Spherical reticulated shell structure node stress analysis method based on refined simulation optimization

Abstract

The invention relates to the technical field of computer aided design, and further relates to a spherical latticed shell structure node stress analysis method based on refined simulation optimization. The method comprises the following steps of: 1, uniformly selecting a plurality of observation points in a target spherical reticulated shell area, correcting a displacement field corresponding to each observation point by using a spherical harmonic function under a spherical coordinate, and constructing a total potential energy functional; 2, discretizing a target spherical reticulated shell area into finite element grids, constructing a discrete residual equation by using a high-order finite element shape function approximation displacement field, and solving to obtain a total degree-of-freedom vector; and 3, calculating the absolute value of the difference between the local error index of each grid unit and the module of the overall degree-of-freedom vector of the observation point closest to the grid unit, and judging that the grid unit is stressed abnormally. According to the method, the calculation precision, the numerical stability and the local anomaly detection capability are remarkably improved.

Description

Technical Field

[0001] The invention belongs to the technical field of computer-aided design, and in particular relates to a spherical lattice shell structure node force analysis method based on refined simulation optimization. Background Art

[0002] As modern engineering construction develops towards large spans and complex structures, spherical lattice shell structures have been widely used in buildings, bridges, stadiums and other fields due to their excellent bearing capacity and space utilization. Traditional spherical structure analysis methods mainly rely on empirical formulas and low-order finite element methods. These methods have high computational efficiency in the early stages of structural design. However, in actual engineering, the external loads borne by spherical lattice shell structures often present multi-directional, multi-modal, and non-uniform distribution characteristics, which leads to stress concentration and abnormal deformation at local nodes. In recent years, with the rapid development of computer technology and sensor technology, researchers have begun to try to apply refined simulation optimization methods to the force analysis of complex structures. By performing high-precision numerical simulations and experimental data collection on the structure, they try to reveal subtle stress changes and local anomalies that are difficult to capture with traditional methods.

[0003] Among the currently disclosed technologies, some studies use high-order finite element methods combined with sensor measurement data to analyze the stress distribution of the entire structure, and use spherical harmonics to correct the displacement field to adapt to the geometric characteristics of the spherical coordinate system. These methods have improved the accuracy of structural force analysis to a certain extent, but there are still some shortcomings. First, the traditional finite element discretization method often uses a fixed density or a rough adaptive strategy in mesh division, which fails to fully consider the complexity of the local geometry and force of the spherical lattice shell structure, resulting in discrete errors in some key areas, which in turn affects the accuracy of the overall simulation results. Secondly, when dealing with large deformation and nonlinear problems, the existing technology usually only uses linear strain theory or simple nonlinear corrections, which cannot accurately capture the additional strain energy caused by local geometric discontinuities and high curvatures, resulting in lag and inaccuracy in the identification of local abnormal stress states. In addition, although some studies have introduced high-order finite element shape functions and modified energy functions to improve 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 local force anomaly predicted by the model under certain complex working conditions. It is difficult to fully match the actual measurement data. For example, the energy storage density functions used in some existing technologies are mostly based on the traditional elastic strain energy expression, ignoring the energy compensation effect caused by curvature changes, local large deformations and nonlinear effects in spherical lattice shell structures, making the judgment criteria for local abnormal stress states at structural nodes vague. At the same time, when using sensors to obtain displacement data, directly using the original measurement data without spatial correction through mathematical tools such as spherical harmonics often leads to significant discreteness and noise effects of the data, which in turn affects the construction of subsequent energy functionals and the accuracy of finite element discretization. Summary of the invention

[0004] The main purpose of the present invention is to provide a spherical lattice shell structure node force analysis method based on refined simulation optimization, by arranging high-precision displacement sensors in the target area to collect discrete displacement data, and using spherical harmonics to correct the data, constructing a total potential energy functional that conforms to the spherical geometric characteristics, and then combining adaptive finite element meshing and high-order finite element shape function approximation displacement field, and then establishing discrete residual equations and local error indicators, 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 force phenomena caused by local curvature changes, large deformations and nonlinear effects of the spherical lattice shell structure, but also significantly improve the calculation accuracy, numerical stability and local anomaly detection capabilities.

[0005] In order to solve the above problems, the technical solution of the present invention is achieved as follows: A spherical lattice shell structure node force analysis method based on refined simulation optimization, the method comprising: 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.

[0006] Furthermore, the total potential energy functional of the observation point is expressed using the following formula: ; 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 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.

[0007] Furthermore, the energy storage density function is expressed using the following formula: in, is the reference radius of the spherical lattice shell; 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.

[0008] Furthermore, the modified 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.

[0009] Further, 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; Furthermore, in step 2, the local error index of each grid unit is calculated by the following formula: ; in, is the distance from the grid cell to the nearest observation point.

[0010] Furthermore, 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.

[0011] Furthermore, the discrete residual equation is expressed using the following formula: ; 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.

