A method for precise calculation of stiffness of irregular cross-section based on feature adaptive sampling and general analytic operator

Abstract

The application discloses a special section stiffness accurate calculation method based on feature adaptive sampling and a general analytic operator, and comprises the following steps: feature adaptive sampling and data semanticization; solving of an explicit spline equation based on topological decoupling; normalized parameter mapping and coefficient vector construction; general analytic calculation based on polynomial algebra; global assembly and stiffness matrix generation; the method retains topological sharp points through feature semantic sampling, unifies an integral domain through a normalized parameter equation, and converts a calculus problem into a linear algebra problem through polynomial algebra operation, so that fast and accurate solving of special section stiffness characteristics is realized.

Description

Technical Field

[0001] This invention relates to the field of engineering mechanics and structural analysis, and more specifically, it is a digital modeling and calculation method for the geometric properties and mechanical stiffness characteristics of complex irregular cross sections in aerospace, civil engineering and mechanical manufacturing. Background Technology

[0002] In engineering fields such as civil, mechanical, and aerospace, irregular cross-section members with complex geometric contours are widely used to meet specific functional, mechanical, and lightweight requirements. The boundaries of such cross-sections are usually composed of curves, broken lines, or other irregular shapes, and may even include cusps and abrupt curvature features. Accurately obtaining the stiffness properties of these cross-sections is the basis for structural analysis, optimization, and safety assessment.

[0003] However, existing technologies have limitations in handling such problems. Traditional analytical methods, while direct and efficient, are only applicable to regular geometries. Traditional spline interpolation methods typically impose global smoothing constraints, leading to geometric distortion when dealing with cusps and abrupt curvature changes common in engineering structures. General-purpose finite element analysis software, commonly used in engineering, while capable of handling complex geometries, is often cumbersome and computationally expensive, requiring the establishment of a complete model for solution, making it difficult to efficiently extract section stiffness parameters. Furthermore, these methods face integration difficulties, struggling to automate and programmatically integrate into parametric design or optimization loops. Additionally, these methods often lack systematic convergence control and error assessment mechanisms, resulting in a trade-off between reliability and efficiency. Existing methods for describing irregular cross-section geometry typically use multi-segment polygonal approximations or high-density discrete point clouds to fit curve boundaries. To ensure fitting accuracy, a large number of coordinate points are often required. This not only increases preprocessing workload but also makes it difficult for optimization algorithms to converge quickly during subsequent cross-section shape optimization design, hindering rapid iteration in the early stages of engineering design. Therefore, engineering practice urgently needs a dedicated method that can balance accuracy, efficiency, and procedural implementation to directly and reliably calculate the stiffness characteristics of arbitrary complex irregular cross sections, providing an effective solution for current technology in rapid analysis and optimization integration in the early stages of design. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of existing technologies and provide a method for accurately calculating the stiffness of irregular cross sections based on feature adaptive sampling and a general analytical operator. This method preserves topological cusps through feature semantic sampling, unifies the integration domain through normalized parametric equations, and transforms the calculus problem into a linear algebra problem through polynomial algebraic operations, thus achieving a fast and accurate solution for the stiffness characteristics of irregular cross sections.

[0005] The technical solution adopted in this invention is as follows: A method for accurately calculating the stiffness of irregular cross sections based on feature adaptive sampling and a general analytical operator includes the following steps: Step 1, Feature Adaptive Sampling and Data Semantics: Obtain discrete measurement points of the irregular cross-sectional profile, and mark the discrete measurement points as topological singularities and morphological process points. Adaptively adjust the sampling density based on the chord height error. Step 2: Solving explicit spline equations based on topological decoupling: Decouple the irregular cross-sectional profile into several independent continuous domains with topological singularities as boundaries. Establish a system of linear equations about the second derivatives (bending moments) of the nodes within each continuous domain. After solving the equations with adaptive boundary conditions, obtain the explicit polynomial coefficients of the cubic spline function on each physical interval, and then obtain the physical explicit equations. Step 3: Normalized parameter mapping and coefficient vector construction: Introducing dimensionless parameters The physical explicit equations obtained in step two are transformed into normalized parametric equations through coordinate mapping, and the X-axis and Y-axis polynomial coefficient vectors applicable to the general analytical operator are extracted. Step 4: General analytical calculation based on polynomial coefficients: Establish a general mapping relationship between physical quantities of irregular cross sections and polynomial coefficients, and directly solve the analytical integral values ​​of each segment boundary using discrete convolution and vector dot product operations; Step 5: Global Assembly and Stiffness Matrix Generation: Calculate the total geometric properties of the irregular section using the principle of vector path superposition, and output the stiffness matrix of the irregular section by combining the material constitutive parameters and spatial coordinate transformation.

