In modern mechanical design, finite element analysis method is an indispensable and important means. 1956, M. J. Turner, R. W. Clough, H. C. Martin and L. J. Topp introduced a new calculation method at the annual meeting of Aviation Society held in new york, and extended the matrix displacement method to solve plane stress problems. They divide the continuous geometric model into triangular and rectangular "elements", give approximate displacement functions to the elements used, and then get the element stiffness matrix of the relationship between node force and node displacement. From 1954 to 1955, J. H. Argyris published a series of papers on energy principle and structural analysis in Journal of Aeronautical Engineering. 1960, Clough put forward the term finite element for the first time in a famous paper named Finite Element in Plane Stress Analysis, which was later widely quoted and became the standard title of this numerical method. At the same time, mathematicians developed approximate solutions of differential equations, including finite difference method, variational principle and weighted residual method, which laid a mathematical and theoretical foundation for the future development of finite element method. 1963, after the work of many people, such as J. F. Besseling, R. J. Melosh, R. E. Jones, R. H. Gallaher, T. H. H. Pian, people realized that the finite element method is a variation of Ritz approximation in variational principles, and thus developed various variational principles. In 1965, O. C. Zienkiewicz and Y. K. Cheung found that all field problems can be solved by the same steps as the finite element method of solid mechanics as long as they can be converted into corresponding variational forms. 1969 B. A. Szabo and G. C. Lee pointed out that the weighted residual method, especially Galerkin method, can be used to derive the standard finite element process for solving unstructured problems. Chinese mechanics workers have made many contributions to the initial development of finite element method, among which Chen Boping (structural matrix method), Qian Lingxi (complementary energy principle), Qian Weichang (generalized variational principle), Hu Haichang (generalized variational principle) and Feng Kang (finite element method theory) are famous.

The basic idea of finite element method is to transform the research object into a mathematical model similar to the original structure through discretization, and then solve the numerical calculation method of stress, displacement, strain and other parameters through a series of standardized steps, as shown in Figure 4- 19. Assuming that a simple function approximately represents the distribution law of displacement in the element, the mechanical characteristic relationship between node force and displacement is established by using mechanical theory (such as variational principle or virtual motion principle) or other methods, and a set of algebraic equations with node displacement as unknown quantity is obtained, so as to solve the displacement component of the node. Then the interpolation function is used to determine the field function on the cell set. Obviously, if the element meets the convergence requirements of the problem, with the decrease of the element size and the increase of the number of elements in the solution area, the approximate degree of the solution will continue to improve, and the approximate solution will eventually converge to the exact solution.

With the rapid development of computer technology, finite element method has become an effective method and means of mechanism analysis. The application field of finite element method has been involved in mechanical engineering, civil engineering, aviation structure, heat conduction, electromagnetic field, geomechanics and many other fields. It is suitable for almost all problems of continuous medium and field, and becomes an indispensable numerical analysis tool for scientific research and engineering design.

Figure 4- 19 General steps of establishing finite element model The calculation steps of finite element method can be summarized as three basic steps: grid division, element analysis and overall analysis.

(1) grid division.

The basic method of finite element method is to replace the original continuum with a group of finite elements. Therefore, it is necessary to simplify the elastic body first, and then divide the elastic body into discrete bodies composed of finite units. These units are connected by nodes. A set composed of cells, nodes and node lines is called a grid, as shown in Figure 4-20.

Figure 4-20 Element Analysis of Finite Element Mesh (2).

For elasticity, element analysis is to establish the relationship between node displacement and node force of each element. Because the node displacement of the element is regarded as the basic variable, it is necessary to determine an approximate expression of the internal displacement of the element, then calculate the strain and stress of the element, and then establish the relationship between the node force and the node displacement in the element.

(3) Overall analysis.

Figure 4-2 1 analyzes the rapid growth of electronic computer capacity as a whole. Nowadays, commercial finite element programs are more and more widely accepted by people, and people don't have to spend a lot of energy on programming. Moreover, the development of commercial finite element program also makes users get rid of manual meshing, simplifies the pre-processing process, and omits the program of inputting node coordinates and element connection information point by point. Moreover, a good man-machine conversation environment can be obtained through the on-screen menu mode, and a clear computer structure analysis can be obtained. Famous commercial limited programs include NASTRAN, ADFNA/ADINAT, ANSTS, Universe /MSAP, etc. The analysis scope and functions of these programs are different, so we should choose a reasonable program according to the different analysis scope when using them.