Design method for determining maximum hoisting weight of ultra-thin hoisting beam

Abstract

This invention relates to the field of electronic digital data processing technology, specifically to computer-aided engineering calculation and numerical simulation data processing technology. This invention discloses a design method for determining the maximum hanging weight of an ultra-thin lifting beam. In incremental iterative solutions, the equivalent stress gradient and strain energy density are extracted in real time to calculate local instability characteristic values. When the rate of change of element strain energy density exceeds a threshold, a mesh reconstruction mechanism is triggered. The nodal displacement field and stress field are extracted as initial boundary conditions. The coarse mesh is split into a fine mesh, and the coordinates of newly added nodes are smoothly interpolated based on the minimum potential energy principle. The local and global mesh degrees of freedom are coupled and assembled to continue nonlinear buckling analysis iterations until the load-displacement curve shows an extreme point. The corresponding load is then extracted as the maximum hanging weight. This invention avoids stiffness matrix singularities and computational divergence caused by distortion of the fixed mesh in local instability regions, ensuring the computational convergence of determining the maximum hanging weight.

Description

Technical Field

[0001] This invention relates to the field of electronic digital data processing technology, specifically to computer-aided engineering calculation and numerical simulation data processing technology. This invention discloses a design method for determining the maximum hanging weight of an ultra-thin lifting beam. Background Technology

[0002] In the design of maximum lifting weight for ultra-thin lifting beams, conventional methods rely on finite element numerical simulation for nonlinear buckling analysis. The specific implementation process is as follows: The geometric model and material property parameters of the ultra-thin lifting beam are obtained, and lifting load boundary conditions are applied at the lifting lug locations to construct an initial finite element model. During the calculation, a fixed-size mesh element is used to divide the geometric model, followed by an incremental iterative solution phase. In each incremental step, the program assembles the overall stiffness matrix and solves for the displacement increment, tracking the correspondence between load and displacement until the extreme point of the load-displacement curve is obtained. The load corresponding to this extreme point is then determined as the maximum lifting weight.

[0003] The wall thickness of the ultra-thin lifting beam is much smaller than its length and width. When approaching the critical instability state, the deformation gradient in the local area exhibits a nonlinear surge. When using the aforementioned fixed mesh for incremental iteration, the mesh elements in the high-curvature deformation region will undergo severe distortion. The distorted mesh elements cause the determinant of their local Jacobian matrix to tend to zero, leading to ill-conditioned or even singular overall stiffness matrix. This causes the iterative solution process of the nonlinear equations to diverge and be interrupted, making it impossible to obtain the extreme points of the post-buckling load-displacement curve that contain the characteristics of large mesh distortion. Summary of the Invention

[0004] The purpose of this invention is to provide a design method for determining the maximum hanging weight of an ultra-thin lifting beam, which can solve the problems in the background art mentioned above.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A design method for determining the maximum hanging weight of an ultra-thin lifting beam includes: obtaining the geometric model and material property parameters of the ultra-thin lifting beam, and applying hanging load boundary conditions to construct an initial finite element model; During the incremental step iterative solution process, the equivalent stress gradient and strain energy density index of each element are extracted in real time, and the local instability characteristic value is calculated. When the strain energy density change rate of a specific unit exceeds a preset physical threshold, a mesh reconstruction mechanism is triggered. The nodal displacement field and stress field of the specific unit and its surrounding adjacent units are extracted as initial boundary conditions. The coarse mesh unit is split into a fine mesh unit, and the coordinates of the newly added nodes are processed by smooth interpolation based on the principle of minimum potential energy. The reconstructed local mesh is coupled and assembled with the global mesh, and the nonlinear buckling analysis iteration continues until the load-displacement curve shows an extreme point. The load corresponding to the extreme point is then extracted as the maximum hanging weight.

[0006] Preferably, the calculation of local instability eigenvalues ​​includes: using the stress tensor and strain tensor of the element nodes in the current increment step as input, constructing a local Jacobian matrix by taking the partial derivative of the stress tensor, and extracting the principal strain energy direction by decomposing the eigenvalues ​​of the local Jacobian matrix; Using the specific unit as the center, search for all adjacent units within a preset physical distance, and perform a vector dot product operation between the principal strain energy direction of the adjacent units and the principal strain energy direction of the specific unit to construct a spatial influence matrix. The equivalent stress gradient of the specific unit is weighted and fused with the spatial influence matrix to output a local instability characteristic value that reflects the stress concentration propagation state, so as to identify potential instability sources of ultra-thin lifting beams under complex lifting conditions.

[0007] Preferably, when the strain energy density change rate of a specific unit exceeds a preset physical threshold, triggering the mesh reconstruction mechanism includes: extracting the current strain energy density of the unit corresponding to the current increment step, and retrieving the historical strain energy density sequence of the corresponding unit in multiple consecutive historical increment steps. Calculate the difference between the current strain energy density and the most recent historical strain energy density in the historical strain energy density sequence, and divide the difference by the time step size of the current increment step to generate the initial rate of change; A low-pass filter operator is applied to the initial rate of change to suppress high-frequency oscillation components caused by rounding errors in numerical calculations, and the smoothed rate of change of strain energy density is output. When the rate of change of strain energy density after the smoothing process is greater than the preset physical threshold, a mesh reconstruction trigger command is generated.

[0008] Preferably, the step of splitting the coarse mesh unit into fine mesh units and performing smooth interpolation processing on the coordinates of the newly added nodes based on the principle of minimum potential energy includes: identifying the topology type of the coarse mesh unit; when the topology type is a four-node quadrilateral unit, inserting new nodes at the midpoints of each edge and the centroid of the coarse mesh unit, and splitting the coarse mesh unit into four eight-node isoparametric fine mesh units. Establish a system total potential energy functional that includes the newly added node coordinate variables, wherein the system total potential energy functional includes strain energy and external force potential energy; Take the first variation of the total potential energy functional of the system with respect to the newly added node coordinate variables and set it to zero to construct a system of linear algebraic equations. By solving the system of linear algebraic equations in conjunction with the initial boundary conditions, the coordinates of the newly added nodes are obtained when the total potential energy functional of the system reaches a minimum value.

[0009] Preferably, the step of coupling and assembling the reconstructed local mesh with the global mesh includes: traversing the outer boundary of the reconstructed local mesh and extracting all boundary nodes on the outer boundary; Search for global nodes in the global grid that coincide with the spatial position of the boundary nodes, and establish a displacement mapping relationship between the boundary nodes and the global nodes; For non-overlapping suspended boundary nodes, a multi-point constraint equation is constructed between the suspended boundary node and the nearest global nodes. The multi-point constraint equation forces the displacement value of the suspended boundary node to be a weighted average of the displacement values ​​of the multiple global nodes. Based on the displacement mapping relationship and the multi-point constraint equation, a degree-of-freedom coupling matrix is ​​established. The local stiffness matrix and the global stiffness matrix are multiplied by the degree-of-freedom coupling matrix to generate the assembled global stiffness matrix.

