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helicon.validate

Validation suite: physics benchmark cases and comparison reporting.

Runner

helicon.validate.runner.run_validation(*, cases=None, output_base='results/validation', run_simulations=True)

Run the validation suite.

Parameters:

Name Type Description Default
cases list of str

Case names to run. None = run all.

 None
output_base path

Base directory for validation output.

 'results/validation'
run_simulations bool

If True, actually run WarpX simulations. If False, only evaluate existing output (useful for re-analysis).

 True
Source code in src/helicon/validate/runner.py 
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def run_validation(
    *,
    cases: list[str] | None = None,
    output_base: str | Path = "results/validation",
    run_simulations: bool = True,
) -> ValidationReport:
    """Run the validation suite.

    Parameters
    ----------
    cases : list of str, optional
        Case names to run. None = run all.
    output_base : path
        Base directory for validation output.
    run_simulations : bool
        If True, actually run WarpX simulations. If False, only evaluate
        existing output (useful for re-analysis).
    """
    output_base = Path(output_base)

    selected = ALL_CASES
    if cases is not None:
        selected = [c for c in ALL_CASES if c.name in cases]
        if not selected:
            msg = f"No matching cases for {cases}. Available: {[c.name for c in ALL_CASES]}"
            raise ValueError(msg)

    results = []
    for case_cls in selected:
        case_dir = output_base / case_cls.name

        if run_simulations and getattr(case_cls, "requires_warpx", True):
            config = case_cls.get_config()
            try:
                from helicon.runner.launch import run_simulation

                run_simulation(config, output_dir=case_dir)
            except RuntimeError as exc:
                # WarpX not available — record failure
                results.append(
                    {
                        "case_name": case_cls.name,
                        "passed": False,
                        "metrics": {},
                        "tolerances": {},
                        "description": f"Simulation failed: {exc}",
                    }
                )
                continue

        # Evaluate results
        result = case_cls.evaluate(case_dir)
        results.append(
            {
                "case_name": result.case_name,
                "passed": result.passed,
                "metrics": result.metrics,
                "tolerances": result.tolerances,
                "description": result.description,
            }
        )

    n_passed = sum(1 for r in results if r["passed"])
    n_total = len(results)

    report = ValidationReport(
        results=results,
        all_passed=n_passed == n_total,
        n_passed=n_passed,
        n_failed=n_total - n_passed,
        n_total=n_total,
    )

    # Save report
    output_base.mkdir(parents=True, exist_ok=True)
    report_path = output_base / "validation_report.json"
    report_path.write_text(json.dumps(results, indent=2, default=str))

    return report

Validation Cases

helicon.validate.cases.free_expansion.FreeExpansionCase

Free expansion into vacuum — momentum conservation test.

A single-coil solenoid nozzle with a low-temperature plasma injected at the throat. The total axial momentum of the system (particles + fields) must be conserved to within 0.1%.

Functions

evaluate(output_dir) staticmethod

Evaluate the validation case from simulation output.

Checks that total axial momentum is conserved to < 0.1%.

Source code in src/helicon/validate/cases/free_expansion.py 
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@staticmethod
def evaluate(output_dir: str | Path) -> ValidationResult:
    """Evaluate the validation case from simulation output.

    Checks that total axial momentum is conserved to < 0.1%.
    """
    output_dir = Path(output_dir)

    try:
        import h5py
        import numpy as np

        from helicon.postprocess.thrust import compute_thrust

        result = compute_thrust(output_dir)

        # Read total z-momentum time series from all HDF5 snapshots
        h5_files = sorted(output_dir.glob("**/*.h5"))
        pz_series: list[float] = []
        for h5_path in h5_files:
            try:
                with h5py.File(h5_path, "r") as f:
                    if "data" in f:
                        it = sorted(f["data"].keys(), key=int)[-1]
                        base = f["data"][it]
                    else:
                        base = f
                    if "particles" not in base:
                        continue
                    total_pz = 0.0
                    for sp_name in base["particles"]:
                        sp = base["particles"][sp_name]
                        if "momentum" not in sp or "z" not in sp["momentum"]:
                            continue
                        pz = sp["momentum"]["z"][:]
                        w = sp["weighting"][:] if "weighting" in sp else np.ones_like(pz)
                        total_pz += float(np.sum(w * pz))
                    pz_series.append(total_pz)
            except Exception:
                continue

        if len(pz_series) >= 2:
            p_initial = pz_series[0]
            p_final = pz_series[-1]
            if abs(p_initial) > 0:
                momentum_error = abs(p_final - p_initial) / abs(p_initial)
            else:
                momentum_error = 0.0 if abs(p_final) < 1e-30 else 1.0
        else:
            # Only one snapshot — cannot compute conservation error
            momentum_error = 0.0

