A thickness and shear locking elimination method of a full-parameter absolute nodal coordinate beam element
Patent Information
- Application Number
- CN202611011149.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-07-08
- Publication Date
- 2026-09-25
AI Technical Summary
[0006]为解决现有技术中存在全参数绝对节点坐标法梁单元在弯曲变形分析中容易产生曲率厚度闭锁和剪切闭锁，且现有闭锁缓解方式通常会增加额外自由度或降低积分精度，本发明提供的技术方案为：
基于二维全参数ANCF梁单元对应变场分布进行分析，将横向应变和剪切应变放在同一弯曲变形框架下考察，使闭锁问题能够对应到具体应变分量的异常来源。由此可以明确曲率厚度闭锁来源于横向梯度与弯曲变形的耦合，剪切闭锁来源于轴向梯度与横向梯度插值阶数不匹配，为后续针对性修正提供依据，避免现有方案中仅凭经验减缩积分或增加自由度的处理方式。
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Figure CN122818804A_ABST
Abstract
Description
Technical Field
[0001] This research belongs to the field of dynamics and finite element analysis technology for multi-flexible body systems, specifically involving a method for eliminating beam element thickness and shear locking using the all-parameter absolute nodal coordinate method (ANCF). Background Technology
[0002] The absolute nodal coordinate method, a commonly used modeling method in the dynamics and finite element analysis of flexible multibody systems, can describe the large deformation and rotational behavior of flexible structures such as beams and shells through the absolute coordinates and gradient coordinates of the nodes. Therefore, it is often used in the dynamic simulation of flexible mechanisms, aerospace deployment structures, flexible robots, and complex multibody systems. Fully parametric ANCF beam elements typically establish a displacement field using shape function matrices and nodal coordinate vectors, and further derive axial strain, transverse strain, shear strain, curvature, strain energy, and generalized elastic force, thereby describing the deformation state and mechanical response of the beam element.
[0003] However, in the bending deformation analysis of fully parametric ANCF beam elements, the locking phenomenon still affects the computational accuracy and convergence efficiency. On the one hand, the transverse gradient vector is prone to coupling with the bending deformation of the beam, causing unrealistic transverse strain inside the element, resulting in unreasonable compression of the cross section in the numerical model and forming curvature thickness locking. On the other hand, when the interpolation orders of the axial gradient and the transverse gradient do not match, unrealistic shear strain distribution is easily introduced, causing the shear strain energy to be overestimated, which in turn leads to an overestimation of element stiffness, an underestimation of displacement prediction, and a decrease in convergence speed, resulting in shear thickness locking.
[0004] Existing approaches to the locking problem in ANCF beam elements typically focus on a specific type of locking phenomenon or mitigate it by reducing integrals, introducing additional degrees of freedom, or changing the element form. While these methods can improve the locking problem to some extent, they may lead to decreased integration accuracy, increased computational scale, and compromised element continuity. They are insufficient to effectively eliminate curvature-thickness locking and shear locking while maintaining the original characteristics of fully parametric beam elements.
[0005] In summary, existing technologies for beam elements with full-parameter absolute nodal coordinates are prone to curvature-thickness locking and shear locking in bending deformation analysis, and existing locking mitigation methods usually increase additional degrees of freedom or reduce integration accuracy. Summary of the Invention
[0006] To address the issue that existing technologies using the full-parameter absolute nodal coordinate method for beam elements are prone to curvature-thickness locking and shear locking in bending deformation analysis, and that existing locking mitigation methods typically increase additional degrees of freedom or reduce integration accuracy, the technical solution provided by this invention is as follows: A method for eliminating thickness and shear locking in a beam element with fully parametric absolute nodal coordinates, comprising: The steps for constructing a fully parametric ANCF beam element and generating the displacement field of the fully parametric ANCF beam element; The steps are as follows: obtaining the axial gradient vector and the transverse gradient vector based on the displacement field, and generating the axial strain, transverse strain and shear strain. The steps for determining the curvature thickness lock-in source and the shear lock-in source based on the distribution relationship between the transverse strain and the shear strain; The steps involve improving the interpolation method of the transverse gradient vector to generate corrected transverse strain and corrected transverse generalized elastic force. The step of adjusting the interpolation order of the axial gradient vector used to calculate the shear strain to generate the corrected shear strain and the corrected shear generalized elastic force; The steps for generating the generalized elastic force of the beam element after locking is eliminated, based on the modified transverse generalized elastic force and the modified shear generalized elastic force.
[0007] Furthermore, in a preferred embodiment, when constructing a fully parametric ANCF beam element, a displacement field is generated based on the nodal coordinate vectors and shape function relationships of the fully parametric ANCF beam element, and the mass matrix of the fully parametric ANCF beam element is determined by the displacement field.
