Post-processing method and system for eliminating grid dependence of finite element stress concentration zone

Abstract

This invention belongs to the field of numerical simulation technology and discloses a post-processing method and system for eliminating mesh dependence in stress concentration zones of finite element methods. The method includes: performing a weighted integral on the displacement values ​​of all nodes in the neighborhood of the point of interest to reconstruct the first derivative of the displacement field at that point; calculating the stress at that point from the displacement derivative based on constitutive relations; and implementing the post-processing based on finite element software. This invention avoids direct numerical differentiation of the displacement field, thereby improving the stability of the solution. The stress at the target point is determined by a large number of node displacements within the finite support domain, and the number of nodes involved in the calculation is tens of times greater than that connected by traditional finite elements. Furthermore, this method eliminates the need for stress extrapolation from Gaussian points to nodes, thus eliminating the uncertainty caused by the fact that the nodal stress in traditional methods is determined only by a few adjacent elements.

Description

Technical Field

[0001] This invention belongs to, but is not limited to, the field of numerical simulation technology, and particularly relates to a post-processing method and system for eliminating mesh dependence in finite element stress concentration zones. Background Technology

[0002] Stress concentration refers to the phenomenon in engineering structural components where sudden changes in geometry or size (such as openings, notches, grooves, etc.) cause localized stress values ​​to significantly exceed the nominal average stress. This phenomenon is one of the main causes of structural fatigue failure and brittle fracture. In engineering practice, the finite element method (FEM) is commonly used for numerical simulation and quantitative evaluation of stress concentration effects. As the most widely used numerical analysis tool, FEM has been extensively applied in academia and industry through mature commercial software (such as ANSYS and ABAQUS), becoming an indispensable key tool in engineering design and scientific research.

[0003] However, when using FEM software to analyze stress concentration problems in specific engineering projects, the stress calculation results often exhibit significant mesh dependence, meaning that the mesh generation method (element size) has a significant impact on the stress calculation results. Taking the heel of a gravity dam as an example, as the mesh becomes increasingly refined, the nodal displacements at this location usually converge to a certain stable value initially. However, the stress derived from the displacements, within a certain range from the corner of the heel, fails to converge as the mesh refinement continues to increase. Furthermore, it may even lead to completely opposite conclusions regarding whether the maximum tensile stress at the same location exceeds the standard value of the concrete tensile strength, while stress solutions outside this range meet the accuracy requirements. This stress distortion phenomenon is not a true reflection of actual physical behavior but rather an inherent numerical defect of the finite element method when simulating high-gradient stress fields.

[0004] To overcome this problem, specialized singular elements were introduced into FEM. At the same time, scholars have also developed post-processing techniques such as the equivalent stress method and the equivalent area averaging method, which use the stress within a certain range of the stress concentration region to replace the directly calculated value and technically correct the distortion of stress values.

[0005] (2) Problems or shortcomings of existing technology

[0006] In dealing with stress concentration problems, besides the mesh dependency of the stress calculation results at the concentration point mentioned above, the current finite element method also has the following two main limitations:

[0007] 1) The theoretical advantages and practical application challenges of singular units

[0008] Singular elements, by introducing singularity into their element shape functions, can more accurately describe the stress-strain field in high-gradient regions such as crack tips, thus effectively reducing the dependence of stress calculation results on the mesh. However, singular elements have not been widely used in practical engineering analysis. The main reasons are their limited applicability and high barrier to entry. On the one hand, highly stress-concentrated regions are often difficult to predict accurately in the early stages of modeling, while singular elements are only suitable for locally defined areas. On the other hand, if singular elements are improperly positioned or excessive in number, it can not only multiply the computational cost but also lead to numerical problems such as ill-conditioned matrices, resulting in difficulty in solving the problem or even distorted results. This dilemma reflects that although singular elements have significant theoretical advantages, it is still difficult to balance applicability, robustness, and computational efficiency in engineering practice.

[0009] 2) Limitations and subjective drawbacks of finite element stress post-processing methods

[0010] In FEM-based engineering analysis, stress concentration is often treated as a localized effect, with its deformation considered constrained and coordinated by adjacent large areas, thus insufficient to threaten overall safety and frequently overlooked in assessments. Furthermore, because the stress value at the concentration point continuously changes with mesh refinement and rarely converges to a stable value, even if analysts intend to conduct in-depth assessments, the uncertainty of numerical results often prevents objective quantitative judgments, ultimately leading to a avoidance of direct assessment of that point. However, this approach not only lacks theoretical rigor but may also mask potential risks in the structure, affecting the reliability and safety of engineering decisions.

[0011] From a numerical perspective, the FEM directly outputs the stress tensor at the Gaussian integration point, while the nodal stress is "extrapolated" from the Gaussian point stress. In stress concentration regions, the Gaussian point stress is prone to local peak values, and these peak values ​​only affect a very small spatial area. The standard strength of a material, however, is a "holistic" index measured based on a standard specimen under uniform uniaxial stress, reflecting the material's macroscopic average load-bearing capacity. If such highly mesh-dependent local peak stresses are directly compared with the material's standard strength, engineering judgments can easily fall into a dilemma: either ignore the potential cumulative effect of local high stresses, creating hidden dangers for long-term structural damage; or overemphasize the impact of local high stresses, taking unnecessary reinforcement measures and thus increasing engineering costs.

[0012] To avoid the evaluation difficulties caused by directly using the local stress peaks output by finite element analysis, engineering practice employs various post-processing methods to obtain relatively reasonable stress values, such as the equivalent stress method, the region-averaged stress method, or the stress region method—empirical post-processing techniques. However, these methods are essentially artificial interventions in the "original" stress field obtained from finite element calculations, and their specific operations lack unified and objective standards. In practical applications, the range, shape, and equivalent calculation method of the stress-averaged region often depend on the engineer's personal experience and subjective judgment, leading to significantly different analytical conclusions from different analysts for the same model. Summary of the Invention

[0013] To address the problems existing in the prior art, this invention provides a post-processing method for eliminating mesh dependence in finite element stress concentration zones.