[0010] Preferably, the step of extracting the load corresponding to the extreme point as the maximum hanging weight until the load-displacement curve reaches an extreme point includes: in each incremental step of the nonlinear buckling analysis iteration, extracting the vertical displacement component of the current load factor and the monitoring node to form the current state point in the load-displacement space; Calculate the determinant of the overall tangent stiffness matrix after the current assembly, and determine whether the sign of the determinant has changed from positive to negative. If a flip occurs, it is determined that the current state point has crossed the extreme point. The search interval is narrowed by a bisection method between the current increment step that has crossed the extreme point and the previous increment step. The nonlinear solution is iteratively executed until the displacement increment is less than the preset convergence tolerance. The product of the load factor at convergence and the applied hanging load boundary condition is output as the maximum hanging weight.

[0011] Preferably, constructing the spatial influence matrix includes: calculating the spatial distance vector from the center node of the adjacent unit to the center node of the specific unit, and normalizing the spatial distance vector to obtain a direction vector; Extract the deformation gradient tensor of the specific unit in the current increment step, and perform polar decomposition on the deformation gradient tensor to obtain the rotation tensor; The rotation tensor is used to perform coordinate rotation mapping on the direction vector to generate a corrected direction vector that follows the large deformation rotation of the ultra-thin lifting beam; Calculate the cosine of the angle between the principal strain energy direction of the adjacent unit and the correction direction vector, assign the cosine of the angle as a weighting factor to the adjacent unit, and summarize the weighting factors of all the adjacent units to generate the spatial influence matrix, so as to eliminate the interference of rigid body rotation on the spatial distribution of instability eigenvalues.

[0012] Preferably, applying a low-pass filter operator to the initial rate of change includes: extracting the historical initial rate of change sequence corresponding to the historical multiple consecutive increment steps; Set a smoothing coefficient, the value of which is inversely proportional to the time step size of the current increment step; The smoothing coefficient is used to apply an exponentially decaying weighted average to the most recent historical initial rate of change in the historical initial rate of change sequence, and then linearly fused with the initial rate of change in the current increment step to generate an intermediate rate of change. The mesh size parameters of the corresponding unit in the initial mesh generation stage are retrieved, the spatial filtering radius is calculated based on the mesh size parameters, the intermediate rate of change is subjected to Gaussian kernel convolution operation within the spatial filtering radius, and the smoothed strain energy density rate of change after eliminating time dimension numerical noise and spatial dimension numerical noise is output.

[0013] Preferably, the construction of the multi-point constraint equation includes: obtaining the coordinates of the suspended boundary node, and finding two global nodes sharing the same global grid edge within a preset search range as interpolation base points; Calculate the distance interpolation coefficient based on the projection position of the suspended boundary node on the line connecting the two global nodes; Construct initial multi-point constraint equations, and express the displacement of the suspended boundary node as a linear combination of the displacement of the two global nodes and the distance interpolation coefficient; The interface stress vector of the local mesh at the suspended boundary node is extracted, and the interface stress vector is decomposed along the normal and tangential directions of the global mesh edge. The normal stress continuity condition is introduced as a penalty term into the initial multi-point constraint equation to generate an enhanced multi-point constraint equation that includes displacement compatibility and stress compatibility.

[0014] Preferably, the step of narrowing the search interval by binary search between the current increment step and the previous increment step after crossing the extreme point includes: extracting the current load factor and the previous load factor when crossing the extreme point, and calculating the arithmetic mean of the two as the trial load factor. Solve the nonlinear equation system under the proposed load factor. If the determinant of the proposed tangent stiffness matrix is ​​positive, then update the previous load factor to the proposed load factor. If the determinant of the proposed tangent stiffness matrix is ​​negative, then update the current load factor to the proposed load factor. After each update, the minimum eigenvalue of the overall tangent stiffness matrix is ​​extracted. When the absolute value of the minimum eigenvalue is less than the preset eigenvalue tolerance, the bisection iteration is terminated, and the trial load factor is directly extracted to calculate the maximum hanging weight, so as to prevent iterative oscillation caused by the tangent stiffness matrix tending to be singular when approaching the extreme point.

[0015] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. This invention calculates local instability characteristic values ​​by extracting equivalent stress gradients and strain energy density indices in real time during incremental iterations. When the strain energy density change rate exceeds a threshold, a mesh reconstruction mechanism is triggered. Using nodal displacement and stress fields as initial boundary conditions, the coordinates of newly generated nodes are smoothed using the minimum potential energy principle. Subsequently, the local and global meshes are coupled and assembled for further iteration. This design, while maintaining the global mesh topology, achieves mesh refinement and field variable mapping in high-curvature deformation regions, avoiding severe distortions caused by fixed meshes in local instability areas. It also eliminates the solution divergence problem caused by singularities in the overall stiffness matrix due to mesh distortion, ensuring that the nonlinear buckling analysis iterative convergence to the extreme point.

[0016] 2. By constructing a spatial influence matrix and introducing a rotation tensor of the deformation gradient tensor for coordinate rotation mapping, the interference of rigid body rotation on the spatial distribution calculation of instability eigenvalues ​​is eliminated. A low-pass filter operator is applied to the initial rate of change to suppress the high-frequency oscillation components in the time and spatial dimensions caused by rounding errors in numerical calculations, thus eliminating misjudgments of mesh reconstruction triggering conditions by non-physical factors. For suspended boundary nodes where the local and global mesh boundaries do not coincide, an enhanced multi-point constraint equation containing a penalty term for the continuity condition of normal stress is constructed to eliminate discontinuities in boundary displacement and stress transmission after local mesh reconstruction. In the extreme point search stage, a bisection strategy combining the sign reversal judgment of the determinant of the tangent stiffness matrix and the termination of the minimum eigenvalue tolerance is adopted to eliminate the iterative oscillation phenomenon caused by the tangent stiffness matrix tending to be singular when approaching the extreme point. Attached Figure Description

[0017] Figure 1 Overall flowchart of the design method for determining the maximum hanging weight of ultra-thin lifting beams; Figure 2 A flowchart illustrating the method for calculating local instability eigenvalues; Figure 3 Flowchart of the method for triggering mesh reconstruction; Figure 4 Flowchart of the method for mesh splitting and node smoothing interpolation; Figure 5 Flowchart of the method for assembling degrees of freedom coupled together; Figure 6A flowchart illustrating the method for extracting extreme points and determining the maximum hanging weight. Detailed Implementation

[0018] Please refer to Figure 1 This embodiment provides a design method for determining the maximum lifting weight of an ultra-thin lifting beam. It obtains the geometric model and material property parameters of the ultra-thin lifting beam and applies lifting load boundary conditions to construct an initial finite element model. Specifically, the geometric model is exported using 3D computer-aided design software. Before exporting, the geometric model is cleaned up, removing redundant features such as chamfers, small holes, and fillets that do not affect the overall mechanical properties, simplifying the geometric topology, and ensuring the continuity and quality of subsequent mesh generation. The geometric model includes core geometric parameters such as the length, width, wall thickness, flange-web connection structure, and installation position and dimensions of the lifting lugs of the ultra-thin lifting beam, adapting to the actual lifting conditions in engineering. The material property parameters are obtained through uniaxial tensile tests, including core mechanical parameters such as the material's elastic modulus, Poisson's ratio, initial yield strength, hardening modulus in the plastic stage, and density. The parameter values ​​correspond to the room temperature mechanical properties of the high-strength low-alloy steel used in the ultra-thin lifting beam, ensuring that the material properties are consistent with the actual working conditions.

