Chapter 1 Introduction
1.1 Historical background
In 1909 Ritz developed an effective method [5] for the approximate solution of problems in the mechanics of deformable solids. It includes an approximation of energy functional by the known functions with unknown coefficients. Minimisation of functional in relation to each unknown leads to the system of equations from which the unknown coefficients may be determined. One from the main restrictions in the Ritz method is that functions used should satisfy to the boundary conditions of the problem.
In 1943 Courant considerably increased possibilities of the Ritz method by introduction of the special linear functions defined over triangular regions and applied the method for the solution of torsion problems [6]. As unknowns, the values of functions in the node points of triangular regions were chosen. Thus, the main restriction of the Ritz functions a satisfaction to the boundary conditions was eliminated. The Ritz method together with the Courant modification is similar with FEM proposed independently by Clough many years later introducing for the first time in 1960 the term finite element in the paper The finite element method in plane stress analysis [7]. The main reason of wide spreading of FEM in 1960 is the possibility to use computers for the big volume of computations required by FEM. However, Courant did not have such possibility in 1943.
An important contribution was brought into FEM development by the papers of Argyris [8], Turner [9], Martin [9], Hrennikov [10] and many others. The first book on FEM, which can be examined as textbook, was published in 1967 by Zienkiewicz and Cheung [11] and called The finite element method in structural and continuum mechanics. This book presents the broad interpretation of the method and its applicability to any general field problems. Although the method has been extensively used previously in the field of structural mechanics, it has been successfully applied now for the solution of several other types of engineering problems like heat conduction, fluid dynamics, electric and magnetic fields, and others.
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