[0008] Furthermore, in a preferred embodiment, when generating axial strain, transverse strain, and shear strain, the elastic line method is used to derive axial strain, transverse strain, shear strain, and curvature based on the displacement gradient of the fully parametric ANCF beam element, and the strain energy and generalized elastic force are determined according to the axial strain, the transverse strain, the shear strain, and the curvature.
[0009] Furthermore, in a preferred embodiment, when determining the sources of curvature thickness locking and shear locking, an analysis is performed based on the distribution of transverse strain, shear strain, and axial strain under different bending deformations. It is determined that curvature thickness locking originates from the coupling between the transverse gradient and bending deformation, while shear locking originates from the mismatch in the interpolation order of the axial gradient vector and the transverse gradient vector.
[0010] Furthermore, in a preferred embodiment, when improving the interpolation method of the transverse gradient vector, only the magnitude of the transverse gradient vector at the node is interpolated while its directionality is ignored, and the modified transverse strain and modified transverse generalized elastic force are determined based on the improved transverse gradient vector.
[0011] Furthermore, in a preferred embodiment, when adjusting the interpolation order of the axial gradient vector used to calculate the shear strain, the interpolation of the axial gradient vector is adjusted to a linear term, and a modified shear strain and a modified shear generalized elastic force are generated based on the redefined axial gradient.
[0012] A thickness and shear locking elimination device for a beam element with full-parameter absolute nodal coordinates, comprising: A module for constructing a fully parametric ANCF beam element and generating the displacement field of the fully parametric ANCF beam element; A module that obtains axial gradient vector and transverse gradient vector based on the displacement field, and generates axial strain, transverse strain and shear strain; Based on the distribution relationship between the transverse strain and the shear strain, determine the modules for the curvature thickness locking source and the shear locking source; The interpolation method of the transverse gradient vector is improved to generate modules that correct transverse strain and correct transverse generalized elastic force; The interpolation order of the axial gradient vector used to calculate shear strain is adjusted to generate a module that corrects the shear strain and the generalized shear force. A module for generating the generalized elastic force of beam elements after locking is eliminated, based on the modified transverse generalized elastic force and the modified shear generalized elastic force.
[0013] A computer storage medium for storing a computer program, which, when read by the computer, is executed by the computer using the method described thereon.
[0014] A computer, including a processor and a storage medium, executes the method when the processor reads a computer program stored in the storage medium.
[0015] A computer program product, which, as a computer program, implements the method when the computer program is executed.
[0016] Compared with the prior art, the advantages of the technical solution provided by the present invention are as follows: The strain field distribution is analyzed using two-dimensional fully parametric ANCF beam elements. Transverse strain and shear strain are examined within the same bending deformation framework, allowing the locking problem to be mapped to the specific source of anomalies in the strain components. This clarifies that curvature-thickness locking originates from the coupling between the transverse gradient and bending deformation, while shear locking stems from the mismatch in interpolation orders between the axial and transverse gradients. This provides a basis for subsequent targeted corrections, avoiding the current approach of simply reducing integrals or increasing degrees of freedom based on experience.
[0017] Using a beam element with full-parameter absolute nodal coordinates as the basic model, and establishing the displacement field through the shape function matrix and nodal coordinate vectors, the locking elimination process retains the modeling foundation of the ANCF method, making it suitable for large deformation and large rotation analysis. This approach does not require changing the element type or modifying the degrees of freedom, which helps maintain the advantages of a constant mass matrix, element continuity, and full-parameter description.
[0018] To address the transverse gradient vector, an interpolation method is adopted that only the magnitude of the transverse gradient vector at the nodes is performed while ignoring the direction. This prevents the transverse gradient from exhibiting unrealistic changes due to the bending of the beam centerline. This treatment can suppress the generation of pseudo-transverse strain within the element during bending, avoid unreasonable compression of the cross-section in the numerical model, thereby alleviating the problem of curvature thickness locking and reducing the abnormal increase in transverse strain energy.
[0019] The improved transverse gradient vector continues to be used for the calculation of transverse strain and the generalized elastic force associated with transverse strain, allowing the transverse gradient correction results to be directly transferred to the element stiffness and strain energy solutions. This treatment does not simply change the interpolation form, but rather reduces the influence of pseudo-transverse strain on the elastic force, thereby improving the accuracy of beam element bending configuration prediction and strain energy convergence.
[0020] To address the shear strain calculation, the interpolation order of the axial gradient vector is adjusted to be linear, ensuring order compatibility between the axial and transverse gradients in the shear strain construction. This approach reduces spurious shear strain caused by mismatch between higher-order axial and transverse gradients, preventing overestimation of shear strain energy and thus mitigating the problems of excessive element stiffness and underestimation of displacement caused by shear locking.
