Biomedical Engineering Reference

In-Depth Information

If your bone is heavily loaded, say due to sport, your bones will gain mass. If your bone is

lightly loaded, say due to being in space and weightless, your bones will lose mass. They

are constantly changing and trying to achieve an optimal solution. In bone this is called

remodeling
.

There is a term for the ultimate objective you are trying to achieve; this is called the
objective

function
. There may be more than one. It is a mathematical expression used to model your

design. It is usually written in the form of an expression

fABC fABC

o
(,,) (,,)

(where
f
o
is the objective function and
f
is any mathematical function of the variables A, B, and C).

Or the objective function of parameters A, B, and C is defined by the equation on the right-hand side.

The simplest form of optimization is
linear
. Often in mathematics you will see the term
linear

programming
. The best way to visualize this is as a graph of two straight lines. Suppose we

have a system where the objective function is

f

o
34

x

y

∫

where

x

0

5

and

y

4

and we need to minimize
f
o
.
Figure 7.1
illustrates the objective function (values of 16, 24,

and 32) and the constraints. The constraints mean that the solution can only lie in the shaded

region. By inspection the minimum lies in the bottom left-hand corner when
x
= 0 and
y
= 4,

giving
f
o
= 16. All other values of
x
and
y
either lie outside of the constraints (outside the

design space) or result in values greater than the minimum (16).

Consider a cylinder of diameter
D
and length
L
made from steel plate of thickness
t
= 5 mm

then we have two possible objective functions: mass of the cylinder itself and the volume it

contains (
Figure 7.2
).

The mass of the cylinder is given by

2

π

D

fDL

o
(,)
1

π

DLt

2

t

ρ

4

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