[0019] During the application of the hanging load boundary conditions, a fixed constraint with full degrees of freedom is applied to the inner surface of the mounting holes of the lifting lugs at both ends of the ultra-thin lifting beam to simulate the constraint effect of the crane hook on the lifting lugs in actual working conditions, thus restricting the translational and rotational degrees of freedom of the mounting holes of the lifting lugs. A reference concentrated load along the direction of gravity is applied at the preset hanging position on the lower flange of the lifting beam. The value of the reference concentrated load is initially set based on the design bearing capacity of the lifting beam. The scaling of the load is achieved through the load factor, which is adapted to the incremental step iterative solution process of nonlinear buckling analysis.

[0020] The initial finite element model was meshed using a mapped meshing method, with separate meshes created for different structural components of the lifting beam, such as the flanges, web, and lugs. Four-node quadrilateral reduced-integral shell elements were selected as the element type to meet the bending deformation calculation requirements of the thin-walled structure of the ultra-thin lifting beam. This reduced-integral scheme avoids the shear locking problem that occurs with shell elements during bending deformation. The average size of the initial elements was set based on the overall geometry of the lifting beam, ensuring that the aspect ratio of the initial mesh elements was less than 3 and the determinant of the element Jacobian matrix was greater than 0.7. This prevented quality defects in the initial mesh that could affect the accuracy and convergence of subsequent nonlinear calculations.

[0021] After constructing the initial finite element model, the incremental iterative solution process begins. The Newton-Raphson iterative method is used for nonlinear buckling analysis. Each incremental step corresponds to an increment of the load factor, with the initial step size set to 0.05 times the load factor. During iteration, the step size is automatically adjusted based on the convergence status. After convergence at each incremental step, the equivalent stress and strain data at the integration point of each element in the model are extracted in real time, and the equivalent stress gradient and strain energy density index of each element are calculated.

[0022] The equivalent stress is the Mises equivalent stress, and the calculation formula is:

[0023] in, This refers to the Mises equivalent stress at the unit integration point; For the deviatoric stress tensor, ; For Cauchy stress tensor; The trace of the stress tensor; A second-order unit tensor; double dot product operator This represents the summation of the corresponding components of two second-order tensors after multiplying them.

[0024] The formula for calculating the equivalent stress gradient is:

[0025] in, This is the equivalent stress gradient vector; , , These are the unit vectors in the three coordinate axes of the global Cartesian coordinate system; the partial derivatives are obtained by interpolating the partial derivatives of the element shape function with respect to the coordinates and the equivalent stress values ​​of the element nodes.

[0026] The formula for calculating strain energy density is:

[0027] in, is the strain energy density at the unit integration point; The Green strain tensor is obtained through displacement field interpolation of the element nodes.

[0028] Based on the extracted equivalent stress gradient and strain energy density indices, the local instability characteristic value of each element is calculated. This characteristic value characterizes the risk of local instability in the element; a higher characteristic value indicates a greater risk of local instability. For each element, the average strain energy density corresponding to the current increment step is extracted. The historical strain energy density sequence of the element over the past five consecutive convergence increment steps is retrieved. The difference between the current strain energy density and the most recent historical strain energy density is calculated. This difference is divided by the time step size of the current increment step to generate the initial rate of change of strain energy density. After smoothing, the initial rate of change is compared with a preset physical threshold. This threshold is based on the strain energy density change rate corresponding to the uniaxial tensile yielding of the material, ensuring that the triggering timing corresponds to the critical state before the element enters plastic deformation. When the smoothed strain energy density change rate exceeds the preset physical threshold, the element is identified as a specific element, triggering the mesh reconstruction mechanism.

[0029] After triggering the mesh reconstruction mechanism, the nodal displacement and stress fields of the specific element and its surrounding adjacent elements are extracted as initial boundary conditions. The surrounding adjacent elements are all elements that share nodes or edges with the specific element, ensuring the continuity of field variables in the local mesh reconstruction region. The coarse mesh elements within the reconstruction region are split into fine mesh elements, maintaining the external boundary topology of the reconstruction region unchanged during the splitting process to avoid affecting the topology of the global mesh. For newly added nodes generated during the splitting process, smooth interpolation is performed on the coordinates of the new nodes based on the principle of minimum potential energy, ensuring uniform coordinate distribution, avoiding distortion in fine mesh elements, and improving mesh quality.

[0030] The total potential energy functional of the system includes strain energy and external force potential energy, and its expression is:

[0031] in, The total potential energy functional of the system; This is the coordinate vector of the newly added node; The volume domain of the split fine mesh cells; Force boundary for fine mesh elements; The surface force vector on the boundary; Let be the displacement vector related to the coordinates of the newly added nodes. Take the first variation of the total potential energy functional of the system with respect to the coordinates of the newly added nodes and set it to zero, i.e. A system of linear algebraic equations is constructed, and the extracted nodal displacement field and stress field are used as initial boundary conditions. The system of linear algebraic equations is solved to obtain the coordinates of the new nodes that minimize the total potential energy functional of the system, and the smooth interpolation processing of the new nodes is completed.

[0032] After reconstructing and smoothing the local mesh, the reconstructed local mesh is coupled and assembled with the global mesh using degrees of freedom. The outer boundary of the reconstructed local mesh is traversed, and all boundary nodes on the outer boundary are extracted. Global nodes that spatially coincide with the boundary nodes are searched in the global mesh, and a displacement mapping relationship is established between the boundary nodes and global nodes to ensure complete coupling of the degrees of freedom and continuous displacement of the coincident nodes. For suspended boundary nodes that do not spatially coincide, multi-point constraint equations are constructed to force the displacements of the suspended boundary nodes to meet compatibility conditions with the displacements of the surrounding global nodes, ensuring continuous displacement transfer between the local and global meshes. Based on the displacement mapping relationship and the multi-point constraint equations, a degree-of-freedom coupling matrix is ​​established. The local stiffness matrix of the local mesh and the global stiffness matrix of the global mesh are multiplied using the degree-of-freedom coupling matrix to generate the assembled global stiffness matrix, completing the degree-of-freedom coupling and assembly of the local and global meshes.

