COMPUTATIONAL MATHEMATICS AND APPLICATIONS Series Editors J. ft WHITEMAN Institute of Computational Mathematics, Brunei University, UK and J. H. DAVENPORT School of Mathematical Sciences, University of Bath, UK E. HINTON and D. R. J. OWEN: Finite Element Programming M. A. JASWON and G. T. SYMM: Integral Equation Methods in Potential Theory and Elastostatics J. R. CASH: Stable Recursions: with applications to the numerical solution of stiff systems H. ENGELS: Numerical Quadrature and Cubature L. M. DELVES and T. L. FREEMAN: Analysis of Global Expansion Methods: weakly asymptotically diagonal systems J. E. AKIN: Application and Implementation of Finite Element Methods J. T. MARTI: Introduction to Sobolev Spaces and Finite Element Solution of Elliptic Boundary Value Problems H. R. SCHWARZ: Finite Element Methods J. DELLA DORA and J. FITCH: Computer Algebra and Parallelism E. TOURNIER: Computer Algebra and Differential Equations A. ZENISEK: Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations M. SINGER: Differential Equations and Computer Algebra G. CHEN and J. ZHOU: Boundary Element Methods B. R. DONALD, D KAPUR and J. L. MUNDY: Symbolic and Numerical Computation for Artificial Intelligence J. E. AKIN: Finite Elements for Analysis and Design Dedication To my family. COMPUTATIONAL MATHEMATICS AND APPLICATIONS Series Editors J. ft WHITEMAN Institute of Computational Mathematics, Brunei University, UK and J. H. DAVENPORT School of Mathematical Sciences, University of Bath, UK E. HINTON and D. R. J. OWEN: Finite Element Programming M. A. JASWON and G. T. SYMM: Integral Equation Methods in Potential Theory and Elastostatics J. R. CASH: Stable Recursions: with applications to the numerical solution of stiff systems H. ENGELS: Numerical Quadrature and Cubature L. M. DELVES and T. L. FREEMAN: Analysis of Global Expansion Methods: weakly asymptotically diagonal systems J. E. AKIN: Application and Implementation of Finite Element Methods J. T. MARTI: Introduction to Sobolev Spaces and Finite Element Solution of Elliptic Boundary Value Problems H. R. SCHWARZ: Finite Element Methods J. DELLA DORA and J. FITCH: Computer Algebra and Parallelism E. TOURNIER: Computer Algebra and Differential Equations A. ZENISEK: Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations M. SINGER: Differential Equations and Computer Algebra G. CHEN and J. ZHOU: Boundary Element Methods B. R. DONALD, D KAPUR and J. L. MUNDY: Symbolic and Numerical Computation for Artificial Intelligence J. E. AKIN: Finite Elements for Analysis and Design Dedication To my family. Finite Elements for Analysis and Design BY J. E. AKIN Department of Mechanical Engineering and Materials Science Rice University, Houston, Texas, USA ACADEMIC PRESS A Harcourt Science and Technology Company San Diego San Francisco New York Boston London Sydney Tokyo This book is printed on acid-free paper. Copyright © 1994 by ACADEMIC PRESS Fourth printing 2000 All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Academic Press A Harcourt Science and Technology Company Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK http://www.academicpress.com Academic Press A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com ISBN 0-12-047653-3 (Hbk) ISBN 0-12-047654-1 (Pbk) A catalogue record for this book is available from the British Library Printed and bound in Great Britain by IBT Global, London 00 01 02 03 04 05 IB 11 10 9 8 7 6 5 4 PREFACE There are several good texts on finite element analysis techniques that each have some special attribute that makes them worthwhile. My personal experience was that I did not really understand the theory until I could successfully implement it in a computer program. Thus, this text includes computational procedures, as well as the basic theory and its mathematical foundations, and several example applications. It is based on courses that I have taught on design and finite element theory. This book is primarily intended for advanced undergraduate engineering students and beginning graduate students. The text contains more material than could be covered in a single quarter or semester course. Therefore, a number of chapters or sections that could be omitted in a first course have been marked with an asterisk (*). Most of the subject matter deals with linear, static, heat transfer and elementary stress analysis. Other novel applications are presented in the later chapters that deal mainly with com- putational concepts. The future of finite element analysis will probably heavily involve adaptive analysis and design methods. One should have a course in Functional Analysis to best understand those techniques. Most undergraduate curriculums do not contain such courses. Therefore, chapters on mathematical preliminaries, error estimators, and adaptive meth- ods are included. A disk is provided that includes all of the source code covered in the text. It