A method for analyzing the force of spherical reticulated shell structure nodes based on refined simulation optimization

Abstract

The present invention relates to the technical field of computer-aided design, and further relates to a method for analyzing the stress of spherical reticulated shell structure nodes based on refined simulation optimization. The method includes: Step 1: Uniformly select a plurality of observation points in the target spherical reticulated shell area, correct the displacement field corresponding to each observation point with spherical harmonic functions in spherical coordinates, and construct a total potential energy functional; Step 2: Discretize the target spherical reticulated shell area into finite element meshes, approximate the displacement field using high-order finite element shape functions, construct a discrete residual equation, and solve to obtain the overall degree-of-freedom vector; Step 3: Calculate the absolute value of the difference between the local error index of each grid element and the modulus of the overall degree-of-freedom vector of the observation point closest to it, and then determine that the grid element has abnormal stress. The present invention significantly improves the calculation accuracy, numerical stability and local anomaly detection ability.

Description

Technical Field

[0001] The present invention belongs to the technical field of computer-aided design, and particularly relates to a method for analyzing the force on nodes of a spherical reticulated shell structure based on refined simulation optimization. Background Art

[0002] With the development of modern engineering construction towards large-span and complex structures, spherical reticulated shell structures have been widely used in fields such as architecture, bridges, and stadiums due to their excellent load-bearing capacity and space utilization rate. Traditional methods for analyzing spherical structures mainly rely on empirical formulas and low-order finite element methods. These methods have high computational efficiency in the initial stage of structural design. However, in actual engineering, the external loads borne by spherical reticulated shell structures often exhibit characteristics such as multi-directional, multi-modal, and non-uniform distribution, resulting in problems such as stress concentration and abnormal deformation at local nodes. In recent years, with the rapid development of computer technology and sensing technology, researchers have begun to attempt to apply refined simulation optimization methods to the force analysis of complex structures. By performing high-precision numerical simulations and experimental data collection on structures, they strive to reveal subtle stress changes and local anomalies that are difficult to capture by traditional methods.

[0003] In the currently disclosed technologies, some studies use the high-order finite element method combined with sensor measurement data to analyze the stress distribution of the overall structure, and use spherical harmonic functions to correct the displacement field to adapt to the geometric characteristics in the spherical coordinate system. These methods have improved the accuracy of structural stress analysis to a certain extent, but there are still several deficiencies. First, traditional finite element discretization methods often adopt fixed density or rough adaptive strategies in mesh generation, failing to fully consider the complexity of the local geometry and stress of the spherical reticulated shell structure, resulting in discrete errors in some key areas, which in turn affect the accuracy of the overall simulation results. Second, when dealing with large deformation and nonlinear problems, existing technologies usually only use linear strain theory or simple nonlinear corrections, unable to accurately capture the additional strain energy caused by local geometric discontinuities and high curvatures, making the identification of local abnormal stress states lag and inaccurate. In addition, although some studies have introduced high-order finite element shape functions and corrected energy functions to improve the model accuracy, there are still problems of inconsistent data processing and insufficient coupling of model parameters in the process of fusing material parameters, geometric factors, and sensor data, resulting in the difficulty of fully matching the local abnormal stress predicted by the model with the actual measurement data under some complex working conditions. For example, the energy storage density functions used in some existing technologies are mostly based on traditional elastic strain energy expressions, ignoring the energy compensation effects caused by curvature changes, local large deformations, and nonlinear effects in the spherical reticulated shell structure, thus making the judgment criteria for local abnormal stress states at the structural nodes relatively vague. At the same time, when using sensors to obtain displacement data, directly using the original measurement data without spatial correction by mathematical tools such as spherical harmonic functions often leads to significant discreteness and noise effects of the data, which in turn affects the construction of the subsequent energy functional and the accuracy of finite element discretization. Summary of the Invention

[0004] The main object of the present invention is to provide a method for analyzing the stress of nodes of a spherical reticulated shell structure based on refined simulation optimization. By arranging high-precision displacement sensors in the target area to collect discrete displacement data, and using spherical harmonic functions for data correction, a total potential energy functional that conforms to the spherical geometric characteristics is constructed. Then, combined with adaptive finite element mesh generation and high-order finite element shape function approximation of the displacement field, discrete residual equations and local error indicators are established to achieve accurate solution of the overall degree-of-freedom vector and local strain state. This method can not only accurately capture the energy correction and abnormal stress phenomena caused by local curvature changes, large deformations, and nonlinear effects in the spherical reticulated shell structure, but also significantly improve the calculation accuracy, numerical stability, and local abnormal detection ability.

[0005] To solve the above problems, the technical solution of the present invention is realized as follows:

[0006] A method for analyzing the stress of spherical reticulated shell structure nodes based on refined simulation optimization, the method comprising:

[0007] Step 1: Uniformly select a plurality of observation points in the target spherical reticulated shell area, deploy displacement sensors at each observation point to obtain the displacement field of the observation points; the distance between adjacent observation points is less than a set distance threshold; correct the displacement field corresponding to each observation point with spherical harmonic functions in spherical coordinates, and construct the total potential energy functional;

[0008] Step 2: Discretize the target spherical reticulated shell area into finite element meshes, combine the total potential energy functional of the observation point closest to each mesh element, and calculate the local error index of each mesh element; approximate the displacement field with high-order finite element shape functions, construct a discrete residual equation, and solve to obtain the overall degree-of-freedom vector;

[0009] Step 3: Calculate the absolute value of the difference between the local error index of each mesh element and the modulus of the overall degree-of-freedom vector of the observation point closest to it. If the absolute value exceeds the set abnormal determination threshold, it is determined that the mesh element has abnormal stress.

[0010] Further, the total potential energy functional of the observation point is expressed by the following formula:

[0011] ;

[0012] Where, is the total potential energy functional; is the energy storage density function of the observation point, is the strain tensor of the observation point; is the corrected displacement field of the observation point ; is the displacement field measured by the displacement sensor; is the gradient operator; is the Laplace operator; is the corrected energy function; L is the order of the spherical harmonic function; is the polar angle of the observation point; is the azimuth angle of the observation point.

[0013] Further, the energy storage density function is expressed by the following formula:

[0014]

[0015]

[0016] Where, is the reference radius of the spherical reticulated shell; is the Young's modulus of the material of the spherical reticulated shell; is the Poisson's ratio of the material of the spherical reticulated shell; Denotes the transpose operation of a vector or matrix; is the trace of the strain tensor.

[0017] Furthermore, the modified energy function is expressed by the following formula:

[0018] ;

[0019] where, is the high-order stiffness parameter, ; Denotes the norm operation; is the thickness of the shell in the target spherical reticulated shell area.

