What every engineer should know about computational techniques of finite element analysis 2nd ed (2009)

Finite Element Analysis The goal of this chapter is to introduce the reader to finite element sis which is the basis for the discussion of the computational methods in theremainder of the

WHAT EVERY ENGINEER SHOULD KNOW ABOUT

COMPUTATIONAL TECHNIQUES OF FINITE ELEMENT

ANALYSIS

Second Edition

LOUIS KOMZSIK

WHAT EVERY ENGINEER SHOULD KNOW ABOUT COMPUTATIONAL TECHNIQUES OF FINITE ELEMENT

ANALYSIS

Second Edition

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To my son, Victor

Preface to the second edition xiii

I Numerical Model Generation 1

1.1 Solution of boundary value problems 3

1.2 Finite element shape functions 6

1.3 Finite element basis functions 9

1.4 Assembly of finite element matrices 12

1.5 Element matrix generation 15

1.6 Local to global coordinate transformation 19

1.7 A linear quadrilateral finite element 20

1.8 Quadratic finite elements 26

References 29

2 Finite Element Model Generation 31 2.1 Bezier spline approximation 31

2.2 Bezier surfaces 37

2.3 B-spline technology 40

2.4 Computational example 43

2.5 NURBS objects 48

2.6 Geometric model discretization 50

2.7 Delaunay mesh generation 51

2.8 Model generation case study 54

References 57

3 Modeling of Physical Phenomena 59 3.1 Lagrange’s equations of motion 59

3.2 Continuum mechanical systems 61

3.3 Finite element analysis of elastic continuum 63

3.4 A tetrahedral finite element 65

3.5 Equation of motion of mechanical system 69

3.6 Transformation to frequency domain 71

vii

References 74

4 Constraints and Boundary Conditions 75 4.1 The concept of multi-point constraints 76

4.2 The elimination of multi-point constraints 79

4.3 An axial bar element 82

4.4 The concept of single-point constraints 85

4.5 The elimination of single-point constraints 86

4.6 Rigid body motion support 88

4.7 Constraint augmentation approach 90

References 92

5 Singularity Detection of Finite Element Models 93 5.1 Local singularities 93

5.2 Global singularities 97

5.3 Massless degrees of freedom 99

5.4 Massless mechanisms 100

5.5 Industrial case studies 102

References 104

6 Coupling Physical Phenomena 105 6.1 Fluid-structure interaction 105

6.2 A hexahedral finite element 106

6.3 Fluid finite elements 109

6.4 Coupling structure with compressible fluid 111

6.5 Coupling structure with incompressible fluid 112

6.6 Structural acoustic case study 113

References 115

II Computational Reduction Techniques 117 7 Matrix Factorization and Linear Systems 119 7.1 Finite element matrix reordering 119

