Advances in Computational Dynamics of Particles, Materials and Structures E-Book

Jason Har

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 913

Veröffentlichungsjahr: 2012

Beschreibung

Computational methods for the modeling and simulation of the dynamic response and behavior of particles, materials and structural systems have had a profound influence on science, engineering and technology. Complex science and engineering applications dealing with complicated structural geometries and materials that would be very difficult to treat using analytical methods have been successfully simulated using computational tools. With the incorporation of quantum, molecular and biological mechanics into new models, these methods are poised to play an even bigger role in the future.

Advances in Computational Dynamics of Particles, Materials and Structures not only presents emerging trends and cutting edge state-of-the-art tools in a contemporary setting, but also provides a unique blend of classical and new and innovative theoretical and computational aspects covering both particle dynamics, and flexible continuum structural dynamics applications.  It provides a unified viewpoint and encompasses the classical Newtonian, Lagrangian, and Hamiltonian mechanics frameworks as well as new and alternative contemporary approaches and their equivalences in [start italics]vector and scalar formalisms[end italics] to address the various problems in engineering sciences and physics.

Highlights and key features

•  Provides practical applications, from a unified perspective, to both particle and continuum mechanics of flexible structures and materials
• Presents new and traditional developments, as well as alternate perspectives, for space and time discretization 
• Describes a unified viewpoint under the umbrella of Algorithms by Design for the class of linear multi-step methods
• Includes fundamentals underlying the theoretical aspects and numerical developments, illustrative applications and practice exercises

The completeness and breadth and depth of coverage makes Advances in Computational Dynamics of Particles, Materials and Structures a valuable textbook and reference for graduate students, researchers and engineers/scientists working in the field of computational mechanics; and in the general areas of computational sciences and engineering.

Leseprobe

Table of Contents

Title Page

Copyright

Dedication

Preface

Acknowledgments

About the Authors

Chapter 1: Introduction

1.1 Overview

1.2 Applications

Chapter 2: Mathematical Preliminaries

2.1 Sets and Functions

2.2 Vector Spaces

2.3 Matrix Algebra

2.4 Vector Differential Calculus

2.5 Vector Integral Calculus

2.6 Mean Value Theorem

2.7 Function Spaces

2.8 Tensor Analysis

Part I: N-Body Dynamical Systems

Chapter 3: Classical Mechanics

3.1 Newtonian Mechanics

3.2 Lagrangian Mechanics

3.3 Hamiltonian Mechanics

Chapter 4: Principle of Virtual Work

4.1 Virtual Work in N-Body Dynamical Systems

4.2 Vector Formalism: Newtonian Mechanics in N-Body Dynamical Systems

4.3 Scalar Formalisms: Lagrangian and Hamiltonian Mechanics in N-Body Dynamical Systems

Chapter 5: Hamilton's Principle and Hamilton's Law of Varying Action

5.1 Introduction

5.2 Variation of the Principal Function

5.3 Calculus of Variations

5.4 Hamilton's Principle

5.5 Hamilton's Law of Varying Action

Chapter 6: Principle of Balance of Mechanical Energy

6.1 Introduction

6.2 Principle of Balance of Mechanical Energy

6.3 Total Energy Representations and Framework in the Differential Calculus Setting

6.4 Appendix: Total Energy Representations and Framework in the Variational Calculus Setting

Chapter 7: Equivalence of Equations

7.1 Equivalence in the Lagrangian Form of D'Alembert's Principle/Principle of Virtual Work

7.2 Equivalence in Hamilton's Principle or Hamilton's Law of Varying Action

7.3 Equivalence in the Principle of Balance of Mechanical Energy

7.4 Equivalence Relations Between Governing Equations

7.5 Conservation Laws

7.6 Noether's Theorem

Part II: Continuous-Body Dynamical Systems

Chapter 8: Continuum Mechanics

8.1 Displacements, Strains and Stresses

8.2 General Principles

8.3 Constitutive Equations in Elasticity

8.4 Virtual Work and Variational Principles

8.5 Direct Variational Methods for Two-Point Boundary-Value Problems

Chapter 9: Principle of Virtual Work: Finite Elements and Solid/Structural Mechanics