[0020] Furthermore, in step 2, when discretizing the target spherical reticulated shell area into finite element meshes, in the circular area constructed with each observation point as the center and the distance between adjacent observation points as the radius, the number of grid elements exceeds the set number threshold;

[0021] Furthermore, in step 2, the local error index of each grid element is calculated by the following formula:

[0022] ;

[0023] where, is the distance from the grid element to the nearest observation point.

[0024] Furthermore, in step 3, the process of constructing the discrete residual equation by approximating the displacement field with high-order finite element shape functions specifically includes: discretizing the total potential energy functional to obtain the discretized total potential energy functional; using high-order finite element shape functions to approximate the displacement field and calculating the variation of the discretized total potential energy functional with respect to the global degree of freedom vector to obtain the discrete residual equation.

[0025] Furthermore, the discrete residual equation is expressed by the following formula:

[0026] ;

[0027] where, is the gradient operator obtained by high-order finite element shape functions; is the global degree of freedom vector, defined as the mean of the displacement fields of adjacent observation points of this observation point; is the second derivative operator of high-order finite element shape functions; is the strain tensor of the global degree of freedom vector.

[0028] Furthermore, the high-order finite element shape function is expressed by the following formula:

[0029] ;

[0030] where, For high-order finite element shape functions.

[0031] Furthermore, the strain tensor of the global degree of freedom vector is calculated using the following formula:

[0032] ;

[0033] where is the number of adjacent observation points of this observation point.

[0034] A method for analyzing the stress of spherical reticulated shell structure nodes based on refined simulation optimization of the present invention has the following beneficial effects: First, an important beneficial effect of the present invention is that by uniformly selecting observation points and arranging high-precision displacement sensors, accurate displacement field data of the spherical reticulated shell structure are obtained. In the prior art, the displacement data directly measured by sensors are usually used for finite element analysis. However, due to the discreteness of the measurement data and the influence of noise, directly using these data will lead to a reduction in the accuracy of the energy functional construction and finite element discretization. The present invention uses spherical harmonic functions to correct the displacement field of the observation points, making the displacement field in the spherical coordinate system smoother and more continuous, thereby improving the data quality. This processing method makes the subsequent energy functional construction more reasonable, the finite element calculation more accurate, and also enhances the robustness of the stress analysis. Second, the present invention optimizes the finite element calculation process through a refined finite element mesh generation method, improving the calculation accuracy and efficiency. In the prior art, a fixed mesh density or a simple adaptive mesh generation strategy is usually adopted, which is difficult to accurately capture the complex stress conditions in the local area of the spherical reticulated shell structure. The present invention constructs local mesh elements centered on each observation point, combines the distances between adjacent observation points, and sets a threshold for the number of finite element meshes to ensure that the mesh generation is fine enough in the key stress areas, while appropriately reducing the mesh density in the areas with relatively uniform stress, thereby improving the calculation efficiency and reducing errors. This optimization method not only ensures the stability of the overall finite element calculation, but also can more finely capture the local stress concentration and abnormal deformation areas, providing higher-precision data support for subsequent error analysis and optimization. In addition, the present invention adopts high-order finite element shape functions in the finite element calculation process, making the discretization calculation more refined and capable of more accurately describing the change of the displacement field. When solving the spherical reticulated shell structure by the traditional low-order finite element method, it is usually difficult to accurately capture the local non-linear deformation and stress concentration phenomena, resulting in insufficient calculation accuracy. The present invention approximates the displacement field by using high-order finite element shape functions and constructs a discrete residual equation on this basis, making the calculation results more in line with the actual stress state. The high-order finite element shape functions can effectively improve the fitting accuracy of the local displacement field, so that in the areas with complex stress, the finite element calculation can still maintain a high calculation accuracy, avoiding the problem of excessive numerical errors in the traditional method in the local high-gradient areas. This method can more realistically reflect the stress distribution of the structure and ensure that more accurate stress analysis results can be obtained under different working conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 FIG. is a schematic flow chart of a method for analyzing the stress of spherical reticulated shell structure nodes based on refined simulation optimization provided by an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0036] To enable those skilled in the art to better understand the solution of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the scope of protection of the present invention.

[0037] Example 1, refer to Figure 1 : A method for analyzing the force on the nodes of a spherical reticulated shell structure based on refined simulation optimization, the method comprising:

[0038] Step 1: Uniformly select a plurality of observation points in the target spherical reticulated shell area, and deploy displacement sensors at each observation point to obtain the displacement field of the observation points; the distance between adjacent observation points is less than a set distance threshold; correct the displacement field corresponding to each observation point with spherical harmonic functions in spherical coordinates, and construct the total potential energy functional;

[0039] Due to the geometric symmetry of the spherical reticulated shell structure, its force characteristics are essentially different from those of planar structures. When analyzing such structures, the traditional Cartesian coordinate system method is difficult to accurately describe its deformation mode. Therefore, this method adopts a spherical coordinate system to represent its displacement field in a more natural way, and optimizes and corrects it through spherical harmonic functions to ensure the mathematical consistency and physical rationality of the data. First, in the target area of the spherical reticulated shell structure, a plurality of observation points need to be uniformly selected, and the selection of these observation points needs to meet certain density conditions so that the deformation characteristics of the entire structure can be fully captured. To ensure the spatial resolution of the observation data, the distance between adjacent observation points must be less than a set distance threshold, and this threshold is usually determined according to the characteristic size and main force mode of the reticulated shell structure. For example, in the case of high-order mode vibration or local stress concentration, appropriately reducing the distance between observation points can improve the fineness of data collection, thereby enhancing the accuracy of the overall simulation optimization. High-precision displacement sensors are deployed at each selected observation point to monitor the displacement response of this point in real time, and these displacement data are important input information for subsequent analysis. Due to the non-uniformity of local deformation caused by the force characteristics of the spherical reticulated shell structure, the deployment method of the sensors should ensure that the displacement components in different directions can be reasonably collected to avoid data errors caused by measurement blind spots.

[0040] However, the directly obtained displacement field data are often affected by measurement errors, environmental disturbances, and discrete distributions, resulting in poor spatial continuity of the data. Therefore, this method uses spherical harmonic functions in the spherical coordinate system to correct the displacement field of the observation points to optimize the data quality. Spherical harmonic functions are orthogonal basis functions suitable for spherical regions and can approximately expand physical quantities distributed in space, thereby constructing a smooth displacement distribution function on the global scale of the spherical reticulated shell structure. Mathematically, the basis function form of spherical harmonic functions depends on the polar angle and azimuth angle in spherical coordinates, and it can effectively capture the displacement change patterns on the spherical structure. By appropriately selecting the order of the spherical harmonic function expansion, the computational complexity can be reduced while ensuring the computational accuracy, so that the corrected displacement field not only retains the basic characteristics of the measurement data but also has higher smoothness and global consistency. On this basis, in order to further quantify the stress state of the entire spherical reticulated shell structure, this method constructs the total potential energy functional to provide more accurate physical constraints. The establishment of the total potential energy functional is based on the variational principle, that is, when the structure is in the equilibrium state, its potential energy function takes the minimum value. The potential energy functional is usually composed of elastic strain energy, external force potential energy, and additional constraint energy terms. Among them, the elastic strain energy is related to the stiffness properties of the structure, while the external force potential energy depends on the distribution of the applied loads. In the spherical reticulated shell structure, since the structural units are arranged in a spherical surface, the traditional Cartesian coordinate system potential energy expression method is difficult to intuitively reflect its true physical characteristics. Therefore, this method adopts the spherical coordinate system and combines the displacement field data corrected by spherical harmonic functions to express the potential energy functional in the form of a spherical surface integral. This can not only ensure that the potential energy functional is consistent with the geometric characteristics of the spherical reticulated shell structure but also better adapt to the deformation characteristics of the spherical structure in terms of calculation, making the final simulation optimization process more stable and reliable.