[0033] After assembling the overall stiffness matrix, incremental iterations of the nonlinear buckling analysis are continued. Upon convergence of each incremental step, the state point of the load-displacement curve is updated. The horizontal axis of the load-displacement curve represents the vertical displacement component of the monitoring node, which is the center node of the lifting beam's hanging position. The vertical axis represents the hanging load value corresponding to the current incremental step. During the iteration process, changes in the load-displacement curve are monitored in real time. When an extreme point appears on the load-displacement curve, the iteration process is terminated, and the load value corresponding to that extreme point is extracted as the maximum hanging weight of the ultra-thin lifting beam. The extreme point is the critical point in the load-displacement curve where the load value changes from rising to falling, corresponding to the critical load at which the lifting beam experiences overall instability, at which point the lifting beam's load-bearing capacity reaches its maximum value.

[0034] Table 1 Initial Finite Element Model Element Topology and Calculation Parameter Settings ; Specifically, the parameters given in Table 1 are set as the basic calculation parameters of the initial finite element model. All parameters are adapted to the nonlinear buckling analysis requirements of the ultrathin lifting beam. The reduction integration scheme and the setting of multi-point integration in the thickness direction ensure the calculation accuracy of the bending deformation of the thin-walled structure. The setting of convergence tolerance balances the calculation accuracy and calculation efficiency. The setting of preset physical threshold ensures the rationality of the mesh reconstruction triggering time and avoids the element distortion problem caused by triggering too late.

[0035] In this embodiment, the equivalent stress gradient and strain energy density indices of the elements are extracted in real time during incremental step iteration to calculate local instability characteristic values. When the strain energy density change rate exceeds a preset physical threshold, a mesh reconstruction mechanism is triggered. Based on the minimum potential energy principle, the coordinates of newly added nodes are smoothed by interpolation. After completing the local mesh reconstruction, it is coupled and assembled with the global mesh to continue nonlinear buckling analysis iteration until the load-displacement curve reaches an extreme point, and the corresponding maximum hanging weight is extracted. This process maintains the global mesh topology unchanged while achieving mesh refinement and field variable mapping in high curvature deformation regions. It avoids severe distortion caused by fixed meshes in local instability regions, eliminates the solution divergence problem caused by singularity of the overall stiffness matrix due to mesh distortion, and ensures that the nonlinear buckling analysis iteration converges to the extreme point.

[0036] refer to Figure 2 In a preferred embodiment, during the incremental step iterative solution process, the stress tensor and strain tensor of the element nodes in the current incremental step are used as input. A local Jacobian matrix is ​​constructed by taking the partial derivative of the stress tensor, and the principal strain energy direction is extracted using the eigenvalue decomposition of the local Jacobian matrix. The local Jacobian matrix is ​​the tangent modulus matrix of the element, and its expression is:

[0037] in, Let be the local Jacobian matrix of the element, representing the rate of change of stress with strain, reflecting the tangential stiffness characteristics of the element under the current deformation state. The local Jacobian matrix is ​​decomposed into eigenvalues, and the decomposition formula is as follows:

[0038] in, The first local Jacobian matrix One eigenvalue; For each eigenvector corresponding to a eigenvalue, the eigenvector corresponding to the largest eigenvalue is selected as the principal strain energy direction of the element. The principal strain energy direction is the direction in which strain energy accumulates the fastest within the element, corresponding to the potential expansion direction of local instability.

[0039] Centered on a specific element whose principal strain energy direction is calculated, a search is performed on all adjacent elements within a predetermined physical distance. This predetermined physical distance is the radius of a spherical region centered on the centroid of the specific element, and the radius is twice the average size of the initial elements, ensuring the search area covers adjacent regions where stress concentration may propagate. The principal strain energy directions of all adjacent elements are then multiplied by the principal strain energy direction of the specific element to construct a spatial influence matrix. The expression for the vector multiplication operation is:

[0040] in, For the first Weighting factors corresponding to each adjacent unit; The unit vector representing the direction of the principal strain energy of a specific element; For the first The principal strain energy directions of each adjacent element are unit vectors. The weighting factor ranges from -1 to 1. When the weighting factor is close to 1, it indicates that the principal strain energy directions of adjacent elements are highly consistent with those of the specific element, and the stress concentration propagates strongly along this direction. When the weighting factor is close to -1, it indicates that the principal strain energy directions of adjacent elements are opposite to those of the specific element, and the stress concentration propagates weakly along this direction. The weighting factors of all adjacent elements are summarized to construct a dimension of... Spatial influence matrix ,in The number of neighboring units obtained from the search.

[0041] The equivalent stress gradient of a specific element is weighted and fused with the spatial influence matrix to output a local instability characteristic value reflecting the stress concentration propagation state. The expression for the weighted fusion is as follows:

[0042] in, This represents a local instability characteristic value. This is the weighted fusion coefficient, with a value range of [0,1]. Let be the magnitude of the equivalent stress gradient; This represents the strain energy density column vector of adjacent elements. By weighted fusion, the stress concentration of the element itself is combined with the strain energy distribution of surrounding elements to comprehensively characterize the risk of local instability of the element and identify potential instability sources of ultra-thin lifting beams under complex lifting conditions.

[0043] In constructing the spatial influence matrix, the spatial distance vector from the center node of an adjacent cell to the center node of a specific cell is calculated. This spatial distance vector is then normalized to obtain the direction vector. The center node of an adjacent cell is the centroid node of the cell. The expression for the spatial distance vector is: ,in For the centroid coordinate vector of a specific unit, For the first The centroid coordinate vectors of adjacent units, after normalization, are the direction vectors. ,in Let be the magnitude of the spatial distance vector.

[0044] Extract the deformed gradient tensor of a specific unit at the current increment step, and perform polar decomposition on the deformed gradient tensor to obtain the rotation tensor. The expression for the deformed gradient tensor is as follows: ,in The displacement gradient tensor is obtained through displacement field interpolation of the element nodes. The deformation gradient tensor is then decomposed into its extreme values, as shown in the following formula:

[0045] in, Let be a rotation tensor, and be an orthogonal tensor, satisfying . This characterizes the rigid body rotation of the element during the deformation process; is the right elongation tensor, and is the symmetric positive definite tensor, which characterizes the pure deformation of the element during the deformation process.

[0046] By using a rotation tensor to perform coordinate rotation mapping on the direction vector, a corrected direction vector is generated that follows the large deformation rotation of the ultra-thin lifting beam. The expression for the coordinate rotation mapping is as follows: ,in This is the corrected direction vector. By mapping the rotation tensor, the direction vector in the global coordinate system is transformed into a direction vector in the local coordinate system that rotates with the rigid body of the element, eliminating the influence of the rigid body rotation on the direction vector. The cosine of the angle between the principal strain energy directions of adjacent elements and the corrected direction vector is calculated, and this cosine is used as a weighting factor for adjacent elements. The weighting factors of all adjacent elements are then aggregated to generate a spatial influence matrix, eliminating the interference of rigid body rotation on the spatial distribution of instability eigenvalues.

