Introduction to the Explicit Finite Element Method for Nonlinear Transient Dynamics

Shen R. Wu, Lei Gu

• Editore: Wiley
• Anno: 2012
• ISBN:
  9781118382073
• Edizione: 1
• Argomento: Mathematics
• Categoria: Discrete Mathematics

A systematic introduction to the theories and formulations of the explicit finite element method

As numerical technology continues to grow and evolve with industrial applications, understanding the explicit finite element method has become increasingly important, particularly in the areas of crashworthiness, metal forming, and impact engineering. Introduction to the Explicit Finite Element Method for Nonlinear Transient Dynamics is the first book to address specifically what is now accepted as the most successful numerical tool for nonlinear transient dynamics. The book aids readers in mastering the explicit finite element method and programming code without requiring extensive background knowledge of the general finite element.

The authors present topics relating to the variational principle, numerical procedure, mechanical formulation, and fundamental achievements of the convergence theory. In addition, key topics and techniques are provided in four clearly organized sections:

• Fundamentals explores a framework of the explicit finite element method for nonlinear transient dynamics and highlights achievements related to the convergence theory

• Element Technology discusses four-node, three-node, eight-node, and two-node element theories

• Material Models outlines models of plasticity and other nonlinear materials as well as the mechanics model of ductile damage

• Contact and Constraint Conditions covers subjects related to three-dimensional surface contact, with examples solved analytically, as well as discussions on kinematic constraint conditions

Throughout the book, vivid figures illustrate the ideas and key features of the explicit finite element method. Examples clearly present results, featuring both theoretical assessments and industrial applications.

Introduction to the Explicit Finite Element Method for Nonlinear Transient Dynamics is an ideal book for both engineers who require more theoretical discussions and for theoreticians searching for interesting and challenging research topics. The book also serves as an excellent resource for courses on applied mathematics, applied mechanics, and numerical methods at the graduate level.

PART I

FUNDAMENTALS

CHAPTER 1

INTRODUCTION

1.1 ERA OF SIMULATION AND COMPUTER AIDED ENGINEERING

1.1.1 A World of Simulation

“Computer simulation” has become a popular terminology in almost all disciplines of science and engineering today. Successful stories of computer simulation on various research projects have been reported in many professional conferences and events. In recent years, many technical journals have emerged dedicating to theories, techniques, and applications of simulations. Simulation shines in almost every aspect of research.

In its final report of 2006, the Blue Ribbon Panel on simulation-based Engineering Science of US National Science Foundation claimed the critical importance of simulation technology in the twenty-first century and considered it as the national priority for tomorrow's engineering and science (available at http://www.nsf.gov/pubs/reports/sbes_final_report.pdf). The working group of scientists of computational mechanics, applied mathematics, and other disciplines has envisioned revolutionizing engineering science through simulation. Simulation is essentially the computational science and engineering. It involves heavily the use of finite element method and other numerical approaches. In the past half century, finite element methods have been used for many engineering applications with the advances of high-speed computing power and software functionality. The evolution of finite element technology has also stimulated the development of computer architectures and technologies. As many physical events are too costly for any type of failure, computer simulation has become a highly desired tool to evaluate the process before carrying out the actual procedure. For example, medical doctors can first perform computer simulation on a bypass surgery procedure for treating disease in aorta and iliac artery to assess the potential results without subjecting any human life to danger. These scientific and engineering applications have placed additional importance on numerical simulation to provide precise and accurate information.

Approximate solutions to the differential and/or integral equations from various engineering problems have been in demand for a long time due to the difficulty in obtaining analytical solutions. Courant (1943) constructed the approximate solution to St. Venant torsion problem by triangulation with linear approximation for the minimum potential energy and the Ritz method. In fact, Courant (1943) demonstrated all the basic concepts of the finite element method. In the mid-1950s, Argyris (1954, 1957) and his colleagues extensively developed certain generalization of the linear theory of structures and presented procedures for analyzing complex discrete structures. Turner et al. (1956) analyzed classical elasticity equation and illustrated the triangular element properties for plane stress. Clough (1960) named such an approximation method the “finite element,” for the first time. Since then, work and research on the finite element method has grown extensively. While many algorithms and applications of linear problems were still under development, nonlinear analysis has been developed at a significantly faster pace. Oden (1972) among others demonstrated significant achievements in nonlinear applications and provided the basic concepts and algorithms of nonlinear finite element methods.

Following the development of the fundamentals, finite element software was quickly commercialized and further propelled engineering applications. The first software program was delivered by Ed Wilson. The subsequent development became SAP and NONSAP. The first nonlinear commercial software MARC led by Pedro Marcal and ADINA led by Jürgen Bathe were among the early software developed for nonlinear structural dynamics. Finite element then started to be introduced into universities’ colloquiums. It is critical that, in accompanying the development of numerical methods and engineering applications, the mathematical theories about interpolation, convergence, and error estimation of the finite element methods have also been heavily developed to provide strong support for the finite element method. The monograph edited by Ciarlet and Lions (1991) is an excellent collection of the mathematical achievements. We agree with the statement by Belytschko (1996) that extending from linear static analysis to nonlinear dynamic analysis greatly increases the l...