[0014] This invention is implemented as follows: a post-processing method for eliminating mesh dependence in finite element stress concentration zones, the method comprising:

[0015] Step 1: Obtain the displacement values ​​of each node within the neighborhood of the point of interest;

[0016] Step 2: Reconstruct the displacement values ​​of each node in the neighborhood by weighted integration to obtain the first derivative of the displacement field of the point of interest;

[0017] Step 3: Calculate the stress at the point of interest using the first derivative of the displacement field based on the constitutive relation.

[0018] Furthermore, the weighted integral reconstruction of the displacement values ​​of each node in the neighborhood to obtain the first derivative of the displacement field of the point of interest is specifically as follows:

[0019] The node positions are represented using three-dimensional Cartesian coordinates, and the displacement field is represented as three displacement components along three coordinate directions.

[0020] For any three-dimensional scalar function, the near-field dynamic differential operator is used to obtain the required derivative by weighted integral of the function values ​​at each point in the neighborhood of the point of interest with the corresponding specific weight function;

[0021] The first derivatives of the three displacement components with respect to the three spatial variables are taken as the first derivatives of the displacement field.

[0022] Furthermore, the specific weight function is a peri-field dynamics function corresponding to the required derivative order. The peri-field dynamics function is constructed based on the orthogonality principle and is expressed in polynomial form.

[0023] The weighting function in the polynomial adopts a Gaussian distribution;

[0024] The undetermined coefficients in the near-field dynamics function are obtained by solving the system of equations established by the orthogonality condition.

[0025] Furthermore, the stress at the point of interest is calculated from the first derivative of the displacement field based on the constitutive relation. Specifically, this involves using the generalized Hooke's law to calculate the three-dimensional stress state at the point of interest based on the elastic modulus and Poisson's ratio of the material.

[0026] Another object of the present invention is to provide a post-processing implementation method, comprising the following steps:

[0027] Modeling, loading, and solving are completed using finite element software;

[0028] In the post-processing stage, parametric design language is used for secondary development;

[0029] In the software interface, there are function buttons for selecting key nodes in stress concentration areas and function buttons for recalculating stress.

[0030] In response to the node selection operation, all nodes in the neighborhood of the key node and their influence range are displayed;

[0031] Based on the coordinates and displacements of the nodes in the neighborhood, the post-processing method is executed to recalculate the stress at the key nodes.

[0032] Furthermore, the neighborhood of the key node is defined as a square or cube-shaped region extending along the coordinate axis with the key node as the center.

[0033] Furthermore, the range of the neighborhood in each coordinate axis direction is determined by subtracting the neighborhood scale from the center coordinates and adding the neighborhood scale to the center coordinates;

[0034] The neighborhood scale is determined by the average node spacing and the neighborhood scale coefficient;

[0035] The order of the displacement derivative is 1, and the neighborhood scaling factor is 3.

[0036] Another object of the present invention is to provide a post-processing system for eliminating mesh dependence in finite element stress concentration zones, comprising:

[0037] The node data acquisition module is used to acquire the coordinates and displacement values ​​of each node in the neighborhood of the point of interest.

[0038] The derivative reconstruction module is used to reconstruct the displacement values ​​of each node in the neighborhood by weighted integration to obtain the first derivative of the displacement field of the point of interest.

[0039] The stress calculation module is used to calculate the stress at the point of interest based on the first derivative of the displacement field according to the constitutive relation.

[0040] The post-processing execution module is used to call the derivative reconstruction module and the stress calculation module during the post-processing stage of the finite element software to recalculate the stress at the stress concentration point.

[0041] Furthermore, the derivative reconstruction module uses a peri-field dynamic differential operator to obtain the derivative, and uses a peri-field dynamic function constructed based on the orthogonality principle as a specific weight function in the weighted integral;

[0042] The near-field dynamics function is expressed in polynomial form, and the weighting function in the polynomial adopts a Gaussian distribution. The undetermined coefficients in the near-field dynamics function are obtained by solving the system of equations established by the orthogonality condition.

[0043] Furthermore, the post-processing execution module is further developed based on the parametric design language of the finite element software;

[0044] The system also includes a node selection and display module, which provides function buttons for selecting key nodes in stress concentration areas and function buttons for recalculating stress in the software interface, and displays all nodes in the neighborhood of the key node and their influence range after the key node is selected.

[0045] The neighborhood of the key node is a square or cube-shaped region centered on the key node and extending along the coordinate axis, with a displacement derivative of order 1 and a neighborhood scale factor of 3.

[0046] Based on the above technical solutions and the technical problems solved, the advantages and positive effects of the technical solution to be protected by this invention are as follows:

[0047] In industrial applications of finite element analysis, significant stress concentrations often occur at locations such as structural abrupt changes, hole edges, welds, notches, and contact areas. Traditional finite element post-processing typically relies on direct numerical differentiation of the displacement field within elements, extrapolation of Gaussian point stress, or averaging of nodal stresses. The calculation results are easily affected by mesh size, element shape, and local mesh quality. When the mesh is coarse or the stress gradient is large, nodal stresses may fluctuate, become distorted, or even exhibit non-physical peaks. This makes it difficult for engineers to accurately determine the true stress level at critical locations, thus affecting the reliability of fatigue life assessment, strength verification, and structural optimization design.

[0048] This method obtains displacement information from multiple nodes within the neighborhood of the point of interest and uses a weighted integral reconstruction method to calculate the first derivative of the displacement field. This avoids direct numerical differentiation of the discrete displacement field and reduces the impact of local mesh distortion and node connectivity on the results. Since the stress at the target point is determined by the displacements of a large number of nodes within the finite support domain, rather than relying solely on extrapolation from a few adjacent elements or Gaussian points, it significantly weakens the mesh dependence of stress concentration areas, improving the stability, continuity, and accuracy of gradient calculation and stress solution. This method can still obtain reliable stress results under relatively sparse mesh conditions, helping to reduce the computational cost of repeatedly refining the mesh, improving the efficiency of finite element analysis, and providing more stable and reliable data support for the safety assessment, lightweight design, and life prediction of engineering structures.

