The displacement-type finite element approach—From art to science

Abstract

The finite element method originated in the aerospace industry in the mid- 1950s to solve practical stress analysis problems associated with the struc- tural design of aerospace vehicles. It is today the most overwhelmingly popular analysis and design tool in structural and solid mechanics and is being extensively applied to a very wide spectrum of engineering science, e.g. fluid mechanics, heat transfer and electromagnetics. In its formative years, the development of the method was guided mostly by engineering intuition, heuristic judgment and trial and error experimentation and vali- dation. Its achievements have been remarkable and there are now very power- ful general-purpose software codes that make a variety of analyses and design tasks routinely simple, that were once considered to be intractable. This is not to say that the progress of the method has been free of hurdles, especi- ally in finding a complete scientific basis for it. Based on the recent stud -ies of this author and his colleagues, this review attempts to provide a more complete paradigmatic understanding of the issues involved. Concepts such as consistency and ( variational) correctness are introduced. These together with the more familiar completeness and continuity requirements are then employed to guide the construction of error-free robust finite elements and also provide procedures to perform a priori error estimates for the quality of approximation. These C-concepts, as we shall call them, are elucidated and their relevance to the design of several key elements commonly found in general-purpose packages used by the aerospace, automobile and mechanical engineering industries is briefly covered. The article reviews what has been achieved in areas where the C-concepts can be applied fruitfully in the study of the displacement type finite element method.