Panel shell element finite element analysis method, system and storage medium for ship structure

Abstract

This invention relates to the field of structural finite element analysis technology, specifically disclosing a finite element analysis method, system, and storage medium for plate and shell elements in ship structures. The method includes: constructing a four-node plate and shell element finite element discrete model of a thin-walled ship structure; determining the block composition of the basic Mindlin plate and shell element stiffness matrix; constructing out-of-plane bending stiffness and correcting the membrane effect stiffness, out-of-plane transverse shear stiffness, and in-plane rotational stiffness respectively; combining the out-of-plane bending stiffness and the corrected membrane effect stiffness, out-of-plane transverse shear stiffness, and in-plane rotational stiffness based on the four-node plate and shell element finite element discrete model of the thin-walled ship structure; and performing finite element analysis of the plate and shell elements of the ship structure based on the corrected plate and shell element stiffness matrix. The plate and shell element finite element analysis method for ship structures provided by this invention can improve the adaptability and computational stability of four-node plate and shell elements in the modeling of complex ship structures.

Description

Technical Field

[0001] This invention relates to the field of structural finite element analysis technology, and in particular to a plate and shell element finite element analysis method for ship structures, a plate and shell element finite element analysis system for ship structures, and a computer-readable storage medium. Background Technology

[0002] In finite element analysis of ship structures, thin-walled structures such as hull plates, bulkheads, and decks are typically modeled using plate and shell finite element elements. Among these, the four-node plate and shell element based on Mindlin theory is widely used in ship structural strength analysis and deformation calculation due to its simplicity and high computational efficiency.

[0003] However, in actual ship structural analysis, shell structures typically exhibit significant curved surface features, complex local geometric variations, and irregular mesh shapes. When using traditional Mindlin four-node shell elements for finite element analysis of these structures, problems such as inaccurate element stiffness calculations, significant shear locking, and decreased numerical stability can easily arise under complex geometric conditions or in regions with high stress gradients. Specifically, in the finite element modeling of ship shell structures, when the element thickness is small or the mesh exhibits a certain degree of geometric distortion, the simplified shear strain assumption in traditional Mindlin shell elements can easily lead to excessively large element stiffness, causing the calculation results to deviate from the theoretical predictions of thin plates. Furthermore, in nonlinear analyses (such as large geometric deformations or material elastoplasticity calculations), these stiffness errors can also lead to a decrease in iterative convergence speed and even cause numerical instability. In addition, in actual ship structural analysis, to improve computational efficiency, the reduced integration method is often used to construct the element stiffness matrix; however, this method may introduce hourglass modes under certain conditions, further affecting the reliability of the calculation results.

[0004] Therefore, in order to meet the actual modeling needs in finite element analysis of ship structures, how to improve the adaptability and computational stability of four-node plate and shell elements in complex ship structure modeling, so as to improve the accuracy and reliability of the overall finite element analysis results, has become a technical problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0005] This invention provides a finite element analysis method for plate and shell elements in ship structures, a finite element analysis system for plate and shell elements in ship structures, and a computer-readable storage medium, which solves the problems of poor adaptability and poor computational stability of four-node plate and shell elements in complex ship structure modeling in related technologies.

[0006] As a first aspect of the present invention, a finite element analysis method for plate and shell elements in ship structures is provided, comprising:

[0007] A four-node plate and shell element finite element discrete model of a thin-walled ship structure is constructed, wherein each node is given three translational degrees of freedom and three rotational degrees of freedom.

[0008] The block composition of the stiffness matrix of the basic Mindlin plate and shell element is determined, and the block composition includes membrane effect stiffness, out-of-plane bending stiffness, out-of-plane transverse shear stiffness, and in-plane rotational stiffness.

[0009] The out-of-plane bending stiffness is constructed, and the membrane effect stiffness, out-of-plane transverse shear stiffness, and in-plane rotational stiffness are respectively modified.

[0010] Based on the four-node plate and shell element finite element discrete model of the thin-walled structure of the ship, the out-of-plane bending stiffness, the modified membrane effect stiffness, the out-of-plane transverse shear stiffness, and the in-plane rotation stiffness are combined to obtain the modified plate and shell element stiffness matrix.

[0011] The finite element method of the plate and shell elements of the ship structure is performed based on the modified plate and shell element stiffness matrix to obtain the displacement and / or stress response of the thin-walled ship structure.

[0012] Furthermore, the stiffness of the thin film effect is corrected by means of:

[0013] An additional displacement term corresponding to the pure bending state is introduced into the displacement mode of the membrane effect stiffness to generate additional degrees of freedom;

[0014] Construct the element equilibrium equations that include the nodal degrees of freedom and the additional degrees of freedom;

[0015] The additional degrees of freedom are statically condensed and eliminated to obtain a corrected membrane effect stiffness matrix including nodal degrees of freedom.

