University of Toronto Aerospace Team · SAE Aero Design · Minerva 2024, Stratus 2025 · wing aerodynamics

The climb angle was free over a quarter of a degree where the file meant fifteen, and that bound is what chose the aerofoil.

A competition wing described by twelve numbers and scored on one: the largest payload the aircraft can lift, which the project measures with a surrogate rather than with payload itself. Judged as sections, one of the two aerofoils is better at every Reynolds number this wing sees. The run picked the other, by 2.8 kg of that surrogate. Re-implementing the loop in Python to check it found the reason: OptimizerMain.m converts an angle that is already in radians, so the climb angle was bounded 57 times tighter than the file intended. Under the corrected bound the two sections do not separate.

Loop: nonlinear lifting line · beam spar sweep · component drag · mission constraints
Record: 500 evaluations · 49 aerodynamic and 25 structural stations · 2 aerofoil tables
Check: objective reproduced against the MATLAB original to the last digit

How big the wing has to be

Before any shape is chosen

SAE Aero Design gives a student team a season, a rulebook and a field. The aircraft has to leave the ground inside a marked distance, fly a circuit and land, and the score is how much payload it carried. UTAT enters the Advanced Class. The wing is one part of that aircraft, and the first question about a wing is not what shape it should be. It is how big.

There is a plain way to answer that. Choose a wing area. Work out the heaviest aircraft that area could still get off the ground in the runway available, the heaviest it could still hold the required climb with, and the heaviest it could still fly the required turn with. Whichever of the three is smallest is what an aircraft with that wing is allowed to weigh. Then do it again for every area. The curves that come out are the wing's whole story before a single aerofoil has been named.

Four numbers come from the rules and the field. The take-off has to finish inside 146 m, which is the Fort Worth field as measured off a map. The climb has to hold ten degrees. The turn has to hold fifty degrees of bank. The motor has 750 W. A fifth constrains the answer from the side: span may not exceed 3.048 m, which is 120 inches, and once an aspect ratio is chosen that span limit becomes a limit on area.

The calculation is the team's. constraintAnalysis.m and its component drag build-up were written between 2021 and 2022 by Neil Farvolden, Daniel MacGregor and Adele Crete-Laurence, and the propeller efficiency is read from the team's own lookup table for the 22x12. Both files were ported to Python line for line, approximations included, and the port was checked against the figure they committed. Seven quantities can be read off that plot and the worst disagreement is .

aspect ratio C_L max

Sizing the wing

01 Three ways to be too heavy

Take-off first, because it is the one with a tape measure on it. At each wing area, the heaviest aircraft that still leaves the ground inside 146 m. More wing lifts sooner, so this line rises, and it rises fast enough that take-off is never what stops this aeroplane. Climb is harsher. Holding ten degrees means carrying the aircraft's own weight uphill on 750 W, and the propeller gives less help the slower it goes, so the line rises slowly and stays low.

02 The one that binds

The turn asks for lift, not thrust: fifty degrees of bank multiplies the lift the wing has to make by about 1.6, and a bigger wing pays that bill more easily. It sits above climb everywhere on this sweep. An aircraft has to satisfy all three at once, so what it can weigh is the lowest line at each area. That envelope is the climb line, from end to end. Climb is what sizes this aeroplane, and take-off, the requirement with a number in the rulebook, is nowhere near binding.

03 Where it stops

The envelope keeps rising, so the design point is wherever the sweep is made to stop, and what stops it is the rulebook: 3.048 m of span at this aspect ratio is of area, and the answer sits on that line. Which is worth being suspicious of. Nothing in these three curves charges anything for a bigger wing: there is no structural mass in the model at all, and the team's own README lists adding one as work still to do. A calculation that gets wing area for free will always ask for the maximum, and this one does.

The wing optimiser supplies that missing cost. It minimises take-off mass less the mass of the spar that holds the wing up, so a wing that buys lift by growing pays for it in the same number it is scored on. The sizing above and the optimisation below are not the same aircraft: the constraint analysis is the team's competition sizing at aspect ratio 6.13 and 1.516 m², and the loop below runs on the case the archived optim_vars.mat holds, 2.000 m² at aspect ratio 4.645. What carries across is the question, not the numbers. The sizing says where an area and an aspect ratio come from; the chapter below takes one such pair and asks what shape to make of it.

