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Aristotle University of Thessaloniki (AUTH) Department of Civil Engineering

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

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Aristotle University of Thessaloniki (AUTH) Department of Civil Engineering

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  1. 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

  2. Contents • Optimization algorithm • Compliance minimization • Desired mechanical behaviour • Stress minimization • Conclusions Applications of FreeFem++ on Structural Optimization

  3. Optimization algorithm • Gradient algorithm with constant step and projection J(un+1) < J(un)‏ Applications of FreeFem++ on Structural Optimization

  4. Applications of FreeFem++ - Compliance minimization 1st Application: Cantilever under horizontal load-Thickness Optimization: Model: Objective function: (compliance-work of the external forces)‏ Admissible set: Applications of FreeFem++ on Structural Optimization

  5. Applications of FreeFem++ - Compliance minimization 100 iterations • h0=0.5 • hmin=0.1 • hmax=1.0 Convergence diagram: Applications of FreeFem++ on Structural Optimization

  6. Applications of FreeFem++ - Compliance minimization 2nd Application: Cantilever under vertical load-Thickness Optimization: Objective function: (compliance)‏ Applications of FreeFem++ on Structural Optimization

  7. Applications of FreeFem++ - Compliance minimization 3rd Application: Cantilever under vertical load-Multiple-loads Optimization: Objective function: (compliance)‏ Applications of FreeFem++ on Structural Optimization

  8. Applications of FreeFem++ - Compliance minimization 4th Application: Cantilever under vertical load-Geometric Optimization: Objective function: Applications of FreeFem++ on Structural Optimization

  9. Applications of FreeFem++ - Compliance minimization Applications of FreeFem++ on Structural Optimization

  10. Applications of FreeFem++ - Compliance minimization 5th Application: Cantilever under vertical load-Coupled Method-(G.O.+P.O.): Objective function: Applications of FreeFem++ on Structural Optimization

  11. Number of Iterations Volume Total Reduction Initialization 25.069 - 5 5.51033 780.2% 20 3.74527 850.6% 100 3.74179 850.7% Applications of FreeFem++ - Desired mechanical behaviour 6th Application: Cantilever under vertical load-Geometric Optimization: Objective function: Applications of FreeFem++ on Structural Optimization

  12. Applications of FreeFem++ - Stress minimization 7th Application: L-shaped structure: Model: Objective function: Norm of a tensor: Applications of FreeFem++ on Structural Optimization

  13. Applications of FreeFem++ L6 norm-Thickness Optimization: Applications of FreeFem++ on Structural Optimization

  14. Applications of FreeFem++ L10 norm-Thickness Optimization: Applications of FreeFem++ on Structural Optimization

  15. Applications of FreeFem++ L6 norm-Geometric Optimization: Applications of FreeFem++ on Structural Optimization

  16. Applications of FreeFem++ L10 norm-Geometric Optimization: Applications of FreeFem++ on Structural Optimization

  17. 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

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