Aristotle University of Thessaloniki AUTH Department of Civil

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

Contents Optimization algorithm Compliance minimization Desired mechanical behaviour Stress minimization Conclusions Applications of Free. Fem++ on Structural Optimization

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

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

Applications of Free. Fem++ - Compliance minimization 100 iterations • h 0=0. 5 • hmin=0. 1 • hmax=1. 0 Convergence diagram: Applications of Free. Fem++ on Structural Optimization

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

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

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

Applications of Free. Fem++ - Compliance minimization Applications of Free. Fem++ on Structural Optimization

Applications of Free. Fem++ - Compliance minimization 5 th Application: Cantilever under vertical load-Coupled Method-(G. O. +P. O. ): Objective function: Initial Compliance 556, 046 G. O. 325, 953 G. O. + P. O. 255, 582 Applications of Free. Fem++ on Structural Optimization

Applications of Free. Fem++ - Desired mechanical behaviour 6 th Application: Cantilever under vertical load-Geometric Optimization: Objective function: Number of Iterations Initialization 5 20 100 Volume 25. 069 5. 51033 3. 74527 3. 74179 Total Reduction 780. 2% 850. 6% 850. 7% Applications of Free. Fem++ on Structural Optimization

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

Applications of Free. Fem++ L 6 norm-Thickness Optimization: Applications of Free. Fem++ on Structural Optimization

Applications of Free. Fem++ L 10 norm-Thickness Optimization: Applications of Free. Fem++ on Structural Optimization

Applications of Free. Fem++ L 6 norm-Geometric Optimization: Applications of Free. Fem++ on Structural Optimization

Applications of Free. Fem++ L 10 norm-Geometric Optimization: Applications of Free. Fem++ 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. Free. Fem++ is a very efficient software to apply optimization algorithms. However, the user of Free. Fem++ 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 Free. Fem++ on Structural Optimization