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The finite element (FE) method is a widely used method for numerically solving differential equations arising in engineering and mathematical modeling. Typical engineering problems include the structural analysis, the heat transfer, the fluid anal-ysis, etc. In this book, the FE analysis for dealing with the static structural analysis is introduced. The finite element method (FEM) is a general numerical approach for solving partial differential equations. To solve a general engineering problem using the FEM, the continuum system is firstly discretized into finite elements, which are connected by nodes, and then a collection of discrete elements is used to represent the original continuum system. In the FE model, the element is defined as the minimum cell/part after discretization and the node is defined as the connection point between elements. It should be noted that in the FE analysis, the forces are passing through the nodes, not the elemental edges, which is different from that in the real engineering structure. Therefore, in the FE calculations, if the forces are applied on the edge, the surface or the body of the elements, the equivalent nodal forces need to be worked out, the details of which will be introduced in this book.

The history of the FEM can be briefly summarized as follows: in 1943, Courant first tried the idea of using several triangular to solve a continuum deformable body. In 1956, in the annual meeting of aerospace in New York, Clough et al. proposed a new idea which is the beginning of the finite element method, i.e., partitioning the structure into triangular and rectangular, forming the global stiffness matrix and then solving the system. In 1960, the word “finite element” was first proposed in the book “The finite element method in plane stress analysis”. In 1968, some mathematicians recog-nized the advantages of the finite element method and further developed the method to a new depth. In 1970s, the first nonlinear FE software—MARC (MSC.Software, Los Angeles, U.S.) was released in the market. In 1975, the nonlinear solver— ADINA (Automatic Dynamic Incremental Nonlinear Analysis) (ADINA R&D Inc., Massachusetts, U.S.) emerged in the market. In 1977, the Mechanical Dynamics Inc. (MDI) released the software ADAMS (MSC.Software, Los Angeles, U.S.), which is mainly used in the mechanical kinematics analysis, the dynamics simulation, etc. In 1978, the major FE software – Abaqus (Dassault SIMULIA, Rhode Island, U.S.) was released in the market, which still serves as one of the main software in the market nowadays. In 1988, Livermore Software Technology Corp. (LSTC) released the soft-ware—LS-DYNA (LSTC Inc., California, U.S.) which is mainly used for solving the nonlinear dynamical problem. In the twenty-first century, another major software— ANSYS(ANSYS Inc., Pennsylvania, U.S.), including the ANAYS Mechanical, the ANSYS Computational Fluid Dynamics (CFD), the ANSYS Workbench, etc., was released, which is a major competitor of Abaqus nowadays. From the history of the development of the FEM, it can be seen that the FEM originates and develops in US, which is partially due to the reason that the third industrial revolution occurred in U.S and consequently the computer programs and relevant software were rapidly developing in U.S. In China, there is also a time period when the national FE software develops very fast, i.e., during the period from 1950 to 1980s. Unfortunately, since the foreign software entered the China market, they start to dominate the market and leave no room/market for the continuous development of the national FE software. Nowadays, due to the barriers among different countries, the national FE software starts to develop again for the national security.