Transcript Applications of FreeFem++ on Structural Optimization

Aristotle University of Thessaloniki (AUTH)
Department of Civil Engineering
“Applications of FreeFem++ on Structural Optimization”
Michailidis Georgios
Civil Engineer
Supervisor: Allaire Gregoire
Professor of Applied Mathematics, Ecole Polytechnique
President of the Department of Applied Mathematics (DMAP)
Co-supervisor: Charalambakis Nicolaos
Professor of Mechanics, AUTH
Chairman of the Institute of Mechanics of Materials
Applications of FreeFem++ on Structural Optimization
Contents
 Optimization algorithm
 Compliance minimization
 Desired mechanical behaviour
 Stress minimization
 Conclusions
Applications of FreeFem++ on Structural Optimization
Optimization algorithm
•Gradient algorithm with constant step and projection
u0
(u n ) nN
inf J ( y)
yV
n

1
n

u

P
(
u


Ju
(n)
)
K
J(un+1) < J(un)
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization
1st Application: Cantilever under horizontal load-Thickness Optimization:
Model:
Objective function:

d
iv
0


2

h
e
()
u

h
tre
(()
uI
)


σ
n
=
g


n0


u
0

inΩ
,
inΩ
,
o
na,
7
o
na
,aaa
, 4, 6,
1
3
o
na,
.
2a
5
Jh
( )gu
 ds (compliance-work of the external forces)
a
7
Admissible set:



U

h

L
(

)
,
s
u
c
h
t
h
a
t
,
h

h
(
x
)

h

0
i
n
Ω
,
h
(
x
)
d
x

h
|

|

a
d
m
a
x
m
i
n
 0



Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization
100 iterations
• h0=0.5
• hmin=0.1
• hmax=1.0
Convergence diagram
Convergence diagram:
Value of the objective function
180.000
170.000
160.000
150.000
140.000
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
Number of iterations
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization
2nd Application: Cantilever under vertical load-Thickness Optimization:
Objective function:
Jh
( )gu
 ds (compliance)
a
7
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization
3rd Application: Cantilever under vertical load-Multiple-loads Optimization:
3*g1
Objective function:
Jh
( )gu
 ds (compliance)
a
7
g1
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization
4th Application: Cantilever under vertical load-Geometric Optimization:
Objective function:
i
n
f{
J
(

)
d
s
}
gu


U
a
d
a
7
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization
5th Application: Cantilever under vertical load-Coupled Method-(G.O.+P.O.):
Objective function:
Initial Compliance
556,046
G.O.
325,953
inf {J ()    gi  ui ds}
U ad
i
a7
G.O. + P.O.
255,582
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Desired mechanical behaviour
6th Application: Cantilever under vertical load-Geometric Optimization:
Objective function:
Number of Iterations
Initialization
5
20
100
Volume
25.069
5.51033
3.74527
3.74179
inf {J ()   (11 ( x)  11des )2 ds}
Uad

Total Reduction
780.2%
850.6%
850.7%
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Stress minimization
7th Application: L-shaped structure:
3
2
3
a
1
Model:
i
n
Ω
,
i
n
Ω
,
o
n
a
,
5
o
n
a
,a
,a
,a
,a
,a
,a
,a
,
2
3
4
6
7
8
9
1
0
o
n
a
.
1
a
7
0
,5
a
6
a
5
2
a
9
a
8
0,2
a
2
0,5
5
a
1
0

d
i
v


0




2

h
e
()
u


h
t
r
(
e
()
u
)
I


σ
n
=
g



n

0


u

0

a
4
a
3
5
Objective function:
J ()   j ( x, ( x))dx

N
Norm of a tensor:
||  ||  
p
i 1
N
| 
j 1
Applications of FreeFem++ on Structural Optimization
ij
|p
Applications of FreeFem++
L6 norm-Thickness Optimization:
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++
L10 norm-Thickness Optimization:
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++
L6 norm-Geometric Optimization:
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++
L10 norm-Geometric Optimization:
Applications of FreeFem++ on Structural Optimization
Conclusions
 The methods presented can prove to be very useful for optimizing parts of a structure
with specific boundary conditions, connections of structural elements, etc., but not the
structure in general.
 The results coming from these methods are superior than our mechanical intuition,
which is very useful and necessary in order to detect possible mistakes in the code.
 FreeFem++ is a very efficient software to apply optimization algorithms. However, the
user of FreeFem++ should first try to understand in depth the details of the programm and
algorithms in simple examples, before applying them to more complicated problems.
Applications of FreeFem++ on Structural Optimization