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Exercises 227

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
17.4.1 Lagrangian elements
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

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.

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.

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



Figure 1.1
The vector a = ae with a > 0.

a c


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

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


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 :
a · b = 0, if φ = . (1.7)
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


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