A beam reduction method for wing aeroelastic design optimisation with detailed stress constraints

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1 A beam reduction method for wing aeroelastic design optimisation with detailed stress constraints O. Stodieck, J. E. Cooper, S. A. Neild, M. H. Lowenberg University of Bristol N.L. Iorga Airbus Operations Ltd. 1

2 Overview Motivation Benefits of analysis-specific structural idealisations (hi-fidelity / low-fidelity) Integration of the beam reduction method into an aeroelastic design optimisation Beam reduction method description Overview Stiffness reduction Examples Rectangular wingbox model University of Bristol Ultra-Green (BUG) aircraft wingbox Conclusions 2

3 Motivation 3

4 Benefits of analysis-specific structural idealisations Why do we use different structural idealisations? Low-Fidelity = Speed Example: Flutter analysis 10 2 DOF* Beam model Wingbox GFEM DC3 wing section * Degrees of freedom ** Global finite element model Hi-Fidelity = Accuracy 4 Example: GFEM** Internal loads analysis 10 4 to 10 6 DOF detailed stress analysis

Gradient based optimization algorithm, iterates the design variables until convergence is reached 29 November 2017 Aeroelastic design optimisation How can we do MDO* with high and low fidelity

5 Gradient based optimization algorithm, iterates the design variables until convergence is reached 29 November 2017 Aeroelastic design optimisation How can we do MDO* with high and low fidelity structural idealisations? Start Multidisciplinary analysis Update of optimization variables values Stress analyses GFEM to Beam model reduction External loads Update of beam model properties and M Aeroelastic analyses Example: Automatic beam reduction integration into a Multidisciplinary Feasible (MDF) architecture. Stress constraint values Aeroelastic constraint values *Multidisciplinary design optimisation End Objective and constraint function values Total derivatives matrix 5

6 Beam reduction method description 6

7 Beam reduction method - Overview GFEM with variables v Thicknesses Ply percentages Stringer orientations Stringer (equivalent area and offset shown) Spar Cap GFEM Front Spar Multi-point Constraint Beam Node Upper Skin Lower Skin Rib Rear Spar 1 Rib-bay 26 0 ply direction in upper and lower skins Beam with properties Ω and [M] C L Beam 7

8 Beam reduction method - Overview CΤ v ΩΤ C ΩΤ v Stiffness reduction Definition of a beam reference axis Beam element flexibility matrix [C] (numerical) 13 physical isotropic beam element stiffness parameters Ω (analytical) Manual input Auto - mated Auto - mated Mass reduction Definition of lumped mass element groups Lumped mass and inertia matrix [M] determination (numerical) MΤ v Manual input Auto - mated 8

Stiffness reduction Example: BUG wing model GFEM Definition of a beam reference axis Beam element flexibility matrix determination (numerical) Calculation of 13 physical isotropic beam element

9 Stiffness reduction Example: BUG wing model GFEM Definition of a beam reference axis Beam element flexibility matrix determination (numerical) Calculation of 13 physical isotropic beam element stiffness parameters Manual input Global model analysis coordinate system Stringer datum Rib datum Stringer (equivalent area and offset shown) Spar Cap Multi-point Constraint Upper Skin Rib Rear Spar C L Fuselage/Wing Interface Beam Element Reference Axis Front Spar y x Beam Node Beam Node Section 1 Section 2 Local beam reference axis coordinate system Section 3 Lower Skin 1 Rib-bay Straight beam sections 9

Stiffness reduction NASTRAN analysis (1s / straight wing section) Definition of a beam reference axis Beam element flexibility matrix determination (numerical) Calculation of 13 physical isotropic

10 Stiffness reduction NASTRAN analysis (1s / straight wing section) Definition of a beam reference axis Beam element flexibility matrix determination (numerical) Calculation of 13 physical isotropic beam element stiffness parameters Auto - mated 1) GFEM analysis with 6 static tip load cases / section 2) Nodal deflection and rotation output 3) Post-process to find 6x6 [C] matrix / beam element Note: This approach is equivalent to the blade property extraction (BPE) method used for wind-turbine blades: Malcolm DJ, Laird DL. Extraction of equivalent beam properties from blade models. Wind Energy Mar 1;10(2):