[0041] Step 2: Discretize the target spherical reticulated shell region into finite element meshes, calculate the local error index of each mesh element by combining the total potential energy functional of the observation point closest to each mesh element; approximate the displacement field using high-order finite element shape functions, construct a discrete residual equation, and solve to obtain the global degree of freedom vector;

[0042] In the specific calculation process, it is first necessary to discretize the target spherical reticulated shell area, that is, divide it into finite element meshes. Since the spherical reticulated shell structure usually has strong geometric inhomogeneity and surface characteristics, the traditional Cartesian coordinate finite element discretization method is difficult to effectively adapt to the mechanical properties of such structures. Therefore, in this method, the division of finite element meshes fully considers the geometric characteristics of the spherical structure and is adaptively adjusted in combination with the distribution density of observation points. During the discretization process, the size of each mesh element should be dynamically optimized according to the key stress areas of the structure. For example, in the stress concentration areas or near the connection nodes, the mesh division should be more refined to ensure the calculation accuracy, while in the areas with relatively uniform stress, the mesh density can be appropriately relaxed to improve the calculation efficiency. Through such a mesh division strategy, the calculation complexity can be reduced and the overall solution efficiency can be improved while ensuring the global calculation accuracy. After completing the finite element mesh division, it is necessary to combine the total potential energy functional of the observation point closest to each mesh element to calculate the local error index. The total potential energy functional is constructed based on the displacement field data of the observation points and includes the elastic strain energy, external force potential energy, and constraint energy terms of the structure. The minimization process can ensure the true mechanical state of the structure under the action of external forces. Therefore, for each finite element mesh element, its theoretical mechanical state should be consistent with the calculation result of the total potential energy functional of the observation point closest to the element. However, due to the inevitable numerical errors in the discretization process and the different scale effects of the elements in the mesh division process, there may be a certain deviation between the local calculated values of each mesh element and the theoretical potential energy functional of the observation point. Therefore, this method constructs a local error index based on this error to measure the calculation error size of each element and provide a basis for further optimization. When calculating the local error index, a key step is how to accurately describe the displacement field of the mesh element. The traditional low-order finite element method uses linear or bilinear interpolation functions to approximate the displacement field. This method can obtain good calculation results in general engineering problems, but in a complex mechanical system such as a spherical reticulated shell structure, due to the existence of its bending characteristics and high-order deformation modes, the low-order interpolation method is difficult to effectively capture the actual displacement changes. Therefore, this method introduces high-order finite element shape functions. By increasing the order of the interpolation function, the displacement field inside each element can more finely approximate the true deformation mode. The advantage of the high-order finite element method is that its shape function not only includes linear terms but also quadratic and even higher-order polynomial terms, so that the deformation behavior of complex surfaces can be more accurately described. In addition, since the displacement distribution of the spherical reticulated shell structure usually has strong continuity and smoothness, the high-order finite element shape function can better adapt to this deformation characteristic, thereby reducing the interpolation error and improving the calculation accuracy. The displacement field constructed based on the high-order finite element shape function can be further used to construct the discrete residual equation. The construction of the discrete residual equation is based on the variational principle, that is, the energy balance equation is established through the principle of minimum potential energy, and then the global degree-of-freedom vector is solved.

[0043] Step 3: Calculate the absolute value of the difference between the local error index of each grid cell and the modulus of the overall degree of freedom vector of the nearest observation point. If the absolute value exceeds the set abnormal determination threshold, it is determined that the grid cell has abnormal stress.

[0044] In this process, the setting of the abnormal determination threshold is an important factor affecting the recognition accuracy. Traditional force analysis methods usually set the error threshold using empirical values or simplified statistical methods, which may lead to misjudgment or missed judgment in some cases. This method fully considers the overall force mode of the spherical reticulated shell structure. Through the error evaluation mechanism based on the total potential energy functional of the observation points, the abnormal determination threshold not only has physical rationality but also can adapt to different types of load conditions. For example, under wind load, the surface force of the spherical reticulated shell structure often shows non-uniform distribution, and there may be large stress gradients in local areas. Therefore, the error threshold should be appropriately increased to avoid misjudging normal local stress concentration phenomena. Under seismic action, due to the relatively intense dynamic response of the structure, sudden displacement anomalies may occur at local nodes. Therefore, the error threshold should be appropriately reduced to improve the sensitivity to abnormal force states. Through this adaptive threshold adjustment strategy, this method can maintain a high abnormal recognition accuracy under different working conditions. When calculating the abnormal determination index, the absolute value of the difference in modulus is a key mathematical measurement method. The calculation of the modulus involves each component of the total degree-of-freedom vector, and these components correspond to different displacement directions of the structure respectively. In the spherical coordinate system, the force state of the spherical reticulated shell structure is mainly determined by the radial, tangential, and normal displacement components. Therefore, the calculation of the degree-of-freedom vector needs to consider the combined action of these components in different directions. For example, at the key connection nodes of the reticulated shell, the change in the radial displacement component may reflect the overall force mode of the structure, while the changes in the tangential and normal displacements may indicate the stress concentration at local nodes. Therefore, by calculating the modulus of the total degree-of-freedom vector, a comprehensive measure of the force state can be obtained, so as to more comprehensively describe the overall deformation characteristics of the structure. Comparing this modulus value with the local error index can effectively identify the abnormal force areas caused by factors such as external forces, defects, or connection loosening. In addition, during the abnormal determination process, the geometric continuity and force transfer characteristics of the reticulated shell structure also need to be considered. Since the spherical reticulated shell structure usually consists of multiple members or panels, its force state has strong spatial correlation, that is, the force changes between adjacent units often have a certain continuity. Therefore, when calculating the force error of each grid unit, not only the local error index of itself needs to be concerned, but also the force states of the surrounding units need to be comprehensively analyzed in combination. For example, if the error value of a certain grid unit slightly exceeds the set abnormal determination threshold, but the error values of its surrounding units are all within the normal range, the abnormal force of this unit may be caused by local grid discretization error rather than a real structural defect. Therefore, in the final abnormal determination process, the abnormal recognition results can be optimized through a certain spatial smoothing algorithm to improve the reliability of the determination.

