium equations for the unit:

[0045] (6)

[0046] Polycondense stress parameters at the unit level Then, using the classic element assembly technique in finite element analysis (the overall structure is composed of element subdivisions), the nonlinear equilibrium equations of the skin panel structure are obtained:

[0047] (7)

[0048] Step 3: Construct a proxy equilibrium equation for the nonlinear equilibrium equation (7) of the skin panel structure, and embed the influence of defects into the proxy equilibrium equation through a post-hoc method.

[0049] The nonlinear equilibrium equation (7) for the skin panel with respect to the displacement increment field Perform a third-order Taylor expansion:

[0050] (8)

[0051] in , and These are the first, second, and third order expansions of the Taylor series of a perfect, defect-free structure, respectively. To introduce defects and the resulting Related additional first-order expansion terms, To introduce defects and the resulting Related additional first-order expansion terms, To introduce defects and the resulting Related additional second-order expansion terms, and These represent the increments of the structural displacement field and the load coefficients, respectively.

[0052] For skin panels with different initial geometric defects, only the additional unfolding terms caused by the defects need to be updated. , and The expansion term of a perfect, flawless structure , and The calculation only needs to be performed once and stored for repeated use. Therefore, the finite element model of a perfect, defect-free skin panel only needs to be built once, and only additional expansion terms need to be updated for different defects. Hence, this method is called the defect post-hoc analysis method.

[0053] Next, we will explain how to establish a proxy equilibrium equation for the nonlinear equilibrium equation (7) of the skin panel, which significantly reduces the computational scale of the nonlinear equilibrium equation (7) and thus enables rapid post-hoc defect analysis.

[0054] Based on the perturbation loads corresponding to each mode and reference external load The right-hand side of the third-order expansion of the nonlinear equilibrium equation (8) for the skin panel can be expressed as:

[0055] (9)

[0056] in, This is a load factor vector, where the first component is the increment of the load factor. The remaining components are 0; This is the load matrix, with the first column representing the reference external load. The rest are listed as perturbation loads, among which For the structure The perturbation load corresponding to the first mode. Perturbation load The structural eigenvalues ​​of the skin panel are determined by solving the structural eigenvalue equations.

[0057] Structural characteristic equation of skin panel:

[0058] (10)

[0059] in, Let be the tangent stiffness matrix of the skin panel. Here is the geometric stiffness matrix of the skin panel. For the structure eigenvalues ​​of order 1 For the structure Eigenvalues ​​and modes are obtained by solving the eigenvalue equations.

[0060] Skin panel perturbation load The calculation method is as follows:

[0061] (11)

[0062] Typically, the first m modes of a structure are selected to construct perturbation loads, where m is the number of critical modes. Critical modes are those whose eigenvalues ​​are within 120% of the magnitude of the first-order eigenvalue.

[0063] load factor vector Expanding on the generalized displacement of the skin panel The third-order expression:

[0064] (12)

[0065] in, , and These are, respectively, the generalized displacement of a perfect, defect-free skin panel. First-order, second-order, and third-order proxy expansion terms, To introduce defects and the resulting Related to generalized displacement Additional first-order proxy expansion term, To introduce defects and the resulting Related to generalized displacement Additional first-order proxy expansion term, To introduce defects and the resulting Related to generalized displacement Additional second-order proxy expansion terms; , and For a perfectly flawless skin panel, only one solution is needed for different initial geometric defects.

[0066] Similarly, the displacement increment field of the structure Expanding on generalized displacement Second-order expression:

[0067] (13)

[0068] in, and These are the first-order and second-order displacement increment fields of the structure, respectively. This refers to the additional displacement increment field caused by the defect. Similarly, and For a perfectly flawless skin panel, only one solution is needed for different initial geometric defects.

[0069] Substituting equations (9), (12), and (13) into the third-order expansion (8) of the nonlinear equilibrium equation of the skin panel, we obtain a new equation. Let the left and right sides of the equation be related to the generalized displacement. Since the coefficients of all terms are equal, a system of six linear algebraic equations can be derived:

[0070] (14)

[0071] (15)

[0072] (16)

[0073] (17)

[0074] (18)

[0075] (19)

[0076] in, for The coefficient matrix, It is the identity matrix. To be displacement increment field Replace with The second-order expansion term after that, To be displacement increment field Replace with The third-order expansion term after that, To be displacement increment field Replace with The additional first-order expansion term after that, for The coefficient matrix, To be displacement increment field Replace with The additional second-order expansion term after that, To be displacement increment field Replace with The addition of first-order expansion terms; Equations (14), (15) and (17) have the same coefficient matrix, so only one decomposition of the coefficient matrix is ​​needed when solving the system of equations, saving computational resources. Equations (16), (18) and (19) only involve matrix multiplication, do not require matrix decomposition, and have a very small computational load.