11 Stiffness reduction Definition of a beam reference axis Beam element flexibility matrix determination (numerical) Calculation of 13 physical isotropic beam element stiffness parameters Auto - mated defines the properties of an isotropic Timoschenko theory beam element. is calculated analytically from C and assumed isotropic material properties E and G. Non-dimensional bend-twist coupling parameters Ψ = K/ EI. GJ can be derived as functions of. z Shear Centre y Elastic axis in Node A (SC A ) x SC AY SCAZ SC BY z y Node A x Shear Centre in Node B (SC B ) SC BZ Bend-twist coupling exists for any SC A SC B Beam element reference axis Node B Effective shear centre offsets at nodes A and B = elastic axis definition Bending and Torsion Stiffness parameters Section centroid offsets from the elastic axis Section Area Timoschenko shear coefficients 11

12 Examples 12

13 1.00m 29 November 2017 Rectangular wingbox model Constant-section rectangular wing-box GFEM: 16.8m span (28 rib bays), 1.68m chord and 1.0m height Baseline model stiffness, with SC A = SC B = 0 N y = N Z = 0 Upper skin Front spar Multi-point Constraint Beam Node Lower skin Rib Stringers, Spar caps Stiffness increase at the root (Y=0m) >> boundary constraint effect Stiffness reduction at the tip (Y=16.8m) >> load introduction effect due to the multipoint constraint (RBE3) assumption here resulting in equally distributed skin forces, not taking into account the section stiffness distribution or the skin/stiffener offsets 13

14 Rectangular wingbox model Effect of incremental stiffness changes on the elastic axis (locus of shear centres). Baseline SC A = 0 SC B = 0 +Thicker front spar + Unbalanced skin laminate + Rotated rib Model 1 Model 2 Model 3 14

15 Gradients of the rectangular wingbox model Stiffness parameter sensitivities to changes in the +45º fibre percentage in the upper skin. Gradients are captured accurately. 15

16 University of Bristol Ultra-Green (BUG) aircraft wingbox (static & dynamic) 29 November ply direction in upper and lower skins 16

University of Bristol Ultra-Green (BUG) aircraft wingbox (static & dynamic) Comparison of the GFEM and equivalent beam displacements and rotations for an applied tip force (linear

17 University of Bristol Ultra-Green (BUG) aircraft wingbox (static & dynamic) Comparison of the GFEM and equivalent beam displacements and rotations for an applied tip force (linear analysis). GFEM beam Z-disp. X-rot. Static deflections captured accurately (tip displacements within 5% error). Upbending Wing Twist* Y-rot. *Twist due to wing sweep + bend-twist coupling 17

University of Bristol Ultra-Green (BUG) aircraft wingbox (static & dynamic) 1.

78 Hz (0.5% Freq. error) Mode 8 16.3 Hz (2.1% Freq.

applied wing tip excitation force of 10 4 N in the Global Z-direction.

18 University of Bristol Ultra-Green (BUG) aircraft wingbox (static & dynamic) 1.26 Hz (-0.1% Freq. error) First up/down bending mode 3.18 Hz (0.1% Freq. error) First FWD/AFT bending mode Hz (0.5% Freq. error) Mode Hz (2.1% Freq. error) Mode 9 29 November 2017 Frequency response plots at the wing tip, for an applied wing tip excitation force of 10 4 N in the Global Z-direction. Mode 8 Mode 9 Dynamic behaviour captured accurately (first 10 modes natural frequencies within 2.1% error). 18

19 Conclusions 19

20 Summary: Developed a numerical method for efficiently and accurately reducing a high-fidelity GFEM to a Timoschenko beam theory model with lumped masses. Suggested an approach for integrating the beam reduction method into a gradient-based aeroelastic design optimisation architecture. Advantages: Speed or Scope faster aeroelastic optimisation more aeroelastic load cases large deflection cases more refined GFEM physical insight into the GFEM design optimization results (e.g. quantify bend-twist coupling in terms of elastic axis rotation or Ψ values) information about the effects of design changes on the overall wing stiffness and mass properties, in the vicinity of the current design 20

21 Publication: O. Stodieck, J. E. Cooper, S. A. Neild, M. H. Lowenberg and N. L. Iorga, Beam reduction method for wing aeroelastic design optimization with detailed stress constraints, submitted to the AIAA Journal in Nov Contact: Dr Olivia Stodieck Acknowledgements Airbus and Innovate UK (AWI project) 21

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