1. Practical parametric geometry for aircraft design
26 May, J. Philip Barnes
Abstract: Practical parametric geometry for aircraft design
J. Philip Barnes, Technical Fellow, Pelican Aero Group
Theory and application of practical methods for aircraft geometry parameterization and visualization are described. The methods, characterizing the surface geometry of complete aircraft, wings, fuselages, ducts, and new or existing airfoils, include fidelities ranging from “rapid visualization” to “high fidelity.” We apply two integrated programming and visualization platforms. The first is EXCEL and Visual Basic and the second is Blender 3D (open-source) with its resident Python programming language. In all cases, we characterize Cartesian coordinates (x,y,z) with parametric coordinates (u,v).
For “rapid synthesis,” we introduce modified trigonometric functions capable of quickly approximating an airfoil, wing, or fuselage with just a handful of parameters. We also introduce a “cubic quadrant” method for for fuselage cross section design. For “good fidelity” modeling of new or existing airfoils, we introduce a “parametric Fourier series” method satisfying specified leading edge radius, max&min vertical coordinates, upper&lower afterbody slopes, and aft-edge thickness. A “fine tuning” parameter allows further subtle adjustments. Upper and lower surfaces can also be modeled separately for greater control.
For “high fidelity,” we describe the theory and application of the cubic spline which is unique in its class by passing through, not just near, all specified points while preserving C2 continuity. Although the cubic spline not C3 continuous, we show that airfoil surface velocity distributions remain smooth with cubic-spline parameterization of the airfoil geometry. We also apply the cubic spline to characterize wings and fuselages. Core algorithms and code blocks are listed or otherwise made available to ensure ready access to the methods.
“Regenosoar” regen-electric aircraft rendered with Blender 3D open-source graphics software and 100% math modeled with parametric equations via Blender’s integrated Python programming language
J. Philip Barnes

2. Practical parametric geometry for aircraft design
Abstract
Practical parametric geometry for aircraft design
J. Philip Barnes, Technical Fellow, Pelican Aero Group
Theory and application of practical methods for aircraft geometry parameterization and visualization are described. The methods, characterizing the surface geometry of complete aircraft, wings, fuselages, ducts, and new or existing airfoils, include fidelities ranging from “rapid visualization” to “high fidelity.” We apply two integrated programming and visualization platforms. The first is EXCEL and Visual Basic and the second is Blender 3D (open-source) with its resident Python programming language. In all cases, we characterize Cartesian coordinates (x,y,z) with parametric coordinates (u,v).
For “rapid synthesis,” we introduce modified trigonometric functions capable of quickly approximating an airfoil, wing, or fuselage with just a handful of parameters. We also introduce a “cubic quadrant” method for for fuselage cross section design. For “good fidelity” modeling of new or existing airfoils, we introduce a “parametric Fourier series” method satisfying specified leading edge radius, max&min vertical coordinates, upper&lower afterbody slopes, and aft-edge thickness. A “fine tuning” parameter allows further subtle adjustments. Upper and lower surfaces can also be modeled separately for greater control.
For “high fidelity,” we describe the theory and application of the cubic spline which is unique in its class by passing through, not just near, all specified points while preserving C2 continuity. Although the cubic spline not C3 continuous, we show that airfoil surface velocity distributions remain smooth with cubic-spline parameterization of the airfoil geometry. We also apply the cubic spline to characterize wings and fuselages. Core algorithms and code blocks are listed or otherwise made available to ensure ready access to the methods.
J. Philip Barnes

3. Presentation Contents ~ Practical parametric geometry
Air vehicle
Applications
Objectives
&Rationale
EXCEL/VB
Blender/Python
Cubic spline
Theory & App.
“Rapid vis”
Trigonometric
Airfoil Geom.
Fourier-series
J. Philip Barnes

4. J. Philip Barnes www.HowFliesTheAlbatross.com
Blender 3D rendering of python-programmed geometry
Python window
Rendering window
J. Philip Barnes

