Fernando Fraternali

research

On the Structural Shape Optimization through Variational Methods and Evolutionary Algorithms
Fernando Fraternali, Andrea Marino, Tamer El Sayed, and Antonio Della Cioppa
We employ the variational theory of optimal control problems and evolutionary algorithms to investigate the form finding of minimum compliance elastic structures. Mathematical properties of ground structure approaches are discussed with reference to arbitrary collections of structural elements. A numerical procedure based on a Breeder Genetic Algorithm is proposed for the shape optimization of discrete structural models. Several numerical applications are presented, showing the ability of the adopted search strategy in avoiding local optimal solutions. The proposed approach is validated against a parade of results available in the literature.
 
   
Figure 1.1: Truss structure with an adoptive node
Figure 1.1: Truss structure with an adoptive node
 
Figure 1.2: (Color online) Strain energy at equilibrium vs vertical position of the loaded node for the structure of Fig. 1.1
Figure 1.2: (Color online) Strain energy at equilibrium vs vertical position of the loaded node for the structure of Fig. 1.1
 
Figure 1.3: (Color online) Local min-max compliance configurations of the structure of Fig. 1.1 (cf. Fig. 1.2)
Figure 1.3: (Color online) Local min-max compliance configurations of the structure of Fig. 1.1 (cf. Fig. 1.2)
 
Figure 4.1: BGA optimization of the truss structure in Fig. 1.1. The horizontal axis shows the generation number and the vertical axis shows the corresponding best fitness.
Figure 4.1: BGA optimization of the truss structure in Fig. 1.1. The horizontal axis shows the generation number and the vertical axis shows the corresponding best fitness.
 
Figure 4.2: (Color online) Physical and design constraints for the optimal design of a truss structure.
Figure 4.2: (Color online) Physical and design constraints for the optimal design of a truss structure.
 
Figure 4.3: BGA optimization of the truss in Fig. 4.2. The horizontal axis shows the generation number and the vertical axis shows the corresponding normalized best fitness J/FL. The best shapes are shown below the corresponding plots.
Figure 4.3: BGA optimization of the truss in Fig. 4.2. The horizontal axis shows the generation number and the vertical axis shows the corresponding normalized best fitness J/FL. The best shapes are shown below the corresponding plots.
 
Figure 4.4: BGA optimization of the truss in Fig. 4.2. The horizontal axis shows the generation number and the vertical axis shows the corresponding normalized best fitness J/FL. The best shapes are shown below the corresponding plots.
Figure 4.4: BGA optimization of the truss in Fig. 4.2. The horizontal axis shows the generation number and the vertical axis shows the corresponding normalized best fitness J/FL. The best shapes are shown below the corresponding plots.
 
Figure 4.5: (Color online) Final best shape of the truss in Fig. 4.2 obtained through the BGA optimization (solid line) in comparison with the optimal shape found by Brain in [12] (dashed line). The horizontal axis shows the generation number and the vertical axis shows the corresponding normalized best fitness J/FL.
Figure 4.5: (Color online) Final best shape of the truss in Fig. 4.2 obtained through the BGA optimization (solid line) in comparison with the optimal shape found by Brain in [12] (dashed line). The horizontal axis shows the generation number and the vertical axis shows the corresponding normalized best fitness J/FL.
 
Figure 4.6: (Color online) Physical and design constraints for the optimal shape problem of a cable network.
Figure 4.6: (Color online) Physical and design constraints for the optimal shape problem of a cable network.
 
Figure 4.7: BGA optimization of the structure in Fig. 4.6: Best shape in correspondence with generation # 0. The horizontal axis shows the generation number and the vertical axis shows the corresponding best fitness J (daNcm). The best shape is shown within the plot.
Figure 4.7: BGA optimization of the structure in Fig. 4.6: Best shape in correspondence with generation # 0. The horizontal axis shows the generation number and the vertical axis shows the corresponding best fitness J (daNcm). The best shape is shown within the plot.
 
Figure 4.8: BGA optimization of the structure in Fig. 4.6: Best shape in correspondence with generation # 2. The horizontal axis shows the generation number and the vertical axis shows the corresponding best fitness J (daNcm). The best shape is shown within the plot.
Figure 4.8: BGA optimization of the structure in Fig. 4.6: Best shape in correspondence with generation # 2. The horizontal axis shows the generation number and the vertical axis shows the corresponding best fitness J (daNcm). The best shape is shown within the plot.
 
