A method and system for vibration analysis of a cracked irregularly shaped plate
By combining NURBS and level set methods, the problem of crack dynamics analysis of irregularly shaped plate structures was solved, achieving high-precision vibration characteristic calculation and reducing computational costs and errors.
Patent Information
- Application Number
- CN202610410195.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-03-31
- Publication Date
- 2026-06-26
AI Technical Summary
Existing technologies struggle to accurately analyze the crack dynamics of irregularly shaped plate structures with complex boundaries. Traditional methods are prone to mesh distortion and a sharp increase in computational costs when dealing with irregular geometric boundaries, and they cannot accurately characterize the three-dimensional stress singularities at the crack tip.
Non-uniform rational B-splines (NURBS) are used to uniformly describe the geometric profile of the plate and the crack topology. By combining the level set method and enrichment function, high-precision geometric expression is directly inherited from the CAD model. Adaptive local refinement technology is used to avoid mesh reconstruction and accurately characterize the stress field at the crack tip.
It significantly reduces the computational cost of crack dynamics modeling under complex boundary conditions, improves the computational accuracy of crack regions, solves the problem of vibration characteristic analysis of irregularly shaped plates, and provides an efficient and robust numerical tool for complex geometry-damage coupled structures in engineering.
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Figure CN122290826A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of structural dynamics, specifically to a vibration analysis method and system for irregularly shaped plates with cracks. Background Technology
[0002] Plate structures, as core load-bearing components in engineering fields such as shipbuilding, aerospace, and machinery, are often designed with irregular geometries, such as curved boundaries and polygonal contours, due to manufacturing limitations or service environment requirements. Under complex alternating loads and corrosive environments, irregularly shaped plates are highly susceptible to the initiation and propagation of microcracks, leading to weakened structural stiffness and distorted vibration characteristics, which in turn can cause safety hazards such as resonance failure or fatigue fracture. Compared to regularly shaped plate structures, irregular geometric features significantly alter the stress field distribution at the crack tip, while existing research is mostly limited to standard-shaped plates such as rectangles or circles, which is insufficient to meet the crack dynamics analysis needs of the numerous complex boundary plate structures present in engineering projects.
[0003] Traditional methods for analyzing the vibration of cracked plates have two limitations: First, discrete modeling based on the finite element method requires fine meshing of the crack tip, which easily leads to mesh distortion when dealing with irregular geometric boundaries, and the computational cost of reconstructing the crack propagation path increases dramatically. Second, analytical methods rely on the assumption of geometric symmetry of the plate, and when using a simplified crack-equivalent spring model, they cannot accurately characterize the three-dimensional stress singularity of the actual crack. In particular, for irregular plates with complex damage morphologies such as curved cracks and multiple crack clusters, existing methods struggle to establish accurate dynamic models of geometry-damage coupling, resulting in large errors in natural frequency prediction and seriously affecting the reliability of subsequent damage assessments.
[0004] Establishing a vibration analysis model for irregularly shaped plates with cracks, and analyzing the influence of crack parameters on the vibration characteristics of the plate structure, can provide a theoretical basis for the development of high-precision online crack monitoring technology. This has significant engineering implications for improving the life prediction capability of key components in major equipment. Therefore, it is essential to establish an effective vibration analysis method applicable to irregularly shaped plates with cracks of various shapes and different crack forms. Summary of the Invention
[0005] To address the aforementioned technical problems, this invention employs Non-Uniform Rational B-Splines (NURBS) to uniformly describe the geometric contours of plates and crack topology. This approach can accurately characterize arbitrarily complex boundary shapes and achieve high-order continuity approximation of the stress field at the crack tip. While maintaining the original CAD-like geometric model, this method introduces adaptive local refinement techniques to ensure accurate calculation results while avoiding the need for redundant geometry definition and mesh re-division.