7.2 Sparse matrix factorization 122

7.3 Multi-frontal factorization 124

7.4 Linear system solution 126

7.5 Distributed factorization and solution 127

7.6 Factorization and solution case studies 130

7.7 Iterative solution of linear systems 134

7.8 Preconditioned iterative solution technique 137

References 139

8 Static Condensation 141

8.1 Single-level, single-component condensation 141

8.2 Computational example 144

8.3 Single-level, multiple-component condensation 147

8.4 Multiple-level static condensation 152

8.5 Static condensation case study 155

References 158

9 Real Spectral Computations 159 9.1 Spectral transformation 159

9.2 Lanczos reduction 161

9.3 Generalized eigenvalue problem 164

9.4 Eigensolution computation 166

9.5 Distributed eigenvalue computation 168

9.6 Dense eigenvalue analysis 172

9.7 Householder reduction technique 175

9.8 Normal modes analysis case studies 177

References 181

10 Complex Spectral Computations 183 10.1 Complex spectral transformation 183

10.2 Biorthogonal Lanczos reduction 184

10.3 Implicit operator multiplication 186

10.4 Recovery of physical solution 188

10.5 Solution evaluation 190

10.6 Reduction to Hessenberg form 191

10.7 Rotating component application 192

10.8 Complex modal analysis case studies 196

References 199

11 Dynamic Reduction 201 11.1 Single-level, single-component dynamic reduction 201

11.2 Accuracy of dynamic reduction 203

11.3 Computational example 206

11.4 Single-level, multiple-component dynamic reduction 208

11.5 Multiple-level dynamic reduction 210

11.6 Multi-body analysis application 212

References 215

12 Component Mode Synthesis 217 12.1 Single-level, single-component modal synthesis 217

12.2 Mixed boundary component mode reduction 219

12.3 Computational example 222

12.4 Single-level, multiple-component modal synthesis 225

12.5 Multiple-level modal synthesis 228

12.6 Component mode synthesis case study 230

References 232

III Engineering Solution Computations 235 13 Modal Solution Technique 237 13.1 Modal solution 237

13.2 Truncation error in modal solution 239

13.3 The method of residual flexibility 241

13.4 The method of mode acceleration 245

13.5 Coupled modal solution application 246

13.6 Modal contributions and energies 247

References 250

14 Transient Response Analysis 251 14.1 The central difference method 251

14.2 The Newmark method 252

14.3 Starting conditions and time step changes 254

14.4 Stability of time integration techniques 255

14.5 Transient response case study 258

14.6 State-space formulation 259

References 262

15 Frequency Domain Analysis 263 15.1 Direct and modal frequency response analysis 263

15.2 Reduced-order frequency response analysis 264

15.3 Accuracy of reduced-order solution 267

15.4 Frequency response case study 268

15.5 Enforced motion application 269

References 271

16 Nonlinear Analysis 273 16.1 Introduction to nonlinear analysis 273

16.2 Geometric nonlinearity 275

16.3 Newton-Raphson methods 278

16.4 Quasi-Newton iteration techniques 282

16.5 Convergence criteria 284

16.6 Computational example 285

16.7 Nonlinear dynamics 287

References 288

17 Sensitivity and Optimization 289 17.1 Design sensitivity 289

17.2 Design optimization 290

17.3 Planar bending of the bar 294

17.4 Computational example 297

17.5 Eigenfunction sensitivities 302

17.6 Variational analysis 304

References 308

18 Engineering Result Computations 309 18.1 Displacement recovery 309

18.2 Stress calculation 311

18.3 Nodal data interpolation 312

18.4 Level curve computation 314

18.5 Engineering analysis case study 316

References 319

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 five years in print, and for herenthusiastic support of the first edition This made the book available to awide range of students and practicing engineers fulfilling 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 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 first edition in his teaching pro-vided me with valuable feedback and confirmation of the approach of the book

instru-A half a decade passed since the original writing of the first edition andthis edition contains numerous noteworthy technical extensions In Part I thefinite element chapter now contains a brief introduction to quadratic finiteelement 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 significant application trend ofthe past years: the use of the technology to couple multiple physical phenom-ena This includes a more detailed description of the fluid-structure interac-tion application, a hexahedral finite element, as well as a structural-acousticscase study

In Part II, a new section (7.7) addressing iterative solutions of linear systemsand specifically the method of conjugate gradients, was also recommended byreaders of the first edition Also in Part II, a new Chapter 10 is dedicated tocomplex spectral computations, a topic briefly mentioned but not elaborated

on in the first 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

(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 computations

in some detail

The application focus has also significantly expanded during the years sincethe publication of the first edition and one of the goals of this edition was toreflect 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 final goal of this edition was to correct some of the typing mistakes andtechnical misstatements of the first edition, which were pointed out to me byreaders While they kindly stated that those were not limiting the usefulness

of 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 first edition

The method of finite 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,significant computational work - in several distinct phases - involved in thesolution of a physical problem with the finite element method The emphasis

of this book is on the computational techniques of this complete process fromthe physical problem to the computed solution

In the first 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 finite element model Applying bound-ary conditions and various constraints to the finite element model results in anumerically solvable form The first part of the book addresses these topics

In the second phase of operations the numerical model is reduced to a putationally more efficient form via various spectral representations Todayfinite 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 efficiently solvable computational model.Finally, the solution of the engineering problem is obtained with specificcomputational 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

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

techniques I have also personally implemented some of these techniques intovarious versions of NASTRAN1, the world’s leading finite element software.Finally, I have also encountered many students during my years of teach-ing whose understanding of these computations would have been significantlybetter with such a book.

