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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: 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? Topology Optimisation Load Cases

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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 Results for 3 load cases Topology Optimisation Model and Results Bending Torsion Transverse bending

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Topology Optimisation Results Iso view: optimization of the barrel for weighted compliance

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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 Topology Optimisation Presence of window openings

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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 L pq is the distance between the points p and q (pq) in the system. 11 Example: A 100-point DOE generated by an optimal Latin hypercube technique

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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: 12 where Z1, Z2, …, Z7 are the design variables. Indications of the quality of fit of the obtained expression into the data:

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

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x y z QzQz Optimisation of the Fuselage Barrel Composite skin and stiffeners 14 An upward gust load case at low altitude and cruise speed Undisturbed anisogrid fuselage barrel Early design stage

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Variables and Constraints Design variables Lower boundUpper bound Skin thickness (h) 0.6 (mm)4.0 (mm) Number of helix rib pairs around the circumference, (n) 50150 Helix rib thickness, (t h ) 0.6 (mm)3.0 (mm) Helix rib height, (H h ) 15.0 (mm)30.0 (mm) Frame pitch, (d) 500.0 (mm)650.0 (mm) Frame thickness, (t f ) 1.0 (mm)4.0 (mm) Frame height, (H f ) 50.0 (mm)150.0 (mm) 15 HfHf tftf W f =20mm HhHh W h =20mm d h =8mm thth Circumferential Frames Helix Ribs Frame Pitch, d Circumf. Helix Rib Pitch, dep. on n 2φ Fuselage Geometry Radius 2m h Barrel Cross Section 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 Variables:

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Results: Summary of parametric optimisation 16 Model Tensile Strain (MS) Compressive Strain (MS) Shear Strain (MS) Buckling (MS) Torsional Stiffness (MS) Bending Stiffness (MS) Normalized mass Prediction I0.020.001.42 --- 0.10 Optimum I0.36-0.091.21 --- 0.11 Prediction II0.030.011.64 --- 0.11 Optimum II0.540.041.54 --- 0.12 Prediction III0.200.231.270.001.210.890.29 Optimum III0.620.081.09-0.071.210.890.29 Comp. Des.1.150.191.31-0.041.250.810.29 Design Skin thickness (h), mm Nr. of helix rib pairs, (n) Helix rib thickness, (t h ), mm Helix rib height, (H h ), mm Frame pitch, (d), mm Frame thickness, (t f ), mm Frame height, (H f ), mm Optimum I2.0860.000.6027.90627.701.0050.00 Optimum II2.2860.000.6627.90627.701.0050.00 Optimum III1.71150.000.6127.80501.701.0050.00 Strength Contraint Stability, Strength, and Stiffness Contraints Optimum III geometry with realistic ply layup: Helical ribs: tall and slender Frames: thin and small 209 mm 628 mm 18.94 ° Optimum II 84 mm 502 mm 9.55 ° Optimum III and Comp. Design (±45,0,45,0,-45,90) s, 14 plies, total thickness = 1.75 mm

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Results: Interpretation of the skin as a laminate, 14 plies 17 Stacking sequence Buckling (MS) Torsional Stiffness Bending Stiffness Normalized mass (±45,0,45,0,-45,90) s -0.041.250.810.29 (±45,0,45,90,-45,0) s 0.041.250.810.29 (±45,90,45,0,-45,0) s 0.131.250.810.29 % of 0° plies% of +/-45° plies% of 90° plies 28.6%57.1%14.3%

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Results: Interpretation of the skin as a laminate, 15 plies 18 Stacking sequence Buckling (MS) Torsional Stiffness Bending Stiffness Normalized mass (±45,0,45,0,-45,90) s,00.121.260.920.30 (±45,0,45,90,-45,0) s,00.201.260.920.30 (±45,90,45,0,-45,0) s,00.281.260.920.30 % of 0° plies% of +/-45° plies% of 90° plies 33.3%53.3%13.3%

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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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20 Thank You for your Attention

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