A method for modifying quadrilateral shell elements in finite element analysis of thin-walled aircraft structures

Abstract

The application belongs to the technical field of aircraft thin-walled structure design, and particularly relates to a correction method of quadrilateral shell element in finite element analysis of aircraft thin-walled structure, which comprises the following steps: step one, establishing an element coordinate system for the quadrilateral shell element; step two, establishing a balance equation of a plane stress element related to in-plane deformation under the element coordinate system, and calculating a plane stress element stiffness matrix; step three, establishing a balance equation of a bending plate element related to out-of-plane deformation under the element coordinate system, and calculating a bending plate element stiffness matrix; step four, applying an in-plane rotation penalty term to the rotational stiffness of the quadrilateral shell element, calculating the influence of the in-plane rotation penalty term on the strain energy function of the quadrilateral shell element under the element coordinate system, and obtaining an in-plane rotation stiffness matrix; step five, calculating the stiffness matrix of the quadrilateral shell element under the element coordinate system; and step six, performing coordinate transformation to obtain the stiffness matrix of the quadrilateral shell element under the overall coordinate system.

Description

Technical Field

[0001] This application belongs to the field of aircraft thin-walled structure design technology, specifically relating to a method for correcting quadrilateral shell elements in finite element analysis of aircraft thin-walled structures. Background Technology

[0002] The main load-bearing structures of aircraft, such as fuselage, wing panels, frames, and beams, are mostly thin-walled structures. These structures are primarily connected by rivets, bolts, or welding, and withstand complex loads such as in-plane pressure and out-of-plane bending. In finite element analysis and design, shell elements are commonly used to simulate thin-walled structures. Among them, quadrilateral shell elements have higher computational accuracy and better directional adaptability than triangular shell elements, making them the preferred choice for simulating thin-walled structures.

[0003] In finite element theory, a node has 6 degrees of freedom, including 3 translational degrees of freedom (UX, UY, UZ) and 3 rotational degrees of freedom (RX, RY, RZ). However, in classical shell element theory, a quadrilateral shell element is composed of plane stress elements describing in-plane deformation and bending plate elements describing out-of-plane deformation. The plane stress element node has 2 degrees of freedom (UX, UY), and the bending plate element node has 3 degrees of freedom (UZ, RX, RY). That is, a quadrilateral shell element node only has 5 degrees of freedom (UX, UY, UZ, RX, RY), lacking one in-plane rotational degree of freedom (RZ). This leads to the following problems when using quadrilateral shell elements to simulate thin-walled structures:

[0004] In-plane torque cannot be accurately transmitted between non-coplanar shell elements;

[0005] When connecting to a beam element with 6 degrees of freedom, a degree-of-freedom mismatch problem may occur;

[0006] Under certain constrained connections or boundary conditions, the stiffness matrix may become singular, leading to abnormal solution results.

[0007] This application is made in view of the aforementioned technical deficiencies. Summary of the Invention

[0008] The purpose of this application is to provide a correction method for quadrilateral shell elements in the finite element analysis of thin-walled structures of aircraft. The method considers the in-plane rotational degrees of freedom to correct the quadrilateral shell elements, making up for the deficiency of missing in-plane rotational degrees of freedom at the nodes of the quadrilateral shell elements, thereby improving the accuracy of the simulation of thin-walled structures and thus improving the accuracy and efficiency of the design of thin-walled structures in aircraft.

[0009] The technical solution of this application is:

[0010] A method for correcting quadrilateral shell elements in finite element analysis of thin-walled aircraft structures includes:

[0011] Step 1: Establish the element coordinate system for the quadrilateral shell element;

[0012] Step 2: In the element coordinate system, establish the equilibrium equations for the plane stress element related to in-plane deformation, and calculate the stiffness matrix of the plane stress element. ;

[0013] Step 3: In the element coordinate system, establish the equilibrium equations for the bending plate element related to out-of-plane deformation, and calculate the stiffness matrix of the bending plate element. ;

[0014] Step 4: Apply an in-plane rotational penalty term to the rotational stiffness of the quadrilateral shell element, and calculate the effect of the in-plane rotational penalty term on the strain energy function of the quadrilateral shell element in the element coordinate system. The in-plane rotational stiffness matrix is ​​obtained. ;

[0015] Step 5: Calculate the stiffness matrix of the quadrilateral shell element in the element coordinate system. ;

[0016] Step 6: Assess the stiffness matrix of the quadrilateral shell element in the element coordinate system. By performing coordinate transformation, the stiffness matrix of the quadrilateral shell element in the global coordinate system is obtained. .

