import aerosandbox.numpy as np
from aerosandbox.geometry.airfoil.airfoil import Airfoil
from aerosandbox.geometry.airfoil.airfoil_families import get_kulfan_parameters
from aerosandbox.modeling.splines.hermite import linear_hermite_patch, cubic_hermite_patch, cosine_hermite_patch
from typing import Union, Dict, List
import warnings
[docs]class KulfanAirfoil(Airfoil):
def __init__(self,
name: str = "Untitled",
lower_weights: np.ndarray = None,
upper_weights: np.ndarray = None,
leading_edge_weight: float = 0.,
TE_thickness: float = 0.,
N1: float = 0.5,
N2: float = 1.0,
):
### Handle the airfoil name
self.name = name
### Check to see if the airfoil is a "known" airfoil, based on its name.
if (
lower_weights is None and
upper_weights is None
): # Try to fall back on parameters from the coordinate airfoil, if it's something from the UIUC database
class IgnoreUserWarnings: # A context manager to ignore UserWarnings
def __enter__(self):
warnings.filterwarnings("ignore", category=UserWarning)
def __exit__(self, exc_type, exc_val, exc_tb):
warnings.resetwarnings()
with IgnoreUserWarnings():
coordinate_airfoil = Airfoil(name)
if coordinate_airfoil.coordinates is None:
raise ValueError("You must either:\n"
"\t* Specify both `lower_weights` and `upper_weights`, at minimum"
"\t* Give an airfoil `name` corresponding to an airfoil in the UIUC database, or a NACA airfoil.")
else:
parameters = get_kulfan_parameters(
coordinates=coordinate_airfoil.coordinates,
n_weights_per_side=8,
N1=N1,
N2=N2,
normalize_coordinates=True,
use_leading_edge_modification=True,
)
lower_weights = parameters["lower_weights"]
upper_weights = parameters["upper_weights"]
leading_edge_weight = parameters["leading_edge_weight"]
TE_thickness = parameters["TE_thickness"]
### Handle the Kulfan parameters
self.lower_weights = lower_weights
self.upper_weights = upper_weights
self.leading_edge_weight = leading_edge_weight
self.TE_thickness = TE_thickness
self.N1 = N1
self.N2 = N2
[docs] def __repr__(self) -> str:
return f"Airfoil {self.name} (Kulfan / CST parameterization)"
@property
[docs] def kulfan_parameters(self):
return {
"lower_weights" : self.lower_weights,
"upper_weights" : self.upper_weights,
"leading_edge_weight": self.leading_edge_weight,
"TE_thickness" : self.TE_thickness,
}
@property
[docs] def coordinates(self) -> np.ndarray:
return self.to_airfoil().coordinates
@coordinates.setter
def coordinates(self, value):
raise TypeError("The coordinates of a `KulfanAirfoil` can't be modified directly, "
"as they're a function of the Kulfan parameters.\n"
"Instead, you can either modify the Kulfan parameters directly, or use the "
"more general (coordinate-parameterized) `asb.Airfoil` class.")
[docs] def to_airfoil(self,
n_coordinates_per_side=200,
spacing_function_per_side=np.cosspace,
) -> Airfoil:
x_upper = spacing_function_per_side(1, 0, n_coordinates_per_side)[:-1]
upper = self.upper_coordinates(x_over_c=x_upper)
x_lower = spacing_function_per_side(0, 1, n_coordinates_per_side)
lower = self.lower_coordinates(x_over_c=x_lower)
return Airfoil(
name=self.name,
coordinates=np.concatenate([
upper,
lower
], axis=0)
)
[docs] def repanel(self,
n_points_per_side: int = 100,
spacing_function_per_side=np.cosspace,
) -> 'Airfoil':
return self.to_airfoil(
n_coordinates_per_side=n_points_per_side,
spacing_function_per_side=spacing_function_per_side
)
[docs] def normalize(
self,
return_dict: bool = False,
) -> Union['KulfanAirfoil', Dict[str, Union['KulfanAirfoil', float]]]:
"""
Returns a copy of the Airfoil with a new set of `coordinates`, such that:
- The leading edge (LE) is at (0, 0)
- The trailing edge (TE) is at (1, 0)
- The chord length is equal to 1
The trailing-edge (TE) point is defined as the midpoint of the line segment connecting the first and last coordinate points (upper and lower surface TE points, respectively). The TE point is not necessarily one of the original points in the airfoil coordinates (`Airfoil.coordinates`); in general, it will not be one of the points if the TE thickness is nonzero.
