Topology and Parametric Optimisation of a Lattice Composite

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Topology and Parametric Optimisation of a Lattice Composite Fuselage Structure Dianzi Liu, Vassili V.

Topology and Parametric Optimisation of a Lattice Composite Fuselage Structure Dianzi Liu, Vassili V. Toropov, Osvaldo M. Querin University of Leeds

Content Introduction Topology Optimisation Parametric Optimisation Conclusion

Content Introduction Topology Optimisation Parametric Optimisation Conclusion

Topology Optimisation Method Topology Optimisation is a computational means of determining the physical domain

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:

Topology Optimisation Load Cases • Topology Optimisation: minimizing the compliance of the structure for

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?

Topology Optimisation Method for weight allocation The following strategy was used: • Do topology

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 •

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

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

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

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

Topology Optimisation Presence of window openings Optimization of the barrel without windows (Top) and

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

Development of the Design Concept by DLR • Reflection on the layout of the

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 •

Multi-parametric Optimisation Method: the multi-parameter global approximation-based approach used to solve the optimization problem

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

Design of Experiments In order to generate the sampling points for approximation building, a

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 11

Genetic Programming (GP) is a symbolic regression technique, it produces an analytical expression that

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 Z 1, Z 2, …, Z 7 are the design variables. Indications of the quality of fit of the obtained expression into the data: 12

FEM Modeling and Simulation Automated Multiparametric Global Barrel FEA Tool: Modeling, Analysis, and Result

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 MSC Patran MSC Nastran PCL Modeling and Analysis PCL Function Post-processing PCL Function Results: Displacement Skin Strains Beam Strains Buckling Results of all analyzed models are summarized in a separate file 13

Optimisation of the Fuselage Barrel Undisturbed anisogrid fuselage barrel Early design stage An upward

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

Variables and Constraints Variables: Design variables Skin thickness (h) Number of helix rib pairs

Variables and Constraints Variables: Design variables Skin thickness (h) Number of helix rib pairs around the circumference, (n) Helix rib thickness, (th) Helix rib height, (Hh) Frame pitch, (d) Frame thickness, (tf) Frame height, (Hf) Lower bound 0. 6 (mm) Upper bound 4. 0 (mm) 50 150 0. 6 (mm) 15. 0 (mm) 500. 0 (mm) 1. 0 (mm) 50. 0 (mm) 30. 0 (mm) 650. 0 (mm) 4. 0 (mm) 150. 0 (mm) h Circumf. Helix Rib Pitch, dep. on n 2φ Frame Pitch, d Fuselage Geometry Wf =20 mm Radius 2 m Wh=20 mm Constraints: • Strength: strains in the skin and in the stiffeners • Stiffness: bending and torsional stiffness Stability: buckling • Normalization Hf Hh th tf Barrel Cross Section dh=8 mm Wf =20 mm Circumferential Frames dh=8 mm Helix Ribs Normalized mass against largest mass • Margin of safety ≥ 0 • Strain • Stiffness • Buckling • 15

Results: Summary of parametric optimisation Tensile Compressive Strain (MS) Model Prediction I Optimum I

Results: Summary of parametric optimisation Tensile Compressive Strain (MS) Model Prediction I Optimum I Prediction II Optimum II Prediction III Optimum III Comp. Des. 0. 02 0. 36 0. 03 0. 54 0. 20 0. 62 1. 15 Optimum III Buckling (MS) Torsional Stiffness (MS) Bending Stiffness (MS) 1. 42 1. 21 1. 64 1. 54 1. 27 1. 09 1. 31 --------- 0. 00 -0. 07 -0. 04 1. 21 1. 25 0. 89 0. 81 0. 00 -0. 09 0. 01 0. 04 0. 23 0. 08 0. 19 Optimum III geometry with realistic ply layup: Design Shear Strain (MS) Skin thickness (h), mm 2. 08 2. 28 1. 71 Optimum II 0. 10 0. 11 0. 12 0. 29 (± 45, 0, -45, 90)s, 14 plies, total thickness = 1. 75 mm Nr. of helix rib pairs, (n) 60. 00 150. 00 Helix rib thickness, (th), mm 0. 60 0. 66 0. 61 628 mm 18. 94 ° Normalized mass Stability, Strength, and Stiffness Contraints Helix rib Frame height, pitch, (d), thickness, height, (Hh), mm mm (tf), mm (Hf), mm 27. 90 627. 70 1. 00 50. 00 27. 80 501. 70 1. 00 502 mm 209 mm Strength Contraint 9. 55 ° Helical ribs: tall and slender Frames: thin and small 84 mm Optimum III and Comp. Design 16

Results: Interpretation of the skin as a laminate, 14 plies Stacking sequence Buckling Torsional

Results: Interpretation of the skin as a laminate, 14 plies Stacking sequence Buckling Torsional Bending (MS) Stiffness Normalized mass (± 45, 0, -45, 90)s -0. 04 1. 25 0. 81 0. 29 (± 45, 0, 45, 90, -45, 0)s 0. 04 1. 25 0. 81 0. 29 (± 45, 90, 45, 0, -45, 0)s 0. 13 1. 25 0. 81 0. 29 % of 0° plies % of +/-45° plies % of 90° plies 28. 6% 57. 1% 14. 3% 17

Results: Interpretation of the skin as a laminate, 15 plies Stacking sequence Buckling Torsional

Results: Interpretation of the skin as a laminate, 15 plies Stacking sequence Buckling Torsional Bending (MS) Stiffness Normalized mass (± 45, 0, -45, 90)s , 0 0. 12 1. 26 0. 92 0. 30 (± 45, 0, 45, 90, -45, 0)s , 0 0. 20 1. 26 0. 92 0. 30 (± 45, 90, 45, 0, -45, 0)s , 0 0. 28 1. 26 0. 92 0. 30 % of 0° plies % of +/-45° plies % of 90° plies 33. 3% 53. 3% 18

Conclusion Multi-parameter global metamodel-based optimization was used for: • Optimization of a composite anisogrid

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

Thank You for your Attention 20

Thank You for your Attention 20