Computational mechanics today
Authors: Zienkiewicz O. C. 
Published: November 2005

Computational Mechanics or Computational Applied Science is today the base on which most of the achievements of engineering and physics are built. Its concern is the solution of complex mathematical theories in numerical terms, without which the translation of these into practical artifacts would be impossible. Indeed, by providing such quantitative measures it enhances the understanding of the physical phenomena and stimulates further development of theory and physical experiment.

Most of the theory underlying physical phenomena is cast in terms of, often involved, differential equations for which closed forms of solution are seldom possible. Numerical approximation or discretization processes are necessary for quantitative solution. Here the first steps were taken at the start of this century by the pioneering work of Richardson introducing finite difference approximations. The invention of relaxation methods by Southwell during the Second World War allowed many practical solutions to be achieved. However, it was the advent of the electronic digital computer that marked the turning point in Computational Mechanics. The dramatic escalation of the power of these machines, which still continues today, allowed the development of the field of Computational Mechanics as we know it.

It is through this computer power that such methods of approximations as finite elements, finite differences, boundary solutions and spectral processes became a practical reality, though each was anticipated in the pre computer area. It is not surprising therefore, that the mathematical foundations and the full development of such methods have been accomplished only relatively recently.

Today we see the field of activity subdivided between those specializing in the development of the various computational approximation processes and those seeking optimal numerical solutions for their particular field of application. It is the objective of this Congress and indeed of the International Association of Computational Mechanics to provide a forum at which an interdisciplinary exchange of information can take place between the various sections and disciplines of the whole field. Indeed, this is the way progress can best be achieved. Recent history indicates that substantial advances are as frequently made due to a method seeking a new application as through a problem requiring a solution.

In recent history we have seen on occasion a liaison of a particular computational approximation method to a field of application occurring through historical accident. Here the intimate association of the finite method and the field of SOLID MECHANICS (CSM or Computational Solid Mechanics) and that of the finite differences with FLUID DYNAMICS (CDF or Computational Fluid Dynamics) can be observed as classical examples. Today the advent of new application fields and a better understanding of the approximation theory are helping to break down the barriers and ensure a more rational matching of objectives and methods. We shall illustrate the lecture with examples of such recent progress and state some possibilities as yet unexplored. Indeed, we are sure that the Congress will achieve in much more detail the same aims.

This presentation stresses the essential unity of the subject and discusses some areas where progress and research are currently active. Two of such, adaptive error controlled analysis and treatment of hyperbolic (fluid) problems, are singled out due to their wide applications.