The leading-edge (LE) point is defined as the coordinate point with the largest Euclidian distance from the trailing edge. (In other words, if you were to center a circle on the trailing edge and progressively grow it, what's the last coordinate point that it would intersect?) The LE point is always one of the original points in the airfoil coordinates.
The chord is defined as the Euclidian distance between the LE and TE points.
Coordinate modifications to achieve the constraints described above (LE @ origin, TE at (1, 0), and chord of 1) are done by means of a translation and rotation.
Args:
return_dict: Determines the output type of the function.
- If `False` (default), returns a copy of the Airfoil with the new coordinates.
- If `True`, returns a dictionary with keys:
- "airfoil": a copy of the Airfoil with the new coordinates
- "x_translation": the amount by which the airfoil's LE was translated in the x-direction
- "y_translation": the amount by which the airfoil's LE was translated in the y-direction
- "scale_factor": the amount by which the airfoil was scaled (if >1, the airfoil had to get
bigger)
- "rotation_angle": the angle (in degrees) by which the airfoil was rotated about the LE.
Sign convention is that positive angles rotate the airfoil counter-clockwise.
All of thes values represent the "required change", e.g.:
- "x_translation" is the amount by which the airfoil's LE had to be translated in the
x-direction to get it to the origin.
- "rotation_angle" is the angle (in degrees) by which the airfoil had to be rotated (CCW).
Returns: Depending on the value of `return_dict`, either:
- A copy of the airfoil with the new coordinates (default), or
- A dictionary with keys "airfoil", "x_translation", "y_translation", "scale_factor", and "rotation_angle".
documentation for `return_tuple` for more information.
"""
### A KulfanAirfoil is already normalized by definition, so we essentially just return the same airfoil.
# (This method serves to overwrite the Airfoil.normalize() method, which is not applicable to KulfanAirfoils.)
if not return_dict:
return self
else:
return {
"airfoil" : self,
"x_translation" : 0,
"y_translation" : 0,
"scale_factor" : 1,
"rotation_angle": 0
}
[docs] def draw(self,
*args,
draw_markers=False,
**kwargs
):
return self.to_airfoil().draw(
*args,
draw_markers=draw_markers,
**kwargs
)
[docs] def get_aero_from_neuralfoil(self,
alpha: Union[float, np.ndarray],
Re: Union[float, np.ndarray],
mach: Union[float, np.ndarray] = 0.,
n_crit: Union[float, np.ndarray] = 9.0,
xtr_upper: Union[float, np.ndarray] = 1.0,
xtr_lower: Union[float, np.ndarray] = 1.0,
model_size: str = "large",
control_surfaces: List["ControlSurface"] = None,
include_360_deg_effects: bool = True,
) -> Dict[str, Union[float, np.ndarray]]:
### Validate inputs
if (
(np.length(self.lower_weights) != 8) or
(np.length(self.upper_weights) != 8)
):
raise NotImplementedError("NeuralFoil is only trained to handle exactly 8 CST coefficients per side.")
if (
self.N1 != 0.5 or
self.N2 != 1.0
):
raise NotImplementedError("NeuralFoil is only trained to handle airfoils with N1 = 0.5 and N2 = 1.0.")
### Set up inputs
if control_surfaces is None:
control_surfaces = []
alpha = np.mod(alpha + 180, 360) - 180 # Enforce periodicity of alpha
##### Evaluate the control surfaces of the airfoil
effective_d_alpha = 0.
effective_CD_multiplier_from_control_surfaces = 1.