9.1 Introduction

9.2 Finite Element Library

9.3 Nonlinear Finite Element Formulations

9.4 Scalar Formalisms: Lagrangian and Hamiltonian Mechanics and Finite Element Formulations in Continuous-Body Dynamical Systems

Chapter 10: Hamilton's Principle and Hamilton's Law of Varying Action: Finite Elements and Solid/Structural Mechanics

10.1 Introduction

10.2 Hamilton's Principle and Hamilton's Law of Varying Action in Elastodynamics

10.3 Lagrangian Mechanics Framework and Finite Element Formulations

10.4 Hamiltonian Mechanics Framework and Finite Element Formulations

Chapter 11: Principle of Balance of Mechanical Energy: Finite Elements and Solid/Structural Mechanics

11.1 Introduction

11.2 Total Energy Representations and Framework in the Differential Calculus Setting and Finite Element Formulations

11.3 Lagrangian Mechanics Framework in the Differential Calculus Setting and Finite Element Formulations

11.4 Hamiltonian Mechanics Framework in the Differential Calculus Setting and Finite Element Formulations

11.5 Appendix: Total Energy Representations and Framework in the Variational Calculus Setting and Finite Element Formulations

Chapter 12: Equivalence of Equations

12.1 Equivalence in the Principle of Virtual Work in Dynamics

12.2 Equivalence in Hamilton's Principle or Hamilton's Law of Varying Action

12.3 Equivalence in the Principle of Balance of Mechanical Energy

12.4 Equivalence of Strong and Weak Forms for Initial Boundary-Value Problems

12.5 Equivalence of the Semi-Discrete Finite Element Equations of Motion

12.6 Equivalence of Finite Element Formulations

12.7 Conservation Laws

Part III: The Time Dimension

Chapter 13: Time Discretization of Equations of Motion: Overview and Conventional Practices

13.1 Introduction

13.2 Single-Step Methods for First-Order Ordinary Differential Equations

13.3 Linear Multistep Methods

13.4 Second-Order Systems and Single Step and/or Equivalent LMS Methods: Brief Overview of Classical Methods from Historical Perspectives and Chronological Developments

13.5 Symplectic-Momentum Conservation and Variational Time Integrators

13.6 Energy-Momentum Conservation and Time Integration Algorithms

Chapter 14: Time Discretization of Equations of Motion: Recent Advances

14.1 Introduction

14.2 Time Discretization and the Total Energy Framework: Linear Dynamic Algorithms and Designs—Generalized Single Step Single Solve [GSSSS] Unified Framework Encompassing LMS Methods

14.3 Time Discretization and the Total Energy Framework: Nonlinear Dynamics Algorithms and Designs - Generalized Single Step Single Solve [GSSSS] Framework Encompassing LMS Methods

14.4 Time Discretization and Total Energy Framework: N-Body Systems

14.5 Time Discretization and Total Energy Framework: Nonconservative/Conservative Mechanical Systems with Holonomic-Scleronomic Constraints

References

Index

© 2012, John Wiley & Sons, Ltd

Registered office

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book's use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.

Library of Congress Cataloguing-in-Publication Data

Har, Jason.

Advances in computational dynamics of particles, materials and structures/Jason Har, Kumar K. Tamma.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-74980-7 (hardback)

1. Dynamics. 2. Dynamics—Data processing. I. Tamma, Kumar K. II. Title.

TA352.H365 2012

531′.163—dc23

2011044208

A catalogue record for this book is available from the British Library.

