WHAT EVERY ENGINEERSHOULD KNOW ABOUTCOMPUTATIONALTECHNIQUES OFFINITE ELEMENTANALYSISSecond Edition

LOUIS KOMZSIK

WHAT EVERY ENGINEERSHOULD KNOW ABOUTCOMPUTATIONALTECHNIQUES OFFINITE ELEMENTANALYSISSecond Edition

CRC Press is an imprint of theTaylor & Francis Group, an informa business

Boca Raton London New York

CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742

2009 by Taylor & Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government worksVersion Date: 20131125

International Standard Book Number-13: 978-1-4398-0295-3 (eBook - PDF)

This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.

Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit-ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.

For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.Visit the Taylor & Francis Web site athttp://www.taylorandfrancis.comand the CRC Press Web site athttp://www.crcpress.com

To my son, Victor

Contents

Preface to the second edition xiii

Preface to the rst edition xv

Acknowledgments xvii

I Numerical Model Generation 1

1 Finite Element Analysis 31.1 Solution of boundary value problems . . . . . . . . . . . . . . 31.2 Finite element shape functions . . . . . . . . . . . . . . . . . 61.3 Finite element basis functions . . . . . . . . . . . . . . . . . . 91.4 Assembly of nite element matrices . . . . . . . . . . . . . . . 121.5 Element matrix generation . . . . . . . . . . . . . . . . . . . . 151.6 Local to global coordinate transformation . . . . . . . . . . . 191.7 A linear quadrilateral nite element . . . . . . . . . . . . . . 201.8 Quadratic nite elements . . . . . . . . . . . . . . . . . . . . 26References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Finite Element Model Generation 312.1 Bezier spline approximation . . . . . . . . . . . . . . . . . . . 312.2 Bezier surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3 B-spline technology . . . . . . . . . . . . . . . . . . . . . . . . 402.4 Computational example . . . . . . . . . . . . . . . . . . . . . 432.5 NURBS objects . . . . . . . . . . . . . . . . . . . . . . . . . . 482.6 Geometric model discretization . . . . . . . . . . . . . . . . . 502.7 Delaunay mesh generation . . . . . . . . . . . . . . . . . . . . 512.8 Model generation case study . . . . . . . . . . . . . . . . . . . 54References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3 Modeling of Physical Phenomena 593.1 Lagranges equations of motion . . . . . . . . . . . . . . . . . 593.2 Continuum mechanical systems . . . . . . . . . . . . . . . . . 613.3 Finite element analysis of elastic continuum . . . . . . . . . . 633.4 A tetrahedral nite element . . . . . . . . . . . . . . . . . . . 653.5 Equation of motion of mechanical system . . . . . . . . . . . 693.6 Transformation to frequency domain . . . . . . . . . . . . . . 71

vii

viii

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 Constraints and Boundary Conditions 754.1 The concept of multi-point constraints . . . . . . . . . . . . . 764.2 The elimination of multi-point constraints . . . . . . . . . . . 794.3 An axial bar element . . . . . . . . . . . . . . . . . . . . . . . 824.4 The concept of single-point constraints . . . . . . . . . . . . . 854.5 The elimination of single-point constraints . . . . . . . . . . . 864.6 Rigid body motion support . . . . . . . . . . . . . . . . . . . 884.7 Constraint augmentation approach . . . . . . . . . . . . . . . 90References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 Singularity Detection of Finite Element Models 935.1 Local singularities . . . . . . . . . . . . . . . . . . . . . . . . 935.2 Global singularities . . . . . . . . . . . . . . . . . . . . . . . . 975.3 Massless degrees of freedom . . . . . . . . . . . . . . . . . . . 995.4 Massless mechanisms . . . . . . . . . . . . . . . . . . . . . . . 1005.5 Industrial case studies . . . . . . . . . . . . . . . . . . . . . . 102References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6 Coupling Physical Phenomena 1056.1 Fluid-structure interaction . . . . . . . . . . . . . . . . . . . . 1056.2 A hexahedral nite element . . . . . . . . . . . . . . . . . . . 1066.3 Fluid nite elements . . . . . . . . . . . . . . . . . . . . . . . 1096.4 Coupling structure with compressible uid . . . . . . . . . . . 1116.5 Coupling structure with incompressible uid . . . . . . . . . . 1126.6 Structural acoustic case study . . . . . . . . . . . . . . . . . . 113References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

II Computational Reduction Techniques 117

7 Matrix Factorization and Linear Systems 1197.1 Finite element matrix reordering . . . . . . . . . . . . . . . . 1197.2 Sparse matrix factorization . . . . . . . . . . . . . . . . . . . 1227.3 Multi-frontal factorization . . . . . . . . . . . . . . . . . . . . 1247.4 Linear system solution . . . . . . . . . . . . . . . . . . . . . . 1267.5 Distributed factorization and solution . . . . . . . . . . . . . 1277.6 Factorization and solution case studies . . . . . . . . . . . . . 1307.7 Iterative solution of linear systems . . . . . . . . . . . . . . . 1347.8 Preconditioned iterative solution technique . . . . . . . . . . 137References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

ix

8 Static Condensation 1418.1 Single-level, single-component condensation . . . . . . . . . . 1418.2 Computational example . . . . . . . . . . . . . . . . . . . . . 1448.3 Single-level, multiple-component condensation . . . . . . . . . 1478.4 Multiple-level static condensation . . . . . . . . . . . . . . . . 1528.5 Static condensation case study . . . . . . . . . . . . . . . . . 155References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

9 Real Spectral Computations 1599.1 Spectral transformation . . . . . . . . . . . . . . . . . . . . . 1599.2 Lanczos reduction . . . . . . . . . . . . . . . . . . . . . . . . 1619.3 Generalized eigenvalue problem . . . . . . . . . . . . . . . . . 1649.4 Eigensolution computation . . . . . . . . . . . . . . . . . . . . 1669.5 Distributed eigenvalue computation . . . . . . . . . . . . . . . 1689.6 Dense eigenvalue analysis . . . . . . . . . . . . . . . . . . . . 1729.7 Householder reduction technique . . . . . . . . . . . . . . . . 1759.8 Normal modes analysis case studies . . . . . . . . . . . . . . . 177References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

10 Complex Spectral Computations 18310.1 Complex spectral transformation . . . . . . . . . . . . . . . . 18310.2 Biorthogonal Lanczos reduction . . . . . . . . . . . . . . . . . 18410.3 Implicit operator multiplication . . . . . . . . . . . . . . . . . 18610.4 Recovery of physical solution . . . . . . . . . . . . . . . . . . 18810.5 Solution evaluation . . . . . . . . . . . . . . . . . . . . . . . . 19010.6 Reduction to Hessenberg form . . . . . . . . . . . . . . . . . . 19110.7 Rotating component application . . . . . . . . . . . . . . . . . 19210.8 Complex modal analysis case studies . . . . . . . . . . . . . . 196References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

11 Dynamic Reduction 20111.1 Single-level, single-component dynamic reduction . . . . . . . 20111.2 Accuracy of dynamic reduction . . . . . . . . . . . . . . . . . 20311.3 Computational example . . . . . . . . . . . . . . . . . . . . . 20611.4 Single-level, multiple-component dynamic reduction . . . . . . 20811.5 Multiple-level dynamic reduction . . . . . . . . . . . . . . . . 21011.6 Multi-body analysis application . . . . . . . . . . . . . . . . . 212References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

12 Component Mode Synthesis 21712.1 Single-level, single-component modal synthesis . . . . . . . . . 21712.2 Mixed boundary component mode reduction . . . . . . . . . . 21912.3 Computational example . . . . . . . . . . . . . . . . . . . . . 22212.4 Single-level, multiple-component modal synthesis . . . . . . . 22512.5 Multiple-level modal synthesis . . . . . . . . . . . . . . . . . . 228

x12.6 Component mode synthesis case study . . . . . . . . . . . . . 230References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

III Engineering Solution Computations 235

13 Modal Solution Technique 23713.1 Modal solution . . . . . . . . . . . . . . . . . . . . . . . . . . 23713.2 Truncation error in modal solution . . . . . . . . . . . . . . . 23913.3 The method of residual exibility . . . . . . . . . . . . . . . . 24113.4 The method of mode acceleration . . . . . . . . . . . . . . . . 24513.5 Coupled modal solution application . . . . . . . . . . . . . . . 24613.6 Modal contributions and energies . . . . . . . . . . . . . . . . 247References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

14 Transient Response Analysis 25114.1 The central dierence method . . . . . . . . . . . . . . . . . . 25114.2 The Newmark method . . . . . . . . . . . . . . . . . . . . . . 25214.3 Starting conditions and time step changes . . . . . . . . . . . 25414.4 Stability of time integration techniques . . . . . . . . . . . . . 25514.5 Transient response case study . . . . . . . . . . . . . . . . . . 25814.6 State-space formulation . . . . . . . . . . . . . . . . . . . . . 259References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

15 Frequency Domain Analysis 26315.1 Direct and modal frequency response analysis . . . . . . . . . 26315.2 Reduced-order frequency response analysis . . . . . . . . . . . 26415.3 Accuracy of reduced-order solution . . . . . . . . . . . . . . . 26715.4 Frequency response case study . . . . . . . . . . . . . . . . . 26815.5 Enforced motion application . . . . . . . . . . . . . . . . . . . 269References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

16 Nonlinear Analysis 27316.1 Introduction to nonlinear analysis . . . . . . . . . . . . . . . . 27316.2 Geometric nonlinearity . . . . . . . . . . . . . . . . . . . . . . 27516.3 Newton-Raphson methods . . . . . . . . . . . . . . . . . . . . 27816.4 Quasi-Newton iteration techniques . . . . . . . . . . . . . . . 28216.5 Convergence criteria . . . . . . . . . . . . . . . . . . . . . . . 28416.6 Computational example . . . . . . . . . . . . . . . . . . . . . 28516.7 Nonlinear dynamics . . . . . . . . . . . . . . . . . . . . . . . . 287References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