[0006] To achieve the above objectives, the present invention provides a vibration analysis method for an irregularly shaped plate containing cracks, comprising the following steps: S1. Based on the obtained geometric parameters, material property parameters and crack parameters of the irregularly shaped plate, establish a NURBS geometric model based on isogeometric analysis; S2. Based on the NURBS geometric model, determine the element type within the model region using the level set method, and construct the corresponding displacement vector function based on the element type. S3. Based on the displacement vector function and the plate and shell deformation theory, obtain the potential energy and kinetic energy expressions of the plate with cracks and irregular shape. S4. Discretize the expressions for potential energy and kinetic energy to obtain the element stiffness matrix and element mass matrix; S5. Based on the element stiffness matrix and element mass matrix, assemble and construct the overall vibration control equation of the structure, and solve it to obtain the vibration characteristics of the irregularly shaped plate with cracks.
[0007] Preferably, in step S2, the method for determining the element type within the model region based on the level set method according to the NURBS geometric model includes: Two mutually perpendicular horizontal set functions Φ(x) and Ψ(x) are introduced to represent the crack surface and crack tip location; Based on the level set function value of the integration point within each element, the element is determined to be a normal element, a crack penetration element, or a crack tip element.
[0008] Preferably, in step S2, the method for constructing the corresponding displacement vector function based on the element type includes: For crack penetration elements, a displacement vector function is constructed that includes the Heaviside function as an enrichment function; For the crack tip element, construct a displacement vector function that includes the asymptotic crack tip function as the enrichment function; For ordinary elements, a displacement vector function consisting only of NURBS basis functions is constructed.
[0009] Preferably, the asymptotic crack tip function is a function that is related to the 3 / 2 power of the polar coordinate radius at the crack tip.
[0010] Preferably, S3 includes: Based on the first-order shear deformation theory, a displacement field is constructed for irregularly shaped plates; Determine the stress and strain fields of the plate structure based on the displacement field and the generalized Hooke's law; Based on the stress field and strain field, the potential energy expression and kinetic energy expression of the structure are constructed.
[0011] Preferably, S4 includes: Substitute the displacement vector function into the potential energy expression and the kinetic energy expression; The displacement of the control points is discretized using NURBS basis functions, and the element stiffness matrix and element mass matrix are derived and calculated.
[0012] Preferably, step S5 includes: assembling the stiffness matrix and mass matrix of all elements to form an overall stiffness matrix and an overall mass matrix, and establishing the characteristic equation of structural vibration: Where M and K are the overall mass and stiffness matrices of the structure, respectively; ω q represents the natural frequency and corresponding mode shape of an irregularly shaped plate containing cracks; Solving the characteristic equation yields the natural frequencies and corresponding mode shapes of the irregularly shaped plate with cracks.
[0013] The present invention also provides a vibration analysis system for an irregularly shaped plate with cracks, the system being used to implement the above method, comprising: The geometric model construction module is used to build a NURBS geometric model based on isogeometric analysis based on the obtained geometric parameters, material property parameters and crack parameters of irregularly shaped plates. The displacement vector construction module is used to determine the element type within the model region based on the level set method according to the NURBS geometric model, and to construct the corresponding displacement vector function based on the element type. The calculation module is used to obtain the potential energy and kinetic energy expressions of a cracked, irregularly shaped plate based on the plate and shell deformation theory, according to the displacement vector function. The discretization module is used to discretize the potential energy and kinetic energy expressions to obtain the element stiffness matrix and element mass matrix; The solver module is used to assemble and construct the overall vibration control equation of the structure based on the element stiffness matrix and element mass matrix, and solve it to obtain the vibration characteristics of the irregularly shaped plate with cracks.
[0014] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention employs Non-Uniform Rational B-Splines (NURBS) to uniformly describe the geometric contours and displacement fields of irregularly shaped plates, directly inheriting high-precision geometric representations from CAD models and completely avoiding geometric approximation errors caused by discrete mesh generation in traditional finite element methods. Addressing the displacement discontinuities and crack tip stress singularities caused by cracks, it innovatively introduces enrichment functions to locally enhance the crack tip and discontinuous interfaces. Combined with the level set method, it independently characterizes the topological location of the crack, completely decoupling the crack description from the isogeometric element mesh. This method eliminates the need to reconstruct the computational mesh due to crack propagation or morphological changes, significantly reducing the computational cost of crack dynamics modeling under complex boundary conditions and improving the computational accuracy of the crack region. Simultaneously, it solves the problem of stress oscillation at the crack tip of irregularly shaped plates, providing an efficient and robust numerical tool for the vibration characteristic analysis of complex geometry-damage coupled structures in engineering. Attached Figure Description
[0015] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.