[0045] Embodiment 2: The total potential energy functional of the observation points is expressed by the following formula:

[0046] ;

[0047] wherein, is the total potential energy functional; is the energy storage density function at the observation point; is the displacement field at the observation point; is the corrected displacement field at the observation point ; is the displacement field measured by the displacement sensor; is the gradient operator; is the Laplace operator; is the corrected energy function.

[0048] Specifically, on the left side of the formula is called the total potential energy functional. The variables involved here not only include the displacement field itself, but also include its gradient and the second-order differential information of the Laplace operator with respect to the displacement field. This multi-level description can capture the deformation characteristics of the structure at both local and global scales simultaneously, reflecting the complexity of the energy distribution within the structure under actual loading conditions. Specifically, represents the energy storage density function at the observation point, where is the strain variable caused by the displacement field and its gradient, which reflects the energy storage of the material during elastic deformation. Traditionally, the energy storage density mainly describes the internal energy of the material. However, in the spherical reticulated shell structure, due to the influence of factors such as curvature and geometric continuity, relying solely on the strain energy often cannot fully reflect the local abnormal stress. Therefore, an additional energy correction term must be introduced. Here, plays this role. It uses the second-order derivative information of the displacement field and captures the bending and diffusion characteristics of the displacement field in space through the Laplace operator to correct the geometric effects that may be ignored in the traditional energy storage description. This term is particularly important for the spherical reticulated shell structure because the stress state of this structure often does not only show linear deformation but is accompanied by complex surface deformation and local stress concentration phenomena. The corrected energy function precisely provides a mathematical basis for this. The corrected displacement field actually takes into account the spherical harmonic characteristics unique to the spherical structure and converts the discrete data measured by the sensor through a normalization factor closely related to the spherical geometry, making the obtained displacement field more in line with the mathematical description in the spherical coordinate system. The factor originates from the orthogonality and normalization requirements of the spherical harmonic function, and the factorial terms in the numerator and denominator reflect the coupling relationship between the measurement data and the geometric parameters of the spherical structure, where is the reference radius of the spherical reticulated shell, and represent the polar angle and the azimuth angle respectively. This correction not only ensures the smoothness and continuity of the displacement field on the spherical surface, but also enables the energy functional constructed based on this displacement field to more realistically reflect the energy state under actual working conditions. In addition, the gradient operator and the Laplace operator in the formula play the roles of capturing the first-order change and the second-order change of the displacement field respectively. The gradient reflects the rate of change of displacement with space, that is, the main source of local strain, while the Laplace operator measures the curvature and diffusion trend of the displacement field, and it can reveal the additional energy effect caused by geometric bending in the spherical reticulated shell structure. Through the combination of the two, the energy functional can not only describe the elastic response of the material, but also take into account the influence brought by geometric nonlinearity, providing rich information for identifying local abnormal forces.

[0049] Example 3: The energy storage density function is expressed by the following formula:

[0050] ;

[0051] where, is the reference radius of the spherical reticulated shell; is the polar angle of the observation point; is the azimuth angle of the observation point; is the Young's modulus of the material of the spherical reticulated shell; is the Poisson's ratio of the material of the spherical reticulated shell; is the strain tensor of the observation point, defined as: ; represents the transpose operation of a vector or matrix.

[0052] Specifically, represents the reference radius of the spherical reticulated shell. It not only describes the size of the structure as a geometric parameter, but also plays a role in scale transformation in the energy storage density function, coupling the local energy with the geometric size of the entire structure. Next, the term reflects the distribution characteristics of the area element in the spherical coordinate system. Because on the spherical surface, the area element is usually weighted by , so that the energy density has different contributions at different polar angles, and further finely describes the energy difference caused by the change of spherical curvature. Following that are and two direction factors, which appear in the first term and the second term respectively to distinguish the energy distribution characteristics at different azimuth angles on the spherical surface. Specifically, appears in the first term, representing the stress state and energy transfer in a specific orientation, while It plays a similar role in the second term. The two together constitute a description of the energy inhomogeneity in all directions of the spherical surface, so that the energy storage density shows a strong anisotropy in space, which is very critical for refined simulation optimization. Regarding the material parameters, It is the Young's modulus of the material used in the spherical lattice shell structure. This parameter determines the stiffness and elastic response of the material when subjected to force. The larger the Young's modulus, the smaller the deformation of the material under the same load, and the more elastic energy is stored. Therefore, it plays a role in directly amplifying or reducing the energy density in the energy expression. At the same time, Poisson's ratio It reflects the proportion of the vertical contraction when the material is stretched in one direction. and This design ensures that the elastic properties of the material can be accurately reflected when describing volume changes and shear deformations. Specifically, the coefficient in the first term is and the coefficient in the second term They correspond to the shear stiffness and bulk modulus of the material, respectively. This division is in line with the principle of energy decomposition in elastic mechanics, that is, the stored energy is divided into the shear energy caused by deformation and the energy caused by volume change. Represents the strain tensor The second-order invariant of , whose physical meaning is to describe the comprehensive effect of isotropic strain in the local area, can accurately capture the energy storage caused by shear deformation and local bending; is the square of the first-order invariant of the strain tensor, which mainly reflects the energy contribution caused by local volume changes. In spherical lattice shell structures, due to their curved surface characteristics, shear and volume deformation often exist at the nodes at the same time. Through this sub-item design, the contribution of the two different deformation modes to the overall energy storage can be analyzed in more detail, thus providing a more accurate theoretical basis for the identification of abnormal node stress.

[0053] In addition, the strain tensor The definition of , and also adds nonlinear terms , which makes it possible to more realistically reflect the strain state in large deformation or local high strain areas. Since the spherical lattice shell structure may experience large local deformation during the actual stress process, this nonlinear correction is very necessary and can effectively improve the adaptability and accuracy of the energy model to actual working conditions. Combined with the definition of the strain tensor, the energy storage density function and The two invariants respectively reflect the energy storage modes under different strain patterns, ensuring a complete mathematical description ability. In the physical meaning of the entire formula, the first half mainly reflects the energy storage of the spherical reticulated shell structure under shear and bending deformations, and its factor not only combines the geometric dimensions of the structure and the distribution of area elements on the sphere, but also weights the energy contributions in each direction; while the second half focuses on describing the energy storage caused by volume changes, and its parameter combination ensures that the energy changes can be accurately characterized when the material undergoes compression or tension. This sub-item description enables the overall energy storage density function to reflect both the inherent elastic properties of the material and fully consider the energy distribution differences caused by geometric and azimuthal angle changes at different positions of the spherical reticulated shell structure, thereby achieving high-precision simulation and evaluation of the stress state of the nodes.