[0012] Furthermore, the high-order finite element shape function is expressed using the following formula: ; in, is a high-order finite element shape function.

[0013] Furthermore, 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.

[0014] A spherical lattice shell structure node force analysis method 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 lattice shell structure is obtained. The prior art usually directly uses the displacement data measured by the sensor for finite element analysis, but due to the discreteness and noise influence of the measurement data, the direct use of these data will lead to a decrease in the accuracy of energy functional construction and finite element discretization. The present invention uses spherical harmonic functions to correct the displacement field of the observation point, so that the displacement field in the spherical coordinate system is smoother and 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 force analysis. Secondly, the present invention optimizes the finite element calculation process through a refined finite element meshing method, and improves the accuracy and efficiency of the calculation. The prior art usually adopts a fixed grid density or a simple adaptive grid division strategy, which is difficult to accurately capture the complex force conditions of the local area of ​​the spherical lattice shell structure. The present invention takes each observation point as the center, constructs local grid units in combination with the distance between adjacent observation points, and sets the threshold of the number of finite element grids to ensure that the grid division in the key stress area is fine enough, and appropriately reduces the grid density in the area where the stress is relatively uniform, thereby improving the calculation efficiency and reducing the error. 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 deformation abnormal areas, and provide more accurate 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, so that the discretization calculation is more refined and can more accurately describe the displacement field changes. When solving the spherical lattice shell structure, the traditional low-order finite element method is usually difficult to accurately capture the local nonlinear deformation and stress concentration phenomenon, 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, so that the calculation results are more in line with the actual stress state. The high-order finite element shape function can effectively improve the fitting accuracy of the local displacement field, so that in the complex stress area, the finite element calculation can still maintain a high calculation accuracy, avoiding the problem of excessive numerical errors in the local high gradient area of ​​the traditional method. 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

[0015] Figure 1 A schematic diagram of a method flow of a spherical lattice shell structure node force analysis method based on refined simulation optimization provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0016] In order to enable those skilled in the art to better understand the scheme of the present invention, the technical scheme in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work should fall within the scope of protection of the present invention.

[0017] Example 1, reference Figure 1 : A spherical lattice shell structure node force analysis method based on refined simulation optimization, the method comprising: 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; Due to its geometric symmetry, the stress characteristics of spherical lattice shell structures are essentially different from those of plane structures. When analyzing this type of structure, the traditional Cartesian coordinate system method is difficult to accurately describe its deformation mode. Therefore, this method uses a spherical coordinate system to characterize 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 lattice shell structure, multiple observation points need to be uniformly selected. 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. In order to ensure the spatial resolution of the observation data, the distance between adjacent observation points must be less than the set distance threshold, which is usually determined based on the characteristic size and main stress mode of the lattice shell structure. For example, in the case of high-order modal vibration or local stress concentration, appropriately reducing the distance between observation points can improve the precision of data acquisition, thereby improving the accuracy of the overall simulation optimization. A high-precision displacement sensor is arranged at each selected observation point to monitor the displacement response of the point in real time. These displacement data are important input information for subsequent analysis. Since the stress characteristics of the spherical lattice shell structure may lead to non-uniform local deformation, the layout of the sensors should ensure that the displacement components in different directions can be reasonably collected to avoid data errors caused by measurement blind areas.

[0018] However, the displacement field data directly obtained are often affected by measurement errors, environmental disturbances and discrete distribution, resulting in poor spatial continuity of the data. Therefore, this method uses spherical harmonics in spherical coordinates to correct the displacement field of the observation point to optimize the data quality. Spherical harmonics is an orthogonal basis function suitable for spherical regions, which can approximately expand the physical quantities distributed in space, thereby constructing a smooth displacement distribution function on the global scale of the spherical lattice shell structure. Mathematically, the basis function form of the spherical harmonic function depends on the polar angle and azimuth angle in the spherical coordinates, which can effectively capture the displacement change pattern on the spherical structure. By appropriately selecting the order of the spherical harmonic expansion, the calculation complexity can be reduced while ensuring the calculation accuracy, so that the corrected displacement field not only maintains the basic characteristics of the measured data, but also has higher smoothness and global consistency. On this basis, in order to further quantify the stress state of the entire spherical lattice 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 equilibrium, its potential energy function takes a minimum value. The potential energy functional is usually composed of elastic strain energy, external force potential energy and additional constraint energy terms, where 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 load. In the spherical lattice shell structure, since the structural units are arranged on a spherical surface, the traditional Cartesian coordinate system potential energy expression method is difficult to intuitively reflect its true physical properties. Therefore, this method adopts a spherical coordinate system and combines the displacement field data corrected by spherical harmonics to express the potential energy functional in the form of a spherical surface integral. This not only ensures that the potential energy functional is consistent with the geometric characteristics of the spherical lattice shell structure, but also can better adapt to the deformation characteristics of the spherical structure in terms of calculation, making the final simulation optimization process more stable and reliable.