[0049] 1) Robust and reliable calculation results. This method fundamentally overcomes the limitations of traditional methods where nodal stress relies solely on information from a few adjacent elements and their limited number of nodes (e.g., stress concentration at an elliptical hole involves only 6 nodes from two adjacent quadrilateral elements). By constructing a support domain containing a large number of nodes for stress integration reconstruction, the robustness and reliability of the calculation results are significantly improved. Taking a neighborhood scale factor of m=3 as an example, in the analysis of a tension plate with an elliptical hole, the support domain contains an average of 345 nodes, nearly 60 times the number of nodes involved in the calculation compared to traditional methods. This greatly enriches the original data foundation upon which stress calculations are based. It is precisely based on this advantage that... Figures 5-7 As shown, within the entire radius of curvature variation, the stress concentration factor calculated by the traditional FEM exhibits extremely strong mesh dependence, and its overall fluctuation range continues to expand with increasing mesh refinement, reaching a maximum amplitude of nearly 30%. The stress concentration factor of the elliptical orifice calculated by the method of this invention remains stable, even under the most extreme conditions, with minimal variation, and highly consistent with the finite element solution and theoretical analytical solution under extremely dense mesh. Furthermore, in the analysis of a tension plate containing a square orifice, as the local mesh at the orifice gradually refines, the number of participating nodes increases from 66 to 161, but the stress concentration factor K calculated by the traditional finite element method... t However, the value fluctuated drastically from 5.62 to 158, failing to converge. In contrast, the K calculated in this invention based on the displacement field integral reconstructed stress method... t The mean is stable at 6.08 and the variance is only 0.024, showing good insensitivity to mesh changes. It effectively suppresses the computational uncertainty introduced by insufficient information in traditional methods, thereby significantly improving the stability of stress solution and the reliability of subsequent analysis.

[0050] 2) Significantly Improved Finite Element Computation Efficiency. Traditional methods often require repeated mesh refinement to obtain high-precision stress, leading to a dramatic increase in computational scale. This method takes a different approach, directly calculating stable, high-precision stress from the displacement field of a coarse mesh through integral reconstruction of a large amount of node information within the support domain. Taking a tension plate with an elliptical hole as an example, the stress concentration factor calculated by this invention based on a coarse mesh (Δ=0.016m) is close to the convergent solution obtained by traditional methods using an extremely dense mesh (Δ=0.004m). This indicates that this invention can significantly save computation time and resources while maintaining the accuracy of stress solution without undergoing repeated iterations of "solve-refine-resolve". In this example, the traditional method requires repeated mesh refinement twice, with the computational cost increasing with each refinement; while this invention only requires one coarse mesh calculation, and the time consumed per calculation is shorter. Overall, the overall computational efficiency is improved by more than 3 times.

[0051] 3) Expanding the application scenarios of finite element methods and lowering the threshold for handling high-gradient problems. This method is fully embedded in the existing finite element analysis workflow, requiring no modification to the preprocessing modeling and solution settings. Simply by clicking a custom button on the existing interface, it can be used as a dedicated post-processing tool to accurately reconstruct high-gradient stress fields that are difficult to capture by traditional methods, breaking the rigid dependence of traditional methods on mesh density. This "non-intrusive" plug-and-play characteristic eliminates the need for engineers to have a deep background in finite element theory and mesh processing experience. It not only significantly lowers the technical threshold for engineers to master high-precision stress analysis but also demonstrates the enormous potential of deep integration between advanced numerical methods and mature commercial software, opening up new technical paths for solving complex frontier mechanical problems.

[0052] 4) Automating and objectifying stress solution. Compared to other post-processing methods, this method is based on fixed mathematical rules, namely, reconstructing stress through integration using a large amount of nodal displacement information within the support domain. This replaces the reliance on manual intervention steps in traditional post-processing, such as stress smoothing, extrapolation strategy selection, or average region definition. Because the calculation rules are unique and pre-defined, different operators running this method on the same model will obtain completely consistent stress results, fundamentally eliminating subjective biases introduced by differences in individual experience. This deterministic calculation process not only eliminates subjective biases caused by differences in experience among different operators, but also enables fully automated batch processing through parametric design language programming, significantly improving processing efficiency and the reproducibility of results.

[0053] The core advantage of this method lies in avoiding direct numerical differentiation of the displacement field, thereby improving the stability of the solution. The stress at the target point is determined by a large number of nodal displacements within the finite support domain, with the number of nodes involved in the calculation being tens of times greater than that of traditional finite elements. Furthermore, this method eliminates the need for stress extrapolation from Gaussian points to nodes, thus removing the uncertainty inherent in traditional methods where nodal stress is determined by only a few adjacent elements. Therefore, even with a relatively sparse mesh, this method can still achieve high-precision gradient calculation and stress solution in stress concentration regions thanks to its extensive data integration capabilities. Attached Figure Description

[0054] Figure 1 This is a flowchart of the post-processing method for eliminating mesh dependence in finite element stress concentration areas provided in an embodiment of the present invention;

[0055] Figure 2 This is a block diagram of the post-processing system for eliminating mesh dependence in finite element stress concentration areas, provided in an embodiment of the present invention.

[0056] Figure 3 This is a calculation flowchart provided in an embodiment of the present invention;

[0057] Figure 4 This is a custom APDL button interface provided in the embodiments of the present invention;

[0058] Figure 5 The tensile plate with an elliptical hole provided in this embodiment of the invention (the ratio of the major axis to the minor axis is a / b=2, ρ=0.01375m).

[0059] Figure 6 This invention provides a mesh division and neighborhood distribution of stress concentration points in a tension plate with elliptical holes, as provided in an embodiment of the invention.

[0060] Figure 7 This is a comparison of the stress concentration factor at the elliptical aperture calculated by the displacement field integral reconstruction stress method of the present invention under different mesh sizes provided in the embodiments of the present invention;

[0061] Figure 8 This is a comparison of the stress concentration factor at the elliptical aperture calculated by the traditional finite element method under different mesh sizes provided in the embodiments of the present invention;

[0062] Figure 9 This invention provides a comparison of two methods for calculating the stress concentration factor Kt at the elliptical orifice as a function of the orifice's radius of curvature.