17 Sensitivity and Optimization 28917.1 Design sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 28917.2 Design optimization . . . . . . . . . . . . . . . . . . . . . . . 29017.3 Planar bending of the bar . . . . . . . . . . . . . . . . . . . . 294

Contents xi

17.4 Computational example . . . . . . . . . . . . . . . . . . . . . 29717.5 Eigenfunction sensitivities . . . . . . . . . . . . . . . . . . . . 30217.6 Variational analysis . . . . . . . . . . . . . . . . . . . . . . . . 304References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

18 Engineering Result Computations 30918.1 Displacement recovery . . . . . . . . . . . . . . . . . . . . . . 30918.2 Stress calculation . . . . . . . . . . . . . . . . . . . . . . . . . 31118.3 Nodal data interpolation . . . . . . . . . . . . . . . . . . . . . 31218.4 Level curve computation . . . . . . . . . . . . . . . . . . . . . 31418.5 Engineering analysis case study . . . . . . . . . . . . . . . . . 316References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Annotation 321

List of Figures 323

List of Tables 325

Index 327

Closing Remarks 331

Preface to the second edition

I am grateful to Taylor & Francis, in particular to Nora Konopka, publisher,for the opportunity to revise this book after ve years in print, and for herenthusiastic support of the rst edition. This made the book available to awide range of students and practicing engineers fullling my original inten-tions. My sincere thanks are also due to Amy Blalock, project coordinator,and Michele Dimont, project editor, at Taylor & Francis.

Mike Gockel, my colleague of many years, now retired, was again instru-mental in clarifying some of the presentation, and he deserves my repeatedgratitude. I would like to thank Professor Duc Nguyen for his proofreading ofthe extensions of this edition. His use of the rst edition in his teaching pro-vided me with valuable feedback and conrmation of the approach of the book.

A half a decade passed since the original writing of the rst edition andthis edition contains numerous noteworthy technical extensions. In Part I thenite element chapter now contains a brief introduction to quadratic niteelement shape functions (1.8). Also in Part I, the geometry modeling chapterhas been extended with three sections (2.3, 2.4 and 2.5) to discuss the B-splinetechnology that has become the de facto industry standard. Several new sec-tions were added to address reader requested topics, such as supporting therigid body motion (4.6), the method of augmenting constraints (4.7) and adiscussion on detecting and eliminating massless mechanisms (5.5).

Still in Part I, a new Chapter 6 describes a signicant application trend ofthe past years: the use of the technology to couple multiple physical phenom-ena. This includes a more detailed description of the uid-structure interac-tion application, a hexahedral nite element, as well as a structural-acousticscase study.

In Part II, a new section (7.7) addressing iterative solutions of linear systemsand specically the method of conjugate gradients, was also recommended byreaders of the rst edition. Also in Part II, a new Chapter 10 is dedicated tocomplex spectral computations, a topic briey mentioned but not elaboratedon in the rst edition. The rotor dynamic application topic and related casestudy examples round up this new chapter.

In Part III, the modal solution chapter has been extended with a new section

xiii

xiv Preface to the second edition

(13.6) describing modal energies and contributions. A new section (14.6) inthe transient response analysis chapter discusses the state-space formulation.The frequency domain analysis chapter has been enhanced with a new sec-tion (15.5) on enforced motion computations. Finally, the nonlinear chapterreceived a new section (16.2) describing geometric nonlinearity computationsin some detail.

The application focus has also signicantly expanded during the years sincethe publication of the rst edition and one of the goals of this edition was toreect these changes. The updated case study sections (2.8, 7.6, 9.8, 10.8,12.6, 14.5, 15.4 and 18.5) state-of-the-art application results demonstrate thetremendously increased computational complexity.

The nal goal of this edition was to correct some of the typing mistakes andtechnical misstatements of the rst edition, which were pointed out to me byreaders. While they kindly stated that those were not limiting the usefulnessof the book, I exercised extreme caution to make this edition as error free andclear as possible.

Louis Komzsik2009

The model in the cover art is courtesy of Pilates Aircraft Corporation,Stans, Switzerland. It depicts the tail wing vibrations of a PC-21 aircraft,computed by utilizing the techniques described in this book.

Preface to the rst edition

The method of nite elements has become a dominant tool of engineeringanalysis in a large variety of industries and sciences, especially in mechanicaland aerospace engineering. In this role, the method enables the engineer orscientist to solve a physical problem or analyze a process. There is, however,signicant computational work - in several distinct phases - involved in thesolution of a physical problem with the nite element method. The emphasisof this book is on the computational techniques of this complete process fromthe physical problem to the computed solution.

In the rst phase the physical problem is described in mathematical form,most of the time by a boundary value problem of some sort. At the same timethe geometry of the physical problem is also approximated by computationalgeometry techniques resulting in the nite element model. Applying bound-ary conditions and various constraints to the nite element model results in anumerically solvable form. The rst part of the book addresses these topics.

In the second phase of operations the numerical model is reduced to a com-putationally more ecient form via various spectral representations. Todaynite element problems are extremely large in industrial applications, there-fore, this is an important step. The subject of the second part of the book isthe reduction techniques to reach an eciently solvable computational model.

Finally, the solution of the engineering problem is obtained with speciccomputational techniques. Both time and frequency domain solutions areused in practice. Advanced computations addressing nonlinearity and opti-mization may also be applied. The third part of the book deals with thesetopics as well as the representation of the computed results.

The book is intended to be a concise, self-contained reference for the topicand aimed at practicing engineers who put the nite element technique topractical use. It may be the subject of specic interest to users of com-mercial nite element analysis products, as those products execute most ofthese computational techniques in various forms. Graduate students of niteelement techniques in any discipline could benet from using the book as well.

The material comes from my three decades of activity in the shipbuild-ing, aerospace and automobile industries, during which I used many of these

xv

xvi Preface to the rst edition

techniques. I have also personally implemented some of these techniques intovarious versions of NASTRAN1, the worlds leading nite element software.

Finally, I have also encountered many students during my years of teach-ing whose understanding of these computations would have been signicantlybetter with such a book.

Louis Komzsik2004

1 - NASTRAN is a registered trademark of the National Aeronautics andSpace Administration

Acknowledgments

I appreciate Mr. Mike Gockels (MSC Software Corporation, retired) technicalevaluation of the manuscript and his important recommendations, especiallythose related to the techniques of Chapters 4 and 5.

I would also like to thank Dr. Al Danial (Northrop-Grumman Corporation)for his repeated and very careful proofreading of the entire manuscipt. Hisclarifying comments representing the application engineers perspective havesignicantly contributed to the readability of the book.

Professor Barna Szabo (Washington University, St. Louis) deserves creditfor his valuable corrections and insightful advice through several revisions ofthe book. His professional inuence in the subject area has reached a widerange of engineers and analysts, including me.

Many thanks are also due to Mrs. Lori Lampert (MSC Software Corpo-ration) for her expertise and patience in producing gures from my hand-drawings.

I also value the professional contribution of the publication sta at Taylorand Francis Group. My sincere thanks to Nora Konopka, publisher, HelenaRedshaw, manager and editor Richard Tressider. They all deserve signicantcredit in the nal outcome.

Louis Komzsik2004

xvii

Part I

Numerical ModelGeneration

1

1Finite Element Analysis

The goal of this chapter is to introduce the reader to nite element analy-sis which is the basis for the discussion of the computational methods in theremainder of the book. This chapter rst focuses on the computational funda-mentals of the method in connection with a simple boundary value problem.These fundamentals will be expanded with the derivation of a practical niteelement and further when dealing with the application of the technique formechanical systems in Chapter 3.

1.1 Solution of boundary value problems

The method of using nite elements for the solution of boundary value prob-lems has almost a century of history. The pioneering paper by Ritz [8] haslaid the foundation for this technology. The most widely used practical tech-nique, however, is Galerkins method [3].

The dierence between the Ritz method and that of Galerkins is in thefact that the rst addresses the variational form of the boundary value prob-lem. Galerkins method minimizes the residual of the dierential equationintegrated over the domain with a weight function, hence it is also called themethod of weighted residuals.

This dierence lends more generality and computational convenience toGalerkins method. Let us consider a linear dierential equation in two vari-ables on a simple domain D:

L(q(x, y)) = 0, (x, y) D,and apply Dirichlet boundary conditions on the boundary B

q(x, y) = 0, (x, y) B.Galerkins method is based on the Ritzs approximate solution idea and

constructs the approximate solution as

3

4 Chapter 1

q(x, y) = q1N1 + q2N2 + ... + qnNn,

where the qi are the yet unknown solution values at discrete points in thedomain (the node points of the nite element mesh) and

Ni, i = 1, ..n,

is the set of the nite element shape functions to be derived shortly. In thiscase, of course there is a residual of the dierential equation

L(q) = 0.Galerkin proposed using the shape functions of the approximate solution alsoas the weights, and requires that the integral of the so weighted residual van-ish.

D

L(q)Nj(x, y)dxdy = 0; j = 1, 2, . . . , n.

This yields a system for the solution of the coecients as

D

L(n

i=1

qiNi(x, y))Nj(x, y)dxdy = 0; j = 1, 2, . . . , n.

This is a linear system and produces the unknown values of qi.

Let us now consider the deformation of an elastic membrane loaded by adistributed force of f(x, y) shown in Figure 1.1. The mathematical model isthe well-known Poissons equation.