[0006] As a preferred embodiment, in step one, when acquiring discrete measurement points of the irregular cross-section profile, the traditional equal-interval sampling is abandoned, and a feature-semantic-based sampling strategy is adopted: the discrete measurement points are divided into three categories: topological singular points (forced selection), morphological control points (selected on demand), and transition points (sparse selection). Morphological control points and transition points are collectively referred to as morphological process points. Topological singular points refer to points on the irregular cross-section where the derivative is discontinuous (such as right-angle corners and sharp points). These points must be explicitly captured and marked during measurement. By capturing these points, the spline interpolation algorithm can be prevented from attempting to smoothly transition the sharp corner, ensuring that the shape is not distorted; shape control points refer to the places where the curve bends most sharply (peaks, troughs) and the inflection points (inflection points), which are marked as... Cubic polynomials excel at describing monotonically changing curves. Therefore, between two topological singularities, the point of maximum curvature and the point of inversion (inflection point) are preferentially selected. This selection method ensures that the concavity and convexity of the curve remain unchanged between any two sampling points, avoiding the phenomenon of multiple peaks and troughs in a single curve segment, which would lead to a deterioration in the fitting effect. Transition points are points collected between topological singularities and shape control points, selected according to sparsity. During the acquisition or preprocessing process, the chord height error between adjacent points is calculated. ,like , For a preset threshold, new sampling points are automatically inserted between two points until the geometric description error meets the requirements. This mechanism ensures that sufficient detail is preserved in areas with drastic curvature changes, while keeping the data sparse in flat areas.

[0007] As a preferred embodiment, step two includes a two-level division. The first level involves constructing an independent continuous domain by traversing the point set. Once a label is encountered... At the point where the curve fitting process is interrupted immediately, the entire closed contour is divided into... Q A single, continuous curve domain, labeled as At points where no first-order derivative continuity constraint is imposed, adjacent curve domains are allowed to have different tangent vectors at the connection points, thus perfectly preserving the sharp corner characteristics. The second level is for constructing basic computational units to solve integration problems. Within each continuous curve domain, each pair of adjacent measurement points is used as a basic computational unit.

[0008] Let the irregular cross-sectional profile be Q A topological singularity partition, forming Q n independent continuous domains. For any nth n... q A continuous field containing n -1 ordered measurement points. In the... i physical sub-intervals Above, the cubic spline function is defined as: in, It is the independent variable; Indicates the first Polynomial coefficients on each segment.

[0009] The above cubic spline function should satisfy the following condition in each subinterval: All of the above are polynomials of degree no higher than three; the spline function, its first derivative, and its second derivative all simultaneously satisfy the continuity condition; to solve for the cubic spline interpolation function, it is only necessary to... Confirm a cubic polynomial cubic polynomial There are 4 unknown coefficients. Determining the expression of a spline function requires 4n conditions. From the conditions satisfied by the spline function, 4n-2 conditions can be determined. The two missing conditions are given by the state requirements of the cubic spline interpolation function at the endpoints of the practical problem; these are also called boundary conditions. Common boundary conditions include: (1) Clamping boundary conditions: Given the first derivative values ​​of the two endpoints, i.e. ; (2) Natural boundary conditions: make the second derivative values ​​at both endpoints zero, i.e. ; (3) Non-knotted boundary conditions: The third derivative value of the first interpolation point is forced to be equal to the third derivative value of the second point, and the third derivative value of the first point is finally equal to the third derivative value of the second point, that is... .

[0010] When the endpoint is connected to the base or a specific geometric feature and the slope of the tangent is known, the clamping boundary condition is selected. When the endpoint is a topological singularity and there is no explicit tangential constraint, the natural boundary condition is selected. When the segment point is only a data encryption point and not a geometric breakpoint, and extreme smoothness is required, the non-knotted boundary condition is selected.

[0011] Based on the above conditions, a general formula is now derived. The calculation formula is as follows: According to the definition of interpolation, the left endpoint of each function segment is equal to its function value: achievable .

[0012] According to the definition of interpolation, the right endpoint of each function segment is equal to its function value: ; make You can get Based on the continuity condition of the first derivative, we can obtain: Equating the two equations above yields the result. .

[0013] According to the continuity condition of the second derivative, we can obtain: Equating the two equations above yields: .

[0014] By organizing the above conditions, we can obtain The above formula yields 4n-2 equations. Adding two boundary conditions completes the solution for all coefficients. However, to improve computational efficiency, elimination is performed first.

