Aristotle University of Thessaloniki (AUTH) Department of Civil Engineering

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


Contents

Optimization algorithm

Compliance minimization

Desired mechanical behaviour

Stress minimization

Conclusions


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++ - 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:



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



2nd Application: Cantilever under vertical load-Thickness Optimization:


( )a

J h g uds (compliance)



g1

3*g1

3rd Application: Cantilever under vertical load-Multiple-loads Optimization:


( )a

J h g uds (compliance)



4th Application: Cantilever under vertical load-Geometric Optimization:


inf { ( ) }adU

a

J g uds





5th Application: Cantilever under vertical load-Coupled Method-(G.O.+P.O.):


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++ - 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%


11 11inf { ( ) ( ( ) ) }ad

des

UJ x ds


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++

L6 norm-Thickness Optimization:



L10 norm-Thickness Optimization:



L6 norm-Geometric Optimization:



L10 norm-Geometric 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.

Documents

Aristotle University of Thessaloniki (AUTH) Department of Civil Engineering