[0016] Figure 1 This is a flowchart of the method of the present invention; Figure 2 A typical model diagram of a cracked, irregularly shaped plate structure for consideration in this invention; Figure 3 A schematic diagram illustrating how the level set function determines the cell type; Figure 4 This is a schematic diagram of the local coordinate system of the crack. Detailed Implementation
[0017] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0018] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.
[0019] Example 1 like Figure 1 The diagram shown is a schematic representation of the method flow in this embodiment, and the steps include: S1. Based on the obtained geometric parameters, material property parameters, and crack parameters of the irregularly shaped plate, establish a NURBS geometric model based on isogeometric analysis.
[0020] Obtain the geometric and material properties of irregularly shaped plates, mainly including the plate structure's geometric dimensions, crack parameters, material properties, and other basic parameters used to solve vibration characteristics; consider two types of irregularly shaped plates, such as... Figure 2 As shown in the figure, the geometric parameter is thickness. h =0.1m, length L 1 = 2.5m L 2 = 2m, height B =1m, boundary conditions are three ends fixed and one end free, the free boundary is the cracked edge, and the crack parameters are... L c / L 1= B c / B =0.5. Material parameters: alloy steel, elastic modulus 201 GPa, density 8166 kg / m³. 3 The Poisson's ratio is 0.28.
[0021] S2. Based on the NURBS geometric model, determine the element type within the model region using the level set method, and construct the corresponding displacement vector function based on the element type.
[0022] like Figure 2 The initial control point coordinates, order, node vectors, and other information of the irregular shape plate on the left are shown in Table 1.
[0023] Table 1 .
[0024] S3. Based on the displacement vector function and the plate and shell deformation theory, obtain the potential energy and kinetic energy expressions of the irregularly shaped plate with cracks.
[0025] The element type within an irregularly shaped plate region containing cracks is determined based on the level set method, including ordinary elements, crack elements, and crack tip elements. In this embodiment, two mutually perpendicular level set domains £=(Φ(x),Ψ(x)) are used to represent crack tips and crack surfaces, and their specific expressions are as follows: in, AB The length of the crack, C Let be any point within the structural region. This indicates finding the absolute value. The direction vector represents the direction of the crack length.
[0026] like Figure 3As shown, points on the blade solid domain of each element are selected, and their level set function values are calculated. Based on the sign of the function value, the crack region can be divided into four parts, thereby determining the different element types: .
[0027] Displacement vector functions that meet displacement characteristics are constructed within different elements, mainly including ordinary elements, crack elements, and crack tip elements.
[0028] Displacement vector function within the crack element: Displacement vector function within the crack tip element: Displacement vector function within a normal element: in, M The number of all control points. H , G These represent the number of control points within the crack penetration element and the crack tip element, respectively; g i These are the standard degrees of freedom, j h and k lt Additional degrees of freedom within the enriched unit; H ( x )and Gl ( x ) are the enrichment functions within the crack element and the crack tip element, respectively, and their expressions are as follows: Where x is the coordinate value of a point within the cell. It is the coordinate of the point on the nearest crack, e n It is the unit normal vector along the crack direction; the Heaviside step function represents a positive value on one side of the crack and a negative value on the other side.
[0029] Using a polar coordinate radius at the crack tip r 3 / 2 related asymptotic crack tip function form Gl ( r , θ ): in The local coordinate system diagram of the crack is shown below, with the polar coordinates of the crack tip as the origin. Figure 4 As shown.
[0030] S4. Discretize the expressions for potential energy and kinetic energy to obtain the element stiffness matrix and element mass matrix.
[0031] Based on the theory of plate and shell deformation, expressions for the potential energy and kinetic energy of a cracked irregularly shaped plate structure are obtained; using the first-order shear deformation theory, the expression for the displacement of the cracked irregularly shaped plate is derived. for: In the formula, Let represent the displacement in the coordinate system of the plate structure. The displacement vector is based on the first-order plate and shell deformation theory. z This represents the displacement along the thickness direction of the plate.