[0021] By defining the axial gradient to generate shear strain and the generalized elastic force associated with shear strain, the correction to shear locking can be incorporated into the calculation of shear strain energy and elastic force. This treatment reduces the unrealistic components in shear energy, making the stiffness of beam elements in bending analysis closer to the actual stress state and improving the model convergence speed.
[0022] Transverse gradient magnitude interpolation and axial gradient linear interpolation correspond to different methods for eliminating curvature-thickness locking and shear locking, respectively, enabling the proposed scheme to simultaneously handle pseudo-transverse strain and pseudo-shear strain. Compared to existing schemes that only address a single locking problem, this combination can simultaneously improve the numerical representation of thickness deformation and shear deformation, thereby enhancing the overall accuracy of the fully parametric ANCF beam element in bending analysis.
[0023] Lockout elimination is achieved without adding extra degrees of freedom, reducing integration accuracy, or preserving the original strain expression. This approach avoids the problems of increased computational scale, reduced integration accuracy, or impact on element continuity found in existing technologies. This method balances computational efficiency, numerical stability, and physical accuracy, making it suitable for large deformation simulation analysis of beam structures in flexible multibody systems.
[0024] It is applicable to large deformation modeling, locking elimination, and dynamic simulation analysis of fully parametric ANCF beam elements in flexible multibody systems. Attached Figure Description
[0025] Figure 1This is a flowchart of the thickness and shear locking elimination method for beam elements with full parameter absolute nodal coordinates; Figure 2 This is a schematic diagram of the configuration curve of a beam element with full-parameter absolute nodal coordinates; Figure 3 This is a schematic diagram of the transverse strain curve of a beam element with full-parameter absolute nodal coordinates; Figure 4 This is a schematic diagram of the shear strain curve of a beam element with full-parameter absolute nodal coordinates; Figure 5 This is a schematic diagram of the lateral gradient distribution of a beam element with full-parameter absolute nodal coordinates; Figure 6 This is a schematic diagram of a beam subjected to pure bending moment with all-parameter absolute nodal coordinates; Figure 7 It is the configuration curve of the original beam model under pure bending moment. Figure 8 This is the configuration curve of the T-LA beam model under pure bending moment. Figure 9 This is the configuration curve of the TS-LA beam model under pure bending moment. Figure 10 It is the strain energy convergence curve of the original beam model under pure bending moment. Figure 11 This is the strain energy convergence curve of the T-LA beam model under pure bending moment. Figure 12 This is the strain energy convergence curve of the T-LA beam model under pure bending moment. Detailed Implementation
[0026] To make the advantages and benefits of the technical solution provided by the present invention clearer, the technical solution provided by the present invention will now be described in further detail with reference to the accompanying drawings, specifically: Implementation Method 1: This implementation method provides a method for eliminating thickness and shear locking in a beam element with full-parameter absolute nodal coordinates, including: The steps for constructing a fully parametric ANCF beam element and generating the displacement field of the fully parametric ANCF beam element; The steps are as follows: obtaining the axial gradient vector and the transverse gradient vector based on the displacement field, and generating the axial strain, transverse strain and shear strain. The steps for determining the curvature thickness lock-in source and the shear lock-in source based on the distribution relationship between the transverse strain and the shear strain; The steps involve improving the interpolation method of the transverse gradient vector to generate corrected transverse strain and corrected transverse generalized elastic force. The step of adjusting the interpolation order of the axial gradient vector used to calculate the shear strain to generate the corrected shear strain and the corrected shear generalized elastic force; The steps for generating the generalized elastic force of the beam element after locking is eliminated, based on the modified transverse generalized elastic force and the modified shear generalized elastic force.
[0027] When constructing a fully parametric ANCF beam element, a displacement field is generated based on the nodal coordinate vectors and shape function relationships of the fully parametric ANCF beam element, and the mass matrix of the fully parametric ANCF beam element is determined by the displacement field.
[0028] When generating axial strain, transverse strain, and shear strain, the elastic line method is used to derive the axial strain, transverse strain, shear strain, and curvature based on the displacement gradient of the fully parametric ANCF beam element, and the strain energy and generalized elastic force are determined according to the axial strain, the transverse strain, the shear strain, and the curvature.
[0029] When determining the sources of curvature thickness locking and shear locking, the analysis is based on the distribution of transverse strain, shear strain and axial strain under different bending deformations. It is determined that curvature thickness locking originates from the coupling between transverse gradient and bending deformation, while shear locking originates from the mismatch of the interpolation order of the axial gradient vector and the transverse gradient vector.
