1. Aristotle University of Thessaloniki (AUTH) Department of Civil Engineering Applications of FreeFem++ on Structural Optimization “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

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

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

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

5. Applications of FreeFem++ on Structural Optimization Applications of FreeFem++ - Compliance minimization 100 iterations Convergence diagram: h 0 =0.5 h min =0.1 h max =1.0

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

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

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

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

10. Applications of FreeFem++ on Structural Optimization Applications of FreeFem++ - Compliance minimization 5 th Application: Cantilever under vertical load-Coupled Method-(G.O.+P.O.): Objective function: Initial ComplianceG.O.G.O. + P.O. 556,046325,953255,582

11. Applications of FreeFem++ on Structural Optimization Applications of FreeFem++ - Desired mechanical behaviour 6 th Application: Cantilever under vertical load-Geometric Optimization: Number of IterationsVolumeTotal Reduction Initialization25.069- 55.51033780.2% 203.74527850.6% 1003.74179850.7% Objective function:

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

13. Applications of FreeFem++ on Structural Optimization Applications of FreeFem++ L 6 norm-Thickness Optimization:

14. Applications of FreeFem++ on Structural Optimization Applications of FreeFem++ L 10 norm-Thickness Optimization:

15. Applications of FreeFem++ on Structural Optimization Applications of FreeFem++ L 6 norm-Geometric Optimization:

16. Applications of FreeFem++ on Structural Optimization Applications of FreeFem++ L 10 norm-Geometric Optimization:

17. 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.