A new arbitrary polygonal element calculation method for structural simulation analysis

Abstract

The present application relates to the technical field of data processing, and more particularly to a novel arbitrary polygonal element calculation method for structural simulation analysis, comprising obtaining a geometric model and extracting node coordinate data, calculating polygonal element center point and edge midpoint coordinates to form an extended coordinate data set, generating quadrilateral sub-element topology through geometric connection operation, establishing strain displacement relationship and fusing equivalent strain matrix data through constraint equation calculation, adopting single point integration algorithm and hourglass control algorithm for numerical integration and stability correction, outputting stiffness matrix data controlled by hourglass, integrating into a finite element solving system for numerical solving, and calculating stress and strain distribution based on node displacement data. The present application overcomes the low efficiency and poor accuracy problem caused by the dependence of the prior art on sub-element decomposition, and improves calculation stability and engineering applicability.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of data processing, and in particular to a novel arbitrary polygonal element calculation method for structural simulation analysis. BACKGROUND

[0002] The polygonal element calculation technology is a widely used discretization method in computational mechanics and computer graphics, which realizes numerical approximation by dividing the continuous geometric domain into polygonal elements with variable number of sides. The advantage of this technology is that it can flexibly handle irregular and complex geometric shapes, because the polygonal elements can be adaptively adjusted according to local features, thereby better capturing boundary details. In finite element analysis, polygonal elements help simplify the mesh generation process, reduce the number of elements while maintaining solution accuracy, logically based on their ability in high-order approximation and adaptive mesh refinement, improving computational efficiency and suitable for multi-scale simulation.

[0003] The existing polygonal element calculation technology has the following technical pain points, specifically: the existing technology relies on a subdivision strategy that decomposes complex polygons into triangular or quadrilateral sub-elements, and uses a sub-element Gaussian integration scheme for stiffness matrix calculation. This decomposition process destroys the overall structural properties of the polygon, not only significantly increasing the computational time and memory consumption due to the multiplication of the number of integration points, but also leading to uncontrollable solution accuracy due to the numerical difference of different subdivision methods, such as Delaunay triangulation or barycentric subdivision, resulting in more than ten percent deviation for the same geometric model with different mesh subdivision. Especially in the case of crack propagation, porous media or granular materials that require high-precision boundary fitting, the displacement coordination at the intersection of sub-elements is difficult to guarantee, which easily leads to stress oscillation and convergence problems, severely restricting the reliability and efficiency of engineering simulation. SUMMARY

[0004] In view of the deficiencies of the prior art, the present application provides a novel arbitrary polygonal element calculation method for structural simulation analysis, which solves the technical problems of low computational efficiency, uncontrollable accuracy and poor numerical stability caused by the existing polygonal element calculation method relying on sub-element decomposition strategy and complex integration scheme.

[0005] To solve the above technical problems, the specific content of the present application is as follows:

[0006] The present application provides a novel arbitrary polygonal element calculation method for structural simulation analysis, comprising:

[0007] Step 1, obtaining the geometric model of the structure to be analyzed, extracting the node coordinate data constituting the polygonal element from the geometric model, and obtaining the node data set including coordinate values;

[0008] Step 2: Receive the node data set, calculate the arithmetic mean of the node coordinates of each polygonal unit in the node data set to generate the center point coordinate data of the polygonal unit, calculate the arithmetic mean of the coordinates of every two adjacent nodes in the polygonal unit to generate the midpoint coordinate data of each side of the polygonal unit, and form an extended coordinate data set including the center point and the midpoint of the side of the polygonal unit.

[0009] Step 3: Read the extended coordinate data set. For each polygonal cell, according to the node connection order of the polygonal cell, perform geometric connection calculations on the center point coordinates, the midpoint coordinates of the adjacent sides, and the corresponding node coordinates to generate geometric connection data describing the topology of the quadrilateral sub-cell.

[0010] Step 4: Obtain geometric connection data. Based on the topology of the quadrilateral sub-unit, establish the strain and displacement relationship data of each sub-unit. Perform fusion calculation on the strain and displacement relationship data through constraint equations to generate equivalent strain matrix data describing the overall strain characteristics of the polygonal unit.

[0011] Step 5: Receive the equivalent strain matrix data, perform numerical integration on the equivalent strain matrix data using a single-point integration algorithm, apply the hourglass control algorithm to correct the stability of the integration result, and output the stiffness matrix data of the polygonal element after hourglass control.

[0012] Step 6: Input the stiffness matrix data of the polygonal elements into the preset finite element solution system, integrate the stiffness matrix data of each polygonal element into the overall stiffness matrix according to the node number, and numerically solve the overall stiffness equation based on the overall stiffness matrix through the preset finite element solution system. Calculate the stress and strain distribution of the structure based on the node displacement data obtained from the solution, and output the structural simulation analysis results.

[0013] Furthermore, in the novel arbitrary polygonal element calculation method for structural simulation analysis of the present invention, step 2 includes:

[0014] Based on the node data set, for a polygonal cell with n nodes, the arithmetic mean of all node coordinates is used to determine the center point coordinates.

[0015] Based on the node data set, the arithmetic mean of the coordinates of two adjacent nodes is used to determine the coordinates of the midpoint of the corresponding edge;

[0016] Based on the coordinates of the center point and the midpoint of the edge, complete the coordinate calculation of all nodes and edges;

[0017] Based on the coordinate calculation results of all nodes, a complete coordinate dataset including the center point, edge midpoints, and original nodes is established.

[0018] Furthermore, in the novel arbitrary polygonal element calculation method for structural simulation analysis of the present invention, step 3 includes:

[0019] Based on the extended coordinate data set, for each edge of the polygonal cell, the original node corresponding to the edge is selected;

[0020] Based on the extended coordinate data set, select the coordinates of the midpoint of the edge corresponding to the edge;

[0021] Based on the extended coordinate data set, select the coordinates of the midpoints of the two sides adjacent to the edge;

[0022] Based on the selected original node coordinates, edge midpoint coordinates, and center point coordinates, connect the center point, edge midpoints, and original nodes to form a quadrilateral sub-unit.

[0023] Furthermore, in the novel arbitrary polygonal element calculation method for structural simulation analysis of the present invention, step 4 includes:

[0024] Based on geometric connection data, the strain-displacement relationship of each quadrilateral sub-unit is established;

[0025] Based on the strain-displacement relationship, the displacement of the midpoint of the edge is expressed as a constraint form of the displacement of the adjacent nodes;

[0026] Based on the constrained strain-displacement relationship, the strain relationships of all quadrilateral sub-elements are recombined according to nodal displacements.

[0027] Based on the recombined strain relationship, an equivalent strain matrix is ​​generated with the center point and the original nodal displacements as variables.

[0028] Furthermore, in the novel arbitrary polygonal element calculation method for structural simulation analysis of the present invention, step 5 includes:

[0029] Based on the equivalent strain matrix data, the linear stiffness term is calculated using a single-point integration scheme.

[0030] The amplitude of the hourglass force is calculated based on the element geometric parameters in the geometric connection data.

[0031] Based on the amplitude of the hourglass force, a mapping relationship between the hourglass force and the nodal displacement is established;

[0032] The stiffness matrix is ​​obtained by superimposing the hourglass stiffness term and the linear stiffness term based on the linear stiffness term and the hourglass stiffness term.

[0033] Furthermore, the novel arbitrary polygonal element calculation method for structural simulation analysis of the present invention includes an hourglass control algorithm comprising:

[0034] Define the hourglass force calculation parameters based on the element geometric dimensions in the geometric connection data;

[0035] Based on the calculation parameters of hourglass force, a mathematical relationship between hourglass force and element deformation mode is established;

[0036] Based on mathematical relationships, the growth of hourglass force is controlled through hyperbolic functions.

[0037] Based on the controlled hourglass force, the hourglass force is discretely distributed to each node of the unit.