Moving either slider recomputes nothing in the browser. Every curve here was solved by the ported calculation before the page was built, on the ten wing areas their script sweeps. Two things are held fixed while you move them: the stall margin, at the 1.15 both of their scripts use, and the aerofoil's two drag constants, at the values in the case they committed. So the sliders ask what a different wing shape would do, not what a different aerofoil would do.

Twelve numbers, and one number to judge them by

What was being designed

SAE Aero Design scores payload, so the wing's job is to let the aircraft take off heavier. That single objective is what makes the problem tractable and it is also what makes it a trap: every discipline can improve its own number while the aircraft gets worse. The team's answer was to stop scoring disciplines and score the aircraft.

The objective the optimiser minimised is -(mtom - m_spar): the largest take-off mass the mission allows, minus the mass of the half-wing spar, both in kilograms. The project's own README calls it a surrogate for payload.

The wing itself is four spanwise stations at a fixed 3.048 m span, which is 120 inches and is not a variable. Each station carries a chord and an incidence; the wing carries an area; and three of the four flight conditions carry an angle of attack. That is the twelve. Two are frozen at the root, which fixes the reference the other chords and incidences are measured against, and the chords are constrained to fall monotonically outboard so the optimiser cannot return a planform nobody would build.

One evaluation, all the way through

The loop

Five hundred times, the optimiser handed twelve numbers to a function and got one back. This is that function, run on the twelve numbers it settled on, with every quantity taken from the arrays it produced. The cards are in call order and the arrows between them are the arguments the code passes: the aerodynamics feeds the spar because sizeSpar takes the lift and moment distributions as arguments, and nothing feeds back, because this loop has no aeroelastic iteration in it.

The framework is the team's, written in 2021 by Neil Farvolden, Andrew Ilersich and Adele Crete-Laurence. What is being shown here is one 2024 run of it and what that run decided. MATLAB was not available, so the modules were ported to Python and the port was checked against what the MATLAB wrote: the bending moment to 9e-15 relative, the deflection to 5e-18 m, and the objective to its last digit.

The evaluation

01 The wing

Four chords and four incidences, scaled to the area asked for. The chord holds to a third of the half-span, then two thirds of it goes outboard. The lifting line iterates on circulation across 49 stations until it stops moving. Close to elliptical inboard, shedding load early at the tip: what the taper buys.

02 The structure

Integrate that load twice from the tip in, apply the 3.3 load factor: 326 N m at the root. The tip bends 9.3 mm against a 152 mm limit. Bending is not what sizes this spar. Twist is. The section's pitching moment runs the whole span, and in the turn the tip goes past its two degree limit before the spar is stiff enough.

03 The aircraft

Parasite drag is a component build-up, done once at stall speed. Induced drag comes from the lifting line. The tail's share arrives last: a nose-down moment has to be trimmed. Four flight conditions, four allowable masses, and the aircraft is the smallest of them. Climb-out, and not close: the next constraint would allow 39 % more.

The more efficient section loses, until the box is right

Choosing the aerofoil

Thirty-one aerofoils were run through the whole loop and nine went to a shortlist. Two of them survive as machine-readable tables, complete with the XFOIL polars the run used, so those two are the ones compared here.

Judged as sections, it is not close. FX76MP120 has the better lift-to-drag ratio at every Reynolds number the wing sees and a smaller nose-down pitching moment. Judged on the aircraft, with the same planform, the same mission and the same spar sweep, it loses 2.8 kg of payload surrogate. A decision matrix scoring efficiency and pitching moment would have picked it. That is what the run found, and it is the first half of the answer.

The second half came out of checking it. The optimiser was allowed to choose an angle of attack for the climb, and OptimizerMain.m converts that bound from degrees to radians twice, so the angle was free over 0.26 degrees where the file meant 15. That box is not neutral between two aerofoils. The section with the lower maximum lift coefficient needs more angle to make the same climb lift, and the bound is exactly what it needed. Below is the same comparison run at every ceiling from 0 to 15 degrees.

Two stacked panels: above, local lift coefficient against spanwise station with two curves close together falling toward the tip; below, the wing planform drawn in plan view with its taper and a spanwise station axis. — 2024 Lift against planform, as the MATLAB drew it, on a different wing. The half-span here is 0.5 m and the root chord about 0.22 m, against this run's 1.524 m and 0.758 m, so this is not the wing the chapter is about; I do not know which wing this plot is from. It is kept because it is the shape of the comparison the loop makes at every evaluation: the solid curve is the wing's own spanwise loading and the red dashed one is an ellipse carrying the same total lift, and the gap between them is the induced drag the taper is trying to buy back. This run's version of that comparison is the drawing at the top of this page.