[0019] 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; In the specific calculation process, the target spherical lattice shell area needs to be discretized first, that is, divided into finite element grids. Since spherical lattice shell structures usually have strong geometric non-uniformity and curved surface characteristics, the traditional Cartesian coordinate finite element discretization method is difficult to effectively adapt to the stress characteristics of this type of structure. Therefore, in this method, the division of the finite element grid fully considers the geometric characteristics of the spherical structure and is adaptively adjusted in combination with the distribution density of the observation points. In the discretization process, the size of each grid unit should be dynamically optimized according to the key stress area of ​​the structure. For example, in the stress concentration area or near the connection node, the grid division should be more refined to ensure the calculation accuracy, while in the relatively uniform stress area, the grid density can be appropriately relaxed to improve the calculation efficiency. Through such a grid 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 grid division, it is necessary to combine the total potential energy functional of the nearest observation point of each grid unit to calculate the local error index. The total potential energy functional is constructed based on the displacement field data of the observation point, which includes the elastic strain energy, external force potential energy and constraint energy terms of the structure. Its minimization process can ensure the real stress state of the structure under the action of external force. Therefore, for each finite element mesh unit, its theoretical stress state should be consistent with the calculation result of the total potential energy functional of the nearest observation point of the unit. However, due to the inevitable numerical errors in the discretization process and the different scale effects of the units in the meshing process, the local calculation value of each mesh unit may have a certain deviation from 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 unit and provide a basis for further optimization. When calculating the local error index, the key step is how to accurately describe the displacement field of the mesh unit. 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 complex force systems such as spherical lattice shell structures, due to its bending characteristics and high-order deformation modes, low-order interpolation methods are difficult to effectively capture the actual displacement changes. Therefore, this method introduces a high-order finite element shape function. By increasing the order of the interpolation function, the displacement field inside each unit can be more accurately approximated to the actual deformation mode. The advantage of the high-order finite element method is that its shape function contains not only linear terms, but also quadratic and even higher-order polynomial terms, which can more accurately describe the deformation behavior of complex surfaces. In addition, since the displacement distribution of spherical lattice shell structures usually has strong continuity and smoothness, high-order finite element shape functions can better adapt to this deformation feature, thereby reducing interpolation errors and improving calculation accuracy. The displacement field constructed based on the high-order finite element shape function can be further used to construct discrete residual equations. 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, so as to solve the overall degree of freedom vector.

[0020] 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.

[0021] In this process, the setting of the abnormality judgment threshold is an important factor affecting the recognition accuracy. Traditional force analysis methods usually use empirical values ​​or simplified statistical methods to set the error threshold, which may lead to misjudgment or missed judgment in some cases. However, this method fully considers the overall force mode of the spherical lattice shell structure. Through the error evaluation mechanism based on the total potential energy functional of the observation point, the abnormality judgment threshold is not only physically reasonable, but also adaptable to different types of load conditions. For example, under the action of wind load, the surface force of the spherical lattice shell structure often presents a non-uniform distribution, and there may be a large stress gradient in the local area. Therefore, the error threshold should be appropriately increased to avoid misjudging the normal local stress concentration phenomenon. Under the action of earthquake, due to the violent dynamic response of the structure, sudden displacement anomalies may occur at local nodes. Therefore, the error threshold should be appropriately reduced to increase the sensitivity to abnormal force states. Through this adaptive threshold adjustment strategy, this method can maintain a high abnormality recognition accuracy under different working conditions. When calculating the abnormality judgment index, the absolute value of the difference in modulus is a key mathematical measurement method. The calculation of the modulus involves the components of the overall degree of freedom vector, and these components correspond to different displacement directions of the structure. In the spherical coordinate system, the stress state of the spherical lattice shell structure is mainly determined by the radial, tangential and normal displacement components, so the calculation of the degree of freedom vector needs to consider the comprehensive effect of these components in different directions. For example, at the key connection nodes of the lattice shell, the change of the radial displacement component may reflect the overall stress mode of the structure, while the change of the tangential and normal displacements may indicate the stress concentration of the local nodes. Therefore, by calculating the modulus of the overall degree of freedom vector, a comprehensive stress state measure can be obtained, thereby more comprehensively describing the overall deformation characteristics of the structure. By comparing this modulus value with the local error index, it is possible to effectively identify those stress abnormal areas caused by external forces, defects or loose connections. In addition, in the process of abnormality judgment, the geometric continuity and force transfer characteristics of the lattice shell structure must also be considered. Since the spherical lattice shell structure is usually composed of multiple rods or panels, its stress state has a strong spatial correlation, that is, the stress changes between adjacent units often have a certain degree of continuity. Therefore, when calculating the force error of each grid unit, it is necessary not only to pay attention to its own local error index, but also to conduct a comprehensive analysis in combination with the stress state of the surrounding units. For example, if the error value of a grid unit slightly exceeds the set abnormality judgment threshold, but the error values ​​of the surrounding units are all within the normal range, the force anomaly of the unit may be caused by the local grid discretization error, and not necessarily a real structural defect. Therefore, in the final abnormality judgment process, the abnormality recognition results can be optimized through a certain spatial smoothing algorithm to improve the reliability of the judgment.