[0016] Furthermore, the out-of-plane transverse shear stiffness is corrected, including:

[0017] Construct the out-of-plane transverse shear stiffness matrix K based on the reduced integral. ts ;

[0018] The geometric correction factor α is determined based on the thickness h of the plate / shell, where 0 < α ≤ 1;

[0019] Based on the geometric correction factor α and the out-of-plane transverse shear stiffness matrix K ts Obtain the corrected out-of-plane transverse shear stiffness matrix K' ts , where K' ts =α*K ts .

[0020] Furthermore, the geometric correction factor decreases as the thickness of the shell decreases and increases as the thickness of the shell increases.

[0021] Furthermore, the out-of-plane transverse shear stiffness matrix K is constructed based on the reduced integral. ts ,include:

[0022] Construct an hourglass to control stiffness;

[0023] Based on the hourglass control stiffness and the reduced integral, construct the external transverse shear stiffness matrix K. ts The hourglass control stiffness can suppress zero-energy modes caused by reduced integrals.

[0024] Furthermore, the in-plane rotational stiffness is corrected, including:

[0025] The reference value of the in-plane rotational stiffness is determined based on the out-of-plane transverse shear stiffness matrix;

[0026] The reference value of the in-plane rotational stiffness is corrected according to the in-plane rotational stiffness correction factor to obtain the corrected in-plane rotational stiffness.

[0027] Furthermore, the out-of-plane bending stiffness includes:

[0028] The geometric relationship between bending strain and nodal degrees of freedom is determined based on the out-of-plane bending displacement mode in order to construct the bending strain matrix;

[0029] Construct the bending stress matrix based on the material's elastic constitutive relation;

[0030] The out-of-plane bending stiffness is constructed based on the bending strain matrix and the bending stress matrix.

[0031] Furthermore, based on the four-node plate and shell element finite element discrete model of the ship's thin-walled structure, the out-of-plane bending stiffness, the corrected membrane effect stiffness, the out-of-plane transverse shear stiffness, and the in-plane rotational stiffness are combined to obtain the corrected plate and shell element stiffness matrix, including:

[0032] The thin film effect stiffness is assembled to two in-plane translational degrees of freedom of each node out of six degrees of freedom. The out-of-plane bending stiffness and out-of-plane transverse shear stiffness are assembled to one out-of-plane translational degree of freedom and two out-of-plane rotational degree of freedom of each node out of six degrees of freedom. The in-plane rotational stiffness is assembled to one in-plane rotational degree of freedom of each node out of six degrees of freedom, so as to form the modified plate and shell element stiffness matrix.

[0033] As another aspect of the present invention, a finite element analysis system for ship structures is provided, comprising:

[0034] Memory is used to store executable instructions for a computer;

[0035] The processor, coupled to the memory, implements the finite element analysis method for plate and shell elements of ship structures as described above when the processor executes the computer-executable instructions.

[0036] As another aspect of the present invention, a computer-readable storage medium is provided having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the finite element analysis method for plate and shell elements for ship structures as described above.

[0037] The finite element analysis method for plate and shell elements used in ship structures provided by this invention effectively alleviates the shear locking problem by introducing an additional displacement term corresponding to the pure bending state into the displacement mode of the membrane effect stiffness and eliminating the additional degrees of freedom through static condensation. By introducing a geometric correction factor that varies with the plate / shell thickness to correct the transverse shear stiffness and constructing an hourglass control stiffness, a continuous transition between thick and thin plate theories is achieved, ensuring numerical stability. Furthermore, by determining and correcting the benchmark value of the in-plane rotational stiffness through the out-of-plane transverse shear stiffness matrix, the singularity problem of the stiffness matrix is ​​avoided, thereby effectively improving the computational accuracy, numerical stability, and solution convergence of the finite element analysis of thin-walled ship structures. Therefore, this finite element analysis method for plate and shell elements used in ship structures constructs a plate and shell element stiffness matrix based on multiple correction strategies, which can effectively improve the accuracy of displacement and stress calculations in the finite element analysis of thin-walled ship structures, ensure numerical stability, and enhance the adaptability and computational stability of the elements in the modeling of complex ship structures, thus improving the accuracy and reliability of the overall finite element analysis results. Attached Figure Description

[0038] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used together with the following detailed description to explain the invention, but do not constitute a limitation thereof.

[0039] Figure 1 The flowchart shows the finite element analysis method for plate and shell elements used in ship structures provided by this invention.

[0040] Figure 2 This is a schematic diagram of a rectangular plate shell unit located in the XY plane provided by the present invention.

[0041] Figure 3 This is a schematic diagram of a model of a partially stiffened deck plate structure provided by the present invention.

[0042] Figure 4 The displacement cloud diagram of the ship stiffening plate structure obtained by the modified four-node plate shell unit provided by the present invention.

[0043] Figure 5 The structural block diagram of the finite element analysis system for ship structures provided by the present invention. Detailed Implementation

[0044] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0045] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments. 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 should fall within the scope of protection of the present invention.

