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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
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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
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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
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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
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B-spline finite elements can be used to loft waveriders
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Start with the Navier Stokes equation
Assume
Inviscid flow
Circumferentially symmetric flow
Irrotational flow
Steady flow
In spherical coordinates:
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The Taylor-MacColl equation models conical flow
222
222
')'1(2
1
)cot'2)('1(2
1'
)("
uuu
uuuuuuu
)(u)(' u
u
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EOT_RT_Template.ppt | 6
Flow streamlines can be generated by solving another ODE
)(')()(
)(')()()('
arctan
uxyxu
xuuxyxy
x
y
00)( yxy
Note: All solutions are self-similar
u
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Each streamline of the waverider is produced this way
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High speed inlets often need to satisfy the shock-on-lip condition
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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
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An implicit ODE describes inlet shape to capture isentropic compression waves
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),( yx)','( yx
0M
),( liplip yx
M
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The contouring problem is a DAE of index 2
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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
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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”
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Boeing also has at least one 2D application of isogeometric analysis
dAxyxdAn
xyxy
SS
),()(
),()()(
21
j
jj vuBtvuS ),()(),(
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Isogeometric analysis makes integral equation domain independent of S
Key result:
The operator enables ready application of Stokes’ Theorem
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Computation of derivatives with respect to design variables now possible
t
tvuSndA
t
ndA
);,()(
)(
ndA
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The method computed Cp effectively across design space
Baseline geometry Cp Deformed geometry Cp
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There is substantial agreement between predicted analysis and actual analysis
Reanalyzed results Predicted results
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EOT_ETS_Sub_Template.ppt | 15
The actual differences really are quite small
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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
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This capability led to interesting discoveries