ISBN: 978-0-470-74980-7

Preface

This book treats the subject matter dealing with advances in computational dynamics from a unified viewpoint and approach, and thereby provides a rigorous treatment and a unique blend of the various underlying mechanics and the numerical aspects to effectively foster modeling and simulation on modern computing environments. In the broader sense, the subject matter under the umbrella of computational dynamics covers the necessary fundamentals associated with particle dynamics; dynamics of materials, structures, deformable continuum media and related applications to include structural/elasto-dynamics; multi-body dynamics dealing with rigid and flexible bodies; contact-impact dynamics; and so on. In particular, this book covers the classical (or traditional) practices to more contemporary aspects which include recent advances dealing with the mathematical, physical, geometrical, as well as computational aspects associated with modeling and simulation as related to numerical discretization in space and/or time. It is designed for engineers, mathematicians, physicists, and students/researchers in allied fields who wish to understand the subject matter with rigor and in a contemporary setting. We intend this book to serve as a multi-semester course at the graduate-level and/or for upper-level undergraduate students (on selected topics), advanced researchers and scientists, and engineers who are keenly interested in the fundamental aspects critical to the computational aspects of the dynamics of particles and rigid bodies, and the computational aspects dealing with structural/elasto-dynamics, continuum mechanics, the finite element method, and time integration schemes for both N-body and continuous-body dynamical systems. This book explores both classical practices as well as new avenues with differing and alternative viewpoints which additionally provide improved physical insight and new computational perspectives. With these considerations in mind, we closely embrace the underlying theme and excerpt due to Gauss as highlighted in Degas (1955): "It is always interesting and instructive to regard the laws of nature from a new and advantageous point of view, so as to solve this or that problem more simply, or to obtain a more precise presentation".

We start with the premise, that in the beginning there were these landmark contributions due to Aristotle (384 BC-322 BC), Archimedes (287 BC-212 BC), Galileo (1564-1642), Kepler (1571-1630), Huygens (1629-1695), Decartes (1596-1650), and the like, and, then there was this thing of beauty, namely, that due to Newton (1643-1727) - the famous Newton's laws of motion. And now there are all these various fields or branches of mechanics and physics with various underlying theoretical pinning's dealing with particle dynamics; dynamics of materials, structures, and deformable continuum media and related applications to include structural/elasto-dynamics; multi-body dynamics dealing with rigid and flexible bodies; contact-impact dynamics; and the like. It is worth noting that the fundamental principles of dynamics have also been abstracted to various other fields and applications to include the theory of relativity, quantum mechanics, economics, robotics, biology, medical and allied applications such as biomechanics, virtual surgery physics based simulations for training medical residents/physicians, and the like.

Keeping the above considerations in perspective, we present an overview of not only the classical developments and the current state-of-the-art, but we also provide new and recent advances dealing with computational aspects related to the dynamics of particles, materials, and structures. In this book, we first highlight the big picture with consistent developments from differing viewpoints not only to derive the governing equations of motion for N-body or continuous-body dynamical systems for a wide class of engineering applications, but also to subsequently enable the discretization in space/time for numerical computations. In particular, we present our viewpoint of the evolution of a variety of numerical developments in the fields encompassing computational dynamics ranging from classical practices to more new and recent advances. Under the umbrella of computational dynamics, at the outset it should be clearly noted that this book is intended to provide a sound and fundamental background on the various theoretical and computational aspects; and we classify the evolution of the various related developments via two principal themes, namely, the mechanics underlying computational dynamics and the associated numerics underlying computational dynamics. Only in selective instances, certain theoretical bases and related considerations dealing with various aspects of classical mechanics have been carefully excerpted and interpreted from several renowned books such as Mach (1907), Pars (1965), Greenwood (1977), Rosenberg (1977), Arnold (1989), Goldstein (2002), and the like which have been some of the primary sources.

