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Notes: 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.

Notes: This note presents a rational basis for a unified finite element algorithm capable of dealing with a wide range of fluid flow in both steady and transient cases. It is hoped that empiricism inherent in many previous approaches can be avoided and a sound basis provided. The algorithm permits the use of equal interpolation for all variables by avoiding the need for the Babuska-Brezzi constraints in regions where the flow is nearly incompressible.The success of the algorithm, which here is written for the non-conservative equation form, is demonstrated on several examples ranging from (nearly) incompressible through transonic regions to supersonic flows. Up to mild shocks such as those occurring in the examples presented in this paper, no 'artificial' viscosity is added at any stage.The algorithm extends some concepts introduced in an earlier paper.

Notes: This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h4) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L2 projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L2 projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.

Notes: In this second part of the paper, the issue of a posteriori error estimation is discussed. In particular, we derive a theorem showing the dependence of the effectivity index for the Zienkiewicz-Zhu error estimator on the convergence rate of the recovered solution. This shows that with superconvergent recovery the effectivity index tends asymptotically to unity. The superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the Zienkiewicz-Zhu error estimator to demonstrate accurate estimation of the exact error attainable. Numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient (stress) error estimation. Several examples of the performance of the error estimator in adaptive mesh refinement are also presented.

Notes: In this paper the necessary requirements for the good behaviour of shear constrained Reissner-Mindlin plate elements for thick and thin plate situations are re-interpreted and a simple explicit form of the substitute shear strain matrix is obtained. This extends the previous work of the authors presented in References 18 and 31. The general methodology is applied to the re-formulation of some well known quadrilateral plate elements and some new triangular and quadrilateral plate elements which show promising features. Some examples of the good behaviour of these elements are given.