[0046] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate for the embodiments of the invention described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0047] This embodiment provides a finite element analysis method for plate and shell elements in ship structures. Figure 1 This is a flowchart of a finite element analysis method for plate and shell elements in ship structures provided by an embodiment of the present invention, as shown below. Figure 1 As shown, it includes:

[0048] S100. Construct a four-node plate and shell element finite element discrete model of a thin-walled ship structure, where each node has three translational degrees of freedom and three rotational degrees of freedom.

[0049] In this embodiment of the invention, thin-walled ship structures such as hull plates, decks, and bulkheads can be discretized into four-node shell elements. Each node has six degrees of freedom, including three translational degrees of freedom u, v, and w, and three rotational degrees of freedom θ. x θ y θ z The degrees of freedom are arranged in the order of node numbers, which is consistent with the basic assumptions of Mindlin's plate and shell theory.

[0050] S200. Determine the block composition of the stiffness matrix of the basic Mindlin plate and shell element, wherein the block composition includes membrane effect stiffness, out-of-plane bending stiffness, out-of-plane transverse shear stiffness, and in-plane rotational stiffness.

[0051] In this embodiment of the invention, the film effect stiffness corresponds to the in-plane translational degrees of freedom u and v, and is used to describe the in-plane tensile and shear deformation behavior of the plate and shell; the out-of-plane bending stiffness corresponds to the out-of-plane translational degrees of freedom w and rotational degrees of freedom θ. x θ y It is used to describe the out-of-plane bending deformation behavior of plates and shells; the out-of-plane transverse shear stiffness corresponds to the out-of-plane translational degree of freedom w and rotational degree of freedom θ. x θ y Used to describe the transverse shear deformation behavior of plates and shells; in-plane rotational stiffness corresponds to the in-plane rotational degree of freedom θ. z , used to describe the in-plane torsional deformation behavior of plates and shells.

[0052] S300, construct out-of-plane bending stiffness, and respectively correct the film effect stiffness, out-of-plane transverse shear stiffness and in-plane rotation stiffness.

[0053] In this embodiment of the invention, the out-of-plane bending stiffness already possesses high computational accuracy within the framework of Mindlin plate theory; therefore, standard Mindlin plate theory is employed for its construction. The membrane effect stiffness, out-of-plane transverse shear stiffness, and in-plane rotational stiffness are specifically modified to meet the analytical requirements of ship structures. Specifically, the membrane effect stiffness is corrected by introducing an additional displacement term corresponding to the pure bending state into the displacement mode and eliminating the additional degrees of freedom using static condensation; the out-of-plane transverse shear stiffness is corrected by reducing the integral, combining a geometric correction factor, and hourglass control stiffness; and the in-plane rotational stiffness is corrected by determining the baseline value based on the diagonal term of the out-of-plane transverse shear stiffness matrix and applying a correction factor.

[0054] S400. Based on the four-node plate and shell element finite element discrete model of the thin-walled ship structure, the out-of-plane bending stiffness, the modified membrane effect stiffness, the out-of-plane transverse shear stiffness, and the in-plane rotational stiffness are combined to obtain the modified plate and shell element stiffness matrix.

[0055] In this embodiment of the invention, the thin film effect stiffness is assembled to two in-plane translational degrees of freedom positions out of the six degrees of freedom of each node, the out-of-plane bending stiffness and out-of-plane transverse shear stiffness are assembled to one out-of-plane translational degree of freedom position and two out-of-plane rotational degree of freedom positions out of the six degrees of freedom of each node, and the in-plane rotational stiffness is assembled to one in-plane rotational degree of freedom position out of the six degrees of freedom of each node, forming a 24×24 four-node plate and shell element modified stiffness matrix.

[0056] S500. Based on the corrected plate and shell element stiffness matrix, perform finite element analysis of the plate and shell elements of the ship structure to obtain the displacement and / or stress response of the thin-walled ship structure.

[0057] In this embodiment of the invention, the modified stiffness matrices of each element are assembled into an overall stiffness matrix according to the node number. Boundary conditions and external loads are applied, and the overall equilibrium equations are solved to obtain the nodal displacements. Then, the element stresses are calculated through the relationship between strain and displacement and the relationship between stress and strain.

[0058] Therefore, the finite element analysis method for plate and shell elements in ship structures provided by this invention effectively alleviates the shear locking problem by introducing an additional displacement term corresponding to the pure bending state into the displacement mode of the membrane effect stiffness and eliminating the additional degrees of freedom through static condensation. By introducing a geometric correction factor that varies with the plate / shell thickness to correct the transverse shear stiffness and constructing an hourglass control stiffness, a continuous transition between thick and thin plate theories is achieved, ensuring numerical stability. Furthermore, by determining and correcting the benchmark value of the in-plane rotational stiffness through the out-of-plane transverse shear stiffness matrix, the singularity problem of the stiffness matrix is ​​avoided, thereby effectively improving the computational accuracy, numerical stability, and solution convergence of the finite element analysis of thin-walled ship structures. Therefore, this finite element analysis method for plate and shell elements in ship structures constructs a plate and shell element stiffness matrix based on multiple correction strategies, which can effectively improve the accuracy of displacement and stress calculations in the finite element analysis of thin-walled ship structures, ensure numerical stability, and enhance the adaptability and computational stability of the elements in the modeling of complex ship structures, thereby improving the accuracy and reliability of the overall finite element analysis results.