Mechanics Underlying Computational Dynamics: The terminology, namely, the mechanics underlying computational dynamics, implies the approach and starting point that is employed as the fundamental axiom via which one can independently derive the governing equations, and the associated strong and/or weak forms that can be readily employed for the associated numerical discretizations. Starting with the premise that in the beginning the well known Newton's law of motion for the dynamics of N-body systems is given, which reflects the statement of the principle of balance of linear momentum, subsequently, using this as a landmark, firstly, the principal relations to various other distinctly different fundamental principles which are of primary interest here are established. This is worth noting. Likewise, for the dynamics of materials, structures, and deformable continuum medium and related applications, under the premise that the governing equations such as the well known Cauchy equations of motion which also reflect the statement of the principle of balance of linear momentum are given, analogous relations as in N-body systems are also established. After first establishing the principal relations to the various fundamental principles, any of the respective principles thenceforth can serve as the standalone starting point for the subsequent theoretical and computational developments for modeling and simulation. In this book, we confine attention primarily to three distinctly different fundamental principles which comprise the pyramid of computational dynamics. Of particular interest are the three distinctly different fundamental principles represented as faces or planes which comprise the pyramid of computational dynamics (see Figure 1), namely: 1) the Principle of Virtual work, 2) Hamilton's Principle, or alternatively, Hamilton's Law of Varying Action (which is not a variational principle), and 3) the Principle of Balance of Mechanical Energy. Each fundamental principle is particularly selected such that it can independently enable the theoretical and computational developments associated with and leading to the strong and/or weak forms and the corresponding numerical discretizations in space/time for applications to computational dynamics. That is, each of the above fundamental principles does not necessarily rely upon the others. However, the pros and cons, limitations of each fundamental principle, and the conditions under which equivalences of the respective formulations amongst the three fundamental principles can be drawn need to be carefully understood to avoid misinterpretation. By no means, we claim that these are the only representations for the classification as various other explanations are also plausible and could be included. Consequently, the present pyramidal structure classification could entertain other faces or planes. However, we confine attention only to the present three fundamental principles with the clear understanding of the restrictions inherent within each fundamental principle.

Figure 1. Pyramid of computational dynamics

Numerics Underlying Computational Dynamics: Subsequently, we also describe the numerics underlying computational dynamics which deals with both classical (or traditional) practices and new avenues for conducting space/time discretizations to find numerical solutions useful for modeling and simulation. The terminology, numerics underlying computational dynamics, refers to the approach and the starting point that is employed by which we address the numerical treatments as related to spatial discretizations in the space domain and temporal discretizations in the time domain. It deals with the numerical aspects and discretization approaches in space and/or time which are necessary ingredients for modeling and simulation. Stemming independently from each of the respective fundamental principles comprising the pyramid of computational dynamics, we describe the various computational developments for the dynamics of N-body systems, and the dynamics of materials, structures, deformable continuum media and related applications. Both classical practices that are customarily followed, as well as other alternative avenues which provide new and different perspectives and/or improved physical insight for the modeling and simulation of computational dynamics applications are described. A unified viewpoint is the end result regardless of which fundamental principle serves as the starting point; and the restrictions and/or limitations associated with each of the respective fundamental principles need to be carefully understood Tamma (2012); Tamma et al. (2011) (DOI10.1007/s11831-011 9060-y).

Outline of this Book: The outline of this book is as follows. Chapter 1 presents an introduction to and an overview of the big picture and our viewpoint of the various theoretical and numerical aspects dealing with computational dynamics. Along the themes, namely, the mechanics underlying computational dynamics and under the umbrella of the pyramid of computational dynamics, and the associated numerics underlying computational dynamics, in this book, we focus attention upon the three fundamental principles comprising the pyramid structure classification, namely: 1) the Principle of Virtual Work, 2) Hamilton's Principle, or alternatively, Hamilton's Law of Varying Action (which is not a variational principle), and 3) the Principle of Balance of Mechanical Energy. Each of the above fundamental principles has a wide range of applicability, and can independently describe the theoretical and numerical developments associated with and leading to the strong and/or weak forms and the corresponding numerical discretizations in space/time for applications to computational dynamics. Chapter 2 provides the basic mathematical background materials necessary for studying classical mechanics, continuum mechanics, finite element theories, and time integration schemes for integrating the equations of motion. Throughout the book, it is very important to have a fundamental grasp of the concepts of sets and functions, and the meaning of the related notations. Vector spaces with numeric entries as well as functions are addressed in this chapter. In the discussion on tensor analysis, we use not only Cartesian tensors but also general tensors, which are crucial for understanding nonlinear continuum mechanics and finite deformation theories for deformable bodies. The book is divided into three parts. Consequently, under the umbrella of the pyramid of computational dynamics, we devote a separate chapter in both N-Body Systems (Part 1) and Continuous-Body Systems (Part 2) to each of these respective principles which independently serve as a starting point for conducting the theoretical and numerical developments associated with and leading to the strong and/or weak forms and the corresponding numerical discretizations in space and/or time. An overview of conventional practices, and in addition, recent advances dealing with a wide variety of Time Discretization (Part 3) approaches and related time integration aspects necessary for appropriately integrating the dynamic equations of motion are finally highlighted.

