A rapid method for assessing the load-bearing capacity of large-scale cyclic symmetric thin-walled structures

Abstract

This invention belongs to the field of aerospace structural computation and provides a rapid method for evaluating the load-bearing capacity of large-scale cyclically symmetric thin-walled structures. Based on the cyclic symmetry characteristics of large-scale cyclically symmetric thin-walled structures, this method establishes a substructure model of the structure. On this basis, by applying cyclic symmetry boundary conditions and Bloch wave boundary conditions, the overall model analysis is transformed into solving the substructure model, which can simultaneously consider the overall and local mechanical behavior of the large-scale cyclically symmetric thin-walled structure and significantly reduce the dimensionality of the analysis problem. Finally, during the nonlinear analysis iteration process, an eigenvalue iteration algorithm based on a combined approximation strategy is used to reasonably predict the increment step size, reducing the number of iterations in the nonlinear analysis, thereby rapidly and accurately evaluating the load-bearing capacity of large-scale cyclically symmetric thin-walled structures.

Description

Technical Field

[0001] This invention belongs to the field of aerospace structural calculation and provides a method for rapid evaluation of the load-bearing capacity of large cyclic symmetric thin-walled structures. Background Technology

[0002] Large-scale cyclic symmetric thin-walled structures are widely used in aerospace equipment such as interstage sections, fuel tanks, common-bottom tanks, and sealed compartments due to their advantages in ease of fabrication and large available space. Their weight can account for up to 80% of the dry weight of the equipment structure. These cyclic symmetric stiffened shells operate in extremely harsh environments and are prone to buckling instability and structural failure under enormous axial or external pressure loads. High-efficiency and high-precision analysis of the load-bearing capacity of cyclic symmetric thin-walled structures is fundamental to their refined design and is recognized as a key technology for improving the carrying capacity, range, and speed performance of aerospace equipment.

[0003] Existing nonlinear analysis methods for large-scale cyclic symmetric thin-walled structures are mainly based on incremental iterative finite element methods. These methods require repeated stiffness matrix assembly and decomposition during nonlinear analysis, which remains challenging even with the rapid improvement in computing power. As large-scale cyclic symmetric thin-walled structures become increasingly larger and more complex, and as design requirements become more demanding, their complexity also increases. This increased computational scale further reduces the efficiency of nonlinear analysis. Common equivalent methods for large-scale cyclic symmetric thin-walled structures mainly include fast buckling analysis methods based on structural equivalence and fast buckling analysis methods based on model order reduction. For the structural characteristics of large-scale cyclically symmetric thin-walled structures, representative rapid buckling analysis methods based on structural equivalence include the equivalent stiffness method and the homogenization method. These methods obtain the equivalent stiffness coefficients of the large-scale cyclically symmetric thin-walled structure by smoothing out the stiffeners and skin, thereby establishing an equivalent model of the structure. This significantly reduces the degrees of freedom in the finite element model analysis, enabling rapid assessment of the load-bearing capacity of the large-scale cyclically symmetric thin-walled structure. However, model equivalence leads to the neglect of many structural details, making it difficult to capture the local mechanical behavior of the structure. Methods based on model order reduction strategies mainly include principal component analysis and combined approximation methods. These methods solve the finite element equations through linear combination approximations of linearly independent basis vectors, effectively improving analysis efficiency while ensuring prediction accuracy. Inspired by the above research, to ensure the design and development cycle of large-scale cyclically symmetric thin-walled structures in the aerospace field, this invention establishes an accurate and efficient method for assessing the load-bearing capacity of large-scale cyclically symmetric thin-walled structures using model order reduction strategies. Summary of the Invention

[0004] This invention primarily addresses the challenges of predicting load-bearing capacity and the surge in computation time caused by the increasing size and complexity of large-scale cyclic symmetric thin-walled structure models. It proposes a rapid load-bearing capacity assessment method suitable for large-scale cyclic symmetric thin-walled structures, ensuring the accuracy of load-bearing capacity prediction while significantly improving the efficiency of load-bearing capacity analysis.