[0047] Table 2. Parameters for constructing the spatial influence matrix of adjacent units ; Specifically, the parameters given in Table 2 are the core calculation parameters for the spatial influence matrix construction process. The setting of the search radius ensures complete coverage of the stress concentration propagation area; the selection of the polar decomposition algorithm ensures the accuracy and stability of the rotation tensor solution; the setting of the weighted fusion coefficient balances the contribution of the element's own state and the state of surrounding elements to the instability eigenvalue; and the activation of the rotation tensor correction switch eliminates the interference of large deformation rigid body rotation on the identification of instability sources.

[0048] In this embodiment, a local Jacobian matrix is ​​constructed and eigenvalue decomposition is performed to extract the principal strain energy direction. A spatial influence matrix is ​​then constructed by combining the principal strain energy directions of adjacent elements. The equivalent stress gradient and the spatial influence matrix are weighted and fused to obtain local instability eigenvalues. Simultaneously, a rotation tensor is obtained through the extreme decomposition of the deformation gradient tensor, and the direction vector is subjected to coordinate rotation mapping to correct the weighting factor of the spatial influence matrix. This process accurately identifies potential instability sources of ultra-thin lifting beams under complex lifting conditions, eliminates the interference of rigid body rotation on the spatial distribution calculation of instability eigenvalues, and improves the accuracy of local instability risk assessment.

[0049] refer to Figure 3In another preferred embodiment, when the strain energy density change rate of a specific element exceeds a preset physical threshold, the mesh reconstruction mechanism is triggered. During this process, the current strain energy density of the element corresponding to the current increment step is extracted, and the historical strain energy density sequence of the corresponding element in the past five consecutive convergence increment steps is retrieved. The historical strain energy density sequence is stored in the solution order of the increment steps. The strain energy density corresponding to each increment step is the average value of the strain energy density at the element integration point, ensuring the temporal consistency of the strain energy density data.

[0050] Calculate the difference between the current strain energy density and the most recent historical strain energy density in the historical strain energy density sequence, and divide the difference by the time step size of the current increment step to generate the initial rate of change. The formula for calculating the initial rate of change is:

[0051] in, For the current increment step The corresponding initial rate of change of strain energy density; This represents the element average strain energy density corresponding to the current increment step. The average strain energy density of the unit corresponding to the most recent increment step in the historical strain energy density sequence; This represents the time step size of the current increment step. The initial rate of change characterizes the rate at which the strain energy density changes with time, reflecting the speed at which unit energy accumulates.

[0052] A low-pass filter operator is applied to the initial rate of change to suppress high-frequency oscillations caused by rounding errors in numerical calculations, outputting a smoothed strain energy density rate of change. The low-pass filter operator consists of two parts: time-dimensional filtering and spatial-dimensional filtering. First, the initial rate of change is subjected to exponential smoothing in the time dimension, extracting the historical initial rate of change sequence corresponding to multiple consecutive increment steps. A smoothing coefficient is set, and its value is inversely proportional to the time step size of the current increment step. The expression for the smoothing coefficient is: ,in The smoothing coefficient corresponding to the current increment step. The baseline smoothing coefficient is 0.2. The base time step is set to the initial step size of the increment step. As the increment step time step decreases, the smoothing coefficient increases, improving the ability to suppress high-frequency oscillation components.

[0053] An exponentially decaying weighted average is applied to the most recent historical initial rate of change in the historical initial rate of change sequence using a smoothing coefficient, and then linearly fused with the initial rate of change in the current increment step to generate an intermediate rate of change. The formula for calculating the exponential smoothing filter is as follows:

[0054] in, This represents the intermediate rate of change corresponding to the current increment step; This represents the intermediate rate of change corresponding to the most recent increment step in the historical initial rate of change sequence. Exponential smoothing filtering suppresses high-frequency oscillations caused by numerical rounding errors in the time dimension, thus eliminating numerical noise in the time dimension.

[0055] The mesh size parameters of the corresponding element during the initial mesh generation stage are retrieved. The spatial filtering radius is calculated based on these parameters. A Gaussian kernel convolution operation is performed on the intermediate rate of change within the spatial filtering radius. The output is the smoothed rate of change of strain energy density after eliminating temporal and spatial numerical noise. The spatial filtering radius is set to 1.5 times the initial mesh size of the element to ensure that the filtering range covers the adjacent areas surrounding the element. The expression for the Gaussian kernel convolution operation is as follows:

[0056] in, The rate of change of strain energy density after smoothing; This is the normalization constant; The spatial domain corresponding to the spatial filtering radius; Location within the spatial domain The rate of change at the intermediate point; The Gaussian kernel function; Let be the standard deviation of the Gaussian kernel. The expression for the Gaussian kernel function is:

[0057] in, The centroid coordinate vector of the target element; Let be the Euclidean distance between two points in space. The standard deviation of the Gaussian kernel. The value is set to 1 / 3 of the spatial filtering radius to ensure that the attenuation characteristics of the Gaussian kernel function are adapted to the spatial filtering range. Through Gaussian kernel convolution, high-frequency oscillation components caused by numerical calculation rounding errors in the spatial dimension are suppressed, thus eliminating numerical noise in the spatial dimension.

[0058] When the change rate of strain energy density after smoothing exceeds a preset physical threshold, a mesh reconstruction trigger command is generated. The preset physical threshold is set based on the change rate of strain energy density of the material's yield strain energy density, and is set to 0.8 times the change rate of strain energy density corresponding to the uniaxial tensile yield of the material. This ensures that the mesh reconstruction is triggered before the elements undergo plastic deformation and severe mesh distortion, avoiding element distortion and solution divergence problems caused by triggering too late.

[0059] Table 3. Strain Energy Density Change Rate Filtering and Trigger Threshold Parameters ; Specifically, the parameters given in Table 3 are the core parameters for calculating the strain energy density change rate and triggering the mesh reconstruction process. Among them, the setting of the number of historical increment steps ensures the stability of the time dimension filtering; the adaptive adjustment mechanism of the smoothing coefficient adapts to the filtering requirements of different increment step sizes; the setting of the spatial filtering radius and the standard deviation of the Gaussian kernel balances the noise suppression capability and signal fidelity; the activation of the dual filtering switch eliminates numerical noise in both the time and spatial dimensions, avoiding misjudgment of the mesh reconstruction triggering conditions by non-physical factors.

[0060] In this embodiment, the initial rate of change is calculated by extracting the historical sequence of strain energy density. A low-pass filter operator, which includes exponential smoothing in the time dimension and Gaussian kernel convolution in the spatial dimension, is applied to the initial rate of change to suppress high-frequency oscillation components caused by rounding errors in numerical calculations. A mesh reconstruction trigger command is generated based on the comparison between the smoothed strain energy density rate of change and a preset physical threshold. This process eliminates misjudgments of mesh reconstruction trigger conditions due to non-physical factors, ensures the accuracy of mesh reconstruction trigger timing, and avoids reduced computational efficiency due to false triggering and element distortion problems caused by late triggering.