[0077] Solving the system of linear algebraic equations (14), (15) and (16) yields the load coefficient vector expansion (12) and the displacement increment field expansion (13) for a perfect, defect-free skin panel.

[0078] The proxy equilibrium equation for a perfectly flawless skin panel is a transformed form of equation (12):

[0079] (20)

[0080] The surrogate equilibrium equation has only m+1 degrees of freedom, where m is the number of key modes contained in the structure, which is usually less than 10 (and much smaller than the number of degrees of freedom in the equilibrium equation (7)). This can significantly reduce the solution scale of the structural nonlinear analysis and quickly obtain the nonlinear response of the structure. Solving the very small-scale surrogate equilibrium equation (20) can yield the generalized displacement of the structure. and load factor Substituting this relationship into the expansion of the displacement increment field of a perfectly flawless skin panel:

[0081] (twenty one)

[0082] The displacement field of the skin panel structure can then be obtained. and load factor The relationship is the load-displacement relationship curve of a perfect, defect-free structure. The load value corresponding to the extreme point of this curve is the ultimate bearing capacity of the structure.

[0083] When considering the initial geometric imperfections of the skin panel, the surrogate equilibrium equation (20) and the displacement increment field expansion (21) for a perfectly defect-free skin panel only need to be constructed once. Solving the equation system (17), (18), and (19) yields the additional surrogate expansion terms of the surrogate equilibrium equation (20) and the incremental displacement expansion (21) after introducing the imperfections. , , and By incorporating this into the surrogate equilibrium equation (20) of a perfect, defect-free skin panel in an a posteriori manner, we can obtain the surrogate equilibrium equation and displacement increment field expansion of the skin panel containing initial geometric defects.

[0084] Surrogate equilibrium equations for skin panels with initial geometric defects:

[0085] (twenty two)

[0086] The displacement increment field expansion of the skin panel containing initial geometric defects is consistent with that of equation (13).

[0087] (twenty three)

[0088] For different initial geometric defects, it is only necessary to resolve the linear algebraic equations (17), (18) and (19) to obtain the additional surrogate expansion terms under the current defect. , , and Therefore, since the coefficient matrix of the linear algebraic equation system (17) is completely consistent with the coefficient matrices of the linear algebraic equation systems (14) and (15) in the perfect and defect-free case, no additional matrix decomposition calculation is required. The generalized displacement can be obtained by solving the surrogate equilibrium equation (22) containing the initial geometric defects. and load factor The relationship between generalized displacement Substituting this into the incremental displacement expansion (23) containing the initial geometric defects, the displacement of the skin panel containing the initial geometric defects can be obtained. and load factor The relationship, namely the load-displacement relationship curve of the defective structure, is the load value corresponding to the extreme point of the curve, which is the ultimate bearing capacity of the defective structure.

[0089] Step 4: For skin panels with different initial geometric defects, first, repeat step 1 to establish the defect displacement field of the structure. Then, in steps 2 and 3, the surrogate equilibrium equation (20) and incremental displacement expansion (21) for the perfect, defect-free skin panel do not need to be recalculated; only the additional surrogate expansion term caused by the initial geometric defect needs to be solved. , , and The surrogate equilibrium equation (20) of the perfect, defect-free skin panel is embedded into it in a posterior manner to obtain the surrogate equilibrium equation (22) of the skin panel with initial geometric defects. Finally, the surrogate equilibrium equation (22) is solved to obtain the nonlinear response curve of the skin panel with initial geometric defects, realizing the posterior analysis of the three-dimensional initial geometric defects of the aircraft skin panel. That is, it is not necessary to re-establish the surrogate equilibrium equation of the perfect, defect-free structure for different defects (this process accounts for more than 50% of the computation of this method). Only a small amount of additional computation is needed to perform a posterior correction of the surrogate equilibrium equation under the current defect. In the process of solving the surrogate equilibrium equation (22), the solution accuracy can be monitored in real time by calculating the residual force of the structure. Once the solution accuracy does not meet the requirements, the solution result needs to be corrected based on the nonlinear equilibrium equation (7) of the skin panel structure, and the surrogate equilibrium equation (22) needs to be re-established and then the solution continues.

