Biomedical Engineering Reference
In-Depth Information
reader is referred to the many excellent texts in the field,
some of which are included in the bibliography.
Overview of the FEM
The essential steps in implementing the FEM follow:
(i)
The region of interest (continuum) is discretized,
that is, subdivided into a smaller number of regions
called elements, interconnected at nodal points.
Nodes may also be placed in the interior of an
element. In one dimension, the elements are line
segments; in two dimensions, they are usually
triangles or quadrilaterals (
Fig. 3.1.3-1
); in three
dimensions, they can be rectangular prisms
(hexahedra) or triangular prisms (tetrahedra), for
example (
Fig. 3.1.3-2
). Elements may be quite
general with the possibility of non-planar faces and
curvilinear sides or edges (
Desai, 1979
;
Zienkiewicz
and Taylor, 1994
).
(ii)
The unknown variables within the continuum
(e.g., displacement, stress, or velocity components)
are defined within each element by suitable
interpolating functions. Interpolating functions are
Fig. 3.1.3-2 3D FE representations of the human femur.
(A) Tetrahedral elements; (B) hexahedral elements. (From
Middleton et al., 1996
, p. 125. Reproduced with permission of
Gordon and Breach Publishers, Overseas Publishers Assn.,
Amsterdam.)
traditionally piece-wise polynomials and are also
known as basis or shape functions. The order of the
interpolating functions (i.e., first, second, or third
order) is usually used to fix the number of nodes in
the elements (
Fig. 3.1.3-3
).
A
C
η
3
2
1
1
ξ
-1
1
1
2
3
-1
4
L
1
= 1
0.25
0.5
0.75
0
B
3
D
5
2
1
5
4
6
9
8
1
2
6
3
7
4
Fig. 3.1.3-3 Examples of two-dimensional elements and their
corresponding local coordinate systems [embedded in (A) and
(C)]. For the rectangles, the local coordinates (
x
,
h
) are referred to
a cartesian system with 1
x
,
h
1; for the triangles, the local
coordinates (L
1
,L
2
,L
3
) are area coordinates satisfying 0 L
1
,L
2
,
L
3
1. Elements with linear interpolating functions (first order)
are shown in (A) and (C). Quadratic elements (second-order
interpolating functions) are shown in (B) and (D).
Fig. 3.1.3-1 (A) Cross-section of an autopsy-retrieved femur
showing a cracked mantle (arrows). (B) Mixed planar quadrilateral/
triangle FE representation of (A). (From Middleton et al., 1996,
p. 35. Reproduced with permission of Gordon and Breach
Publishers, Overseas Publishers Assn., Amsterdam.)