[0061] refer to Figure 4 In another preferred embodiment, during the process of splitting the coarse mesh unit into fine mesh units and performing smooth interpolation processing on the coordinates of the newly added nodes based on the principle of minimum potential energy, the topology type of the coarse mesh unit is identified. When the topology type is a four-node quadrilateral unit, new nodes are inserted at the midpoints of each edge and the centroid of the coarse mesh unit, splitting the coarse mesh unit into four eight-node isoparametric fine mesh units. Specifically, the four edges of the four-node quadrilateral unit are the four topological edges of the unit, and the two endpoints of each edge are the original nodes of the coarse mesh unit. A midpoint node is inserted at the midpoint of each edge, and an internal node is inserted at the centroid of the unit. A total of 5 new nodes are added, including 4 midpoint nodes and 1 centroid node. By adding nodes, the original four-node quadrilateral unit is split into four eight-node isoparametric fine mesh units. Each eight-node isoparametric element contains 2 original nodes of the original coarse mesh unit, 2 newly added midpoint nodes, and 1 newly added centroid node. Eight-node isoparametric elements are quadratic elements, which have higher fitting accuracy for high curvature deformation than four-node linear elements. They can effectively adapt to the large deformation calculation requirements of local unstable regions and avoid element distortion.

[0062] Establish a system total potential energy functional that includes the coordinate variables of the newly added nodes. The system total potential energy functional contains strain energy and external force potential energy. Take the first variation of the system total potential energy functional with respect to the coordinate variables of the newly added nodes and set it to zero to construct a system of linear algebraic equations. ,in The stiffness matrix is ​​given by the coordinates of the newly added node. The nodal force vectors, relating to the coordinates of newly added nodes, are constructed based on the shape functions and material property parameters of the split fine mesh elements. A system of linear algebraic equations is solved using initial boundary conditions. These initial boundary conditions consist of the nodal displacement and stress fields of the extracted specific element and its adjacent elements. The coordinates and displacements of the original nodes are substituted into the linear algebraic equations as fixed boundary conditions, eliminating the rigid body displacement degrees of freedom in the equations. The optimal coordinates of the newly added nodes are then obtained. These coordinates minimize the total potential energy functional of the system, ensuring uniform node distribution and good mesh compliance in the fine mesh elements, preventing distortion in the fine mesh elements, and improving mesh quality.

[0063] refer to Figure 5 During the assembly of the reconstructed local mesh and the global mesh with degrees of freedom, the outer boundary of the reconstructed local mesh is traversed, and all boundary nodes on the outer boundary are extracted. The outer boundary of the local mesh is the shared boundary between the reconstructed region and the global non-reconstructed region, and the boundary nodes are all nodes located on this shared boundary. Global nodes that coincide with the spatial position of the boundary nodes are searched in the global mesh. During the search, a position tolerance is set, which is set to 1 × 10⁻⁶ times the average size of the initial elements. -6 To avoid misjudgment of node overlap due to numerical calculation errors, when the spatial distance between a boundary node and a global node is less than the position tolerance, the two nodes are determined to be spatially coincident. A displacement mapping relationship is established between boundary nodes and global nodes. This displacement mapping relationship forces the coincident boundary nodes and global nodes to have completely equal displacement values ​​in all degrees of freedom, achieving full coupling of the degrees of freedom of the coincident nodes.

[0064] For non-overlapping suspended boundary nodes, multi-point constraint equations are constructed between the suspended boundary node and its nearest global nodes. These multi-point constraint equations force the displacement of the suspended boundary node to be a weighted average of the displacements of the global nodes. During the construction of the multi-point constraint equations, the coordinates of the suspended boundary node are obtained. Within a preset search range, two global nodes sharing the same global mesh edge are found as interpolation base points. The preset search range is set to twice the average size of the initial element to ensure that the interpolation base points are located in the surrounding area of ​​the suspended boundary node. Based on the projection position of the suspended boundary node onto the line connecting the two global nodes, the distance interpolation coefficient is calculated. The expression for the distance interpolation coefficient is: , ,in Let be the coordinate vector of the suspended boundary node. , These are the coordinate vectors of the two interpolation base points. , These are the distance interpolation coefficients corresponding to the two interpolation base points, and the sum of the distance interpolation coefficients is 1 to ensure the linearity of the interpolation.

[0065] An initial multi-point constraint equation is constructed, representing the displacement of the suspended boundary node as a linear combination of the displacement and distance interpolation coefficients of two global nodes. The interface stress vector at the suspended boundary node is extracted from the local mesh. This interface stress vector is decomposed along the normal and tangential directions of the global mesh edges. The normal stress continuity condition is introduced as a penalty term into the initial multi-point constraint equation, generating an enhanced multi-point constraint equation that includes displacement and stress compatibility. The expression for the enhanced multi-point constraint equation is:

[0066] in, For the displacement components of the suspended boundary nodes; , These are the corresponding displacement components of the two interpolation base points; The penalty factor has a value of 1×10. -3 The material's elastic modulus is times that of the material itself. For the normal stress components at the suspended boundary node; , These represent the normal stress components at the two interpolation base points. By introducing a penalty term for the normal stress continuity condition, the continuity of stress transfer between the local and global meshes is ensured, eliminating the discontinuity error in boundary displacement and stress transfer after local mesh reconstruction.

[0067] refer to Figure 6 Based on the displacement mapping relationship and multi-point constraint equations, a degree-of-freedom coupling matrix is ​​established. The local stiffness matrix and the global stiffness matrix are multiplied using the degree-of-freedom coupling matrix to generate the assembled global stiffness matrix. The degree-of-freedom coupling matrix is ​​a combination of a Boolean matrix and a constraint coefficient matrix. The number of rows in the matrix represents the total number of degrees of freedom of the overall model, and the number of columns represents the number of degrees of freedom of the local mesh. The elements in the matrix correspond to the coupling relationship and constraint coefficients between the nodal degrees of freedom. The local stiffness matrix is ​​obtained by assembling the element stiffness matrices of the reconstructed local mesh, and the global stiffness matrix is ​​obtained by assembling the element stiffness matrices of the non-reconstructed region. The local stiffness matrix is ​​mapped to the global degree-of-freedom space using the degree-of-freedom coupling matrix and superimposed with the global stiffness matrix to generate the assembled global stiffness matrix, thus completing the degree-of-freedom coupling assembly of the local and global meshes.

[0068] Until the load-displacement curve reaches an extreme point, the load corresponding to the extreme point is extracted as the maximum suspended weight. In each incremental step of the nonlinear buckling analysis iteration, the current load factor and the vertical displacement component of the monitoring node are extracted to form the current state point in the load-displacement space. The monitoring node is the center node of the lifting beam's suspension position, the vertical displacement component is the displacement component along the gravity direction, the load factor is the ratio of the load value of the current incremental step to the initially applied reference load value, the horizontal axis of the load-displacement space is the vertical displacement component of the monitoring node, and the vertical axis is the product of the load factor and the reference load value, i.e., the suspended load value corresponding to the current incremental step.