[0015] Introducing intermediate variables Indicates the curve at the th i The second derivative of each node, with respect to Taking the derivative twice, we get ,when When can be obtained Combined with the definition of the second derivative Relationships can be established .

[0016] for Although Only can be retrieved But when The above equation still holds true, and we can obtain... ,in For the curve at the th Second derivative at each node.

[0017] for You will then receive: ; After organizing the coefficients, we can obtain: for Substituting the above coefficients, we can obtain: After sorting, we can obtain ; We received information about of A system of equations, and exist There are 1 unknowns, among which For the curve at the th Second derivative at each node.

[0018] The equations under the three boundary conditions can be expressed as follows: (1) Clamping boundary conditions: , After sorting, we can obtain: , After sorting, we can obtain: (2) Natural boundary conditions: ; (3) Non-knotted boundary conditions: , Summarized as follows: ; , Summarized as follows: Solve the above equations simultaneously to get Substituting these values ​​into the above equation allows us to solve for the unknown coefficients of all segments within the continuous domain. This allows us to solve for its spline function. The invention first solves for the global bending moment based on all sampling points within a continuous domain, and then discretizes the entire domain into independent cubic polynomial function segments defined on each pair of adjacent measurement intervals.

[0019] As a preferred embodiment, in step three, to adapt to general calculation algorithms, the physical measurement points need to be transformed into a standardized mathematical model by introducing dimensionless normalization parameters. The physical explicit equations obtained in step two are transformed into normalized parametric equations. and Define step size ,make and Satisfies linear mapping: Therefore, the transformation factor can be obtained: .

[0020] The normalized parametric equation can be expressed as: coefficient vector for: .

[0021] Substituting the transformation factor into the original explicit equation yields: coefficient vector for: .

[0022] For the Segment boundary, let Parametric equations for: As a preferred option, in step four, the calculation of the physical quantities of the irregular cross-section does not employ the traditional numerical integration method, but rather an analytical solution mechanism based on polynomial coefficient algebraic operations. The specific implementation steps are as follows: For any planar physical quantity Its definition is usually as According to Green's theorem By selecting a specific field function and It can perform dimensionality reduction calculations on cross-sectional stiffness characteristics. Under a normalized parameter system, for any parameterized boundary segment (parameter... ), the differential term is transformed into .

[0023] Therefore, a general calculation formula is constructed for the th Segment boundary: .

[0024] in, Green's constant factor; For power-base polynomials ( or ); It is the power order; For differential polynomials ( or ).

[0025] make , but in, Let be the highest order of the integrand mixture polynomial (for cubic splines, ...). ); yes and The combination; , .

[0026] Ultimately, we can obtain the first i The formula for calculating the physical contribution value of the segment boundary is: Depending on the physical quantity to be determined, the corresponding power-base polynomial in the general calculation formula Poly 1. Power order γ Differential polynomial Poly 2. Green's constant factor K and the highest order of a mixed polynomial M The possible values ​​are configured as follows: When the physical quantity to be determined is the cross-sectional area ( A When taking Poly 1 is x ( t ), γ =1, Poly 2 is y ( t ),at this time K =1, M =5; when calculating the static moment of the cross section about the X-axis ( S x When taking Poly 1 is y ( t ), γ =2, Poly 2 is x ( t ),at this time K =-1 / 2, M =8; when calculating the static moment of the section about the Y-axis ( S y When taking Poly 1 is x ( t ), γ =2, Poly 2 is y ( t ),at this time K =1 / 2, M =8; when calculating the moment of inertia of the cross section about the X-axis ( I xx When taking Poly 1 is y ( t ), γ =3, Poly 2 is x ( t ),at this time K =-1 / 3, M =11; When calculating the moment of inertia of the cross section about the Y-axis ( I yy When taking Poly 1 is x ( t ), γ =3, Poly 2 is y ( t ),at this time K =1 / 3, M =11; when calculating the cross-sectional inertia ( I xy When taking Poly 1 is y ( t ), γ =2, Poly 2 is x2 ( t ) / 2, at this time K =-1 / 2, M =8.

[0027] As a preferred embodiment, in step five, the system traverses all segment boundaries of the cross section and calculates the scalar result for each segment based on the general analytical operator. Accumulated to global variable In this process, due to the directionality of the integration path, the negative contribution value generated by the hole boundary is automatically subtracted from the positive contribution value of the outer contour, thus enabling a single traversal solution for any multi-connected cross-section.

[0028] The total physical quantities of the cross section are: .