[0038] Furthermore, in the novel arbitrary polygonal element calculation method for structural simulation analysis of the present invention, step 6 includes:

[0039] Based on the hourglass-controlled stiffness matrix data of polygonal elements, assemble the stiffness matrix of all polygonal elements.

[0040] Based on the preset boundary condition data, the boundary conditions are applied to the overall stiffness equation after assembly and then solved.

[0041] Based on the nodal displacement data obtained from the solution, the stress and strain distribution of the structure is calculated and output.

[0042] The results of structural stress-strain distribution are visualized.

[0043] Furthermore, in the novel arbitrary polygonal element calculation method for structural simulation analysis of the present invention, step 6 further includes:

[0044] Based on the hourglass-controlled polygonal element stiffness matrix data, equilibrium iterative calculations are performed in an implicit solution scheme.

[0045] Internal forces are calculated in an explicit solution scheme based on the stiffness matrix data of polygonal elements controlled by an hourglass.

[0046] Based on the hourglass control algorithm, stability control parameters are generated in both implicit and explicit solution processes;

[0047] Based on the single-point integration algorithm and the hourglass control algorithm, the volumetric self-locking and shear self-locking effects in the unit strain data are corrected.

[0048] Furthermore, in the novel arbitrary polygonal element calculation method for structural simulation analysis of the present invention, step 6 further includes:

[0049] Based on the grid density distribution data, polygonal cell grid data is generated in the transition region of different grid densities;

[0050] Processing element boundary displacement field data based on polygonal element mesh data and strain-displacement relationship;

[0051] Based on boundary displacement field data and constraint equations, nodal displacement data is fused and processed.

[0052] Based on the fused nodal displacement data and equivalent strain matrix data, stress and strain calculations are performed in the mesh transition region.

[0053] Furthermore, in the novel arbitrary polygonal element calculation method for structural simulation analysis of the present invention, step 6 further includes:

[0054] Based on the crack and hole feature data in the geometric model, polygonal unit mesh data adapted to complex geometry is generated.

[0055] Based on polygonal cell mesh data and node data sets, granular materials and porous media are discretized to generate discretized model data.

[0056] Based on the discretized model data and the equivalent strain matrix data, the stress concentration factor distribution data is calculated;

[0057] Based on stress concentration factor distribution data and single-point integration algorithm, simulation calculations of complex structures are performed.

[0058] Beneficial effects of this invention;

[0059] The beneficial effects of this invention are as follows: This invention constructs multiple quadrilateral units by using the center point, side midpoint, and nodes of arbitrary node polygonal units, and constructs a new type of shape function based on quadrilateral units. It also proposes a new hourglass control method, which greatly improves the computational efficiency and significantly overcomes the volume self-locking and hourglass phenomenon, thus greatly improving the computational stability.

[0060] This invention constructs a novel hourglass control method based on the constraint equation of the center point of polygonal elements. It makes full use of the essential problem of the hourglass phenomenon and is a very effective method for handling hourglass control. It is applicable to implicit and explicit calculations in structural simulation. Attached Figure Description

[0061] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on the drawings without creative effort.

[0062] Figure 1 A flowchart illustrating a novel arbitrary polygonal element calculation method for structural simulation analysis provided in this embodiment of the invention.

[0063] Figure 2 This is a schematic diagram of a polygonal unit provided in an embodiment of the present invention.

[0064] Figure 3 A schematic diagram of a hexagonal unit provided in an embodiment of the present invention.

[0065] Figure 4The deformation cloud diagram of the cantilever beam provided in the embodiment of the present invention, wherein (a) is a polygonal element. (b) is a polygonal unit. (c) represents a polygonal unit. (d) is a quadrilateral unit. (e) is a quadrilateral element (with one non-coordinated region); (f) is a quadrilateral element (with two non-coordinated regions); (g) is an analytical solution. Detailed Implementation

[0066] To make the technical solution of the present invention clearer, the present invention will be clearly and completely described below with reference to specific embodiments and corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. The present invention provided by various embodiments will be described in detail below with reference to the accompanying drawings. To better understand the purpose of the present invention, the present invention will be described in further detail below.

[0067] Firstly, please refer to Figure 1 This invention provides a novel method for calculating arbitrary polygonal elements for structural simulation analysis, comprising:

[0068] Step 1: Obtain the geometric model of the structure to be analyzed, extract the node coordinate data of the polygonal units from the geometric model, and obtain a set of node data including coordinate values.

[0069] Step 2: Receive the node data set, calculate the arithmetic mean of the node coordinates of each polygonal unit in the node data set to generate the center point coordinate data of the polygonal unit, calculate the arithmetic mean of the coordinates of every two adjacent nodes in the polygonal unit to generate the midpoint coordinate data of each side of the polygonal unit, and form an extended coordinate data set including the center point and the midpoint of the side of the polygonal unit.

[0070] Step 3: Read the extended coordinate data set. For each polygonal cell, according to the node connection order of the polygonal cell, perform geometric connection calculations on the center point coordinates, the midpoint coordinates of the adjacent sides, and the corresponding node coordinates to generate geometric connection data describing the topology of the quadrilateral sub-cell.

[0071] Step 4: Obtain geometric connection data. Based on the topology of the quadrilateral sub-unit, establish the strain and displacement relationship data of each sub-unit. Perform fusion calculation on the strain and displacement relationship data through constraint equations to generate equivalent strain matrix data describing the overall strain characteristics of the polygonal unit.

[0072] Step 5: Receive the equivalent strain matrix data, perform numerical integration on the equivalent strain matrix data using a single-point integration algorithm, apply the hourglass control algorithm to correct the stability of the integration result, and output the stiffness matrix data of the polygonal element after hourglass control.

[0073] Step 6: Input the stiffness matrix data of the polygonal elements into the preset finite element solution system, integrate the stiffness matrix data of each polygonal element into the overall stiffness matrix according to the node number, and numerically solve the overall stiffness equation based on the overall stiffness matrix through the preset finite element solution system. Calculate the stress and strain distribution of the structure based on the node displacement data obtained from the solution, and output the structural simulation analysis results.

[0074] This invention provides a novel method for calculating arbitrary polygonal elements for structural simulation analysis, comprising the following steps: Step 1 involves acquiring the geometric model of the structure to be analyzed and extracting the node coordinate data of the polygonal elements from the geometric model, thereby obtaining a node data set including coordinate values. The geometric model is typically derived from a computer-aided design system or a 3D scanning device. By parsing the boundary representation or mesh file of the model, vertex information of the polygonal elements is extracted. The node coordinate data set includes the coordinate values ​​of each node in a two-dimensional or three-dimensional coordinate system. This step provides the basic geometric data for subsequent calculations.

[0075] Step 2 involves receiving a set of node data and calculating the arithmetic mean of the node coordinates of each polygonal cell in the set to generate the center point coordinates of the polygonal cell. Simultaneously, the arithmetic mean of the coordinates of every two adjacent nodes in the polygonal cell is calculated to generate the midpoint coordinates of each side of the polygonal cell, thus forming an extended coordinate data set including the center point and edge midpoints of the polygonal cell. The center point coordinates are obtained by calculating the arithmetic mean of all node coordinates and represent the geometric center of the polygon. The edge midpoint coordinates are obtained by calculating the arithmetic mean of the coordinates of adjacent nodes and represent the midpoint of each edge. The extended coordinate data set integrates the original node, center point, and edge midpoint coordinates, enhancing the geometric description capability of the cell.

[0076] Step 3 involves reading the extended coordinate data set and, for each polygonal cell, performing geometric connection operations on the center point coordinates, adjacent side midpoint coordinates, and corresponding node coordinates according to the node connection order of the polygonal cell. This generates geometric connection data describing the topology of the quadrilateral sub-cells. The geometric connection operation, based on the connection order of the polygonal nodes, connects the center point to each side midpoint and related nodes, forming multiple quadrilateral sub-cells. Each quadrilateral sub-cell consists of a center point, two adjacent side midpoints, and one node. The geometric connection data defines the topological relationships of these sub-cells, including the vertex connection order and edge information.

