Multi-Objective Design Optimization of the Reinforced Composite Roof in a Solar Vehicle
<p>The multi-occupant solar vehicle: (<b>a</b>) general layout; (<b>b</b>) roof support structure.</p> "> Figure 2
<p>Shape definition: four geometries were considered: (<b>a</b>) rounded rectangles and (<b>b</b>) ellipses holes, or (<b>c</b>) rectangular and (<b>d</b>) triangles in quadridirectional grids.</p> "> Figure 3
<p>Different scales of analysis: basic unit (500 × 500 m), experimental mock-up (2500 × 800 mm) and half (front) section of the roof (~5200 × 1.600 mm).</p> "> Figure 4
<p>Details in sandwich discretization (in the case of rectangular grid): (<b>a</b>) the 500 × 500 mm base section; (<b>b</b>) vertical beams over respect to the horizontal ones; (<b>c</b>) sides where the thickness of the sandwich must decrease; (<b>d</b>) final mesh (also showing the thickness changes).</p> "> Figure 5
<p>Numerical evaluation of (<b>a</b>) resonance frequency, (<b>b</b>) flexural and (<b>c</b>) torsional stiffness.</p> "> Figure 6
<p>Modal analysis with results from (<b>a</b>) experiment and (<b>b</b>) simulation.</p> "> Figure 7
<p>Numerical modal analysis of (<b>a</b>) front, (<b>b</b>) back section of the roof.</p> "> Figure 8
<p>Solar roof manufacturing and installation on the solar vehicle.</p> ">
Abstract
:Featured Application
Abstract
1. Introduction
- The optimization problem aims to find the optimal solution from all feasible solutions,
- Where variables can be continuous or discrete, creating a divergent solution space;
- It assumes that, thanks to classification mechanisms, it is possible to find the optimal solution without checking all the possible combinations, one by one;
- The optimization criteria, at the same time, must be expressible in terms of functions of different variables, including relative minimum and maximum limits;
- It is usually advisable to standardize and make comparable these outputs by factoring them (e.g., by max value) in the way to provide unitary indexes (scores between 0 and 1);
- In the simplest case the problem is scaled down in maximizing (or minimizing) a real function by choosing the input values from a set allowed and calculating that function.
2. Materials and Methods
2.1. Composite Materials and Structures
2.2. Optimization Parameters
2.3. Overall Methodology
- Preliminary shape definition when different geometrical shapes were considered, specifically rectangles, ellipsis and triangles, in terms of their ability to minimize the surface.
- Manufacturing techniques taken into consideration since the very beginning, identifying two practical solutions in production and four geometrical options (Figure 2):
- -
- A composite laminate made in a single piece where rectangular holes, with rounded edges (a), or elliptical ones (b) where shaped through;
- -
- A grid-based structure, made up of a series of intersecting straight (vertical, horizontal and angular) lines (grid lines), forming a rectangular (c) or triangular (d) texture of beams.
- Geometric shape (topology) optimization performed by automatic algorithms using screening and response surface method respect to the most relevant geometric parameters (e.g., axes for ellipsis, lengths for rectangle, distances between grids (as reported in Table 4)).
- Score criteria, as flexural and torsional stiffness or first resonance frequency, derived by finite element analyses (FEA) using commercial codes in static and modal simulations.
- Valuations performed, at first, respect to a basic unit, dimensionally set to a 500 × 500 mm section (roughly equivalent to 4 × 4 solar cells).
- An overall structure for the roof made up by a repetition of this basic unit.
- The basic unit solution, as here optimized, adopted to produce a larger section, a 2500 × 800 mm (~2 m2) flat mock-up, to be used for experimental (modal) test and numerical model validation.
- This design solution also applied to the roof shape in accordance with its real double curvature and the 3D model (~5200 × 1.600 mm) then analyzed by FEA respect to static and dynamic loads.
- The roof definite structure manufactured by autoclave composite techniques, installed on the solar car and finally examined in real operative conditions.