[0059] In this embodiment of the invention, to avoid the use of bilinear shape functions for interpolation of in-plane displacements in traditional Mindlin shell elements, which is difficult to accurately describe in-plate deformation under pure bending conditions and prone to shear locking when the shell thickness is small or the element mesh is distorted, this embodiment of the invention corrects the film effect stiffness by including:

[0060] (1) Introduce an additional displacement term corresponding to the pure bending state into the displacement mode of the membrane effect stiffness to generate additional degrees of freedom;

[0061] In this embodiment of the invention, the correction of the membrane effect stiffness can be achieved by introducing an additional displacement term corresponding to the pure bending state into the displacement mode of the membrane effect stiffness, thereby generating additional degrees of freedom. By introducing an additional secondary displacement term corresponding to the pure bending state on the basis of the traditional displacement mode, the displacement mode is extended to include both nodal degrees of freedom and additional degrees of freedom.

[0062] (2) Construct the element equilibrium equations including nodal degrees of freedom and the additional degrees of freedom;

[0063] Based on the aforementioned additional degrees of freedom, an element equilibrium equation is further constructed, which includes nodal degrees of freedom and additional degrees of freedom. This equilibrium equation contains the stiffness coupling relationship between nodal degrees of freedom and additional degrees of freedom.

[0064] (3) Static condensation of the additional degrees of freedom and elimination of the additional degrees of freedom to obtain a modified film effect stiffness matrix including nodal degrees of freedom.

[0065] Specifically, the additional degrees of freedom are statically condensed and eliminated to obtain a modified membrane effect stiffness matrix that includes only nodal degrees of freedom, thereby enhancing the element's ability to represent pure bending states and reducing the influence of shear locking.

[0066] In this embodiment of the invention, the correction process for the out-of-plane transverse shear stiffness includes:

[0067] (1) Construct the out-of-plane transverse shear stiffness matrix K based on the reduced integral. ts ;

[0068] In this embodiment of the invention, the out-of-plane transverse shear stiffness matrix K is constructed based on the reduced integral. ts ,include:

[0069] (11) Construct an hourglass to control stiffness;

[0070] (12) Construct the external transverse shear stiffness matrix K based on the hourglass control stiffness and the reduced integral. ts The hourglass control stiffness can suppress zero-energy modes caused by reduced integrals.

[0071] (2) Determine the geometric correction factor α based on the thickness h of the plate and shell, where 0 < α ≤ 1;

[0072] It should be noted that the geometric correction factor decreases as the plate / shell thickness decreases and increases as the plate / shell thickness increases. Specifically, when the plate / shell thickness approaches the thin plate limit, α approaches a small value, and the corrected out-of-plane transverse shear stiffness approaches the Kirchhoff thin plate theoretical value; when the plate / shell thickness approaches the thick plate limit, α approaches 1, and the corrected out-of-plane transverse shear stiffness approaches the Mindlin thick plate theoretical value.

[0073] (3) Based on the geometric correction factor α and the out-of-plane transverse shear stiffness matrix K ts Obtain the corrected out-of-plane transverse shear stiffness matrix K' ts , where K' ts =α*K ts .

[0074] In this embodiment of the invention, for the correction of out-of-plane transverse shear stiffness, the out-of-plane transverse shear stiffness matrix K is first constructed based on the reduced integral. ts To mitigate shear locking under thin plate conditions, a single Gaussian point integration method is used to calculate the shear strain energy. In this embodiment, to address the potential introduction of a zero-energy mode (hourglass mode) while reducing the integral, an hourglass control stiffness is constructed. This hourglass control stiffness is orthogonal to the normal strain energy and provides restoring force only for the zero-energy mode, without affecting the calculation accuracy of the normal deformation mode. Then, a geometric correction factor α is determined based on the plate thickness h. This geometric correction factor α increases with increasing plate thickness and decreases with decreasing plate thickness, with 0 < α ≤ 1. Finally, the geometric correction factor α and the out-of-plane transverse shear stiffness matrix K are used to calculate the shear strain energy. ts Obtain the corrected out-of-plane transverse shear stiffness matrix K' ts =α*K ts After the above modifications, when the plate thickness is large, the modified shear stiffness retains the characteristics of Mindlin's thick plate theory, and when the plate thickness is small, the modified shear stiffness tends to the characteristics of Kirchhoff's thin plate theory, thus effectively suppressing the shear locking phenomenon in thin plate analysis.

[0075] In this embodiment of the invention, the in-plane rotational stiffness is corrected by:

[0076] (1) Determine the reference value of the in-plane rotational stiffness based on the out-of-plane transverse shear stiffness matrix;

[0077] In this embodiment of the invention, the correction process for in-plane rotational stiffness is based on the out-of-plane transverse shear stiffness matrix K. ts The diagonal terms are used to determine the baseline value of the in-plane rotational stiffness, specifically using the out-of-plane transverse shear stiffness matrix K. ts Half of the diagonal value is used as the baseline.