[0022] Example 2: The total potential energy functional of the observation point is expressed using the following formula: ; in, is the total potential energy functional; is the energy storage density function of the observation point; is the displacement field 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 modified energy function.

[0023] Specifically, the left side of the formula It is called the total potential energy functional. The variables involved here include not only the displacement field itself, including its gradient and the Laplacian operator The second-order differential information of the displacement field. This multi-level description can simultaneously capture the deformation characteristics of the structure at both the local and global scales, reflecting the complexity of the energy distribution within the structure under actual stress conditions. Specifically, represents the energy storage density function at the observation point, where It is the strain caused by the displacement field and its gradient, which reflects the energy storage of the material during elastic deformation. Traditionally, energy storage density mainly describes the internal energy of the material. However, in spherical lattice shell structures, due to the influence of factors such as curvature and geometric continuity, strain energy alone often cannot fully reflect local force anomalies, so additional energy correction terms must be introduced. Here, This is the role played by using the second-order derivative information of the displacement field through the Laplace operator Capturing the bending and diffusion characteristics of the displacement field in space, thus correcting the geometric effects that may be ignored in the traditional energy storage description. This is particularly important for spherical lattice shell structures, because the stress state of the structure is often not just linear deformation, but is accompanied by complex surface deformation and local stress concentration. The modified energy function provides a mathematical basis for this. In fact, the spherical harmonic characteristics unique to the spherical structure are taken into account, and the discrete data measured by the sensor are converted into The resulting displacement field is transformed by a normalization factor that is closely related to spherical geometry. It is more in line with the mathematical description of the spherical coordinate system. The factor here is is derived from the orthogonality and normalization requirements of spherical harmonics, while the factorial terms in the numerator and denominator reflect the coupling relationship between the measured data and the geometric parameters of the spherical structure, where is the radius of the sphere, and Represent the polar angle and azimuth angle respectively. This correction not only ensures the smoothness and continuity of the displacement field on the sphere, but also enables the subsequent energy functional constructed based on the displacement field to more realistically reflect the energy state under actual working conditions. In addition, the gradient operator in the formula and the Laplacian operator They play the role of capturing the first-order and second-order changes of the displacement field respectively. reflects the rate at which displacement changes with space, which is the main source of local strain, while the Laplace operator The energy functional measures the curvature and diffusion trend of the displacement field, which can reveal the additional energy effect caused by geometric bending in the spherical lattice 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 of geometric nonlinearity, providing rich information for identifying local force anomalies.

[0024] Example 3: The energy storage density function is expressed using the following formula: ; in, is the reference radius of the spherical lattice shell; is the polar angle of the observation point; is the azimuth of the observation point; 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; is the strain tensor of the observation point, defined as: ; Represents the transpose operation of a vector or matrix.

[0025] Specifically, represents the base radius of the spherical lattice shell, which not only describes the size of the structure as a geometric parameter, but also plays a role of 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 sphere, the area element is usually As weight, the energy density has different contributions at different polar angles, which can accurately describe the energy difference caused by the change of spherical curvature. and Two directional factors, which appear in the first and second terms respectively, are used 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.

[0026] 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 method under different strain modes, ensuring that they have complete mathematical description capabilities. In the physical meaning of the entire formula, the first half It mainly reflects the energy storage of the spherical lattice shell structure under shear and bending deformation. It not only combines the geometric size of the structure and the distribution of surface elements on the spherical surface, but also weights the energy contribution in each direction; The energy storage caused by volume change is described in detail, and its parameter combination ensures that the energy change can be accurately described when the material is compressed or stretched. This sub-item description allows the overall energy storage density function to reflect the inherent elastic properties of the material and fully consider the energy distribution differences caused by geometric and azimuth changes at different positions of the spherical lattice shell structure, thereby achieving high-precision simulation and evaluation of the node stress state.

[0027] Embodiment 4: The modified 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.

