Properties of materials - Biomedical Engineering Desk Reference - page 117

equivalence of the weighted residual and variational
methods for this particular example. We note that the
weighting function is required to be zero on those parts of
the boundary where the unknown function is specified
( G 1 , in our example) and that there can be other choices
of the weighting function w. Choosing weighting func-
tions to be the same as interpolating functions defines the
Galerkin FEM ( Strang and Fix, 1973 ; Zienkiewicz and
Taylor, 1994 ).

nodes that define the element. The requirement for
''local support'' means that within an element,

N i ð x ; h Þ¼ 1 at node i (3.1.3.11)

0 at all other nodes

as shown in Fig. 3.1.3-5 . This is the single most important
property of the interpolating functions. This property
makes it possible for the contributions of all the elements
to be summed up to give the response of the whole
domain.
The notation P m is conventionally used to indicate the
degree m of the interpolating polynomial. The notation
C n is used to indicate that all derivatives of the in-
terpolating function, up to and including n 1 , exist and
are continuous. By convention, the notation P m C n is
therefore used to indicate the order and smoothness
properties of the interpolating polynomials.

Properties of interpolating functions

The process of discretizing the continuum into smaller
regions means that the global shape functions N j ( x , y )are
replaced by local shape functions N j ð x ; h Þ; defined within
each element e ,wherex, h are the local coordinates within
the element ( Fig. 3.1.3-3 ). In the FEM, interpolating
functions are usually piecewise polynomials that are re-
quired to have (a) the minimum degree of smoothness,
(b) continuity between elements, and (c) ''local support.''
The minimum degree of smoothness is dictated by the
highest derivative of the unknown function that occurs in
the ''weak'' or variational form of the continuum prob-
lem. The requirement for continuity between elements
can always be satisfied by an appropriate choice of the
approximating polynomial and number of boundary

Examples from biomechanics

The following are examples of FEA applications in bio-
materials science and biomechanics.

Analysis of commonplace maneuvers
at risk for total hip dislocation

Dislocation is a frequent complication of total hip arthro-
plasty (THA). In this FE study (Nadzadi et al., 2003),
a motion tracking system and a recessed force plate were
used to capture the kinematics and ground reaction forces
from several trials of realistic dislocation-prone maneu-
vers performed by actual subjects. Kinematics and kinetic
data associated with the experiments were imported into
a FEmodel of THAdislocation. The FEmodel was used to
compute stresses developed within the implant, given the
observed angular motion of the hip and contact force in-
ferred frominversedynamics. The FEmesh ( Fig. 3.1.3-6A )
was created using PATRAN version 8.5 and the simula-
tions were executed with ABAQUS version 5.8. In the
FEA, the resultant resisting moment developed around
the hip-cup center was tracked, as a function of hip angle.
The peak of this resistive moment was a key outcome
measure used to estimate the relative risk of dislocations
from the motions. All seven maneuvers studied led to
frequent instances of computationally predicted disloca-
tion ( Fig. 3.1.3-6B ). The authors conclude that this library
of dislocation-prone maneuvers appear to substantially
extend the information base previously available to study
this important complication of THA. Additionally the
hope is that their results will contribute to improvements
in implant design and surgical technique and reduce in
vivo incidence.

ξ= −1

2

η=1

6

1 5

N 3 =

η

3

9

1

ξ

7

η= −1

8

ξ = 1

4

N 1 ( ξ, η) = ξη ( 1+ξ) ( 1+η)/4
N 2 ( 1+η)/4

N 3 ( ξ, η) = ξη ( 1−ξ) ( 1−η)/4
N 4 (

ξ, η) = −ξη ( 1−ξ) (

1−η)/4
N 5 ( ξ, η) = η ( 1−ξ 2 ) ( 1+η)/2
N 6 (

ξ, η) = −ξη ( 1+ξ) (

1−η 2 )/2
N 7 ( ξ, η) = −η ( 1−ξ 2 ) ( 1−η)/2
N 8 (

1−ξ 2 )

ξ, η) = −ξ ( (

1−η 2 )/2
N 9 ( ξ, η) = ξ ( 1−ξ 2 ) ( 1−η 2 )

1+ξ 2 )

ξ, η) = ξ ( (

Fig. 3.1.3-5 Sample shape functions for a nine-noded rectan-
gular element. Shape functions are defined in terms of local
coordinates x and h where 1 x , h 1; N 3 ð x ; h Þ is shown in
the plot. It can be checked that N i ¼ 1 at node i and zero at all
other nodes (compact support) as required.