An Introduction to the Finite Element Method

The Finite Element Method (FEM) can be seen as a numerical computation method for the solution of problems of mathematical physics. The concept of mathematical physics must clarify the universality of the method. To the problems of mathematical physics, which nowadays to a large extend are solved with the help of FEM, belong primarily:

• problems of structural mechanics (statics and dynamics), where we understand here under the term structure a structural framework in the broadest sense, not only special types of structures such as trusses and frames, surface or body structures, but also complex structures composed of different types of other structures,
• stationary and unsteady field problems from the theory of heat conduction, fluid mechanics, electromagnetism, and acoustic wave theory.

The equations, which the problems of mathematical physics describe, are two- and three-dimensional problems of partial differential equations. The FEM can be used for the solution of such differential equations. When applied to problems of structural mechanics, the basic idea of FEM is that the entire structure is decomposed into a multitude of small elements whose mechanical behavior is known either approximately or exactly. This defines the unknowns of the problem at a multitude of discrete points on the structure rather than having continuous functions. From the differential equations arise mostly linear systems of equations, and it is possible, to leave to the computer not only the solution of the resulting systems of equations but also their generation, and thus to automate the whole process.

The starting point of the historical development of the modern structural mechanics is the theory of rod and frame structures, which was developed over the last two centuries (Maxwell, Castigliano, Mohr, etc.) This theory also forms the cornerstone of the matrix method of rod statics, which in turn became the starting point for the FEM. Until the beginning of the last century, the practical structure calculation focused on the force method, in which only the forces as unknown in the equation occurred. Around this time, the displacement method was theoretically formulated so far that already in 1926 by Ostenfeld a textbook with the title "The deformation method" was published. The displacement quantities form the unknown parameters, just like with the FEM. It can be said that the deformation method developed at that time for rods and beams can be regarded as the forerunner of FEM.

In the above simulation, Gaea is used as solver and Paraview is used for the visualization.

In the period shortly after the World War II, a main focus of development was on the methodology for reducing the number of unknowns and for solving systems of equations using structural methods. The calculation of complex structures was always limited by the computational effort required to solve the linear systems of equations. At the beginning of the 50s, the first practical digital computers appeared, their application in the structure calculation was characterized by the fact that the "hand" calculation methods available so far were directly translated without taking into consideration the special possibilities of the computers. The latter was only achieved by the consistent application of the matrix calculation already in the problem formulation and not only in the solution of the equation systems.

Two publications can be viewed as starting points for the development of the FEM: Argyris, Kelsey, "Energy Theorems and Structural Analysis" and Turner, Clough, Martin, and Topp, "Stiffness and Deflection Analysis of Complex Structures". In both, the matrix calculation is used to discretize not only frameworks but also continuum problems in a computer-appropriate way.

The subsequent development of the FEM was characterized by the work dealing with the formulation of various types of elements. After the applications in the field of stress and strain analysis in the linear elastic range under static loads, the extensions of the method came to problems of linear elastodynamics and stability theory, as well as the general field problems of mathematical physics (heat conduction, fluid mechanic, etc.).

During the 1960s, the commercial development of the first large multipurpose program systems took place, the very existence of which enabled the widespread use of FEM in almost all branches of industry. In the following period, the development of FEM went in the direction of non-linear problems in terms of material (plasticity, creep) and geometry (large deformations that are no small compared to the geometrical dimensions of the structure). Moreover, much research has been done to date with improvements in elemental representation, numerical methods, and the use of FEM in interdisciplinary fields.