[0028] Specifically, the modified energy function It is a detailed description of the local energy correction part of the spherical lattice shell structure. Its core is to quantify the local curvature effect of the structure by introducing the second-order differential information of the displacement field, and thereby reflect the additional energy contribution caused by geometric nonlinearity and large deformation. In this formula, Represents the displacement field The result of the Laplace operator can also be regarded as the norm of the second-order gradient of the displacement field, which captures the curvature or bending degree of the displacement field in the local area. In the spherical lattice shell structure, due to the complex curvature characteristics of the structural surface itself, it is not only necessary to rely on the first-order displacement change to describe the strain state, but also to use the second-order differential term to characterize the energy change caused by bending, arching or local instability. Specifically, It can be regarded as a secondary penalty for the local curvature distribution of the entire structure. It reflects the areas with larger curvature or severe bending at a higher energy cost, thereby giving these areas a higher weight in the overall energy calculation, which is of great significance for refined simulation optimization and local anomaly determination. It is defined by the basic physical parameters and geometric parameters of the material, which can be expressed as .in, is the Young's modulus of the material, which directly reflects the tensile and compressive resistance of the material; is the shell thickness of the target spherical lattice shell area, and its square term shows that the bending stiffness increases significantly with the increase of thickness; while Poisson's ratio It reflects the relationship between the lateral and longitudinal deformation of the material under stress, which appears in the denominator, taking into account the volume effect of the material and its actual performance under plane stress state. The high-order mechanical properties of the spherical lattice shell in bending and shear are accurately characterized, which not only affects the numerical value of the overall energy functional, but also determines the resistance of the local area to geometric discontinuities or mutations. In other words, a higher The value means that when there is a drastic change in curvature locally, the structure needs to consume more energy to maintain stability, which is in sharp contrast to the simplified treatment of traditional low-order models that ignore the curvature effect. The modified energy function essentially introduces high-order energy terms. This design concept originates from the expression of bending energy in thin shell theory and plate bending theory. In traditional energy models, only first-order strain energy is considered in most cases. However, for spherical lattice shell structures with complex geometric shapes, local bending or arching behavior often has an important impact on the safety of the overall structure. By adding This term, the model can more comprehensively describe the additional energy changes caused when a significant curvature change occurs in a local area, and impose a corresponding penalty on this change. In this way, in the simulation optimization process, not only can the intrinsic properties of materials and structures be reflected more accurately, but also local abnormal deformations caused by external loads, manufacturing errors or environmental effects can be timely identified and warned. Combined with the overall idea of ​​the present invention, that is, the spherical lattice shell structure node force analysis method based on refined simulation optimization, this corrected energy function plays a vital role. By utilizing the second-order differential information, the entire energy model takes into account the energy storage effect brought by the elastic deformation of the material when describing the stress state of the local node, and also takes into account the energy correction caused by the structural geometric discontinuity or the local curvature mutation. Especially in spherical structures, there are obvious differences in the geometric curvatures of different regions. The traditional first-order model is difficult to capture these subtle changes, and the introduction of high-order energy terms enables 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, thus providing a solid theoretical and numerical basis for subsequent anomaly determination, local error analysis, and optimal design.

[0029] Example 5: 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 threshold value.

[0030] Specifically, this method first uses uniformly distributed observation points to form a direct measurement of structural deformation. Each observation point not only provides displacement field data, but also represents an important sampling point of the local stress state of the spherical lattice shell. In the discretization process, a circular area with the observation point as the center and the distance between adjacent observation points as the radius is used to ensure that each sampling area covers the main geometric and mechanical features in the local area, and ensures that the finite element grid in the area is dense enough to capture the slight differences in local stress and deformation. When the number of finite element grids exceeds the set number threshold, it means that the grid density in the area has reached or exceeded the accuracy standard required by the optimization design. At this time, it can not only more accurately reflect the complex stress distribution and deformation gradient in the area, but also help to reduce the uncertainty caused by discretization errors in subsequent numerical calculations. On the other hand, if the number of grids is lower than the set threshold, it may indicate that the sampling density in the area is insufficient, and important local stress concentration or deformation features may be missed, thereby affecting the accuracy of the solution of the overall degree of freedom vector. Therefore, the adaptive meshing strategy takes full account of the particularity of the stress analysis of the spherical lattice shell structure in its design. On the one hand, it uses the observation points to construct the local area, ensuring the high matching between the sampled data and the finite element model. On the other hand, by setting the threshold of the number of grids, the local inhomogeneity that may exist in the discretization process is dynamically regulated, so that the entire simulation model has higher accuracy and robustness when reflecting the real working conditions. Since the spherical lattice shell structure itself has complex surface geometric characteristics and highly nonlinear stress response, the traditional finite element meshing often relies on a fixed grid density, which makes it difficult to take into account the accurate description of the overall structure and local details. The strategy in this embodiment uses the spatial information of the observation points as the key basis, and dynamically adjusts the number of grids to achieve accurate capture of the local abnormal stress area, thereby effectively reducing the numerical error caused by the rough discretization in the process of finite element discretization, and then provides a solid data foundation for the subsequent error index calculation, the overall degree of freedom vector solution and the final abnormal stress judgment.

[0031] Example 6: In step 2, the local error index of each grid unit is calculated by the following formula: ; in, is the distance from the grid cell to the nearest observation point.

