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The first edition of this book, published in 1991, focused on the conceptual and algorithmic development of the finite element method from the perspective of solution verification, that is, estimation and control of the errors of approximation in terms of the quantities of interest. Since that time the importance of solution verification became widely recognized. It is a key constituent of predictive computational science, the branch of computational science concerned with the prediction of physical events.

Predictive computational science embraces the formulation of mathematical models, definition of the quantities of interest, code and solution verification, definition of statistical sub‐models, calibration and validation of models, and forecasting physical events with quantified uncertainty. The second edition covers the main conceptual and algorithmic aspects of predictive computational science pertinent to solid mechanics. The formulation and application of design rules for mechanical and structural components subjected to cyclic loading are used for illustration.

Another objective in writing the first edition was to make some fundamentally important results of research in the field of applied mathematics accessible to the engineering community. Speaking generally, engineers and mathematicians view the finite element method very differently. Engineers see the method as a way to construct a numerical problem the solution of which is expected to provide quantitative information about the response of some physical system, for example a structural shell, to some form of excitation, such as the application of loads. Their view tends to be element‐oriented: They tend to believe that sufficiently clever formulation of elements can overcome the various shortcomings of the method.

Mathematicians, on the other hand, view the finite element method as a method for approximating the exact solution of differential equations cast in a variational form. Mathematicians focus on a priori and a posteriori error estimation and error control. In the 1970s adaptive procedures were developed for the construction of sequences of finite element mesh such that the corresponding solutions converged to the exact solution in energy norm at optimal or nearly optimal rates. An alternative way of achieving convergence in energy norm through increasing the polynomial degrees of the elements on a fixed mesh was proven in 1981. The possibility of achieving exponential rates of convergence in energy norm for an important class of problems, that includes the problem of elasticity, was proven and demonstrated in 1984. This required the construction of sequences of finite element mesh and optimal assignment of polynomial degrees.

Superconvergent methods of extraction of certain quantities of interest (such as the stress intensity factor) from finite element solutions were developed by 1984.

These developments were fundamentally important milestones in a journey toward the emergence of predictive computational science. Our primary objective in publishing this second edition is to provide engineering analysts and software developers a comprehensive account of the conceptual and algorithmic aspects of verification, validation and uncertainty quantification illustrated by examples.

Quantification of uncertainty involves the application of methods of data analysis. A brief introduction to the fundamentals of data analysis is presented in this second edition.

We recommend this book to students, engineers and analysts who seek Professional Simulation Engineer (PSE) certification.

We would like to thank Dr. Ricardo Actis for many useful discussions, advice and assistance provided over many years; Dr. Börje Andersson for providing valuable convergence data relating to the solution of an interesting model problem of elasticity, and Professor Raul Tempone for guidance in connection with the application of data analysis procedures.

Barna Szabó and Ivo Babuška