[0063] Figure 10 This invention provides a tensile thin plate with a central square hole (global Δ=0.016m, local Δ at the corner of the hole). k =1E-4m);

[0064] Figure 11 This invention provides a tension-bearing thin plate with a triangular hole in the center (global Δ=0.016m, local Δ at the corner of the hole). k =1E-4m);

[0065] Figure 12 This is a curve showing the variation of the stress concentration factor in the y-direction at the corner of a square aperture with the local mesh size, provided in an embodiment of the present invention.

[0066] Figure 13 This is a curve showing the variation of the stress concentration factor in the y-direction at the corner of a triangular aperture with the local mesh size, provided in an embodiment of the present invention. Detailed Implementation

[0067] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0068] like Figure 1 As shown, the post-processing method for eliminating mesh dependence in finite element stress concentration zones provided by this embodiment of the invention includes:

[0069] S101, obtain the displacement values ​​of each node in the neighborhood of the point of interest;

[0070] S102, the displacement values ​​of each node in the neighborhood are reconstructed by weighted integration to obtain the first derivative of the displacement field of the point of interest;

[0071] S103, Based on the constitutive relation, the stress at the point of interest is calculated from the first derivative of the displacement field.

[0072] The embodiment of the present invention provides a weighted integral reconstruction of the displacement values ​​of each node in the neighborhood to obtain the first derivative of the displacement field of the point of interest, specifically as follows:

[0073] The node positions are represented using three-dimensional Cartesian coordinates, and the displacement field is represented as three displacement components along three coordinate directions.

[0074] For any three-dimensional scalar function, the near-field dynamic differential operator is used to obtain the required derivative by weighted integral of the function values ​​at each point in the neighborhood of the point of interest with the corresponding specific weight function;

[0075] The first derivatives of the three displacement components with respect to the three spatial variables are taken as the first derivatives of the displacement field.

[0076] The specific weight function provided in this embodiment of the invention is a peri-field dynamics function corresponding to the required derivative order. The peri-field dynamics function is constructed based on the orthogonality principle and is expressed in polynomial form.

[0077] The weighting function in the polynomial adopts a Gaussian distribution;

[0078] The undetermined coefficients in the near-field dynamics function are obtained by solving the system of equations established by the orthogonality condition.

[0079] The present invention provides a method for calculating the stress at the point of interest based on constitutive relations and the first derivative of the displacement field. Specifically, it employs the generalized Hooke's law to calculate the three-dimensional stress state at the point of interest based on the elastic modulus and Poisson's ratio of the material.

[0080] This invention provides a post-processing implementation method based on finite element software, comprising the following steps:

[0081] Modeling, loading, and solving are completed using finite element software;

[0082] In the post-processing stage, parametric design language is used for secondary development;

[0083] In the software interface, there are function buttons for selecting key nodes in stress concentration areas and function buttons for recalculating stress.

[0084] In response to the node selection operation, all nodes in the neighborhood of the key node and their influence range are displayed;

[0085] Based on the coordinates and displacements of the nodes in the neighborhood, the post-processing method described in claim 1 is executed to recalculate the stress at the key node.

[0086] The neighborhood of a key node provided in this embodiment of the invention is defined as a square or cube-shaped region extending along the coordinate axis with the key node as the center.

[0087] The range of the neighborhood in each coordinate axis direction provided in the embodiments of the present invention is determined by subtracting the neighborhood scale from the center coordinates and adding the neighborhood scale to the center coordinates;

[0088] The neighborhood scale is determined by the average node spacing and the neighborhood scale coefficient;

[0089] The order of the displacement derivative is 1, and the neighborhood scaling factor is 3.

[0090] like Figure 2 As shown, an embodiment of the present invention provides a post-processing system for eliminating mesh dependence in finite element stress concentration zones, comprising:

[0091] The node data acquisition module is used to acquire the coordinates and displacement values ​​of each node in the neighborhood of the point of interest.

[0092] The derivative reconstruction module is used to reconstruct the displacement values ​​of each node in the neighborhood by weighted integration to obtain the first derivative of the displacement field of the point of interest.

[0093] The stress calculation module is used to calculate the stress at the point of interest based on the first derivative of the displacement field according to the constitutive relation.

[0094] The post-processing execution module is used to call the derivative reconstruction module and the stress calculation module during the post-processing stage of the finite element software to recalculate the stress at the stress concentration point.

[0095] The derivative reconstruction module provided in this embodiment of the invention uses a peri-field dynamic differential operator to obtain the derivative, and uses a peri-field dynamic function constructed based on the orthogonality principle as a specific weight function in the weighted integral;

[0096] The near-field dynamics function is expressed in polynomial form, and the weighting function in the polynomial adopts a Gaussian distribution. The undetermined coefficients in the near-field dynamics function are obtained by solving the system of equations established by the orthogonality condition.

[0097] The post-processing execution module provided in this embodiment of the invention is based on secondary development using the parametric design language of finite element software;

[0098] The system also includes a node selection and display module, which provides function buttons for selecting key nodes in stress concentration areas and function buttons for recalculating stress in the software interface, and displays all nodes in the neighborhood of the key node and their influence range after the key node is selected.

[0099] The neighborhood of the key node is a square or cube-shaped region centered on the key node and extending along the coordinate axis, with a displacement derivative of order 1 and a neighborhood scale factor of 3.

[0100] Figure 3 This is a calculation flowchart provided in an embodiment of the present invention;

[0101] Figure 4 This is a custom APDL button interface provided in the embodiments of the present invention;

[0102] This invention provides a post-processing method for eliminating mesh dependence in finite element stress concentration zones, the method comprising:

[0103] S1: Perform a weighted integral on the displacement values ​​of all nodes in the neighborhood of the point of interest to reconstruct the first derivative of the displacement field at that point;

[0104] S2: Calculate the stress at this point using the displacement derivative based on the constitutive relation;

[0105] S3: Post-processing implementation based on finite element software.