for surf in control_surfaces:
effectiveness = 1 - np.maximum(0, surf.hinge_point + 1e-16) ** 2.751428551177291
# From XFoil-based study at `/AeroSandbox/studies/ControlSurfaceEffectiveness/`
effective_d_alpha += surf.deflection * effectiveness
effective_CD_multiplier_from_control_surfaces *= (
2 + (surf.deflection / 11.5) ** 2 - (1 + (surf.deflection / 11.5) ** 2) ** 0.5
)
# From fit to wind tunnel data from Hoerner, "Fluid Dynamic Drag", 1965. Page 13-13, Figure 32,
# "Variation of section drag coefficient of a horizontal tail surface at constant C_L"
##### Use NeuralFoil to evaluate the incompressible aerodynamics of the airfoil
import neuralfoil as nf
nf_aero = nf.get_aero_from_kulfan_parameters(
kulfan_parameters=dict(
lower_weights=self.lower_weights,
upper_weights=self.upper_weights,
leading_edge_weight=self.leading_edge_weight,
TE_thickness=self.TE_thickness,
),
alpha=alpha + effective_d_alpha,
Re=Re,
n_crit=n_crit,
xtr_upper=xtr_upper,
xtr_lower=xtr_lower,
model_size=model_size
)
CL = nf_aero["CL"]
CD = nf_aero["CD"] * effective_CD_multiplier_from_control_surfaces
CM = nf_aero["CM"]
# Cpmin_0 = nf_aero["Cpmin"]
Cpmin_0 = np.softmin(
*[
1 - nf_aero[f"upper_bl_ue/vinf_{i}"] ** 2
for i in range(len(nf.bl_x_points))
],
*[
1 - nf_aero[f"lower_bl_ue/vinf_{i}"] ** 2
for i in range(len(nf.bl_x_points))
],
softness=0.01
)
Top_Xtr = nf_aero["Top_Xtr"]
Bot_Xtr = nf_aero["Bot_Xtr"]
##### Extend aerodynamic data to 360 degrees (post-stall) using wind tunnel behavior here.
if include_360_deg_effects:
from aerosandbox.aerodynamics.aero_2D.airfoil_polar_functions import airfoil_coefficients_post_stall
CL_if_separated, CD_if_separated, CM_if_separated = airfoil_coefficients_post_stall(
airfoil=self,
alpha=alpha
)
import aerosandbox.library.aerodynamics as lib_aero
# These values are set relatively high because NeuralFoil extrapolates quite well past stall
alpha_stall_positive = 20
alpha_stall_negative = -20
# This will be an input to a tanh() sigmoid blend via asb.numpy.blend(), so a value of 1 means the flow is
# ~90% separated, and a value of -1 means the flow is ~90% attached.
is_separated = np.softmax(
alpha - alpha_stall_positive,
alpha_stall_negative - alpha
) / 3
CL = np.blend(
is_separated,
CL_if_separated,
CL
)
CD = np.exp(np.blend(
is_separated,
np.log(CD_if_separated + lib_aero.Cf_flat_plate(Re_L=Re, method="turbulent")),
np.log(CD)
))
CM = np.blend(
is_separated,
CM_if_separated,
CM
)
"""
Separated Cpmin_0 model is a very rough fit to Figure 3 of:
Shademan & Naghib-Lahouti, "Effects of aspect ratio and inclination angle on aerodynamic loads of a flat
plate", Advances in Aerodynamics.
https://www.researchgate.net/publication/342316140_Effects_of_aspect_ratio_and_inclination_angle_on_aerodynamic_loads_of_a_flat_plate
"""
Cpmin_0 = np.blend(
is_separated,
-1 - 0.5 * np.sind(alpha) ** 2,
Cpmin_0
)
Top_Xtr = np.blend(
is_separated,
0.5 - 0.5 * np.tanh(10 * np.sind(alpha)),
Top_Xtr
)
Bot_Xtr = np.blend(
is_separated,
0.5 + 0.5 * np.tanh(10 * np.sind(alpha)),
Bot_Xtr
)
###### Add compressibility effects
### Step 1: compute mach_crit, the critical Mach number
"""
Below is a function that computes the critical Mach number from the incompressible Cp_min.
It's based on a Laitone-rule compressibility correction (similar to Prandtl-Glauert or Karman-Tsien,
but higher order), together with the Cp_sonic relation. When the Laitone-rule Cp equals Cp_sonic, we have reached
the critical Mach number.
This approach does not admit explicit solution for the Cp0 -> M_crit relation, so we instead regress a
relationship out using symbolic regression. In effect, this is a curve fit to synthetic data.