2q

x2

2q

y2= f(x, y),

where q(x, y) is the vertical displacement of the membrane at (x, y) and f(x, y)is the distributed load on the surface of the membrane. Assume the membraneoccupies the D domain in the xy plane with a boundary B. We assume thatthe membrane is clamped manifested by a Dirichlet boundary condition. Itshould be noted that in practical problems the boundary is not necessarily assmooth as shown on the Figure 1.1, in fact it is usually only piecewise analytic.

Let us now apply Galerkins method to this problem. D

( 2q

x2+

2q

y2+ f(x, y))Njdxdy = 0, j = 1, . . . , n.

Substituting the approximate solution yields

D

(n

i=1

qi2Nix2

+n

i=1

qi2Niy2

+ f(x, y))Njdxdy = 0, j = 1, . . . , n.

Finite Element Analysis 5

z

q(x, y)

y

D

B

q = 0

x

FIGURE 1.1 Membrane model

The left hand side terms may be integrated by parts and after employing theboundary condition they simplify as

D

(2Nix2

+2Niy2

)Njdxdy =

D

(Nix

Njx

+Niy

Njy

)dxdy.

Substituting and regrouping yields

D

(n

i=1

qiNix

Njx

+n

i=1

qiNiy

Njy

f(x, y)Nj)dxdy = 0, j = 1, . . . , n.

Unrolling the sums and reordering we get the Galerkin equations: ((q1

N1x

+ ...+ qnNnx

)Njx

+ (q1N1y

+ ...+ qnNny

)Njy

)dxdy =

f(x, y)Njdxdy

for j = 1, .., n. Introducing the notation

Kij = Kji =

(Nix

Njx

+Niy

Njy

)dxdy

6 Chapter 1

andFj =

(f(x, y)Nj)dxdy

the Galerkin equations may be written as a matrix equation

Kq = F.

The system matrix is

K =

K1,1 K1,2 . . . K1,nK2,1 K2,2 . . . K2,n. . . . . . . . . . . .Kn,1 Kn,2 . . . Kn,n

,

with solution vector of

q =

q1q2. . .qn

,

and right hand side vector of

F =

F1F2. . .Fn

.

The assembly process is addressed in more detail in Section 1.4 after intro-ducing the shape functions. The K matrix is usually very sparse as many Kijbecome zero. This equation is known as the linear static analysis problem,where K is called the stiness matrix, F is the load vector and q is the vectorof displacements, the solution of Poissons equation. Other dierential equa-tions could lead to similar form as demonstrated in, for example [2].

The concept, therefore, is generally contributing to its wide-spread appli-cation success. For the mathematical theory see [6]; the matrix algebraicfoundation is thoroughly discussed in [7]. More details may be obtained fromthe now classic text of [11].

1.2 Finite element shape functions

To interpolate inside the elements piecewise polynomials are usually used.For example a triangular discretization of a two dimensional domain may be

Finite Element Analysis 7

approximated by bilinear interpolation functions of form

q(x, y) = a + bx+ cy.

In order to nd the coecients let us consider the triangular region (element)of the x y plane in a specically located local coordinate system and thenotation shown in Figure 1.2.

3 q3(x3, y3)

q2(x2, y2)(x2, 0)

q1(x1, y1)(0, 0)

q(x, y )

2x

y

1

FIGURE 1.2 Local coordinates of triangular element

The usage of a local coordinate system in Figure 1.2 does not limit the gen-erality of the following discussion. The arrangement can always be achievedby appropriate coordinate transformations on a generally located triangle.Using the notation and assignments on Figure 1.2 and by evaluating at eachnode of the triangle

8 Chapter 1

qe =

q1q2q3

=

1 0 01 x2 01 x3 y3

abc

.

The triangular system of equations is easily solved for the unknown coe-cients as

abc

=

1 0 0 1

x21x2

0x3x2x2y3

x3x2y3

1y3

q1q2q3

.

By back-substituting into the approximation equation we get

q(x, y) = N

q1q2q3

= [N1 N2 N3 ]

q1q2q3

.

Here N contains the N1, N2, N3 shape functions (more precisely the traces ofshape functions inside an element). With these we are now able to describethe relationship between the solution value inside an element in terms of thesolutions at the corner node points

q(x, y) = N1q1 +N2q2 +N3q3.

The values of Ni are

N1 = 1 1x2

x+x3 x2x2y3

y,

N2 =1x2

x x3x2y3

y,

andN3 =

1y3

y.

These clearly depend on the coordinates of the corner node of the particulartriangular element of the domain. It is easy to see that at every node only oneof the shape functions is nonzero. Specically at node 1: N2 and N3 vanish,while N1 = 1. At node 2: N2 = 1, both N1 and N3 are zero. Finally at node3: N3 takes a value of one and the other two vanish. It is also easy to verifythat the

N1 +N2 +N3 = 1

equation is satised.

The nonzero shape functions at a certain node point reduce to zero at theother two nodes, respectively. The interpolations are continuous across theneighboring elements. On an edge between two triangles, the approximation

Finite Element Analysis 9

is linear. It is the same when it is approached from either element.

Specically along the edge between nodes 1 and 2 the shape function N3is zero. The shape functions N1 and N2 along this edge are the same whencalculated from an element on either side of that edge.

Naturally, additional computations are required to reect to the fact whenthe triangle is generally located, i.e. none of its sides is collinear with anyaxes. This issue of local-global coordinate transformations will be discussedshortly.

1.3 Finite element basis functions

There is another (sometimes misinterpreted) component of nite element tech-nology, the basis functions. They are sometimes used in place of shape func-tions by engineers, although as shown below, they are distinctly dierent. Theapproximation of

q(x, y) = Nqe

may also be written as

q(x, y) = Mce

where M is the matrix of basis functions and ce is the vector of basis coe-cients.

Clearly for our exampleM =

[1 x y

]and

ce =

abc

.

The family of basis functions for two-dimensional elements may be writtenfrom the terms shown on Table 1.1.

Depending on how the basis functions are chosen, various two-dimensionalelements may be derived. Naturally a higher order basis function family re-quires more node points. For example, a quadratic (order= 2) triangularelement, often used in industry, is based on introducing midpoint nodes oneach side of the triangle. This enables the use of the following interpolation

10 Chapter 1

TABLE 1.1

Basis function terms fortwo-dimensional elementsOrder Terms

0 11 x y2 x2 xy y23 x3 x2y xy2 y3

function

q(x, y) = a+ bx+ cy + dx2 + ey2 + fxy

in each triangle. The six coecients are again easily established by a proce-dure similar to the linear triangular element above. The interpolation acrossquadratic element boundaries is also continuous, however, now it is parabolicalong an edge. Nevertheless, the parabola produced by the neighboring ele-ments is the same from both sides. Quadratic nite elements will be discussedin section 1.8.

For a rst order rectangular element the interpolation may be of the form

q(x, y) = a + bx+ cy + dxy.

In this case, all the rst-order basis functions were used as well as one com-ponent of the second-order basis function family. We will derive a practicalrectangular element in Section 1.7. Similarly a second-order (eight noded)rectangular element is approximated as

q(x, y) = a+ bx+ cy + dxy + ex2 + fx2y + gxy2 + hy2.

This is again the use of the complete 2nd order family plus two components ofthe 3rd order family to accommodate additional node points. The latter areusually located on the midpoints of each side, as they were on the quadratictriangle.

For a three-dimensional domain, the four noded tetrahedron is one of themost commonly used nite elements. The interpolation inside a tetrahedralelement is of form

q(x, y, z) = a+ bx+ cy + dz.

The basis function terms for three-dimensional elements is shown in Table 1.2.

Quadratic interpolation of the tetrahedron is also possible; the related ele-ment is called the 10-noded tetrahedron. The extra node points are located

Finite Element Analysis 11

TABLE 1.2

Basis function terms forthree-dimensional elementsOrder Terms

0 11 x y z2 x2 xy y2 xz yz z23 x3 ... xyz ... y3 z3

on the midpoints of the edges.

q(x, y, z) = a+ bx+ cy + dz + ex2 + fxy + gy2 + hxz + iyz + jz2.

The third-order three-dimensional basis function family introduces another 10terms, some of them are shown on Table 1.2.

Finally, additional volume elements are also frequently used. The hexahe-dron is one of the most widely accepted. Its rst order version consists ofeight node points at the corners of the hexahedron and therefore, it is denedwith specically chosen basis functions as

q(x, y, z) = a+ bx+ cy + dz + exy + fxz + gyz + hxyz.

The quadratic hexahedral element consists of 20 nodes, the eight corner nodesand the 12 mid points on the edges. A 3rd order hexahedral element with 27nodes is also used, albeit not widely. The additional seven nodes come fromthe mid-point of the six faces and from the center of the volume.

Finally, higher order polynomial (p-version) elements are also used in theindustry. These elements introduce side shape functions in addition to thenodal shape functions mentioned earlier. The side shape functions, as theirname indicates, are assigned to the sides of the elements. They are formu-lated in terms of some orthogonal, most often Legendre, polynomial of orderp, hence the name. There are clearly advantages in computational accuracywhen applying such elements. On the other hand, they introduce extra com-putational costs, so they are mainly used in specic applications and notgenerally. The method and some applications are described in detail in thebook of the pioneering authors of the technique [9].

The gradual widening of the nite element technology may be assessed byreviewing the early articles of [10] and [1], as well as from the reference ofthe rst general purpose and still premier nite element analysis tool [4].

12 Chapter 1

1.4 Assembly of nite element matrices

The repeated application of general triangles may be used to cover the D pla-nar domain as shown in Figure 1.3. The process is called meshing. The pointsinside the domain and on the boundary are the node points. They span thenite element mesh. There may be small gaps between the boundary and the

y

x

FIGURE 1.3 Meshing the membrane model

sides of the triangles adjacent to the boundary. This issue contributes to theapproximation error of the nite element method. The gaps may be lled byprogressively smaller elements or those triangles may be replaced by triangleswith curved edges. Nevertheless, all the elemental matrices contribute to theglobal nite element matrices and the process of computing these contribu-tions is the nite element matrix assembly process.