[0106] S1 specifically includes:

[0107] Let the displacement field be ,in Let be the position vector of a node, and be the displacement vector in three-dimensional Cartesian space. It contains three displacement components, denoted as follows: , and To facilitate consistent description, index symbols are introduced: x1, x2, and x3 represent the x, y, and z coordinates, respectively. Add subscript , , Let x, y, and z represent the components of the displacement, respectively; thus, the first derivative of the displacement, for example... , and , can be uniformly represented as , and Similarly, other derivatives can be expressed in the same way.

[0108] Based on the peri-field dynamical differential operator PDDO proposed by Madenci, for any three-dimensional scalar function Its derivatives up to order N can be obtained by weighted integration of the function values ​​at all points in its neighborhood using a specific weight function, namely the PD function:

[0109] (1)

[0110] in, Point The near-field neighborhood, For point Any point within the near-field neighborhood of , Let be the relative position vectors of the two. Let x1, x2, and x3 be the orders of their partial derivatives with respect to the independent variables x1, x2, and x3, respectively. ,and , ; The required derivative order The corresponding PD function is at the point The function constructed based on the orthogonality principle within the neighborhood of the function is also the core of the derivative calculation method, and it must satisfy the following conditions:

[0111] (2)

[0112] in, , The Kroneck symbol, i.e., when hour, ; and when hour, Equation (2) in matrix form can be expressed as:

[0113] (3)

[0114] The PD function constructed in this way is usually expressed in polynomial form as follows:

[0115] (4)

[0116] In the formula, And satisfy ; For undetermined coefficients, Is with The weighting function related to the terms represents the degree of interaction. This invention uses a Gaussian distribution, making all... ;

[0117] The matrix expression of equation (4) is as follows:

[0118] (5)

[0119] Substituting equation (5) into equation (3) yields:

[0120] (6)

[0121] Equation (6) can be simplified to

[0122] (7)

[0123] In the formula The shape coefficient matrix, The vector of undetermined coefficients. Let a known vector be constructed using orthogonality;

[0124] Multiplying equation (7) by the inverse of the shape coefficient matrix on the left yields the vector of undetermined coefficients, which in turn allows us to obtain the PD function, thus enabling us to express the function. partial derivatives ;

[0125] (8)

[0126] Therefore, the displacement For spatial variables first derivative In this regard, we can obtain from equation (1)

[0127] (9)

[0128] In the formula, the PD function Equation (5) can be expressed as

[0129] (10)

[0130] In equation (10), the PD function is used to represent... Undetermined coefficients The system of equations formed by equation (6) can be solved, i.e.

[0131] (11)

[0132] Equation (11) can be simplified to

[0133] (12)

[0134] The PD function can be obtained by solving the system of equations (12). All undetermined coefficients Thus, by integrating the displacement values ​​of all points in its neighborhood using equation (9), the displacement can be obtained. For spatial variables first derivative ;

[0135] Similarly, by integrating the displacement values ​​of all points in the neighborhood using a specific weighting function, three displacement components can be obtained. For three spatial variables first derivative .

[0136] S2 specifically includes:

[0137] The inherent constitutive relations of materials serve as a physical bridge connecting deformation and stress; based on this, the internal stress of an object can be calculated from the displacement derivative describing deformation; specifically in engineering applications, the generalized Hooke's law is the most commonly used core equation for solving such three-dimensional elastic stress problems.

[0138] (13)

[0139] In the formula, E is the elastic modulus of the material. Given its Poisson's ratio, the three-dimensional stress state at the stress concentration point can be obtained by using the displacement field integral stress reconstruction method.

[0140] S3 specifically includes:

[0141] Based on the standard analysis workflow of the finite element software ANSYS, including modeling, loading, and solving, secondary development was performed using the software's built-in parametric design language APDL in the post-processing stage. Two new function buttons, "HORIZON" and "RE-STRESS," were added to the "Toolbar" of the ANSYS interface. The specific implementation steps are as follows: First, based on the analysis requirements, click the "HORIZON" button in the stress concentration area to select the node requiring focus. The system then displays all nodes within the critical node's region and their influence range. Next, click the "RE-STRESS" button. The program will then recalculate the stress at the stress concentration point using the displacement field integral stress reconstruction method based on the coordinates and displacement results of the nodes within that region.

[0142] The neighborhood of a key point is defined as a square or cube region centered at that point and extending along the coordinate axes, i.e., x∈[x0-m]. , x0+m The same applies to the y and z directions; here, The average node spacing is given by the neighborhood scale coefficient m, which typically ranges from N ≤ m ≤ N + 2, where N is the order of the displacement derivative, i.e., N = 1. To ensure computational efficiency and sufficient material points in the neighborhood, m = N + 2 = 3 is preferred.

[0143] (1) Constructing the structural geometric model

[0144] In finite element software, a corresponding geometric model is established for structural components containing holes, material parameters (including elastic modulus, Poisson's ratio, etc.) are set, and displacement and load boundary conditions are applied.

[0145] (2) Mesh generation and displacement field solution

[0146] By selecting appropriate structural analysis element types, different global or local grid sizes such as coarse grid, dense grid, and ultra-dense grid are used to discretize the structure, and then the displacement field distribution under different grid scales is obtained by solving them respectively.

[0147] (3) Determine the area of ​​stress concentration points

[0148] Set the neighborhood radius parameter. Based on the finite element stress analysis results, focus on the orifice location with the highest stress concentration. By clicking the "HORIZON" button in the custom toolbar, select the node that needs to be focused on. The system will automatically identify and display all nodes in the neighborhood of the key node and their influence range according to the preset neighborhood size.

[0149] (4) Calculation of stress based on displacement field integral reconstruction stress method

[0150] Click the “RE-STRESS” button in the custom toolbar, set the weight function form, perform a weighted integral of the displacement field in the neighborhood of the stress concentration point, obtain the first spatial derivative of the displacement, and then calculate the stress at that point based on the constitutive relation.

[0151] (5) Compare and analyze the stress calculation results of the two methods.

[0152] The stress results obtained by the displacement field integral reconstruction stress method are compared with the stress values ​​obtained by the finite element method under different mesh densities. Combined with analytical solutions or theoretical reference values, the solution accuracy and computational stability of the two methods in typical stress concentration regions are systematically evaluated.