2.4. Optimization Space Reducing
- -
- Decoupling the effects related to the geometric characteristics of the basic shapes (i.e., rectangle, triangle, etc.) from those related to their recurrence and rearrangement. It was obtained repeating the optimization procedure acting on different levels. Specifically, in Figure 3 these different scales of analysis are displayed, showing the basic unit (500 × 500 m), used for geometrical optimization of shapes (Figure 2), and the mock-up (2500 × 800 mm), used for grid optimization, experiments and model validation. It should be noted that the figure shows half (front) section of the vehicle roof (with ~5200 × 1.600 mm as overall dimension);
- -
- Limiting the (full) parametric (and automated) analysis to an optimization only based on the structural outputs as first; the most promising solutions from this topologic optimization are then (individually) verified in terms of impacts on the performance outputs.
2.5. Design and Simulation Tools
2.6. Simulation Procedures
2.6.1. Resonance Frequency
2.6.2. Flexural Stiffness
2.6.3. Torsional Stiffness
3. Results and Discussion
3.1. Shape Topological Optimisation
- -
- All beams in each specific grid have the same dimensions (i.e., width and thickness);
- -
- These dimensions can vary (in general) between the upper and lower grids;
- -
- The thickness, however, is fixed by the specific composite layout;
- -
- The width of beams in the upper grid is fixed (with the scope to permit to sustain the solar cells);
- -
- The width of beams in the lower grid is related to their number (since a constant weight).
3.2. Grid Topological Optimisation
3.3. Multi-Objective Optimization Results at a Glace
- -
- Minor free surface for heat transfer, not permitting an efficient the solar cells cooling;
- -
- Greater complexity in fabrication related to a larger number of basic elements (beams).
3.4. Validation
3.5. Results Implementation
3.6. Further Considerations and Novelty
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
Basic Unit | Parameter | Unit | Range | Type | |
Overall Dimension | mm | 500 × 500 | Fixed | ||
Thickness | mm | 5.2 | Fixed | ||
Holes distance | mm | 125 | Fixed | ||
Hole Shape | Parameter | Unit | Range | Type | Simulations |
Rectangular | Width (W) | mm | 50–95 | Modified | 15 |
Height (H) | mm | 50–95 | Modified | ||
Fillet radius (r) | mm | 1–24 | Modified | (Table 5) | |
Elliptical | 1st axis (A) | mm | 125 | Fixed | 9 |
2nd axis (B) | mm | 50–95 | Modified | (Table 5) | |
Rectangular grid | 1st Width (a) | mm | 30–50 | Modified | 5 |
Quadridirectional | 1st Width (a) | mm | 30–40 | Modified | 11 |
Grid | 2nd Width (b) | mm | 14, 23.5, 25–30 | Modified | |
Crosses | 1, 1 ½ | Modified | |||
Spacing | Mm | 250, 500 | Modified | (Table 5 and Table 6) | |
Quadridirectional | 1st Width (a) | mm | 30 | Fixed | 9 |
Grid | 2nd Width (b) | mm | 10.5, 20, 23.5, 28, 30 | Modified | |
Number of crosses | 1, 1 ⅓, 1 ½, 2, 4 | Modified | (Table 5 and Table 6) |
Mock-up | Parameter | Unit | Range | Type | |
Overall Dimension | mm | 2500 × 800 | Fixed | ||
Thickness | mm | 5.2 | Fixed | ||
Holes distance | mm | 125 | Fixed | ||