[0054] Example 4: The modified energy function is expressed by the following formula:

[0055] ;

[0056] where is the high-order stiffness parameter, ; represents the norm operation; is the shell thickness of the target spherical reticulated shell area.

[0057] Specifically, the modified energy function is a refined description of the local energy correction part of the spherical reticulated shell structure. Its core lies in quantifying the local curvature effect of the structure by introducing the second-order differential information of the displacement field, and thereby reflecting the additional energy contributions caused by geometric nonlinearity and large deformations. In this formula, represents the result of the Laplace operator acting on the displacement field , and can also be regarded as the norm of the second-order gradient of the displacement field. This operation captures the curvature or bending degree of the displacement field in the local area. In the spherical reticulated shell structure, due to the complex curvature characteristics of the structure surface itself, it not only depends on the first-order displacement changes to describe the strain state, but more requires the second-order differential term to characterize the energy changes caused by bending, arching or local instability. Specifically, can be regarded as a quadratic penalty for the local curvature distribution of the entire structure. It reflects the regions with larger curvature or severe bending at a higher energy cost, thereby giving these regions higher weights in the overall energy calculation, which is of great significance for refined simulation optimization and local anomaly determination. The introduced high-order stiffness parameter is defined by the basic physical parameters and geometric parameters of the material, and is specifically expressed as . Where is the Young's modulus of the material, which directly reflects the tensile and compressive capacities of the material; is the thickness of the shell in the target spherical reticulated shell region, and its square term indicates that with the increase of the thickness, the flexural stiffness is significantly improved; while the Poisson's ratio reflects the relationship between the transverse and longitudinal deformations of the material under the stress state. It appears in the denominator, considering the volume effect of the material and its actual performance under the plane stress state. Through this combination, accurately characterizes the high-order mechanical properties of the spherical reticulated shell in terms of bending and shear. It not only affects the numerical value of the overall energy functional, but also determines the resistance ability shown by the local area in the face of geometric discontinuities or mutations. In other words, a higher value means that when there are significant curvature changes locally, the structure needs to consume more energy to maintain stability, which is in sharp contrast to the simplified treatment that ignores the curvature effect in traditional low-order models. The modified energy function essentially introduces high-order energy terms, and this design concept is derived from the bending energy expression in the thin shell theory and the plate bending theory. In traditional energy models, most cases only consider the first-order strain energy, but for the spherical reticulated shell structure with complex geometric shapes, local bending or arching behaviors often have an important impact on the overall structural safety. By adding this term to the energy expression, the model can more comprehensively describe the additional energy changes caused by obvious curvature changes in the local area and give corresponding penalty effects to such changes. In this way, in the simulation optimization process, it can not only more accurately reflect the internal properties of the material and the structure, but also timely identify and warn of local abnormal deformations caused by external loads, manufacturing errors or environmental effects. Combining with the overall idea of the present invention, that is, the method for analyzing the force on the nodes of the spherical reticulated shell structure based on refined simulation optimization, this modified energy function plays a crucial role. By using the second-order differential information, the entire energy model, when describing the force state of local nodes, takes into account both the energy storage effect brought by the elastic deformation of the material and the energy correction caused by the geometric discontinuity or local curvature mutation of the structure. Especially in the spherical structure, there are obvious differences in the geometric curvature of different regions, and it is difficult for traditional first-order models to capture these subtle changes, while the introduced high-order energy terms enable the model to more finely display the local deformation characteristics in the global energy distribution. Furthermore, by calculating the overall energy functional, the stress state of each node can be accurately evaluated, thereby providing a solid theoretical and numerical basis for subsequent anomaly determination, local error analysis and optimization design.

[0058] Example 5: In step 2, when discretizing the target spherical reticulated shell region into finite element meshes, in the circular region constructed with each observation point as the center and the distance between adjacent observation points as the radius, the number of mesh elements exceeds the set number threshold.

[0059] Specifically, this method first uses evenly distributed observation points to directly measure the structural deformation. Each observation point not only provides displacement field data but also represents an important sampling point for the local stress state of the spherical reticulated shell. During the discretization process, a circular area centered at the observation point with the distance between adjacent observation points as the radius is adopted, which can ensure that each sampling area covers the main geometric and mechanical characteristics within the local area and ensure that the finite element mesh within this area is dense enough to capture the minute differences in local stress and deformation. When the number of finite element meshes exceeds the set number threshold, it means that the mesh density within this area has reached or exceeded the accuracy standard required by the optimized design. At this time, it can not only more accurately reflect the complex stress distribution and deformation gradient within this area but also helps to reduce the uncertainty caused by discretization error in subsequent numerical calculations. On the other hand, if the number of meshes is lower than the set threshold, it may indicate insufficient sampling density within this area, possibly missing local important stress concentrations or deformation characteristics, thus affecting the solution accuracy of the overall degree-of-freedom vector. Therefore, when designing this adaptive mesh generation strategy, the particularity of the stress analysis of the spherical reticulated shell structure is fully considered. On the one hand, it uses observation points to construct local areas, ensuring a high match between the sampling data and the finite element model. On the other hand, by setting the threshold of the number of meshes, it dynamically regulates the possible local non-uniformity during the discretization process, enabling the entire simulation model to have higher accuracy and robustness when reflecting the actual working conditions. Due to the complex curved surface geometric characteristics and highly nonlinear stress response of the spherical reticulated shell structure itself, traditional finite element mesh generation often relies on a fixed mesh density and is difficult to balance the accurate description of the overall structure and local details. However, the strategy in this embodiment uses the spatial information of the observation points as the key basis and realizes the precise capture of local abnormally stressed areas by dynamically adjusting the number of meshes, thereby effectively reducing the numerical error caused by rough discretization during the finite element discretization process, and further providing a solid data basis for subsequent error index calculation, overall degree-of-freedom vector solution, and final abnormal stress judgment.

[0060] Example 6: In step 2, the local error index of each grid element is calculated through the following formula:

[0061] ;

[0062] where is the distance from the grid element to the nearest observation point.

