cfd-optimization

name	cfd-optimization
description	CFD shape optimization — parametric vs. adjoint, surrogate models (Kriging/RBF), design of experiments (LHS), mesh morphing, multi-objective NSGA-II, RANS-based optimization, Pareto front.
metadata	{"priority":7,"promptSignals":{"phrases":["CFD optimization","shape optimization CFD","surrogate model CFD","aerodynamic optimization","design of experiments CFD","NSGA-II CFD"],"minScore":3}}

CFD Shape Optimization — Complete Skill

Optimization Framework Overview

Design variable space: α = [α₁, α₂, ..., αₙ] — shape parameters (FFD control points, CAD dimensions, Bézier coefficients)

Objective(s): J = [J₁(α), J₂(α), ...] — e.g., drag, pressure drop, heat transfer, noise

Constraints: g_i(α) ≤ 0 — e.g., volume ≥ V_min, stress ≤ σ_allow, Cl ≥ Cl_min

Problem: min J(α) subject to g_i(α) ≤ 0, α_L ≤ α ≤ α_U

Shape Parameterization

Free-Form Deformation (FFD)

Lattice of control points surrounds geometry; deform lattice → geometry moves x_new = Σ B_ijk(s,t,u) × P_ijk [trivariate Bernstein polynomial; P = control points] Degrees of freedom: n_x × n_y × n_z × 3 (3D displacement) = design variables Advantage: differentiable; adjoint compatible; any geometry

CAD Parameters

Directly vary geometry dimensions: chord, twist, chord position, camber Fewer variables (5–50); physical meaning; constrains design space Disadvantage: re-meshing required for each design; costly

NACA Profile Parameters (Airfoil)

4-digit NACA (m, p, t): 4 parameters t = max thickness; m = max camber; p = position of max camber Combined with CST or Hicks-Henne bumps for finer control

Hicks-Henne Bumps

Surface perturbation: Δy(x) = Σ aₙ × sin^n(π x^c) [aₙ = bump amplitude; c = bump location exponent] n_bumps per surface (upper + lower); n = 10–30 per surface typical

Design Space Exploration

Latin Hypercube Sampling (LHS)

Space-filling DoE: ensures uniform coverage of n-dimensional design space N_samples = 3–10 × n_variables (minimum for surrogate building) For 20 variables: 60–200 CFD runs at space-filling points

Randomized vs. optimal LHS: maximize minimum distance between points Tools: pyDOE (Python), Minitab, JMP

Central Composite Design (CCD)

Face-centered CCD: factorial + axial + center points N_runs = 2^k + 2k + 1 [k = number of variables; 3 variables → 15 runs] Good for quadratic surrogate building; limited variables (< 7)

Surrogate Models

Kriging (Gaussian Process)

Interpolating surrogate: ŷ(x) = f(x)^T β + Z(x) [trend + stochastic process] Z(x) ~ Gaussian Process with covariance: cov(Z(x_i), Z(x_j)) = σ² × R(|x_i - x_j|)

Kriging prediction: ŷ(x*) = F(x*)β̂ + r(x*)^T R⁻¹ (y - Fβ̂) r(x*) = [R(x*, x_1), ..., R(x*, x_N)] — correlation vector Advantage: provides prediction uncertainty σ̂²(x*) → guides next sample point

Hyperparameter optimization: maximize log-likelihood to find θ in R(d) = exp(-Σθ_i |d_i|^p)

Radial Basis Function (RBF)

Surrogate: ŷ(x) = Σ w_i × φ(||x - x_i||) [φ = basis function; w = weights from training data] φ types: multiquadric φ = √(r² + c²); thin plate: r² ln(r); Gaussian: exp(-r²/σ²)

Training: solve linear system [φ(||x_i - x_j||)] w = y_training

Advantage: fast; simple; good for smooth functions; less robust to noise than Kriging

Neural Network Surrogate

Multi-layer perceptron (MLP): Input: α → output: J; trained on CFD dataset Advantage: handles nonlinear high-dimensional; requires large dataset (N > 500 typically) Tools: TensorFlow, PyTorch; scikit-learn (small networks)

Optimization Algorithms

Gradient-Based (with Adjoint)

SLSQP (Sequential Least SQuares Programming): min J(α) using gradients from adjoint; fast convergence (10–50 iterations) Best for: smooth objective, single objective, large n

BFGS / L-BFGS: Quasi-Newton; approximate Hessian; 20–100 iterations to convergence PyOpt, scipy.optimize.minimize; gradient from adjoint or finite difference

Multi-Objective: NSGA-II

Non-dominated Sorting Genetic Algorithm: Population: 50–200 individuals Generations: 50–200 (total CFD runs = pop × gen = 5000–40,000 → use surrogate) Selection: tournament + crowding distance Output: Pareto front of J₁ vs. J₂ trade-off

NSGA-III: extended NSGA-II for > 2 objectives MOEA/D: decomposition approach; best for many objectives (4+)

Bayesian Optimization (EGO — Efficient Global Optimization)

Sequential infill strategy:

1. Build Kriging surrogate on initial LHS samples
2. Maximize Expected Improvement: EI(x) = E[max(J_best - ŷ(x), 0)]
3. Run CFD at x* = argmax EI(x)
4. Update Kriging; repeat Optimal allocation: ~10–50 CFD runs per variable (much fewer than gradient-free DoE)

Mesh Morphing

RBF mesh morphing: Move boundary nodes per shape change; interpolate interior nodes via RBF Interior displacement: Δx_int = Σ w_j × φ(||x_int - x_j||) [w from boundary constraint] Advantage: no re-meshing; fast; differentiable (for adjoint compatibility) Tools: Sculptor, RBF Morph, ANSA morphing, Altair HyperMorph

Limitation: large deformations cause poor mesh quality → check mesh quality metrics (min orthogonality, aspect ratio) after each morphing step

CFD Solver Integration

Automated parametric loop:

1. Update parameters → CAD/FFD deformation
2. Mesh generation (if re-mesh) or morph
3. CFD solve (RANS; k-ω SST typical for aerodynamic optimization)
4. Extract objective and constraint values
5. Update surrogate/optimizer
6. Repeat

Tools: Pointwise, ICEM CFD (meshing); OpenFOAM, Fluent, STAR-CCM+ (CFD); Dakota, OpenMDAO (optimization framework)

OpenMDAO (NASA): Unified framework for CFD + adjoint + gradient-based optimization MACH-Aero: full aircraft aero-structural optimization; production use

Output

Provide: design variable count and type (FFD/CAD/Hicks-Henne), DoE method (LHS/CCD) and sample count N, surrogate type (Kriging/RBF/NN) and cross-validation error [%], optimizer (BFGS/NSGA-II/EGO), objective improvement ΔJ [%], Pareto front description (if multi-objective), design variable range [% of nominal], CFD runs total (DoE + infill), mesh morphing quality check, and software stack (mesher/solver/optimizer).