Louis Komzsik2004

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

I appreciate Mr Mike Gockel’s (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 engineer’s perspective havesignificantly 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 influence 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 ration) for her expertise and patience in producing figures from my hand-drawings

Corpo-I also value the professional contribution of the publication staff at Taylorand Francis Group My sincere thanks to Nora Konopka, publisher, HelenaRedshaw, manager and editor Richard Tressider They all deserve significantcredit in the final outcome

Louis Komzsik2004

xvii

Part I

Numerical Model

Generation

1

Finite Element Analysis

The goal of this chapter is to introduce the reader to finite element sis which is the basis for the discussion of the computational methods in theremainder of the book This chapter first 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 finiteelement and further when dealing with the application of the technique formechanical systems in Chapter 3

analy-1.1 Solution of boundary value problems

The method of using finite elements for the solution of boundary value 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 Galerkin’s method [3]

prob-The difference between the Ritz method and that of Galerkin’s is in thefact that the first addresses the variational form of the boundary value prob-lem Galerkin’s method minimizes the residual of the differential equationintegrated over the domain with a weight function, hence it is also called themethod of weighted residuals

This difference lends more generality and computational convenience toGalerkin’s method Let us consider a linear differential 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

Galerkin’s method is based on the Ritz’s approximate solution idea andconstructs the approximate solution as

3

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 finite element mesh) and

Ni, i = 1, n,

is the set of the finite element shape functions to be derived shortly In thiscase, of course there is a residual of the differential equation

L(q)= 0

Galerkin proposed using the shape functions of the approximate solution also

as the weights, and requires that the integral of the so weighted residual ish

van- van-DL(q)Nj(x, y)dxdy = 0; j = 1, 2, , n

This yields a system for the solution of the coefficients as

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 Poisson’s 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 x−y 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 Galerkin’s method to this problem

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

Substituting the approximate solution yields

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

Kij= Kji=

 (∂Ni

Fj=

 (f (x, y)Nj)dxdythe Galerkin equations may be written as a matrix equation

Kn,1Kn,2 Kn,n

⎤

⎥

⎦ ,with solution vector of

qn

⎤

⎥

⎦ ,and right hand side vector of

intro-of displacements, the solution intro-of Poisson’s equation Other differential tions could lead to similar form as demonstrated in, for example [2]

equa-The concept, therefore, is generally contributing to its wide-spread 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]

appli-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 7approximated by bilinear interpolation functions of form

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

In order to find the coefficients let us consider the triangular region (element)

of the x− y plane in a specifically located local coordinate system and thenotation shown in Figure 1.2

(x3, y3)

q2

(x2, y2) (x2, 0)

q1

(x1, y1) (0, 0)

q(x, y)

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 erality of the following discussion The arrangement can always be achieved

gen-by 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

q(x, y) = N1q1+ N2q2+ N3q3.The values of Ni are

N1+ N2+ N3= 1equation is satisfied

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

Specifically along the edge between nodes 1 and 2 the shape function N3

is 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 reflect 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 finite element nology, the basis functions They are sometimes used in place of shape func-tions by engineers, although as shown below, they are distinctly different Theapproximation of

tech-q(x, y) = N qemay also be written as

q(x, y) = M cewhere M is the matrix of basis functions and ceis the vector of basis coeffi-cients

Clearly for our example

com-q(x, y) = a + bx + cy + dxy + ex2+ f x2y + 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 finite 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

Finally, additional volume elements are also frequently used The dron is one of the most widely accepted Its first order version consists ofeight node points at the corners of the hexahedron and therefore, it is definedwith specifically chosen basis functions as

hexahe-q(x, y, z) = a + bx + cy + dz + exy + f xz + 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 order

p, 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 specific 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 finite element technology may be assessed byreviewing the early articles of [10] and [1], as well as from the reference ofthe first general purpose and still premier finite element analysis tool [4]

1.4 Assembly of finite element matrices

The repeated application of general triangles may be used to cover the D 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 thefinite element mesh There may be small gaps between the boundary and the

pla-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 finite element method The gaps may be filled byprogressively smaller elements or those triangles may be replaced by triangleswith curved edges Nevertheless, all the elemental matrices contribute to theglobal finite element matrices and the process of computing these contribu-tions is the finite 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 function