[0063] Specifically, in the factor at the beginning of the formula, represents the reference radius of the spherical reticulated shell, serving as a representative of the geometric scale of the entire structure; and ​is the distance from the current grid cell to the nearest observation point. This scaling factor is used to reflect that in regions closer to the observation point, the accuracy and representativeness of the measurement data are higher, while in more distant regions, there may be greater uncertainties, thus amplifying or reducing the sensitivity of that region in the error metric. In other words, when is small, the value is large, meaning that this grid cell is more sensitive to error changes, thus requiring a more precise description of the energy in this region; conversely, when is large, the error metric is relatively weakened, reflecting a reduced impact of local measurement information on the overall energy state. Next, the term within the curly brackets combines the partial derivative of the energy storage density function with respect to the displacement field and the second - derivative terms involved in the modified energy function. Among them, represents the sensitivity or gradient of the energy storage density with respect to displacement, and its role is to quantify the energy change caused by displacement changes, which is of great significance for capturing the uneven distribution of local strain energy; while multiplying by emphasizes the influence in the polar - angle direction in the spherical coordinate system, enabling a reasonable weighting of the energy contributions at different polar - angle positions. On the other hand, the term introduces the product of the high - order stiffness parameter and the Laplacian of the displacement field . Here, is defined as , which combines parameters such as the Young's modulus of the material, the shell thickness and the Poisson's ratio , reflecting the resistance ability of the spherical reticulated shell under bending and local curvature changes; while measures the second - order change of the displacement field, that is, the curvature or bending degree within the local region. Multiplying by further considers the influence of errors in the azimuth - angle direction in the spherical coordinate system, comprehensively reflecting the differences in energy correction caused by geometric and force non - uniformity in different directions. In this way, the entire term within the curly brackets combines first - order displacement changes and second - order curvature changes, providing a composite measure for the local error metric that reflects both the change in material energy storage and captures the bending sensitivity of the structure.

[0064] The next product term in the formula mainly reflects the regulatory role of material properties on local errors. The Young's modulus represents the rigidity of the material. The larger the value, the more difficult it is for the material to deform, so it has higher sensitivity in energy storage; the Poisson's ratio It is used to describe the relationship between the transverse and longitudinal deformations of the material when it is stressed. Its expression form in the denominator ensures the balance of the energy response of the material under different stress states. The introduction of this coefficient ensures that the local error index can match the actual elastic characteristics of the material, thus theoretically ensuring the physical rationality and numerical stability of the calculation results. Finally, the term at the end of the formula is the normalization of the proportional relationship between the shell thickness and the geometric parameters. Here, is the shell thickness in the spherical reticulated shell area, and its square reflects the quadratic effect of the thickness on the local stiffness and energy storage. While represents the effective geometric scale obtained by subtracting the distance from the grid element to the observation point from the reference radius. Such normalization can eliminate the influence caused by size differences, making the local error index comparable among different regions. Through the square form of this ratio, the local error value can be effectively amplified or reduced, and thus during the finite element discretization process, the capture of local abnormal stress states is more accurate. The design of this part not only emphasizes the role of the structural geometric characteristics in error transmission, but also incorporates the influence of the thickness on the stress state into the calculation scope of the error index, making the sensitivity of the overall model to local deformations significantly improved.

[0065] Example 7: In step 3, the process of constructing the discrete residual equation by approximating the displacement field using the high-order finite element shape function specifically includes: discretizing the total potential energy functional to obtain the discretized total potential energy functional; using the high-order finite element shape function to approximate the displacement field, and calculating the variation of the discretized total potential energy functional with respect to the global degree-of-freedom vector to obtain the discrete residual equation.

[0066] Specifically, the discretization process requires constructing a fine and locally geometric feature-adaptive finite element mesh within the entire spherical reticulated shell region, approximately integrating the energy over the continuous domain using the shape functions of each mesh element, and transforming the energy functional originally defined on the continuous domain into a function of the degrees of freedom of each node. Here, the application of high-order finite element shape functions is particularly crucial because, due to the curvature changes, local non-linear deformations, and complex stress states in the spherical reticulated shell structure, traditional low-order shape functions often struggle to fully represent the high-order information of the actual displacement field. In contrast, high-order shape functions can capture local subtle deformation characteristics by increasing the polynomial order, enabling the discretized total potential energy functional to more accurately reflect the actual physical response of the structure. Next, the displacement field is approximated using high-order finite element shape functions, and the discretized total potential energy functional is variational calculated with respect to the global degrees of freedom vector. This variational process is essentially a necessary condition for solving the minimum potential energy problem, that is, by taking the partial derivatives of the discretized energy functional with respect to the degrees of freedom of each node and setting the partial derivatives to zero, the equilibrium equations or residual expressions between each node are established. During the variational calculation process, the fine approximation provided by the high-order shape functions ensures that the stress state of each node is not only affected by the energy contributions within its immediate neighborhood but also reflects the energy corrections caused by bending and curvature effects in more distant regions, thus constructing a relatively complete and accurate discrete residual equation system as a whole. In this residual equation system, each equation corresponds to the equilibrium state of a node's degree of freedom, representing the deviation of the node within the overall energy field. This deviation is the local residual, and its numerical value reflects the degree of finite element approximation error and local stress anomaly. In subsequent iterative solution processes, by solving these discrete residual equations, an accurate solution for the global degrees of freedom vector can be obtained, thereby achieving high-precision prediction and anomaly determination of the stress state of the nodes in the spherical reticulated shell structure. It should be noted that high-order finite element shape functions not only improve the discretization accuracy here but also significantly enhance the numerical stability and convergence of the discrete residual equations because they can more fully capture the high-order energy effects caused by curvature, thickness changes, and non-linear deformations in the structure.

[0067] Example 8: The discrete residual equation is expressed using the following formula:

[0068] ;

[0069] where, is the gradient operator obtained from the high-order finite element shape function; is the global degrees of freedom vector, defined as the mean of the displacement fields of adjacent observation points at this observation point; is the second derivative operator of the high-order finite element shape function; is the strain tensor of the global degrees of freedom vector.

[0070] Specifically, the leading coefficient in the formula reflects the influence of spherical geometric factors: is the reference radius of the spherical reticulated shell, and represent the polar angle and azimuth angle of the observation point respectively. Such a combination can appropriately weight the energy and mechanical effects in the spherical coordinate system, enabling the physical quantities at different positions to reflect the non-uniformity of the spherical distribution. Next, the factor is directly related to the basic mechanical properties of the material, where is the Young's modulus of the material, representing the rigidity of the material, and the Poisson's ratio characterizes the characteristics of the lateral deformation of the material when it is in tension or compression. The appearance of in the denominator is a common elastic constant correction factor to ensure the physical consistency of the energy expression. Immediately following in the formula, is the transpose of the gradient operator matrix constructed by high-order finite element shape functions, which establishes a differential relationship between the degrees of freedom at discrete nodes and the local displacement field. Next, the term inside the parentheses is a comprehensive expression of the linear and non-linear parts of the strain tensor, where represents the linear strain tensor corresponding to the overall degree of freedom vector (defined as the average value of the displacement fields of this observation point and its adjacent observation points), and its construction follows the small strain theory, capable of capturing the local deformation information around the nodes; while considers the geometric non-linear effect, that is, under large deformation or local high strain conditions, the square term of the displacement gradient can more realistically reflect the actual deformation of the structure. The sum of the two takes into account both the basic description of linear elastic theory and the correction effect under large deformation, enabling the residual equation to more comprehensively depict the energy change and stress state.