Part 1: N-Body Systems With the above considerations in mind, in Part 1 which deals with N-body Systems, Chapter 3 covers classical mechanics including Newtonian, Lagrangian and Hamiltonian mechanics. In Chapter 4, after first establishing the relation between Newton's second law and the principle of virtual work (which is a restatement of the Lagrangian form of D'Alembert's principle), we directly show the subsequent theoretical and numerical developments starting from this principle. The Lagrangian form of D'Alembert's principle (or equivalently, the principle of virtual work in dynamics) is the key principle leading to analytical mechanics and descriptive scalar function formalism, in contrast to the Newtonian mechanics framework and vector formalism. Alternatively, Chapter 5 describes both Hamilton's principle and Hamilton's law of varying action for N-body dynamical systems. We draw attention in the book to the fact that Hamilton's law of varying action is equivalent to the integral form of the principle of virtual work. Consequently, it is a descriptive scalar function representation of the principle of virtual work, which naturally contains the weighted residual form in time for N-body dynamical systems. In contrast, Chapter 6 describes the principle of balance of mechanical energy as the starting point, and the corresponding formulations associated with the Total Energy representation of the equation of motion and framework in the differential calculus setting which is valid for holonomic-scleronomic systems with a new, measurable, and built-in descriptive scalar function, namely, the Total Energy (and in addition, the variational calculus setting which is valid for holonomic systems is also highlighted in the Appendix). As a descriptive scalar function analogous to the Lagrangian and the Hamiltonian, the Total Energy defined on the velocity phase space is yet another alternative and it offers good physical insight and computationally attractive features. There exist various subject areas in mechanics and physics where it is desirable to have a direct measurable descriptive scalar function such as the Total Energy. The related developments also readily enable the theoretical and numerical formulations for computational dynamics just as those obtained from the other two fundamental principles. Next, Chapter 7 describes equivalence relations between governing equations for N-body dynamical systems subject to holonomic constraints within the three frameworks, namely, the Lagrangian, Hamiltonian and Total Energy frameworks. Noether's Theorem for N-body dynamical systems, and the invariant properties, namely, the conservation of Linear Momentum, Angular Momentum and Total Energy of the descriptive scalar functions, such as the Lagrangian, Hamiltonian and Total Energy are also highlighted.

