Biomedical Engineering Reference
In-Depth Information
Fig. 5.35  Types of finite elements. 1D elements are lines that can represent springs, beams, pipes.
2D elements can be triangles that represent plates, shells
whole computational domain. Boundary conditions are then imposed to solve
the equations.
The discretized finite elements can be one, two or three dimensions, and also special
elements with zero dimensionality (lumped springs or point masses). Lower di-
mensional elements can be expanded to build a model in 2D or 3D space. Nodes
are usually located at the corners or end points of the elements (Fig. 5.35 ). Most
elements used in practice have fairly simple geometries. In one-dimension, ele-
ments are straight lines or curved segments, in two dimensions they are triangular
or quadrilateral, and in three dimensions they can be tetrahedra, pentahedra (wedges
or prisms), and hexahedra (cuboids or “bricks”).
Unlike the finite volume method, a finite element is not a differential element of
size dx × dy but rather is an element at which the value of a field variable is explic-
itly calculated. This quantity is approximated over each element using polynomial
interpolation to produce a very large set of simultaneous equations. In this section,
we present simplified examples to allow the reader to appreciate the discretisation
process in FEM. For a more complete coverage of the FEM method notable topics
by Hutton (2005), Zienkiewicz and Taylor (2000), and Stasa (1985) are recom-
mended.
One-Dimensional Finite Element Example The simplest finite element is a linear
one-dimensional element shown in Fig. 5.36 which is a bar that obeys Hooke's law,
with forces applied at its ends. This model is a two-force member where the forces
exerted on the ends are equal in magnitude, and opposite in sense. The element has
length L , and we denote the axial displacement at any position along the length of
the bar as u (  x ). The nodes are placed at each end as nodes-1 and − 2, with nodal
displacements u ( x = 0 ) = u 1 and u ( x = L ) = u 2 . The continuous displacement field vari-
able u (  x ) is expressed (approximately) in terms of the two nodal variables u 1 and u 2
through discretisation of the bar into a finite element given as
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