[0029] Since the initial moment of inertia calculated in step four is based on a measurement coordinate system and cannot be directly used for mechanical analysis, the centroid coordinates of the cross-section are first calculated based on the obtained static moment. : in, These are the static moments of the cross section about the X-axis and about the Y-axis, respectively. This represents the total cross-sectional area.

[0030] Subsequently, the centroidal inertial tensor is constructed using the parallel axis shift theorem: in, Let x be the moment of inertia of the cross section about the centroid X-axis. Let be the moment of inertia of the cross section about the centroidal Y-axis. Let the product of inertia of the cross section about the centroidal axis be . The total moment of inertia of the cross section about the X-axis of the measurement coordinate system is given by... The total moment of inertia of the cross section about the Y-axis of the measurement coordinate system is given by... Let be the total product of inertia of the cross section with respect to the measurement coordinate system.

[0031] For irregular cross-sections (usually asymmetrical), the centroidal coordinate axes are not necessarily the directions of strongest / weakest bending resistance. We can now find the principal axes of inertia by rotating the coordinate system. This results in inertial product Maximum principal moment of inertia and minimum principal moment of inertia It can be represented as: The angle between the principal inertial axis and the horizontal direction of the centroid coordinate axis satisfy: Axial tensile and compressive stiffness of irregular cross sections and spindle bending stiffness The calculation formula is as follows: .

[0032] in, This is the elastic modulus of the material.

[0033] The beneficial effects of this invention are: 1. Achieves efficient utilization and lightweight storage of sampled data: Based on an adaptive sampling strategy using chord height error, this invention only requires data encryption in regions with drastic curvature changes and sparse sampling in flat regions. Combined with the high-order fitting capability of cubic splines, complex contours can be accurately described with only a very small number of key feature points. Compared to traditional methods that rely on high-density point cloud approximation, this significantly reduces data storage requirements and transmission bandwidth, providing effective support for lightweight management of engineering data.

[0034] 2. Breakthroughs in geometric reconstruction fidelity and topological adaptability: Compared to the problem of geometric model distortion caused by the forced global smoothing of traditional spline interpolation, this invention introduces a topological singularity identification mechanism and adopts a domain decoupling strategy to remove the first-order derivative continuity constraint at singularities. This modeling method based on topological semantics can accurately reproduce common sharp edges, creases, and abrupt changes in engineering cross-sections (such as chemically milled steps and stiffener corners). Compared to the finite element method, which requires the establishment of a complete three-dimensional solid model and mesh generation, this invention directly provides a parametric description of the cross-sectional contour, fundamentally eliminating geometric fitting errors.

[0035] 3. Significantly improved computational accuracy: This invention abandons traditional infinitesimal discretization and numerical integration methods, and innovatively constructs a general analytical operator based on polynomial algebra operations. By transforming Green's theorem integral into discrete convolution and dot product operations of polynomial coefficient vectors, the truncation error caused by numerical approximation is eliminated. Its calculation results are theoretically exact solutions, achieving a qualitative leap in computational accuracy.

[0036] 4. Significant advantages in computational efficiency and algorithm complexity: This invention utilizes normalized parametric equations to uniformly map boundary segments of arbitrary physical dimensions to standard unit intervals, transforming integral operations into dot product operations of fixed-dimensional vectors, maintaining a constant computational complexity. (Constant level). Compared to traditional infinitesimal methods that require a significant increase in the number of partitions to achieve convergence, this invention requires no iteration or mesh refinement, and can obtain accurate results in a single calculation, improving the calculation speed by 1-2 orders of magnitude. It is particularly suitable for embedding into high-frequency calculation scenarios such as structural topology optimization.

[0037] 5. Possesses extremely high algorithm versatility and automated processing capabilities: This invention constructs a unified general-purpose calculation model, allowing the same core code to be reused to solve for area, static moment, and moment of inertia through simple parameter configuration (changing the power and constant factors). Simultaneously, utilizing the vector path superposition principle of Green's formula, the algorithm can automatically achieve the algebraic superposition of solid and hole attributes, automatically processing complex cross-sections of arbitrarily connected domains containing holes without additional Boolean geometric operations. Attached Figure Description

[0038] The invention will now be further described with reference to the accompanying drawings, in which: Figure 1 Here is the flowchart of the core algorithm; Figure 2 This is a schematic diagram of three-level feature adaptive sampling; Figure 3 This is a schematic diagram of domain decoupling based on topological singularities; Figure 4 This is a schematic diagram of vector path overlay and hole processing; Figure 5 This is a schematic diagram for an example; Detailed Implementation

[0039] The following detailed description of embodiments of the present invention, in conjunction with the accompanying drawings, does not constitute a limitation of the invention and is merely illustrative. Through these examples, the advantages of the present invention will become clearer and easier to understand.