See fits at: /AeroSandbox/studies/MachFitting/CriticalMach/
"""
Cpmin_0 = np.softmin(
Cpmin_0,
0,
softness=0.001
)
mach_crit = (
1.011571026701678
- Cpmin_0
+ 0.6582431351007195 * (-Cpmin_0) ** 0.6724789439840343
) ** -0.5504677038358711
mach_dd = mach_crit + (0.1 / 320) ** (1 / 3) # drag divergence Mach number
# Relation taken from W.H. Mason's Korn Equation
### Step 2: adjust CL, CD, CM, Cpmin by compressibility effects
gamma = 1.4 # Ratio of specific heats, 1.4 for air (mostly diatomic nitrogen and oxygen)
beta_squared_ideal = 1 - mach ** 2
beta = np.softmax(
beta_squared_ideal,
-beta_squared_ideal,
softness=0.5 # Empirically tuned to data
) ** 0.5
CL = CL / beta
# CD = CD / beta
CM = CM / beta
# Prandtl-Glauert
Cpmin = Cpmin_0 / beta
# Karman-Tsien
# Cpmin = Cpmin_0 / (
# beta
# + mach ** 2 / (1 + beta) * (Cpmin_0 / 2)
# )
# Laitone's rule
# Cpmin = Cpmin_0 / (
# beta
# + (mach ** 2) * (1 + (gamma - 1) / 2 * mach ** 2) / (1 + beta) * (Cpmin_0 / 2)
# )
### Step 3: modify CL based on buffet and supersonic considerations
# Accounts approximately for the lift drop due to buffet.
buffet_factor = np.blend(
50 * (mach - (mach_dd + 0.04)), # Tuned to RANS CFD data empirically
np.blend(
(mach - 1) / 0.1,
1,
0.5
),
1,
)
# Accounts for the fact that theoretical CL_alpha goes from 2 * pi (subsonic) to 4 (supersonic),
# following linearized supersonic flow on a thin airfoil.
cla_supersonic_ratio_factor = np.blend(
(mach - 1) / 0.1,
4 / (2 * np.pi),
1,
)
CL = CL * buffet_factor * cla_supersonic_ratio_factor
# Step 4: Account for wave drag
t_over_c = self.max_thickness()
CD_wave = np.where(
mach < mach_crit,
0,
np.where(
mach < mach_dd,
80 * (mach - mach_crit) ** 4,
np.where(
mach < 1.1,
cosine_hermite_patch(
mach,
x_a=mach_dd,
x_b=1.1,
f_a=80 * (0.1 / 320) ** (4 / 3),
f_b=0.8 * t_over_c,
dfdx_a=0.1,
dfdx_b=-0.8 * t_over_c * 8,
),
np.blend(
8 * 2 * (mach - 1.1) / (1.2 - 0.8),
0.8 * 0.8 * t_over_c,
1.2 * 0.8 * t_over_c,
)
)
)
)
CD = CD + CD_wave
# Step 5: If beyond M_crit or if separated, move the airfoil aerodynamic center back to x/c = 0.5 (Mach tuck)
has_aerodynamic_center_shift = (mach - (mach_dd + 0.06)) / 0.06
if include_360_deg_effects:
has_aerodynamic_center_shift = np.softmax(
is_separated,
has_aerodynamic_center_shift,
softness=0.1
)
CM = CM + np.blend(
has_aerodynamic_center_shift,
-0.25 * np.cosd(alpha) * CL - 0.25 * np.sind(alpha) * CD,
0,
)
N = len(nf.bl_x_points)
return {
"analysis_confidence": nf_aero["analysis_confidence"],
"CL" : CL,
"CD" : CD,
"CM" : CM,
"Cpmin" : Cpmin,
"Top_Xtr" : Top_Xtr,
"Bot_Xtr" : Bot_Xtr,
"mach_crit": mach_crit,
"mach_dd" : mach_dd,
"Cpmin_0" : Cpmin_0,
**{f"upper_bl_theta_{i}": nf_aero[f"upper_bl_theta_{i}"] for i in range(N)},
**{f"upper_bl_H_{i}": nf_aero[f"upper_bl_H_{i}"] for i in range(N)},
**{f"upper_bl_ue/vinf_{i}": nf_aero[f"upper_bl_ue/vinf_{i}"] for i in range(N)},
**{f"lower_bl_theta_{i}": nf_aero[f"lower_bl_theta_{i}"] for i in range(N)},
**{f"lower_bl_H_{i}": nf_aero[f"lower_bl_H_{i}"] for i in range(N)},
**{f"lower_bl_ue/vinf_{i}": nf_aero[f"lower_bl_ue/vinf_{i}"] for i in range(N)},
}
[docs] def upper_coordinates(self,
x_over_c: Union[float, np.ndarray] = np.linspace(1, 0, 101),
) -> np.ndarray:
x_over_c = np.array(x_over_c)
# Class function
C = (x_over_c) ** self.N1 * (1 - x_over_c) ** self.N2
from scipy.special import comb
def shape_function(w):
# Shape function (Bernstein polynomials)
N = np.length(w) - 1 # Order of Bernstein polynomials
K = comb(N, np.arange(N + 1)) # Bernstein polynomial coefficients
dims = (np.length(w), np.length(x_over_c))
def wide(vector):
return np.tile(np.reshape(vector, (1, dims[1])), (dims[0], 1))
def tall(vector):