5. Getting started: EXCEL as a scientific spreadsheet
Purpose (typical):
Read input and/or data from spreadsheet
Edit & run algorithm; generate new data
Write to spreadsheet cells & plot results
Copy all data & plots as new sheet; re-run
One-time setup:
EXCEL Options ~ Formulas ~ R1C1 ...
Trust Ctr. ~ settings ~ macro ~ enable & trust
Toolbar ~ more... ~ all ... ~ Visual Basic ~ Add
Set VB editor window to float on spreadsheet
Typical operations:
Type in the column headers, i.e. t, x, y, z
VB ~ insert ~ module ~ Type: sub example
Enter or edit code ~ save file as *.xlsm
Click run icon (note: module stays with the file)
Highlight applicable columns & plot the results
New case: Copy sheet, revise inputs, repeat 4)
In the older days of programming, we would read input from one file and write output to another. Here, we read from and write to the spreadsheet. With an engineering perspective, it is imperative that the rows and columns be numbered, not lettered. In this way, working for example by column with a time history of 200 parameters, we can readily find and process the 125th parameter. Toward this end, we’ll select (by a one-time operation) the “R1C1” preference. And, so as not to be limited by "click-based" versus "code-based" access to methods and information, we’ll program our own methods within the Visual Basic editor.

6. Powerful Parametrics for Airfoil Geometry J. Philip Barnes June, 2015
Cubic Spline
Herein we introduce new and practical methods for airfoil geometric design. These methods characterize the airfoil as mathematical formulas which, for the convenience of the airfoil designer, are computed within EXCEL based on a handful of key inputs which dominate the aerodynamics and shape. The methods can also be used to smooth existing airfoils.
The airfoil designer has the choice of characterizing the new airfoil with the resulting tabulated points, or far more efficiently, as we recommend, with the compact formulas and/or computed data blocks as discussed herein. The mathematical approach assures the same smooth geometry throughout for all disciplines using the airfoil.
EXCEL files are included herein for those readers interested in further study of the example airfoils chosen, or in the design of new airfoils. Although the examples exhibit a good match with existing designs, our main purpose is not so much to match existing designs, but instead to show that the new methods can design any airfoil within a range of airfoil classes.
This project is a much-improved derivative of the methods introduced in the author’s 1996 paper SAE , “Math Modeling of Airfoil Geometry.”
J. Philip Barnes

7. Airfoil parametric geometry
Objectives and Applications
Closely match/smooth existing airfoils
Geometric design of new airfoils
Option: modest-fidelity rapid vizualization
Three methods herein
Trigonometric (“Rapid viz”)
Fourier Series (good fidelity)
Parametric cubic spline (high fidelity)
Common approach
One or two parametric surfaces
Set LE radius, 1-to-3 midpoints, aft slope
X(W) parametric, 0 ≤ W ≤ 1, front to back
Z(U) Fourier, or Z(W) polynomial or spline
“Fine tuning” via one or more aux. params.
EXCEL files included herein, each method
Traditionally, airfoils have been described and shared as a lengthy tabulation of coordinates. This leaves open the method of interpolation between the points, including as a worst case a “faceted” result with linear interpolation. More importantly, today’s airfoil designer runs an aerodynamic module in parallel with the geometry definition module. Here, the designer does not have time to “push and pull” on numerous closely-spaced points to obtain a desirable velocity or pressure distribution. Experience shows this inevitably yields a “zig-zag” velocity distribution. Much more efficient is the variation of one or more strategically-placed, but also relatively-widely-spaced, points connected by a smooth, continuous, and accurate math model. In parallel with upper forward coordinate manipulation can be the variation of the leading edge radius which, to avoid discontinuity, should be the same for the upper and lower surfaces, since the typical stagnation point resides below the leading edge. Sudden change of curvature can lead to early transition.
The obvious key parameters representing airfoil geometry are leading-edge radius, maximum thickness, and afterbody shape. Indeed, the last few percent of afterbody camber will almost entirely define the pitching moment and zero-lift angle. Forward camber can be added with almost no impact on low-speed pitching moment.
Herein, we offer powerful methods to characterize the airfoil (high or low speed) smoothly and efficiently with a small handful of basic and “fine-tuning” parameters which go straight to the heart of the matter. We show two methods supporting either “matching” or “design” over a wide range of geometries, with the cubic spline offering the largest design space, but at the expense of a more cumbersome “airfoil sharing package.” All the methods ensure that all disciplines sharing a given airfoil will obtain precisely the same coordinates everywhere, while providing mathematical continuity to promote maximum laminar flow and/or minimum drag.
J. Philip Barnes