The spar is sized by twist, in the turn

Structure

The spar is not optimised. It is swept: six cap widths by six cap thicknesses, each one integrated for bending deflection and for torsional twist under that flight condition's own loading, and the lightest one that stays inside both limits is charged to the objective. A candidate that misses a limit is not rejected, it is charged a penalty, which is how a hard requirement is turned into something an optimiser can walk down.

Two stacked panels against spanwise station: bending moment falling to zero at the tip with deflection rising, and torsion with torsional deflection, each on twin axes in blue and orange. — 2024 The shape of the argument, on a different wing. This is the sizing script's own plot, kept in the project directory, and its axes are not this run's: the span runs to 0.5 m against this wing's 1.524 m half-span, and the moment, deflection and twist are one to four orders of magnitude below the numbers in the chapter above. I do not know which case it is. What it does show, and what the chapter is arguing, is the shape: bending falls to nothing at the tip while torsional deflection keeps growing, which is why the twist and not the bending is what sizes this spar. This run's own arrays are the dark chart beside it.

Five hundred evaluations, and where they stopped

The search

MATLAB's surrogateopt was given a budget of 500 evaluations and used all of it; it stopped because the budget ran out, not because it converged. The run kept one objective value and one constraint violation per evaluation. Everything that produced them was discarded.

Re-running the whole history through the ported loop recovers the rest: which flight condition bound each candidate, and whether its spar sweep was penalised. It is also five hundred independent checks on the port instead of one.

The wing was built and the aircraft flew

Design, build, fly

The chord distribution above is a stack of laser-cut ribs, the spar caps are carbon over foam, and two airframes, a season apart, each went to competition. A wing that exists is the strongest available statement that the numbers were used.

The aircraft, and the parts it was cut from

Everything above is an optimiser talking to itself. These are what came out the other end. Two airframes, two seasons: UT-24 Minerva at SAE Aero Design West 2024, and UT-25 Stratus, the one this loop sized, at SAE Aero Design East 2025.

Me on a grass field, holding the finished competition aircraft level at chest height: a white high-wing airframe with two tractor motors on the wing, the number 213 on the wing and on the fin, and UT-25 STRATUS lettered on the fuselage side. — Stratus **UT-25 Stratus, competition number 213**, lettered on the fuselage and marked on the wing and the fin. Two tractor motors on the wing, payload pod under the fuselage; a tiltrotor. This is the airframe the loop sized. Stratus took first place in the Advanced Class design report at SAE Aero Design East, Fort Worth, 2025. Photographed by the team.

A different competition aircraft on grass, seen from above and behind: blue and white, single nose engine, the number 216 on the wing and on the fin, UT-24 MINERVA lettered on the fuselage, tail assembly detached and the equipment bay open. — Minerva **UT-24 Minerva, competition number 216**, at tech inspection at SAE Aero Design West 2024. A different airframe and the season before Stratus: single nose engine, blue and white, tail assembly off and the equipment bay open for the inspector. Minerva took first place in the Advanced Class written design report. An aircraft only gets a score after someone with a checklist agrees it is the aircraft that was declared, which is the part of the process no optimiser models.

Stratus part-assembled on a picnic table at a flying field in late winter: wing joined and covered in white film with both motors mounted on the leading edge, fuselage open at the avionics and payload bay, nose cone, a mainwheel, a hex key set and a tape measure on the table beside it. — Stratus The same airframe part-assembled on a picnic table at the flying field, snow still on the ground. Wing joined and covered, both motors on their leading-edge mounts, the fuselage open at the avionics and payload bay, nose cone and a mainwheel still off. Under that film the chord distribution the optimiser chose is a stack of laser-cut ribs, and the spar it sized is carbon over foam. Rib pitch, motor-mount alignment and the fit of the wing to the fuselage are all set at this table, not in the loop.

The two airframes in the air

Photographed by the team at the two competitions. Everything above ends in a chord distribution and a spar cap thickness. This is the same wing at the condition it was sized for.