return np.tile(np.reshape(vector, (dims[0], 1)), (1, dims[1]))
S_matrix = (
tall(K) * wide(x_over_c) ** tall(np.arange(N + 1)) *
wide(1 - x_over_c) ** tall(N - np.arange(N + 1))
) # Bernstein polynomial coefficients * weight matrix
S_x = np.sum(tall(w) * S_matrix, axis=0)
# Calculate y output
y = C * S_x
return y
y_upper = shape_function(self.upper_weights)
# Add trailing-edge (TE) thickness
y_upper += x_over_c * self.TE_thickness / 2
# Add Kulfan's leading-edge-modification (LEM)
y_upper += self.leading_edge_weight * (x_over_c) * (1 - x_over_c) ** (np.length(self.upper_weights) + 0.5)
return np.stack((
np.reshape(x_over_c, (-1)),
y_upper,
), axis=1)
[docs] def lower_coordinates(self,
x_over_c: Union[float, np.ndarray] = np.linspace(0, 1, 101),
) -> np.ndarray:
x_over_c = np.array(x_over_c)
# Class function
C = (x_over_c) ** self.N1 * (1 - x_over_c) ** self.N2
from scipy.special import comb
def shape_function(w):
# Shape function (Bernstein polynomials)
N = np.length(w) - 1 # Order of Bernstein polynomials
K = comb(N, np.arange(N + 1)) # Bernstein polynomial coefficients
dims = (np.length(w), np.length(x_over_c))
def wide(vector):
return np.tile(np.reshape(vector, (1, dims[1])), (dims[0], 1))
def tall(vector):
return np.tile(np.reshape(vector, (dims[0], 1)), (1, dims[1]))
S_matrix = (
tall(K) * wide(x_over_c) ** tall(np.arange(N + 1)) *
wide(1 - x_over_c) ** tall(N - np.arange(N + 1))
) # Bernstein polynomial coefficients * weight matrix
S_x = np.sum(tall(w) * S_matrix, axis=0)
# Calculate y output
y = C * S_x
return y
y_lower = shape_function(self.lower_weights)
# Add trailing-edge (TE) thickness
y_lower -= x_over_c * self.TE_thickness / 2
# Add Kulfan's leading-edge-modification (LEM)
y_lower += self.leading_edge_weight * (x_over_c) * (1 - x_over_c) ** (np.length(self.lower_weights) + 0.5)
return np.stack((
np.reshape(x_over_c, (-1)),
y_lower
), axis=1)
[docs] def local_camber(self,
x_over_c: Union[float, np.ndarray] = np.linspace(0, 1, 101),
) -> Union[float, np.ndarray]:
upper = self.upper_coordinates(x_over_c=x_over_c)
lower = self.lower_coordinates(x_over_c=x_over_c)
if np.isscalar(x_over_c):
return (upper[0, 1] + lower[0, 1]) / 2
else:
return (upper[:, 1] + lower[:, 1]) / 2
[docs] def local_thickness(self,
x_over_c: Union[float, np.ndarray] = np.linspace(0, 1, 101),
) -> Union[float, np.ndarray]:
upper = self.upper_coordinates(x_over_c=x_over_c)
lower = self.lower_coordinates(x_over_c=x_over_c)
if np.isscalar(x_over_c):
return (upper[0, 1] - lower[0, 1])
else:
return (upper[:, 1] - lower[:, 1])
[docs] def LE_radius(self, relative_softness: float = 0.03):
# LE_radius_upper = np.where(
# self.upper_weights[0] > 0,
# self.upper_weights[0] ** 2,
# 0
# ) / 2
# LE_radius_lower = np.where(
# self.lower_weights[0] < 0,
# self.lower_weights[0] ** 2,
# 0
# ) / 2
return np.softmin_scalefree(
np.where(
self.upper_weights[0] > 0,
self.upper_weights[0],
0
) ** 2,
np.where(
self.lower_weights[0] < 0,
self.lower_weights[0],
0
) ** 2,
relative_softness=relative_softness
) / 2
[docs] def TE_angle(self):
return np.degrees(
np.arctan(self.upper_weights[-1]) -
np.arctan(self.lower_weights[-1]) +
np.arctan(self.TE_thickness)
)
[docs] def area(self):
def get_area_of_side(weights):
from scipy.special import beta, comb
N = np.length(weights) - 1
i = np.arange(N + 1)
area_of_each_mode = comb(N, i) * beta(
self.N1 + i + 1,
self.N2 + N - i + 1
)
return np.sum(
area_of_each_mode * weights
)
return (
get_area_of_side(self.upper_weights) -
get_area_of_side(self.lower_weights) +
(self.TE_thickness / 2)
)
[docs] def set_TE_thickness(self,
thickness: float = 0.,
) -> 'KulfanAirfoil':
"""
Creates a modified copy of the KulfanAirfoil that has a specified trailing-edge thickness.