[0153] (6) Verify the numerical stability of stress calculation in the high stress concentration zone at the orifice.

[0154] At the orifice where the stress concentration is highest, observe whether the stress result calculated by the method of this invention remains stable as the mesh is refined, and compare it with the corresponding result of the finite element method to verify its mesh insensitivity characteristics.

[0155] Example 1: Stress Concentration Factor Analysis of a Tension Plate with an Elliptical Hole

[0156] Step (1) A two-dimensional plane stress finite element model with an elliptical hole in the center is constructed using the finite element software ANSYS. The plate is a 1.0m × 1.0m square thin plate with an elliptical hole in the center. The major axis 2a of the ellipse is fixed at 0.11m, and the radius of curvature ρ=b at the endpoint of the major axis is controlled by changing the length of the minor axis 2b. 2 / a, Figure 5 This is a schematic diagram of a tension plate when the ratio of its major and minor axes, a / b, is 2. Material parameters are: elastic modulus 70 GPa, Poisson's ratio 0.3, and the plate is subjected to a uniform tensile load p0 = 1.0 N / m at its upper and lower boundaries. 2 .

[0157] Figure 6 A tension plate with an elliptical hole (major-minor axis ratio a / b=2, ρ= 0.01375m)

[0158] Step (2) Discretize using PLANE42 quadrilateral elements. Solve using three different global mesh sizes: coarse mesh (Δ=0.016m), dense mesh (Δ=0.008m), and extremely dense mesh (Δ=0.004m) to compare stress solutions under different mesh densities.

[0159] Step (3) After the ANSYS conventional solution is completed, click the "HORIZON" button in the ANSYS toolbar, select the stress concentration critical node at the elliptical aperture, and the system will then display all nodes within the range of that node and their influence areas, such as... Figure 6As shown. The neighborhood radius is set to δ=3Δ to ensure that the stress concentration point will be supported by sufficient family nodes. Then, click the "RE-STRESS" button, and the program will automatically calculate the stress value at that point using the displacement field integral stress reconstruction method based on the displacement solutions and coordinate information of the nodes in the neighborhood.

[0160] Step (4) Calculate the y-direction stress σ at the endpoint of the major axis for elliptical holes with different radii of curvature ρ. max Then calculate the stress concentration factor. .

[0161] Step (5) Combine the calculation results of the two methods with the analytical solution ( (Compare)

[0162] like Figure 6 As shown, with a selected mesh size m=3, the method of this invention has an average of 345 nodes involved in the stress calculation at the stress concentration point at the orifice under three different mesh sizes. In contrast, the traditional finite element method typically relies only on the two elements containing that point and its six internal nodes, a difference of approximately 60 times. The introduction of a large amount of node information greatly enriches the original data foundation for stress analysis and effectively suppresses the uncertainty caused by insufficient computational information in traditional methods. Further analysis is as follows... Figure 7 and Figure 8 As shown, when the radius of curvature ρ of the orifice gradually decreases from 0.055m to 0.006m, the stress gradient around the orifice increases significantly. During this process, the stress concentration factor calculated by traditional FEM exhibits extremely strong mesh dependence, and its overall fluctuation range continues to expand with increasing mesh refinement, reaching a maximum amplitude of nearly 30%. Taking the most extreme case of ρ=0.006m as an example, the K0 of the FEM... t The value increased from 5.20 in the coarser grid (Δ=0.016m) to 6.75 in the extremely dense grid (Δ=0.004m). The calculation results remained stable throughout the entire process of decreasing radius of curvature, demonstrating strong robustness to local numerical fluctuations. Even at the minimum radius of curvature at the orifice (ρ=0.006m), the variation was minimal.

[0163] Figure 7 Comparison of stress concentration factors at elliptical orifices calculated by the displacement field integral reconstruction stress method of the present invention under different grid sizes; Figure 1 Comparison of stress concentration factor at elliptical aperture calculated by traditional finite element method under different mesh sizes.

[0164] Figure 7The results of calculating the stress concentration factor at the elliptical orifice as a function of the radius of curvature using the displacement field integral reconstruction method under different mesh sizes are presented. It can be seen that under three mesh scales with Δ of 0.016, 0.008, and 0.004 m, the curves basically overlap and are highly consistent with the analytical solution results. This indicates that the displacement derivative obtained by weighted integral reconstruction of the displacement values ​​of neighboring nodes has good stability and consistency, and can effectively suppress numerical fluctuations caused by differences in mesh division. Especially in the region of strong stress concentration with a small radius of curvature, the curves still maintain good fit, indicating that the method remains reliable in regions with large stress gradients.

[0165] Figure 8 The results of calculations using the traditional finite element method at different mesh sizes are presented. It can be seen that when the mesh is coarse, the calculated stress concentration factor is significantly lower than the analytical solution; as the mesh becomes finer, the results gradually approach the analytical solution, but significant differences still exist between different meshes, exhibiting typical mesh dependence. This difference is particularly pronounced in regions with small radii of curvature.

[0166] By comparison Figure 7 and Figure 8 It is clear that this invention reconstructs stress by integrating displacement fields, transforming stress calculation from local element values ​​to a comprehensive expression of neighborhood information. This makes the stress results insensitive to changes in grid scale and can stably approximate analytical solutions under different grid conditions, significantly improving the reliability and consistency of calculation results in stress concentration regions.

[0167] Figure 9 The stress calculation method based on coarse-grid displacement field reconstruction of this invention was further compared with the calculation results of traditional finite element methods at stress concentration points in elliptical orifices under three different grid sizes. Throughout the entire range of curvature radius variation, this method exhibits excellent accuracy and robustness. Specifically, when the stress concentration is low (large orifice curvature radius), the results obtained by this method are highly consistent with the FEM solutions and analytical solutions under the three grid sizes. However, as the orifice curvature radius decreases and stress concentration intensifies, the coarse-grid solution (Δ=0.016m) of the traditional FEM gradually deviates from the analytical solution. While the dense grid (Δ=0.008m) shows improvement, it still requires refining to an extremely dense grid (Δ=0.004m) to obtain a convergent and stable solution. It is worth emphasizing that even when calculated based on a coarse-grid displacement field, the stress results of this method are consistently highly consistent with the FEM solution under an extremely dense grid and continuously approach the theoretical analytical solution. This indicates that the method of this invention can stably capture high-gradient stress fields without mesh refinement, fully verifying its engineering practical value in high-precision stress analysis.