[0032] Specifically, the first factor of the formula middle, represents the base radius of the spherical lattice shell, which is representative of the geometric scale of the entire structure; and is the distance between the current grid cell and the nearest observation point. This scaling factor is used to reflect that the closer the area is to the observation point, the more accurate and representative the measurement data will be, while the farther away the area may have greater uncertainty, thus amplifying or reducing the sensitivity of the area in the error index. In other words, when When smaller, A larger value means that the grid unit is more sensitive to the error change, which requires a more accurate description of the energy in the area; conversely, when When it is larger, the error index is relatively weakened, reflecting the weakening influence of local measurement information on the overall energy state. Next, the terms in the curly brackets Then the energy storage density function Displacement Field The partial derivative of is combined with the second-order derivative term involved in the modified energy function. Among them, It represents the sensitivity or gradient of energy storage density with displacement. Its function is to quantify the energy change caused by displacement change, which is important for capturing the uneven distribution of local strain energy. It emphasizes the influence of the polar angle direction in the spherical coordinate system, so that the energy contribution of different polar angle positions is reasonably weighted. The higher-order stiffness parameters are introduced Laplace operator with displacement field Here, According to the definition , which combines the Young's modulus of the material , Shell thickness and Poisson's ratio Parameters such as reflect the resistance of the spherical lattice shell to bending and local curvature changes; The second-order change in the displacement field is measured, which is the curvature or bending degree in the local area. The influence of the error in the azimuth direction in the spherical coordinate system is further considered, which comprehensively reflects the energy correction difference caused by the uneven geometry and force in different directions of the structure. In this way, the entire term in the brackets organically combines the first-order displacement change and the second-order curvature change, providing a composite measure for the local error index that reflects the change in material energy storage and captures the bending sensitivity of the structure.

[0033] The next product term in the formula It mainly reflects the regulatory effect of material properties on local errors. Young's modulus Represents the rigidity of the material. The larger the value, the more difficult it is to deform the material, and therefore the more sensitive it is in terms of energy storage. Poisson's ratio It is used to describe the relationship between the lateral and longitudinal deformation of the material when it is subjected to stress. Its expression in the denominator ensures the balance of the energy response of the material under different stress conditions. The introduction of this coefficient ensures that the local error index can match the actual elastic properties of the material, thereby theoretically ensuring the physical rationality and numerical stability of the calculation results. Finally, the end of the formula The term is the shell thickness With geometric parameters Here, is the shell thickness of the spherical lattice shell region, the square of which reflects the quadratic effect of thickness on local stiffness and energy storage, while It represents the effective geometric scale after subtracting the distance from the grid unit to the observation point from the reference radius. Such normalization can eliminate the influence caused by size differences and make the local error index comparable between different regions. Through the square form of this ratio, the local error value can be effectively enlarged or reduced, thereby making it more accurate to capture the local abnormal stress state during the finite element discretization process. The design of this part not only emphasizes the role of structural geometric characteristics in error transmission, but also includes the influence of thickness on the stress state into the calculation range of the error index, which significantly improves the sensitivity of the overall model to local deformation.

[0034] Example 7: 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.

[0035] Specifically, the discretization process requires the construction of a fine finite element mesh that adapts to the local geometric characteristics in the entire spherical lattice shell area, and the shape function of each mesh unit is used to approximate the energy on the continuous domain, and the energy functional originally defined on the continuous domain is converted into a function about the degrees of freedom of each node. Here, the application of high-order finite element shape functions is particularly critical, because the spherical lattice shell structure often makes it difficult for traditional low-order shape functions to fully express the high-order information of the actual displacement field due to its curvature change, local nonlinear deformation and complex stress state. High-order shape functions can capture local subtle deformation characteristics by increasing the order of polynomials, so that the discretized total potential energy functional can more accurately reflect the actual physical response of the structure. Next, the displacement field is approximated by high-order finite element shape functions, and the discretized total potential energy functional is calculated by variation with respect to the total degree of freedom vector. This variation process is essentially a necessary condition for solving the minimum potential energy problem, that is, by calculating the partial derivatives of the degrees of freedom of each node of the discretized energy functional and setting the partial derivatives to zero, the equilibrium equation or residual expression between each node is established. During the variational calculation process, the fine approximation provided by the high-order shape function makes the stress state of each node not only affected by the energy contribution in its direct neighborhood, but also reflects the energy correction caused by bending and curvature effects in distant areas, thereby constructing a relatively complete and accurate discrete residual equation group as a whole. In this residual equation group, each equation corresponds to the equilibrium state of a node's degree of freedom, expressing the deviation of the node in the overall energy field. This deviation is the local residual, and its numerical value reflects the finite element approximation error and the degree of local stress anomaly. In the subsequent iterative solution process, by solving these discrete residual equations, the exact solution of the overall degree of freedom vector can be obtained, thereby realizing high-precision prediction and abnormal judgment of the stress state of the nodes of the spherical lattice shell structure. It is worth noting that the high-order finite element shape function here not only improves the discretization accuracy, but also significantly improves the numerical stability and convergence of the discrete residual equation, because it can more fully capture the high-order energy effects caused by curvature, thickness changes and nonlinear deformation in the structure.

[0036] Embodiment 8: The discrete residual equation is expressed using the following formula: ; 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.