[0040] Figure 1 This is a flowchart of the present invention; A method for accurately calculating the stiffness of irregular cross sections based on feature adaptive sampling and a general analytical operator includes the following steps: S1: Feature Adaptive Sampling and Data Semantics: Obtain discrete measurement points of the cross-sectional contour and mark the measurement points as topological singularities and morphological process points, and adaptively adjust the sampling density based on chord height error; S2: Explicit spline equation solution based on topological decoupling: The cross-sectional profile is decoupled into several independent continuous domains by using topological singular points as boundaries. A system of linear equations about the second derivatives (bending moments) of the nodes is established in each continuous domain. After solving the equations with adaptive boundary conditions, the explicit polynomial coefficients of the cubic spline function on each physical interval are obtained. S3: Normalized parameter mapping and coefficient vector construction: Introducing dimensionless parameters The physical explicit equations obtained in step two are transformed into normalized parametric equations through coordinate mapping, and the X-axis and Y-axis polynomial coefficient vectors applicable to the general analytical operator are extracted. S4: General analytical calculation based on polynomial algebra: Establish a general mapping relationship between cross-sectional physical quantities and polynomial coefficients, and directly solve the analytical integral values ​​of each segment boundary by using discrete convolution and vector dot product operations; S5: Global Assembly and Stiffness Matrix Generation: Calculates the total geometric properties of the cross section using the principle of vector path superposition, and outputs the cross section stiffness matrix by combining material constitutive parameters and spatial coordinate transformation.

[0041] This embodiment selects, as follows: Figure 5 The verification was performed using a symmetrical irregular cross-section with complex curvature and cusp features. This cross-section has a bottom width of approximately 4.0m, a top width of approximately 2.0m, and a height of 3.0m. y There is a tip where the derivative is discontinuous at 1.8m.

[0042] Step one involves adaptive feature sampling and data semanticization, the principle of which is illustrated in the diagram below. Figure 2 As shown, in order to minimize the data volume while ensuring geometric high fidelity, a hierarchical sampling strategy is adopted: right control point 1 (2.0, 0.0), right control point 3 (1.5, 1.8), right control point 5 (1.0, 3.0), left control point 5 (-1.0, 3.0), left control point 3 (-1.5, 1.8), and left control point 1 (-2.0, 0.0) are marked as topological singularities. T break The right control point 2 (0.8, 1.0), right control point 4 (1.1, 2.3), left control point 4 (-1.1, 2.3), and left control point 2 (-0.8, 1.0) are marked as shape control points. T shape ), preset threshold E threshold Set the value to 5mm and insert 20 transition points on the left and right boundaries respectively.

[0043] Step two involves solving the explicit spline equations based on topological decoupling. Figure 3 This demonstrates the process of dividing the cross-sectional profile into independent continuous domains based on topological singularities. In this example, using right control point 3 and left control point 3 as boundaries, the cross-sectional profile is decoupled into four independent continuous domains: upper left, lower left, upper right, and lower right. For each continuous domain, boundary conditions are adaptively configured based on its physical connectivity. Bottom left / bottom right segment (base connection): Apply a clamping boundary at the bottom and a natural boundary at the tip.

[0044] Top left / top right segment (free end): Natural boundary conditions are applied to both the tip and the top.

[0045] By solving the tridiagonal matrix, the global bending moment vector and its piecewise explicit expressions are obtained. The global bending moment vector is: Bottom left section: M=[62.90, -19.78, 2.38, -3.56, -1.97, -2.40, -2.28, -2.31, -2.30, -2.31, -2.30, -2.31, -2.29, -2.35, -2.14, -2.92, 0.00]; Top left section: M =[0.00, -1.39, -1.02, -1.12, -1.08, -1.12, -1.02, -1.39, 0.00]; Bottom right section: M =[-62.90, 19.78, -2.38, 3.56, 1.97, 2.40, 2.28, 2.31, 2.30, 2.31, 2.30, 2.31, 2.29, 2.35, 2.14, 2.92, 0.00]; Top right section: M =[0.00, 1.39, 1.02, 1.12, 1.08, 1.12, 1.02, 1.39, 0.00].