[0090] In this embodiment, for Figure 5The three-dimensional initial geometric defect shown is used to create three different initial geometric defects of sizes (0.001t, 0.1t, and 0.2t) by adjusting the magnitude of the defect size. The nonlinear mechanical response curves of the skin panel obtained using conventional methods and the method of this invention for these three defects are shown below. Figure 5 As shown, it can be seen that as the defect size increases, the ultimate load-bearing capacity of the skin panel (i.e., the extreme point of the curve) gradually decreases. Figure 5 It can be seen that, for this embodiment, the nonlinear response curves obtained by the method of the present invention and the conventional method have good consistency. Taking one initial geometric defect size (0.001t) as an example, the method of the present invention takes 23.2 seconds to construct the surrogate equilibrium equation for a perfect defect-free skin panel, 4.7 seconds to introduce the defect posteriorly, 0.4 seconds to solve the surrogate equilibrium equation, and 16.2 seconds for the correction stage, for a total computation time of 44.5 seconds. For different initial geometric defect conditions (defect size (0.1t), since the method of the present invention has posterior characteristics for defects, it does not need to repeatedly construct the surrogate equilibrium equation for a perfect defect-free skin panel, thus saving 23.2 seconds of time, and the computation time is only 21.3 seconds. Therefore, it can be seen that in order to obtain Figure 5 The three curves shown indicate a total analysis time of 87.1 seconds (44.5 seconds + 21.3 seconds + 21.3 seconds) for the three initial geometric defects. In contrast, the conventional method takes an average of approximately 380 seconds for each initial geometric defect, with a total calculation time of 1140 seconds (380 seconds × 3) for all three defects, significantly longer than the method of this invention. In summary, the posterior analysis method for three-dimensional initial geometric defects in aircraft skin panels proposed in this invention significantly improves computational efficiency by 98.13% while maintaining computational accuracy.

[0091] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A method for post-hoc analysis of three-dimensional initial geometric defects in aircraft skin panels, characterized in that: Includes the following steps: Step 1: Obtain the initial three-dimensional geometric defects of the aircraft skin panels and construct the defect displacement field for structural finite element analysis. ; Step 2: Construct hybrid solid shell elements. Use hybrid solid shell elements to construct a three-dimensional finite element model of the defect-free skin panel and establish the nonlinear equilibrium equations of the skin panel structure. Step 3: Construct a surrogate equilibrium equation for the nonlinear equilibrium equation of the skin panel structure, and embed the influence of defects into the surrogate equilibrium equation through a posterior method to obtain the surrogate equilibrium equation for the skin panel with initial geometric defects. By solving the surrogate equilibrium equation for the skin panel with initial geometric defects, the nonlinear response curve of the skin panel with initial geometric defects is obtained, thus realizing the posterior analysis of the three-dimensional initial geometric defects of the aircraft skin panel.

2. The method for post-hoc analysis of three-dimensional initial geometric defects in aircraft skin panels according to claim 1, characterized in that: The surrogate equilibrium equations for the skin panel with initial geometric defects include first-order, second-order, and third-order surrogate expansions of the defect-free skin panel with respect to the generalized displacement, as well as additional surrogate expansions of the generalized displacement after the introduction of defects.

3. The method for post-hoc analysis of three-dimensional initial geometric defects in aircraft skin panels according to claim 2, characterized in that: For skin panels with different initial geometric defects, when performing posterior analysis, step 1 is first repeated to establish the defect displacement field of the structure. Then, in steps 2 and 3, we only need to solve the additional surrogate expansion terms of the generalized displacement after introducing the new initial geometric defects, and combine them with the first, second and third order surrogate expansion terms of the generalized displacement of the defect-free skin panel that have been calculated, to obtain and solve the surrogate equilibrium equation of the skin panel with the new initial geometric defects, and then obtain the nonlinear response curve of the skin panel with the new initial geometric defects.

4. The method for post-hoc analysis of three-dimensional initial geometric defects in aircraft skin panels according to claim 1, characterized in that: In step 2, the specific process of establishing the nonlinear equilibrium equations for the skin panel structure is as follows: Step 2.1: Establish the stress formula for the hybrid solid shell element; Step 2.2: Calculate the three transverse strain components of the hybrid solid shell element using the ANS method; Step 2.3: Combine the three transverse strain components of the element with the in-plane strain components to form the element strain vector E, and then combine the defect displacement field... Introduce the element strain expression; Step 2.4: Based on the stress and strain calculation formulas of the element, establish the Hellinger-Reissner energy functional of the three-dimensional hybrid solid shell element; Step 2.5: Apply variational principles to the energy functional to obtain the nonlinear equilibrium equations of the elements, reduce stress parameters at the element level, and assemble to obtain the nonlinear equilibrium equations of the skin panel structure.