[0069] Calculate the determinant of the current assembled global tangent stiffness matrix and determine if the sign of the determinant has reversed from positive to negative. The global tangent stiffness matrix is ​​the global stiffness matrix under the current deformation state during the nonlinear iteration process. The sign of its determinant reflects the positive definiteness of the stiffness matrix. When the determinant is positive, the global tangent stiffness matrix is ​​positive definite, and the structure is in a stable equilibrium state; when the determinant is negative, the global tangent stiffness matrix is ​​non-positive definite, and the structure is in an unstable equilibrium state; when the determinant is zero, the global tangent stiffness matrix is ​​singular, and the structure is in a critical equilibrium state, corresponding to the extreme point of the load-displacement curve. During the iteration process, record the sign of the determinant of the global tangent stiffness matrix after convergence at each increment step. When the sign of the determinant changes from positive to negative between two adjacent increment steps, it is determined that the current state point has crossed the extreme point.

[0070] If a flip occurs, the current state point is determined to have crossed an extreme point. A bisection method is used to narrow the search interval between the current increment step and the previous increment step after crossing the extreme point. Nonlinear solutions are iteratively executed until the displacement increment is less than the preset convergence tolerance. The product of the load factor at convergence and the applied hanging load boundary conditions is output as the maximum hanging weight. During the bisection search, the current load factor and the previous load factor at the time of crossing the extreme point are extracted, and the previous load factor is set as the lower limit of the search interval. The current load factor is set to the upper limit of the search interval. The arithmetic mean of the two is calculated as the trial loading factor. The formula for calculating the trial loading factor is:

[0071] Solve the nonlinear equations under the experimental load factor to obtain the global tangent stiffness matrix under the experimental deformation state. Calculate the determinant of the experimental tangent stiffness matrix. If the determinant of the experimental tangent stiffness matrix is ​​positive, update the previous load factor to the experimental load factor. If the determinant of the trial tangent stiffness matrix is ​​negative, then the current load factor is updated to the trial load factor, i.e. After each update, the minimum eigenvalue of the overall tangent stiffness matrix is ​​extracted. When the absolute value of the minimum eigenvalue is less than the preset eigenvalue tolerance, the bisection iteration is terminated, and the trial load factor is directly extracted to calculate the maximum suspended weight. The preset eigenvalue tolerance is set to 1×10. -1 To prevent iterative oscillations caused by the tangent stiffness matrix becoming singular when approaching extreme points, the displacement increment of the monitoring node is calculated synchronously during the bisection method iteration. The iteration terminates when the displacement increment is less than a preset convergence tolerance, which is set to 1×10⁻⁶. -6 The maximum lifting weight of the ultra-thin lifting beam is calculated as the product of the load factor at convergence and the reference hanging load, which is times the total length of the lifting beam.

[0072] Table 4. Parameter Recording Table for the Iterative Convergence Process of the Bisection Method ; Specifically, Table 4 records the complete process of the bisection method iterative search for the extreme point. The initial search interval has a lower limit of 0.85 and an upper limit of 0.9, corresponding to the load factors of two adjacent increment steps where the determinant sign changes from positive to negative. As the number of iterations increases, the search interval gradually shrinks, the trial load factor gradually approaches the critical load factor corresponding to the extreme point, the absolute value of the minimum eigenvalue of the overall tangent stiffness matrix gradually decreases, and the displacement increment of the monitoring node gradually decreases. When iterating to the 6th time, the absolute value of the minimum eigenvalue is less than the preset eigenvalue tolerance, and the displacement increment is less than the preset convergence tolerance, indicating that the iteration has converged. The trial load factor at convergence, 0.8828125, is output, and the maximum lifting weight of the ultra-thin lifting beam is calculated by combining it with the benchmark hanging load. This bisection method iterative process combines the determination of the determinant sign of the tangent stiffness matrix with the termination condition of the minimum eigenvalue tolerance, avoiding iterative oscillations caused by the tangent stiffness matrix tending to be singular when approaching the extreme point, and ensuring the convergence and accuracy of the extreme point search.

[0073] In this embodiment, by splitting coarse mesh elements into eight-node isoparametric fine mesh elements, and performing smooth interpolation on the coordinates of the newly added nodes based on the minimum potential energy principle, the quality of the reconstructed mesh is ensured. By constructing an enhanced multi-point constraint equation that includes displacement and stress compatibility, and establishing a degree-of-freedom coupling matrix to assemble the local and global meshes, the continuity of displacement and stress transfer is guaranteed. The extreme point crossing state is determined by flipping the sign of the determinant of the tangent stiffness matrix, and a bisection method combining the minimum eigenvalue tolerance termination condition is used to search for the critical load factor, accurately obtaining the extreme points of the load-displacement curve. This process eliminates the discontinuity error in boundary displacement and stress transfer after local mesh reconstruction, avoids the iterative oscillation phenomenon caused by the tangent stiffness matrix tending towards singularity when approaching extreme points, and ensures the accuracy and convergence of the maximum suspended weight calculation.

Claims

1. A design method for determining the maximum hanging weight of an ultra-thin lifting beam, characterized in that, include: Obtain the geometric model and material property parameters of the ultrathin lifting beam, and apply the hanging load boundary conditions to construct the initial finite element model; During the incremental step iterative solution process, the equivalent stress gradient and strain energy density index of each element are extracted in real time, and the local instability characteristic value is calculated. When the strain energy density change rate of a specific unit exceeds a preset physical threshold, a mesh reconstruction mechanism is triggered. The nodal displacement field and stress field of the specific unit and its surrounding adjacent units are extracted as initial boundary conditions. The coarse mesh unit is split into a fine mesh unit, and the coordinates of the newly added nodes are processed by smooth interpolation based on the principle of minimum potential energy. The reconstructed local mesh is coupled and assembled with the global mesh, and the nonlinear buckling analysis iteration continues until the load-displacement curve shows an extreme point. The load corresponding to the extreme point is then extracted as the maximum hanging weight.

2. The design method for determining the maximum hanging weight of an ultra-thin lifting beam according to claim 1, characterized in that, The calculation of local instability eigenvalues ​​includes: taking the stress tensor and strain tensor of the unit nodes in the current increment step as input, constructing a local Jacobian matrix by taking the partial derivative of the stress tensor, and extracting the principal strain energy direction by decomposing the eigenvalues ​​of the local Jacobian matrix; Using the specific unit as the center, search for all adjacent units within a preset physical distance, and perform a vector dot product operation between the principal strain energy direction of the adjacent units and the principal strain energy direction of the specific unit to construct a spatial influence matrix. The equivalent stress gradient of the specific unit is weighted and fused with the spatial influence matrix to output a local instability characteristic value that reflects the stress concentration propagation state.