Finite Element Analysis 13

One way to view the assembly of the K matrix is by way of the shapefunctions. For the triangular element discussed in the last section a shapefunction associated with a node describes a plane going through the othertwo nodes and having a height of unity above the associated node. On theother hand, in an adjacent element the shape function associated with thesame node describes another plane, and so on. In general, a shape functionNi will dene a pyramid over node i.

This geometric interpretation explains the sparsity of the K matrix. Onlythose NiNj products will exist, and in turn produce a Kij entry in the Kmatrix, where the two pyramids of Ni and Nj overlap.

A computationally more practical method is based on summing up theenergy contributions from each element to the global matrix. The strainenergy (a component of the potential energy) of a certain element is

Ee =12

[(q

x)2 + (

q

y)2]dxdy.

Introducing the strain vector

=

[qxqy

].

the strain energy of the element is

Ee =12

T dxdy.

Considering our simple triangular element, dierentiating and using a matrixnotation yields

=

[qxqy

]=[bc

]=[ 1x2 1x2 0

x3x2x2y3

x3x2y3

1y3

] q1q2q3

= Bqe,

where

qe =

q1q2q3

.

In the above, B is commonly called the strain-displacement matrix. The qxand qy terms are the strain components of our element, in essence the rate ofchange of the deformation of the element in the coordinate directions. The Bmatrix relates the strains to the nodal displacements on the right, hence thename.

14 Chapter 1

Note, that the structure of B depends on the physical model, in our casehaving only one degree of freedom per node point for the membrane element.Elements representing other physical phenomena, for example, triangles hav-ing two in-plane degrees of freedom per node point, have dierent B matrix,as they have more possible strain components. This issue will be addressed inmore detail in Section 1.7 and in Chapter 3. Here we stay on a mathematicalfocus.

With this the element energy contribution is

Ee =12

qTe B

TBqedxdy.

Since the node point coordinates are constant with respect to the integrationwe may write

Ee =12qTe (

BTBdxdy)qe =12qTe keqe.

Here ke is the element matrix whose entries depend only on the shape of theelement. If our element is the element described by nodes 1, 2 and 3, then theterms in ke contribute to the terms of the 1st, 2nd and 3rd columns and rowsof the global K matrix. The actual integration for computing ke is addressedin the next section.

Let us assume that another element is adjacent to the 2-3 edge, its othernode being 4. Then by similar arguments, the 2nd elements matrix terms(depending on that particular elements shape) will contribute to the 2nd, 3rdand 4th columns and rows of the global matrix. This process is continued forall the elements contained in the nite element mesh.

Note, that in the case of quadratic or quadrilateral shape elements the ac-tual element matrices are again of dierent sizes. This fact is due to thedierent number of node points describing the element geometry. Neverthe-less, the matrix generation and assembly process is conceptually the same.

Furthermore, in the case of three-dimensional elements the energy formu-lation is even more complex. These issues will be discussed in more detail inChapter 3.

Finite Element Analysis 15

1.5 Element matrix generation

Let us now focus on calculating the element matrix integrals. Since for ourmodel B is constant (function of only the coordinates of the node points ofthe element), this may be simplied to

ke = BTB

dxdy = BTBAe,

where Ae is the surface area of the element as

Ae =

dxdy.

In order to evaluate this integral, the element is usually represented in para-metric coordinates. Let us consider again the local coordinates of the tri-angular element, now shown in Figure 1.4 with two specic coordinate axesrepresenting the parametric system. The axis (coincident with the local xaxis) going through node points 1 and 2 is the rst parametric axis u. Denethe other axis going from node 1 through node 3 as v. If we dene the (0, 0)parametric location to be node 1, the (1, 0) to be node 2 and the (0, 1) to benode 3, then the parametric transformation is of form

u =1x2

x x3x2y3

y

andv =

1y3

y.

Here we took advantage of the local coordinates of the nodes as shown onFigure 1.2. Note, that this transformation may also be written

u = N2

andv = N3.

Furthermore, the points inside one element may be written as

x = N1x1 +N2x2 +N3x3,

andy = N1y1 +N2y2 +N3y3.

Since we describe the coordinates of a point inside an element with the sameshape functions that were used to approximate the displacement eld, thisis called an iso-parametric representation and our element is called an iso-parametric element. Applying the local coordinates of our element of Figure

16 Chapter 1

x, u

y

3

12

v

u = 0, v = 1

u = 1v = 0

u = 0v = 0

FIGURE 1.4 Parametric coordinates of triangular element

1.2 yields

x = x2u+ x3v

andy = y3v.

The integral with this parameterization is dxdy =

det[

(x, y)(u, v)

]dudv.

Here the Jacobian matrix

(x, y)(u, v)

=[

xu

xv

yu

yv

]=[x2 x30 y3

].

With this resultAe = x2y3

dudv.

In practice the parametric integral for each element is executed numerically,most commonly via Gaussian numerical integration, quadrature for two di-mensions and cubature for three dimensions. Note, that this is in essence a

Finite Element Analysis 17

reduction type computation, main focus of Part II, as opposed to the analyticintegration over the continuum domain.

Gaussian numerical integration has become the industry standard tool forintegration of the element matrices by virtue of its higher accuracy than theNewton-Cotes type methods such as Simpsons. In general an integral overa specic continuous interval is approximated by a sum of weighted functionvalues at some specic locations.

11

f(t)dt = ni=1cif(ti).

Here n is the number of integration points used. The specic sampling loca-tions are the zeroes of the n-th Legendre polynomial:

ti : Pn(t) = 0

and the recursive denition of Legendre polynomials is

(k + 1)Pk+1(t) = (2k + 1)tPk(t) kPk1(t).Starting from P0(t) = 1 and P1(t) = t the recurrence form produces

P2(t) =12(3t2 1),

P3(t) =12(5t3 3t),

and so on. The ci weights are computed as

ci = 11

Ln1,i(t)dt

where

Ln,i =n

j=1,j =i

t tjti tj

is the i-th n-th order Lagrange polynomial with roots of the Legendre poly-nomials described above. For the most commonly occurring cases Table 1.3shows the values of ci and ti.

Now integrating over the parametric domain of our element, the integralhas the following boundaries

1u=0

1uv=0

dvdu.

This is clear when looking at Figure 1.4. One needs to transform above in-tegral boundaries to the standard [1, 1] interval required by the Gaussian

18 Chapter 1

TABLE 1.3

Gauss weights andlocationsn ti ci

1 0 22 1

3, 1

31, 1

3

35 , 0,

35

59 ,

89 ,

59

numerical integration. This may be done with the transformation

v =1 u2

+1 u2

r,

anddv =

1 u2

dr,

as well asu =

12+

12s,

anddu =

12ds.

The transformed integral amenable to Gauss quadrature is 1s=1

12

1r=1

(14 1

4s)drds.

Using the 1-point Gauss formula this is

122(

142) =

12.

With this the surface area of the element is

Ae =x2y32

,

which agrees with the geometric computation based on the triangles localcoordinates. This is a rather roundabout way of computing the area of atriangle. Note, however, that the discussion here is aimed at introducing gen-erally applicable principles.

Naturally, there is a wealth of element types used in various industries.Even for the simple triangular geometry there are other formulations. Theextensions are in both the number of node points describing the triangularelement as well as in the number of degrees of freedom associated with a nodepoint.

Finite Element Analysis 19

1.6 Local to global coordinate transformation

When the element matrix assembly issue was addressed earlier, the elementmatrix had been developed in terms of local (x, y, z) coordinates. In the caseof multiple elements, all the elements have their respective local coordinatesystem chosen on the same principle of the local x axis being collinear withone of the element sides and another one perpendicular.

Thus before assembling any element, the element matrix must be trans-formed to the global coordinate system common to all the elements. Let usdenote the elements local coordinate systems with (x, y, z) and the globalcoordinate system with (X,Y, Z). The unit direction vectors of the two coor-dinate systems are related as

ijk

= T

IJK

,

where the terms of the transformations are easily obtained from the geometricrelation between the local and global systems. Specically

T =

t11 t12 t13t21 t22 t23t31 t32 t33

,

where the tmn term is the cosine of the angle between the mth local coor-dinate axis and the nth global coordinate axis. The same transformation isapplicable to the nodal degrees of freedom of any element

qxqyqz

= T

qXqYqZ

.

Hence, the element displacements in the two systems are related as

qe = Glgqge ,

where the upper left and the lower right 3 3 blocks of the 6 6 Glg matrixare the same as the T matrix, the other blocks are zero. The qge notationrefers to the element displacements in the global coordinate system.

Considering the element energy contribution

Ee =12qTe keqe

20 Chapter 1

and substituting above we get

Ee =12qg,Te G

lg,T keGlgqge

orkge = G

lg,T keGlg .

This transformation follows the element matrix generation and precedes theassembly process. Naturally, the solution qge is also represented in global co-ordinates, which is the subject of the interest of the engineer anyways.

This issue will not be further discussed, the elements introduced later willbe generated either in terms of local or global coordinates for simplifying theparticular discussion. Commercial nite element analysis systems have spe-cic rules for the denition of local coordinates for various element types.

1.7 A linear quadrilateral nite element

So far we have discussed the rather limited triangular element formulation,mainly to provide a foundation for presenting the integration and assemblycomputations. We continue this chapter with the discussion of a more practi-cal quadrilateral or rectangular element, but rst we focus on the linear case.Quadrilateral elements are the most frequently used elements of industrialnite element analysis when analyzing topologically two-dimensional models,such as the body of an automobile or an airplane fuselage.

