Development of a Fabric-Reinforced Porous Graft for Vascular Tissue Engineering Using Finite Element Methods and Genetic Algorithms - Cardiovascular and Cardiac Therapeutic Devices - page 41

In Eq. ( 19 ), x represents either axial stress r A , localized axial strain e A ,or
localized transverse strain e T . This ensured that numerical models which did not
run to completion were weighted accordingly. Tests that failed to reach 10 % axial
strain were given a lower ranking to those which reached an axial strain of 20 %,
similarly for 20 and 30 %, thus favoring those models which ran to completion.

Global Objective, Penalty and Fitness Function

The objective function was represented in terms of the partial objective functions
by

obj / r War A þ w r Weft

/ ¼ w r Warp

obj / r Wef A þ w e Warp obj / e War A þ w e Weft obj / e Wef A þ w e Warp obj / e War T þ w e Weft

obj / e Weft

A

A A A T T

T

;

w r Warp

obj þ w r Weft obj þ w e Warp obj þ w e Weft

obj þ w e Warp obj þ w e Weft

A A A A T T

obj

ð 20 Þ

where w i obj are pre-defined weightings and i represents the partial objective
function to which the weightings apply. These weightings were used to bias the
objective value to either the warp or weft direction, to stresses, to localized axial
and or transverse strains. From Eq. ( 20 ), / tends to unity as / r Warp , / r Weft , / e Warp ,

A A A

/ e Weft , / e Warp and / e Weft

tend to unity. The pre-defined values of the weightings are

A T T

w r Warp ¼ w r Weft ¼ 200 ; w e Warp ¼ w e Weft

obj ¼ 10 and w e Warp

¼ w e Weft

obj ¼ 100.

A
obj

A
obj

A
obj

A T

obj

T

The global penalty functions, p, were defined as

if r A ; 10 % 0 then p ¼ 1 else p ¼ 1 ;

ð 21 Þ

if r T ; 10 % 0 then p ¼ 1 else p ¼ 1 ; ð 22 Þ

if e T ; 10 % 0 then p ¼ 1 else p ¼ 1 ;

ð 23 Þ

and ensure that the models run and the results obtained are reasonable and
acceptable.
The fitness function, f , for the comparison of model solutions and physical data
for warp and weft uniaxial tension was:

f ¼ p / : ð 24 Þ

2.3.2 Optimised Model Solutions

For the four fabric samples, convergence of GA1 on a set of single solutions was
found after 47, 42, 39 and 45 generations, respectively. Table 1 provides the model
coefficients and the fitness, objective and partial objective values obtained from
GA1 for each fabric. The fitness values coincided with the objective values, all of
which closely approached unity. The average partial objective values obtained for