3. The design method for determining the maximum hanging weight of an ultra-thin lifting beam according to claim 1, characterized in that, When the strain energy density change rate of a specific unit exceeds a preset physical threshold, the mesh reconstruction mechanism is triggered by: extracting the current strain energy density of the unit corresponding to the current increment step, and retrieving the historical strain energy density sequence of the corresponding unit in multiple consecutive historical increment steps. Calculate the difference between the current strain energy density and the most recent historical strain energy density in the historical strain energy density sequence, and divide the difference by the time step size of the current increment step to generate the initial rate of change; A low-pass filter operator is applied to the initial rate of change to suppress high-frequency oscillation components caused by rounding errors in numerical calculations, and the smoothed rate of change of strain energy density is output. When the rate of change of strain energy density after the smoothing process is greater than the preset physical threshold, a mesh reconstruction trigger command is generated.

4. The design method for determining the maximum hanging weight of an ultra-thin lifting beam according to claim 1, characterized in that, The step of splitting the coarse mesh unit into fine mesh units and performing smooth interpolation processing on the coordinates of the newly added nodes based on the principle of minimum potential energy includes: identifying the topology type of the coarse mesh unit; when the topology type is a four-node quadrilateral unit, inserting new nodes at the midpoints of each edge and the centroid of the coarse mesh unit, and splitting the coarse mesh unit into four eight-node isoparametric fine mesh units. Establish a system total potential energy functional that includes the newly added node coordinate variables, wherein the system total potential energy functional includes strain energy and external force potential energy; Take the first variation of the total potential energy functional of the system with respect to the newly added node coordinate variables and set it to zero to construct a system of linear algebraic equations. By solving the system of linear algebraic equations in conjunction with the initial boundary conditions, the coordinates of the newly added nodes are obtained when the total potential energy functional of the system reaches a minimum value.

5. The design method for determining the maximum hanging weight of an ultra-thin lifting beam according to claim 1, characterized in that, The process of coupling and assembling the reconstructed local mesh with the global mesh includes: traversing the outer boundary of the reconstructed local mesh and extracting all boundary nodes on the outer boundary; Search for global nodes in the global grid that coincide with the spatial position of the boundary nodes, and establish a displacement mapping relationship between the boundary nodes and the global nodes; For non-overlapping suspended boundary nodes, a multi-point constraint equation is constructed between the suspended boundary node and the nearest global nodes. The multi-point constraint equation forces the displacement value of the suspended boundary node to be a weighted average of the displacement values ​​of the multiple global nodes. Based on the displacement mapping relationship and the multi-point constraint equation, a degree-of-freedom coupling matrix is ​​established. The local stiffness matrix and the global stiffness matrix are multiplied by the degree-of-freedom coupling matrix to generate the assembled global stiffness matrix.

6. The design method for determining the maximum hanging weight of an ultra-thin lifting beam according to claim 1, characterized in that, The step of extracting the load corresponding to the extreme point as the maximum hanging weight until the load-displacement curve reaches an extreme point includes: in each incremental step of the nonlinear buckling analysis iteration, extracting the vertical displacement component of the current load factor and the monitoring node to form the current state point in the load-displacement space. Calculate the determinant of the overall tangent stiffness matrix after the current assembly, and determine whether the sign of the determinant has changed from positive to negative. If a flip occurs, it is determined that the current state point has crossed the extreme point. The search interval is narrowed by a bisection method between the current increment step that has crossed the extreme point and the previous increment step. The nonlinear solution is iteratively executed until the displacement increment is less than the preset convergence tolerance. The product of the load factor at convergence and the applied hanging load boundary condition is output as the maximum hanging weight.

7. The design method for determining the maximum hanging weight of an ultra-thin lifting beam according to claim 2, characterized in that, The construction of the spatial influence matrix includes: calculating the spatial distance vector from the center node of the adjacent unit to the center node of the specific unit, and normalizing the spatial distance vector to obtain the direction vector; Extract the deformation gradient tensor of the specific unit in the current increment step, and perform polar decomposition on the deformation gradient tensor to obtain the rotation tensor; The rotation tensor is used to perform coordinate rotation mapping on the direction vector to generate a corrected direction vector that follows the large deformation rotation of the ultra-thin lifting beam; Calculate the cosine of the angle between the principal strain energy direction of the adjacent unit and the correction direction vector, assign the cosine of the angle as a weighting factor to the adjacent unit, and summarize the weighting factors of all the adjacent units to generate the spatial influence matrix.

8. The design method for determining the maximum hanging weight of an ultra-thin lifting beam according to claim 3, characterized in that, The step of applying a low-pass filter operator to the initial rate of change includes: extracting the historical initial rate of change sequence corresponding to the historical multiple consecutive increment steps; Set a smoothing coefficient, the value of which is inversely proportional to the time step size of the current increment step; The smoothing coefficient is used to apply an exponentially decaying weighted average to the most recent historical initial rate of change in the historical initial rate of change sequence, and then linearly fused with the initial rate of change in the current increment step to generate an intermediate rate of change. The mesh size parameters of the corresponding unit in the initial mesh generation stage are retrieved, the spatial filtering radius is calculated based on the mesh size parameters, the intermediate rate of change is subjected to Gaussian kernel convolution operation within the spatial filtering radius, and the smoothed strain energy density rate of change after eliminating time dimension numerical noise and spatial dimension numerical noise is output.

9. The design method for determining the maximum hanging weight of an ultra-thin lifting beam according to claim 5, characterized in that, The construction of the multi-point constraint equation includes: obtaining the coordinates of the suspended boundary node, and finding two global nodes sharing the same global grid edge within a preset search range as interpolation base points; Calculate the distance interpolation coefficient based on the projection position of the suspended boundary node on the line connecting the two global nodes; Construct initial multi-point constraint equations, and express the displacement of the suspended boundary node as a linear combination of the displacement of the two global nodes and the distance interpolation coefficient; The interface stress vector of the local mesh at the suspended boundary node is extracted, and the interface stress vector is decomposed along the normal and tangential directions of the global mesh edge. The normal stress continuity condition is introduced as a penalty term into the initial multi-point constraint equation to generate an enhanced multi-point constraint equation that includes displacement compatibility and stress compatibility.

10. The design method for determining the maximum hanging weight of an ultra-thin lifting beam according to claim 6, characterized in that, The step of narrowing the search interval by using a bisection method between the current increment step and the previous increment step after crossing the extreme point includes: extracting the current load factor and the previous load factor when crossing the extreme point, and calculating the arithmetic mean of the two as the trial load factor. Solve the nonlinear equation system under the proposed load factor. If the determinant of the proposed tangent stiffness matrix is ​​positive, then update the previous load factor to the proposed load factor. If the determinant of the proposed tangent stiffness matrix is ​​negative, then update the current load factor to the proposed load factor. After each update, the minimum eigenvalue of the overall tangent stiffness matrix is ​​extracted. When the absolute value of the minimum eigenvalue is less than the preset eigenvalue tolerance, the bisection iteration is terminated, and the trial load factor is directly extracted to calculate the maximum hanging weight.

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