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

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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


Cost function for a multi point design

Cost Function for a Multi-point Design

Gradients


Multi point design process

Multi-point Design Process


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


Optimization and design using sensitivities calculated by the finite difference method

Optimization and Design using Sensitivities Calculated by the Finite Difference Method

f(x)


Disadvantage of the finite difference method

Disadvantage of 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 (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

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

See paper for more detail


Test case

Test Case

  • 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 at Cruise condition


Isolated shock free theorem

Mach .90

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

  • 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 (Planform fixed)


Successive 2 point design for sections planform fixed

Successive 2-Point Design for Sections(Planform fixed)

MDD is dramatically improved


Lift to drag ratio of the final design

Lift-to-Drag Ratio of the Final Design


C p at mach 0 78 0 79 0 92

Cp at Mach 0.78, 0.79, …, 0.92

  • Shock free solution no longer exists.

  • But overall performance is significantly improved.


Conclusion

Conclusion

  • 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

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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