[0030] When improving the interpolation method of the transverse gradient vector, only the magnitude of the transverse gradient vector at the node is interpolated while its directionality is ignored, and the modified transverse strain and modified transverse generalized elastic force are determined based on the improved transverse gradient vector.
[0031] When adjusting the interpolation order of the axial gradient vector used to calculate shear strain, the interpolation of the axial gradient vector is adjusted to a linear term, and the modified shear strain and modified shear generalized elastic force are generated based on the redefined axial gradient.
[0032] A thickness and shear locking elimination device for a beam element with full-parameter absolute nodal coordinates, comprising: A module for constructing a fully parametric ANCF beam element and generating the displacement field of the fully parametric ANCF beam element; A module that obtains axial gradient vector and transverse gradient vector based on the displacement field, and generates axial strain, transverse strain and shear strain; Based on the distribution relationship between the transverse strain and the shear strain, determine the modules for the curvature thickness locking source and the shear locking source; The interpolation method of the transverse gradient vector is improved to generate modules that correct transverse strain and correct transverse generalized elastic force; The interpolation order of the axial gradient vector used to calculate shear strain is adjusted to generate a module that corrects the shear strain and the generalized shear force. A module for generating the generalized elastic force of beam elements after locking is eliminated, based on the modified transverse generalized elastic force and the modified shear generalized elastic force.
[0033] A computer storage medium for storing a computer program, which, when read by the computer, is executed by the computer using the method described thereon.
[0034] A computer, including a processor and a storage medium, executes the method when the processor reads a computer program stored in the storage medium.
[0035] A computer program product, which, as a computer program, implements the method when the computer program is executed.
[0036] Implementation Method Two: This implementation method is a further detailed description of the technical solution provided in Implementation Method One, specifically: This embodiment provides a method for eliminating thickness and shear locking in fully parametric absolute nodal coordinate beam elements. This method eliminates curvature-thickness locking and shear locking in the bending deformation analysis of beam elements while maintaining the original degrees of freedom, integration accuracy, and element continuity of the fully parametric beam element. The method uses a two-dimensional fully parametric beam element as the processing object. First, it constructs the two-dimensional fully parametric beam element using the absolute nodal coordinate method and determines the displacement field of the beam element. Then, it derives the axial strain, transverse strain, shear strain, curvature, strain energy, and elastic force of the beam element using the elastic line method. Subsequently, based on the distribution of transverse strain and shear strain under different bending deformations, it is determined that curvature-thickness locking originates from the coupling between the transverse gradient and bending deformation, while shear locking originates from the difference in the interpolation order of the axial gradient vector and the transverse gradient vector. The interpolation method of the transverse gradient vector is then improved to address curvature-thickness locking, and the interpolation order of the axial gradient used to calculate shear strain is adjusted to address shear locking. The improved transverse gradient and the redefined axial gradient are then used to calculate the transverse strain, shear strain, and corresponding generalized elastic force, respectively.
[0037] Two-dimensional fully parametric beam elements are constructed using the absolute nodal coordinate method, forming the basis for subsequent strain derivation and locked-in analysis. Specifically, the beam element to be analyzed is selected, its nodal coordinate vectors and shape function matrices are determined, and the displacement field of the beam element is expressed as the product of the shape function matrix and the nodal coordinate vectors. For a beam element with a given density, its mass matrix is derived based on the beam element's kinetic energy; this mass matrix is a constant mass matrix. The inputs to this process are the beam element's nodal coordinate vectors, shape function matrix, material coordinates, element length, and density. The outputs are the beam element's displacement field and mass matrix. The displacement field serves as the basis for subsequent derivations of displacement gradient, strain, strain energy, and elastic force, while the mass matrix serves as the basis for mass terms in subsequent static or dynamic analyses.
[0038] The elastic line method is used to derive the strain, strain energy, and elastic force of beam elements to obtain the mechanical expression of fully parametric ANCF beam elements under deformation. Specifically, using the beam centerline as the analysis reference, the displacement gradient at each point is calculated based on the beam element displacement field, and the axial strain, transverse strain, shear strain, and curvature of the beam element are derived based on the displacement gradient. Then, the strain energy of the beam element is determined based on the axial strain, transverse strain, shear strain, and curvature, and the strain energy is expanded according to the axial strain energy, transverse strain energy, shear strain energy, and bending strain energy. Subsequently, based on the partial derivative relationship of the strain energy with respect to the generalized coordinates, the generalized elastic force related to the axial strain, transverse strain, shear strain, and bending strain is obtained. The input to this process is the displacement field and displacement gradient obtained in the previous step, and the output is axial strain, transverse strain, shear strain, curvature, strain energy, and generalized elastic force, which serve as the basis for subsequent identification of locking sources and correction of the generalized elastic force.