[0078] (2) Correct the reference value of the in-plane rotation stiffness according to the in-plane rotation stiffness correction factor to obtain the corrected in-plane rotation stiffness.

[0079] In this embodiment of the invention, the reference value is corrected according to a correction factor to obtain the corrected in-plane rotational stiffness. Determining the in-plane rotational stiffness in this way avoids the singularity problem in the stiffness matrix that may occur when using arbitrary non-zero values ​​in traditional methods.

[0080] In this embodiment of the invention, the out-of-plane bending stiffness includes:

[0081] (1) Determine the geometric relationship between bending strain and nodal degrees of freedom based on the out-of-plane bending displacement mode in order to construct the bending strain matrix;

[0082] (2) Construct the bending stress matrix based on the material's elastic constitutive relation;

[0083] (3) Construct out-of-plane bending stiffness based on the bending strain matrix and the bending stress matrix.

[0084] It should be understood that, for the construction of out-of-plane bending stiffness, the standard Mindlin plate theory is directly adopted. The geometric relationship between bending strain and nodal degrees of freedom is determined according to the out-of-plane bending displacement mode, and the bending strain matrix is ​​constructed. The bending stress matrix is ​​constructed according to the material elastic constitutive relation. The transpose of the bending strain matrix is ​​multiplied by the bending stress matrix and integrated to construct the out-of-plane bending stiffness matrix.

[0085] In this embodiment of the invention, the out-of-plane bending stiffness, the corrected membrane effect stiffness, the out-of-plane transverse shear stiffness, and the in-plane rotational stiffness are combined according to the four-node plate and shell element finite element discrete model of the thin-walled ship structure to obtain the corrected plate and shell element stiffness matrix, including:

[0086] The thin film effect stiffness is assembled to two in-plane translational degrees of freedom of each node out of six degrees of freedom. The out-of-plane bending stiffness and out-of-plane transverse shear stiffness are assembled to one out-of-plane translational degree of freedom and two out-of-plane rotational degree of freedom of each node out of six degrees of freedom. The in-plane rotational stiffness is assembled to one in-plane rotational degree of freedom of each node out of six degrees of freedom, so as to form the modified plate and shell element stiffness matrix.

[0087] It should be understood that the stiffnesses of the above-mentioned parts are combined in the order of nodal degrees of freedom to form a complete plate and shell element stiffness matrix, which is then used for element assembly and overall solution in the finite element analysis of ship structures.

[0088] The following is a detailed description of the specific implementation process of the finite element analysis method for plate and shell elements used in ship structures according to the present invention.

[0089] To verify the effectiveness of this invention, a partial shell structure of the deck in a typical thin-walled structure of a ship was used as the test object. A single four-node shell element was used for discretization to analyze the displacement and deformation response of the structure under external loads. This embodiment can be used to simulate the structural response of hull plates, bulkheads, decks, and other thin-walled curved structures under complex stress conditions. The embodiments of this invention are described in detail below. This embodiment uses the modified method of this invention to calculate the stiffness of the shell element. The scope of protection of this invention includes, but is not limited to, the following embodiments. Discretization is performed using a single four-node shell element. Figure 2The model shown is a rectangular plate shell element located in the XY plane. The displacement of the nodes under the action of force is analyzed. The geometric parameters of the model are: length l=5, width w=1, thickness h=0.1; material parameters are: Young's modulus E=1e8, Poisson's ratio ν=0.3; mesh parameters are: mesh size 5, divided into 1 element; boundary and load: one end is fixed, and the other end is subjected to a concentrated force along the Z+ direction.

[0090] Step S1: Establish a four-node plate and shell element degree-of-freedom system for finite element analysis of ship structures.

[0091] This embodiment uses four-node plate and shell elements to discretize the thin-walled structure of the ship. Each node has six degrees of freedom, including three translational degrees of freedom u, v, and w, and three rotational degrees of freedom. , , Where u and v represent the in-plane translational degrees of freedom of the plate and shell; w represents the out-of-plane translational degrees of freedom of the plate and shell. , Represents the rotational degrees of freedom about the local coordinate axes; This represents the degree of freedom of rotation within the plane.

[0092] Based on the above degrees of freedom, construct the overall degree-of-freedom vector of the unit:

[0093] ,

[0094] The four-node plate and shell element has a total of 24 degrees of freedom. These degrees of freedom are arranged sequentially according to the node numbers, ultimately forming a 24×24 overall stiffness matrix for the plate and shell element. Based on the mechanical behavior corresponding to different degrees of freedom, the overall stiffness is divided into: membrane effect stiffness; out-of-plane bending stiffness; out-of-plane transverse shear stiffness; and in-plane rotational stiffness. Each part of the stiffness is then constructed separately.

[0095] Step S2: Introduce a quadratic term of the shape function corresponding to the pure bending state into the traditional thin film stiffness algorithm to construct the thin film stiffness.

