NNFoil

KulfanParameters{T,V}

Parameter container for the Kulfan (CST) airfoil shape parameterization.

This type stores the upper and lower Bernstein polynomial weights, along with leading-edge and trailing-edge scalar parameters, all sharing a common floating-point element type T. The coordinate arrays are stored with a concrete vector type V<:AbstractVector{T} (e.g. Vector{T}, SVector{N,T}).

Type parameters

• T<:Real: floating-point element type.
• V<:AbstractVector{T}: concrete vector type used for the weight arrays.

Fields

• upper_weights::V: weights for the upper surface (commonly length 8).
• lower_weights::V: weights for the lower surface (commonly length 8).
• leading_edge_weight::T: scalar parameter controlling leading-edge thickness/rounding.
• trailing_edge_thickness::T: scalar trailing-edge thickness parameter.

KulfanParameters(upper_weights, lower_weights, leading_edge_weight,
    trailing_edge_thickness)

Converting constructor for KulfanParameters that promotes all inputs to a common floating-point type T and returns a KulfanParameters{T, V} where V matches the concrete vector type of the provided weights arrays (after promotion).

Arguments

• upper_weights::AbstractVector{<:Real}: upper weights.
• lower_weights::AbstractVector{<:Real}: lower weights.
• leading_edge_weight::Real: LE scalar parameter.
• trailing_edge_thickness::Real: TE thickness parameter.

NeuralNetworkOutput{V}

Stores the aerodynamic coefficients predicted by the neural network.

Type Parameters

• V<:AbstractVector{<:Real}: numeric vector type used for all output quantities.

Fields

• analysis_confidence::V: confidence level of the neural network prediction.
• CL::V: lift coefficient values.
• CD::V: drag coefficient values.
• CM::V: moment coefficient values.
• Top_Xtr::V: transition location on the upper surface.
• Bot_Xtr::V: transition location on the lower surface.

NeuralNetworkParameters{R, V, M, W, B}

Stores parameters of the pretrained neural network model.

Type Parameters

• R<:Real: numeric type used for all elements.
• V<:AbstractVector{R}: vector type (for input means and biases).
• M<:AbstractMatrix{R}: matrix type (for covariances and weight matrices).
• W<:AbstractVector{M}: collection of weight matrices, one per layer.
• B<:AbstractVector{V}: collection of bias vectors, one per layer.

Fields

• mean_inputs_scaled::V: mean values of the scaled input features.
• cov_inputs_scaled::M: covariance matrix of the scaled inputs.
• inv_cov_inputs_scaled::M: inverse of the covariance matrix.
• weights::W: vector of weight matrices for each layer.
• biases::B: vector of bias vectors for each layer.

__evaluate_aero(network_parameters, x) -> NeuralNetworkOutput

Evaluate the neural network for an input x using the pretrained parameters network_parameters, performing symmetry fusion and applying post-processing transformations to produce physically meaningful aerodynamic coefficients.

Arguments

• network_parameters: Pretrained parameters of the neural network.
• x::AbstractMatrix: Input features for the original flow condition. Each column corresponds to one input sample.

Returns

A NeuralNetworkOutput struct with fields:

• analysis_confidence: confidence level (0–1)
• CL: lift coefficient
• CD: drag coefficient
• CM: moment coefficient
• Top_Xtr: upper-surface transition location (0–1)
• Bot_Xtr: lower-surface transition location (0–1)

__flip_x!(x)

Flip the input vector or matrix x in-place, creating a geometrically mirrored version of the input features.

Arguments

• x::AbstractVecOrMat{<:Real}: Input vector or matrix where each column represents a sample. The flipping is applied across specific rows.

Returns

• x: The modified input matrix, flipped in-place.

airfoil_cst(x, parameters, x_split_id; N1=0.5, N2=1.0)

Reconstruct an airfoil surface from Kulfan (CST) parameters.

Arguments

• x: Vector of nondimensional chordwise coordinates (0–1)
• parameters::AbstractVector: upper and lower weights, leading-edge weight, trailing-edge thickness
• x_split_id::Int: Index separating upper and lower surface coordinates
• N1::Real: Leading-edge exponent (default: 0.5)
• N2::Real: Trailing-edge exponent (default: 1.0)

Returns

• Vector: Airfoil surface y-coordinates corresponding to x

airfoil_cst_zero_trailing_edge(x, parameters, x_split_id; N1=0.5, N2=1.0)

Reconstruct an airfoil surface from Kulfan (CST) parameters with a zero trailing-edge gap.

Arguments

• x: Vector of nondimensional chordwise coordinates (0–1)
• parameters::AbstractVector: upper and lower weights, and leading-edge weight
• x_split_id::Int: Index separating upper and lower surface coordinates
• N1::Real: Leading-edge exponent (default: 0.5)
• N2::Real: Trailing-edge exponent (default: 1.0)

Returns

• Vector: Airfoil surface y-coordinates corresponding to x

bernstein(x, v, n)

Evaluate the Bernstein basis polynomial of degree n and index v at x.

Arguments

• x: Evaluation points (scalar, vector, or array).
• v::Signed: Index of the basis polynomial.
• n::Signed: Degree of the polynomial.

Returns

• Array of the same shape as x: Values of the Bernstein polynomial.

class_function(x, N1, N2)

Evaluate the class function used in Kulfan’s parametrization.

