Biomedical Engineering Reference
In-Depth Information
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resort to approximate or numerical methods. The most
popular numerical method for solving problems in contin-
uum mechanics is the finite element method (FEM), also
referred to as finite element analysis (FEA).
FEA is a computational approach widely used in solid
and fluid mechanics in which a complex structure is di-
vided into a large number of smaller parts, or elements,
with interconnecting nodes, each with geometry much
simpler than that of the whole structure. The behavior of
the unknown variable within the element and the shape
of the element are represented by simple functions that
are linked by parameters that are shared between the
elements at the nodes. By linking these simple elements
together, the complexity of the original structure can be
duplicated with good fidelity. After boundary conditions
are taken into account, a large system of equations for the
unknown nodal parameters always results; these equa-
tions are solved simultaneously by a computer, using
indirect or iterative means.
FEA is extremely versatile. The size and configuration
of the elements can be adjusted to best suit the problem;
complex geometries can be discretized and solutions can
be stepped through time to analyze dynamic systems.
Very often, simple analytical methods are used to make
a first approximation to the design of the device, and FEA
is subsequently used to further refine the design and
identify potential stress concentrations. FEA can be ap-
plied to both solids and fluids or, with additional com-
plexity, to systems containing both. FEA software is very
mature and computing power is now sufficiently cheap to
allow FEMs to be applied to a wide range of problems. In
fluid flow, FEA has been applied to weather forecasting
and supersonic flow around aircraft and within engines,
and in the medical field, to optimizing blood pumps and
cannulas. In solids, FEA has been used to design, build,
and crash automobiles, estimate the impact of earth-
quakes, and reconstruct crime scenes. In biomaterials,
FEA has been applied to almost every implantable device,
ranging from artificial joints to pacemaker leads. Although
originally developed to help structural engineers analyze
stress and strain, FEA has been adopted by basic scientists
and biologists to study the dynamic environment within
arteries, muscles and even cells.
We now hope to introduce the reader to FEMs without
digressing into detailed discussion of some of the more
difficult concepts that are often required to properly
define and execute a real-world problem. For that, the
3.1.3 Finite element analysis
Ivan Vesely and Evelyn Owen Carew
Introduction
The reader may be familiar with the concepts of elasticity,
stress, and strain. Estimations of material stress and strain
are necessary during the course of device design to mini-
mize the chance of device failure. For example, artificial
hip joints need to be designed to withstand the loads that
they are expected to bear without fracture or fatigue.
Stress analysis is therefore required to ensure that all
components of the device operate below the fatigue limit.
For deformable structures such as diaphragms for artificial
hearts, an estimate of strains or deformations is required
to ensure that during maximal deformation, components
do not contact other structures, potentially causing in-
terference and unexpected failure modes
such as
abrasion.
For simple calculations, such as the sizing of a bolt to
connect two components that bear load, simple analytical
calculations usually suffice. Often, these calculations are
augmented by reference to engineering tables that can be
used to refine the stress estimates based on local geometry,
such as the pitch of the threads. Such analyticalmethods are
preferred because they are exact and can be supported by
a wealth of engineering experience. Unfortunately, analyt-
ical solutions are usually limited to linear problems and
simple geometries governedby simple boundaryconditions.
The boundary conditions can be considered input data or
constraints on the solution that are applied at the bound-
aries of the system. Most practical engineering problems
involve some combination of material or geometrical non-
linearity, complex geometry, and mixed boundary condi-
tions. In particular, all biological materials have nonlinear
elastic behavior and most experience large strains when
deformed. As a result, nonlinearities of one form or the
other are usually present in the formulation of problems in
biomechanics. These nonlinearities are described by the
equations relating stress to strain and strain todisplacement.
Applying analytical methods to such problems would re-
quire so many assumptions and simplifications that the
results would have poor accuracy and would thus be of little
engineering value. There is therefore no alternative but to
