Applications of Isogeometric Analysis at Boeing … · 1988 – First FORTRAN version [Grandine] 1989 – Automatic generation of Gauss points [Epton] 1991 – Improved adaptivity

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Applications of Isogeometric Analysis at Boeing Thomas A. Grandine Senior Technical Fellow Michael A. Epton Associate Technical Fellow Flight Sciences Technology

EOT_RT_Template.ppt | 8/1/2011 1

Engineering, Operations & Technology | Boeing Research & Technology


FaST | Flight Sciences Technology

EOT_RT_Template.ppt | 2

Boeing has made substantial use of B-spline finite elements in one variable

Use B-splines as finite elements

Approximate ODE solution by solving

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j

jjBu

uufu

)()(

))('),(,()("

i

iiii uufu

))('),(,()("

are the Gauss-Legendre points

over each polynomial piece




Boeing’s ODE solver has a long history

Major improvements

1988 – First FORTRAN version [Grandine]

1989 – Automatic generation of Gauss points [Epton]

1991 – Improved adaptivity algorithm [Bieterman]

1995 – LAPACK’s banded solver instead of SolveBlok [Klein]

2001 – Code rewritten in C [Pierce]

2004 – Improved Newton convergence [Ettinger]

2009 – Leveraged decoupling for IVPs [Klein]

2011 – Automatic spline space determination [Hogan]

2011 – Continuation method for Newton solve [Grandine]

Many minor improvements along the way, too

10 formal revisions in 2011

50 formal revisions since 2001

Current version dated 8 June, 2012





A waverider is a lifting surface which rides on the shock wave created by its own leading edge in supersonic flight

First developed by Terence Nonweiler in 1951

Only production design to use waveriders was XB-70 in 1960s


B-spline finite elements can be used to loft waveriders




Start with the Navier Stokes equation

Assume

Inviscid flow

Circumferentially symmetric flow

Irrotational flow

Steady flow

In spherical coordinates:


The Taylor-MacColl equation models conical flow

222

222

')'1(2

1

)cot'2)('1(2

1'

)("

uuu

uuuuuuu

)(u)(' u

u





Flow streamlines can be generated by solving another ODE

)(')()(

)(')()()('

arctan

uxyxu

xuuxyxy

x

y

00)( yxy

Note: All solutions are self-similar

u





Each streamline of the waverider is produced this way





High speed inlets often need to satisfy the shock-on-lip condition


Example: Isentropic high speed inlets

Flow speed decreases

and pressure increases

as it turns

Direction of flow

Shock and Mach wave angles

increase as flow speed decreases

Cowl lip point

The engine should capture all the compression waves without spillage




An implicit ODE describes inlet shape to capture isentropic compression waves


),( yx)','( yx

0M

),( liplip yx

M




The contouring problem is a DAE of index 2


Earliest and most extensive use has been solving contouring problems

0))((

0)(")('

uxf

uxux

Curve intersection Curve projection Horizon line

determination

Tool path

determination

Variable radius

filleting

Rolling ball

filleting

Envelope calculation




Starting in 1989, Boeing developed an application to enable aeroservoelastic design based on the boundary element method

Given the integral equation

surface S depends on a design parameter t

S is given as a tensor product B-spline series

Work appears in the PACAM III conference proceedings, 1993: “The Boundary Integral Equation for the parametric derivative of the solution to the Prandtl-Glauert equation”


Boeing also has at least one 2D application of isogeometric analysis

dAxyxdAn

xyxy

SS

),()(

),()()(

21

j

jj vuBtvuS ),()(),(




Isogeometric analysis makes integral equation domain independent of S

Key result:

The operator enables ready application of Stokes’ Theorem


Computation of derivatives with respect to design variables now possible

t

tvuSndA

t

ndA

);,()(

)(

ndA





The method computed Cp effectively across design space

Baseline geometry Cp Deformed geometry Cp





There is substantial agreement between predicted analysis and actual analysis

Reanalyzed results Predicted results




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The actual differences really are quite small




Shape sensitivities can be accurately computed using isogeometric analysis. In this case the accuracy was better than 10 digits.

Working in the parameter space of the geometry eliminates many computational difficulties

Most of the benefits of isogeometric analysis claimed in academic studies can be achieved in an industrial setting


This capability led to interesting discoveries

Documents

Applications of Isogeometric Analysis at Boeing … · 1988 – First FORTRAN version [Grandine] 1989 – Automatic generation of Gauss points [Epton] 1991 – Improved adaptivity