Part 2: Continuous-Body Systems Part 2 focuses upon Continuous-body Systems, and the continuum mechanics aspects associated with deformations, strains, and stresses in solid/structural applications. In Chapter 8, we start with basic continuum mechanics materials necessary for developing finite element formulations. Chapter 8 describes displacements, strains, and stresses with general tensors. We then discuss five fundamental principles dealing with thermo-mechanical motion which continuous bodies must obey; these include, the principle of conservation of mass, the principle of balance of linear momentum, the principle of balance of angular momentum, the principle of balance of energy, and the principle of entropy inequality. Chapter 8 also includes constitutive equations in elasticity, fundamentals of virtual work and variational principles, and direct variational methods for two-point boundary-value problems such as the Rayleigh-Ritz method, the Bubnov-Galerkin weighted residual method, and the modified Bubnov-Galerkin weighted residual method. As in N-body systems described in Part 1, we next devote a separate chapter dealing with continuous-body systems to each of the three fundamental principles comprising the pyramid of dynamics which independently serve as the starting point for developing the related theoretical and numerical formulations. In this regard, Chapter 9 comprehensively deals with the first of the three principles outlined earlier, namely, the principle of virtual work in dynamics; and consequently, describes conventional finite element formulations and vector formalism for continuous-body dynamical systems. We additionally describe a variety of structural members including axial bar, rotating circular bar, Euler-Bernoulli beam, Timoshenko beam, Kirchhoff-Love thin plate, and Reissner-Mindlin plate. The weak forms for continuum and structural members are derived from the weighted residual form. With regards to structural members, we first set up a free-body diagram and count on D'Alembert's principle to obtain the governing equations of motion. Then, we establish the weighted residual statement to derive the weak form by performing integration by parts and imposing natural boundary conditions. Finally, the resulting weak form is spatially discretized by using appropriate trial and test functions. In addition to the finite element formulations, we additionally describe a variety of finite elements including axial bar element, plane stress/strain two-dimensional triangular and quadrilateral elements, three-dimensional tetrahedral and hexahedral brick elements, Euler-Bernoulli beam element, Timoshenko beam element, Kirchhoff-Love plate element and Reissner-Mindlin plate element. Lastly, we highlight nonlinear finite element formulations including total and updated Lagrangian formulations. Scalar formalisms with respect to the Lagrangian and the Hamiltonian are briefly referenced. In Chapter 10, in contrast to the traditional practices described in Chapter 9, we present finite element formulations using descriptive scalar functions via Hamilton's Principle or Hamilton's Law of Varying Action as the starting point which also yield the same and/or equivalent finite element representations from another viewpoint. As yet another alternative with several computationally attractive features and good physical insight, in Chapter 11 we describe other related developments via the Total Energy representations and framework for developing the finite element formulations using the various descriptive scalar functions. This is via the theorem of power expended, and consequently the principle of balance of mechanical energy with differential calculus setting valid for holonomic-scleronomic systems. In the Appendix, we briefly also highlight the variational calculus setting in the context of the Total Energy representations and framework which is valid for holonomic systems. Chapter 12 discusses the equivalences between the strong forms and also between the weak forms which are respectively obtained via each of the three distinctly different fundamental principles, namely, the principle of virtual work, Hamilton's principle or equivalently, Hamilton's law of varying action, and the theorem of power expended and consequently the principle of balance of mechanical energy. We also present a brief discussion on Noether's Theorem for continuous-body dynamical systems, wherein after the spatial discretization they lead to finite dimensional systems analogous to the discussion highlighted in Chapter 7 for N-body systems.

Part 3: Time Discretization Finally, Part 3 is devoted to the Time Dimension and the numerical aspects that are necessary for properly dealing with the time integration of the equations of motion in both single-field and two-field forms of representation Tamma (2012); Tamma et al. (2011 (DOI 10.1007/s11831-011-9060-y). For the time discretization, an overview of the big picture and specific guidelines for developing algorithms by design that meet targeted objectives are provided and discussed. In Chapter 13 we present the following: (i) We first show starting from the standard representation of the linear semi-discretized equations of motion, the various classical and chronological developments in time integration of linear dynamical systems from historical perspectives that appear in the open literature over the past fifty years or so, (ii) Next, we highlight variational integrators stemming from the so-called Discrete Euler-Lagrange representations that inherit features which are symplectic-momentum conserving, and (iii) Following this, we highlight the so-called energy-momentum conserving/dissipating algorithm designs for finite dimensional systems following the original methods of development (classical practices) through enforcing energy constraints. Lastly, in Chapter 14, in contrast to all the previously mentioned classical and/or traditional practices described in Chapter 13, we focus special attention upon and highlight the more recent developments directly emanating from the new Total Energy framework and representations as a starting point (unlike traditional practices) in conjunction with a generalized time weighted residual approach. In particular, we provide new perspectives, a unified viewpoint, and in addition, the underlying theoretical basis on how to properly provide appropriate extensions of the parent linear dynamics algorithm designs to nonlinear dynamics applications for developing practical algorithms by design useful for integrating the equations of motion; and the associated computationally attractive features are that the developments are based upon symplectic-momentum conservation or energy-momentum conservation aspects, respectively. These latter developments via the Total Energy representations and framework, and the generalized time weighted residual approach also cover most of the developments that have been previously derived from various other classical viewpoints as mentioned in (i), (ii), and (iii) previously in Chapter 13. In summary, for both single-field and two-field forms of representation, we first describe linear dynamics algorithms by design for integrating the equations of motion, and we then provide the necessary theoretical basis for proper extensions to nonlinear dynamics algorithms by design. The overall developments are generally applicable to a wide variety of applications encompassing linear and nonlinear structural/elasto-dynamics applications in continuous-body dynamics, N-body systems, and conservative/nonconservative mechanical systems with holonomic-scleronomic constraints such as those encountered in multi-body dynamics applications.