[0096] In traditional Mindlin plate and shell elements, in-plane displacements are interpolated using bilinear shape functions:

[0097] ,

[0098] in, Representing the unit shape function:

[0099] ;

[0100] in, This represents the coordinates in the natural coordinate system of the quadrilateral shell element.

[0101] It should be understood that the traditional bilinear displacement mode is difficult to accurately describe the deformation within the plate under pure bending conditions. Especially in the analysis of thin-walled ship structures, when the plate thickness is small or the element mesh is distorted, shear self-locking is likely to occur, resulting in excessive element stiffness and underestimation of structural displacement.

[0102] To address the aforementioned problems, this invention introduces an additional secondary displacement term corresponding to the pure bending state, based on the traditional displacement mode:

[0103] ,

[0104] The displacement mode then becomes:

[0105] ,

[0106] in, Let represent the additional degrees of freedom. Then, the equilibrium equation for the element in the thin film direction is:

[0107] ,

[0108] in, This represents the vector of degrees of freedom of a unit node. This indicates that the element has an additional internal degree of freedom vector. This represents the element's equivalent nodal load vector. This represents the stiffness matrix corresponding to the degrees of freedom of the nodes. This represents the stiffness matrix corresponding to the additional degrees of freedom. This represents the coupling stiffness matrix between nodal degrees of freedom and additional degrees of freedom.

[0109] By using static condensation technology, the introduced additional degrees of freedom are condensed away, and the final film stiffness is:

[0110] ,

[0111] in, This indicates the stiffness of the thin film.

[0112] By employing the above methods, the ability of the element to represent pure bending states can be enhanced, and the adaptability of the element under complex curved shell plate and distorted mesh conditions can be improved, thereby reducing the shear self-locking problem in traditional Mindlin plate and shell elements and improving the computational accuracy in the analysis of thin-walled ship structures.

[0113] Step S3: Using the traditional Mindlin theory, construct the out-of-plane bending stiffness.

[0114] The displacement mode of out-of-plane bending is as follows:

[0115] ,

[0116] The strain matrix for out-of-plane bending is:

[0117] ,

[0118] The stress matrix is:

[0119] ,

[0120] The out-of-plane bending stiffness is:

[0121] .

[0122] Step S4: Introduce geometric correction factors and hourglass control stiffness to construct out-of-plane transverse shear stiffness based on Mindlin theory.

[0123] The strain matrix for out-of-plane transverse shear is:

[0124] ,

[0125] The stress matrix is:

[0126] ,

[0127] The out-of-plane transverse shear stiffness is:

[0128] .

[0129] To address the accuracy issues of Mindlin theory in solving thin plate problems, this invention introduces a thickness-related geometric factor. Used for correction When the shell thickness is large, the corrected Approaching the Mindlin theoretical value for thick plates; when the plate / shell thickness is small, the corrected value is... It is close to the theoretical value of Kirchhoff for thin plates.

[0130] An hourglass stiffness is introduced to solve the hourglass problem when calculating the stiffness of plate and shell elements using reduced integrals.

[0131] ,

[0132] ,

[0133] The final transverse shear stiffness is:

[0134] .

[0135] Step S5: Construct the in-plane rotational stiffness by multiplying half of the diagonal value of the out-of-plane transverse shear stiffness by a correction factor.

[0136] ,

[0137] in, This represents the correction factor. It represents half the diagonal value of the out-of-plane transverse shear stiffness.

[0138] Step S6: Based on the degrees of freedom corresponding to different stiffnesses, assemble the membrane stiffness, out-of-plane bending stiffness, out-of-plane transverse shear stiffness, and in-plane rotational stiffness to finally form the modified plate and shell unit stiffness matrix of this invention.

[0139] .

[0140] The following section, using a typical stiffened plate structure of a ship as an example, further illustrates the application effect of the modified four-node plate and shell unit proposed in this invention in actual engineering.

[0141] This invention selects a locally stiffened deck structure as the analysis object. The structure consists of a panel and multiple longitudinal stiffeners. Both the panel and the stiffeners are discretized using the modified four-node shell element proposed in this invention. This type of structure is widely used in thin-walled structural areas of ships, such as decks, bulkheads, and double bottoms, and exhibits typical stress characteristics of thin-walled shells.

[0142] In this embodiment of the invention, the stiffened plate structure has a panel length of 10.8m, a width of 6.75m, and a panel thickness of 10mm; the stiffeners are evenly arranged along the width of the plate, with a spacing of 0.75m and a thickness of 6mm. The structural material is steel commonly used in ship structures, with a Young's modulus of 2.1 × 10⁻⁶. 11 Pa, Poisson's ratio is 0.3, and the material density is 7850 kg / m³. 3 To simulate the support connections in an actual ship structure, fixed supports are applied to the edges of the panels; simultaneously, a uniformly distributed pressure load perpendicular to the plane of the panels is applied to the intermediate grid area to simulate the external loads experienced by the ship structure under actual service conditions, such as... Figure 3 As shown.

