14 Solution of the one-dimensional diffusion equation

by means of the Finite Element Method 232

14.1 Introduction 232

14.2 The diffusion equation 233

14.3 Method of weighted residuals and weak form of the model

problem 235

14.4 Polynomial interpolation 237

ix Contents

14.5 Galerkin approximation 239

14.6 Solution of the discrete set of equations 246

14.7 Isoparametric elements and numerical integration 246

14.8 Basic structure of a ¬nite element program 250

14.9 Example 253

Exercises 256

15 Solution of the one-dimensional convection-diffusion

equation by means of the Finite Element Method 264

15.1 Introduction 264

15.2 The convection-diffusion equation 264

15.3 Temporal discretization 266

15.4 Spatial discretization 269

Exercises 273

16 Solution of the three-dimensional convection-diffusion

equation by means of the Finite Element Method 277

16.1 Introduction 277

16.2 Diffusion equation 278

16.3 Divergence theorem and integration by parts 279

16.4 Weak form 280

16.5 Galerkin discretization 280

16.6 Convection-diffusion equation 283

16.7 Isoparametric elements and numerical integration 284

16.8 Example 288

Exercises 291

17 Shape functions and numerical integration 295

17.1 Introduction 295

17.2 Isoparametric, bilinear quadrilateral element 297

17.3 Linear triangular element 299

17.4 Lagrangian and Serendipity elements 302

303

17.4.1 Lagrangian elements

304

17.4.2 Serendipity elements

17.5 Numerical integration 305

Exercises 309

18 In¬nitesimal strain elasticity problems 313

18.1 Introduction 313

18.2 Linear elasticity 313

Contents

x

18.3 Weak formulation 315

18.4 Galerkin discretization 316

18.5 Solution 322

18.6 Example 322

Exercises 324

References 329

Index 331

About the cover

The cover contains images re¬‚ecting biomechanics research topics at the

Eindhoven University of Technology. An important aspect of mechanics is exper-

imental work to determine material properties and to validate models. The

application ¬eld ranges from microscopic structures at the level of cells to larger

organs like the heart. The core of biomechanics is constituted by models for-

mulated in terms of partial differential equations and computer models to derive

approximate solutions.

• Main image: Myogenic precursor cells have the ability to differentiate and fuse to form

multinucleated myotubes. This differentiation process can be in¬‚uenced by means of

mechanical as well as biochemical stimuli. To monitor this process of early differentia-

tion, immunohistochemical analyses are performed to provide information concerning

morphology and localization of characteristic structural proteins of muscle cells. In the

illustration, the sarcomeric proteins actin (red), and myosin (green) are shown. Nuclei

are stained blue. Image courtesy of Mrs Marloes Langelaan.

• Left top: To study the effect of a mechanical load on the damage evolution of skeletal

tissue an in-vitro model system using tissue engineered muscle was developed. The

image shows this muscle construct in a set-up on a confocal microscope. In the device

the construct can be mechanically deformed by means of an indentor. Fluorescent iden-

ti¬cation of both necrotic and apoptotic cells can be established using different staining

techniques Image courtesy of Mrs Debby Gawlitta.

• Left middle: A three-dimensional ¬nite element mesh of the human heart ventricles is

shown. This mesh is used to solve the equations of motion for the beating heart. The

model was used to study the effect of depolarization waves and mechanics in the paced

heart. Image courtesy of Mr Roy Kerckhoffs.

• Left bottom: The equilibrium equations are derived from Newton™s laws and describe

(quasi-)static force equilibrium in a three-dimensional continuum. Chapter 9 of the

present book.

Preface

In September 1997 an educational programme in Biomedical Engineering, unique

in the Netherlands, started at the Eindhoven University of Technology, together

with the University of Maastricht, as a logical step after almost two decades of

research collaboration between both universities. This development culminated in

the foundation of the Department of Biomedical Engineering in April 1999 and

the creation of a graduate programme (MSc) in Biomedical Engineering in 2000

and Medical Engineering in 2002.

Already at the start of this educational programme, it was decided that a com-

prehensive course in biomechanics had to be part of the curriculum and that this

course had to start right at the beginning of the Bachelor phase. A search for

suitable material for this purpose showed that excellent biomechanics textbooks

exist. But many of these books are very specialized to certain aspects of biome-

chanics. The more general textbooks are addressing mechanical or civil engineers

or physicists who wish to specialize in biomechanics, so these books include chap-

ters or sections on biology and physiology. Almost all books that were found are

at Masters or post-graduate level, requiring basic to sophisticated knowledge of

mechanics and mathematics. At a more fundamental level only books could be

found that were written for mechanical and civil engineers.

We decided to write our own course material for the basic training in mechan-

ics appropriate for our candidate biomedical engineers at Bachelor level, starting

with the basic concepts of mechanics and ending with numerical solution proce-

dures, based on the Finite Element Method. The course material assembled in the

current book, comprises three courses for our biomedical engineers curriculum,

distributed over the three years of their Bachelor studies. Chapters 1 to 6 mostly

treat the basic concepts of forces, moments and equilibrium in a discrete context

in the ¬rst year. Chapters 7 to 13 in the second year discuss the basis of continuum

mechanics and Chapters 14 to 18 in the third year are focussed on solving the ¬eld

equations of mechanics using the Finite Element Method.