8. “Rapid viz” airfoil shaping: Hybrid Cartesian & trig. functions
X = 1 - sin(pu) ; Z = c sin(2pu)
DZ = c sin2(2pu)
1. simple wave, Z(u)
4. add camber
X = 1 - (1-g) sin(pu) + g sin(3pu)
DZ = c sin (X3p)
2. reshape X(u)
5. lower negative cusp
DZ = c sin (X3p)
DZ = c sin (X3p)
3. add aftbody cusp
6. opposite-sign cusps
J. Philip Barnes

9. “Higher” fidelity ~ Parametric Fourier-series airfoilW U
Fourier Series
Fourier Series terms z(u)
Best used for one curve Z(U), not two Z(W)
Add 8 sinusoidal terms plus aft-edge width
Single L.E. rad.(R), max/min (X,Z) , two aft (b)
Use upper & lower fine-tune parameters (g)
Continuous in all derivatives
Solve eight eqns. for Fourier amplitudes
Satisfy end slopes (dW/dZ) & max/min
Compact “airfoil-sharing” formula
Airfoil construction sequence:
U = 0 to 1 ; W = if(U < 0.5, 1 - 2U, 2U - 1)
g = gb + (gt - gb) cos2 (pU/2)
X = 1 - (1-g) cos(pW/2) – g cos(3pW/2)
Z = S m=1 to 8 {am sin(mpU)} + Za(1-2U) W 1 X g
“fine-tune”
parameter
Parameterization
for X(W)
Curves which have a vertical coordinate (Z) which both begins and ends with zero are well suited to modeling with a Fourier series (see sketch, upper right). The solution involves one lower order than that required to represent a curve which does not both start and end at Z=zero. The (1-to-n) terms of the Fourier series represent wave amplitudes, where the first term is one “halfwave” and the last term has (n) halfwaves. These waveforms are added to represent Z(u). The amplitudes are solved with (n) simultaneous equations, usually representing point coordinates, but sometimes instead representing slopes (dZ/dW) at the ends or anywhere along the curve.
The parametric Fourier-series airfoil borrows from the classic aerodynamic method of characterizing the spanwise distribution of lift on a wing. Here, we characterize the airfoil (Z) coordinate parametrically with the “graphics coordinate” (u). A common leading edge radius is applied to the upper and lower surfaces, but each surface has its own “fine-tuning” parameter (g).
Of the methods herein, the Fourier-series method offers the best overall combination of simplicity, geometry control, and smoothness. U Z
First 4 terms
of the series
J. Philip Barnes

10. Parametric Fourier-Series airfoil ~ NLF(1)-0416 ~ match
J. Philip Barnes

11. Parametric Fourier-Series airfoil ~ PCS-001 ~ new design
Notice the extraordinary smoothness of the first and second derivatives (bottom plots).
J. Philip Barnes