Note that the trailing-edge thickness is given nondimensionally (e.g., as a fraction of chord).
Args:
thickness: The target trailing-edge thickness, given nondimensionally (e.g., as a fraction of chord).
Returns: The modified KulfanAirfoil.
"""
return KulfanAirfoil(
name=self.name,
lower_weights=self.lower_weights,
upper_weights=self.upper_weights,
leading_edge_weight=self.leading_edge_weight,
TE_thickness=thickness,
N1=self.N1,
N2=self.N2,
)
[docs] def scale(self,
scale_x: float = 1.,
scale_y: float = 1.,
) -> "KulfanAirfoil":
"""
Scales a KulfanAirfoil about the origin.
Args:
scale_x: Amount to scale in the x-direction. Note: not supported by KulfanAirfoil due to inherent
limitations of parameterization; only given here so that argument symmetry to Airfoil.scale() is
retained. Raises a ValueError if modified, along with instructions to use `Airfoil` if needed.
scale_y: Amount to scale in the y-direction. Scaling by a negative y-value will result in `lower_weights`
and `upper_weights` being flipped as appropriate.
Returns: A copy of the KulfanAirfoil with appropriate scaling applied.
"""
if scale_x != 1:
raise ValueError("\n".join([
"Scaling a KulfanAirfoil in the x-direction is not supported due to inherent limitations of the",
"Kulfan parameterization. If you need to scale in the x-direction: "
"\t- Convert to an Airfoil first (`KulfanAirfoil.to_airfoil()`)"
"\t- Scale it (`Airfoil.scale()`)"
"\t- Convert back to a KulfanAirfoil (`Airfoil.to_kulfan_airfoil()`)"
]))
if scale_y >= 0:
return KulfanAirfoil(
name=self.name,
lower_weights=self.lower_weights * scale_y,
upper_weights=self.upper_weights * scale_y,
leading_edge_weight=self.leading_edge_weight * scale_y,
TE_thickness=self.TE_thickness * scale_y,
N1=self.N1,
N2=self.N2,
)
else:
return KulfanAirfoil(
name=self.name,
lower_weights=self.upper_weights * scale_y,
upper_weights=self.lower_weights * scale_y,
leading_edge_weight=self.leading_edge_weight * scale_y,
TE_thickness=self.TE_thickness * (-1 * scale_y),
N1=self.N1,
N2=self.N2,
)
[docs] def blend_with_another_airfoil(self,
airfoil: Union["KulfanAirfoil", Airfoil],
blend_fraction: float = 0.5,
) -> "KulfanAirfoil":
if not isinstance(airfoil, KulfanAirfoil):
try:
airfoil = airfoil.to_kulfan_airfoil()
except AttributeError:
raise TypeError("The `airfoil` argument should be either a `KulfanAirfoil` or an `Airfoil`.\n"
f"You gave an object of type \"{type(airfoil)}\".")
foil_a = self
foil_b = airfoil
a_fraction = 1 - blend_fraction
b_fraction = blend_fraction
### Determine parameters for the blended airfoil
name = f"{a_fraction * 100:.0f}% {self.name}, {b_fraction * 100:.0f}% {airfoil.name}"
return KulfanAirfoil(
name=name,
lower_weights=a_fraction * foil_a.lower_weights + b_fraction * foil_b.lower_weights,
upper_weights=a_fraction * foil_a.upper_weights + b_fraction * foil_b.upper_weights,
leading_edge_weight=a_fraction * foil_a.leading_edge_weight + b_fraction * foil_b.leading_edge_weight,
TE_thickness=a_fraction * foil_a.TE_thickness + b_fraction * foil_b.TE_thickness,
N1=a_fraction * foil_a.N1 + b_fraction * foil_b.N1,
N2=a_fraction * foil_a.N2 + b_fraction * foil_b.N2,
)