[0005] To achieve the above research objectives, the technical solution adopted in this invention is as follows:

[0006] A rapid method for assessing the load-bearing capacity of large-scale cyclically symmetric thin-walled structures includes the following steps:

[0007] Step 100: Establish a substructure model of the large-scale cyclically symmetric thin-walled structure for the load-bearing capacity analysis of the large-scale thin-walled structure in Step 300, including the following sub-steps:

[0008] Step 101: The geometry of the large thin-walled structure exhibits periodic changes along the circumference, such as... Figure 1 As shown. Therefore, it can be divided into several sub-structure models with the same geometric shape along the circumference. One of them is selected to build a sub-structure model of a large-scale cyclic symmetric thin-walled structure, such as... Figure 2 As shown.

[0009] Step 102: Mesh the large-scale cyclic symmetric thin-walled structure substructure model. Ensure that the left and right nodes of the large-scale cyclic symmetric thin-walled structure substructure model correspond one-to-one, and extract the corresponding node numbers.

[0010] Step 200: Based on the loads and constraints of the large-scale cyclically symmetric thin-walled structure, apply corresponding loads and boundary conditions to the substructure model to ensure that the working conditions of the substructure model are consistent with those of the overall model. This is then applied to the analysis of the load-bearing capacity of the large-scale thin-walled structure in Step 300, and includes the following sub-steps:

[0011] Step 201: Constrain the 6 degrees of freedom of the lower boundary of the large cyclic symmetric thin-walled substructure model established in step 100, including 3 displacement degrees of freedom and 3 rotational degrees of freedom; relax the axial displacement degree of freedom of the upper boundary of the substructure model; constrain the remaining 5 degrees of freedom of the upper boundary; and apply axial force load or displacement load.

[0012] Step 202: First, based on the coordinate transformation relationship between the left and right nodes of the substructure model, establish the constraint equations for the cyclic symmetric boundary conditions of the substructure model. These equations are used in the linear and nonlinear analysis processes of the substructure model in step 300. Based on this, establish the constraint equations for the Bloch wave boundary conditions of the substructure model. These equations are used in the buckling analysis of the substructure model under predetermined loads in step 300.

[0013] The constraint equations for the cyclically symmetric boundary conditions of the substructure model are expressed in the following form:

[0014] u B =Tu A

[0015]

[0016] Among them, u A It is the displacement vector of the left node of the substructure model, u B is the displacement vector of the right node of the substructure model. T is the coordinate transformation matrix of the left and right boundaries of the substructure model, which is composed of the coordinate transformation matrix T0 of the corresponding node, and μ is the angle between the left and right boundaries of the substructure model.

[0017] The constraint equations for the Bloch wave boundary conditions are expressed as follows:

[0018]

[0019] in, and The modal deformation of the left and right boundary nodes of the substructure model is represented by l, where l is the wave number.

[0020] Step 300: Establish an eigenvalue iteration process based on a combined approximation strategy to accurately trace the equilibrium path of a large-scale cyclically symmetric structure and evaluate the load-bearing capacity of the large-scale cyclically symmetric thin-walled structure. This includes the following sub-steps:

[0021] Step 301: First, based on the constraint equations of the cyclic symmetric boundary conditions established in Step 202, the prestress of the substructure model is calculated, and then the geometric stiffness matrix of the substructure model is established. On this basis, based on the constraint equations of the Bloch wave boundary conditions established in Step 202, buckling analysis of the substructure model is performed, solving for the first-order buckling eigenvalues ​​and first-order buckling modes corresponding to different wave numbers. The smallest first-order buckling load corresponds to the critical buckling load of the large thin-walled structure, and the critical buckling load and the corresponding buckling mode are saved. Then, the incremental load of the nonlinear iteration is predicted by multiplying the critical buckling load by the load factor. The specific iterative process is as follows... Figure 3 As shown, the expression is as follows:

[0022] λ i+1 =λ i +η i Δλ i ,i=1,2,...,n (3)

[0023] Where i is the load step, η i (0<η i <1) is the loading factor. λ iThis is the load value at the i-th load step. Δλ i η1 is the critical buckling load at the i-th load step, and n is the total number of load steps. Here, η1 is typically taken as 0.7 to 0.8, and η... i The value of i > 1 is generally taken as 0.5. Next, based on the constraint equations of the cyclic symmetric boundary conditions established in step 202, the state of the substructure model under incremental load is obtained by nonlinear analysis using the Newton-Raphson algorithm.

[0024] Step 302: Based on the calculation of the first two load steps in step 301, construct the basis vectors in this state using a combined approximation strategy. Considering that the tangent stiffness matrix and geometric stiffness matrix of the substructure model will change under load, construct a recursive form for solving the basis vectors:

[0025] r i+1 =(K0) -1 (-ΔK0-G0-ΔG0)r i i = 1, 2, ..., s (4)

[0026] Where K0 is the tangent stiffness matrix and G0 is the geometric stiffness matrix, with dimensions M×M, where M is the number of degrees of freedom of the finite element model. ΔK0 and ΔG0 are the changes in the tangent stiffness matrix and geometric stiffness matrix, respectively, and s is the number of basis vectors. The basis vectors can be constructed recursively, and r1 is chosen as the first-order buckling mode obtained before the changes to the tangent stiffness matrix and geometric stiffness matrix. After assembling the changed tangent stiffness matrix K and geometric stiffness matrix G, the buckling mode can be obtained through independent basis vectors Φ = [r1, r2, ..., r...]. s The linear combination of ] is approximately obtained, and its specific form is as follows:

[0027]

[0028] in, The matrix has dimensions of s×s. Solving the reduced-order buckling analysis equation yields the first-order eigenvalue Δλ and the first-order buckling mode α. The dimension of the buckling analysis is reduced from M×M to s×s, significantly improving the efficiency of the buckling analysis.

[0029] Step 303: The third load step begins, and step 301 is repeated, except that the buckling analysis is calculated using equation (5) from step 302, until the convergence criterion is met, at which point the iteration stops. The convergence condition is...

[0030]

[0031] ε1 is a predefined tolerance, typically chosen between 1e-3 and 5e-3.

[0032] The beneficial effects of this invention are as follows:

[0033] (1) This invention addresses the problem of the increasing size and complexity of large-scale cyclic symmetric thin-walled structures leading to a surge in model degrees of freedom. By utilizing the cyclic symmetry characteristics of large-scale cyclic symmetric thin-walled structures, the nonlinear analysis of cyclic symmetric thin-walled structures is transformed into the analysis of representative substructure models, which greatly reduces the degrees of freedom in model analysis and improves the analysis efficiency of the load-bearing capacity of large-scale cyclic symmetric structures.

[0034] (2) The present invention uses an eigenvalue iterative algorithm based on a combined approximation strategy to reasonably predict the incremental step size, which significantly reduces the number of iterations in nonlinear analysis and avoids the extra cost of buckling analysis during the iteration process, thereby further improving the analysis efficiency of the load-bearing capacity of large cyclic symmetric thin-walled structures.

[0035] (3) The algorithm proposed in this invention has a simple calculation process and is easy to integrate into general nonlinear analysis programs. Attached Figure Description

[0036] Figure 1 A schematic diagram of the overall model of a large-scale cyclic symmetric thin-walled structure;

[0037] Figure 2 A schematic diagram of a substructure model of a large-scale cyclic symmetric thin-walled structure;

[0038] Figure 3 A schematic diagram of the eigenvalue iterative algorithm for a large cyclic symmetric thin-walled structure;

[0039] Figure 4 A flowchart of a method for rapid evaluation of the load-bearing capacity of large cyclic symmetric thin-walled structures provided in this embodiment of the invention;

[0040] Figure 5 A schematic diagram of the buckling modes of a large cyclically symmetric thin-walled structure under a given load: Figure 5 (a) is the overall model; Figure 5 (b) is a substructure model; Detailed Implementation

[0041] To make the problem solved by the present invention, the method adopted, and the effect achieved by the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the present invention and not intended to limit the invention. Furthermore, it should be noted that, for ease of description, only the parts relevant to the present invention are shown in the accompanying drawings, not all of them.