[0017] Optionally, in the above-mentioned correction method for quadrilateral shell elements in the finite element analysis of thin-walled aircraft structures, the four nodes of the quadrilateral shell element are G1, G2, G3, and G4.

[0018] In step one, the origin of the coordinate system is the intersection point o of the line connecting G1 and G3 and the line connecting G2 and G4, and the bisector of the angle between the line connecting oG3 and oG2 is the x-axis.

[0019] The y-axis of the unit coordinate system lies in the plane of the quadrilateral shell unit, and the z-axis is perpendicular to the plane containing the x-axis and y-axis. The positive directions conform to the right-hand rule according to G1, G2, G3, and G4.

[0020] Optionally, in the above-mentioned method for correcting quadrilateral shell elements in the finite element analysis of thin-walled aircraft structures, step two specifically comprises:

[0021] ;

[0022] in,

[0023] , , , The force loads on the four nodes of the plane stress element;

[0024] , , , The displacements of the four nodes of the plane stress element are given.

[0025] Optionally, in the above-mentioned method for correcting quadrilateral shell elements in the finite element analysis of thin-walled aircraft structures, step three specifically involves:

[0026] ;

[0027] in,

[0028] , , , The force loads on the four nodes of the bending plate element;

[0029] , , , The displacements of the four nodes of the bending plate element are given.

[0030] Optionally, in the above-mentioned method for correcting quadrilateral shell elements in the finite element analysis of thin-walled aircraft structures, step four specifically involves:

[0031] ;

[0032] ;

[0033] in,

[0034] The shear modulus of the quadrilateral shell element;

[0035] Let be the area of ​​the quadrilateral shell unit;

[0036] Let be the in-plane rotation angle of the quadrilateral shell element;

[0037] , The displacement of the quadrilateral shell element along the x and y axes;

[0038] The thickness of the quadrilateral shell unit;

[0039] The strain is that of the quadrilateral shell element.

[0040] Optionally, in the above-mentioned method for correcting quadrilateral shell elements in the finite element analysis of thin-walled aircraft structures, step six specifically involves:

[0041] ;

[0042] ;

[0043] ;

[0044] in,

[0045] , For intermediate calculation matrix;

[0046] This is the direction cosine matrix between the unit coordinate system and the global coordinate system.

[0047] This application has at least the following beneficial technical effects:

[0048] This paper presents a method for correcting quadrilateral shell elements in the finite element analysis of thin-walled aircraft structures. By introducing a minimal in-plane rotation penalty term, an in-plane rotation stiffness matrix is ​​obtained, which corrects the stiffness of the quadrilateral shell elements. This completes the in-plane rotation mechanical properties of the quadrilateral shell elements without substantially interfering with their main in-plane and bending mechanical properties. It can effectively compensate for the deficiency of missing in-plane rotation degrees of freedom at the nodes of quadrilateral shell elements, improve the accuracy of thin-walled structure simulation, and thus enhance the accuracy and efficiency of thin-walled structure design in aircraft. Attached Figure Description

[0049] Figure 1 This is a flowchart of the method for correcting quadrilateral shell elements in the finite element analysis of thin-walled aircraft structures provided in this application embodiment;

[0050] Figure 2 This is a schematic diagram of establishing an element coordinate system for a quadrilateral shell element provided in an embodiment of this application.