[0046] After calculating the bending moment at each node in the independent domain, the spline function expression between any two points can be obtained using the following formula: The spline functions for some key segments are as follows: Bottom left boundary (L_Lower, 16 segments in total): Top left boundary (L_Upper, 8 segments in total): Bottom right boundary (R_Lower, 16 segments in total): Top right boundary (R_Upper, 16 segments in total): Step three involves normalizing the parameter mapping and constructing the coefficient vector. To adapt to the general analytical operator and satisfy the counterclockwise closed-loop requirement of Green's formula, it is necessary to construct the normalized coefficient vector for each boundary segment. Introducing dimensionless parameters t The above explicit equations are transformed into a system of parametric equations through a mapping transformation:

[0047] Step four involves a general analytical calculation based on polynomial algebra. The system iterates through each set of coefficient vectors generated in step three and calls the corresponding analytical operator based on the physical quantity to be determined (area, static moment, moment of inertia). For the... i The formula for calculating the physical contribution value of a segment boundary is as follows: Taking the area calculation of the first segment of the lower right boundary in the embodiment as an example, the values ​​of each part in the calculation formula are as follows: W = [0.000e+00, 0.000e+00, 2.691e-02, -6.142e-02, 0.000e+00, 2.500e-01]; V int = [1.667e-01, 2.000e-01, 2.500e-01, 3.333e-01, 5.000e-01, 1.000e+00]; Green's constant factor when the physical quantity is the cross-sectional area. K =1; the calculated contribution of this segment to the area is 2.3625e-01m. 2 .

[0048] Step five involves global assembly and the formation of the stiffness matrix. Using the principle of vector path superposition, the contribution values ​​of all segments obtained in step four are algebraically summed to obtain the total geometric properties of the cross-section relative to the measurement coordinate system: Total cross-sectional area static moment (about the X-axis) Static moment (about the Y-axis) Moment of inertia (about the X-axis) Moment of inertia (about the Y-axis) Inertial product .

[0049] Centroid coordinates for The centroidal inertial tensor can be solved using the parallel axis shift formula as follows: The calculation results show that the product of inertia is zero (within the double-precision error range), verifying the symmetry assumption of the cross section. This indicates that the currently established measurement coordinate system is consistent with the principal stiffness direction of the cross section, and the principal moment of inertia can be obtained without additional coordinate rotation.

[0050] If the cross-section contains a porous structure, such as Figure 4 As shown, the total geometric properties of the cross section can be calculated using the principle of vector path superposition.

[0051] Combining the constitutive parameters of the material (such as elastic modulus) The final stiffness is calculated as follows: Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for accurately calculating the stiffness of irregular cross-sections based on feature adaptive sampling and a general analytical operator, characterized in that, Includes the following steps, Step 1, Feature Adaptive Sampling and Data Semantics: Obtain discrete measurement points of the irregular cross-sectional profile, and mark the discrete measurement points as topological singularities and morphological process points. Adaptively adjust the sampling density based on the chord height error. Step 2: Solving explicit spline equations based on topological decoupling: Decouple the irregular cross-sectional profile into several independent continuous domains with topological singular points as boundaries. Establish a system of linear equations about the second derivatives of the nodes within each continuous domain. After solving the equations with adaptive boundary conditions, obtain the explicit polynomial coefficients of the cubic spline function on each physical interval, and then obtain the physical explicit equations. Step 3: Normalized parameter mapping and coefficient vector construction: Introducing dimensionless parameters The physical explicit equations obtained in step two are transformed into normalized parametric equations through coordinate mapping, and the X-axis and Y-axis polynomial coefficient vectors applicable to the general analytical operator are extracted. Step 4: General analytical calculation based on polynomial coefficients: Establish a general mapping relationship between physical quantities of irregular cross sections and polynomial coefficients, and directly solve the analytical integral values ​​of each segment boundary using discrete convolution and vector dot product operations; Step 5: Global Assembly and Stiffness Matrix Generation: Calculate the total geometric properties of the irregular section using the principle of vector path superposition, and output the stiffness matrix of the irregular section by combining the material constitutive parameters and spatial coordinate transformation.

2. The method for accurately calculating the stiffness of irregular cross-sections based on feature adaptive sampling and a general analytical operator as described in claim 1, characterized in that, In step one, when acquiring discrete measurement points of the irregular cross-sectional profile, a feature-semantic-based sampling strategy is adopted: the discrete measurement points are divided into three categories: topological singularities, morphological control points, and transition points. Morphological control points and transition points are collectively referred to as morphological process points. Topological singularities refer to points on the irregular cross-section where the derivative is discontinuous. These points must be explicitly captured and marked during measurement. Shape control points refer to the points where the curve bends most severely and the points where it reverses, and are marked as follows: Cubic polynomials excel at describing monotonically changing curves. Therefore, between two topological singularities, the point of maximum curvature and the point of inverse curvature are preferentially selected. This selection method ensures that the concavity and convexity of the curve remain unchanged between any two sampling points, avoiding the phenomenon of multiple peaks and troughs in a single curve segment, which would lead to a deterioration in the fitting effect. Transition points are points collected between topological singularities and shape control points, selected according to sparsity. During the acquisition or preprocessing process, the chord height error between adjacent points is calculated. ,like , For a preset threshold, new sampling points are automatically inserted between two points until the geometric description error meets the requirements.