[0077] Step 4 involves acquiring geometric connection data and establishing strain-displacement relationship data for each sub-element based on the quadrilateral sub-element topology. The strain-displacement relationship data is then fused and calculated using constraint equations to generate an equivalent strain matrix describing the overall strain characteristics of the polygonal element. The strain-displacement relationship data is derived based on the shape functions and displacement fields of the quadrilateral sub-element, and the strain matrix is ​​established using the finite element method. The constraint equations represent the midpoint displacement as a linear combination of adjacent node displacements, achieving displacement continuity. The fusion calculation, by superimposing the strain contributions of all sub-elements and applying constraint conditions, derives the equivalent strain matrix with the center point and original node displacements as variables.

[0078] Step 5 involves receiving the equivalent strain matrix data and performing numerical integration on it using a single-point integration algorithm. An hourglass control algorithm is then applied to correct the stability of the integration result, outputting the stiffness matrix data of the polygonal element after hourglass control. The single-point integration algorithm performs numerical integration at the element center or a specific point to calculate the linear portion of the stiffness matrix. The hourglass control algorithm detects and suppresses hourglass patterns, enhancing stability by adding hourglass stiffness terms. The hourglass force is calculated based on the element's geometric parameters and deformation patterns, and its growth is controlled using a mathematical function.

[0079] Step 6 involves inputting the stiffness matrix data of the polygonal elements into a pre-defined finite element solution system, integrating the stiffness matrix data of each polygonal element into the overall stiffness matrix according to node number, and numerically solving the overall stiffness equation based on the overall stiffness matrix using the pre-defined finite element solution system. Based on the obtained nodal displacement data, the stress and strain distribution of the structure is calculated, and the structural simulation analysis results are output. The finite element solution system assembles the stiffness matrices of all elements to form the overall stiffness matrix, and solves the system equations after applying boundary conditions. Nodal displacement data is used to calculate strain and stress distribution using constitutive relations such as Hooke's law. The output results include stress contour plots, strain distributions, and deformation animations for engineering analysis and design verification.

[0080] Specifically, the novel arbitrary polygon element calculation method for structural simulation analysis of the present invention includes step 2 as follows:

[0081] Based on the node data set, for a polygonal cell with n nodes, the arithmetic mean of all node coordinates is used to determine the center point coordinates.

[0082] Based on the node data set, the arithmetic mean of the coordinates of two adjacent nodes is used to determine the coordinates of the midpoint of the corresponding edge;

[0083] Based on the coordinates of the center point and the midpoint of the edge, complete the coordinate calculation of all nodes and edges;

[0084] Based on the coordinate calculation results of all nodes, a complete coordinate dataset including the center point, edge midpoints, and original nodes is established.

[0085] In step 2, geometric data expansion processing of the polygonal unit is performed based on the node data set. The node data set includes the coordinate values ​​of each node of the polygonal unit. First, for a polygonal unit with n nodes, the arithmetic mean of all node coordinates is calculated as the center point coordinates, which represent the geometric center position of the polygonal unit. Subsequently, based on the node data set, for each edge of the polygonal unit, the arithmetic mean of the coordinates of two adjacent nodes is calculated as the midpoint coordinates of the corresponding edge. The midpoint coordinates are used to describe the center position of each edge. Through the calculation of the center point coordinates and the midpoint coordinates, the coordinate calculation of all nodes and edges is completed, achieving the integrity of the geometric data. Finally, based on the coordinate calculation results of all nodes, a complete coordinate dataset including the center point, edge midpoints, and original nodes is established. This dataset provides a comprehensive geometric information foundation for subsequent steps. Each step is executed sequentially. The calculation of the center point coordinates and edge midpoint coordinates depends on the node data set, while the establishment of the complete coordinate dataset integrates all calculated coordinate values, forming a coherent data processing flow.

[0086] Specifically, the novel arbitrary polygon element calculation method for structural simulation analysis of the present invention includes step 3 as follows:

[0087] Based on the extended coordinate data set, for each edge of the polygonal cell, the original node corresponding to the edge is selected;

[0088] Based on the extended coordinate data set, select the coordinates of the midpoint of the edge corresponding to the edge;

[0089] Based on the extended coordinate data set, select the coordinates of the midpoints of the two sides adjacent to the edge;

[0090] Based on the selected original node coordinates, edge midpoint coordinates, and center point coordinates, connect the center point, edge midpoints, and original nodes to form a quadrilateral sub-unit.

[0091] Step 3's core lies in constructing geometric connection data describing the topology of the quadrilateral sub-unit based on the extended coordinate data set. First, for each edge of the polygonal unit, the original node coordinates corresponding to that edge are selected from the extended coordinate data set. These original nodes are the basic geometric feature points on the polygon boundary. Next, the midpoint coordinates of that edge are selected from the extended coordinate data set. These midpoint coordinates are obtained by calculating the arithmetic mean of the coordinates of adjacent original nodes, representing the geometric center position of the edge. Then, the midpoint coordinates of the two adjacent edges are further selected from the extended coordinate data set. These midpoint coordinates of adjacent edges are used to construct the remaining vertices of the quadrilateral sub-unit. Finally, based on the selected original node coordinates, midpoint coordinates, and center point coordinates, geometric connection operations are used to sequentially connect the center point coordinates, the midpoint coordinates corresponding to the current edge, the midpoint coordinates corresponding to adjacent edges, and the original node coordinates corresponding to the current edge, forming a quadrilateral sub-unit with a defined vertex sequence, thus generating complete geometric connection data describing the sub-unit's topological relationships.

[0092] Specifically, step 4 of the novel arbitrary polygon element calculation method for structural simulation analysis of the present invention includes:

[0093] Based on geometric connection data, the strain-displacement relationship of each quadrilateral sub-unit is established;

[0094] Based on the strain-displacement relationship, the displacement of the midpoint of the edge is expressed as a constraint form of the displacement of the adjacent nodes;

[0095] Based on the constrained strain-displacement relationship, the strain relationships of all quadrilateral sub-elements are recombined according to nodal displacements.

[0096] Based on the recombined strain relationship, an equivalent strain matrix is ​​generated with the center point and the original nodal displacements as variables.

[0097] Based on the geometric connectivity data, a strain-displacement relationship is established for each quadrilateral sub-element using the finite element method. The strain matrix is ​​calculated by the partial derivative of the shape function with respect to displacement, describing the linear mapping relationship between strain and nodal displacement within the sub-element. This process involves defining the displacement field function of the sub-element and deriving the shape function based on the nodal coordinates in the geometric connectivity data, thereby deriving the elements of the strain matrix.

[0098] Based on the strain-displacement relationship, constraint equations are introduced to express the midpoint displacement of an edge as the arithmetic mean of the displacements of adjacent original nodal points. Specifically, linear constraints are used to couple the midpoint displacement with the original nodal displacements, thereby reducing the number of independent degrees of freedom and maintaining the displacement compatibility of the element boundary. The application of constraint equations ensures that the midpoint displacement is no longer an independent variable but depends on the original nodal displacements, simplifying subsequent calculations.

[0099] Based on the constrained strain-displacement relationship, the strain matrices of all quadrilateral sub-elements are recombined according to the original nodal displacements and center point displacements. By summarizing the contribution terms to the same nodal displacements in the strain matrices of each sub-element and integrating the matrix elements according to the node numbers, a global strain expression is formed. The recombination process involves weighted summation of matrix elements to reflect the superposition effect of each sub-element on the global strain.