[0032] Based on the stress-displacement relationship and the generalized Hooke's law, the stress in a plate structure can be obtained as follows: In the formula, σ and ε represent the stress and strain of the structure, respectively. E The elastic modulus of the material, μ Poisson's ratio, express u 0 displacement pair x The first partial derivative of the variable, D B D represents the constitutive matrix of a bending material. S Represents the constitutive equation for shear materials. kn This is the shearing correction factor, which is set to 5 / 6 in this embodiment.
[0033] This yields the potential energy of the structure. U and kinetic energy T expression: in, t Indicates time, ρ For the density of the plate structure, V It is the total volume of the structure.
[0034] Based on the expressions for potential and kinetic energy, the energy functional of the structure is obtained using Hamilton's principle and variational principle. The element stiffness and mass matrix are then discretized using analytical basis functions. The displacement vector expression, obtained from the analytical basis functions and displacements at control points, is then presented. In the formula, u is the displacement vector at any point, u A It is the displacement vector at the control point. M The number of control points. N A (x) is the NURBS basis function, and its form differs in different units: Crack element: Crack tip unit: Ordinary unit: By substituting the kinetic and potential energy equations into the element discretization, the discretized element stiffness and mass matrix can be obtained.
[0035] S5. Based on the element stiffness matrix and element mass matrix, assemble and construct the overall vibration control equation of the structure, and solve it to obtain the vibration characteristics of the irregularly shaped plate with cracks.
[0036] The system is assembled cyclically according to unit numbers to construct the overall vibration control equation of the structure, which is expressed as follows: Where M and K are the overall mass and stiffness matrices of the structure, respectively. ω Let ν be the vibration frequency of the irregular structure containing cracks, and q be the vibration vector of the structure. The calculation steps for the stiffness matrix and mass matrix are as follows: in, V The total volume of the structure. N A,x , N A,y To analyze basis functions N A right x and y The first derivative.
[0037] By solving the characteristic matrix, the vibration characteristics of the cracked irregular-shaped plate, including structural frequencies and mode shapes, are obtained. The calculated frequency results are shown in Table 2. Comparing the frequency results obtained in this paper with those obtained by traditional finite element simulation software, it can be seen that the difference between the two is not significant, thus proving the accuracy of the method proposed in this invention. It is worth noting that when the finite element method solves the vibration characteristics of the two types of cracked irregular-shaped plates, the number of meshes is more than ten times that of the method in this invention due to the mesh refinement at the cracks.
[0038] Table 2 .
[0039] In summary, this invention employs an isogeometric method for direct calculation based on the CAD model, eliminating the need for meshing as in traditional finite element methods and fundamentally avoiding errors caused by geometric simplification. Addressing the critical challenge of crack handling, this method innovatively combines enrichment functions and level set techniques. Enrichment functions accurately capture the singularity and discontinuous displacement at the crack tip, while level set methods independently mark the crack location, completely decoupling the crack description from the computational mesh. The advantages of this method are: firstly, even if the crack propagates during the calculation, there is no need to re-mesh, significantly saving computation time; secondly, it maintains high accuracy for plates with arbitrarily complex shapes, improving the solution accuracy for cracked regions; and thirdly, it is simple to operate, allowing direct analysis using existing CAD models from engineering designs, significantly reducing the conversion cost from design to analysis.
[0040] Example 2 This embodiment also provides a vibration analysis system for an irregularly shaped plate with cracks, including: a geometric model construction module, used to establish a NURBS geometric model based on isogeometric analysis according to the obtained geometric parameters, material property parameters and crack parameters of the irregularly shaped plate; a displacement vector construction module, used to determine the element type in the model region based on the level set method according to the NURBS geometric model, and construct the corresponding displacement vector function based on the element type; a calculation module, used to obtain the potential energy and kinetic energy expressions of the irregularly shaped plate with cracks based on the plate and shell deformation theory according to the displacement vector function; a discretization module, used to discretize the potential energy and kinetic energy expressions to obtain the element stiffness matrix and element mass matrix; and a solution module, used to assemble and construct the overall vibration control equation of the structure based on the element stiffness matrix and element mass matrix, and solve to obtain the vibration characteristics of the irregularly shaped plate with cracks.