Ni will define 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

Ee= 12

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 different 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

of the global K matrix The actual integration for computing keis addressed

in 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 element’s matrix terms(depending on that particular element’s 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 finite element mesh

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

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

formu-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 simplified 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 metric coordinates Let us consider again the local coordinates of the tri-angular element, now shown in Figure 1.4 with two specific coordinate axesrepresenting the parametric system The axis (coincident with the local xaxis) going through node points 1 and 2 is the first parametric axis u Definethe other axis going from node 1 through node 3 as v If we define 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

x2x− x3

x2y3yand

v = N3.Furthermore, the points inside one element may be written as

x = N1x1+ N2x2+ N3x3,and

y = 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 field, this

is called an parametric representation and our element is called an parametric element Applying the local coordinates of our element of Figure

y = y3v.

The integral with this parameterization is

 dxdy =

 det[∂(x, y)

Ae= 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 17reduction 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 Simpson’s In general an integral over

a specific continuous interval is approximated by a sum of weighted functionvalues at some specific locations

ci=

 1

−1Ln −1,i(t)dtwhere

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

 1u=0

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

which agrees with the geometric computation based on the triangle’s 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 case

of 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 formed to the global coordinate system common to all the elements Let usdenote the element’s 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

trans-⎡

⎣jik

⎤

⎦ = T

⎡

⎣JIK

qe= Glgqeg,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 qg notationrefers to the element displacements in the global coordinate system

Considering the element energy contribution

Ee=1

2qT

ekeqe

and substituting above we get

Ee= 1

2q

g,T

e Glg,TkeGlgqgor

keg= Glg,TkeGlg.This transformation follows the element matrix generation and precedes theassembly process Naturally, the solution qg 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 will

be generated either in terms of local or global coordinates for simplifying theparticular discussion Commercial finite element analysis systems have spe-cific rules for the definition of local coordinates for various element types

1.7 A linear quadrilateral finite 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 first we focus on the linear case.Quadrilateral elements are the most frequently used elements of industrialfinite 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 x− y plane as shown in Figure 1.5, but pose

no other restriction on its location Based on the principles we developed

in connection with the simple triangular element, we introduce shape 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 For

Finite Element Analysis 21

FIGURE 1.5 A planar quadrilateral element

element to the parametric coordinates is the following counterclockwise tern:

pat-(x1, y1)→ (−1, −1),(x2, y2)→ (1, −1),(x3, y3)→ (1, 1),and

N4=1

4(1− u)(1 + v)

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 mulation The above selection of the Ni functions obviously again satisfies

for-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

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) = N qe.Since

a point inside the element is approximated again with the same four shapefunctions as the displacement field:

x = N1x1+ N2x2+ N3x3+ N4x4,and

y = N1y1+ N2y2+ N3y3+ N4y4.Here xi, yiis the location of the i-th node of the element in the x, y directions.Using the shape functions defined above with the element coordinates and bysubstituting we get

discus-the first two components are discus-the rates of changes in distances between points

of the element in the appropriate directions The third component is a bined rate of change with respect to the other variable in the plane, defining

com-an com-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

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,

the strain components required for the element are

Taking advantage of the components of J and using the adjoint-based inverse

Finite Element Analysis 25From the displacement field approximation equations we obtain

ke=

 

BTBdet[∂(x, y)

∂(u, v)]dudv.

By the fortuitous choice of the parametric coordinate system this integral now

is directly amenable to Gaussian quadrature as the limits are−1, +1 ducing

Intro-f (u, v) = BTBdet(J)the element integral becomes

loca-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 lateral element In practice the quadratic version is much preferred and will

quadri-be descriquadri-bed in the following

1.8 Quadratic finite elements

We view the element in the x− y plane as shown in Figure 1.5, but add nodes

on 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 tions as

equa-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 finally

1 + u + v = 0,connecting nodes 8 and 1, completing the loop For j = 1, , 8 we seekfunctions Nithat are unit at the ithe node and vanish at the others: