The method of weighted residuals and variational principles, with application in fluid mechanics, heat and mass transfer

This classic book covers the solution of differential equations in science and engineering in such as way as to provide an introduction for novices before progressing toward increasingly more difficult problems. The Method of Weighted Residuals and Variational Principles describes variational principles, including how to find them and how to use them to construct error bounds and create stationary principles. The book also illustrates how to use simple methods to find approximate solutions, shows how to use the finite element method for more complex problems, and provides detailed information on error bounds. Problem sets make this book ideal for self-study or as a course text.

Recent years have seen a growing trend to derive models of macroscopic phenomena encountered in the fields of engineering, physics, chemistry, ecology, self-organisation theory and econophysics from various variational or extremum principles. Through the link between the integral extremum of a functional and the local extremum of a function (explicit, for example, in the Pontryagin's maximum principle variational and extremum principles are mutually related. Thus it makes sense to consider them within a common context. The main goal of Variational and Extremum Principles in Macroscopic Systems is to collect various mathematical formulations and examples of physical reasoning that involve both basic theoretical aspects and applications of variational and extremum approaches to systems of the macroscopic world. The first part of the book is focused on the theory, whereas the second focuses on applications. The unifying variational approach is used to derive the balance or conservation equations, phenomenological equations linking fluxes and forces, equations of change for processes with coupled transfer of energy and substance, and optimal conditions for energy management. - A unique multidisciplinary synthesis of variational and extremum principles in theory and application - A comprehensive review of current and past achievements in variational formulations for macroscopic processes - Uses Lagrangian and Hamiltonian formalisms as a basis for the exposition of novel approaches to transfer and conversion of thermal, solar and chemical energy

The Finite Element Method: Fundamentals and Applications demonstrates the generality of the finite element method by providing a unified treatment of fundamentals and a broad coverage of applications. Topics covered include field problems and their approximate solutions; the variational method based on the Hilbert space; and the Ritz finite element method. Finite element applications in solid and structural mechanics are also discussed. Comprised of 16 chapters, this book begins with an introduction to the formulation and classification of physical problems, followed by a review of field or continuum problems and their approximate solutions by the method of trial functions. It is shown that the finite element method is a subclass of the method of trial functions and that a finite element formulation can, in principle, be developed for most trial function procedures. Variational and residual trial function methods are considered in some detail and their convergence is examined. After discussing the calculus of variations, both in classical and Hilbert space form, the fundamentals of the finite element method are analyzed. The variational approach is illustrated by outlining the Ritz finite element method. The application of the finite element method to solid and structural mechanics is also considered. This monograph will appeal to undergraduate and graduate students, engineers, scientists, and applied mathematicians.

Nuclear Science and Technology, Volume 10: Variational Methods in Nuclear Reactor Physics presents the mathematical methods of a variational origin that are useful in obtaining approximate solutions to science and engineering problems. This book is composed of five chapters and begins with a discussion on the variation principles for physical systems described by both inhomogeneous and homogeneous equations to develop a generalized perturbation theory. Chapter 2 deals with the applications of variational estimates and generalized perturbation theory to neutron transport problems. Chapter 3 covers the variation principles of the Lagrangian form that are constructed for a general, linear- time-dependent process and for the specific case of the P1 neutron kinetics equations. Chapter 4 presents the general procedure for the variational derivation of synthesis approximations and their applications to problems in reactor physics. This chapter also examines the relationship of the spatial synthesis and finite-element method and a hybrid method that combines features of both methods. Chapter 5 describes the relationship of variation theory with the Hamilton-Jacobi theory and with the optimization theories of the maximum principle and dynamic programming. Nuclear physicists and researchers will find this text invaluable.

Variational Methods in the Mechanics of Solids contains the proceedings of the International Union of Theoretical and Applied Mechanics Symposium on Variational Methods in the Mechanics of Solids, held at Northwestern University in Evanston, Illinois, on September 11-13, 1978. The papers focus on advances in the application of variational methods to a variety of mathematically and technically significant problems in solid mechanics. The discussions are organized around three themes: thermomechanical behavior of composites, elastic and inelastic boundary value problems, and elastic and inelastic dynamic problems. This book is comprised of 58 chapters and opens by addressing some questions of asymptotic expansions connected with composite and with perforated materials. The following chapters explore mathematical and computational methods in plasticity; variational irreversible thermodynamics of open physical-chemical continua; macroscopic behavior of elastic material with periodically spaced rigid inclusions; and application of the Lanczos method to structural vibration. Finite deformation of elastic beams and complementary theorems of solid mechanics are also considered, along with numerical contact elastostatics; periodic solutions in plasticity and viscoplasticity; and the convergence of the mixed finite element method in linear elasticity. This monograph will appeal to practitioners of mathematicians as well as theoretical and applied mechanics.

This is a textbook written for use in a graduate-level course for students of mechanics and engineering science. It is designed to cover the essential features of modern variational methods and to demonstrate how a number of basic mathematical concepts can be used to produce a unified theory of variational mechanics. As prerequisite to using this text, we assume that the student is equipped with an introductory course in functional analysis at a level roughly equal to that covered, for example, in Kolmogorov and Fomin (Functional Analysis, Vol. I, Graylock, Rochester, 1957) and possibly a graduate-level course in continuum mechanics. Numerous references to supplementary material are listed throughout the book. We are indebted to Professor Jim Douglas of the University of Chicago, who read an earlier version of the manuscript and whose detailed suggestions were extremely helpful in preparing the final draft. We also gratefully acknowedge that much of our own research work on va ri at i ona 1 theory was supported by the U. S. Ai r Force Offi ce of Scientific Research. We are indebted to Mr. Ming-Goei Sheu for help in proofreading. Finally, we wish to express thanks to Mrs. Marilyn Gude for her excellent and painstaking job of typing the manuscript. This revised edition contains only minor revisions of the first. Some misprints and errors have been corrected, and some sections were deleted, which were felt to be out of date.

This book provides a comprehensive survey of analytic and approximate solutions of problems of applied mechanics, with particular emphasis on nonconservative phenomena. Include

"A longtime classic text in applied mathematics, this volume also serves as a reference for undergraduate and graduate students of engineering. Topics include real variable theory, complex variables, linear analysis, partial and ordinary differential equations, and other subjects. Answers to selected exercises are provided, along with Fourier and Laplace transformation tables and useful formulas. 1978 edition"--

The Mathematics of Finite Elements and Applications provides information pertinent to the mathematics of finite elements, applications, algorithms, and computational techniques. This book discusses the developments in the mathematics of finite elements. Organized into 32 chapters, this book begins with an overview of the basis of the finite element process as a general approximation tool. This text then examines the methods for obtaining bounds on the errors in finite element solutions to two-dimensional elliptic boundary value problems defined on simply connected polygonal regions. Other chapters consider the practical implementation of the Galerkin and the Rayleigh–Ritz methods to equations of importance to physics and engineering. This book discusses as well a fundamental investigation into the problem of convergence in the finite element method. The final chapter deals with an algorithm that is applicable to the analysis of arbitrary plane stress or plane strain configurations. This book is a valuable resource for numerical analysts, mathematical physicist, applied mathematicians, computer scientists, and engineers.