Let us place the element in the xy plane as shown in Figure 1.5, but poseno other restriction on its location. Based on the principles we developedin connection with the simple triangular element, we introduce shape func-tions. As we have four nodes in a quadrilateral element, we will have fourshape functions, each of whose values vanish at any other node but one. Forj = 1, 2, 3, 4, we dene

Ni ={1 when i = j,0 when i = j.

We create an element parametric coordinate system u, v originated in the in-terior of the element and having the following denition:

Ni =14(1 + uui)(1 + vvi).

Such a coordinate system is shown in Figure 1.6. The mapping of the general

Finite Element Analysis 21

3

1

2

4

q3y

q3x

q2y

q2x

q1x

q1y

q4y

q4x

p x y,( )

y

x

FIGURE 1.5 A planar quadrilateral element

element to the parametric coordinates is the following counterclockwise pat-tern:

(x1, y1) (1,1),(x2, y2) (1,1),(x3, y3) (1, 1),

and(x4, y4) (1, 1).

The corresponding four shape functions are:

N1 =14(1 u)(1 v),

N2 =14(1 + u)(1 v),

N3 =14(1 + u)(1 + v),

andN4 =

14(1 u)(1 + v).

22 Chapter 1

1 2

34

1 1,( ) 1 1,( )

u

v

1 1,( ) 1 1,( )

0 0,( )

FIGURE 1.6 Parametric coordinates of quadrilateral element

These so-called Lagrangian shape functions will be used for the element for-mulation. The above selection of the Ni functions obviously again satises

N1 +N2 +N3 +N4 = 1.

The element deformations, however, will not be vertical to the plane of theelement as in the earlier triangular membrane element. This element willhave deformation in the plane of the element. Hence, there are eight nodaldisplacements of the element as

qe =

q1xq1yq2xq2yq3xq3yq4xq4y

.

The displacement at any location inside this element is approximated withthe help of the matrix of shape functions as

Finite Element Analysis 23

q(x, y) = Nqe.

Since

q(x, y) =[qx(x, y)qy(x, y)

]the N matrix of the four shape functions is organized as

N =[N1 0 N2 0 N3 0 N4 00 N1 0 N2 0 N3 0 N4

].

Following the iso-parametric principle also introduced earlier, the location ofa point inside the element is approximated again with the same four shapefunctions as the displacement eld:

x = N1x1 +N2x2 +N3x3 +N4x4,

andy = N1y1 +N2y2 +N3y3 +N4y4.

Here xi, yi is the location of the i-th node of the element in the x, y directions.Using the shape functions dened above with the element coordinates and bysubstituting we get

x =14[(1u)(1 v)x1+(1+u)(1 v)x2 +(1+u)(1+ v)x3+(1u)(1+ v)x4]

=14[(x1 + x2 + x3 + x4) + u(x1 + x2 + x3 x4)+

v(x1 x3 + x3 + x4) + uvu(x1 x2 + x3 x4)].Similarly

y =14[(y1 + y2 + y3 + y4) + u(y1 + y2 + y3 y4)+

v(y1 y3 + y3 + y4) + uvu(y1 y2 + y3 y4)].To calculate the element energy and the element matrix, the strain com-ponents and the B strain-displacement matrix need to be computed. Theelement has three constant strains dened from the possible six used in three-dimensional continuum. They are

=

qxxqyy

qxy +

qyx

.

We still make an eort here to stay on the mathematical side of the discus-sion; this will be expanded when modeling a physical phenomenon. Clearly

24 Chapter 1

the rst two components are the rates of changes in distances between pointsof the element in the appropriate directions. The third component is a com-bined rate of change with respect to the other variable in the plane, deningan angular deformation.

The relationship to the nodal displacements is described in matrix form as

= Bqe.

Since the shape functions are given in terms of the parametric coordinates weneed again the Jacobian as

J =(x, y)(u, v)

=[

xu

xv

yu

yv

]=

14

[j11 j12j21 j22

].

The terms are

j11 = (1 v)x1 + (1 v)x2 + (1 + v)x3 (1 + v)x4,j12 = (1 u)x1 (1 + u)x2 + (1 + u)x3 + (1 u)x4,j21 = (1 v)y1 + (1 v)y2 + (1 + v)y3 (1 + v)y4,

andj22 = (1 u)y1 (1 + u)y2 + (1 + u)y3 + (1 u)y4.

Since

[quqv

]= J

[qxqy

],

the strain components required for the element are[qxxqxy

]= J1

[ qxuqxv

],

and [qyxqyy

]= J1

[qyuqyv

].

Taking advantage of the components of J and using the adjoint-based inversewe compute

=

qxxqyy

qxy +

qyx

= 1

det(J)

j22 j12 0 00 0 j21 j11j21 j11 j22 j12

qxuqxvqyuqyv

.

Finite Element Analysis 25

From the displacement eld approximation equations we obtain

qxuqxvqyuqyv

=

14

(1 v) 0 (1 v) 0 (1 + v) 0 (1 + v) 0(1 u) 0 (1 + u) 0 (1 + u) 0 (1 u) 0

0 (1 v) 0 (1 v) 0 (1 + v) 0 (1 + v)0 (1 u) 0 (1 + u) 0 (1 + u) 0 (1 u)

qe.

The last two equations produce the B matrix of size 3 12 that is now notconstant, it is linear in u and v. Recall that the energy of the element is

Ee =12

T dxdy.

With substitution of = Bqe we obtain

Ee =12qTe

BTBdxdyqe =

12qTe keqe.

The element matrix is

ke =

BTBdet[(x, y)(u, v)

]dudv.

By the fortuitous choice of the parametric coordinate system this integral nowis directly amenable to Gaussian quadrature as the limits are 1,+1. Intro-ducing

f(u, v) = BTBdet(J)

the element integral becomes

ke = 1u=1

1v=1

f(u, v)dudv = ni=1cinj=1cjf(ui, vj).

Here ui, vj are not the nodal point displacements, but the Gauss point loca-tions (shown as ti is Table 1.3) in those directions. With applying the twopoint (n = 2) formula

ke = c21f(u1, v1) + c1c2f(u1, v2) + c2c1f(u2, v1) + c22f(u2, v2)

and ci are listed in Table 1.3 also.

This concludes the computation techniques of the linear 2-dimensional quadri-lateral element. In practice the quadratic version is much preferred and willbe described in the following.

26 Chapter 1

1.8 Quadratic nite elements

We view the element in the x y plane as shown in Figure 1.5, but add nodeson the middle of the sides of the square shown in Figure 1.6 depicting theparametric plane of the element. The locations of these new node points ofthe quadratic element are:

(x5, y5) (0,1),(x6, y6) (1, 0),(x7, y7) (0, 1),

and(x8, y8) (1, 0).

Connecting these points are four interior lines, described by parametric equa-tions as

1 u+ v = 0,connecting nodes 5 and 6,

1 u v = 0,connecting nodes 6 and 7,

1 + u v = 0,connecting nodes 7 and 8, and nally

1 + u+ v = 0,

connecting nodes 8 and 1, completing the loop. For j = 1, . . . , 8 we seekfunctions Ni that are unit at the ithe node and vanish at the others:

Ni ={1 when i = j,0 when i = j.

Let us consider for example node 3. N3 must vanish along the opposite sidesof the rectangle

u = 1,and

v = 1.

Finite Element Analysis 27

That will account for nodes 1, 2, 4, 5, 8. Furthermore it must also vanish atnodes 6 and 7, represented by the line

1 u v = 0.Hence the form of the corresponding shape function is

N3 = nc(1 + u)(1 + v)(1 u v).where the normalization coecient nc for the corner shape functions may beestablished from the condition of N3 becoming unit at node 3

N3 = nc(1 + 1)(1 + 1)(1 1 1) = nc(4) = 1,yielding

nc = 14 .This is in part identical to the N3 shape function of the linear element, apartfrom the last term. The shape functions corresponding to the corner nodes,based on similar considerations, are of form

N1 = 14(1 u)(1 v)(1 + u+ v),

N2 = 14(1 + u)(1 v)(1 u+ v).

N3 = 14(1 + u)(1 + v)(1 u v),and

N4 = 14(1 u)(1 + v)(1 + u v).To dene the shape functions at the mid-points, we consider node 6 rst. N6must vanish at 3 edges

v = 1,

v = 1,and

u = 1.Hence it will be of form

N6 = nm(1 + u)(1 + v)(1 v).Substituting the last two terms with the well known algebraic identity, weobtain

N6 = nm(1 + u)(1 v2).

28 Chapter 1

This form now demonstrates the quadratic nature of the element. The nor-malization constant of the mid-side nodes nm is established by using thecoordinates (1, 0) of node 6

N6 = nm(1 + 1)(1 02) = nm 2 = 1implying

nm =12.

Hence, the mid-side shape functions are:

N5 =12(1 u2)(1 v),

N6 =12(1 + u)(1 v2).

N7 =12(1 u2)(1 + v).

N8 =12(1 u)(1 v2).

From here on, the process established in connection with the linear elementis directly applicable. The nodal displacement vector will, of course, con-sist of 16 components and the N matrix of shape functions will also double incolumn size. The steps of the element matrix generation process are identical.

A similar ow of operations, in connection with the triangular element, re-sults in a quadratic triangular element, the six-noded triangle. Let us considerthe element depicted in Figure 1.4 and place mid-side nodes as follows:

(x4, y4) (1/2, 1/2),(x5, y5) (0, 1/2),

and(x6, y6) (1/2, 0).

Following above, the shape functions of the corner nodes will be

N1 = (1 u v)2,N2 = u(2u 1),

andN3 = v(2v 1).

The mid-side nodes are represented by

N4 = 4uv,

Finite Element Analysis 29

N5 = 4(1 u v)v,and

N6 = 4u(1 u v).They are of unit value in their respective locations and zero otherwise.