[0100] Based on the recombined strain relationship, the strain components of all sub-elements are integrated to generate an equivalent strain matrix with the center point displacement and the original nodal displacements as variables. This matrix, by rearranging the recombined strain expression into matrix form, where the column vectors include the displacements of the center point and all original nodes, and the row vectors correspond to the strain components, fully describes the overall strain characteristics of the polygonal element. The equivalent strain matrix provides the foundation for subsequent stiffness matrix calculations.

[0101] Specifically, step 5 of the novel arbitrary polygon element calculation method for structural simulation analysis of the present invention includes:

[0102] Based on the equivalent strain matrix data, the linear stiffness term is calculated using a single-point integration scheme.

[0103] The amplitude of the hourglass force is calculated based on the element geometric parameters in the geometric connection data.

[0104] Based on the amplitude of the hourglass force, a mapping relationship between the hourglass force and the nodal displacement is established;

[0105] The stiffness matrix is ​​obtained by superimposing the hourglass stiffness term and the linear stiffness term based on the linear stiffness term and the hourglass stiffness term.

[0106] Based on the equivalent strain matrix data, a single-point integration scheme is used to calculate the linear stiffness term. This scheme involves applying a numerical integration algorithm at the center point of the polygonal element to integrate the equivalent strain matrix and the material constitutive matrix, thus obtaining the linear stiffness term. The linear stiffness term characterizes the stiffness properties of the element under elastic deformation. The integration process involves calculating the strain energy density function, thereby deriving the stiffness matrix elements of the linear portion.

[0107] The hourglass force amplitude is calculated based on the element geometric parameters in the geometric connectivity data. These geometric parameters include dimensional information such as the circumradius and incircle radii of the polygonal elements, extracted from the geometric connectivity data. The hourglass force amplitude is calculated using the geometric parameters and element deformation modes, where the deformation modes are evaluated based on nodal displacement deviations, and the amplitude calculation reflects the magnitude of the hourglass deformation.

[0108] Based on the hourglass force amplitude, a mapping relationship between the hourglass force and nodal displacements is established. This mapping relationship is achieved by defining a linear or nonlinear association between the hourglass force vector and the nodal displacement vector, where the hourglass force amplitude serves as the input parameter to generate the distribution of the hourglass force at the element nodes. Specifically, the mapping relationship is represented by a transformation matrix that maps nodal displacements to equivalent hourglass forces, thus providing a foundation for the derivation of the hourglass stiffness term.

[0109] Based on linear and hourglass stiffness terms, the stiffness matrix is ​​obtained by superimposing the hourglass stiffness terms with the linear stiffness terms. The hourglass stiffness terms are derived from the mapping relationship between hourglass force and nodal displacement, and the stiffness contribution is obtained by taking partial derivatives or similar operations. The superposition process involves matrix addition, merging the linear and hourglass stiffness terms to form the final polygonal element stiffness matrix, which possesses enhanced numerical stability.

[0110] Specifically, the novel arbitrary polygon element calculation method for structural simulation analysis of the present invention includes an hourglass control algorithm comprising:

[0111] Define the hourglass force calculation parameters based on the element geometric dimensions in the geometric connection data;

[0112] Based on the calculation parameters of hourglass force, a mathematical relationship between hourglass force and element deformation mode is established;

[0113] Based on mathematical relationships, the growth of hourglass force is controlled through hyperbolic functions.

[0114] Based on the controlled hourglass force, the hourglass force is discretely distributed to each node of the unit.

[0115] The hourglass force calculation parameters are defined based on the element geometric dimensions in the geometric connectivity data. Element geometric dimensions include characteristic lengths such as the circumcircle and incircle radii of polygonal elements; these dimensions are obtained directly from the geometric connectivity data or derived through calculation. The hourglass force calculation parameters are defined based on these geometric dimensions and are used to quantify the calculation benchmark and proportional relationship of the hourglass force.

[0116] Based on the hourglass force calculation parameters, a mathematical relationship between the hourglass force and the element deformation mode is established. The element deformation mode is described by the deviation between the nodal displacement vector and the reference geometric configuration. The mathematical relationship associates the hourglass force calculation parameters with the deformation mode, forming a calculation expression for the hourglass force. This mathematical relationship characterizes the variation of the hourglass force with the element deformation mode.

[0117] Based on mathematical relationships, the growth of the hourglass force is controlled using a hyperbolic function. This hyperbolic function, applied to the mathematical relationships, provides nonlinear control over the magnitude of the hourglass force's growth, preventing excessive increase during deformation. This control method allows the hourglass force to grow slowly in the early stages of deformation and moderately increase when deformation becomes significant, thus balancing computational stability and accuracy.

[0118] Based on the controlled hourglass force, the hourglass force is discretized and distributed to each node of the element. The distribution process decomposes the overall hourglass force into equivalent nodal forces on each node according to the geometric position and weight coefficient of the node in the element. The discretized nodal forces will participate in the construction of the element equilibrium equations and are finally integrated into the system's stiffness matrix.

[0119] Specifically, in the novel arbitrary polygonal element calculation method for structural simulation analysis of the present invention, step 6 includes:

[0120] Based on the hourglass-controlled stiffness matrix data of polygonal elements, assemble the stiffness matrix of all polygonal elements.

[0121] Based on the preset boundary condition data, the boundary conditions are applied to the overall stiffness equation after assembly and then solved.

[0122] Based on the nodal displacement data obtained from the solution, the stress and strain distribution of the structure is calculated and output.

[0123] The results of structural stress-strain distribution are visualized.

[0124] Based on the hourglass-controlled stiffness matrix data of polygonal elements, the stiffness matrices of all polygonal elements are assembled. The assembly process involves integrating the stiffness matrix elements of each polygonal element into the corresponding positions of the overall stiffness matrix according to the global node number, and summing the matrix elements corresponding to shared nodes to finally form the overall stiffness matrix describing the mechanical properties of the entire structure.

[0125] Based on preset boundary condition data, boundary conditions are applied to the assembled global stiffness equation and then solved. The boundary condition data includes displacement constraints and external load information, and the application process is achieved by modifying the equation terms corresponding to the constraint degrees of freedom in the global stiffness equation. The solution is obtained by numerically calculating the modified linear equations to obtain the nodal displacement solutions of the system.

[0126] Based on the nodal displacement data obtained from the solution, the stress-strain distribution of the structure is calculated and output. The calculation process uses the element strain matrix to convert nodal displacements into element strains, and then uses the material constitutive relation to convert element strains into element stresses. The stress-strain distribution results are output in the form of a data file, including the stress and strain values ​​for each element and node.

[0127] Based on the structural stress-strain distribution results, a visualization is performed. The visualization process uses graphics rendering technology to map the stress-strain data into color cloud maps or contour maps, which are then overlaid on the structural geometric model. The visualization results can show the distribution patterns and numerical magnitudes of stress and strain, providing an intuitive basis for structural performance evaluation.

[0128] Specifically, in the novel arbitrary polygonal element calculation method for structural simulation analysis of the present invention, step 6 further includes:

[0129] Based on the hourglass-controlled polygonal element stiffness matrix data, equilibrium iterative calculations are performed in an implicit solution scheme.

[0130] Internal forces are calculated in an explicit solution scheme based on the stiffness matrix data of polygonal elements controlled by an hourglass.

[0131] Based on the hourglass control algorithm, stability control parameters are generated in both implicit and explicit solution processes;

[0132] Based on the single-point integration algorithm and the hourglass control algorithm, the volumetric self-locking and shear self-locking effects in the unit strain data are corrected.

[0133] Based on the hourglass-controlled stiffness matrix data of polygonal elements, the stiffness matrices of all polygonal elements are assembled. The assembly process involves integrating the stiffness matrix elements of each polygonal element into the corresponding positions of the overall stiffness matrix according to the global node number, and summing the matrix elements corresponding to shared nodes to finally form the overall stiffness matrix describing the mechanical properties of the entire structure.

