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

Topology OptimisationMethod

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:

eVd

pEE

FuEKtsuFC

ee

pee

e

T

,10,

,1,

:..min

0

N

LoadLoadLoadTot cc

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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 OptimisationLoad Cases

Topology OptimisationMethod 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 Results for 3 load cases

Topology OptimisationModel and Results

BendingTorsion

Transverse bending

Topology OptimisationResults

Iso view: optimization of the barrel for weighted compliance

Optimization of the barrel without windows (Top) and with windows (Bottom)

Two backbones on top and bottom of the barrelNearly +-45° stiffening on the side panel

Result: beam structure for the barrel

Note: SIMP approach does not consider buckling

Topology OptimisationPresence of window openings

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

Bearing Skin Stiffener grid

Frame

Aerodynamic skin

Foam core

Bearing Skin Stiffener grid

Frame

Aerodynamic skin

Foam core

Multi-parametric OptimisationMethod: 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

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Design of ExperimentsIn 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.

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min1

1 12

p

p

p

pq pqLU

Example: A 100-point DOE generatedby an optimal Latin hypercube technique

Genetic ProgrammingGenetic 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:

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where Z1, Z2, …, Z7 are the design variables.

0.975381 )Z Z Z Z/(Z Z Z 15.5318

)Z Z Z Z/(Z 660.152 -)Z Z/(Z Z Z 0.202164 +

)Z Z/(Z Z 163.814 +)Z Z Z/(Z 4143.98-

Z Z Z Z 06045610.00000000+ Z/Z 603.316+ Z/Z 2.93847+

Z Z 0.00132105+Z 1.76206-Z 1.26902= )Z;Z;Z;Z;Z;Z;f(Z

542

34

24

762

1

742

323

12

7562

42

5243642

22

1

72

5315323

53317654321

Indications of the quality of fit of the obtained expression into the data:

FEM Modeling and Simulation

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

Displacement Skin Strains Beam Strains BucklingResults:

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 PatranMSC Nastran

PCL

xy

z Qz

Optimisation of the Fuselage Barrel

Composite skin and stiffeners

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An upward gust load case at low altitude and cruise speed

Undisturbed anisogrid fuselage barrelEarly design stage

Variables and Constraints

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) 0.6 (mm) 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) 4.0 (mm)Frame height, (Hf) 50.0 (mm) 150.0 (mm)

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Hf

tf

Wf =20mm

Wf =20mm

Hh

Wh=20mm

dh=8mm dh=8mm

th

Circumferential Frames Helix Ribs

Frame Pitch, d

Circumf. Helix Rib Pitch, dep. on n

2φ

Fuselage Geometry

Radius2m 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:

Results: Summary of parametric optimisation

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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 --- --- --- 0.10Optimum I 0.36 -0.09 1.21 --- --- --- 0.11Prediction II 0.03 0.01 1.64 --- --- --- 0.11Optimum II 0.54 0.04 1.54 --- --- --- 0.12Prediction III 0.20 0.23 1.27 0.00 1.21 0.89 0.29Optimum III 0.62 0.08 1.09 -0.07 1.21 0.89 0.29Comp. Des. 1.15 0.19 1.31 -0.04 1.25 0.81 0.29

DesignSkin

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), mmOptimum I 2.08 60.00 0.60 27.90 627.70 1.00 50.00Optimum II 2.28 60.00 0.66 27.90 627.70 1.00 50.00Optimum III 1.71 150.00 0.61 27.80 501.70 1.00 50.00

Strength Contraint

Stability, Strength,

and Stiffness

ContraintsOptimum III geometry with realistic ply layup:

Helical ribs: tall and slender Frames: thin and small

209 mm

628 mm

18.94 °

Optimum II84 mm

502 mm

9.55 °

Optimum III and Comp. Design

(±45,0,45,0,-45,90)s, 14 plies, total thickness = 1.75 mm

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

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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 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%

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

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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 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% 13.3%

ConclusionMulti-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°

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