For higher order (so-called p-version) or physically more elaborate (non-planar) element formulations the reader is referred to [9] and [5], respectively.

The computational process of adding shape functions to mid-side nodes willeasily generalize to three dimensions. The linear three dimensional elements,such as the linear tetrahedral element introduced in Section 3.4 and the lin-ear hexahedral element, subject of Section 6.2 may be extended to quadraticelements by the same procedure.

The foundation established in this chapter should carry us into the mod-eling of a physical phenomenon, where one more generalization of the niteelement technology will be done by addressing three-dimensional domains. Be-fore this issue is explored, however, the generation of a nite element modelis discussed.

References

[1] Clough, R. W.; The nite element method in plane stress analysis, Pro-ceedings of 2nd Conference of electronic computations, ASCE, 1960

[2] Courant, P.; Variational methods for the solution of problems of equi-librium and vibration, Bulletin of American Mathematical Society, Vol.49, pp. 1-23, 1943

[3] Galerkin, B. G.; Stabe und Platten: Reihen in gewissen Gleichgewicht-sproblemen elastischer Stabe und Platten, Vestnik der Ingenieure, Vol.19, pp. 897-908, 1915

[4] MacNeal, R. H.; NASTRAN theoretical manual, The MacNeal-Schwendler Corporation, 1972

[5] MacNeal, R. H.; Finite elements: Their design and performance, MarcelDekker, New York, 1994

[6] Oden, J. T. and Reddy, J. N.; An introduction to the mathematicaltheory of nite elements, Wiley, New York, 1976

30 Chapter 1

[7] Przemieniecki, J. S.; Theory of matrix structural analysis, McGraw-Hill,New York, 1968

[8] Ritz, W.; Uber eine neue Methode zur Losung gewisser Variationsprob-leme der Mathematischen Physik, J. Reine Angewendte Mathematik,Vol. 135, pp. 1-61, 1908

[9] Szabo, B. and Babuska, I.; Finite element analysis, Wiley, New York,1991

[10] Turner, M. J. et al; Stiness and deection analysis of complex struc-tures, Journal of Aeronautical Science, Vol. 23, pp. 803-823, 1956

[11] Zienkiewicz, O. C.; The nite element method, McGraw-Hill, New York,1968

2Finite Element Model Generation

Finite element model generation involves two distinct components. First, thereal life geometry of the physical phenomenon is approximated by geometricmodeling. Second, the computational geometry model is discretized produc-ing the nite element model. These issues are addressed in this chapter.

2.1 Bezier spline approximation

The rst step in modeling the geometry of a solid object involves approxi-mating its surfaces and edges with splines. Note, that this step also embodiesa certain reduction as the real life continuum geometry is approximated bya nite number of computational geometry entities. The most popular andpractical geometric modeling tools are based on parametric splines.

Let us rst consider an edge of a physical model described by a curve whoseequation is

r(t) = x(t)i + y(t)j + z(t)k.

The original curve will be approximated by a set of cubic parametric splinesegments of form

S(t) = a + bt+ ct2 + dt3,

where t ranges from 0.0 to 1.0. Let us assume a set of points Pj , j = 1...m,representing the geometric object we are to model. For simplicity let us focuson the rst segment of the curve dened by four points P0, P1, P2, P3. Thesefour points dene a Bezier [1] polygon as shown in Figure 2.1. The curve willgo through the end-points P0 and P3. The tangents of the curve at the endpoints will be dened by the two intermediate (control) points P1, P2.

The Bezier spline segment is formed from these four points as

S(t) = 3i=0PiJ3,i(t).

31

32 Chapter 2

P2

P3

P1

Po

FIGURE 2.1 Bezier polygon

Here

J3,i(t) =(3i

)ti(1 t)3i

are binomial polynomials. Using the boundary conditions of the Bezier curve(S(0), S(1), S(0), S(1)) the matrix form of the Bezier spline segment may bewritten as

S(t) = TMP.

Here the matrix P contains the Bezier vertices

P =

P0P1P2P3

,

and the matrix M the interpolation coecients

Finite Element Model Generation 33

M =

1 0 0 03 3 0 03 6 3 01 3 3 1

.

T is a parametric row vector:

T =[1 t t2 t3

].

A very important generalization of this form is to introduce weight functions.The result is the rational parametric Bezier spline segment of form

S(t) =3i=0wiPiJ3,i(t)3i=0wiJ3,i(t)

,

or in matrix notation

S(t) =TMP

TMW.

Here

P =

w0P0w1P1w2P2w3P3

is the vector of weighted point coordinates and

W =

w0w1w2w3

is the array of weights. The weights have the eect of moving the curve closerto the control points, P1, P2, as shown in Figure 2.2.

The location of a specied point on the curve, Ps in Figure 2.2, denes threeweights, while the remaining weight is covered by specifying the parametervalue t to which the specied point should belong on the spline. Most com-monly t = 12 is chosen for such a point. The weights enable us to increasethe delity of the approximation of the original curves.

The curve segment is nally approximated by

r(t) =TMX

TMWi+

TMY

TMWj +

TMZ

TMWk.

Here

X =

w0x0w1x1w2x2w3x3

, Y =

w0y0w1y1w2y2w3y3

, Z =

w0z0w1z1w2z2w3z3

,

34 Chapter 2

PS

P1 P2

P3

Po

FIGURE 2.2 The eect of weights on the shape of spline

where xi, yi, zi are the coordinates of the i-th Bezier point. An additional ad-vantage of using rational Bezier splines is to be able to exactly represent conicsections and quadratic surfaces. These are common components of industrialmodels, for manufacturing as well as esthetic reasons.

In practice the geometric boundary is likely to be described by many pointsand therefore, a collection of spline segments. Consider the collection of pointsdescribing multiple spline segments shown in Figure 2.3. The most importantquestion arising in this regard is the continuity between segments. Since theBezier splines are always tangential to the rst and last segments of the Bezierpolygon, clearly a rst order continuity exists only if the Pi1, Pi, Pi+1 pointsare collinear.

The presence of weights further species the continuity. Computing

S

t(t = 0) = 3

w1w0

(P1 P0)

andS

t(t = 1) = 3

w2w3

(P3 P2).

Finite Element Model Generation 35

Po

Pi

Pi+3

P3n+1

FIGURE 2.3 Multiple Bezier segments

Here (Pi Pj) is a vector pointing to Pj from Pi. Focusing on the adjoiningsegments of splines in Figure 2.4 the rst order continuity condition is

wi1wi0

(Pi Pi1) = wi+1wi+0

(Pi+1 Pi).

There is a rather subtle but important distinction here. There is a geometriccontinuity component that means that the tangents of the neighboring splinesegments are collinear. Then there is an algebraic component resulting in thefact that the magnitude of the tangent vectors is also the same. The notationwi+0, wi0 manifests the fact that the weights assigned to a control point inthe neighboring segments do not have to be the same. If they are, a simpliedrst order continuity condition exists when

wi1wi+1

=(Pi+1 Pi)(Pi Pi1) .

Enforcing such a continuity is important in the delity of the geometry ap-proximation and in the discretization to be discussed later.

For the same reasons a second order continuity is also desirable. By deni-tion

36 Chapter 2

Pi2

Pi3

Pi1

Pi +1

Pi+2

Pi+3Pi

FIGURE 2.4 Continuity of spline segments

2S

t2(t = 0) = (6

w1w0

+ 6w2w0

18w21

w20)(P1 P0) + 6w2

w0(P2 P1)

and

2S

t2(t = 1) = (6

w1w3

+ 6w2w3

18w22

w23)(P2 P3) + 6w1

w3(P1 P2).

Generalization to the boundary of neighboring segments, assuming that theweights assigned to the common point between the segments is the same,yields the second order continuity condition as

wi2(Pi2 Pi) 3w2i1wi

(Pi1 Pi) = wi+2(Pi+2 Pi) 3w2i+1wi

(Pi+1 Pi).

This is a rather strict condition requiring that the two control points prior andafter the common point (ve points in all) are coplanar with some additionalweight relations.

Finite Element Model Generation 37

v

v = v2

v = v1

u = u2u = u1

w12

w11

w22w02

w01

w00 w10 w20

w21

u

FIGURE 2.5 Bezier patch denition

2.2 Bezier surfaces

The method discussed above is easily generalized to surfaces. A Bezier sur-face patch is dened by a set of points on the surface of the physical model(plus the control points and weights) as shown in Figure 2.5. The rationalparametric Bezier patch is described as

S(u, v) =3i=0

3j=0wijJ3,i(u)J3,j(v)Pij

3i=03j=0wijJ3,i(u)J3,j(v)

or in matrix form

S(u, v) =UMPMTV

UMWMTV.

38 Chapter 2

The computational components are the matrix of weighted point coordinates

P =

w00P00 w01P01 w02P02 w03P03w10P10 w11P11 w12P12 w13P13w20P20 w21P21 w22P22 w23P23w30P30 w31P31 w32P32 w33P33

,

the parametric row vector of

U =[1 u u2 u3

],

and column vector of

V =

1vv2

v3

.

The weights form a matrix of:

W =

w00 w01 w02 w03w10 w11 w12 w13w20 w21 w22 w23w30 w31 w32 w33

.

The geometric surface of the physical model now is approximated by the patchof

r(u, v) =UMXMTV

TMWMTVi+

TMYMTV

TMWMTVj +

TMZMTV

TMWMTVk.

Here X,Y , Z, contain the weighted x, y, z point coordinates, respectively.Again, in a complex physical domain a multitude of these patches is usedto completely cover the surface. The earlier continuity discussion generalizesfor surface patches. The derivatives

S(u, v)u

,

andS(u, v)

v

will be the cornerstones of such relations. Similar arithmetic expressions usedfor the spline segments produce the rst order continuity condition across thepatch boundaries as shown in Figure 2.6.

wi+1,j+1wi1,j+1

=(Pi1,j+1 Pi,j+1)(Pi+1,j+1 Pi,j+1) .