[0037] Specifically, the first coefficient in the formula It reflects the influence of spherical geometry factors: is the base radius of the spherical lattice shell, and Represent the polar angle and azimuth of the observation point respectively. This combination can properly weight the energy and mechanical effects in the spherical coordinate system, so that the physical quantities at different positions can reflect the uneven distribution of the spherical surface. Next, the factor is directly related to the basic mechanical properties of the material, among which is the Young's modulus of the material, which indicates the rigidity of the material, and Poisson's ratio The denominator describes the lateral deformation characteristics of the material when it is under tension or compression. The appearance of is the common elastic constant correction factor, which is used to ensure the physical consistency of energy expression. is the transpose of the gradient operator matrix constructed by the high-order finite element shape function, which establishes a differential relationship between the degrees of freedom at discrete nodes and the local displacement field. is a comprehensive expression of the linear and nonlinear parts of the strain tensor, where Represented by the total degree of freedom vector The linear strain tensor corresponding to (defined as the average displacement field of the observation point and its adjacent observation points) follows the small strain theory and can capture the local deformation information around the node. The geometric nonlinear effect is taken into account, 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 addition of the two not only takes into account the basic description of linear elastic theory, but also integrates the correction effect under large deformation, so that the residual equation can more comprehensively describe the energy change and stress state.

[0038] The terms in the second row of the formula It mainly reflects the contribution of volume deformation to energy. Here, is the trace of the strain tensor, representing the volume change in the local area, that is, the volume strain, which is closely related to the bulk modulus of the material in elastic mechanics. Poisson's ratio appears again in the numerator and is related to as well as Together they form the standard volume energy correction factor, which can correctly reflect the intrinsic energy changes of the material when it is under compression or tension when calculating the volume deformation energy. Finally, this part of the residual reflects the mechanical equilibrium condition of the node caused by volumetric strain, which complements the construction of the overall discrete residual equation. The term introduces higher-order curvature effects in the modified energy function. is a high-order stiffness parameter, which is defined as ,in represents the thickness of the lattice shell, which reflects the physical properties of the structural bending stiffness; is the second-order derivative operator obtained by the high-order finite element shape function, which can capture the second-order variation of the displacement field, that is, the local curvature or bending degree. The combined operation of is equivalent to a quadratic penalty on the second-order differential term, which reflects that when a large curvature change occurs in the local area, the system needs to consume additional energy to maintain balance. This term directly acts on the overall degree of freedom vector , a numerical description of the bending effect is provided in the discrete residual equation, ensuring that the model not only considers the strain energy in the plane when dealing with the spherical lattice shell structure, but also takes into account the high-order energy correction caused by curvature. Superimposing the above items and setting the entire expression equal to zero is the equilibrium condition of the discrete residual equation. Its physical meaning is that when the system is in equilibrium, the residual force generated by the combined action of material energy storage, volume strain and bending energy at each discrete node described by the high-order finite element shape function must be zero. This is not only to solve the overall degree of freedom vector It is also the key to ensure that the simulation model can accurately reflect the actual stress state. By solving this nonlinear equation group, the displacement distribution of each node in the spherical lattice shell structure can be obtained, and the local strain, stress and energy distribution can be further derived, which provides a solid theoretical basis for subsequent abnormal force determination and structural optimization design. It is worth mentioning that in this discrete residual equation, the geometric factor in the spherical coordinate system is combined and , so that the formula can fully reflect the inhomogeneity of the spherical structure in different directions, so that in the numerical simulation process, the local errors caused by the change of curvature and the difference in the distance between nodes can be accurately characterized. At the same time, the application of high-order finite element shape functions not only improves the approximation accuracy of the displacement field, but also makes the strain tensor This is crucial to capture energy discontinuities caused by large deformations or local geometric nonlinearities. It is particularly important, as it makes up for the shortcomings of traditional low-order finite element models in large deformation analysis to a certain extent.

[0039] Example 9: The high-order finite element shape function is expressed using the following formula: ; in, is a high-order finite element shape function.

[0040] Specifically, the variables in the formula Usually represents a local coordinate variable, and its range in the reference domain is often mapped to The interval, and the denominator In fact, this interval boundary information is reflected in a normalized way to ensure that the shape function satisfies specific zero values ​​or other constraints at the boundary. Specifically, this denominator can be regarded as a constraint on the function value at the two endpoints in the reference coordinate system (usually corresponding to the boundaries of the unit), which ensures that when Take 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, we can see that it is composed of three factors: , as well as Among them, the first factor With the second factor The appearance of and These two positions must have zero values. This design is usually to set internal nodes or intermediate interpolation points in the unit, so that the entire interpolation polynomial has a higher order and can meet the precise node matching requirements at these key points. By selecting these two symmetrical zero points, the smoothness of the shape function is guaranteed, and it is also helpful to more finely characterize the gradient changes and bending effects of the local displacement field during finite element discretization. The third factor is more special, as it directly introduces material properties into the construction of shape functions. is the Young's modulus of the spherical lattice shell material, and represents Poisson's ratio, both of which are basic parameters for describing the mechanical properties of materials. and by Form and variables Subtraction actually introduces a nonlinear modulation mechanism into the shape function, so that the shape function not only depends on the geometric distribution, but also can sensitively reflect the local response differences caused by the material stiffness and lateral deformation characteristics. This processing method is particularly suitable for spherical lattice shell structures. Due to the complex stress state and significant changes in local geometric curvature, it is often difficult to capture the slight differences caused by material nonlinearity by simply relying on traditional low-order shape functions. The method of embedding material parameters into the interpolation function can compensate for this deficiency to a certain extent, making the high-order finite element model more accurate in reflecting node deformation and energy distribution.

