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Aristotle University of Thessaloniki (AUTH)
Department of Civil Engineering
Applications of FreeFem++ on Structural Optimization
“Applications of FreeFem++ on Structural Optimization”
Michailidis GeorgiosCivil Engineer
Supervisor: Allaire GregoireProfessor of Applied Mathematics, Ecole PolytechniquePresident of the Department of Applied Mathematics (DMAP)
Co-supervisor: Charalambakis NicolaosProfessor of Mechanics, AUTHChairman 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
0u( )n n Nu
inf ( )y VJ y
J(un+1) < J(un)
1 ( ( ))n n nKu P u J u
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization
1st Application: Cantilever under horizontal load-Thickness Optimization:
7
1 3 4 6
0 in Ω,
2 ( ) ( ( )) in Ω,
σ n=g on a ,
0 on a , , , ,
0
div
he u htr e u I
n a a a
u
2 5 on a , .a
Model:
Objective function:7
( )a
J h g uds (compliance-work of the external forces)
max min 0( ), such that, h ( ) 0 in Ω, ( ) | |adU h L h x h h x dx h
Admissible set:
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization
100 iterations
Convergence diagram:
• h0=0.5• hmin=0.1• hmax=1.0
Convergence diagram
140.000
150.000
160.000
170.000
180.000
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Number of iterations
Va
lue
of
the
ob
jec
tiv
e f
un
cti
on
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization
2nd Application: Cantilever under vertical load-Thickness Optimization:
Objective function:7
( )a
J h g uds (compliance)
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization
g1
3*g1
3rd Application: Cantilever under vertical load-Multiple-loads Optimization:
Objective function:7
( )a
J h g uds (compliance)
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization
4th Application: Cantilever under vertical load-Geometric Optimization:
Objective function:7
inf { ( ) }adU
a
J g uds
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:7
inf { ( ) }ad
i iU
i a
J g u ds
Initial Compliance G.O. G.O. + P.O.556,046 325,953 255,582
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Desired mechanical behaviour
6th Application: Cantilever under vertical load-Geometric Optimization:
Number of Iterations Volume Total ReductionInitialization 25.069 -5 5.51033 780.2%20 3.74527 850.6%100 3.74179 850.7%
Objective function:2
11 11inf { ( ) ( ( ) ) }ad
des
UJ x ds
Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Stress minimization
7th Application: L-shaped structure:
2
5
3
5
2 3
0,5
0,2
a1
a2
a3
a4
a5
a6
a7a8a9
a10
0,5
Model: 5
2 3 4 6 7 8 9 10
0 in Ω,
2 ( ) ( ( )) in Ω,
σ n=g on a ,
0 on a , , , , , , , ,
0
div
he u htr e u I
n a a a a a a a
u
1 on a .
Objective function: ( ) ( , ( ))J j x x dx
1 1
|| || | |N N
p pij
i j
Norm of a tensor:
Applications of FreeFem++ on Structural Optimization
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.