A similar treatment is applied to the v parametric direction. The second order

Finite Element Model Generation 39

i1, j+1

i1, j

i+1, j+1

i+1, j

i, j+1

S1

i, j

u

v

u

FIGURE 2.6 Patch continuity denition

continuity is based on

2P (u, v)u v

computed at the corners and the mathematics is rather tedious, albeit straight-forward. The strictness of this condition is now almost overbearing, requiringnine control points to be the coplanar. Therefore, it is seldom enforced ingeometric modeling for nite element applications. It mainly contributes tothe esthetic appearance of the surface created and as such it is preferred byshape designers.

The technique also generalizes to volumes of the physical model as

S(u, v, t) =3i=0

3j=0

3k=0wijkJ3,i(t)J3,j(u)J3,k(v)Pijk

3i=03j=0

3k=0wijkJ3,i(t)J3,j(u)J3,k(v)

.

The matrix form and the nal approximation form may be developed alongthe same lines as above for splines or patches. The result is

S(u, v, t) =3k=0J3,i(t)UMP kM

TV

3k=0J3,i(t)UMWkMTV.

40 Chapter 2

The k layers of the volume are individual spline patches and the weights aredened as earlier. The formulation enables the modeling of volumes of revo-lutions or extrusions, details of those are beyond our needs here.

The points corresponding to equi-parametric values of the splines, surfacepatches and volumes are of course not equally separated in space. Sometimesit is necessary to re-parameterize one of these objects to smoothen the para-metric distribution in a geometric sense. Nevertheless, the equi-parametriclocations of these objects may constitute a basis for the discretization dis-cussed in the next section.

The Bezier objects industrial popularity is due to the following reasons:

1. The convex hull property: All Bezier curves, surface patches or volumesare contained inside of the hull of their control points,2. The variation diminishing property: The number of intersection pointsbetween a Bezier curve and an innite plane is the same as the number ofintersections between the plane and the control polygon,3. All derivatives and products of Bezier functions are easily computed Bezierfunctions.

These properties are exploited in industrial geometric modeling computations.

2.3 B-spline technology

An alternative to the Bezier spline technology is based on the B-splines. Thetechnology allows a set of input points to be either interpolated or approxi-mated, providing much more exibility. The curves are still directed by controlpoints, however, they are not given a priori, they are computed as part of theprocess. The technology, therefore, is more exible than the Bezier technologyand is preferred in the industry.

A general non-uniform, non-rational B-spline is described by

S(t) =n

i=0

Bi,k(t)Qi,

where Qi are the yet unknown control points and Bi,k are the B-spline basisfunctions of degree k. They are computed based on a certain parameteriza-tion inuencing the shape of the curve. Note that for now we are focusing on

Finite Element Model Generation 41

non-rational, non-uniform B-splines.

The basis functions are initiated by

Bi,0(t) ={

1, ti t < ti+10, t < ti, t ti+1

and higher order terms are recursively computed:

Bi,k(t) =t ti

ti+k tiBi,k1(t) +ti+k+1 t

ti+k+1 ti+1Bi+1,k1(t).

The parameter values for the spline may be assigned via various methods. Thesimplest, and most widely used method is the uniform spacing. The methodfor n + 1 points is dened by the parameter vector

t =[0 1 2 . . . n

].

When the input points are geometrically somewhat equidistant this is provento be a good method for parameterization. When the input points are spacedin widely varying intervals, a parameterization based on the chord length mayalso be used.

The parameter vector is commonly normalized as

t =[0 1/n 2/n . . . 1

].

Such normalization places all the parameter values in the interval (0, 1) easingthe complexity of the evaluation of the basis functions.

First we seek to interpolate a given set of points

Pj = (Pxj , Pyj, P zj); j = 0, . . . ,m,

requiring that the B-spline (S(t) at parameter value tj passes through thegiven point Pj . This results in the equation

P0P1. . .Pm

=

B0,k(t0) B1,k(t0) B2,k(t0) . . . Bn,k(t0)B0,k(t1) B1,k(t1) B2,k(t1) . . . Bn,k(t1)

. . . . . . . . . . . . . . .B0,k(tm) B1,k(tm) B2,k(tm) . . . Bn,k(tm)

Q0Q1. . .Qn

.

Using a matrix notation, the problem is

P = BQ,

where the P column matrix contains m + 1 terms and the Q column matrixcontains n+1 terms, resulting in a rectangular system matrix B with (m+1)rows and (n + 1) columns. This problem may not be solved in general when

42 Chapter 2

m < n, the case when the number of points given is less than the number ofcontrol points. The problem may also be only solved in a least squares sensewhen m > n, having more input points than control points.

The problem has a unique solution for the case of m = n and in this casethe sequence of unknown control points is obtained in the form of

Q = B1P,

where the inverse is shown for the sake of simplicity, it is not necessarily com-puted. In fact, the B matrix exhibits a banded pattern that is dependent onthe degree k of the spline chosen. Specically, the semi-bandwidth is less thanthe order k.

Bi,k(tj) = 0; for|i j| >= k.This fact should be exploited to produce an ecient solution.

The second approach is to approximate the input points in a least squaressense, resulting in a distinctly dierent curve. This may be obtained by nd-ing a minimum of the squares of the distances between the spline and thepoints.

mj=0

(S(tj) Pj)2.

Substituting the B-spline formulation and the basis functions results in

mj=0

(n

i=0

Bi,k(tj)Qi Pj)2.

The derivative with respect to an unknown control point Qp is

2m

j=0

Bp,k(tj)(n

i=0

Bi,k(tj)Qi Pj) = 0,

where p = 0, 1, . . . , n. This results in a system of equations, with n + 1 rowsand columns, in the form:

BTBQ = BTP

with the earlier introduced B matrix. The solution of this system producesan approximated, not interpolated solution.

The technology may also be extended to include smoothing considerationsand directional constraints to the splines, topics that are discussed at lengthin [3].

Finite Element Model Generation 43

2.4 Computational example

Considering that the problem is given in 3-space, the solution for the x, y, zcoordinates may be obtained simultaneously.

Qx0 Qy0 Qz0Qx1 Qy1 Qz1. . .Qxn Qyn Qzn

= B1

Px0 Py0 Pz0Px1 Py1 Pz1. . .Pxn Pyn Pzn

.

For a xed degree, say k = 3, and uniformly parameterized B-spline segmentsthe basis functions may be analytically computed as:

B0,3 =16(1 t)3,

B1,3 =16(3t3 6t2 + 4),

B2,3 =16(3t3 + 3t2 + 3t+ 1),

and

B3,3 =16t3.

For the case of 4 points (n = 3) the uniform parameter vector becomes:

t =[0 1 2 3

].

For this case the interpolation system matrix is easily computed by hand as

B =16

1 4 1 00 1 4 11 4 5 88 31 44 27

.

The matrix is positive denite and its inverse is:

B1 =16

21 28 17 44 5 4 11 8 1 00 1 8 1

.

The solution for the control points is obtained as

Q = B1P,

44 Chapter 2

where P is the vector of input points. For example for the points

P =

0 01 12 13 0

,

the control points obtained are

Q =

1 11/60 1/61 7/62 7/6

.

Figure 2.7 shows the curve generated from the control points interpolatingthe given input points, while spanning the parameter range from 0 to 3.

FIGURE 2.7 B spline interpolation

To evaluate the spline curve as function of any parameter value in the span,the following matrix formula (conceptually similar to the Bezier form) may

Finite Element Model Generation 45

be used:

S(t) = TCQ,

with

C =16

1 4 1 03 0 3 03 6 3 01 3 3 1

where the C matrix is gathered from the coecients of the analytic basis func-tions above, and

T =[1 t t2 t3

].

This formula enables the validation of the spline going through the inputpoints. For example

St=1 = TCQ =[1 1],

which of course agrees with the second input point.

For demonstration of the approximation computation, we add anotherinput point to the above set. The given set of 5 input points are:

P =

0 01 12 13 02 1

.

For the case of 5 points (n = 4) the parameter vector becomes:

t =[0 1 2 3 4

].

For this case the B matrix is

B =16

1 4 1 00 1 4 11 4 5 88 31 44 2727 100 131 64

.

The solution for the control points in this case is obtained as

Q = (BTB)1BTP.

Figure 2.8 shows the curve generated from these control points approximat-ing the given input points, while spanning the parameter range from 0 to 4.The evaluation yields the approximation points

46 Chapter 2

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-0.5 0 0.5 1 1.5 2 2.5 3

input.datcontrol.dat

x(t), y(t)

FIGURE 2.8 B spline approximation

Sapp =

0.028571 0.0142860.885714 1.0571432.171429 0.9142862.885714 0.0571432.028571 1.014286

,

which reasonably well approximate the input points, while producing a smoothcurve.

The selection of the parameter values enables interesting and useful shapevariations of the spline around the same set of given points. For example, re-peated values of the parameter values at both ends enforce a clamped bound-ary condition, forcing the curve through the end points. Figure 2.9 shows thecurve generated from these control points and for the same input points, stillapproximating them.

Note that the number of pre-assigned parameter values, in this case be-comes 9 and the parameter vector will be

t =[0 0 0 1/4 1/2 3/4 1 1 1

].

Finite Element Model Generation 47

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2 2.5 3

inputx1(t), y1(t)x2(t), y2(t)x5(t), y5(t)x6(t), y6(t)x3(t), y3(t)x4(t), y4(t)

FIGURE 2.9 Clamped B spline approximation

Also note that the number of sections of the spline increased to 6 in this case.The gure depicts the sections of the spline with dierent line patterns asshown on the legend, while spanning the parameter range from 0 to 1.