UT-24 Minerva rolling on a runway with tall grass behind it: a white and blue high-wing airframe on fixed tricycle gear, UTAT lettered on the wing. In the same frame, to the left and closer to the camera, a second much smaller airframe on its own wheel, with a fin carrying the Canadian flag and the number 216. — Minerva **UT-24 Minerva at SAE Aero Design West, 2024**, on the runway, with one of the deployable gliders it carries in the same frame: a separate airframe, on its own wheel, marked with the same competition number 216 and the same flag. This is the season before the one the loop sized. Photographed by the team.

The same aircraft airborne against blue sky and broken cloud, banked hard enough that the wing is nearly edge-on to the camera, with the glider still carried on a slim boom beside the fuselage. — Minerva The same airframe airborne and banked hard, the glider still on it. Nothing in the loop above sees this: every evaluation in it is a wing in steady, trimmed flight, and the roll a pilot uses to make a competition course is outside the model the wing was scored in. Photographed by the team.

UT-25 Stratus standing on a taxiway: a white high-wing airframe with two motors mounted on the wing, a pod fuselage on fixed tricycle gear, sponsor decals across the wing, and competition marquees far out of focus behind it. — Stratus **UT-25 Stratus, competition number 213**, on the taxiway at SAE Aero Design East, Fort Worth, 2025. This is the airframe the loop above sized. Two motors out on the wing, a pod fuselage on fixed tricycle gear, and the tail carried back on a slim boom. Under the covering film, the chord distribution from chapter 03 is a stack of laser-cut ribs. Photographed by the team.

Stratus airborne against clear sky, seen from below and to the left, with the number 213 and rows of sponsor decals legible on the underside of the wing, both motors turning and the V-tail visible at the end of its boom. — Stratus The same aircraft flying, with **213** legible on the wing. The spar in chapter 05 was sized by how far the wing twists in a turn, and this is that wing carrying its own load in the air. Photographed by the team.

Stratus on another pass, seen from almost directly underneath: the full span of the wing across the frame with both motor mounts on it, the fuselage pod below, and the tail boom running back to a V-tail with a red and white marking. — Stratus Another pass, from almost directly underneath, which is the only view here that shows the whole span at once. The motors are out on the wing, and the tail sits well behind it. The loop sized a wing; it did not choose that arrangement, and the arrangement is what the wing had to be good inside. Photographed by the team.

The two mission patches

The team draws one per season. They are the shortest statement anyone on it makes of what the aircraft was built to do, and both of them describe a payload rather than an aeroplane.

The UT-24 Minerva mission patch: a circular badge reading UT-24 MINERVA and UTAT SAE AERO DESIGN, with a great horned owl over a treeline at dusk, the aircraft in the foreground, a second smaller aircraft climbing away at the upper left, curved yellow tracks across the ground, and a red and white target marked on it. — 2024 **UT-24 Minerva.** An owl over a treeline, the aircraft below it, a second smaller aircraft leaving on a curved yellow track, and a target marked on the ground. The glider in the runway photograph above is the object those tracks belong to. The sponsor marks inside the circle are the team's.

The UT-25 Stratus mission patch: a circular badge reading UTAT SAE AERO DESIGN and UT-25 STRATUS, with the twin-motor aircraft drawn on short final, gear down, over a large black and white fiducial marker laid flat in grass. — 2025 **UT-25 Stratus.** The aircraft on short final, gear down, over a black and white fiducial marker laid flat in the grass. The marker is the thing the aircraft has to find, and it is drawn at the centre of the badge rather than the aeroplane.

The cut sheets

The optimiser's output is a set of numbers. What a laser cutter needs is a nested sheet, and the translation between them is where a geometry either survives manufacture or does not.

A laser cut sheet drawn in red line: dozens of nested wing rib profiles of decreasing chord with lightening holes and spar cutouts, plus aileron and control surface parts, packed to fill a rectangular sheet. — 2024 Ribs and control surfaces, nested for one sheet. Every rib is a different chord because the wing tapers, and every one carries the same two spar cutouts, which is what makes the spar sizing above a manufacturing constraint and not just a structural one.

A second laser cut sheet of long narrow spar webs and stiffeners in red line, nested end to end across the sheet. — 2024 Spar webs and stiffeners. The bending and torsion curves in the chapter above end here, as a set of parts with a length and a grain direction.

What I would do differently

Earlier engagement with manufacturing. The cleanest aerodynamic choice is not the right one if the build schedule cannot absorb it, and I would now spend the first design week in the shop with the structures and build leads before the first lifting-line run. The cut sheets above are why: every rib on them is the consequence of a number chosen weeks earlier by someone looking at a drag polar.