[0051] To better illustrate this embodiment, some content in the accompanying drawings may be omitted, enlarged, or reduced. They are for illustrative purposes only and should not be construed as limiting the scope of this application. Detailed Implementation

[0052] To make the technical solution and advantages of this application clearer, the technical solution of this application will be described in a clearer and more complete manner below with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are only some embodiments of this application, and are only used to explain this application, not to limit this application. It should be noted that, for ease of description, only the parts related to this application are shown in the accompanying drawings, and other related parts can be referred to the general design.

[0053] Furthermore, unless otherwise defined, the technical or scientific terms used in this application description shall have the ordinary meaning understood by one of ordinary skill in the art to which this application pertains. The word "comprising" as used in this application description indicates that the concept preceding the word encompasses the concepts listed following the word and their equivalents, without excluding other related concepts.

[0054] A method for correcting quadrilateral shell elements in finite element analysis of aircraft thin-walled structures is proposed. Based on the stiffness matrix of the quadrilateral shell element, an in-plane rotational degree of freedom is introduced. Using a stiffness matrix with small parameter penalties, the quadrilateral shell element is corrected. Figure 1 As shown.

[0055] Step 1: Establish the element coordinate system for the quadrilateral shell element.

[0056] The four nodes of the quadrilateral shell element are G1, G2, G3, and G4, as follows: Figure 2 As shown.

[0057] Establish a unit coordinate system with the intersection point o of the line connecting G1 and G3 and the line connecting G2 and G4 as the origin, and the angle bisector of the angle between the line connecting oG3 and oG2 as the x-axis.

[0058] The y-axis of the unit coordinate system lies in the plane of the quadrilateral shell unit, and the z-axis is perpendicular to the plane containing the x-axis and y-axis. The positive directions conform to the right-hand rule according to G1, G2, G3, and G4.

[0059] Shape function of quadrilateral shell element , , , It can be represented as:

[0060] ;

[0061] ;

[0062] ;

[0063] ;

[0064] in, , Let be the natural coordinates defined on the interval [−1, 1].

[0065] Step 2: In the element coordinate system, establish the equilibrium equations for the plane stress element related to in-plane deformation, and calculate the stiffness matrix of the plane stress element. .

[0066] ;

[0067] in,

[0068] , , , The force loads at the four nodes of the plane stress element include the loads along the x-axis. Load along the y-axis ;

[0069] , , , The displacements of the four nodes of the plane stress element include the displacements along the x-axis. Displacement along the y-axis .

[0070] Step 3: In the element coordinate system, establish the equilibrium equations for the bending plate element related to out-of-plane deformation, and calculate the stiffness matrix of the bending plate element. .

[0071] ;

[0072] in,

[0073] , , , The force loads on the four nodes of the bending plate element include the loads along the z-axis. Bending moment about the x-axis Bending moment about the y-axis ;

[0074] , , , The displacements of the four nodes of the bending plate element include the displacements along the z-axis. Rotation angle around the x-axis Rotation angle around the y-axis .

[0075] Step 4: Apply an in-plane rotational penalty term to the rotational stiffness of the quadrilateral shell element, and calculate the effect of the in-plane rotational penalty term on the strain energy function of the quadrilateral shell element in the element coordinate system. The in-plane rotational stiffness matrix is ​​obtained. .

[0076] ;

[0077] ;

[0078] in,

[0079] The shear modulus of the quadrilateral shell element;

[0080] Let be the area of ​​the quadrilateral shell unit;

[0081] Let be the in-plane rotation angle of the quadrilateral shell element;

[0082] , The displacement of the quadrilateral shell element along the x and y axes;

[0083] The thickness of the quadrilateral shell unit;

[0084] The strain is that of the quadrilateral shell element.

[0085] Step 5: Calculate the stiffness matrix of the quadrilateral shell element in the element coordinate system. .

[0086] Step 6: Assess the stiffness matrix of the quadrilateral shell element in the element coordinate system. By performing coordinate transformation, the stiffness matrix of the quadrilateral shell element in the global coordinate system is obtained. This completes the correction of the quadrilateral shell unit.