12. Cubic spline ~ Parametric u(t) or Cartesian y(x)
Get smooth curve passing through (1_to_n) points
VB array dim. (n) elements: 0_to_n ~ ignore 0th elem.
1st & 2nd derivative Continuity (3rd is not continuous)
Independently control L/R-end slope or 2nd derivative
Internal-node continuity yields tri-diagonal system
End constraints are applied in first and last rows
Parametric x(t) ; v  “velocity”; a  “acceleration” t x 1 2 n 3
The cubic spline joins 1_to_n points with a cubic curve which preserves continuity of the first and second derivatives, while allowing independent specification of end constraints such as curvature or slope. The cubic spline can be applied in either Cartesian or parametric form. In the sketch above, we show a coordinate (x) as parametric with “time” (t). The first derivative is then in effect a velocity, and the second derivative is the acceleration. The figure at bottom center reveals the discontinuity of the third derivative. The matrix representation at lower right represents the tri-diagonal system which preserves “C2” continuity at the internal “knots.” Perhaps contrary to our expectations, we do not solve for the cubic coefficients; instead we solve for the accelerations (a) at the internal knots. This solution is subject to end-point constraints which, for x(t), include either a “stiff” (zero acceleration), “flexible” (acceleration linearly interpolated), “flat” (zero velocity), or “slope set” (velocity specified) condition. The first three of these are illustrated next for a cubic spline passing through four points (we can fit a cubic spline to as few as three points).
Set ends; Solve linear EQs. for internal-knot accelerations (a)

13. Parametric cubic spline ~ Various end constraints
In each of the three graphical columns above, we plot the coordinate (x), its velocity, and its acceleration, all versus time, but with specific end constraints for each plot column. The zero-acceleration for the “stiff-end” condition is apparent at lower left. The “flexible-end” condition is evident at bottom center where the end-point accelerations are linearly interpolated from neighboring accelerations. Finally, the “flat-end” condition is seen at upper right and middle right.
Along the bottom row of plots we see the discontinuity of the third derivative. If we intend to use the cubic spline to model an aerodynamic surface, an immediate question arises: “Does the geometric surface appear smooth to the air flowing by?” We will soon address this question, but first we’ll characterize airfoil geometry with a cubic spline.
“Stiff” ends
“Flexible” ends
“Flat” ends

14. Parametric cubic spline airfoilW U
Cubic Spline
Cubic spline(s) pass through all set points
Wider design space including “unusual”
Match 0th, 1st, 2nd derivatives, ea. node
Discontinuous 3rd derivative
Input LE rad.(R), 3 pts. (X,Z) , aft slope (b)
g can be varied but is normally fixed (0.1)
Solves 5 eqns. spline-knot 2nd derivatives
Gauss-Seidel in lieu of Gaussian Diag.
3 midpoints versus single midpoint
Any position, not necessarily max/min
Less compact “airfoil-sharing package”
U = 0 to 1 ; W = if(U < 0.5, 1 - 2U, 2U - 1)
X = 1 - (1-g) cos(pW/2) – g cos(3pW/2)
EXCEL solves for cubic splines, Z(W)
Package: sol’n data block & interpolator W 1 X g
The parametric cubic spline airfoil shares many similarities with the Fourier series method, but also substantial differences. In exchange for greater design space, we accept somewhat more cumbersome “airfoil-sharing package” consisting of a datablock of computed results, and short but necessary interpolation module (found in the VB code).
The cubic spline preserves continuity up to the second derivative at all spline internal knots, with first derivatives at the end knots set by the calculus related to the leading-edge radius and afterbody trailing-edge slope.
Although more knots can be used for the purpose of “matching,” our primary goal here is design. Accordingly, just three internal knots, in conjunction with two powerful end-knot constraints, are found to offer the best combination of smoothness and shape control for each upper or lower surface. Relative to the other methods herein, the cubic spline provides the greatest geometric control. W Z + -
J. Philip Barnes

15. Parametric cubic spline airfoil Sample Gauss-Seidel convergence
The mathematics of the cubic spline yield a “tri-diagonal” system to which the Gauss-Seidel solution is ideally suited, typically converging within five iterations, as shown above. Both code and computation time are dramatically shortened in comparison to those of the Gaussian Diagonalization method used for the PP method covered earlier herein.
J. Philip Barnes