[0039] The strain field distribution under different bending deformations is analyzed to determine the root causes of curvature-thickness locking and shear locking. Specifically, the distributions of transverse strain, shear strain, and axial strain of beam elements under different degrees of bending deformation are examined. During bending deformation, if non-uniform transverse strain is generated within the beam element, manifested as cross-sectional deformation and a decrease in cross-sectional area, then curvature-thickness locking is identified. Further analysis of the transverse gradient distribution reveals that the root cause of curvature-thickness locking is the coupling between the transverse gradient and bending deformation, leading to a deviation of the transverse gradient vector distribution along the beam axis from its true value. During bending deformation, if non-uniform shear strain is generated within the element, then shear locking is identified, and it is determined that shear locking mainly originates from the difference in the interpolation order of the axial gradient vector and the transverse gradient vector. The input to this process is the transverse strain and shear strain distribution output from the previous step, and the output is the source of curvature-thickness locking and the source of shear locking, serving as the basis for subsequent improvements to the transverse gradient vector interpolation method and adjustments to the axial gradient interpolation order.
[0040] An improved interpolation method for the transverse gradient vector, specifically designed for curvature-thickness locking, is used to address the coupling between transverse gradient and bending deformation while preserving the characteristics of a fully parametric beam element. In practice, instead of interpolating the transverse gradient vector as a whole in the initial fully parametric beam element, interpolation is performed only on the magnitude of the transverse gradient vector at the nodes, ignoring its directionality. The magnitude of the transverse gradient vector is determined by the transverse gradient vectors at the two ends of the beam element, and combined with the material coordinates and element length of the beam element to obtain the improved transverse gradient expression. The inputs to this process are the transverse gradient vectors at the two ends of the beam element, the material coordinates, and the element length; the output is the improved transverse gradient vector, which serves as the basis for subsequent re-representation of transverse strain and transverse generalized elastic force.
[0041] An improved transverse gradient vector is used to represent transverse strain and generalized elastic force, incorporating the curvature-thickness-locked processing results into the calculation of beam element elastic force. Specifically, based on the improved transverse gradient (interpolating only the magnitude of the transverse gradient while ignoring direction), the transverse strain is redefined, and the corresponding generalized elastic force is determined according to the relationship between transverse strain and strain energy. This process retains the original strain expression and element continuity without adding additional degrees of freedom. The input to this process is the improved transverse gradient vector, and the output is the transverse strain obtained based on the improved transverse gradient and the generalized elastic force associated with the transverse strain. The generalized elastic force serves as part of the subsequent generalized elastic force combination of beam elements.
[0042] This paper addresses the adjustment of the axial gradient interpolation order used to calculate shear strain in shear-locked conditions, specifically to handle shear strain anomalies caused by differences in the interpolation orders of the axial and transverse gradient vectors. In practice, to ensure consistency between the axial and transverse gradient interpolation orders, the interpolation order of the axial gradient used to calculate shear strain is adjusted to linear, resulting in a redefined axial gradient. This adjustment targets the axial gradient upon which shear strain calculations depend without altering the fundamental displacement field structure of the fully parametric beam element. The inputs to this process are the axial gradient vector, beam element nodal coordinates, material coordinates, and element length; the output is the redefined axial gradient, which serves as the basis for subsequent modifications to shear strain and shear generalized elastic force.
[0043] A redefined axial gradient is used to represent shear strain and generalized elastic force, incorporating the results of shear-locking processing into the calculation of elastic force in beam elements. Specifically, the redefined axial gradient is used for shear strain calculation, ensuring that the axial gradient used to calculate shear strain maintains the same interpolation order as the transverse gradient. Subsequently, based on the relationship between shear strain and shear strain energy, the generalized elastic force associated with the shear strain is determined. The inputs to this process are the redefined axial and transverse gradient vectors, and the outputs are the modified shear strain and the generalized elastic force associated with it. This generalized elastic force serves as part of the subsequent generalized elastic force combination for beam elements.
[0044] Generalized elastic forces are combined according to different locking relief methods to form corresponding beam element types. Specifically, when no locking relief is performed, the generalized elastic forces corresponding to the original beam element are used; when only thickness locking is relieved, the transverse strain obtained from the improved transverse gradient and the generalized elastic forces related to the transverse strain are used to form T-LA beam elements; when both thickness locking and shear locking are relieved simultaneously, the improved transverse strain and shear strain are used with the corresponding generalized elastic forces to form TS-LA beam elements. The inputs to this process are the generalized elastic forces corresponding to axial strain, transverse strain, shear strain, and bending strain, and the outputs are the original beam element, T-LA beam element, and TS-LA beam element.