[0041] The entire shape function The structure of can be understood as a fractional polynomial, whose numerator determines the zero points and characteristic points of the shape function at each key position in the unit through the product of three factors, while the denominator serves as a normalization factor to ensure numerical stability at the reference domain boundary and to meet the necessary interpolation conditions. This expression is actually a carefully designed high-order interpolation function. The higher 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 conditions. In specific finite element analysis, the role of high-order finite element shape functions is to discretize continuous physical fields into discrete variables at each node by interpolating the displacement field in the unit. Due to the complex surface characteristics and non-uniform force distribution of the spherical lattice shell structure, traditional low-order interpolation functions often cannot meet the requirements of refined simulation, which easily leads to numerical errors and insufficient local response. The high-order shape function in this embodiment not only ensures that the actual displacement value can be accurately matched at different key positions (such as internal nodes and boundary nodes) in the unit by introducing multiple key factors in the interpolation process, but also integrates the elastic properties of the material into the shape function through special factors, so that the numerical simulation can fully consider the physical properties of the material while reflecting the geometric shape. In addition, the construction of this shape function also has good smoothness and continuity, and it is continuously differentiable in the entire reference domain, which is crucial for solving complex nonlinear residual equations and performing high-precision energy functional variation calculations. Since the high-order finite element shape function can approximate the real displacement field in a higher dimension, it can more accurately describe the local deformation and strain gradient when constructing the discrete residual equation, so that the overall degree of freedom vector obtained by the subsequent solution has higher numerical reliability and physical rationality. This is also the reason why the present invention uses high-order finite element shape functions as the core tool in the spherical lattice shell structure node force analysis method based on refined simulation optimization. From the perspective of practical engineering applications, spherical lattice shell structures are often used to bear complex and multi-directional loads, and the stress state at their local nodes may have significant nonlinearity and local high gradient phenomena. Traditional finite element methods are difficult to capture this complexity due to the low order of shape functions, while high-order finite element shape functions can achieve more sophisticated interpolation approximation within the unit by increasing the number of polynomial terms, thereby effectively improving the accuracy and stability of the model. The high-order shape function can not only realize the fine discretization of the displacement field, but also maintain consistency and high precision in subsequent steps such as constructing energy functionals and discrete residual equations, thereby ensuring that the stress state and local energy distribution of each node can be fully reflected in the overall simulation process, providing a solid numerical foundation for structural health monitoring, anomaly judgment and optimization design.

[0042] Example 10: 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.

[0043] Specifically, in the formula represents the total degrees of freedom vector The gradient operation is the rate of change of the displacement of each node in the local area with the spatial coordinates. Usually, in the traditional small strain theory, the linear part of the strain tensor is given by It is defined by , which reflects the contribution of displacement gradient to strain when the structure is subjected to small deformation. However, in spherical lattice shell structures, due to their curved geometry and possible local large deformation, simple linear approximation is often insufficient to describe the actual stress state. Therefore, this formula not only includes This symmetric part also introduces nonlinear terms To compensate for the effects caused by large deformation or geometric nonlinearity. This nonlinear 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, thereby improving the model's ability to capture nonlinear behavior. In the entire expression in brackets, the first two terms and They represent the gradient changes of the displacement field in different directions. The sum of these two terms constitutes a symmetric tensor, whose physical meaning is to describe the average change of the strain in all directions in the material. By taking the symmetric part, the false strain caused by local rotation can be eliminated, so that only the real deformation information is retained. Then, the additional nonlinear term represents the product of the displacement field gradient. This term plays a corrective role in large deformation situations. Its physical significance is to capture the additional strain energy caused by geometric nonlinearity. Especially in spherical lattice shell structures with large curvature and complex deformation, 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 to calculate the total degrees of freedom vector The strain calculation is normalized. It is composed of the displacement field average of the adjacent observation points of the observation point. Different observation points may have different numbers of adjacent sampling points. In order to ensure that the strain tensors calculated in different regions are comparable, the The number of local neighboring points is averaged so that the strain tensor does not have numerical deviations due to different sampling densities. Through this normalization process, the model can maintain consistency on a global scale while accurately reflecting the true stress-strain state of the structure in local areas.

[0044] As described above, the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the same. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that the technical solutions described in the aforementioned embodiments may still be modified, or some of the technical features thereof may be replaced by equivalents. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.

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 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 reference radius of the spherical lattice shell; 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 using the following formula: ； 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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