Jason Har and Kumar K. Tamma

Acknowledgments

Professor Kumar K. Tamma is particularly grateful to Dr. Jason Har for his steadfast commitment to embrace the original ideas and concepts that are put forth in this book, learn, and contribute during the five and a half year period he served as a post-doctoral associate and research assistant professor under Professor Tamma's supervision in the Department of Mechanical Engineering at the University of Minnesota. Special thanks are due to Mr. Masao Shimada, graduate Ph.D research student in the Department of Mechanical Engineering at the University of Minnesota and working under the supervision of Professor Kumar K. Tamma, for his valuable technical comments and contributions; in particular on the time integration aspects.

About the Authors

Professor Kumar K. Tamma is a highly recognized researcher and distinguished scholar, and is Professor in the Department of Mechanical Engineering, College of Science and Engineering, at the University of Minnesota, Minneapolis, Minnesota. Professor Tamma has published over 200 research papers in archival journals and book chapters, and over 300 research papers in refereed conference proceedings/abstracts. Professor Tamma's primary areas of research encompass: Computational mechanics with emphasis on multi-scale and multi-physics aspects in space and time, and on design and development of novel numerical methods and computational algorithms by design for the modeling and simulation of time dependent problems and High Performance Computing applications; multi-disciplinary computational fluid-thermal-structural interactions; structural dynamics and large deformation and large strain contact-impact-penetration-damage; multi-body dynamics of rigid and flexible bodies; computational aspects of macroscale/microscale/nanoscale heat transfer; advanced and lightweight composites and multifunctional materials manufacturing processes, and solidification. Professor Tamma serves on the editorial boards for over 15 national and international journals and is the co-editor-in-chief of an online journal. Professor Tamma is the recipient of numerous research awards including the George Taylor Research Award for significant and exceptional contributions to research at the University of Minnesota. Professor Tamma is also the recipient of numerous Outstanding Teacher of the Year and other national and university related awards. Professor Tamma has presented several Plenary/Semi-Plenary/Keynote lectures and various invited lectures in national/international conferences, and across various government and industrial agencies, and academic institutions. Professor Tamma is a fellow of various related societies in his field, and is also listed in various Who's Who of organizations and professionals.

Dr. Jason Har is a Senior Software Developer with ANSYS, Inc., in Canonsburg, Pennsylvania. Dr. Har worked under the supervision of Professor Kumar K. Tamma at the University of Minnesota in Minneapolis, Minnesota, where Dr. Har learned and embraced the original ideas and concepts being pursued by Professor Tamma that are put forth in this textbook; and contributed to the various developments for a period of five and a half years as a post doctoral associate and as a research assistant professor in the Department of Mechanical Engineering at the University of Minnesota. Dr. Jason Har has extensive industrial experience in finite element technology of structures and structural components, contact-impact, and parallel computations for over 15 years, and worked at the Korea Institute of Aerospace Technology where Dr. Har also served as Managing Research Director. Dr. Har has presented various invited and special lectures at various organizations and national/international conferences.