[0071] The term in the second line of the formula mainly reflects the contribution of volume deformation to the energy. Here, is the trace of the strain tensor, representing the volume change within the local area, that is, the volume strain, and this term is closely related to the bulk modulus of the material in elastic mechanics. The Poisson's ratio appears again in the numerator and, together with and forms a standard volume energy correction factor, so as to correctly reflect the internal energy change of the material when it is in compression or tension during the calculation of volume deformation energy. After multiplying by , this part of the residual reflects the nodal mechanical equilibrium condition caused by volume strain and plays a supplementary role in the construction of the overall discrete residual equation. Finally, the term in the formula introduces the high-order bending effect in the correction energy function. The parameter is a high - order stiffness parameter, which is defined as , where represents the thickness of the reticulated shell, and this parameter reflects the physical property of the structural flexural stiffness; while is the second - order derivative operator obtained through the high - order finite - element shape function, and this operator can capture the second - order variation of the displacement field, that is, the local curvature or bending degree. Through 's combined operation, it is equivalent to a quadratic penalty on the second - order differential term, reflecting that when there is a large curvature change in the local area, additional energy needs to be consumed within the system to maintain balance. This term directly acts on the global degree - of - freedom vector , provides a numerical description of the bending effect in the discrete residual equation, ensuring that the model not only considers the in - plane strain energy when dealing with the spherical reticulated shell structure, but also takes into account the high - order energy correction caused by curvature. Adding up the above - mentioned terms and setting the whole expression equal to zero is the equilibrium condition of the discrete residual equation, and its physical meaning is that when the system is in equilibrium, the residual forces generated by the combined action of material energy storage, volumetric strain, and bending energy at each discrete node described by the high - order finite - element shape function must be zero. This is not only a necessary condition for solving the global degree - of - freedom vector , but also the key to ensuring that the simulation model can accurately reflect the actual stress state. By solving this non - linear equation set, the displacement distribution of each node in the spherical reticulated shell structure can be obtained, and then the local strain, stress, and energy distribution can be further derived, providing a solid theoretical basis for subsequent abnormal stress determination and structural optimization design. It is worth mentioning that in this discrete residual equation, the geometric factors and in the spherical coordinate system are combined, enabling the formula to fully reflect the non - uniformity of the spherical structure in different directions, so that during the numerical simulation process, local errors caused by curvature changes, node - to - node distance differences, etc. can be accurately characterized. At the same time, the application of the high - order finite - element shape function not only improves the approximation accuracy of the displacement field, but also makes the calculation of the strain tensor more accurate, which is crucial for capturing the energy discontinuity caused by large deformations or local geometric non - linearities. Thus, the non - linear term appearing in the discrete residual equation is particularly important, as it makes up for the possible deficiencies of traditional low - order finite - element models in large - deformation analysis to a certain extent.

[0072] Example 9: The high - order finite - element shape function is expressed by the following formula:

[0073] ;

[0074] where is the high - order finite - element shape function.

[0075] Specifically, the variables in the formula usually represent local coordinate variables, and the range of their variation in the reference domain is often mapped to an interval, while the in the denominator actually reflects this interval boundary information in a normalized manner, ensuring that the shape function satisfies specific zero values or other constraint conditions at the boundaries. Specifically, this denominator can be regarded as the constraint of the function values by two endpoints in the reference coordinate system (usually corresponding to the boundaries of the element), which guarantees that when takes the boundary value or , the denominator is not zero, and the entire function form can correctly reflect the weight distribution and interpolation requirements of the nodes. Entering the numerator part, it can be seen that it consists of three factors: , and . Among them, the appearance of the first factor and the second factor indicates that the shape function must take zero values at these two positions of and . This design is usually to set internal nodes or intermediate interpolation points within the element, so that the entire interpolation polynomial has a higher order and can meet the exact node matching requirements at these key points. By selecting these two symmetric zeros, not only the smoothness of the shape function is guaranteed, but also it helps to more finely depict the gradient change and bending effect of the local displacement field during finite element discretization. The third factor is more special, and it directly introduces the material properties into the construction of the shape function. Here, is the Young's modulus of the spherical reticulated shell material, and represents the Poisson's ratio. Both are basic parameters describing the mechanical properties of the material. By subtracting and from the variable in the form of , a non-linear modulation mechanism is actually introduced into the shape function, making the shape function not only depend on the geometric distribution, but also sensitive to the local response differences caused by the material stiffness and transverse deformation characteristics. This treatment method is particularly applicable to the spherical reticulated shell structure. Due to the complex stress state and significant local geometric curvature changes, it is often difficult to capture the weak differences caused by material non-linearity solely relying on traditional low-order shape functions. And this approach of embedding material parameters into the interpolation function can, to a certain extent, compensate for this deficiency, thus making the high-order finite element model more accurate in reflecting node deformations and energy distributions.

[0076] The entire shape function The structure can be understood as a fractional polynomial. The numerator determines the zeros and characteristic points of the shape function at key positions within the element through the product of three factors, while the denominator serves as a normalization factor to ensure numerical stability at the boundary of the reference domain and satisfy the necessary interpolation conditions. This expression is actually a carefully designed high-order interpolation function. The high order of the polynomial enables it to capture more subtle changes in the displacement field, thereby providing sufficient numerical accuracy for simulation optimization under complex stress states. In a specific finite element analysis, the role of the high-order finite element shape function is to discretize the continuous physical field into discrete variables at each node by interpolating the displacement field within the element. Due to the complex curved surface characteristics and non-uniform stress distribution of the spherical reticulated shell structure, traditional low-order interpolation functions often fail to meet the requirements of refined simulation, easily leading to numerical errors and insufficient local responses. The high-order shape function in this embodiment incorporates multiple key factors during the interpolation process, ensuring accurate matching of the actual displacement values at different key positions within the element (such as internal nodes and boundary nodes), and integrating the elastic properties of the material into the shape function through special factors, enabling the numerical simulation to fully consider the physical properties of the material while reflecting the geometric shape. In addition, the construction of this shape function also exhibits good smoothness and continuity, being continuously differentiable throughout the reference domain, which is crucial for solving complex non-linear residual equations and performing high-precision energy functional variational calculations. Since the high-order finite element shape function can approximate the true displacement field in a higher dimension, when constructing the discrete residual equation, it can more accurately describe local deformation and strain gradients, making the subsequent obtained global degree of freedom vector have higher numerical reliability and physical rationality. This is precisely why the high-order finite element shape function is adopted as the core tool in the method for analyzing the stress of spherical reticulated shell structure nodes based on refined simulation optimization in this invention. From the perspective of engineering practical applications, spherical reticulated shell structures are often used to bear complex and multi-directional loads, and there may be significant non-linearity and local high-gradient phenomena in the stress state at local nodes. Due to the low order of the shape function in traditional finite element methods, it is difficult to capture this complexity, while the high-order finite element shape function can achieve a more refined interpolation approximation within the element by increasing the number of polynomial terms, thereby effectively improving the accuracy and stability of the model. By adopting a high-order shape function similar to the above formula not only can the displacement field be finely discretized, but also consistency and high precision can be maintained in subsequent steps such as constructing the energy functional and discrete residual equation, thereby ensuring that the stress state and local energy distribution of each node are fully reflected during the overall simulation process, providing a solid numerical foundation for structural health monitoring, anomaly determination, and optimization design.