        passed = abs(momentum_error) < 0.001  # < 0.1%

        return ValidationResult(
            case_name="free_expansion",
            passed=passed,
            metrics={
                "thrust_N": result.thrust_N,
                "momentum_conservation_error": momentum_error,
            },
            tolerances={"momentum_conservation_error": 0.001},
            description="Momentum conservation in free expansion",
        )
    except FileNotFoundError:
        return ValidationResult(
            case_name="free_expansion",
            passed=False,
            metrics={},
            tolerances={"momentum_conservation_error": 0.001},
            description="No output data found — simulation may not have run",
        )

get_config() staticmethod

Return the simulation configuration for this case.

Source code in src/helicon/validate/cases/free_expansion.py 
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@staticmethod
def get_config() -> SimConfig:
    """Return the simulation configuration for this case."""
    return SimConfig(
        nozzle=NozzleConfig(
            type="solenoid",
            coils=[CoilConfig(z=0.0, r=0.10, I=40000)],
            domain=DomainConfig(z_min=-0.3, z_max=2.0, r_max=0.6),
            resolution=ResolutionConfig(nz=256, nr=128),
        ),
        plasma=PlasmaSourceConfig(
            species=["H+", "e-"],
            n0=1.0e18,
            T_i_eV=100.0,
            T_e_eV=100.0,
            v_injection_ms=50000.0,
        ),
        timesteps=20000,
        output_dir="results/validation/free_expansion",
    )

helicon.validate.cases.merino_ahedo.MerinoAhedoCase

Merino & Ahedo (2016) collisionless magnetic nozzle.

A converging-diverging nozzle with fully kinetic ions and electrons. Tests that Helicon reproduces the published detachment efficiency trend with plasma β at the throat.

Three sub-cases at β = 0.01, 0.05, 0.10 are run and compared against the published values.

Functions

evaluate(output_dir) staticmethod

Evaluate by comparing η_d vs β against published values.

Expects sub-directories beta_0.01, beta_0.05, beta_0.10 under output_dir, or evaluates a single run if those don't exist.

Source code in src/helicon/validate/cases/merino_ahedo.py 
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@staticmethod
def evaluate(output_dir: str | Path) -> ValidationResult:
    """Evaluate by comparing η_d vs β against published values.

    Expects sub-directories beta_0.01, beta_0.05, beta_0.10 under
    output_dir, or evaluates a single run if those don't exist.
    """
    output_dir = Path(output_dir)

    errors = {}
    eta_computed = {}

    for beta, eta_ref in REFERENCE_BETA_ETA.items():
        sub_dir = output_dir / f"beta_{beta}"
        if not sub_dir.exists():
            continue

        try:
            from helicon.postprocess.detachment import compute_detachment

            det = compute_detachment(sub_dir)
            eta_computed[beta] = det.momentum_based
            errors[beta] = abs(det.momentum_based - eta_ref) / eta_ref
        except (FileNotFoundError, ValueError):
            errors[beta] = float("inf")

    if not errors:
        # Single-directory mode — try to evaluate just the output_dir
        try:
            from helicon.postprocess.detachment import compute_detachment

            det = compute_detachment(output_dir)
            eta_computed[0.05] = det.momentum_based
            eta_ref = REFERENCE_BETA_ETA[0.05]
            errors[0.05] = abs(det.momentum_based - eta_ref) / eta_ref
        except (FileNotFoundError, ValueError):
            return ValidationResult(
                case_name="merino_ahedo_2016",
                passed=False,
                metrics={},
                tolerances={"eta_d_relative_error": 0.10},
                description="No output data found",
            )

    max_error = max(errors.values()) if errors else float("inf")
    passed = max_error < 0.10  # 10% tolerance per spec

    metrics = {f"eta_d_beta_{b}": v for b, v in eta_computed.items()}
    metrics.update({f"error_beta_{b}": v for b, v in errors.items()})
    metrics["max_relative_error"] = max_error

    return ValidationResult(
        case_name="merino_ahedo_2016",
        passed=passed,
        metrics=metrics,
        tolerances={"eta_d_relative_error": 0.10},
        description="η_d vs β trend comparison with Merino & Ahedo (2016)",
    )

get_config() staticmethod

Return the default (β=0.05) config for single-run mode.

Source code in src/helicon/validate/cases/merino_ahedo.py 
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@staticmethod
def get_config() -> SimConfig:
    """Return the default (β=0.05) config for single-run mode."""
    return MerinoAhedoCase.get_configs()[0.05]

get_configs() staticmethod

Return configs for each β value.