[0134] Based on preset boundary condition data, boundary conditions are applied to the assembled global stiffness equation and then solved. The boundary condition data includes displacement constraints and external load information, and the application process is achieved by modifying the equation terms corresponding to the constraint degrees of freedom in the global stiffness equation. The solution is obtained by numerically calculating the modified linear equations to obtain the nodal displacement solutions of the system.

[0135] Based on the nodal displacement data obtained from the solution, the stress-strain distribution of the structure is calculated and output. The calculation process uses the element strain matrix to convert nodal displacements into element strains, and then uses the material constitutive relation to convert element strains into element stresses. The stress-strain distribution results are output in the form of a data file, including the stress and strain values ​​for each element and node.

[0136] Based on the structural stress-strain distribution results, a visualization is performed. The visualization process uses graphics rendering technology to map the stress-strain data into color cloud maps or contour maps, which are then overlaid on the structural geometric model. The visualization results can show the distribution patterns and numerical magnitudes of stress and strain, providing an intuitive basis for structural performance evaluation.

[0137] Specifically, in the novel arbitrary polygonal element calculation method for structural simulation analysis of the present invention, step 6 further includes:

[0138] Based on the grid density distribution data, polygonal cell grid data is generated in the transition region of different grid densities;

[0139] Processing element boundary displacement field data based on polygonal element mesh data and strain-displacement relationship;

[0140] Based on boundary displacement field data and constraint equations, nodal displacement data is fused and processed.

[0141] Based on the fused nodal displacement data and equivalent strain matrix data, stress and strain calculations are performed in the mesh transition region.

[0142] Based on hourglass-controlled polygonal element stiffness matrix data, equilibrium iterative calculations are performed in an implicit solution scheme. The equilibrium iterative calculation process assembles the polygonal element stiffness matrix data into the global stiffness matrix, applies preset boundary conditions, and solves the nonlinear equilibrium equation using the Newton-Raphson method, progressively updating nodal displacements and stress fields until convergence to an equilibrium state. During the iteration process, the linearized form of the stiffness matrix, combined with residual calculations and tangent stiffness updates, achieves stable convergence of the numerical solution.

[0143] Internal forces are calculated using hourglass-controlled polygonal element stiffness matrix data in an explicit solution format. The explicit solution employs the central difference method for time integration, utilizing the stiffness matrix data to calculate element internal forces. These internal force calculations are based on the current strain state and material constitutive relations, deriving nodal forces from the strain matrix and stress tensor of the polygonal elements. The internal force data is used to update nodal accelerations and velocities, enabling explicit time-stepping in dynamic analysis, suitable for high-speed or transient problems.

[0144] Based on the hourglass control algorithm, stability control parameters are generated in both implicit and explicit solution processes. The algorithm defines stability control parameters, such as the hourglass force coefficient and stiffness correction factor, by calculating the hourglass force amplitude and hourglass stiffness term. These parameters are dynamically adjusted according to the geometry and deformation mode of the polygonal element, suppressing zero-energy modes, enhancing iterative convergence in implicit solutions, controlling numerical oscillations in explicit solutions, and improving overall solution stability.

[0145] Based on the single-point integration algorithm and the hourglass control algorithm, volumetric and shear locking effects in element strain data are corrected. The single-point integration algorithm calculates strain energy through a single integration point, reducing the risk of integration lock-in, while the hourglass control algorithm compensates for potential numerical errors by introducing an hourglass stiffness term. The correction process involves adaptive adjustment of the strain matrix, employing selective subtraction integration techniques to handle volumetric and shear strains, ensuring that the strain field accurately reflects actual deformation behavior and avoiding numerical inaccuracies caused by excessive constraints.

[0146] Specifically, in the novel arbitrary polygonal element calculation method for structural simulation analysis of the present invention, step 6 further includes:

[0147] Based on the crack and hole feature data in the geometric model, polygonal unit mesh data adapted to complex geometry is generated.

[0148] Based on polygonal cell mesh data and node data sets, granular materials and porous media are discretized to generate discretized model data.

[0149] Based on the discretized model data and the equivalent strain matrix data, the stress concentration factor distribution data is calculated;

[0150] Based on stress concentration factor distribution data and single-point integration algorithm, simulation calculations of complex structures are performed.

[0151] Based on crack and hole feature data from the geometric model, polygonal element mesh data adapted to complex geometries is generated. The geometric model is acquired through a computer-aided design system or 3D scanning equipment, including crack contour information and hole boundary data. During processing, the size and distribution of polygonal elements are adaptively adjusted according to the high stress gradient region at the crack tip and the geometric irregularities around the holes, enabling the mesh data to accurately capture these features. The generation of polygonal element mesh data avoids existing triangulation or quadrilateral subdivision decomposition strategies, directly utilizing the flexibility of arbitrary polygonal elements to achieve a high degree of fit between the mesh and complex geometric boundaries.

[0152] Based on polygonal element mesh data and node datasets, granular materials and porous media are discretized to generate discretized model data. The node dataset includes vertex coordinates of the polygonal elements. Combining this with the particle size distribution of the granular material or the pore structure of the porous medium, each particle or pore element is treated as an independent polygonal entity. Discretization calculates the center point and edge midpoints using arithmetic means, expanding the coordinate dataset to construct a topological relationship describing interparticle contact or porous connections. The discretized model data integrates geometric and material properties, providing a foundation for subsequent strain analysis.

[0153] Based on the discretized model data and equivalent strain matrix data, the stress concentration factor distribution data is calculated. The equivalent strain matrix data, obtained from step four, describes the overall strain characteristics of the polygonal elements. The stress concentration factor calculation converts the strain field into a stress field by applying the principles of elasticity and identifies high-stress regions, such as crack tips or hole edges. During the calculation, constraint equations are used to ensure displacement continuity, avoid stress oscillations, and output stress concentration factor distribution data, quantifying the local stress amplification effect.

[0154] Based on stress concentration factor distribution data and a single-point integration algorithm, complex structural simulation calculations are performed. The single-point integration algorithm performs numerical integration at the element center points, reducing computational load, while an hourglass control algorithm is combined to maintain stability. The simulation calculation uses stress concentration factors as input parameters and solves the global stiffness equation through a finite element solution system to obtain nodal displacement data. Finally, based on the nodal displacement data, the stress-strain distribution of the structure is calculated, outputting the simulation analysis results of the complex structure for engineering evaluation and design optimization. The entire process, from geometric processing to simulation output, is logically coherent and progressively advanced, achieving efficient and high-precision analysis of complex structures.

[0155] This invention divides polygonal elements into multiple quadrilaterals based on the element center point and edge midpoint for strain reconstruction, constructing a novel calculation format. The specific steps are as follows:

[0156] Step 1: Construct n quadrilateral units based on an n-node polygon;

[0157] Based on an n-node polygonal element, point n+1 is selected from inside the polygon, the midpoints of edges I-1 and I are selected, and node I is selected to construct a quadrilateral element, as follows: Figure 2 As shown;

[0158] Figure 2 This is a schematic diagram of constructing arbitrary polygonal units according to the present invention. The polygon has n nodes (X1, X2, ..., Xn). First, the center point Xc is calculated according to equation (1). Then, the midpoints of each side (Xn+1, Xn+2, ...) are calculated according to equation (2). For side I, the center point Xc, node I, and the midpoints of the sides (corresponding to sides I and I-1) together form a quadrilateral subunit eI (shown as shaded in the figure).

[0159] The coordinates of the center point c are selected as follows:

[0160] ; (1)

[0161] In the formula Let be the coordinates of the i-th node, and let the coordinates of the midpoint of the edge be represented as ;

[0162] ; (2)

[0163] In the formula, when I equals n, I+1 represents node 1. During the construction process, we found that constraining the midpoints of element edges increases self-locking but significantly improves the hourglass phenomenon. On the other hand, constraining the center nodes leads to severe volumetric and shear self-locking; increasing the center point degree of freedom significantly improves this phenomenon. Therefore, in subsequent element construction, we only added the element center nodes as additional degrees of freedom, while the midpoints of the edges were constrained to release the degrees of freedom.