Hole Shape | Parameter | Unit | Range | Type | Simulations |
Laminate | Width (W) | mm | 95 | Fixed | 1 |
Rectangle | Height (H) | mm | 93 | Fixed | |
Fillet radius (r) | mm | 23.5 | Fixed | (Table 7 and Table 8) | |
Quadridirectional | 1st Width (a) | mm | 30 | Fixed | 3 |
grid | 2nd Width (b) | mm | 14, 23.5, 30 | Modified | |
Crosses Number | -- | 1, 1 ½ | Modified | ||
Spacing | mm | 250, 500 | Modified | (Table 7 and Table 8) | |
Quadridirectional | 1st Width (a) | mm | 30–35 | Modified | 47 |
grid | 2nd Width (b) | mm | 14, 22–52.5 | Modified | |
Spacing | mm | 250, 375, 500, 625, 750, 875, 1000 | Modified | (Table 8) |
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Property | Unit | T1000 | T800 | PVC |
---|---|---|---|---|
Type | - | Unidirectional | Twill | Foam |
Density | Kg/m3 | 1490 | 1420 | 100 |
Young’s Modulus | MPa | 121,000, 8600, 8600 | 61,340, 61,340, 6900 | 125 |
Poisson’s Ratio | - | 0.27, 0.4, 0.27 | 0.04, 0.3, 0.3 | 0.40 |
Tensile Stress Limit | Mpa | 2231, 29, 29 | 805, 805, 50 | 2.5 |
Layer | Material | Thickness | Angle |
---|---|---|---|
1 | T800 | 0.30 mm | 0°/90° |
2 | T1000 | 0.15 mm | 0° |
3 | T1000 | 0.15 mm | 0° |
4 | PVC | 4.0 mm | -- |
5 | T1000 | 0.15 mm | 0° |
6 | T1000 | 0.15 mm | 0° |
7 | T800 | 0.30 mm | 0°/90° |
Features | Output | Unit | Target | Weighting |
---|---|---|---|---|
Mechanical | Flexural stiffness | N/mm | Highest | 0.15 |
Torsional stiffness | N·mm/rad | Highest | 0.15 | |
Resonance frequency | Hz | Highest | 0.30 | |
Functional | Heat transfer surface | mm2 | Highest | 0.40 |
Weight | Kg | Lowest | (fixed) |
Rectangular Hole | Elliptical Hole | Rectangular Grid | Quadridirectional Grid |
---|---|---|---|
Width (W) | 1st axis (A) | 1st Width (a) | 1st Width (a) |
Height (H) | 2nd axis (B) | 2nd Width (b) | 2nd Width (b) |
Fillet radius (r) | Angle (α) | ||
Configuration | Laminate | Grid | |||||
---|---|---|---|---|---|---|---|
Rectangle | Ellipsis | Orthogrid | Cross | Double cross | Quadridirection | ||
Hole Size (mm) | 94.7 | 95.0 | |||||
92.8 | 95.0 | ||||||
23.7 | |||||||
Grid Width Size | Upper grid (mm) | 40.7 | 30.0 | 30.0 | 30.0 | ||
Lower grid (mm) | 23.5 | 14 | |||||
Lowest Frequency (Hz) | 84.2 | 73.6 | 64.3 | 88.9 | 97.3 | 111.9 | |
Flexural stiffness (N/mm) | 209 | 103 | 204 | 137 | 114 | 122 | |
Torsional stiffness (N mm/rad) | 5247 | 3193 | 4391 | 5170 | 5319 | 6623 | |
Element Weight (kg) | 0.263 | 0.263 | 0.264 | 0.263 | 0.265 | 0.264 | |
Performance Index | 0.61 | 0.10 | 0.32 | 0.48 | 0.53 | 0.79 |
Configuration | Number of Crosses | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | 1 ⅓ | 1 ⅓ | 1 ½ | 1 ½ | 2 | 2 | 4 | 4 | ||
Width Size | Upper grid (mm) | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 |
Lower grid (mm) | 30 | 28 | 28 | 23.5 | 23.5 | 20 | 20 | 10.5 | 10.5 | |
Load Direction | x or y | x | y | x | y | x | y | x | y | |
Lowest Frequency (Hz) | 88.9 | 83.5 | 83.6 | 97.3 | 89.8 | 84.7 | 84.9 | 62.7 | 63.9 | |
Flexural stiffness (N/mm) | 137.1 | 127.9 | 121.4 | 114.0 | 116.2 | 169.8 | 113.9 | 194.4 | 115.4 | |
Torsional stiffness (N mm/rad) | 5170 | 4676 | 4248 | 5319 | 4604 | 4786 | 4459 | 3675 | 3219 | |