5. The method for post-hoc analysis of three-dimensional initial geometric defects in aircraft skin panels according to claim 4, characterized in that: In step 2, the stress formula for the established hybrid solid shell unit is: in For element stress vectors, Here is the stress interpolation matrix. These are stress parameters; The three transverse strain components of the hybrid solid shell element were calculated using the ANS method as follows: in, The transverse normal strain at a point in the element. and The transverse shear strain at a point in the element. , , and The element shape function that includes the coordinates of a point within the element. and These are two coordinates in the unit natural coordinate system; Represents coordinates in the natural coordinate system of the unit The transverse normal strain at the location is given, where the origin of the element's natural coordinate system is established at the geometric center of the element, and the coordinates in the element's natural coordinate system are expressed as follows: ; The element strain vector E is in, The geometric matrix of the linear strain of the element is obtained from the three transverse strain components and the in-plane strain components. The geometric matrix of the nonlinear strain of the element is obtained from the three transverse strain components and the in-plane strain components. For the nodal displacement field of the element, An additional matrix is ​​added to the element strain introduced by the defect; Based on the stress and strain calculation formulas of the element, the Hellinger-Reissner energy functional of the three-dimensional hybrid solid shell element is established as follows: in, Unit energy, For unit volume, For the element constitutive matrix, The structural load factor is... For reference external load, , , All are intermediate matrices in, For the element constitutive matrix, The structural load factor is... For reference external load; Applying the variational principle to the energy functional yields the nonlinear equilibrium equations for the element: Polycondense stress parameters at the unit level The nonlinear equilibrium equations of the skin panel structure are obtained through assembly: 。 6. The method for post-hoc analysis of three-dimensional initial geometric defects in aircraft skin panels according to claim 1, characterized in that: In step 3, the process of constructing the surrogate equilibrium equation for the nonlinear equilibrium equation of the skin panel structure and embedding the influence of defects into the surrogate equilibrium equation through a posterior approach is as follows: The nonlinear equilibrium equations for the skin panel with respect to the displacement increment field Perform a third-order Taylor expansion: in , and These are the first, second, and third order expansion terms of the Taylor expansion of a defect-free structure, respectively. To introduce defects and the resulting Related additional first-order expansion terms, To introduce defects and the resulting Related additional first-order expansion terms, To introduce defects and the resulting Related additional second-order expansion terms, and These are the increments of the structural displacement field and the load factor, respectively. Based on the perturbation loads corresponding to each mode and reference external load The right-hand side of the third-order expansion of the nonlinear equilibrium equation for the skin panel is expressed as: in, This is a load factor vector, where the first component is the increment of the load factor. The remaining components are 0; This is the load matrix, with the first column representing the reference external load. The rest are listed as perturbation loads, among which For the structure The perturbation load corresponding to the first mode; Structural characteristic equation of skin panel: in, Let be the tangent stiffness matrix of the skin panel. Here is the geometric stiffness matrix of the skin panel. For the structure eigenvalues ​​of order 1 For the structure First mode; Skin panel perturbation load The calculation method is as follows: The first m modes of the structure are selected to construct the perturbation load, where m is the number of critical modes; Load coefficient vector Expanding on the generalized displacement of the skin panel The third-order expression: in, , and These are the generalized displacements of defect-free skin panels. First-order, second-order, and third-order proxy expansion terms, To introduce defects and the resulting Related to generalized displacement Additional first-order proxy expansion term, To introduce defects and the resulting Related to generalized displacement Additional first-order proxy expansion term, To introduce defects and the resulting Related to generalized displacement Additional second-order proxy expansion terms; , and For a defect-free skin panel, only one solution is needed for different initial geometric defects. The displacement increment field of the structure Expanding on generalized displacement Second-order expression: in, and These are the first-order and second-order displacement increment fields of the structure, respectively. This refers to the additional displacement increment field caused by the defect; similarly... and For defect-free skin panels, only one solution is needed for different initial geometric defects; Further, six systems of linear algebraic equations are derived: in, for The coefficient matrix, It is the identity matrix. To be displacement increment field Replace with The second-order expansion term after that, To be displacement increment field Replace with The third-order expansion term after that, To be displacement increment field Replace with The additional first-order expansion term after that, for The coefficient matrix, To be displacement increment field Replace with The additional second-order expansion term after that, To be displacement increment field Replace with The additional first-order expansion term; The load coefficient vector expansion and displacement increment field expansion of the defect-free skin panel are obtained by solving. The surrogate equilibrium equation for a defect-free skin panel is: Solving the surrogate equilibrium equations of the defect-free skin panel yields the generalized displacement of the structure. and load factor Substituting this relationship into the displacement increment field expansion of the defect-free skin panel: The load-displacement relationship curve of the defect-free structure is obtained. The load value corresponding to the extreme point of the curve is the ultimate bearing capacity of the structure. When considering the initial geometric imperfections of the skin panel, the surrogate equilibrium equation for the skin panel containing initial geometric imperfections is as follows: The displacement increment field expansion of a skin panel with initial geometric defects: The generalized displacement can be obtained by solving the surrogate equilibrium equations containing the initial geometric imperfections. and load factor The relationship between generalized displacement Substituting the incremental displacement expansion with initial geometric defects into the equation yields the load-displacement relationship curve of the defective structure. The load value corresponding to the extreme point of this curve is the ultimate bearing capacity of the defective structure.

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