[0164] Step 2: Construct the equivalent transformation matrix of the n-node polygonal element based on the n quadrilateral elements.

[0165] Taking the first quadrilateral unit (c, n+n, 1, n+1) as an example, the coordinates of any point inside the quadrilateral can be represented as:

[0166] (3)

[0167] Substituting the constraint equation of the midpoint of the edge into equation (3) yields the following:

[0168] (4)

[0169] The result can be obtained by rearranging;

[0170] (5)

[0171] Similarly, for any quadrilateral element eI, the strain can be obtained as follows:

[0172] (6)

[0173] The displacement of the edge nodes can be constrained by the node displacement;

[0174] (7)

[0175] Among them, when hour, When; when hour, The strain matrix of the quadrilateral element eI can be rearranged as follows:

[0176] (8)

[0177] The result can be obtained by arranging the matrix.

[0178] ; (9)

[0179] According to equation (9), here is given a when Consider the following hexagon;

[0180] Figure 3 This is a schematic diagram of the hexagonal unit constructed according to the present invention. The hexagon has 6 nodes (X1, X2, ..., X6). First, the center point Xc is calculated according to equation (1). Then, the midpoints of each side (X7, X8, ..., X12) are calculated according to equation (2). For side 1, the center point Xc, node 1, and the midpoints of the side (corresponding to sides 7 and 12) together form a quadrilateral subunit eI (shown as shaded in the figure).

[0181] Strain matrix of 6 quadrilateral elements;

[0182] (10a)

[0183] (10b)

[0184] (10c)

[0185] (10d)

[0186] (10e)

[0187] (10f)

[0188] Considering the general case of arbitrary polygons, the strain matrix and displacement vector of these n quadrilateral elements are rearranged to obtain:

[0189] (11)

[0190] in;

[0191] (12)

[0192] (13)

[0193] (14)

[0194] (15)

[0195] (16)

[0196] Step 3: Construct a new hourglass pattern;

[0197] Within each polygon, quadrilateral elements can be constructed using a one-point integration method. Hourglass stiffness is then constructed within each quadrilateral to maintain computational stability. However, this construction method is extremely time-consuming. This invention analyzes the root cause of hourglass formation in polygonal elements. The fundamental reason for hourglass formation is that strain and deformation modes are not in a one-to-one correspondence. In polygonal elements, we only need to constrain the central node c to limit the deformation mode. Based on this, this invention defines a new hourglass strain.

[0198] (17)

[0199] in , For ease of representation, the central node c is introduced into the node sequence. The present invention redefines hourglass force according to the following formula;

[0200] (18)

[0201] in It is the radius of the circumcircle of the polygon. It is the radius of the inscribed circle. It is the linear hourglass stiffness, and its magnitude is on the same order of magnitude as Young's modulus. and These are the control parameters for the nonlinear hourglass. ,and The hourglass force discretized to the element nodes can be written as:

[0202] (19)

[0203] The hourglass modulus matrix can be obtained by taking the partial derivative of the hourglass force.

[0204] (20)

[0205] Therefore, the element stiffness matrix can be expressed as:

[0206] ; (twenty one)

[0207] Finally, the internal force and stiffness matrices of this element can be obtained:

[0208] ;(twenty two)

[0209] ;(twenty three)

[0210] in, The strain matrix of the element, Represents the stress tensor. Let be the area of ​​the polygonal unit. Let be the stiffness matrix of the polygonal element.

[0211] Embodiment of the present invention: Simulation of cantilever beam bending;

[0212] The method of this invention is used to perform static bending analysis on a cantilever beam. The cantilever beam model is discretized using both the polygonal elements of this invention and progressively refined quadrilateral elements (where the regions with varying mesh density are non-coordinated mesh regions, which can be naturally handled by the polygonal elements constructed using this invention). The analytical solution to this problem is:

[0213] ;

[0214] ;

[0215] in, It is the moment of inertia, analyzed using a plane stress state model, with the following mechanical parameters: , , , and .

[0216] Final stress The calculation results are as follows Figure 4 As shown, the polygonal unit of the present invention can solve solid mechanics problems well and does not exhibit hourglass phenomenon.

[0217] The technical features of this invention are explained below:

[0218] Polygonal elements are basic discrete units in structural simulation analysis, formed by connecting multiple nodes, and can flexibly adapt to complex geometric shapes. In this invention, polygonal elements are constructed by extracting node coordinate data from the geometric model. The node coordinate data set contains the coordinate values ​​of each node, providing basic geometric information for subsequent calculations.

[0219] The center point is obtained by calculating the arithmetic mean of the coordinates of all nodes in a polygonal element, and represents the geometric center of the polygon. In the application of this invention, the center point coordinate data is used to generate an extended coordinate data set and participates as an additional degree of freedom in the construction of the strain matrix, which helps to improve the calculation accuracy and stability.

[0220] The midpoint of an edge is a point obtained by calculating the arithmetic mean of the coordinates of every two adjacent nodes of a polygonal element, representing the center position of each edge. The coordinate data of the midpoints are used to form an extended coordinate data set, which is combined with the center point and nodes in the geometric connection operation to form a quadrilateral sub-element.

[0221] Quadrilateral sub-elements are sub-elements formed by geometrically connecting the center point, side midpoints, and corresponding nodes, and are used to describe the topological structure of polygonal elements. In this invention, each quadrilateral sub-element is constructed based on the node connection sequence, and its strain-displacement relationship is used to derive the overall strain characteristics.

[0222] The strain matrix is ​​a matrix established through the strain-displacement relationship of quadrilateral sub-elements, describing the linear mapping relationship between strain and displacement. In this invention, the strain matrix is ​​calculated by fusing constraint equations to generate equivalent strain matrix data, which is used for subsequent stiffness matrix calculations.

[0223] The stiffness matrix is ​​obtained by numerically integrating the equivalent strain matrix using a single-point integration algorithm and an hourglass control algorithm, and it describes the stiffness characteristics of the element. In this invention, the stiffness matrix is ​​corrected using hourglass control and then used in the finite element solution system for structural analysis.

[0224] The hourglass control algorithm is based on element geometry and deformation patterns and is used to suppress the hourglass phenomenon in numerical calculations. In this invention, hourglass control uses a hyperbolic function to control the growth of the hourglass force and discretizes the hourglass force to each node of the element, thereby enhancing computational stability.

[0225] The finite element method (FEM) solution system is a pre-defined numerical solution platform used to integrate the stiffness matrices of polygonal elements and solve the global stiffness equation. In this invention, the system calculates the stress-strain distribution based on nodal displacement data and outputs structural simulation analysis results.

[0226] Visualization is the process of presenting stress and strain distribution results in graphical form to intuitively demonstrate the analysis results. In this invention, visualization uses graphics rendering technology to generate cloud maps or contour maps to assist in engineering evaluation and design verification.

[0227] The technical features of this invention are explained below:

[0228] The polygonal unit model is constructed by acquiring the geometric model of the structure to be analyzed and extracting the node coordinate data that makes up the polygonal unit. The node coordinate data set includes the coordinate values ​​of each node. In the technical solution of this invention, the polygonal unit model is used to form the basic geometric data. The coordinates of the center point and the midpoints of the edges are generated by calculating the arithmetic mean of the node coordinates, thereby expanding the coordinate data set. The data processing involves coordinate calculation and set establishment, and the output is a complete coordinate dataset including the center point, the midpoints of the edges, and the original nodes, providing a geometric foundation for the subsequent construction of sub-units.