[0042] Figure 1 This is a flowchart illustrating the implementation of a rapid load-bearing capacity assessment method for large-scale cyclically symmetric thin-walled structures, provided by an embodiment of the present invention. The invention will be further described in detail below with reference to embodiments, specifically including:

[0043] Step 100, taking a typical metal orthogonal stiffened shell model in engineering as an example, such as... Figure 2 As shown, the radius R of the orthogonally stiffened shell is 1500 mm, the height H is 1000 mm, and the skin thickness t is... s The diameter is 4 mm, and the number of circumferential unit cells is N. a The value is 90, and the number of axial unit cells N c The value is 25. The reinforced shell material is an aluminum-lithium alloy with a Young's modulus E of 70.0 GPa, a Poisson's ratio μ of 0.33, and a density of 2.7 × 10⁻⁶. -6 kg / mm 3 The mesh size was determined to be 30mm through mesh size convergence analysis. The number of meshes in the stiffener height direction of the stiffened shell was fixed at 2, the number of elements in the finite element model was 48950, and the number of degrees of freedom was 270204. The boundary conditions of the overall model were set as follows: the bottom was fixed, the top was constrained except for axial displacement, and all nodes on the top surface were rigidly coupled to a reference point. An axial compressive load was applied to the reference point until the structure collapsed. The substructure model of the stiffened shell is now being built:

[0044] Step 101: Based on the cyclic symmetry characteristics of the stiffened shell, divide it into several sub-structure models along the circumferential direction, select one of them to establish the sub-structure model of the stiffened shell. The sub-structure model can be 1 / 90, 1 / 45, 1 / 30, etc. of the overall model.

[0045] Step 102: Taking a 1 / 30th scale substructure model of the overall structural model as an example, mesh generation is performed. The mesh size is 30mm, the number of elements is 1050, and the number of degrees of freedom is 6436. The number of degrees of freedom of the substructure model is reduced by two orders of magnitude compared to the overall detailed model. During mesh generation, it is ensured that the left and right nodes of the stiffened shell substructure model correspond one-to-one, and the corresponding node numbers are extracted to establish a node set.

[0046] Step 200: Based on the obtained mesh model of the stiffened shell structure model, apply loads and constraints to ensure that the working conditions of the substructure model are consistent with those of the overall structure. This includes the following steps:

[0047] Step 201: Constrain the six degrees of freedom of the lower boundary of the stiffened shell structure model, including displacement and rotation degrees of freedom; relax the axial displacement degree of freedom of the upper boundary of the substructure model; constrain the remaining five degrees of freedom; and apply axial force load, thereby ensuring that the boundaries and constraints of the substructure model and the overall structure model are consistent.

[0048] Step 202: Based on the coordinate relationship between the left and right nodes of the substructure model, establish the constraint equations for the cyclic symmetric boundary conditions and apply them to the nonlinear analysis of the substructure model. On this basis, establish the constraint equations for the Bloch wave boundary conditions and apply them to the buckling analysis of the substructure model under a given load.