Arguments

• x: Nondimensional chordwise coordinates [0–1].
• N1::Real: Leading-edge exponent.
• N2::Real: Trailing-edge exponent.

Returns

• Array of the same shape as x: Values of the class function

cst(x, coefficients, leading_edge_weight, trailing_edge_thickness; N1=0.5, N2=1.0)

CST (Class–Shape Transformation) airfoil parametrization.

Arguments

• x: Nondimensional chordwise coordinates [0, 1]
• coefficients::AbstractVector: Shape function weights
• leading_edge_weight::Real: Leading-edge modification term
• trailing_edge_thickness::Real: Trailing-edge thickness parameter
• N1::Real: Leading-edge exponent (default: 0.5)
• N2::Real: Trailing-edge exponent (default: 1.0)

Returns

• Same shape as x: Airfoil surface coordinates defined by the CST parametrization

get_aero_from_kulfan_parameters(
    network_parameters,
    kulfan_parameters,
    alpha,
    Reynolds;
    n_crit=9,
    xtr_upper=1,
    xtr_lower=1
) -> NeuralNetworkOutput

Compute aerodynamic coefficients from Kulfan airfoil parameters using a pretrained neural network.

This function supports scalar and vector inputs for both the angle of attack alpha and the Reynolds number. If either is a scalar while the other is a vector, the scalar will be broadcasted to match the length of the vector.

Arguments

• network_parameters::NeuralNetworkParameters: Pretrained neural network parameters.
• kulfan_parameters::KulfanParameters: Kulfan shape parameters describing the airfoil.
• alpha: Angle(s) of attack in degrees (Real or AbstractVector{<:Real}).
• Reynolds: Reynolds number(s) corresponding to each alpha (Real or AbstractVector{<:Real}).
• n_crit::Real=9: Critical amplification factor for transition prediction.
• xtr_upper::Real=1: Forced transition location on the upper surface.
• xtr_lower::Real=1: Forced transition location on the lower surface.

Returns

• NeuralNetworkOutput: Predicted aerodynamic coefficients.

get_kulfan_parameters(coordinates; num_coefficients=8, N1=0.5, N2=1.0)

Fit Kulfan (CST) parameters to airfoil coordinates.

Arguments

• coordinates::AbstractMatrix: Airfoil coordinates with columns [x, y]
• num_coefficients::Int: Number of Bernstein polynomial coefficients per surface (default: 8)
• N1::Real: Leading-edge exponent (default: 0.5)
• N2::Real: Trailing-edge exponent (default: 1.0)

Returns

• KulfanParameters: Fitted upper and lower weights, leading-edge weight, and trailing-edge thickness

load_network_parameters(; model_size=:xlarge, T=Float64)

Load and convert the pretrained neural network parameters used by NeuralFoil.

Arguments

• model_size::Symbol=:xlarge: Size of the pretrained model parameters to load.
• T::Type=Float64: Numerical type to which all loaded arrays will be converted.

net(network_parameters::NetworkParameters, x::AbstractMatrix{<:Real})

Evaluate the neural network using the pretrained network parameters on the given input x.

Arguments

• network_parameters::NetworkParameters: pretrained network weights and biases.
• x::AbstractArray{<:Real}: Input data.

Returns

• AbstractMatrix{<:Real}

normalize_coordinates!(coordinates)

Normalize the input coordinates in place so that the x values lie within the unit interval [0, 1].

Arguments

• coordinates::AbstractMatrix: Matrix of airfoil coordinates with columns representing the x and y values.

shape_function(x, coefficients)

Kulfan shape function defined as a weighted sum of Bernstein polynomials.

Arguments

• x: Nondimensional chordwise coordinates [0, 1].
• coefficients::AbstractVector: Weights for the Bernstein polynomials.

Returns

• Same shape as x: Values of the shape function.

sigmoid(x::T; ln_eps=log(10 / floatmax(eltype(T)))) where {T}

Apply the sigmoid activation function elementwise. Also employ input clipping for numerical stability.

Arguments

• x::T: Input value.
• ln_eps::Real=log(10 / floatmax(eltype(T))): Logarithmic bound used to clip input values for stability.

Returns

• Scalar or Array of the same shape as x: Values in the range (0, 1).

split_upper_lower_surfaces(coordinates)

Split airfoil coordinates into upper and lower surfaces.

Arguments

• coordinates::AbstractMatrix: Matrix of airfoil coordinates with columns representing the x and y values.

Returns

• (upper, lower): Two matrices containing the coordinates of the upper and lower surfaces, respectively.

squared_mahalanobis_distance(network_parameters::NetworkParameters,
    x::AbstractMatrix{<:Real})

Compute the squared Mahalanobis distance between input samples and the mean of the scaled input distribution.

Arguments

• network_parameters::NetworkParameters: pretrained neural network parameters containing the mean and inverse covariance of the scaled input distribution.
• x::AbstractArray{<:Real}: Input samples.

Returns

• AbstractArray{<:Real}

swish(x; beta=1)

Apply the Swish activation function elementwise.

Arguments

• x: Input value.
• beta::Real=1: Slope parameter controlling smoothness.

Returns

• Scalar or Array of the same shape as x: Activated values.