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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
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Contents

• Optimization algorithm
• Compliance minimization
• Desired mechanical behaviour
• Stress minimization
• Conclusions

Applications of FreeFem on Structural
Optimization
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Optimization algorithm

• Gradient algorithm with constant step and
  projection

J(un1) lt J(un)?
Applications of FreeFem on Structural
Optimization
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Applications of FreeFem - Compliance
minimization
1st Application Cantilever under horizontal
load-Thickness Optimization
Model
Objective function
(compliance-work of the external forces)?
Admissible set
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Optimization
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Applications of FreeFem - Compliance
minimization
100 iterations

• h00.5
• hmin0.1
• hmax1.0

Convergence diagram
Applications of FreeFem on Structural
Optimization
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Applications of FreeFem - Compliance
minimization
2nd Application Cantilever under vertical
load-Thickness Optimization
Objective function
(compliance)?
Applications of FreeFem on Structural
Optimization
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Applications of FreeFem - Compliance
minimization
3rd Application Cantilever under vertical
load-Multiple-loads Optimization
Objective function
(compliance)?
Applications of FreeFem on Structural
Optimization
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Applications of FreeFem - Compliance
minimization
4th Application Cantilever under vertical
load-Geometric Optimization
Objective function
Applications of FreeFem on Structural
Optimization
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Applications of FreeFem - Compliance
minimization
Applications of FreeFem on Structural
Optimization
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Applications of FreeFem - Compliance
minimization
5th Application Cantilever under vertical
load-Coupled Method-(G.O.P.O.)
Objective function
Initial Compliance G.O. G.O. P.O.
556,046 325,953 255,582
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Optimization
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Applications of FreeFem - Desired mechanical
behaviour
6th Application Cantilever under vertical
load-Geometric Optimization
Objective function
Applications of FreeFem on Structural
Optimization
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Applications of FreeFem - Stress minimization
7th Application L-shaped structure
Model
Objective function
Norm of a tensor
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Optimization
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Applications of FreeFem
L6 norm-Thickness Optimization
Applications of FreeFem on Structural
Optimization
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Applications of FreeFem
L10 norm-Thickness Optimization
Applications of FreeFem on Structural
Optimization
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Applications of FreeFem
L6 norm-Geometric Optimization
Applications of FreeFem on Structural
Optimization
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Applications of FreeFem
L10 norm-Geometric Optimization
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Optimization
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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