also has the source, data and output files for most of the examples, and typical applications presented in the text. Most of the computational details are included in the last third of the text, but the basic computational needs are introduced early in the text. For simplicity, most of the theoretical examples are presented in terms of relatively simple linear elements or quadratic elements. The corresponding numerical examples are given on the disk. Example extensions to higher order elements are often included on the disk but not covered in the text. Thus, after completing a chapter, the reader should also review the corresponding chapter files in the example files directory. A complete UNIX version of MODEL is available via Email to soft lib© cs .rice .edu with the message, "send catalogue". I would like to thank many current and former students, colleagues, and friends at Rice University for their constructive criticisms and comments during the evolution of this book. My son, Jeffrey, prepared many of the drawings. The text was prepared in a camera-ready form by Mr. Don Schroeder employing the UNIX Troff systems. xii Chapter 1 INTRODUCTION 1.1 Finite Element Methods In modern engineering design it is rare to find a project that does not require some type of finite element analysis (FEA). When not actually required, FEA can usually be utilized to improve a design. The practical advantages of FEA in stress analysis and structural dynamics have made it the accepted design tool for the last two decades. It is also heavily employed in thermal analysis, especially in connection with thermal stress analysis. Its use in Computational Fluid Dynamics (CFD) is rapidly becoming commonplace. Clearly, the greatest advantage of FEA is its ability to handle truly arbitrary geometry. Probably its next most important features are the ability to deal with general boundary conditions and to include nonhomogeneous materials. These features alone mean that we can treat systems of arbitrary shape that are made up of numerous different material regions. Each material could have constant properties or the properties could vary with spatial location. To these very desirable features we can add a large amount of freedom in prescribing the loading conditions and in the postprocessing of items such as the stresses and strains. For elliptical boundary value problems the FEA procedures offer significant computational and storage efficiencies that further enhance its use. These classes of problems include stress analysis, heat conduction, electrical fields, magnetic fields, ideal fluid flow, etc. FEA also gives us an important solution technique for other problem classes such as the nonlinear Navier-Stokes equations for fluid dynamics, and for plasticity in nonlinear solids. Here we will show what FEA has to offer the designer and illustrate some of its theoretical formulations and practical applications. The modern designer should study finite element methods in more detail than we can consider here. It is still an active area of research. The current trends are toward the use of error estimators and automatic adaptive FEA procedures that give the maximum accuracy for the minimum computational cost. This is also closely tied to shape modification and optimization procedures. 1.2 Capabilities of FEA There are many commercial and public-domain finite element systems that are available to the designer. To summarize the typical capabilities, several of the most widely used software systems have been compared to identify what they have in 1 2 J. E. Akin common. Often we find about 90% of the options are available in all the systems. Some offer very specialized capabilities such as aeroelastic flutter or hydroelastic lubrication. The mainstream capabilities to be listed here are found to be included in the majority of the commercial systems of ABACUS, ANSYS, MARC, and NASTRAN. The newer adaptive systems like ADAPT, PHLEX, PROBE, and RASNA may have fewer options installed but they are rapidly adding features common to those given above. Most of these systems are available on engineering workstations and personal computers as well as mainframes and supercomputers. The extent of the usefulness of a FEA system is directly related to the extent of its element library. The typical elements found within a single system usually include membrane, solid, and axisymmetric elements that offer linear, quadratic, and cubic approximations with a fixed number of unknowns per node. The new hierarchical elements have relatively few basic shapes but they do offer a potentially large number of unknowns per node (up to 81 for a solid). Thus, the actual effective element library size is extremely large. In the finite element method, the boundary and interior of the region are subdivided by lines (or surfaces) into a finite number of discrete sized subregions or