Figure 4.9: BGA optimization of the structure in Fig. 4.6: Best shape in correspondence with generation # 3. The horizontal axis shows the generation number and the vertical axis shows the corresponding best fitness J (daNcm). The best shape is shown within the plot.
Figure 4.9: BGA optimization of the structure in Fig. 4.6: Best shape in correspondence with generation # 3. The horizontal axis shows the generation number and the vertical axis shows the corresponding best fitness J (daNcm). The best shape is shown within the plot.
 
Figure 4.10: BGA optimization of the structure in Fig. 4.6: Best shape in correspondence with generation # 100. The horizontal axis shows the generation number and the vertical axis shows the corresponding best fitness J (daNcm). The best shape is shown within the plot.
Figure 4.10: BGA optimization of the structure in Fig. 4.6: Best shape in correspondence with generation # 100. The horizontal axis shows the generation number and the vertical axis shows the corresponding best fitness J (daNcm). The best shape is shown within the plot.
 
Figure 4.11: (Color online) Funicular polygon corresponding to the minimum compliance shape of the structure in Fig. 4.6.
Figure 4.11: (Color online) Funicular polygon corresponding to the minimum compliance shape of the structure in Fig. 4.6.
 
Figure 4.12: Examined dome model.
Figure 4.12: Examined dome model.
 
Figure 4.13: Compliance optimization of a spherical dome.
Figure 4.13: Compliance optimization of a spherical dome.
 
Figure 4.14: Compliance optimization of a baroque dome.
Figure 4.14: Compliance optimization of a baroque dome.
 
Figure 4.15: Compliance optimization of a gothic dome.
Figure 4.15: Compliance optimization of a gothic dome.
 
Figure 4.16: (Color online, adapted from [55]) BGA evolution of the funicular profile of St. Peter’s cupola: Generations 0 and 1 (red/light solid line: Poleni’s funicular polygon; dark/marked solid line: current best BGA shape).
Figure 4.16: (Color online, adapted from [55]) BGA evolution of the funicular profile of St. Peter’s cupola: Generations 0 and 1 (red/light solid line: Poleni’s funicular polygon; dark/marked solid line: current best BGA shape).
 
Figure 4.17: (Color online, adapted from [55]) BGA evolution of the funicular profile of St. Peter’s cupola: Generations 4 and 8 (red/light solid line: Poleni’s funicular polygon; dark/marked solid line: current best BGA shape).
Figure 4.17: (Color online, adapted from [55]) BGA evolution of the funicular profile of St. Peter’s cupola: Generations 4 and 8 (red/light solid line: Poleni’s funicular polygon; dark/marked solid line: current best BGA shape).
 
Figure 4.18: (Color online, adapted from [55]) BGA evolution of the funicular profile of St. Peter’s cupola: Generations 14 and 115 (red/light solid line: Poleni’s funicular polygon; dark/ marked solid line: current best BGA shape).
Figure 4.18: (Color online, adapted from [55]) BGA evolution of the funicular profile of St. Peter’s cupola: Generations 14 and 115 (red/light solid line: Poleni’s funicular polygon; dark/ marked solid line: current best BGA shape).
 
Figure 4.19: 3D finite element model of S. Peter’s cupola (1237 shell elements and 1297 nodes).
Figure 4.19: 3D finite element model of S. Peter’s cupola (1237 shell elements and 1297 nodes).
 
Figure 4.20: BGA evolution of a 3D finite element model of St. Peter’s cupola: Generations 0, 10, 15 and 20.
Figure 4.20: BGA evolution of a 3D finite element model of St. Peter’s cupola: Generations 0, 10, 15 and 20.
 
Figure 4.21: BGA evolution of a 3D finite element model of St. Peter’s cupola: Generations 25, 100, 150 and 250.
Figure 4.21: BGA evolution of a 3D finite element model of St. Peter’s cupola: Generations 25, 100, 150 and 250.
 
Figure 4.22: Comparison between the actual shape (left, U = 1.911 Umin) Nmm) and the virtualminimumcompliance shape (right, U = Umin, generation # 300) of St. Peter’s cupola.
Figure 4.22: Comparison between the actual shape (left, U = 1.911 Umin) Nmm) and the virtualminimumcompliance shape (right, U = Umin, generation # 300) of St. Peter’s cupola.
 
 
 

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keywords: Shape Optimization, Optimal Control, Ground Structures, Compliance Optimization, Evolutionary Algorithms.