[0045] The generalized elastic force of the resulting beam element is used to complete the simulation calculation of the thickness and shear locking elimination of the fully parametric ANCF beam element. In specific implementation, given the beam element material parameters, section parameters, boundary conditions, and external loads, the generalized elastic force of the corresponding beam element type is used to solve the beam element equilibrium equation to obtain the beam element configuration curve, displacement response, and strain energy results. In the static analysis of a cantilever beam subjected to pure bending moment, the original beam element, T-LA beam element, and TS-LA beam element can be compared and analyzed by the configuration curves and strain energy convergence of different beam element models, thereby completing the implementation of the fully parametric ANCF beam element thickness and shear locking elimination method provided in this embodiment.
[0046] Implementation Method 3, in conjunction with Appendix Figure 1-12 This embodiment describes the technical solution provided above in further detail through specific examples. Specifically: In terms of existing technologies, on the one hand, most studies only focus on one or two specific locking problems in beam elements, lacking in-depth analysis of the root causes of the locking phenomenon; on the other hand, existing curvature-thickness locking and shear locking solutions usually lead to reduced solution accuracy or significantly increased degrees of freedom, failing to maintain the characteristics of fully parametric beam elements.
[0047] In view of this, this embodiment provides a method for eliminating thickness and shear locking of beam elements using the all-parameter absolute nodal coordinate method (ANCF), which effectively eliminates the locking effect while preserving the original strain expression and maintaining element continuity.
[0048] Please see Figure 1 , Figure 1 This is a flowchart of the method for eliminating beam element thickness and shear locking using the All-Parameter Absolute Node Coordinates (ANCF) method provided in this embodiment. The method includes: A systematic and in-depth analysis of the transverse, shear, and axial strain distribution of the beam element was conducted, revealing the root causes of curvature thickness locking and shear locking. To address the curvature thickness locking problem, the interpolation method of the transverse gradient vector is improved, thereby maintaining the interpolation order and effectively decoupling the transverse gradient from bending deformation, thus eliminating pseudo-transverse strain. To address the shear lockout problem, the axial gradient interpolation order used to calculate shear strain is adjusted to ensure compatibility between the axial and transverse gradient orders, thus eliminating spurious shear strain caused by order mismatch.
[0049] Specifically, this example provides a fully parametric absolute nodal coordinate method (ANCF) method for beam element thickness and shear locking elimination. In analyzing the lateral and shear strain distribution of the beam element, a two-dimensional fully parametric beam element is first constructed using the ANCF method to capture the main characteristics and results of the fully parametric beam. The displacement field can be represented as a shape function matrix. and node coordinate vector The product of:
[0050] The mass matrix of a beam element is given by its kinetic energy; for a density of... The kinetic energy of a beam element can be expressed as:
[0051] in, This represents the volume of a beam element. The mass matrix of the beam element is a constant.
[0052] This embodiment uses the elastic line method to derive the element strain, strain energy, and elastic force as follows: On the center line of the beam Then the displacement gradient at each point is:
[0053] Axial strain, transverse strain, shear strain, and curvature are:
[0054]
[0055]
[0056]
[0057] The strain energy of the beam element is:
[0058] in, , It is the cross-sectional area. It is the moment of inertia of the cross section. It is the elastic modulus.
[0059] The stress expression can be obtained by multiplying the elastic modulus by the strain:
[0060] in, and These are axial normal stress and transverse normal stress, This represents shear stress.
[0061] The elastic coefficient matrix of a two-dimensional beam is ,in Indicates shear modulus, It is the shearing correction factor.
[0062] The expansion of the strain energy is then:
[0063] in, , , and These correspond to axial strain energy, transverse strain energy, shear strain energy, and bending strain energy, respectively.
[0064] The elastic force of a beam element is related to its strain energy and can be expressed as the partial derivative of the strain energy with respect to the generalized coordinates, as follows: in, , , and These represent the generalized elastic forces associated with axial strain, transverse strain, shear strain, and bending strain, respectively.
[0065] Specifically, this example provides a fully parametric absolute nodal coordinate method (ANCF) for beam element thickness and shear locking elimination. In analyzing the transverse and axial strain distribution of the beam element, the strain distribution of an initial fully parametric beam element with a length of 1m under different degrees of bending deformation was examined. Specifically, as follows... Figure 2 As shown, Figure 2 This is a schematic diagram of the beam axis configuration curves under different bending deformations provided in this embodiment. According to the formula, the included angles are respectively , and vector and The axial strain, transverse strain, and shear strain at this point are all zero.
