Multi point wing planform optimization via control theory
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Multi-point Wing Planform Optimization via Control Theory. Kasidit Leoviriyakit and Antony Jameson Department of Aeronautics and Astronautics Stanford University, Stanford CA 43 rd Aerospace Science Meeting and Exhibit January 10-13, 2005 Reno Nevada.

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Multi point wing planform optimization via control theory

Multi-point Wing Planform Optimizationvia Control Theory

Kasidit Leoviriyakit

and

Antony Jameson

Department of Aeronautics and Astronautics

Stanford University, Stanford CA

43rd Aerospace Science Meeting and Exhibit

January 10-13, 2005

Reno Nevada


Typical drag break down of an aircraft
Typical Drag Break Down of an Aircraft

Mach .85 and CL .52

Induced Drag is the largest component


Cost function

Wing planform modification can yield larger

improvements BUT affects structural weight.

Cost Function

Simplified Planform Model

Can be thought

of as constraints


Choice of weighting constants
Choice of Weighting Constants

Minimizing

Maximizing

Range

using


Structural model for the wing
Structural Model for the Wing

  • Assume rigid wing

  • (No dynamic interaction between Aero and Structure)

  • Use fully-stressed wing box to estimate the structural weight

  • Weight is calculated based on material of the skin


Trend for planform modification
“Trend” for Planform Modification

Increase L/D without any penalty on structural weight by

  • Stretching span to reduce vortex drag

  • Decreasing sweep and thickening wing-section to reduce structural wing weight

  • Modifying the airfoil section to minimize shock

Suggested

Baseline

Boeing 747 -Planform Optimization


Redesign of section and planform using a single point optimization
Redesign of Section and Planformusing a Single-point Optimization

Redesign

Baseline

Flight Condition (cruise): Mach .85 CL .45


The need of multi point design

Undesired characteristics

The Need of Multi-Point Design

Designed Point




Review of single point design using an adjoint method
Review of Single-Point designusing an Adjoint method

Design Variables

Using 4224 mesh points

on the wing as design variables

Boeing 747

  • Plus 6 planform variables

    • -Sweep

    • -Span

    • -Chord at 3span –stations

    • -Thickness ratio



Disadvantage of the finite difference method
Disadvantage of the Finite Difference Method the Finite Difference Method

The need for a number of flow calculations proportional to the number of design variables

Using 4224 mesh points

on the wing as design variables

4231 flow calculations

~ 30 minutes each (RANS)

Too Expensive

Boeing 747

Plus 6 planform variables


Application of control theory adjoint
Application of Control Theory the Finite Difference Method(Adjoint)

GOAL : Drastic Reduction of the Computational Costs

Drag Minimization

Optimal Control of Flow Equations

subject toShape(wing) Variations

(for example CDat fixed CL)

(RANS in our case)


Application of control theory
Application of Control Theory the Finite Difference Method

4230 design

variables

One Flow Solution + One Adjoint Solution


Sobolev gradient

Key issue for successful implementation of the Continuous adjoint method.

Sobolev Gradient

Continuous descent path


Design using the navier stokes equations
Design using the Navier-Stokes Equations adjoint method.

See paper for more detail


Test case
Test Case adjoint method.

  • Use multi-point design to alleviate the undesired characteristics arising form the single-point design result.

  • Minimizing at multiple flight conditions;

    I = CD +a CW at fixed CL

    (CD and CW are normalized by fixed reference area)

    a is chosen also to maximizing the Breguet range equation

  • Optimization: SYN107

    Finite Volume, RANS, SLIP Schemes,

    Residual Averaging, Local Time Stepping Scheme,

    Full Multi-grid


Single point redesign using at cruise condition
Single-point Redesign using adjoint method.at Cruise condition


Isolated shock free theorem

Mach .90 adjoint method.

Mach .84

Mach .85

Isolated Shock Free Theorem

“Shock Free solution is an isolated point, away from the point shocks will develop”

Morawetz 1956


Design approach
Design Approach adjoint method.

  • If the shock is not too strong, section modification alone can alleviate the undesired characteristics.

  • But if the shock is too strong, both section and planform will need to be redesigned.


3 point design for sections alone planform fixed
3-Point Design for Sections alone adjoint method.(Planform fixed)


Successive 2 point design for sections planform fixed
Successive 2-Point Design for Sections adjoint method.(Planform fixed)

MDD is dramatically improved



C p at mach 0 78 0 79 0 92
C adjoint method.p at Mach 0.78, 0.79, …, 0.92

  • Shock free solution no longer exists.

  • But overall performance is significantly improved.


Conclusion
Conclusion adjoint method.

  • Single-point design can produce a shock free solution, but performance at off-design conditions may be degraded.

  • Multi-point design can improve overall performance, but improvement is not as large as that could be obtained by a single optimization, which usually results in a shock free flow.

  • Shock free solution no longer exists.

  • However, the overall performance, as measured by characteristics such as the drag rise Mach number, is clearly superior.


Acknowledgement
Acknowledgement adjoint method.

This work has benefited greatly from the support of Air Force Office of Science Research under grant No. AF F49620-98-2005

Downloadable Publicationshttp://aero-comlab.stanford.edu/http://www.stanford.edu/~kasidit/


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