application are solid and fluid mechanics. The former includes structures which,
for obvious reasons,
are fabricated with solids. Computational solid mechanics takes an applied
sciences approach,
whereas computational structural mechanics emphasizes techn

the cause-andeffect sense. For example: if the applied forces are doubled, the
displacements and internal stresses
also double. Problems outside this domain are classified as nonlinear.
1 Except that their function may not be clear to us. The usual approa

the reasons behind the power and acceptance of the method. Historically the
Physical FEM was the
first one to be developed to model complex physical systems such as aircraft, as
narrated in 1.7.
The Mathematical FEM came later and, among other things, pro

activity in error estimation and
mesh adaptivity is fostered by better understanding of the mathematical
foundations [237].
Commercial FEM codes gradually gain importance. They provide a reality check
on what works in the real
world and what doesnt. By th

On the left Figure 1.4 shows an ideal physical system. This may be presented as a
realization of
the mathematical model. Conversely, the mathematical model is said to be an
idealization of this
system. E.g., if the mathematical model is the Poissons PDE,

aircraft) it makes no sense. Indeed Physical FEM discretizations may be
constructed and adjusted
without reference to mathematical models, simply from experimental
measurements.
The concept of error arises in the Physical FEM in two ways. These are known

Topics 15 belong to what may be called Advanced Linear FEM, whereas 67
pertain to Nonlinear FEM.
Topics 810 fall into advanced applications, whereas 11 is an interdisciplinary
topic that interweaves with
computer science.
1.7. *Historical Sketch and Bibli

instance of multiscale analysis.
joint
Physical System
Idealized and
Discrete System
support
member
IDEALIZATION
Figure 1.6. The idealization process for a simple structure.
The physical system here a roof truss is directly idealized
by the mathematical m

1.1.1. Computational Mechanics
Several branches of computational mechanics can be distinguished according to
the physical scale
of the focus of attention:
Computational Mechanics
Nanomechanics and micromechanics
Continuum mechanics Solids and Structures F

Continuum mechanics problems may be subdivided according to whether inertial
effects are taken
into account or not:
Continuum mechanics Statics Dynamics (1.3)
In dynamics actual time dependence must be explicitly considered, because the
calculation of ine

The pioneers were structural engineers, schooled in classical mechanics. They
followed a century of tradition
in regarding structural elements as a device to transmit forces. This element as
force transducer was the
standard view in pre-computer structura

as discussed later. For now the underlying concept will be partly illustrated through
a truly ancient
problem: find the perimeter L of a circle of diameter d. Since L = d, this is
equivalent to
obtaining a numerical value for .
Draw a circle of radius r a

many sequences. See, e.g, [273].
16
17 1.3 THE FEM ANALYSIS PROCESS
disconnecting nodes, a process called disassembly in the FEM. Upon disassembly a
generic element
can be defined, independently of the original circle, by the segment that connects
two nod

studies on circle rectification by 250 B.C. But comparison with the modern FEM,
as covered in
following Chapters, shows this to be a stretch. The example does not illustrate the
concept of degrees
of freedom, conjugate quantities and local-global coordina

the roof as a a substructure of a building.
110
111 1.4 INTERPRETATIONS OF THE FINITE ELEMENT METHOD
1.4. Interpretations of the Finite Element Method
Just like there are two complementary ways of using the FEM, there are two
complementary interpretations

Displacement
Equilibrium
Mixed
Hybrid
FEM Solution Stiffness Flexibility Mixed (a.k.a. Combined) (1.6)
Using the foregoing classification, we can state the topic of this book more
precisely: the computational analysis of linear static structural problems

boost by the invention of the
isoparametric formulation and related tools (numerical integration, fitted natural
coordinates, shape functions,
patch test) by Irons and coworkers [144148]. Low order displacement models
often exhibit disappointing
performan

inter-component glue. The multilevel decomposition process is diagramed in
Figure 1.5, in which
intermediate levels are omitted for simplicity
Remark 1.2. More intermediate decomposition levels are used in systems such as
offshore and ship structures, whi

based on the minimum potential energy principle. This influential paper marks the
confluence of three lines of research: Argyris dual
formulation of energy methods [8], the Direct Stiffness Method (DSM) of Turner
[256258], and early ideas
of interelement

approximation to
mathematical problems. Both are reviewed below to introduce terminology used in
the sequel.
1.3.1. The Physical FEM
A canonical use of FEM is simulation of
physical systems. This requires models.
Consequenty the methodology is often
calle

apparent were new branches
of mathematics (operational calculus, distribution theory and piecewiseapproximation theory, respectively)
constructed to justify that success. In the case of the finite element method, the
development of a formal
mathematical t

As used today, FEM represents the confluence of three ingredients: Matrix
Structural Analysis (MSA), variational approximation theory, and the digital
computer. These came together in the early 1950. The reader
should not think, however, that they simulta

separation, decomposition) of a complex mechanical system into simpler, disjoint
components called
finite elements, or simply elements. The mechanical response of an element is
characterized in terms
of a finite number of degrees of freedom. These degrees

problem cannot be expressed in a standard variational form.
Remark 1.4. In the mathematical interpretation the emphasis is on the concept of
local (piecewise) approximation. The concept of element-by-element breakdown
and assembly, while convenient in the

the FEM in everyday use? there is no question in the writers mind: M. J. (Jon)
Turner at Boeing over the
period 19501962. He generalized and perfected the Direct Stiffness Method, and
forcefully got Boeing to
commit resources to it while other aerospace c

resemble spiders, who make cobwebs out of their own substance. But the bee takes
the middle course: it gathers its
material from the flowers of the garden and field, but transforms and digests it by a
power of its own. (Francis Bacon).
12 Of course at the

energy of the system is the
sum of element energies. But for Archimedes to reach modern FEM long is the
way, and hard, since
physical energy calculations require derivatives and Calculus would not be
invented for 20 centuries.
In his studies leading to t