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**Topology and Parametric Optimisation of a Lattice Composite Fuselage Structure**

Dianzi Liu, Vassili V. Toropov, Osvaldo M. Querin University of Leeds

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Content Introduction Topology Optimisation Parametric Optimisation Conclusion

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**Topology Optimisation Method**

Topology Optimisation is a computational means of determining the physical domain for a structure subject to applied loads and constraints. The method used in this research is the Solid Isotropic Material with Penalization (SIMP). It works by minimising the compliance (maximising global stiffness) of the structure by solving the following optimization problem: for a single load case, or by minimising the weighted compliance for multiple (N) load cases:

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**Topology Optimisation Load Cases**

Topology Optimisation: minimizing the compliance of the structure for 3 load cases Load cases consist of distributed loads over the length and loads at the barrel end (shear forces, bending moments and torque) Question: what are the appropriate weight coefficient values?

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**Topology Optimisation Method for weight allocation**

The following strategy was used: Do topology optimization separately for each load case, obtain the corresponding compliance values Allocate the weights to the individual compliance components (that correspond to the individual load cases) in the same proportion The logic behind this is as follows: if for a particular load case topology optimization produced a relatively high compliance value, then this load case is a critical one and hence it should be taken with a higher weight in the total weighted compliance optimization problem

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**Topology Optimisation Model and Results**

Topology Optimisation Results for 3 load cases Bending Transverse bending Torsion

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**Topology Optimisation Results**

Iso view: optimization of the barrel for weighted compliance

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**Topology Optimisation Presence of window openings**

Optimization of the barrel without windows (Top) and with windows (Bottom) Two backbones on top and bottom of the barrel Nearly +-45° stiffening on the side panel Result: beam structure for the barrel Note: SIMP approach does not consider buckling

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**Development of the Design Concept by DLR**

Reflection on the layout of the “ideal” structure from the topology optimization it in the aircraft design context Consideration of airworthiness and manufacturing requirements Fuselage design concept developed by DLR High potential for weight savings achievable due to new material for stiffeners and non-rectangular skin bays Due to large number of parameters in the obtained concept a multi-variable optimisation should be performed

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**Multi-parametric Optimisation**

Method: the multi-parameter global approximation-based approach used to solve the optimization problem Problem: optimize an anisogrid composite fuselage barrel with respect to weight and stability, strength, and stiffness using 7 geometric design variables, one of which is an integer variable. Procedure: develop a set of numerical experiments (FEA runs) where each corresponds to a different combinations of the design variables. The concept of a uniform Latin hypercube Design of Experiments (DOE) with 101 experiments (points in the variable space) was used. FE analysis of these 101 fuselage geometries was performed global approximations built as explicit expressions of the design variables using Genetic Programming (GP) parametric optimisation of the fuselage barrel by a Genetic Algorithm (GA) verification of the optimal solution by FE simulation 10

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Design of Experiments In order to generate the sampling points for approximation building, a uniform DOE (optimal Latin hypercube design) is proposed. The main principles in this approach are as follows: The number of levels of factors (same for each factor) is equal to the number of experiments and for each level there is only one experiment; The points of experiments are distributed as uniformly as possible in the domain of factors, which are achieved by minimizing the equation: where Lpq is the distance between the points p and q (p≠q) in the system. Example: A 100-point DOE generated by an optimal Latin hypercube technique 1. Why a 101 DOE design? The size of DOE (the number of FE runs) was selected as 101, this coincides with the range for the number of helical ribs. 2. Each design variable was discretised into 101 equally spaced values in the range. 11

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Genetic Programming Genetic Programming (GP) is a symbolic regression technique, it produces an analytical expression that provides the best fit of the approximation into the data from the FE runs. Example: a approximation for the shear strain obtained from the 101 FE responses: where Z1, Z2, …, Z7 are the design variables. Indications of the quality of fit of the obtained expression into the data: 12

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**FEM Modeling and Simulation**

Automated Multiparametric Global Barrel FEA Tool: Modeling, Analysis, and Result Summary User Defined Parameters: -Geometry -Loads -Materials -Mesh seed Session file: List of Models to be Analyzed Modeling and Analysis PCL Function Post-processing MSC Patran MSC Nastran PCL Displacement Skin Strains Beam Strains Buckling Results: Results of all analyzed models are summarized in a separate file 13

