EXPERIMENT NOTE2026-09-02 · 8 MIN READDOMAIN: AEROSPACE COMPUTATION

NUMERICAL STABILITY AND CONVERGENCE BOUNDARIES IN REAL-TIME AIRFOIL SOLVERS

ABSTRACT / SUMMARY

Empirical failure analysis of WebAssembly-compiled geometric constraint solvers under high aerodynamic twist angle iterations (Experiment 01).

01. Hypothesis and Computational Scope

In Experiment 01 (Camber Lattice), we evaluated whether parametric constraint solvers can calculate aerodynamic camber modifications at sub-100ms latencies. Full 3D finite element mesh updates require multiple seconds per iteration, rendering interactive aerodynamic tailoring impossible for human-in-the-loop engineering.

02. 2D Parametric Slice Decomposition

By decomposing the 3D wing volume into continuous chordwise 2D parametric spline constraints interconnected via spanwise shear relations, we reduced degrees of freedom by two orders of magnitude. The solver was compiled to WebAssembly with SIMD vectorization enabled.

Wasm-targeted spline solver inner iteration looprust

#[inline(always)]
pub fn solve_chord_slice(
    coords: &mut [f64; 64],
    target_camber: f64,
    max_iter: usize,
) -> Result<f64, SolverError> {
    let mut residual = f64::MAX;
    for _ in 0..max_iter {
        residual = compute_residual(coords, target_camber);
        if residual < 1e-6 { return Ok(residual); }
        apply_gradient_step(coords, residual);
    }
    Err(SolverError::ConvergenceStall { final_residual: residual })
}

03. Boundary Singularities at >18° Angle of Attack

Across 92% of continuous deformation trials, numerical stability was maintained within 14ms average solve times. However, when the geometric twist angle exceeded 18°, the Jacobian matrix developed non-linear singularities, causing Newton-Raphson iterations to oscillate indefinitely.

Twist Angle	Mean Latency (ms)	Max Residual	Convergence Status
0° to 6°	11.2 ms	4.2e-7	PASS (Stable)
6° to 12°	14.8 ms	8.9e-7	PASS (Stable)
12° to 18°	26.4 ms	2.1e-5	PASS (Marginal)
> 18.2°	102.6 ms	1.4e-1	FAIL (Oscillating Singularity)

Fig 2: Solver Latency and Residual Variance Across Sweep Angles

04. Convergence Telemetry & Next Steps

The failure boundary was isolated to abrupt gradient shifts in the airfoil trailing-edge spline. Rather than forcing convergence with artificial dampening that distorts physical lift predictions, the solver was patched with adaptive trust-region step-clamping. The experiment is ongoing to validate iteration cycle 05.

CITATIONS & PRIMARY SOURCES

Theory of Wing Sections: Including a Summary of Airfoil Data, Abbott & Von Doenhoff (1959)
Non-Linear Geometric Constraints in Interactive CAD, ACM Transactions on Graphics (2024)

01. Hypothesis and Computational Scope

02. 2D Parametric Slice Decomposition

Wasm-targeted spline solver inner iteration looprust

#[inline(always)]
pub fn solve_chord_slice(
    coords: &mut [f64; 64],
    target_camber: f64,
    max_iter: usize,
) -> Result<f64, SolverError> {
    let mut residual = f64::MAX;
    for _ in 0..max_iter {
        residual = compute_residual(coords, target_camber);
        if residual < 1e-6 { return Ok(residual); }
        apply_gradient_step(coords, residual);
    }
    Err(SolverError::ConvergenceStall { final_residual: residual })
}

03. Boundary Singularities at >18° Angle of Attack

Twist Angle	Mean Latency (ms)	Max Residual	Convergence Status
0° to 6°	11.2 ms	4.2e-7	PASS (Stable)
6° to 12°	14.8 ms	8.9e-7	PASS (Stable)
12° to 18°	26.4 ms	2.1e-5	PASS (Marginal)
> 18.2°	102.6 ms	1.4e-1	FAIL (Oscillating Singularity)

Fig 2: Solver Latency and Residual Variance Across Sweep Angles

04. Convergence Telemetry & Next Steps

01. Hypothesis and Computational Scope

02. 2D Parametric Slice Decomposition

03. Boundary Singularities at >18° Angle of Attack

04. Convergence Telemetry & Next Steps

CITATIONS & PRIMARY SOURCES

CONNECTED QUANTRIX SYSTEMS

01. Hypothesis and Computational Scope

02. 2D Parametric Slice Decomposition

03. Boundary Singularities at >18° Angle of Attack

04. Convergence Telemetry & Next Steps

CITATIONS & PRIMARY SOURCES

CONNECTED QUANTRIX SYSTEMS