[0077] Example 10: The strain tensor of the global degree of freedom vector is calculated using the following formula:

[0078] ;

[0079] Among them, is the number of adjacent observation points of this observation point.

[0080] Specifically, in the formula represents the gradient operation of the total degree of freedom vector , that is, the rate of change of the displacements of each node with respect to the spatial coordinates within the local area. Usually, in the traditional small strain theory, the linear part of the strain tensor is defined by , which reflects the contribution of the displacement gradient to the strain when the structure is subjected to small deformations. However, in the spherical reticulated shell structure, due to its curved surface geometry and possible local large deformations, a simple linear approximation is often insufficient to describe the actual stress state. Therefore, this formula not only includes this symmetric part, but also introduces a non - linear term to compensate for the effects caused by large deformations or geometric non - linearities. This non - linear term reflects the coupling effect between displacement gradients. When the local deformation is large, the contribution of this term to the strain will increase significantly, thus improving the model's ability to capture non - linear behavior. In the entire expression within the parentheses, the first two terms and respectively represent the gradient changes of the displacement field in different directions. The sum of these two terms forms a symmetric tensor, whose physical meaning is to describe the average change of strains in all directions within the material. By taking the symmetric part, the spurious strains caused by local rotations can be eliminated, thus only retaining the real deformation information. Then, the additional non - linear term represents the product of the displacement field gradients. This term plays a corrective role in the case of large deformations, and its physical meaning is to capture the additional strain energy caused by geometric non - linearities. Especially in a spherical reticulated shell structure with a large curvature and complex deformations, this term can more accurately reflect the strain enhancement effect caused by local curvature changes and uneven node displacements. In addition, the factor before the fraction is for normalizing the strain calculation of the total degree of freedom vector . Since is composed of the average values of the displacement fields of the adjacent observation points of this observation point, different observation points may have different numbers of adjacent sampling points. To ensure the comparability of the strain tensors calculated in different regions, is introduced to average the number of local adjacent points, so that the strain tensor does not show numerical deviations due to different sampling densities. Through this normalization process, the model can maintain consistency on a global scale and accurately reflect the true stress - strain state of the structure in the local area.

Claims

1. A method for analyzing the node stress of a spherical lattice shell structure based on refined simulation optimization, characterized in that: The method comprises: Step 1: uniformly select multiple observation points in the target spherical lattice shell area, and deploy displacement sensors at each observation point to obtain the displacement field of the observation point; the distance between adjacent observation points is less than the set distance threshold; the displacement field corresponding to each observation point is corrected using spherical harmonics in spherical coordinates, and the total potential energy functional is constructed; Step 2: Discretize the target spherical lattice shell region into finite element grids, and calculate the local error index of each grid unit by combining the total potential energy functional of each grid unit's nearest observation point; use high-order finite element shape functions to approximate the displacement field, construct a discrete residual equation, and solve it to obtain the overall degree of freedom vector; Step 3: Calculate the absolute value of the difference between the local error index of each grid unit and the modulus of the total degree of freedom vector of the nearest observation point. If the absolute value exceeds the set abnormality judgment threshold, it is judged that the grid unit has a force abnormality.

2. The method for analyzing the node stress of a spherical lattice shell structure based on refined simulation optimization according to claim 1, characterized in that: The total potential energy functional at the observation point is expressed as follows: ； in, is the total potential energy functional; is the energy storage density function of the observation point, is the strain tensor of the observation point; is the corrected displacement field of the observation point , is the reference radius of the spherical lattice shell; is the displacement field measured by the displacement sensor; is the gradient operator; is the Laplace operator; is the modified energy function; L is the order of the spherical harmonic function; is the polar angle of the observation point; is the azimuth of the observation point.

3. The method for analyzing the node stress of a spherical lattice shell structure based on refined simulation optimization according to claim 2 is characterized in that: The energy storage density function is expressed using the following formula: in, is the Young's modulus of the material of the spherical lattice shell; is the Poisson's ratio of the material of the spherical lattice shell; Represents the transpose operation of a vector or matrix; is the trace of the strain tensor.

4. The method for analyzing the node stress of a spherical lattice shell structure based on refined simulation optimization according to claim 3 is characterized in that: The corrected energy function is expressed using the following formula: ； in, is a high-order stiffness parameter, ; represents norm operation; is the shell thickness of the target spherical lattice shell area.

5. The method for analyzing the node stress of a spherical lattice shell structure based on refined simulation optimization according to claim 4 is characterized in that: In step 2, when the target spherical lattice shell region is discretized into a finite element grid, the number of grid cells in a circular region constructed with each observation point as the center and the distance between adjacent observation points as the radius exceeds a set number threshold.

6. The method for analyzing the node stress of a spherical lattice shell structure based on refined simulation optimization according to claim 5, characterized in that: In step 2, the local error index of each grid cell is calculated by the following formula: ； in, is the distance from the grid cell to the nearest observation point.

7. The method for analyzing the node stress of a spherical lattice shell structure based on refined simulation optimization according to claim 6, characterized in that: In step 3, the process of using high-order finite element shape functions to approximate the displacement field and constructing a discrete residual equation specifically includes: discretizing the total potential energy functional to obtain a discretized total potential energy functional; using high-order finite element shape functions to approximate the displacement field, calculating the variation of the discretized total potential energy functional with respect to the total degree of freedom vector, and obtaining a discrete residual equation.

8. The method for analyzing the node stress of a spherical lattice shell structure based on refined simulation optimization according to claim 7, characterized in that: The discrete residual equation is expressed as follows: ； in, Gradient operators obtained for high-order finite element shape functions; is the total degree of freedom vector, defined as the mean of the displacement fields of the neighboring observation points of the observation point; is the second-order derivative operator of the high-order finite element shape function; is the strain tensor of the overall degree of freedom vector.

9. The method for analyzing the node stress of a spherical lattice shell structure based on refined simulation optimization according to claim 8, characterized in that: The high-order finite element shape function is expressed using the following formula: ； in, is a high-order finite element shape function.

10. The method for analyzing the node stress of a spherical lattice shell structure based on refined simulation optimization according to claim 9, characterized in that: The strain tensor of the overall degree of freedom vector is calculated using the following formula: ； in, is the number of adjacent observation points of this observation point.

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