16. Parametric cubic spline airfoil ~ 13-point match
(blue) closely matches
target (white points)
Notice the “zig-zag” of the second derivative at lower right. Although the second derivative is continuous, the third is not. Later, we re-visit the question of whether such discontinuity adversely affects the smoothness of the surface velocity profile.
J. Philip Barnes

17. Parametric cubic-spline airfoil ~ NLF(1)-0416 ~ 9-pt match
J. Philip Barnes

18. Parametric cubic-spline airfoil ~ PCS-001 ~ new design
J. Philip Barnes

19. Laminar airfoil study ~ integrated geometric/aero design
Theodorsen Angle (f)
Velocity ratio
Parametric cubic spline
Discontinuous 3rd-deriv.
of cubic spline does not
disrupt smooth airflow
Pressure coefficient
Earlier herein, we questioned whether airfoil geometry defined by a cubic spline would appear smooth to the air flowing by, given the cubic spline's third-derivative discontinuity. Here, we hope to show that the surface indeed appears smooth.
The study shown above pertains to the integrated geometric and aerodynamic design of a laminar airfoil at low speed. However, in this case the geometry spline is set by the (z/c) coordinate parameterized with the “Theodorsen Angle*” (f), taken clockwise from the lower to the upper trailing edge. The (x/c) coordinate is then automatically parameterized as (f) sweeps out a circle centered at 50% chord [x/c = 0.5(1+cosf)].
In this integrated design and analysis process,** the points along the forward upper surface are manually varied to attain the “rooftop” (flat) local-to-freestream velocity ratio (“vortex density” |G|=v/vo) profile at the design-point angle of attack (a=3o ; note: the zero-lift angle is 6.2o). Examination of the various aerodynamic properties including velocity ratio and pressure coefficient (cp) suggests that the geometry appears smooth to the air flowing over the airfoil.
Theodorsen used this as an airfoil coordinate ; Glauert used it as wing spanwise coordinate.
**The related analysis package is in development ; the related EXCEL file is not included herein.

20. J. Philip Barnes www.HowFliesTheAlbatross.com
Parametric fuselage section: “cubic quadrant” method
Four parameters per quadrant
One cubic curve per quadrant
J. Philip Barnes

21. Parametric Fuselage – cubic spline & trig. compared
cubic-spline basis
Trig. functions provide 99% desired
result with just 1% of computation
J. Philip Barnes

22. J. Philip Barnes www.HowFliesTheAlbatross.com
Parametric wing: cubic spline throughout (EXCEL/VB)
J. Philip Barnes

23. Application: Dynamic soaring in the jet stream
The formula above
Energy From an Atmosphere in Motion - Dynamic Soaring and Regen-electric Flight Compared J. Philip Barnes

24. J. Philip Barnes www.HowFliesTheAlbatross.com
Application: Regen of electrical power in ridge lift
J. Philip Barnes

25. J. Philip Barnes www.HowFliesTheAlbatross.com
About the Author
Phil Barnes has a Master’s Degree in Aerospace Engineering from Cal Poly Pomona and BSME from the University of Arizona. He is a Principal Engineer and 34-year veteran of air vehicle and subsystems performance analysis at Northrop Grumman, where he presently supports both mature and advanced tactical aircraft programs. Author of several SAE and AIAA technical papers, and often invited to lecture at various universities, Phil is presently leading several Northrop Grumman-sponsored university research projects including an autonomous thermal soaring demonstration, passive bleed-and-blow airfoil wind-tunnel test, and application of Blender 3D software for flight simulation. This presentation includes highlights of one such collaboration (public domain) using EXCEL/Visual Basic and Blender 3D with its resident Python programming language to parameterize and visualize aircraft geometry. Outside of work, Phil is a leading expert on dynamic soaring, and he is pioneering the science of regen-electric flight.
J. Philip Barnes