[0143] The modified four-node plate-shell element proposed in this invention is used to perform finite element discretization and numerical calculation on the entire stiffened plate structure. Figure 4 The image shows the displacement cloud diagram of a ship stiffening plate structure obtained using the modified four-node plate shell unit of this invention.

[0144] Calculation results show that the modified four-node shell element proposed in this invention can effectively suppress the shear locking phenomenon of traditional Mindlin shell elements in the analysis of thin-walled structures. In the bending analysis of thin-walled stiffened plate structures, this invention can effectively reduce the problem of excessive structural stiffness caused by shear locking in traditional Mindlin shell elements, thereby improving the accuracy of structural displacement and deformation analysis. For local bending regions in stiffened plate structures, this invention can more accurately describe the bending deformation and stress distribution of the shell structure. When the structure has elements with a large aspect ratio or local mesh distortion, this invention can still maintain good numerical stability and calculation accuracy. Furthermore, this invention maintains high computational efficiency while ensuring calculation accuracy, making it suitable for finite element analysis of large-scale ship shell structures.

[0145] The above engineering structural examples demonstrate that the modified four-node plate and shell element proposed in this invention can effectively improve the calculation accuracy and engineering applicability in the analysis of thin-walled stiffened ship structures, and is especially suitable for finite element analysis and engineering calculation of complex ship plate and shell structures.

[0146] In summary, the finite element analysis method for plate and shell elements in ship structures provided by this invention effectively alleviates the shear locking problem by introducing additional displacement terms into the membrane effect stiffness-displacement mode and eliminating additional degrees of freedom through static condensation; by introducing a geometric correction factor that varies with the plate and shell thickness to correct the transverse shear stiffness and constructing an hourglass control stiffness, a continuous transition between thick and thin plate theories is achieved, and numerical stability under reduced integrals is guaranteed; by constructing in-plane rotational stiffness based on the diagonal terms of the transverse shear stiffness matrix, the problem of stiffness matrix singularity is avoided; and by clarifying the correspondence between each stiffness component and nodal degrees of freedom, a clear and regular combination method of plate and shell element stiffness matrices is formed.

[0147] As another embodiment of the present invention, a finite element analysis system for ship structures is provided, comprising:

[0148] Memory is used to store executable instructions for a computer;

[0149] A processor, coupled to the memory, implements, when the processor executes the computer-executable instructions, the plate and shell element finite element analysis method for ship structures as described in any of the preceding descriptions.

[0150] like Figure 5As shown, the electronic device 10 may include: at least one processor 11, such as a CPU (Central Processing Unit), at least one communication interface 13, a memory 14, and at least one communication bus 12. The communication bus 12 is used to enable communication between these components. The communication interface 13 may include a display screen or a keyboard; optionally, the communication interface 13 may also include a standard wired interface or a wireless interface. The memory 14 may be high-speed RAM (Random Access Memory) or non-volatile memory, such as at least one disk drive. Optionally, the memory 14 may also be at least one storage device located remotely from the aforementioned processor 11. The memory 14 stores application programs, and the processor 11 calls the program code stored in the memory 14 to execute any of the aforementioned method steps.

[0151] The communication bus 12 can be a peripheral component interconnect (PCI) bus or an extended industry standard architecture (EISA) bus, etc. The communication bus 12 can be divided into an address bus, a data bus, a control bus, etc. For ease of representation, Figure 5 The bus is represented by a single thick line, but this does not mean that there is only one bus or one type of bus.

[0152] The memory 14 may include volatile memory, such as random-access memory (RAM); the memory may also include non-volatile memory, such as flash memory, hard disk drive (HDD) or solid-state drive (SSD); the memory 14 may also include a combination of the above types of memory.

[0153] The processor 11 can be a central processing unit (CPU), a network processor (NP), or a combination of CPU and NP.

[0154] The processor 11 may further include a hardware chip. This hardware chip may be an application-specific integrated circuit (ASIC), a programmable logic device (PLD), or a combination thereof. The PLD may be a complex programmable logic device (CPLD), a field-programmable gate array (FPGA), a generic array logic (GAL), or any combination thereof.

[0155] Optionally, memory 14 is also used to store program instructions. Processor 11 can invoke program instructions to implement the present invention. Figure 1 The embodiment illustrates a finite element analysis method for plate and shell elements in ship structures.

[0156] As another embodiment of the present invention, a computer-readable storage medium is provided having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the finite element analysis method for plate and shell elements for ship structures as described in any of the preceding claims.

[0157] In this embodiment of the invention, a non-transitory computer-readable storage medium is provided. The computer-readable storage medium stores computer-executable instructions that can execute the finite element analysis method for plate and shell elements of ship structures in any of the above method embodiments. The storage medium can be a magnetic disk, optical disk, read-only memory (ROM), random access memory (RAM), flash memory, hard disk drive (HDD), or solid-state drive (SSD), etc.; the storage medium may also include combinations of the above types of memory.