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**Optimisation of the Fuselage Barrel**

Undisturbed anisogrid fuselage barrel Early design stage An upward gust load case at low altitude and cruise speed x y z Qz Composite skin and stiffeners 14

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**Variables and Constraints**

Frame Pitch, d Circumf. Helix Rib Pitch, dep. on n 2φ Fuselage Geometry Variables: Design variables Lower bound Upper bound Skin thickness (h) 0.6 (mm) 4.0 (mm) Number of helix rib pairs around the circumference, (n) 50 150 Helix rib thickness, (th) 3.0 (mm) Helix rib height, (Hh) 15.0 (mm) 30.0 (mm) Frame pitch, (d) 500.0 (mm) 650.0 (mm) Frame thickness, (tf) 1.0 (mm) Frame height, (Hf) 50.0 (mm) 150.0 (mm) Radius 2m h Barrel Cross Section Hf tf Wf =20mm Hh Wh=20mm dh=8mm th Circumferential Frames Helix Ribs Constraints: Strength: strains in the skin and in the stiffeners Stiffness: bending and torsional stiffness Stability: buckling Normalization Normalized mass against largest mass Margin of safety ≥0 Strain Stiffness Buckling 15

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**Results: Summary of parametric optimisation**

Model Tensile Strain (MS) Compressive Strain (MS) Shear Strain (MS) Buckling (MS) Torsional Stiffness (MS) Bending Stiffness (MS) Normalized mass Prediction I 0.02 0.00 1.42 --- 0.10 Optimum I 0.36 -0.09 1.21 0.11 Prediction II 0.03 0.01 1.64 Optimum II 0.54 0.04 1.54 0.12 Prediction III 0.20 0.23 1.27 0.89 0.29 Optimum III 0.62 0.08 1.09 -0.07 Comp. Des. 1.15 0.19 1.31 -0.04 1.25 0.81 Strength Contraint Stability, Strength, and Stiffness Contraints Optimum III geometry with realistic ply layup: (±45,0,45,0,-45,90)s, 14 plies, total thickness = 1.75 mm Design Skin thickness (h), mm Nr. of helix rib pairs, (n) Helix rib thickness, (th), mm Helix rib height, (Hh), mm Frame pitch, (d), mm Frame thickness, (tf), mm Frame height, (Hf), mm Optimum I 2.08 60.00 0.60 27.90 627.70 1.00 50.00 Optimum II 2.28 0.66 Optimum III 1.71 150.00 0.61 27.80 501.70 209 mm 628 mm 18.94 ° Optimum II 84 mm 502 mm 9.55 ° Optimum III and Comp. Design Helical ribs: tall and slender Frames: thin and small 16

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**Results: Interpretation of the skin as a laminate, 14 plies**

Stacking sequence Buckling (MS) Torsional Stiffness Bending Stiffness Normalized mass (±45,0,45,0,-45,90)s -0.04 1.25 0.81 0.29 (±45,0,45,90,-45,0)s 0.04 (±45,90,45,0,-45,0)s 0.13 % of 0° plies % of +/-45° plies % of 90° plies 28.6% 57.1% 14.3% 17

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**Results: Interpretation of the skin as a laminate, 15 plies**

Stacking sequence Buckling (MS) Torsional Stiffness Bending Stiffness Normalized mass (±45,0,45,0,-45,90)s ,0 0.12 1.26 0.92 0.30 (±45,0,45,90,-45,0)s ,0 0.20 (±45,90,45,0,-45,0)s ,0 0.28 % of 0° plies % of +/-45° plies % of 90° plies 33.3% 53.3% 13.3% 18

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Conclusion Multi-parameter global metamodel-based optimization was used for: Optimization of a composite anisogrid fuselage barrel with respect to weight, stability, strength, stiffness using 7 design variables, 1 being an integer 101-point uniform design of numerical experiments, i.e. 101 designs analysed Automated Multiparametric Global Barrel FEA Tool generates responses global approximations built using Genetic Programming (GP) parametric optimization on global approximations optimal solution verified via FE Overall, the use of the global metamodel-based approach has allowed to solve this optimization problem with reasonable accuracy as well as provided information on the structural behavior of the anisogrid design of a composite fuselage. There is a good correspondence of the obtained results with the analytical estimates of DLR, e.g. the angle of the optimised triangular grid cell is 9.55° whereas the DLR value is 12° 19

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