[0087] ;

[0088] ;

[0089] ;

[0090] in,

[0091] , For intermediate calculation matrix;

[0092] This is the direction cosine matrix between the unit coordinate system and the global coordinate system.

[0093] Coordinates in a unit coordinate system Corresponding to coordinates in the global coordinate system Then we have:

[0094] .

[0095] The above-described embodiment discloses a method for correcting quadrilateral shell elements in the finite element analysis of thin-walled aircraft structures. By introducing a minimal in-plane rotation penalty term, an in-plane rotation stiffness matrix is ​​obtained, which corrects the stiffness of the quadrilateral shell elements. This completes the in-plane rotation mechanical properties of the quadrilateral shell elements without substantially interfering with their main in-plane and bending mechanical properties. It can effectively compensate for the deficiency of missing in-plane rotation degrees of freedom at the nodes of quadrilateral shell elements, improve the accuracy of thin-walled structure simulation, and thus enhance the accuracy and efficiency of thin-walled structure design in aircraft.

[0096] The technical solution of this application has been described in conjunction with the preferred embodiments shown in the accompanying drawings. Those skilled in the art should understand that the scope of protection of this application is obviously not limited to these specific embodiments. Without departing from the principles of this application, those skilled in the art can make equivalent changes or substitutions to the relevant technical features, and the technical solutions after these changes or substitutions will all fall within the scope of protection of this application.

Claims

1. A method for modifying quadrilateral shell elements in finite element analysis of thin-walled structures of an aircraft, characterized in that, include: Step 1: Establish the element coordinate system for the quadrilateral shell element; Step two, in the unit coordinate system, establish the balance equation of the plane stress element related to the in-plane deformation, calculate the plane stress element stiffness matrix ; Step three, in the unit coordinate system, the balance equation of the curved plate unit related to the out-of-plane deformation is established, and the stiffness matrix of the curved plate unit is calculated ; Step four, a in-plane rotation penalty term is imposed on the rotational stiffness of the quadrilateral shell element, and the influence of the in-plane rotation penalty term on the strain energy function of the quadrilateral shell element is calculated in the element coordinate system , to obtain an in-plane rotation stiffness matrix ​ Step five, calculate the stiffness matrix of the quadrilateral shell element in the element coordinate system ; Step six, stiffness matrix of quadrilateral shell element in element coordinate system Perform coordinate transformation to obtain stiffness matrix of quadrilateral shell element in global coordinate system ; Step two is as follows: ； in, , , , Fijare the force loads on the four nodes of the plane stress element; , , , are the displacements of the four nodes of the plane stress element; Step three specifically involves: ； in, , , , Fijare the force loads on the four nodes of the curved plate element. , , , are the displacements of the four nodes of the curved plate element; Step four is as follows: ； ； in, G is the shear modulus for a quadrilateral shell element; A is the area of the quadrilateral shell element; is the in-plane rotation angle for the quadrilateral shell element; , are the displacements of the quadrilateral shell element along the x, y axes; The thickness of the quadrilateral shell unit; The strain is that of the quadrilateral shell element.

2. The method for correcting quadrilateral shell elements in the finite element analysis of thin-walled aircraft structures according to claim 1, characterized in that, The four nodes of the quadrilateral shell element are G1, G2, G3, and G4; In step one, the origin of the coordinate system is the intersection point o of the line connecting G1 and G3 and the line connecting G2 and G4, and the bisector of the angle between the line connecting oG3 and oG2 is the x-axis. The y-axis of the unit coordinate system lies in the plane of the quadrilateral shell unit, and the z-axis is perpendicular to the plane containing the x-axis and y-axis. The positive directions conform to the right-hand rule according to G1, G2, G3, and G4.

3. The method for correcting quadrilateral shell elements in the finite element analysis of thin-walled aircraft structures according to claim 2, characterized in that, Step six specifically involves: ； ； ； in, , For intermediate calculation matrix; This is the direction cosine matrix between the unit coordinate system and the global coordinate system.

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