Chapter 1

Introduction

The present book encompasses classical (or traditional practices) as well as advances in computational dynamics for computer modeling and simulation of applications in science and engineering. The highlights of this book are outlined in Chapter 1. The targeted objectives are towards a wide variety of science and engineering problems in particle dynamics; dynamics of materials, structures and deformable continuum media; and related applications which fall under this class of applications. We first introduce in this book the big picture and a unified viewpoint, and the various approaches which follow for the modeling and simulation in the broad field encompassing computational dynamics. In the broader sense, in this book the subject matter under the umbrella of computational dynamics covers the necessary fundamentals associated with particle dynamics; dynamics of materials and deformable continuum media and related applications to include structural/elasto-dynamics; multi-body dynamics dealing with rigid and flexible bodies; contact-impact dynamics; and so on. We classify the evolution of the various related developments under the umbrella of computational dynamics via two principal themes, namely, the mechanics underlying computational dynamics and the numerics underlying computational dynamics.

1.1 Overview

An overview of the “big picture” follows next. With regards to the mechanics underlying computational dynamics, we start with the premise that in the beginning, the well known Newton's law of motion for N-body systems is given, which reflects the statement of the principle of balance of linear momentum. Subsequently using this as a landmark, firstly, the principal relations to three distinctly different fundamental principles, that comprise the pyramid of computational dynamics, and are of primary interest here are established. Likewise, for the dynamics of materials, structures and deformable continuum media and related applications, under the premise that the governing equations such as the Cauchy's equations of motion which reflect the statement of the principle of balance of linear momentum are given, analogous developments are also established. Once the principal relations to the three fundamental principles are established, any of the respective principles can thenceforth serve as the standalone starting point for the subsequent theoretical and numerical developments because of their wide range of applicability. The overall developments provide a fundamental understanding and improved insight into the mathematical equations governing the dynamic motion for N-body and continuous body systems, and the consequent numerical discretization in space and/or time. Stemming from the three distinctly different fundamental principles, we present recent advances in both vector and scalar formalisms for N-body dynamical systems and also continuous-body dynamical systems with focus upon the numerical aspects related to space/time discretization. The three distinctly different fundamental principles which comprise the pyramid of computational dynamics are the following: the Principle of Virtual Work in Dynamics, Hamilton's Principle and as an alternative (due to inconsistencies associated with Hamilton's principle), Hamilton's Law of Varying Action, and the Principle of Balance of Mechanical Energy. Essentially, the aforementioned three fundamental principles have been particularly highlighted and selected as each of these principles can be independently employed to derive the governing equations of motion for N-body dynamical systems, and the strong and weak forms for continuous-body dynamical systems. However, of importance and noteworthy are the various formalisms and the different ways by which one can describe the theoretical and computational developments; and there exist fundamental differences in the three distinctly different fundamental principles and their underlying axioms.

Customarily in the literature, the equations of motion, which govern the mechanical behavior of N-body or continuous-body dynamical systems for a wide class of engineering applications, have been represented by vectorial quantities in the Newtonian mechanics framework (which is referred to in this book as the vector formalism). Alternatively, in the Lagrangian or Hamiltonian mechanics framework (which is referred to in this book as the scalar formalism), they have been described by generalized or canonical coordinates with descriptive scalar functions such as the Lagrangian or the Hamiltonian; this is mostly in the sense of applications to N-body dynamical systems (Greenwood 1977; Pars 1965). This has been the traditional paradigm. It is a matter of convenience and preferred choice of the analyst in the particular selection of either vector or scalar formalism, and the corresponding framework. Although it is not customary in the classical mechanics setting, other alternative descriptive scalar functions exist and can also be employed. . There exist various subject areas in mechanics and physics where it is desirable to have a direct measurable descriptive scalar function such as the Total Energy. It provides a new and different perspective with good physical insight and computationally attractive and convenient features in contrast to the classical mechanics setting. The end result is that any of the three previously mentioned fundamental principles can independently be employed to derive the governing equations of motion for N-body dynamical systems, and the strong and weak forms for continuous-body dynamical systems. Also, both the vector and scalar formalisms indeed can be shown to be identical and/or equivalences can be drawn. Furthermore, each respective framework has its own pros and cons which need to be carefully understood in developing the numerical discretizations in space and/or time. In summary, we describe both classical practices that are customarily followed and new avenues for conducting space/time discretizations to find numerical solutions.