[0041] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.
Claims
1. A vibration analysis method for an irregularly shaped plate with cracks, characterized in that, Includes the following steps: S1. Based on the obtained geometric parameters, material property parameters and crack parameters of the irregularly shaped plate, establish a NURBS geometric model based on isogeometric analysis; S2. Based on the NURBS geometric model, determine the element type within the model region using the level set method, and construct the corresponding displacement vector function based on the element type. S3. Based on the displacement vector function and the plate and shell deformation theory, obtain the potential energy and kinetic energy expressions of the plate with cracks and irregular shape. S4. Discretize the expressions for potential energy and kinetic energy to obtain the element stiffness matrix and element mass matrix; S5. Based on the element stiffness matrix and element mass matrix, assemble and construct the overall vibration control equation of the structure, and solve it to obtain the vibration characteristics of the irregularly shaped plate with cracks.
2. The vibration analysis method for a cracked, irregularly shaped plate according to claim 1, characterized in that, In S2, the method for determining the element type within the model region based on the level set method according to the NURBS geometric model includes: Two mutually perpendicular horizontal set functions Φ(x) and Ψ(x) are introduced to represent the crack surface and crack tip location; Based on the level set function value of the integration point within each element, the element is determined to be a normal element, a crack penetration element, or a crack tip element.
3. The vibration analysis method for a cracked, irregularly shaped plate according to claim 2, characterized in that, In S2, the method for constructing the corresponding displacement vector function based on the element type includes: For crack penetration elements, a displacement vector function is constructed that includes the Heaviside function as an enrichment function; For the crack tip element, construct a displacement vector function that includes the asymptotic crack tip function as the enrichment function; For ordinary elements, a displacement vector function consisting only of NURBS basis functions is constructed.
4. The vibration analysis method for a cracked, irregularly shaped plate according to claim 3, characterized in that, The asymptotic crack tip function is a function that is related to the 3 / 2 power of the polar coordinate radius at the crack tip.
5. The vibration analysis method for a cracked, irregularly shaped plate according to claim 1, characterized in that, S3 includes: Based on the first-order shear deformation theory, a displacement field is constructed for irregularly shaped plates; Determine the stress and strain fields of the plate structure based on the displacement field and the generalized Hooke's law; Based on the stress field and strain field, the potential energy expression and kinetic energy expression of the structure are constructed.
6. The vibration analysis method for a cracked, irregularly shaped plate according to claim 5, characterized in that, S4 includes: Substitute the displacement vector function into the potential energy expression and the kinetic energy expression; The displacement of the control points is discretized using NURBS basis functions, and the element stiffness matrix and element mass matrix are derived and calculated.
7. The vibration analysis method for an irregularly shaped plate with cracks according to claim 1, characterized in that, S5 includes: assembling the stiffness and mass matrices of all elements to form an overall stiffness matrix and an overall mass matrix, and establishing the characteristic equations for structural vibration: Where M and K are the overall mass and stiffness matrices of the structure, respectively; ω q represents the natural frequency and corresponding mode shape of an irregularly shaped plate containing cracks; Solving the characteristic equation yields the natural frequencies and corresponding mode shapes of the irregularly shaped plate with cracks.
8. A vibration analysis system for an irregularly shaped plate with cracks, said system being used to implement the method according to any one of claims 1-7, characterized in that, include: The geometric model construction module is used to build a NURBS geometric model based on isogeometric analysis based on the obtained geometric parameters, material property parameters and crack parameters of irregularly shaped plates. The displacement vector construction module is used to determine the element type within the model region based on the level set method according to the NURBS geometric model, and to construct the corresponding displacement vector function based on the element type. The calculation module is used to obtain the potential energy and kinetic energy expressions of a cracked, irregularly shaped plate based on the plate and shell deformation theory, according to the displacement vector function. The discretization module is used to discretize the potential energy and kinetic energy expressions to obtain the element stiffness matrix and element mass matrix; The solver module is used to assemble and construct the overall vibration control equation of the structure based on the element stiffness matrix and the element mass matrix, and solve it to obtain the vibration characteristics of the irregularly shaped plate with cracks.