[0049] Step 300: Apply an eigenvalue iterative algorithm based on a combined approximation strategy to analyze the load-bearing capacity of a large-scale cyclically symmetric thin-walled structure, including the following sub-steps:

[0050] Step 301: First, based on the constraint equations of the cyclic symmetric boundary conditions established in step 202 (1), the stress of the substructure model is calculated, and then the geometric stiffness matrix of the substructure model is constructed. On this basis, based on the constraint equations of the Bloch wave boundary conditions established in step 202 (2), the buckling analysis of the substructure model is performed, and the first-order buckling eigenvalues ​​and first-order buckling modes corresponding to different wave numbers are solved. The smallest first-order buckling load corresponds to the critical buckling load of the large thin-walled structure. The critical buckling load and the buckling mode corresponding to the critical buckling load are saved. Then, the incremental load of the nonlinear iteration is predicted by multiplying the critical buckling load by the load factor. The specific iteration process is referred to in equation (3). In this example, η1 is generally taken as 0.8, η i i > 1 takes the value of 0.5. Next, based on the constraint equation of the cyclic symmetric boundary condition established in step 202 (1), the state of the substructure model under incremental load is obtained by nonlinear analysis through the Newton-Raphson algorithm.

[0051] Step 302: Based on the calculation of the first two load steps in step 301, in this state, construct the reduced-order basis vector Φ = [r1, r2, ..., r] using the recursive formula (4) of the combined approximation strategy. s ].

[0052] Step 303, the third load step begins, and step 301 is repeated, where the buckling analysis is calculated using equation (5) from step 302, until the convergence criterion is met, and the iteration stops. ε1 is predefined as 5e-3. Table 1 shows the comparison of the load-bearing capacity prediction results and analysis efficiency between the stiffened shell substructure model and the overall structure model. The overall model is analyzed nonlinearly using the Riks algorithm. The results show that the Bloch wave method can significantly improve the analysis efficiency of the load-bearing capacity of the cyclically symmetric stiffened shell. The substructure model has a 96.2% higher analysis efficiency than the overall model, and the error in load-bearing capacity prediction is less than 0.1%. Figure 5 A comparison of the buckling modes of the overall model and the substructure model under a given load is presented. The results show that the Bloch wave method can accurately predict the buckling load and buckling modes of the substructure model under a given load.

[0053] Table 1 Comparison of load-bearing capacity prediction results and analysis efficiency between the overall model and the substructure model.

[0054]

[0055] Finally, it should be noted that the above embodiments are only used to illustrate the method solutions of the present invention, and are not intended to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications to the method solutions described in the foregoing embodiments, or equivalent substitutions for some or all of the method features, do not cause the essence of the corresponding method solutions to deviate from the scope of the method solutions of the embodiments of the present invention.

Claims

1. A rapid method for evaluating the load-bearing capacity of large-scale cyclically symmetric thin-walled structures, characterized in that, Includes the following steps: Step 100: Establish a substructure model of the large-scale cyclic symmetric thin-walled structure for the analysis of the load-bearing capacity of the large-scale thin-walled structure in Step 300; Step 200: Based on the loads and constraints of the large cyclic symmetric thin-walled structure, apply corresponding loads and boundary conditions to the substructure model to ensure that the working conditions of the substructure model are consistent with those of the overall model, and apply them to the analysis of the load-bearing capacity of the large thin-walled structure in step 300. Step 300: Establish an eigenvalue iteration process based on a combined approximation strategy, trace the equilibrium path of a large cyclic symmetric structure, and evaluate the load-bearing capacity of a large cyclic symmetric thin-walled structure. Step 300 includes the following sub-steps: Step 301: First, based on the constraint equations of the cyclic symmetric boundary conditions, the prestress of the substructure model is calculated, and then the geometric stiffness matrix of the substructure model is established. Next, based on the constraint equations of the Bloch wave boundary conditions, buckling analysis of the substructure model is performed, solving for the first-order buckling eigenvalues ​​and first-order buckling modes corresponding to different wave numbers. The smallest first-order buckling load corresponds to the critical buckling load of the large thin-walled structure, and the critical buckling load and the corresponding buckling mode are saved. Then, the incremental load of the nonlinear iteration is predicted by multiplying the critical buckling load by the load factor, expressed as follows: (3) Where i is the load step, It is the loading factor; It is the load value of the i-th load step; Here, is the critical buckling load of the i-th load step, and n is the total number of load steps; The value is typically between 0.7 and 0.