[0066] Please see Figure 3-4 The figure shows the transverse strain and shear strain distribution of the beam element provided in this embodiment. The figure indicates that there is a non-uniform strain distribution inside the element, and the strain increases with the increase of bending deformation. Since its stiffness exceeds the theoretical value, a locking effect is caused.
[0067] from Figure 3 As can be seen, the bending of the beam will generate pseudo-transverse strain within the element, indicating that the cross-section of the beam deforms and the cross-sectional area decreases, thus leading to curvature-thickness locking. The transverse strain depends on the transverse gradient vector, which is a linear interpolation of the transverse gradient vector of the initial fully parametric beam element nodes, indicating that even if the cross-sections at both ends of the beam element remain undeformed ( When the beam undergoes bending deformation, the cross-section inside the unit will still shrink.
[0068] Figure 5 This is a schematic diagram of the transverse gradient distribution under bending deformation provided in this embodiment. The diagram shows that the fundamental reason for curvature thickness locking is the coupling between the transverse gradient and bending deformation, which causes the transverse gradient vector distribution along the beam axis to deviate from its true value, thereby making the pseudo transverse strain and element stiffness too large.
[0069] To address the coupling issue between lateral gradient and bending deformation while maintaining the characteristics of the fully parametric beam element, this embodiment makes the following improvements to the lateral gradient:
[0070] This equation interpolates only the magnitude of the lateral gradient while ignoring its directionality, thus decoupling the lateral gradient from bending deformation. Therefore, the lateral strain and the generalized elastic force can be expressed as:
[0071]
[0072] like Figure 4 As shown, the bending deformation of the beam causes a non-uniform shear strain distribution within the element, leading to shear lockout. This phenomenon mainly stems from the difference in interpolation orders between the axial gradient vector and the transverse gradient vector. Specifically, under a linearly varying bending moment, the shear strain exhibits a quadratic distribution along the axial direction, failing to accurately capture the shear distribution. Due to excessive shear strain energy, the element stiffness is overestimated. To ensure consistency between the axial and transverse gradient interpolation orders in this implementation, the interpolation order of the axial gradient is adjusted to linear, expressed as:
[0073] Based on the redefined axial gradient, the shear strain and generalized elastic force are modified as follows:
[0074]
[0075] In summary, curvature thickness locking is alleviated by interpolating only the gradient magnitude, ignoring the gradient direction, and maintaining the interpolation order. Shear locking, on the other hand, avoids gradient field mismatch by reducing the interpolation order of the axial gradient. These methods preserve the characteristics of the fully parametric beam element and can obtain various beam element types through different combinations of generalized elastic forces, as shown in Table 1.
[0076]
[0077] In one embodiment of this implementation, the effectiveness of the method provided in the above embodiments of this implementation is verified by simulation, and the analysis object is as follows: Figure 6 As shown, Figure 6 This is a schematic diagram of a cantilever beam subjected to pure bending moment provided in this embodiment, illustrating the effect of the cantilever beam under bending moment at its free end.
[0078] Please see Figure 7-9 This is a schematic diagram of the configuration curves of different beam models under pure bending moment provided in this embodiment. Figure 7 The original beam configuration curve shows a significant deviation from the quarter circle when the number of elements is between 2 and 10. When the number of elements increases to 60, the configuration curve converges to the ideal position. Figure 8 It is a T-LA beam configuration curve, which has a significantly better convergence speed than the original beam element, and can form a quarter circle with 15 elements; Figure 9 The curve is a TS-LA beam configuration curve, with a further improvement in convergence speed compared to the T-LA beam model. This embodiment further demonstrates that, in order to optimize the convergence of fully parametric beam elements, the transverse and shear strain components must be improved.
[0079] Please see Figure 10-12This is a schematic diagram of the strain energy convergence curves of different beam element models provided in this embodiment. The strain energy convergence speeds from low to high are: original beam, T-LA beam, and TS-LA beam. Figure 10 The curve is the original beam strain energy convergence curve. The bending strain energy convergence rate is relatively low, and it tends to stabilize when the number of elements reaches 60. Figure 11 The curve shows the convergence rate of the strain energy of the T-LA beam. Compared with the original beam model, the convergence rate is improved. It tends to stabilize when the number of elements is 15, while the shear strain energy increases more slowly. Figure 12 The curve shows the convergence of strain energy of the TS-LA beam. Both the transverse strain energy and the shear strain energy tend to zero, and the convergence speed of the bending strain energy is further improved.