[0158] It is understood that the above embodiments are merely exemplary implementations used to illustrate the principles of the present invention, and the present invention is not limited thereto. For those skilled in the art, various modifications and improvements can be made without departing from the spirit and essence of the present invention, and these modifications and improvements are also considered to be within the scope of protection of the present invention.

Claims

1. A finite element analysis method for plate and shell elements in ship structures, characterized in that, include: A four-node plate and shell element finite element discrete model of a thin-walled ship structure is constructed, wherein each node is given three translational degrees of freedom and three rotational degrees of freedom. The block composition of the stiffness matrix of the basic Mindlin plate and shell element is determined, and the block composition includes membrane effect stiffness, out-of-plane bending stiffness, out-of-plane transverse shear stiffness, and in-plane rotational stiffness. The out-of-plane bending stiffness is constructed, and the membrane effect stiffness, out-of-plane transverse shear stiffness, and in-plane rotational stiffness are respectively modified. Based on the four-node plate and shell element finite element discrete model of the thin-walled structure of the ship, the out-of-plane bending stiffness, the modified membrane effect stiffness, the out-of-plane transverse shear stiffness, and the in-plane rotation stiffness are combined to obtain the modified plate and shell element stiffness matrix. The finite element method of the plate and shell elements of the ship structure is performed based on the modified plate and shell element stiffness matrix to obtain the displacement and / or stress response of the thin-walled ship structure.

2. The finite element analysis method for plate and shell elements in ship structures according to claim 1, characterized in that, The correction process for the stiffness of the thin film effect includes: An additional displacement term corresponding to the pure bending state is introduced into the displacement mode of the membrane effect stiffness to generate additional degrees of freedom; Construct the element equilibrium equations that include the nodal degrees of freedom and the additional degrees of freedom; The additional degrees of freedom are statically condensed and eliminated to obtain a corrected membrane effect stiffness matrix including nodal degrees of freedom.

3. The finite element analysis method for plate and shell elements in ship structures according to claim 1, characterized in that, The out-of-plane transverse shear stiffness is corrected by: Construct the out-of-plane transverse shear stiffness matrix K based on the reduced integral. ts ; The geometric correction factor α is determined based on the thickness h of the plate / shell, where 0 < α ≤ 1; Based on the geometric correction factor α and the out-of-plane transverse shear stiffness matrix K ts Obtain the corrected out-of-plane transverse shear stiffness matrix K' ts , where K' ts =α*K ts .

4. The finite element analysis method for plate and shell elements in ship structures according to claim 3, characterized in that, The geometric correction factor decreases as the thickness of the shell decreases and increases as the thickness of the shell increases.

5. The finite element analysis method for plate and shell elements in ship structures according to claim 3, characterized in that, Construct the out-of-plane transverse shear stiffness matrix K based on the reduced integral. ts ,include: Construct an hourglass to control stiffness; Based on the hourglass control stiffness and the reduced integral, construct the external transverse shear stiffness matrix K. ts The hourglass control stiffness can suppress zero-energy modes caused by reduced integrals.

6. The finite element analysis method for plate and shell elements in ship structures according to claim 1, characterized in that, The correction process for the in-plane rotational stiffness includes: The reference value of the in-plane rotational stiffness is determined based on the out-of-plane transverse shear stiffness matrix; The reference value of the in-plane rotational stiffness is corrected according to the in-plane rotational stiffness correction factor to obtain the corrected in-plane rotational stiffness.

7. The finite element analysis method for plate and shell elements in ship structures according to claim 1, characterized in that, Out-of-plane bending stiffness includes: The geometric relationship between bending strain and nodal degrees of freedom is determined based on the out-of-plane bending displacement mode in order to construct the bending strain matrix; Construct the bending stress matrix based on the material's elastic constitutive relation; The out-of-plane bending stiffness is constructed based on the bending strain matrix and the bending stress matrix.

8. The finite element analysis method for plate and shell elements in ship structures according to claim 1, characterized in that, Based on the four-node plate and shell element finite element discrete model of the thin-walled ship structure, the out-of-plane bending stiffness, the corrected membrane effect stiffness, the out-of-plane transverse shear stiffness, and the in-plane rotational stiffness are combined to obtain the corrected plate and shell element stiffness matrix, including: The thin film effect stiffness is assembled to two in-plane translational degrees of freedom of each node out of six degrees of freedom. The out-of-plane bending stiffness and out-of-plane transverse shear stiffness are assembled to one out-of-plane translational degree of freedom and two out-of-plane rotational degree of freedom of each node out of six degrees of freedom. The in-plane rotational stiffness is assembled to one in-plane rotational degree of freedom of each node out of six degrees of freedom, so as to form the modified plate and shell element stiffness matrix.

9. A finite element analysis system for ship structures, characterized in that, include: Memory is used to store executable instructions for a computer; A processor, coupled to the memory, implements the finite element analysis method for plate and shell elements for ship structures as described in any one of claims 1 to 8 when the processor executes the computer-executable instructions.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the finite element analysis method for plate and shell elements for ship structures as described in any one of claims 1 to 8.

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