8. The value is typically taken as 0.

5. Next, based on the constraint equations of the cyclic symmetric boundary conditions established in step 202, the state of the substructure model under incremental load is obtained through nonlinear analysis using the Newton-Raphson algorithm. Step 302: Based on the calculation of the first two load steps in step 301, construct basis vectors in this state using a combined approximation strategy; considering that the tangent stiffness matrix and geometric stiffness matrix of the substructure model will change under load, construct a recursive form for solving the basis vectors: (4) in, It is the tangent stiffness matrix and It is the geometric stiffness matrix with dimension . , It is the number of degrees of freedom of the finite element model; and These are the changes in the tangent stiffness matrix and the geometric stiffness matrix, and s is the number of basis vectors; basis vectors can be constructed through recursive relations, and the selection... The first-order buckling mode is obtained before the tangent stiffness matrix and geometric stiffness matrix are changed; the tangent stiffness matrix is ​​obtained after the assembly is modified. and geometric stiffness matrix Buckling modes can be obtained through independent basis vectors The linear combination approximation is obtained as follows: (5) in, and Dimension is The matrix; solve the reduced-order buckling analysis equation to obtain the first-order eigenvalues. and first-order buckling mode The dimensions of buckling analysis are: Reduce to The efficiency of buckling analysis is significantly improved; Step 303, the third load step begins, repeating step 301, where the buckling analysis is changed to the formula in step 302. The calculation continues until the convergence criterion is met, at which point the iteration stops.

2. The method for rapid evaluation of the load-bearing capacity of large-scale cyclic symmetric thin-walled structures according to claim 1, characterized in that, Step 100 includes the following sub-steps: Step 101: Divide the large thin-walled structure into several sub-structure models with the same geometric shape along the circumferential direction, and select one of them to establish the sub-structure model of the large cyclic symmetric thin-walled structure. Step 102: Mesh the large-scale cyclic symmetric thin-walled structure substructure model to ensure that the left and right nodes of the large-scale cyclic symmetric thin-walled structure substructure model correspond one-to-one, and extract the corresponding node numbers.

3. The method for rapid evaluation of the load-bearing capacity of large-scale cyclic symmetric thin-walled structures according to claim 1, characterized in that, Step 200 includes the following sub-steps: Step 201: Constrain the 6 degrees of freedom of the lower boundary of the large cyclic symmetric thin-walled substructure model established in step 100, including 3 displacement degrees of freedom and 3 rotational degrees of freedom; relax the axial displacement degree of freedom of the upper boundary of the substructure model; constrain the remaining 5 degrees of freedom of the upper boundary; and apply axial force load or displacement load. Step 202: First, based on the coordinate transformation relationship between the left and right nodes of the substructure model, establish the constraint equations for the cyclic symmetric boundary conditions of the substructure model, which are used for the linear and nonlinear analysis processes of the substructure model in step 300; on this basis, establish the constraint equations for the Bloch wave boundary conditions of the substructure model, which are used for the buckling analysis of the substructure model under the predetermined load in step 300. The constraint equations for the cyclically symmetric boundary conditions of the substructure model are expressed in the following form: (1) in, It is the displacement vector of the left node of the substructure model. It is the displacement vector of the right-hand node of the substructure model; It is the coordinate transformation matrix of the left and right boundaries of the substructure model, derived from the coordinate transformation matrix of the corresponding nodes. composition, It is the angle between the left and right boundaries of the substructure model; The constraint equations for the Bloch wave boundary conditions are expressed as follows: (2) in, and This represents the modal deformation of the left and right boundary nodes of the substructure model. It is the wave number.

4. The method for rapid evaluation of the load-bearing capacity of large-scale cyclic symmetric thin-walled structures according to claim 1, characterized in that, The convergence condition for step 303 is: (6) in, It is a predefined tolerance, selected between 1e-3 and 5e-3.

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