3. The method for accurately calculating the stiffness of irregular cross-sections based on feature adaptive sampling and a general analytical operator as described in claim 1, characterized in that, Step two involves a two-level division. The first level is constructing an independent continuous domain by traversing the point set and, once a label is encountered... At the point where the curve fitting process is interrupted immediately, the entire closed contour is divided into... Q A single, continuous curve domain, labeled as At the point, no first derivative continuity constraint is imposed, allowing adjacent curve domains to have different tangent vectors at the connection point, thus perfectly preserving the sharp corner feature; The second level is to construct basic computational elements to solve integration problems. Within each continuous curve domain, each pair of adjacent measurement points is a basic computational unit. Let the irregular cross-sectional profile be Q A topological singularity partition, forming Q n independent continuous domains; for any nth... q A continuous field containing n -1 ordered measurement points; in the... i physical sub-intervals Above, the cubic spline function is defined as: in, It is the independent variable; Indicates the first Polynomial coefficients on each segment; The above cubic spline function should satisfy the following condition in each subinterval: All of the above are polynomials of degree no higher than three; the spline function, its first derivative, and its second derivative all simultaneously satisfy the continuity condition; to solve for the cubic spline interpolation function, it is only necessary to... Confirm a cubic polynomial cubic polynomial There are 4 unknown coefficients. Determining the expression of a spline function requires 4n conditions. From the conditions satisfied by the spline function, 4n-2 conditions can be determined. The two missing conditions are given by the state requirements of the cubic spline interpolation function at the endpoints of the practical problem; these are also called boundary conditions. Common boundary conditions include: (1) Clamping boundary conditions: Given the first derivative values ​​of the two endpoints, i.e. ; (2) Natural boundary conditions: make the second derivative values ​​at both endpoints zero, i.e. ; (3) Non-knotted boundary conditions: The third derivative value of the first interpolation point is forced to be equal to the third derivative value of the second point, and the third derivative value of the first point is finally equal to the third derivative value of the second point, that is: ； When the endpoint is connected to the base or a specific geometric feature and the tangent slope is known, the clamping boundary condition is selected. When the endpoint is a topological singularity and there is no explicit tangential constraint, the natural boundary condition is selected. When the segment point is only a data encryption point and not a geometric break point, and extreme smoothness is required, the non-knotted boundary condition is selected. Based on the above conditions, a general formula is now derived. The calculation formula is as follows: According to the definition of interpolation, the left endpoint of each function segment is equal to its function value: achievable ; According to the definition of interpolation, the right endpoint of each function segment is equal to its function value: ； make You can get Based on the continuity condition of the first derivative, we can obtain: Equating the two equations above yields the result. ; According to the continuity condition of the second derivative, we can obtain: Equating the two equations above yields: ; By organizing the above conditions, we can obtain The above formula yields 4n-2 equations. Adding two boundary conditions completes the solution for all coefficients. However, to improve computational efficiency, elimination is performed first. Introducing intermediate variables Indicates the curve at the th i The second derivative of each node, with respect to Taking the derivative twice, we get ,when When can be obtained Combined with the definition of the second derivative Relationships can be established ; for Although Only can be retrieved But when The above equation still holds true, and we can obtain... ,in For the curve at the th Second derivative at each node; for You will then receive: ； After organizing the coefficients, we can obtain: for Substituting the above coefficients, we can obtain: After sorting, we can obtain ; We received information about of A system of equations, and exist There are 1 unknowns, among which For the curve at the th Second derivative at each node; The equations under the three boundary conditions can be expressed as follows: (1) Clamping boundary conditions: ， After sorting, we can obtain: ， After sorting, we can obtain: (2) Natural boundary conditions: ； (3) Non-knotted boundary conditions: ， Summarized as follows: ; ， Summarized as follows: Solve the above equations simultaneously to get Substituting these values ​​into the above equation allows us to solve for the unknown coefficients of all segments within the continuous domain. Thus, its spline function can be solved.