finite elements. A number of nodal points are established with the mesh. These nodal points can lie anywhere along, or inside, the subdividing mesh, but they are usually located at intersecting mesh lines (or surfaces). The elements may have straight boundaries and thus, some geometric approximations will be introduced in the geometric idealization if the actual region of interest has curvilinear boundaries. The nodal points and elements are assigned identifying integer numbers beginning with unity and ranging to some maximum value. The assignment of the nodal numbers and element numbers will have a significant effect on the solution time and storage requirements. The analyst assigns a number of generalized degrees of freedom to each and every node. These are the unknown nodal parameters that have been chosen by the analyst to govern the formulation of the problem of interest. Common nodal parameters are displacement components, temperatures, and velocity components. The nodal parameters do not have to have a physical meaning, although they usually do. For example, the hierarchical elements typically use the derivatives up to order six as the midside nodal parameters. This idealization procedure defines the total number of degrees of freedom associated with a typical node, a typical element, and the total system. Data must be supplied to define the spatial coordinates of each nodal point. It is common to associate some code to each node to indicate which, if any, of the parameters at the node have boundary constraints specified. In the new adaptive systems the number of nodes, elements, and parameters per node usually all change with each new iteration. Another important concept is that of element connectivity, i.e., the list of global node numbers that are attached to an element. The element connectivity data defines the topology of the (initial) mesh, which is used, in turn, to assemble the system algebraic equations. Thus, for each element it is necessary to input, in some consistent order, the node numbers that are associated with that particular element. The list of node numbers connected to a particular element is usually referred to as the element incident list for that element. We usually also associate a material code with each element. Finite element analysis can require very large amounts of input data. Thus, most FEA systems, and some CAD systems, offer the user significant data generation or supplementation capabilities. The common data generation and validation options include the generation and/or replication of coordinate systems, node locations, element connectivity, loading sets, restraint conditions, etc. The verification of such extensive Finite Elements 3 Table 1.1 Typical variables in finite element analysis Application Primary Associated Secondary Stress analysis Displacement, Force, Stress, Rotation Moment Failure criterion Error estimates Heat transfer Temperature Flux Interior flux Error estimates Potential flow Potential function Normal velocity Interior velocity Error estimates Navier-Stokes Velocity Pressure Error estimates amounts of input and generated data is greatly enhanced by the use of computer graphics. In the hierarchical methods we must also compute the error indicators, error estimators, and various energy norms. All these quantities are output at 1 to 27 points in each of thousands of elements. Thus, stress file editors are usually provided to allow the designer to selectively extract such data. Most of the output options from an FEA system are available in graphical form. The most commonly needed information in the design process is the state of temperatures or stresses and displacements. Thus, almost every system offers linear static stress analysis capabilities, and linear thermal analysis capabilities for conduction and convection that are often needed to provide temperature distributions for thermal stress analysis. Usually the same mesh geometry is used for the temperature analysis and the thermal stress analysis. Of course, some designs require information on the natural frequencies of vibration or the response to dynamic forces or the effect of frequency driven excitations. Thus, dynamic analysis options are usually available. Today efficient utilization of materials in the design processes often requires us to employ nonlinear material properties and/or nonlinear equations. Such resources require a more experienced and sophisticated user. The usual nonlinear stress analysis features in large commercial FEA systems include buckling, creep, large deflections, and plasticity. Mechanical design may also require the use of computational fluid dynamics. There are a small number of FEA systems that offer such analysis and design aids. For example, the FIDAP product for CFD FEA offers numerous practical incompressible flow features, such as isothermal Newtonian and