Method for constructing stiffness matrix of finite element unit

Abstract

The invention discloses a method for constructing a finite element unit stiffness matrix, which is suitable for constructing a unit stiffness matrix of a finite element plane polygon unit and a three-dimensional polyhedron unit, and can directly construct the unit stiffness matrix for different unit types without firstly constructing a shape function on a standard unit. And then a real element shape function is obtained through coordinate transformation, so that the construction method for constructing the element stiffness matrix by different types of elements is simplified, the applicability of the element stiffness matrix construction method is improved, and the method is of great significance to finite element theoretical research and engineering application.

Description

Technical Field

[0001] The invention relates to the technical field of finite element structure calculation, and is particularly applicable to a method for constructing a finite element unit stiffness matrix. Background Art

[0002] The finite element method originated from the matrix displacement method of bar structures developed in the 1940s and 1950s. In 1956, Turner and others extended the idea of ​​the matrix displacement method of bar structures to solve plane problems in elasticity mechanics. In 1960, Clough named this method for solving plane problems in elasticity mechanics the "finite element method". Since then, the finite element method has developed rapidly and gradually matured. The finite element method is considered to be the most effective numerical method at present, with its outstanding advantages such as solid theoretical foundation, versatility and strong practicality.

[0003] When the finite element method is used to solve elastic mechanics problems, the steps generally include structural discretization, unit analysis, overall analysis and numerical solution. One of the main tasks of unit analysis is to construct the relationship between node force and node displacement, that is, to construct the unit stiffness matrix.

[0004] The current method for constructing the unit stiffness matrix is ​​to first construct the unit shape function, then derive the unit shape function to obtain the unit strain, obtain the unit stress from the unit strain through the constitutive relationship, and then construct the unit stiffness matrix based on the virtual displacement principle. In order to ensure the convergence of the finite element calculation, the shape function is required to be continuous on the boundaries of adjacent units; in order to obtain the unit strain through the derivation method, the shape function is required to be differentiable inside the unit.

[0005] For simple triangular three-node elements, simple linear shape functions can achieve the shape function's requirements for continuity and differentiability. For other types of elements, linear shape functions are difficult to apply. The commonly used method is to first construct a shape function on a standard element and then obtain the shape function of the real element through coordinate transformation. Therefore, the construction method and process of the element stiffness matrix are complex and not very universal. Summary of the invention

[0006] The purpose of the present invention is to provide a method for constructing a finite element unit stiffness matrix to solve the problem that it is difficult to directly construct a shape function that meets the requirements for other types of units other than triangular three-node units. It is necessary to first construct a shape function on a standard unit and then obtain the shape function of the real unit through coordinate transformation before continuing to construct the unit stiffness matrix.

[0007] To achieve the above object, the present invention adopts the following technical solutions: The method for constructing the finite element unit stiffness matrix of the present invention comprises the following steps: S1, using a plane quadrilateral four-node element, determine the elastic modulus E and Poisson's ratio v of the element; S2, number the nodes in a counterclockwise direction and determine the node coordinates; S3, construct the unit geometry matrix B; S4, construct the unit elasticity matrix D; S5, according to the formula K=B T DB constructs the element stiffness matrix K.

[0008] Furthermore, the geometry matrix B in step S3 is constructed as follows: in A is the unit area; and They are the x-coordinates and y-coordinates of node 1, node 2, node 3, and node 4 of the unit respectively.

[0009] Furthermore, the unit elasticity matrix D in step S4 is constructed as follows: in E is the unit elastic modulus, is the unit Poisson's ratio.

[0010] Furthermore, for plane problems in elasticity mechanics, the unit may adopt other types of units besides quadrilateral four-node units, and establish a unit stiffness matrix based on the idea of ​​establishing the unit stiffness matrix; for three-dimensional problems in elasticity mechanics, the unit may adopt polyhedral units, and establish a unit stiffness matrix based on the idea of ​​establishing the unit stiffness matrix.

[0011] The advantage of the present invention is that when finite elements are used for elastic mechanics analysis, unit stiffness matrices can be directly constructed for different types of units, without the need to first construct shape functions on standard units and then obtain real unit shape functions through coordinate transformation. This simplifies the unit stiffness matrix construction method for different types of units and improves the applicability of the unit stiffness matrix construction method. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 Schematic diagram of a quadrilateral four-node unit in the method of the present invention. DETAILED DESCRIPTION

[0013] The technical solutions in the embodiments of the present invention are described clearly and completely below. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0014] Taking a quadrilateral four-node unit as an example, the specific implementation process of the method for constructing the finite element unit stiffness matrix of the present invention is explained.

[0015] like Figure 1 As shown, this is a quadrilateral four-node element with an elastic modulus of 210 GPa, a Poisson's ratio of 0.3, and element node coordinates of (1, 0), (4, 1), (3, 4), and (0, 2).

[0016] For the quadrilateral four-node unit in this embodiment, the traditional method needs to first construct the unit shape function on the standard unit, and then obtain the shape function of the real unit through coordinate transformation. Then, the shape function of the real unit is derived to obtain the unit strain, and the unit stress is obtained from the unit strain through the constitutive relationship, and the unit stiffness matrix is ​​constructed by the virtual displacement principle. The whole process is complicated and not very applicable.

[0017] According to the unit stiffness matrix construction method described in this application, according to the construction formula of the geometric matrix B, the unit geometric matrix B can be obtained as follows: According to the construction formula of the unit elastic matrix D, the unit elastic matrix D can be obtained as: According to the construction formula of the unit stiffness matrix K, the unit stiffness matrix K can be obtained as: The method for constructing the finite element unit stiffness matrix of the present invention can be applied to the construction of the unit stiffness matrix of the finite element plane polygonal unit and the three-dimensional polyhedral unit. The method has a solid theoretical basis and strong applicability. For different types of unit types, the unit stiffness matrix can be directly constructed without first constructing the shape function on the standard unit and then obtaining the real unit shape function through coordinate transformation. This simplifies the construction method of the unit stiffness matrix constructed by different types of units, improves the applicability of the unit stiffness matrix construction method, and is of great significance to finite element theory research and engineering applications.

Claims

1. A method for constructing a finite element unit stiffness matrix, characterized in that: The following steps are involved: S1, using a plane quadrilateral four-node element, determine the elastic modulus E and Poisson's ratio v of the element; S2, number the nodes in a counterclockwise direction and determine the node coordinates; S3, construct the unit geometry matrix B; S4, construct the unit elasticity matrix D; S5, according to the formula K=B T DB constructs the element stiffness matrix K.

2. A method for constructing a finite element unit stiffness matrix according to claim 1, characterized in that: The unit geometry matrix B is constructed according to the following formula: in A is the unit area; and They are the x-coordinates and y-coordinates of node 1, node 2, node 3, and node 4 of the unit respectively.

3. The method for constructing a finite element unit stiffness matrix according to claim 1, characterized in that: In step S4, the unit elasticity matrix D is constructed according to the following formula: in E is the unit elastic modulus, is the unit Poisson's ratio.

4. The method for constructing a finite element unit stiffness matrix according to claims 1-4, characterized in that: For plane problems of elastic mechanics, the unit may adopt other types of units other than quadrilateral four-node units, and establish a unit stiffness matrix based on the idea of ​​establishing the unit stiffness matrix; For three-dimensional problems of elasticity mechanics, the unit can adopt a polyhedral unit, and establish a unit stiffness matrix based on the idea of ​​establishing the unit stiffness matrix.

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