Element Weight (kg) | 0.263 | 0.266 | 0.266 | 0.265 | 0.265 | 0.266 | 0.266 | 0.266 | 0.266 | |
Performance Index | 0.68 | 0.52 | 0.45 | 0.75 | 0.56 | 0.68 | 0.47 | 0.30 | 0.02 |
Configuration | Laminate | Quadridirectional Grid | ||
---|---|---|---|---|
Rectangle | Low | Medium | High | |
Lowest Frequency (Hz) | 11.4 | 12.3 | 12.9 | 13.8 |
Flexural stiffness (N/mm) | 7.7 | 4.4 | 3.5 | 3.6 |
Torsional stiffness (N mm/rad) | 1851 | 1960 | 2090 | 2665 |
Element Weight (kg) | 2.090 | 2.047 | 2.031 | 2.028 |
Performance Index | 0.25 | 0.27 | 0.38 | 0.76 |
Configuration | Quadridirectional Grid | |||||||
---|---|---|---|---|---|---|---|---|
Crosses | 2 × 1 | 2 ⅓ × 1 ⅓ | 2⅔ × 1⅓ | 3½ × 1¾ | 4 × 2 | 5¼ × 2¼ | 8 × 4 | |
Spacing (mm) | 1000 × 1000 | 875 × 875 | 750 × 750 | 625 × 625 | 500 × 500 | 375 × 375 | 250 × 250 | |
Width Size | Upper grid (mm) | 30 | 30 | 30 | 30 | 30 | 30 | 30 |
Lower grid (mm) | 50 | 42 | 38 | 33 | 30 | 22 | 14 | |
Lowest Frequency (Hz) | 14.5 | 13.5 | 14.7 | 14.0 | 12.9 | 15.9 | 13.8 | |
Flexural stiffness (N/mm) | 3.4 | 4.5 | 4.3 | 4.3 | 3.5 | 3.7 | 3.6 | |
Torsional stiffness (N mm/rad) | 2587 | 2252 | 2379 | 2358 | 2090 | 2156 | 2665 | |
Element Weight (kg) | 2.031 | 2.031 | 2.031 | 2.031 | 2.031 | 2.033 | 2.028 | |
Performance Index | 0.49 | 0.42 | 0.64 | 0.50 | 0.03 | 0.60 | 0.46 |
Configuration | Laminate | Quadridirectional Grid | |||||
---|---|---|---|---|---|---|---|
Weight | Target | Best | Rectangle | Low | Medium | High | |
Flexural stiffness (N/mm) | 0.25 | ← | 7.7 | 7.7 | 4.4 | 3.5 | 3.6 |
Torsional stiffness (N mm/rad) | 0.10 | ← | 2665 | 1851 | 1960 | 2090 | 2665 |
Max displacement (1st frequency) (mm) | 0.10 | ↓ | 61.6 | 61.6 | 63.0 | 62.6 | 70.4 |
Lowest Frequency (Hz) | 0.10 | ← | 13.8 | 11.4 | 12.3 | 12.9 | 13.8 |
Heat transfer area (m^2) | 0.15 | ← | 1.1 | 1.07 | 0.83 | 0.83 | 0.85 |
Producibility | 0.30 | ↓ | 1.0 | 0.75 | 0.80 | 1.00 | 0.50 |
Element Weight (kg) | --- | ↓ | 2.090 | 2.090 | 2.047 | 2.031 | 2.028 |
Performance Index | 1.00 | 0.57 | 0.50 | 0.39 | 0.57 | 0.37 |
Mode | Frequency (Hz) | Difference (%) | |
---|---|---|---|
Measure | Simulation | ||
I | 16.7 | 15.26 | −8.6% |
II | 20.1 | 18.69 | −7.0% |
III | 42.6 | 40.13 | −5.8% |
IV | 46.5 | 41.67 | −10.4% |
V | 68.6 | 65.40 | −4.7% |
Section | Dimensions [m] | Surface [m2] |
---|---|---|
Front | 1.714 × 2.344 × 0.142 | 4.567 |
Back | 1.540 × 2.056 × 0.510 | 3.522 |
Total | 1.714 × 2.344 × 0.510 | 8.089 |
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Pavlovic, A.; Sintoni, D.; Fragassa, C.; Minak, G. Multi-Objective Design Optimization of the Reinforced Composite Roof in a Solar Vehicle. Appl. Sci. 2020, 10, 2665. https://doi.org/10.3390/app10082665
Pavlovic A, Sintoni D, Fragassa C, Minak G. Multi-Objective Design Optimization of the Reinforced Composite Roof in a Solar Vehicle. Applied Sciences. 2020; 10(8):2665. https://doi.org/10.3390/app10082665
Chicago/Turabian StylePavlovic, Ana, Davide Sintoni, Cristiano Fragassa, and Giangiacomo Minak. 2020. "Multi-Objective Design Optimization of the Reinforced Composite Roof in a Solar Vehicle" Applied Sciences 10, no. 8: 2665. https://doi.org/10.3390/app10082665