min l u
ð Þ
η ∗ x e
ð Þ, i¼1, ..., n
¼
ð
Ω
fu dx þ
ð
Γ t
tu ds
ð6Þ
subjected to constraints:
Problem Definition
Analysis Proprocessor
Analysis Solver
Optimisation Solver
Design definition
Material definition
CAD Model
Usable Material M 0
Creating FEM Model
Applying B.C.s on Ω and Γ
Applying Loads, f and t
FEM Model Input
Normalised Material Density r i,j
Evaluating Design Variable r i,j
Objective function:
Sensitivity: -2a . ESE i,j /r i,j
f
t
u d +
j
.
Termination criterion
Optimisation Model
Topology Optimisation Result Output
Normalized Material Density r i,final
True
False
min
0
0
1
s. t.
≤
≤
≤
M 0
P 0
–
j=j+1
Evaluating Displacement u j
Evaluating Element Strain Energy ESE i,j
Evaluating Element Volume/area v i
Computing Material Property E i,j
=
E i,j
E 0
Normalised Material Density r i,0
r i,j
a
Material Property E 0 , r 0 , v 0
Design Domain, Ω
Ω
∫
∫ u d
j
.
Γ
f
t
u d +
j
.
Ω
dΩ
Ω
Ω
∫
∫
∫ u d
j
.
Γ
Γ
Ω
Γ
r j
r j
Fig. 1 A general topological optimization process
6
Henrique A. Almeida and Paulo J. Ba ´ rtolo