[0168] Example 2: Stress concentration factor analysis of a thin plate with a square or equilateral triangular hole in the center under tension.

[0169] Step (1) Using the finite element software ANSYS, two two-dimensional plane stress finite element models with high stress concentration were established. Both models were 1.0m × 1.0m square thin plates, with a square hole and an equilateral triangular hole respectively opened at the center of the plate, as shown below. Figure 10 (a) and Figure 11 As shown in (a), the material parameters are uniformly set as follows: elastic modulus E = 70 GPa, Poisson's ratio ν = 0.3, and a uniform tensile load p0 = 1.0 N / m is applied at the upper and lower boundaries of the plate. 2 To ensure geometric comparability, the circumcircle radius of the square hole and the equilateral triangular hole is kept consistent and set to 0.055m.

[0170] Step (2) For both square and triangular hole models, PLANE42 triangular elements are used uniformly, and the global mesh size is uniformly set to Δ=0.016m. To study the influence of local mesh refinement in the hole opening area on the stress solution, the software's built-in KESIZE function is used to gradually refine the mesh near the hole opening. The element size Δ near the hole opening is... k The initial value was 1E-3m, then 1E-4m, until it was reduced to 1E-4m. In addition, more extreme element sizes of 2E-5m and 2E-6m were also considered to comprehensively evaluate the impact of mesh refinement on stress calculation results.

[0171] Step (3): After setting the mesh size and completing the ANSYS solution, click the "HORIZON" button in the ANSYS toolbar, select the orifice corner point, and the system will then display all nodes within the area of ​​that corner point and their influence range, such as... Figure 10 (b) and Figure 11 As shown in (b).

[0172] Figure 10 A thin sheet under tension with a square hole in the center (global Δ = 0.016 m, local Δ at the corner of the hole). k =1E-4m)

[0173] Figure 2 A thin sheet under tension with a triangular hole in the center (global Δ = 0.016 m, local Δ at the corner of the hole). k =1E-4m)

[0174] Step (4): The neighborhood radius is still taken as δ=3Δ. Then, click the "RE-STRESS" button in the toolbar. The program will automatically calculate the stress value at the point based on the displacement solution and coordinate information of the nodes in the neighborhood, using the displacement field integral stress reconstruction method. Thus, the stress σ at the corners of the square and triangular orifices is calculated respectively. y and the corresponding stress concentration factor K t =σ y / p0.

[0175] Step (5) Compare and analyze the stress results calculated by the method of the present invention with the stress results of the traditional finite element method.

[0176] Figure 12 and Figure 13 The results show that as the local mesh size at the orifice corner is gradually refined, the stress concentration factor K in the y-direction obtained by the traditional finite element method at that point decreases. t It exhibits an upward trend with a continuously increasing slope, demonstrating significant numerical instability. Specifically, for the square hole model, when the local mesh size decreases, K... t The value jumped from 5.62 to 158; for the triangular hole model, K t The value increased from 7.11 to 132. This drastic numerical fluctuation reflects the limitations of the traditional finite element method in handling regions with high local stress gradients. In contrast, the displacement field integral reconstructed stress method exhibits excellent numerical stability during mesh refinement. Specifically, the K value for the square hole model... t The mean is 6.08, and the variance is only 0.024; K for the triangular hole model t The mean is 5.89 and the variance is 0.053. It is worth noting that although the number of nodes within a 3Δ distance from the orifice increases from 66 to 161 after the local mesh size is reduced, and the local discretization density is significantly improved, the stress solution obtained by the displacement field integral reconstruction stress method does not fluctuate significantly as a result.

[0177] The above results demonstrate that the displacement field integral reconstruction stress method, based on the traditional finite element displacement solution, reconstructs the first derivative field of the displacement by introducing a nonlocal displacement integral, thereby completing the stress calculation. This method effectively overcomes the mesh dependency problem of the traditional finite element method in stress concentration regions. It not only provides stable numerical solutions but also achieves a reasonable physical estimate of the stress concentration at the orifice, thus providing a reliable computational approach for the strength analysis of perforated structural members.

[0178] Example 1 illustrates the tensile analysis of a two-dimensional plate with a circular hole. First, modeling, loading, and solving are completed. Then, a stress concentration point is selected at the edge of the hole. Using the displacement values ​​of the surrounding nodes as input, a weighted integration is performed using a near-field dynamic differential operator to reconstruct the first derivative of the displacement at that point. Finally, the stress is calculated according to the generalized Hooke's law. This example demonstrates the completeness and feasibility of the basic process and shows that it can replace conventional mesh-sensitive post-processing results at typical stress concentration locations.

[0179] Example 2 presents a three-dimensional elastic analysis of a beam with a notch. Key points at the root of the notch are selected, a cubic neighborhood is established, and the coordinates and displacements of all nodes within the neighborhood are retrieved. The first derivatives of the three displacement components with respect to the three spatial variables are obtained using a polynomial near-field dynamics function with Gaussian weights, and then the three-dimensional stress state is calculated. This example demonstrates that the solution also applies in three-dimensional cases and is not limited to planar problems.

[0180] Example 3 presents a comparative analysis of multiple meshes for the same model. For the same perforated or notched structure, coarse, medium, and fine meshes were created. Under the same load and boundary conditions, the stress calculation for the same point of interest was repeatedly performed using integral reconstruction, and the results were compared with the stress values ​​obtained from direct post-processing in the software. This example demonstrates that the proposed scheme exhibits better stability in the face of mesh variations, directly addressing the technical objective of "eliminating mesh dependence in stress concentration areas."

[0181] Example 4 is a graphical implementation of post-processing. Based on the finite element software's post-processing workflow, a node selection and stress recalculation function is set in the toolbar. The operator first selects a key point, then displays its neighboring nodes and influence range, and then executes the recalculation program to output the stress at that point. This example does not change the core mechanism, but rather strengthens the implementation path of this solution in engineering software.