[0229] The quadrilateral sub-unit model is constructed based on an extended coordinate data set. Multiple quadrilateral sub-units are formed by connecting the coordinates of the center point, the midpoints of the sides, and the corresponding node coordinates through geometric connection operations. In this invention, the quadrilateral sub-unit model is used to describe the topological structure of polygonal units. Each sub-unit consists of a center point, two midpoints of adjacent sides, and one node. The data processing includes selecting coordinate data and performing connection operations to generate geometric connection data. The output is geometric connection data describing the topological relationships of the sub-units, used for strain relationship derivation.

[0230] The strain-displacement relationship model is established based on the quadrilateral sub-element topology, and the strain matrix of each sub-element is derived through shape functions and displacement fields. In the technical solution of this invention, the strain-displacement relationship model is used to define the linear mapping between strain and nodal displacements, and the midpoint displacement of the edge is expressed as the arithmetic mean of the displacements of adjacent nodes using constraint equations. The data processing involves matrix calculations and constraint applications, outputting strain-displacement relationship data to ensure displacement compatibility and reduce the number of degrees of freedom.

[0231] The equivalent strain matrix model is constructed by integrating the strain relationships of all quadrilateral sub-elements, recombining and integrating the sub-element strain matrices according to nodal displacements. In the technical solution of this invention, the equivalent strain matrix model is used to describe the overall strain characteristics of the polygonal element, with the center point and original nodal displacements as variables. The data processing includes matrix superposition and variable integration to generate equivalent strain matrix data, outputting strain characteristics in matrix form, providing input for stiffness calculation.

[0232] The stiffness matrix model employs a single-point integration algorithm to numerically integrate the equivalent strain matrix data, and applies an hourglass control algorithm for stability correction. In this invention, the stiffness matrix model is used to calculate the element stiffness characteristics; single-point integration is performed at the element center point; and hourglass control suppresses zero-energy modes by adding an hourglass stiffness term. The data processing involves integration operations and hourglass force calculations, outputting hourglass-controlled polygonal element stiffness matrix data for finite element solution.

[0233] The hourglass control model, based on element geometry and deformation mode definitions, controls the growth of the hourglass force and discretizes its distribution to nodes using hyperbolic functions. In this invention, the hourglass control model enhances numerical stability by calculating the hourglass force amplitude and establishing a mapping relationship with nodal displacements. The data processing includes parameter definition and mathematical relationship establishment, outputting the hourglass force distribution and stiffness correction terms, which are then integrated into the stiffness matrix.

[0234] The finite element method (FEM) solution system model integrates the stiffness matrix data of all polygonal elements into the global stiffness matrix and solves the global stiffness equation. In this invention, the FEM solution system model is used to perform numerical solutions, apply boundary condition data, and calculate nodal displacements. The data processing includes matrix assembly and equation solving, calculating stress and strain distributions based on nodal displacement data, and outputting structural simulation analysis results, such as stress contour maps and strain distributions.

[0235] The cantilever beam model is used as an application example. Static bending analysis is performed through polygonal element discretization, and the stress distribution is calculated and compared with the analytical solution. In the technical solution of this invention, the cantilever beam model is used to verify the effectiveness of the method, and simulation is performed based on geometric parameters and mechanical inputs. The data processing involves element construction, strain calculation, and integration, outputting stress and strain distribution results, demonstrating the absence of hourglass phenomenon and high accuracy.

[0236] The algorithms for calculating the center point and edge midpoint employ an arithmetic mean method to process the node coordinates. In this technical solution, the algorithms calculate the arithmetic mean of all node coordinates for each polygonal element to obtain the center point coordinates, and calculate the arithmetic mean of the coordinates of every two adjacent nodes to obtain the edge midpoint coordinates. The data processing procedure performs numerical calculations based on the node dataset to generate an extended coordinate dataset. The output is a complete coordinate dataset containing the original nodes, center point, and edge midpoints, enhancing the geometric description capability.

[0237] The strain matrix fusion algorithm handles the strain-displacement relationship of quadrilateral sub-elements through constraint equations. In this technical solution, the algorithm establishes the strain-displacement relationship of each sub-element based on geometric connection data, representing the midpoint displacement as a constraint form of the adjacent node displacement. The data processing involves matrix calculation, constraint application, and recombination of strain relationships. The output is equivalent strain matrix data with the center point and original node displacements as variables, describing the overall strain characteristics of the polygonal element.

[0238] Single-point integration and hourglass control algorithms are used to numerically integrate and correct the equivalent strain matrix. In this technical solution, the single-point integration algorithm calculates the linear stiffness term by numerical integration at the element center point, while the hourglass control algorithm calculates the hourglass force amplitude based on the element's geometric parameters and establishes a mapping relationship. The data processing includes integration, hourglass force calculation, and stiffness term superposition. The output is the stiffness matrix data of the polygonal element after hourglass control, exhibiting enhanced numerical stability.

[0239] The finite element method (FEM) algorithm integrates all element stiffness matrices and performs numerical solutions. In this technical solution, the FEM algorithm integrates the polygonal element stiffness matrix data into the overall stiffness matrix according to node numbers, and applies boundary conditions to solve the overall stiffness equation. The data processing includes matrix assembly, equation solving, and stress-strain calculation. The output is structural stress-strain distribution data, which can be visualized as contour maps or isoline maps.

[0240] Polygonal elements are fundamental discrete units in structural simulation analysis, formed by connecting multiple nodes, and can flexibly adapt to complex geometries. In this invention, polygonal elements are constructed by extracting node coordinate data from the geometric model. This node coordinate dataset contains the coordinate values ​​of each node, providing basic geometric information for subsequent calculations. The data processing involves parsing the CAD model or mesh file, extracting vertex coordinate information, and the output is a node coordinate dataset that forms the basis of the finite element analysis.

[0241] The center point is obtained by calculating the arithmetic mean of the coordinates of all nodes in the polygonal element, representing the geometric center of the polygon. In this invention, the center point coordinate data is used to generate an extended coordinate data set and participates as an additional degree of freedom in the construction of the strain matrix, which helps improve computational accuracy and stability. The data processing procedure calculates the arithmetic mean based on the node data set and outputs the center point coordinate data.

[0242] The midpoint of an edge is obtained by calculating the arithmetic mean of the coordinates of every two adjacent nodes of a polygonal element, representing the center position of each edge. The midpoint coordinates are used to form an extended coordinate dataset, which is combined with the center point and nodes in geometric connection operations to form quadrilateral sub-elements. The data processing also involves calculating the arithmetic mean based on the node dataset, outputting the midpoint coordinates.

[0243] Quadrilateral sub-elements are sub-elements formed by geometric connection operations of the center point, side midpoints, and corresponding nodes, and are used to describe the topological structure of polygonal elements. In this invention, each quadrilateral sub-element is constructed based on the node connection sequence, and its strain-displacement relationship is used to derive the overall strain characteristics. The data processing procedure includes selecting coordinate data and performing connection operations, outputting geometric connection data describing the topological relationships of the sub-elements.

[0244] The strain matrix is ​​a matrix established through the strain-displacement relationship of quadrilateral sub-elements, describing the linear mapping relationship between strain and displacement. In this invention, the strain matrix is ​​calculated by fusing constraint equations to generate equivalent strain matrix data, which is used for subsequent stiffness matrix calculations. The data processing involves matrix calculations and constraint applications, outputting an equivalent strain matrix with the center point and original nodal displacements as variables.

[0245] The stiffness matrix is ​​obtained by numerically integrating the equivalent strain matrix using a single-point integration algorithm and an hourglass control algorithm, and it describes the stiffness characteristics of the element. In this invention, the stiffness matrix is ​​corrected using hourglass control and used for structural analysis in a finite element solution system. The data processing includes integration operations and stability correction, outputting hourglass-controlled stiffness matrix data for polygonal elements.