Preface

xiv

What makes this book different from other basic mechanics or biomechanics

treatises? Of course there is the usual attention, as in standard books, focussed on

kinematics, equilibrium, stresses and strains. But several topics are discussed that

are normally not found in one single textbook or only described brie¬‚y.

• Much attention is given to large deformations and rotations and non-linear constitutive

equations (see Chapters 4, 9 and 10).

• A separate chapter is devoted to one-dimensional visco-elastic behaviour (Chapter 5).

• There is special attention to long slender ¬bre-like structures (Chapter 4).

• The similarities and differences in describing the behaviour of solids and ¬‚uids and

aspects of diffusion and ¬ltration are discussed (Chapters 12 to 16).

• Basic concepts of mechanics and numerical solution strategies for partial differential

equations are integrated in one single textbook (Chapters 14 to 18).

Because of the usually rather complex geometries (and non-linear aspects)

found in biomechanical problems hardly any relevant analytical solutions can be

derived for the ¬eld equations and approximate solutions have to be constructed.

It is the opinion of the authors that at Bachelor level at least the basis for these

numerical techniques has to be addressed.

In Chapters 14 to 18 extensive use is made of a ¬nite element code written in

Matlab by one of the authors, which is especially developed as a tool for students.

Applying this code requires that the user has a licence for the use of Matlab, which

can be obtained via MathWorks (www.mathworks.com). The ¬nite element code,

which is a set of Matlab scripts, including manuals, is freely available and can be

downloaded from the website: www.mate.tue.nl/biomechanicsbook.

1 Vector calculus

1.1 Introduction

Before we can start with biomechanics it is necessary to introduce some basic

mathematical concepts and to introduce the mathematical notation that will be

used throughout the book. The present chapter is aimed at understanding some of

the basics of vector calculus, which is necessary to elucidate the concepts of force

and momentum that will be treated in the next chapter.

1.2 Definition of a vector

A vector is a physical entity having both a magnitude (length or size) and a

direction. For a vector a it holds, see Fig. 1.1:

a = ae. (1.1)

The length of the vector a is denoted by |a| and is equal to the length of the

arrow. The length is equal to a, when a is positive, and equal to ’a when a is

negative. The direction of a is given by the unit vector e combined with the sign

of a. The unit vector e has length 1. The vector 0 has length zero.

1.3 Vector operations

Multiplication of a vector a = ae by a positive scalar ± yields a vector b having

the same direction as a but a different magnitude ±|a|:

b = ±a = ±ae. (1.2)

This makes sense: pulling twice as hard on a wire creates a force in the wire

having the same orientation (the direction of the wire does not change), but with

a magnitude that is twice as large.

Vector calculus

2

a

e

Figure 1.1

The vector a = ae with a > 0.

a c

b

Figure 1.2

Graphical representation of the sum of two vectors: c = a + b.

The sum of two vectors a and b is a new vector c, equal to the diagonal of the

parallelogram spanned by a and b, see Fig. 1.2:

c = a + b. (1.3)

This may be interpreted as follows. Imagine two thin wires which are attached

to a point P. The wires are being pulled at in two different directions according

to the vectors a and b. The length of each vector represents the magnitude of the

pulling force. The net force vector exerted on the attachment point P is the vector

sum of the two vectors a and b. If the wires are aligned with each other and the

pulling direction is the same, the resulting force direction is clearly coinciding

with the direction of the two wires and the length of the resulting force vector is

the sum of the two pulling forces. Alternatively, if the two wires are aligned but

the pulling forces are in opposite directions and of equal magnitude, the resulting

force exerted on point P is the zero vector 0.

The inner product or dot product of two vectors is a scalar quantity, de¬ned

as

a · b = |a||b| cos( φ) , (1.4)

where φ is the smallest angle between a and b, see Fig. 1.3. The inner product is

commutative, i.e.

a · b = b · a. (1.5)

3 1.3 Vector operations

b

φ

a

Figure 1.3

De¬nition of the angle φ.

The inner product can be used to de¬ne the length of a vector, since the inner

product of a vector with itself yields (φ = 0):

a · a = |a||a| cos( 0) = |a|2 . (1.6)

If two vectors are perpendicular to each other the inner product of these two

vectors is equal to zero, since in that case φ = π :

2

π

a · b = 0, if φ = . (1.7)

2

The cross product or vector product of two vectors a and b yields a new vector

c that is perpendicular to both a and b such that a, b and c form a right-handed

system. The vector c is denoted as

c=a—b. (1.8)

The length of the vector c is given by

|c| = |a||b| sin( φ) , (1.9)

where φ is the smallest angle between a and b. The length of c equals the area of

the parallelogram spanned by the vectors a and b. The vector system a, b and c

forms a right-handed system, meaning that if a corkscrew is used rotating from a