[0182] Example 5 compares the different neighborhood scale values. Different neighborhood scale coefficients are used to calculate the same key point, with a focus on demonstrating a good balance between computational efficiency and sufficiency of neighborhood information when the displacement derivative order is 1 and the neighborhood scale coefficient is 3. This example is still based on the original technical mechanism, but helps to further highlight the rationality of the parameter settings and its engineering applicability.

[0183] It should be noted that embodiments of the present invention can be implemented in hardware, software, or a combination of both. The hardware portion can be implemented using dedicated logic; the software portion can be stored in memory and executed by a suitable instruction execution system, such as a microprocessor or dedicated-design hardware. Those skilled in the art will understand that the above-described devices and methods can be implemented using computer-executable instructions and / or included in processor control code, for example, such code provided on a carrier medium such as a disk, CD, or DVD-ROM, a programmable memory such as read-only memory (firmware), or a data carrier such as an optical or electronic signal carrier. The devices and modules of the present invention can be implemented by hardware circuitry such as very large-scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, or programmable hardware devices such as field-programmable gate arrays, programmable logic devices, etc., or by software executed by various types of processors, or by a combination of the above-described hardware circuitry and software, such as firmware.

[0184] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications, equivalent substitutions, and improvements made by those skilled in the art within the scope of the technology disclosed in the present invention, and within the spirit and principles of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A post-processing method for eliminating mesh dependence in finite element stress concentration zones, characterized in that, Includes the following steps: Step 1: Obtain the displacement values ​​of each node within the neighborhood of the point of interest; Step 2: Reconstruct the displacement values ​​of each node in the neighborhood by weighted integration to obtain the first derivative of the displacement field of the point of interest; Step 3: Calculate the stress at the point of interest using the first derivative of the displacement field based on the constitutive relation.

2. The post-processing method according to claim 1, characterized in that, The first derivative of the displacement field of the point of interest is obtained by reconstructing the displacement values ​​of each node in the neighborhood by weighted integration, specifically as follows: The node positions are represented using three-dimensional Cartesian coordinates, and the displacement field is represented as three displacement components along three coordinate directions. For any three-dimensional scalar function, the near-field dynamic differential operator is used to obtain the required derivative by weighted integral of the function values ​​at each point in the neighborhood of the point of interest with the corresponding specific weight function; The first derivatives of the three displacement components with respect to the three spatial variables are taken as the first derivatives of the displacement field.

3. The post-processing method according to claim 2, characterized in that, The specific weight function is a peri-field dynamics function corresponding to the required derivative order. The peri-field dynamics function is constructed based on the orthogonality principle and is expressed in polynomial form. The weighting function in the polynomial adopts a Gaussian distribution; The undetermined coefficients in the near-field dynamics function are obtained by solving the system of equations established by the orthogonality condition.

4. The post-processing method according to claim 1, characterized in that, The stress at the point of interest is calculated based on the constitutive relation using the first derivative of the displacement field. Specifically, the generalized Hooke's law is used to calculate the three-dimensional stress state at the point of interest based on the elastic modulus and Poisson's ratio of the material.

5. A post-processing implementation method based on finite element software to implement the post-processing method according to any one of claims 1 to 4, characterized in that, Includes the following steps: Modeling, loading, and solving are completed using finite element software; In the post-processing stage, parametric design language is used for secondary development; In the software interface, there are function buttons for selecting key nodes in stress concentration areas and function buttons for recalculating stress. In response to the node selection operation, all nodes in the neighborhood of the key node and their influence range are displayed; Based on the coordinates and displacements of the nodes in the neighborhood, the post-processing method described in claim 1 is executed to recalculate the stress at the key node.

6. The post-processing implementation method according to claim 5, characterized in that, The neighborhood of the key node is defined as a square or cube-shaped region centered on the key node and extending along the coordinate axis.

7. The post-processing method according to claim 6, characterized in that, The extent of the neighborhood in each coordinate axis direction is determined by subtracting the neighborhood scale from the center coordinates and adding the neighborhood scale to the center coordinates; The neighborhood scale is determined by the average node spacing and the neighborhood scale coefficient; The order of the displacement derivative is 1, and the neighborhood scaling factor is 3.

8. A post-processing system for eliminating mesh dependence of finite element stress concentration zones for implementing the post-processing method according to any one of claims 1 to 4 or the post-processing implementation method according to any one of claims 5 to 7, characterized in that, include: The node data acquisition module is used to acquire the coordinates and displacement values ​​of each node in the neighborhood of the point of interest. The derivative reconstruction module is used to reconstruct the displacement values ​​of each node in the neighborhood by weighted integration to obtain the first derivative of the displacement field of the point of interest. The stress calculation module is used to calculate the stress at the point of interest based on the first derivative of the displacement field according to the constitutive relation. The post-processing execution module is used to call the derivative reconstruction module and the stress calculation module during the post-processing stage of the finite element software to recalculate the stress at the stress concentration point.

9. The post-processing system for eliminating mesh dependence in finite element stress concentration zones according to claim 8, characterized in that, The derivative reconstruction module uses a peri-field dynamic differential operator to obtain the derivative and uses a peri-field dynamic function constructed based on the orthogonality principle as a specific weight function in the weighted integral. The near-field dynamics function is expressed in polynomial form, and the weighting function in the polynomial adopts a Gaussian distribution. The undetermined coefficients in the near-field dynamics function are obtained by solving the system of equations established by the orthogonality condition.

10. The post-processing system for eliminating mesh dependence in finite element stress concentration zones according to claim 8, characterized in that, The post-processing execution module is developed based on the parametric design language of finite element software. The system also includes a node selection and display module, which provides function buttons for selecting key nodes in stress concentration areas and function buttons for recalculating stress in the software interface, and displays all nodes in the neighborhood of the key node and their influence range after the key node is selected. The neighborhood of the key node is a square or cube-shaped region centered on the key node and extending along the coordinate axis, with a displacement derivative of order 1 and a neighborhood scale factor of 3.

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