Finally, the curve adhering to the same set of input points may also beclosed by repeating an input point. The starting point repeated at the endresults in the closed curve shown in Figure 2.10, in this case with 5 sections.

This example also demonstrated that a level continuity between the seg-ments of the B-spline is automatically assured depending on the degree k ofthe spline. There is no need for special considerations when large number ofinput points are given.

48 Chapter 2

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2 2.5 3

inputx1(t), y1(t)x2(t), y2(t)x3(t), y3(t)x4(t), y4(t)x5(t), y5(t)

FIGURE 2.10 Closed B spline approximation

2.5 NURBS objects

As in the Bezier technology, it is also possible to use weights in the B-splinetechnology, resulting in rational B-splines. When a non-uniform parameteriza-tion is also used, the splines become Non-Uniform, Rational B-splines, knownas NURBS.

Introducing weights associated with each control point results in the NURBScurve of form

S(t) =n

i=0 wiBi,k(t)Qini=0 wiBi,k(t)

.

The geometric meaning of the weights is similar to that of the Bezier tech-nology, they will pull the curve closer to the input points. It is, however,important to point out that changing one single weight value will result onlyin a local shape change in the segment related to the point. This local controlis one of the advantages of the B-spline technology over the Bezier approach.

Finite Element Model Generation 49

The formulation extends quite easily to surfaces:

S(u, v) =

ni=0

mj=0 wi,jBi,k(u)Bj,l(v)Qi,jn

i=0

mj=0 wi,jBi,k(u)Bj,l(v)

.

Note that the degree of the v directional parametric curve may be dierentthan that of the u curve, denoted by l. Similarly the parameterization in bothdirections may be dierent. This gives tremendous exibility to the method.

Geometric modeling operations are enabled by these objects. Consider gen-erating a swept surface by moving a curve C(u) along a trajectory T (v). Thisis conceptually similar to generating a cylinder by dening a circle and theaxis perpendicular to the plane of the circle. In general, the surface generatedby this process may be described as

S(u, v) = C(u) + T (v).

Assume that the curves are NURBS of the same order

C(u) =n

i=0 wCi Bi,k(u)Q

Cin

i=0 wCi Bi,k(u)

and

T (v) =

mj=0 w

Tj Bj,k(v)Q

Tjm

j=0 wTj Bj,k(v)

.

Then the swept NURBS surface is of form

S(u, v) =

ni=0

mj=0 wi,jBi,k(u)Bj,l(v)Qi,jn

i=0

mj=0 wi,jBi,k(u)Bj,k(v)

,

whereQi,j = QCi +Q

Tj

andwi,j = wCi w

Tj .

Similar considerations may be used to generate NURBS surfaces of revolutionaround a given axis.

Finally, the NURBS also generalize to three dimensions for modeling vol-umes:

S(u, v, t) =

ni=0

mj=0

qp=0 wi,j,pBi,k(u)Bj,k(v)Bp,k(t)Qi,j,pn

i=0

mj=0

qp=0 wi,j,pBi,k(u)Bj,k(v)Bp,k(t)

.

The form is written with the assumption of the curve degree being the same(k) in all three parametric directions, albeit that is not necessary.

50 Chapter 2

Finally, it is important to point out that the surface representations via ei-ther Bezier or B-splines may produce non-rectangular surface patches. Such,for example triangular, patches are very important in the nite element dis-cretization step to be discussed next. They may easily be produced fromabove formulations by collapsing a pair of points into one and will not bediscussed further [3].

2.6 Geometric model discretization

The foundation of many general methods of discretization (commonly calledmeshing) is the classical Delaunay triangulation method [4]. The Delau-

FIGURE 2.11 Voronoi polygon

nay triangulation technique in turn is based on Voronoi polygons [8] . TheVoronoi polygon, assigned to a certain point of a set of points in the plane,

Finite Element Model Generation 51

contains all the points that are closer to the selected point than to any otherpoint of the set.

Let us dene the set of points S R2 and Pi S be the points of the seti = 1, 2, ..n. The points Q(x, y) R2 that satisfy

Q(x, y) Pi Q(x, y) Pj, Pj S,constitute the Voronoi polygon V (Pi) of point Pi. The Voronoi polygon is aconvex polygon.

The inequalities represent half planes between point Pi and every point Pj .The intersection of these half planes produces the Voronoi polygon. For ex-ample consider the set of points shown in Figure 2.11. The irregular hexagoncontaining one point in the middle (the Pi point) is the Voronoi polygon ofpoint Pi.

It is easy to see that the points inside the polygon (Q(x, y)) are closer toPi than to any other points of the set. It is also quite intuitive that the edgesof the Voronoi polygon are the perpendicular bisectors of the line segmentsconnecting the points of the set.

The union of the Voronoi polygons of all the points in the set completelycovers the plane. It follows that the Voronoi polygon of two points of theset do not have common interior points; at most they share points on theircommon boundary.

The denition and process generalizes to three dimensions very easily. Ifthe set of points are in space, S R3, the points Q(x, y, z) R3 that satisfy

Q(x, y, z) Pi Q(x, y, z) Pj, Pj S,dene the Voronoi polyhedron V (Pi) of Pi.

Every inequality denes a half-space and the Voronoi polyhedron V (Pi) isthe intersection of all the half-spaces dened by the point set. The Voronoipolyhedron is a convex polyhedron.

2.7 Delaunay mesh generation

The Delaunay triangulation process is based on the Voronoi polygons as fol-lows. Let us construct Delaunay edges by connecting points Pi and Pj when

52 Chapter 2

their Voronoi polygons V (Pi) and V (Pj) have a common edge. Constructingall such possible edges will result in the covering of the planar region of ourinterest with triangular regions, the Delaunay triangles.

FIGURE 2.12 Delaunay triangle

Figure 2.12 shows a Delaunay triangle. The dotted lines are the edges of theVoronoi polygons and the solid lines depict the Delaunay edges. The processextends quite naturally and covers the plane as shown in Figure 2.13 with6 Delaunay triangles. It is known that under the given denitions no twoDelaunay edges cross each other.

On the other hand it is possible to have a special case when four (or evenmore) Voronoi polygons meet at a common point. This degenerate case willresult in the Delaunay edges producing a quadrilateral. As the discretizedregions are the nite elements for our further computations, this case is nocause for panic. We can certainly have quadrilateral nite elements as wasshown earlier. There are also remedies to preserve a purely triangular mesh;slightly moving one of the points participating in the scenario will eliminate

Finite Element Model Generation 53

FIGURE 2.13 Delaunay triangularization

the special case.

Finally, in three dimensions the Delaunay edges are dened as lines con-necting points that share a common Voronoi facet (a face of a Voronoi poly-hedron). Furthermore, the Delaunay facets are dened by points that sharea common Voronoi edge (an edge of a Voronoi polyhedron). In general eachedge is shared by exactly three Voronoi polyhedron, hence the Delaunay re-gions facets are going to be triangles.

The Delaunay regions connect points of Voronoi polyhedra that share acommon vertex. Since in general the number of such polyhedra is four, thegenerated Delaunay regions will be tetrahedra. The Delaunay method gener-alized into three dimensions is called Delaunay tessellation [6].

There are many automatic methods to discretize a two-dimensional, notnecessarily planar, domain. [2] describes such a method for surface meshingwith rectangular elements. There are also other methods in the industry topartition a three-dimensional domain into a collection of non-overlapping ele-ments that covers the entire solution domain, see for example [7]. The mostsuccessful techniques are the proprietary heuristic algorithms used in commer-

54 Chapter 2

cial software. The quality of the mesh heavily inuences the nite elementsolution results. A good quality mesh has elements close to equal in size withshapes that are not too distorted. In the case of hexahedron elements thismeans element shapes that approach cubes. Gross inequality in the ratios ofthe sides (called aspect ratio in the industry) results in less accurate solutions.

The nal topic of the nite element model generation is the assignment ofnode numbers. This step will inuence the topology of the assembled niteelement matrices, and as such, it inuences the computational performance.The nite element matrix reordering is discussed in Section 7.1.

The assignment of the node numbers usually starts at a corner or an edgeof the geometric model, now meshed, and proceeds inward towards the inte-rior of the model while at the same time considering the element connectivity.The goal of this is that nodes of an element should have neighboring num-bers. It is not necessary to achieve that, but is it useful as a pre-processingfor reordering and assuring that operations eciency.

2.8 Model generation case study

To demonstrate the model generation process we consider a simple engineeringcomponent of a bracket. This example will be used in the last section alsoas a case study for a complete engineering analysis. The process in todaysengineering practice is almost exclusively executed in a computer aided design(CAD) software environment. The advantage of working in such environmentis that the engineer is able to immediately analyze the model, since the modelis created in a computer.

Still, the engineer starts by creating a design sketch, such as shown in Fig-ure 2.14. The role of the design sketch is to specify the contours of the desiredshape that will accommodate the kinematic relationships between the com-ponent and the rest of the product. Since this is an interactive process, theengineer could easily modify the sketch until it satises the goals.

The next step in the design process is to ll out the details of the geom-etry. The model may be extruded from two dimensional contour elements ina certain direction, or blended between contour curves. The interior volumesmay be lled with standard geometrical components like cylinders or cones.The process usually entails generating the faces and interior volumes of themodel from many components. Figure 2.15 depicts the geometric model of

Finite Element Model Generation 55

FIGURE 2.14 Design sketch of a bracket

the bracket example.

The geometric modeling software environment facilitates the easy executionof coordinate transformations, such as rotations, translations of reections,enabling the engineer to try various scenarios. Earlier designs may be reusedand modications easily made to produce a variant product. Since shape isreally independent of frame of reference, this approach encapsulates the shapein a parametric form, not in a xed reference frame of the blue-prints of thepast. Another ad