170 likes | 353 Views
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
E N D
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
Contents • Optimization algorithm • Compliance minimization • Desired mechanical behaviour • Stress minimization • Conclusions Applications of FreeFem++ on Structural Optimization
Optimization algorithm • Gradient algorithm with constant step and projection J(un+1) < J(un) Applications of FreeFem++ on Structural Optimization
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
Applications of FreeFem++ - Compliance minimization 100 iterations • h0=0.5 • hmin=0.1 • hmax=1.0 Convergence diagram: Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization 2nd Application: Cantilever under vertical load-Thickness Optimization: Objective function: (compliance) Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization 3rd Application: Cantilever under vertical load-Multiple-loads Optimization: Objective function: (compliance) Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization 4th Application: Cantilever under vertical load-Geometric Optimization: Objective function: Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ - Compliance minimization Applications of FreeFem++ on Structural Optimization
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
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
Applications of FreeFem++ - Stress minimization 7th Application: L-shaped structure: Model: Objective function: Norm of a tensor: Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ L6 norm-Thickness Optimization: Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ L10 norm-Thickness Optimization: Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ L6 norm-Geometric Optimization: Applications of FreeFem++ on Structural Optimization
Applications of FreeFem++ L10 norm-Geometric Optimization: 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. Applications of FreeFem++ on Structural Optimization