non-Newtonian flows, free, forced, or mixed convection, flows in saturated porous media, advection-diffusion problems, etc. Similar adaptive codes like ADAPT and PHLEX offer the incompressible Navier-Stokes equations, porous media flow, etc. There are certain features of finite element systems which are so important from a practical point of view that, essentially, we cannot get along without them. Basically we have the ability to handle completely arbitrary geometries, which is essential to practical engineering design. Almost all the structural analysis, whether static, dynamic, linear or 4 J. E. Akin Table 1.2 Typical given variables and corresponding reactions Application Given Reaction Stress analysis Displacement Force Rotation Moment Force Displacement Couple Rotation Heat transfer Temperature Heat flux Heat flux Temperature Potential flow Potential Normal velocity Normal velocity Potential nonlinear, is done by finite element techniques on large problems. The other abilities provide a lot of flexibility in specifying loading and restraints (support capabilities). Typically, we will have several different materials at different arbitrary locations within an object and we automatically have the capability to handle these nonhomogeneous materials. Just as importantly, the boundary conditions that attach one material to another are automatic, and we don't have to do anything to describe them unless it is possible for gaps to open between materials. Most important, or practical, engineering components are made up of more than one material, and we need an easy way to handle that. What takes place less often is the fact that we have anisotropic materials (one whose properties vary with direction, instead of being the same in all directions). There is a great wealth of materials that have this behavior, although at the undergraduate level, anisotropic materials are rarely mentioned. Many materials, such as reinforced concrete, plywood, any filament-wound material, and composite materials, are essentially anisotropic. Likewise, for heat-transfer problems, we will have thermal conductivities that are directionally dependent and, therefore, we would have to enter two or three thermal conductivities that indicate how this material is directionally dependent. Thus, these things mean that for practical use in design, finite element analysis is very important to us. The biggest disadvantage of the finite element method is that it has so much power there is the potential that large amounts of data and computation will be required. On small problems with about two thousand unknowns, many personal computers are available that can run an FEA system. For moderate problems with 10,000 to 15,000 equations, we need an engineering workstation or superminicomputer. Above 25,000 unknowns, we usually need to use a mainframe. A supercomputer or mini- supercomputer is usually necessary when we have more than 100,000 unknowns. All these systems should provide access to good graphical displays. All components employed in a design are three-dimensional but several common special cases have been defined that allow two-dimensional studies to provide useful design procedures. The most common examples in solid mechanics are the states of plane stress (covered in undergraduate mechanics of materials) and plane strain, the axisymmetric solid model, the thin-plate model, and the thin-shell model. The latter is Finite Elements 5 defined in terms of two parametric surface coordinates even though the shell exists in three dimensions. The thin beam can be thought of as a degenerate case of the thin-plate model. Similar concepts are used in CFD to avoid three-dimensional models. We often encounter plane and axisymmetric flows in various design projects. Even though today's solid modellers can generate three-dimensional meshes relative easily one should learn to approach such problems carefully. A well planned series of two-dimensional approximations can provide important insight into planning a good three-dimensional model. They also provide good "ballpark" checks on the three-dimensional answers. Of course, use of basic handbook calculations in estimating the answer before approaching a FEA system is also recommended. 1.3 Outline of Finite Element Procedures From the mathematical point of view the finite element method is an integral formulation. Modern finite element integral formulations are usually obtained by either of two different procedures: variational formulations or weighted residual formulations. The following sections briefly outline the common procedures for establishing finite element models. It is fortunate that all these techniques use the same bookkeeping operations to generate the final assembly of algebraic equations that must be solved for the unknowns. a) Mesh ra m b) Equation SD = C - 0 v.U j A LD6. Figure 1.3.1 Graphical assembly