[0246] The hourglass control algorithm, based on element geometry and deformation mode definitions, is used to suppress the hourglass phenomenon in numerical calculations. In the application of this invention, hourglass control manages the hourglass force growth through a hyperbolic function and discretizes the hourglass force to each node of the element, enhancing computational stability. The data processing involves parameter definition and the establishment of mathematical relationships, outputting the hourglass force distribution and stiffness correction terms.

[0247] The finite element method (FEM) solution system is a pre-defined numerical solution platform used to integrate the stiffness matrices of polygonal elements and solve the global stiffness equation. In this application, the system calculates stress-strain distribution based on nodal displacement data and outputs structural simulation analysis results. The data processing includes matrix assembly and equation solving, outputting stress-strain distribution data.

[0248] Visualization is the process of presenting stress-strain distribution results in graphical form to intuitively demonstrate the analysis results. In this invention, visualization uses graphics rendering technology to generate contour maps or isoline maps to assist in engineering evaluation and design verification. The data processing involves data mapping and rendering operations, outputting visualized graphical results.

Claims

1. A novel method for calculating arbitrary polygonal elements for structural simulation analysis, characterized in that, include: Step 1: Obtain the geometric model of the structure to be analyzed, extract the node coordinate data of the polygonal units from the geometric model, and obtain a set of node data including coordinate values. Step 2: Receive the node data set, calculate the arithmetic mean of the node coordinates of each polygonal unit in the node data set to generate the center point coordinate data of the polygonal unit, calculate the arithmetic mean of the coordinates of every two adjacent nodes in the polygonal unit to generate the midpoint coordinate data of each side of the polygonal unit, and form an extended coordinate data set including the center point and the midpoint of the side of the polygonal unit. Step 3: Read the extended coordinate data set. For each polygonal cell, according to the node connection order of the polygonal cell, perform geometric connection calculations on the center point coordinates, the midpoint coordinates of the adjacent sides, and the corresponding node coordinates to generate geometric connection data describing the topology of the quadrilateral sub-cell. Step 4: Obtain geometric connection data. Based on the topology of the quadrilateral sub-unit, establish the strain and displacement relationship data of each sub-unit. Perform fusion calculation on the strain and displacement relationship data through constraint equations to generate equivalent strain matrix data describing the overall strain characteristics of the polygonal unit. Step 5: Receive the equivalent strain matrix data, perform numerical integration on the equivalent strain matrix data using a single-point integration algorithm, apply the hourglass control algorithm to correct the stability of the integration result, and output the stiffness matrix data of the polygonal element after hourglass control. Step 6: Input the stiffness matrix data of the polygonal elements into the preset finite element solution system, integrate the stiffness matrix data of each polygonal element into the overall stiffness matrix according to the node number, and numerically solve the overall stiffness equation based on the overall stiffness matrix through the preset finite element solution system. Calculate the stress and strain distribution of the structure based on the node displacement data obtained from the solution, and output the structural simulation analysis results.

2. The novel arbitrary polygonal element calculation method for structural simulation analysis according to claim 1, characterized in that, Step 2 includes: Based on the node data set, for a polygonal cell with n nodes, the arithmetic mean of all node coordinates is used to determine the center point coordinates. Based on the node data set, the arithmetic mean of the coordinates of two adjacent nodes is used to determine the coordinates of the midpoint of the corresponding edge; Based on the coordinates of the center point and the midpoint of the edge, complete the coordinate calculation of all nodes and edges; Based on the coordinate calculation results of all nodes, a complete coordinate dataset including the center point, edge midpoints, and original nodes is established.

3. The novel arbitrary polygonal element calculation method for structural simulation analysis according to claim 2, characterized in that, Step 3 includes: Based on the extended coordinate data set, for each edge of the polygonal cell, the original node corresponding to the edge is selected; Based on the extended coordinate data set, select the coordinates of the midpoint of the edge corresponding to the edge; Based on the extended coordinate data set, select the coordinates of the midpoints of the two sides adjacent to the edge; Based on the selected original node coordinates, edge midpoint coordinates, and center point coordinates, connect the center point, edge midpoints, and original nodes to form a quadrilateral sub-unit.

4. The novel arbitrary polygonal element calculation method for structural simulation analysis according to claim 3, characterized in that, Step 4 includes: Based on geometric connection data, the strain-displacement relationship of each quadrilateral sub-unit is established; Based on the strain-displacement relationship, the displacement of the midpoint of the edge is expressed as a constraint form of the displacement of the adjacent nodes; Based on the constrained strain-displacement relationship, the strain relationships of all quadrilateral sub-elements are recombined according to nodal displacements. Based on the recombined strain relationship, an equivalent strain matrix is ​​generated with the center point and the original nodal displacements as variables.

5. The novel arbitrary polygonal element calculation method for structural simulation analysis according to claim 4, characterized in that, Step 5 includes: Based on the equivalent strain matrix data, the linear stiffness term is calculated using a single-point integration scheme. The amplitude of the hourglass force is calculated based on the element geometric parameters in the geometric connection data. Based on the amplitude of the hourglass force, a mapping relationship between the hourglass force and the nodal displacement is established; The stiffness matrix is ​​obtained by superimposing the hourglass stiffness term and the linear stiffness term based on the linear stiffness term and the hourglass stiffness term.

6. The novel arbitrary polygonal element calculation method for structural simulation analysis according to claim 5, characterized in that, The hourglass control algorithm includes: Define the hourglass force calculation parameters based on the element geometric dimensions in the geometric connection data; Based on the calculation parameters of hourglass force, a mathematical relationship between hourglass force and element deformation mode is established; Based on mathematical relationships, the growth of hourglass force is controlled through hyperbolic functions. Based on the controlled hourglass force, the hourglass force is discretely distributed to each node of the unit.

7. The novel arbitrary polygonal element calculation method for structural simulation analysis according to claim 6, characterized in that, Step 6 includes: Based on the hourglass-controlled stiffness matrix data of polygonal elements, assemble the stiffness matrix of all polygonal elements. Based on the preset boundary condition data, the boundary conditions are applied to the overall stiffness equation after assembly and then solved. Based on the nodal displacement data obtained from the solution, the stress and strain distribution of the structure is calculated and output. The results of structural stress-strain distribution are visualized.

8. The novel arbitrary polygonal element calculation method for structural simulation analysis according to claim 7, characterized in that, Step 6 also includes: Based on the hourglass-controlled polygonal element stiffness matrix data, equilibrium iterative calculations are performed in an implicit solution scheme. Internal forces are calculated in an explicit solution scheme based on the stiffness matrix data of polygonal elements controlled by an hourglass. Based on the hourglass control algorithm, stability control parameters are generated in both implicit and explicit solution processes; Based on the single-point integration algorithm and the hourglass control algorithm, the volumetric self-locking and shear self-locking effects in the unit strain data are corrected.

9. The novel arbitrary polygonal element calculation method for structural simulation analysis according to claim 8, characterized in that, Step 6 also includes: Based on the grid density distribution data, polygonal cell grid data is generated in the transition region of different grid densities; Processing element boundary displacement field data based on polygonal element mesh data and strain-displacement relationship; Based on boundary displacement field data and constraint equations, nodal displacement data is fused and processed. Based on the fused nodal displacement data and equivalent strain matrix data, stress and strain calculations are performed in the mesh transition region.

10. The novel arbitrary polygonal element calculation method for structural simulation analysis according to claim 9, characterized in that, Step 6 also includes: Based on the crack and hole feature data in the geometric model, polygonal unit mesh data adapted to complex geometry is generated. Based on polygonal cell mesh data and node data sets, granular materials and porous media are discretized to generate discretized model data. Based on the discretized model data and the equivalent strain matrix data, the stress concentration factor distribution data is calculated; Based on stress concentration factor distribution data and single-point integration algorithm, simulation calculations of complex structures are performed.

Images