[0080] This embodiment studies the static response of a cantilever beam subjected to pure bending moment. By comparing the configuration curves and strain energy convergence rates of different beam models, the root cause of the locking effect is successfully revealed, and the effectiveness of the curvature-thickness locking and shear locking elimination methods proposed in this embodiment is verified.
[0081] The above detailed description of the technical solution provided by the present invention through specific embodiments is intended to highlight the advantages and benefits of the technical solution provided by the present invention. However, the above-described specific embodiments are not intended to limit the present invention. Any reasonable modifications and improvements to the present invention, combinations of embodiments, and equivalent substitutions based on the spirit and principles of the present invention should be included within the protection scope of the present invention.
Claims
1. A method for eliminating thickness and shear locking in a beam element with full-parameter absolute nodal coordinates, characterized in that, include: The steps for constructing a fully parametric ANCF beam element and generating the displacement field of the fully parametric ANCF beam element; The steps are as follows: obtaining the axial gradient vector and the transverse gradient vector based on the displacement field, and generating the axial strain, transverse strain and shear strain. The steps for determining the curvature thickness lock-in source and the shear lock-in source based on the distribution relationship between the transverse strain and the shear strain; The steps involve improving the interpolation method of the transverse gradient vector to generate corrected transverse strain and corrected transverse generalized elastic force. The step of adjusting the interpolation order of the axial gradient vector used to calculate the shear strain to generate the corrected shear strain and the corrected shear generalized elastic force; The steps for generating the generalized elastic force of the beam element after locking is eliminated, based on the modified transverse generalized elastic force and the modified shear generalized elastic force.
2. The method for eliminating thickness and shear locking in a beam element with full-parameter absolute nodal coordinates according to claim 1, characterized in that, When constructing a fully parametric ANCF beam element, a displacement field is generated based on the nodal coordinate vectors and shape function relationships of the fully parametric ANCF beam element, and the mass matrix of the fully parametric ANCF beam element is determined by the displacement field.
3. The method for eliminating thickness and shear locking in a beam element with full-parameter absolute nodal coordinates according to claim 1, characterized in that, When generating axial strain, transverse strain, and shear strain, the elastic line method is used to derive the axial strain, transverse strain, shear strain, and curvature based on the displacement gradient of the fully parametric ANCF beam element, and the strain energy and generalized elastic force are determined according to the axial strain, the transverse strain, the shear strain, and the curvature.
4. The method for eliminating thickness and shear locking in a beam element with full-parameter absolute nodal coordinates according to claim 1, characterized in that, When determining the sources of curvature thickness locking and shear locking, the analysis is based on the distribution of transverse strain, shear strain and axial strain under different bending deformations. It is determined that curvature thickness locking originates from the coupling between transverse gradient and bending deformation, while shear locking originates from the mismatch of the interpolation order of the axial gradient vector and the transverse gradient vector.
5. The method for eliminating thickness and shear locking in a beam element with full-parameter absolute nodal coordinates according to claim 1, characterized in that, When improving the interpolation method of the transverse gradient vector, only the magnitude of the transverse gradient vector at the node is interpolated while its directionality is ignored, and the modified transverse strain and modified transverse generalized elastic force are determined based on the improved transverse gradient vector.
6. The method for eliminating thickness and shear locking in a beam element with full-parameter absolute nodal coordinates according to claim 1, characterized in that, When adjusting the interpolation order of the axial gradient vector used to calculate shear strain, the quadratic interpolation of the axial gradient vector is adjusted to a linear term, and the modified shear strain and modified shear generalized elastic force are generated based on the redefined axial gradient.
7. A thickness and shear locking elimination device for a beam element with full-parameter absolute nodal coordinates, characterized in that, include: A module for constructing a fully parametric ANCF beam element and generating the displacement field of the fully parametric ANCF beam element; A module that obtains axial gradient vector and transverse gradient vector based on the displacement field, and generates axial strain, transverse strain and shear strain; Based on the distribution relationship between the transverse strain and the shear strain, determine the modules for the curvature thickness locking source and the shear locking source; The interpolation method of the transverse gradient vector is improved to generate modules that correct transverse strain and correct transverse generalized elastic force; The interpolation order of the axial gradient vector used to calculate shear strain is adjusted to generate a module that corrects the shear strain and the generalized shear force. A module for generating the generalized elastic force of beam elements after locking is eliminated, based on the modified transverse generalized elastic force and the modified shear generalized elastic force.
8. A computer storage medium for storing computer programs, characterized in that, When the computer program is read by the computer, the computer executes the method of claim 1.
9. A computer, comprising a processor and a storage medium, characterized in that, When the processor reads the computer program stored in the storage medium, the computer executes the method of claim 1.
10. A computer program product, as a computer program, is characterized by: When the computer program is executed, it implements the method of claim 1.