4. The method for accurately calculating the stiffness of irregular cross-sections based on feature adaptive sampling and a general analytical operator as described in claim 1, characterized in that, In step three, to adapt to general calculation algorithms, the physical measurement points need to be transformed into standardized mathematical models by introducing dimensionless normalization parameters. The physical explicit equations obtained in step two are transformed into normalized parametric equations. and Define step size ,make and Satisfies linear mapping: Therefore, the transformation factor can be obtained: ; The normalized parametric equation can be expressed as: coefficient vector for: ; Substituting the transformation factor into the original explicit equation yields: coefficient vector for: ； For the Segment boundary, let Parametric equations for: 。 5. The method for accurately calculating the stiffness of irregular cross-sections based on feature adaptive sampling and a general analytical operator as described in claim 1, characterized in that, In step four, the specific implementation steps for calculating the physical quantities of the irregular cross-section are as follows: For any planar physical quantity Its definition is usually as According to Green's theorem By selecting a specific field function and It can perform dimensionality reduction calculations on cross-sectional stiffness characteristics. Under a normalized parameter system, for any parameterized boundary segment, the differential term is transformed into... ; Therefore, a general calculation formula is constructed for the th Segment boundary: ； in, Green's constant factor; It is a power-base polynomial; It is the power order; It is a differential polynomial; make , but in, Let be the highest order of the integrand polynomial; yes and The combination; , ; Ultimately, we can obtain the first i The formula for calculating the physical contribution value of the segment boundary is: 。 6. The method for accurately calculating the stiffness of irregular cross-sections based on feature adaptive sampling and a general analytical operator as described in claim 5, characterized in that, Depending on the physical quantity to be determined, the corresponding power-base polynomial in the general calculation formula Poly 1. Power order γ Differential polynomial Poly 2. Green's constant factor K and the highest order of a mixed polynomial M The possible values ​​are configured as follows: When the physical quantity to be determined is the cross-sectional area A At that time, take Poly 1 is x ( t ), γ =1, Poly 2 is y ( t ),at this time K =1, M =5; when calculating the static moment of the cross section about the X-axis S x At that time, take Poly 1 is y ( t ), γ =2, Poly 2 is x ( t ),at this time K =-1 / 2, M =8; when calculating the static moment of the cross section about the Y-axis S y At that time, take Poly 1 is x ( t ), γ =2, Poly 2 is y ( t ),at this time K =1 / 2, M =8; When calculating the moment of inertia of the cross section about the X-axis I xx At that time, take Poly 1 is y ( t ), γ =3, Poly 2 is x ( t ),at this time K =-1 / 3, M =11; When calculating the moment of inertia of the cross section about the Y-axis I yy At that time, take Poly 1 is x ( t ), γ =3, Poly 2 is y ( t ),at this time K =1 / 3, M =11; when calculating the cross-sectional inertia area I xy At that time, take Poly 1 is y ( t ), γ =2, Poly 2 is x 2 ( t ) / 2, at this time K =-1 / 2, M =8.

7. The method for accurately calculating the stiffness of irregular cross-sections based on feature adaptive sampling and a general analytical operator as described in claim 1, characterized in that, In step five, the system traverses all segment boundaries of the cross section and calculates the scalar result for each segment based on the general analytical operator. Accumulated to global variable In the middle; due to the directionality of the integration path, the negative contribution value generated by the hole boundary will be automatically deducted from the positive contribution value of the outer contour, thereby realizing a single traversal solution for any multi-connected cross section; The total physical quantities of the cross section are: ; Since the initial moment of inertia calculated in step four is based on a measurement coordinate system and cannot be directly used for mechanical analysis, the centroid coordinates of the cross-section are first calculated based on the obtained static moment. : in, These are the static moments of the cross section about the X-axis and about the Y-axis, respectively. This represents the total cross-sectional area. Subsequently, the centroidal inertial tensor is constructed using the parallel axis shift theorem: in, Let x be the moment of inertia of the cross section about the centroid X-axis. Let be the moment of inertia of the cross section about the centroidal Y-axis. Let the product of inertia of the cross section about the centroidal axis be . The total moment of inertia of the cross section about the X-axis of the measurement coordinate system is given by... The total moment of inertia of the cross section about the Y-axis of the measurement coordinate system is given by... The product of inertia of the cross section with respect to the measurement coordinate system; For irregular cross-sections, the centroidal coordinate axis is not necessarily the direction of strongest / weakest bending resistance. We can now find the principal inertial axes by rotating the coordinate system. This results in inertial product Maximum principal moment of inertia and minimum principal moment of inertia It can be represented as: The angle between the principal inertial axis and the horizontal direction of the centroid coordinate axis satisfy: Axial tensile and compressive stiffness of irregular cross sections and spindle bending stiffness The calculation formula is as follows: ； in, This is the elastic modulus of the material.

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