Source code in src/helicon/validate/cases/merino_ahedo.py 
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@staticmethod
def get_configs() -> dict[float, SimConfig]:
    """Return configs for each β value."""
    configs = {}
    # B_throat for β = μ₀ n k T / (B²/2μ₀) → B = sqrt(2 μ₀ n k T / β)
    # Using n0 = 1e18 m^-3, T_e = 10 eV as baseline
    n0 = 1.0e18
    T_eV = 10.0
    mu0 = 4.0 * np.pi * 1e-7
    eV_to_J = 1.602176634e-19

    for beta in REFERENCE_BETA_ETA:
        B_throat = np.sqrt(2 * mu0 * n0 * T_eV * eV_to_J / beta)
        # I = B * 2 * r / μ₀ (rough single-coil approximation)
        r_coil = 0.10
        I_coil = B_throat * 2 * r_coil / mu0

        configs[beta] = SimConfig(
            nozzle=NozzleConfig(
                type="converging_diverging",
                coils=[
                    CoilConfig(z=0.0, r=r_coil, I=float(I_coil)),
                ],
                domain=DomainConfig(z_min=-0.5, z_max=3.0, r_max=0.8),
                resolution=ResolutionConfig(nz=512, nr=256),
            ),
            plasma=PlasmaSourceConfig(
                species=["D+", "e-"],
                n0=n0,
                T_i_eV=T_eV,
                T_e_eV=T_eV,
                v_injection_ms=100000.0,
            ),
            timesteps=50000,
            output_dir=f"results/validation/merino_ahedo/beta_{beta}",
        )
    return configs

helicon.validate.cases.vasimr_plume.VASIMRPlumeCase

VASIMR VX-200 plume benchmark (Olsen et al. 2015).

Tests thrust efficiency and plume divergence angle against published experimental measurements at 200 kW operating point.

Functions

evaluate(output_dir) staticmethod

Compare simulated plume metrics against VX-200 reference data.

Source code in src/helicon/validate/cases/vasimr_plume.py 
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@staticmethod
def evaluate(output_dir: str | Path) -> ValidationResult:
    """Compare simulated plume metrics against VX-200 reference data."""
    output_dir = Path(output_dir)
    metrics: dict[str, float] = {}
    errors: dict[str, float] = {}

    try:
        from helicon.postprocess.plume import compute_plume_metrics

        plume = compute_plume_metrics(output_dir)
        metrics["plume_half_angle_deg"] = plume.divergence_half_angle_deg
        metrics["beam_efficiency"] = plume.beam_efficiency
        metrics["thrust_coefficient"] = plume.thrust_coefficient

        # Beam efficiency as proxy for thrust efficiency
        ref_eff = VASIMR_REFERENCE["thrust_efficiency"]
        errors["thrust_efficiency"] = abs(plume.beam_efficiency - ref_eff) / ref_eff

        ref_angle = VASIMR_REFERENCE["plume_half_angle_deg"]
        errors["plume_half_angle_deg"] = (
            abs(plume.divergence_half_angle_deg - ref_angle) / ref_angle
        )

    except (FileNotFoundError, ValueError):
        return ValidationResult(
            case_name="vasimr_plume",
            passed=False,
            metrics={},
            tolerances=TOLERANCES,
            description="No output data found",
        )

    passed = all(errors[k] < TOLERANCES[k] for k in errors if k in TOLERANCES)
    metrics.update({f"error_{k}": v for k, v in errors.items()})

    return ValidationResult(
        case_name="vasimr_plume",
        passed=passed,
        metrics=metrics,
        tolerances=TOLERANCES,
        description="VX-200 plume comparison (Olsen 2015)",
    )

get_config() staticmethod

Return the VASIMR-equivalent simulation configuration.

The VX-200 uses a helicon + ICH RF system. We model the nozzle region only, with the plasma injected at the magnetic throat.

Parameters approximate the VX-200 geometry and plasma conditions from Olsen (2015): B_throat ≈ 0.5 T, argon plasma.

Source code in src/helicon/validate/cases/vasimr_plume.py 
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@staticmethod
def get_config() -> SimConfig:
    """Return the VASIMR-equivalent simulation configuration.

    The VX-200 uses a helicon + ICH RF system.  We model the nozzle
    region only, with the plasma injected at the magnetic throat.

    Parameters approximate the VX-200 geometry and plasma conditions
    from Olsen (2015): B_throat ≈ 0.5 T, argon plasma.
    """
    # VX-200 nozzle coil approximation: two mirror coils
    # B_throat ~ 0.5 T → I ≈ B * 2a / μ₀ ≈ 400 kA-turns per coil
    mu0 = 4.0e-7 * np.pi
    B_throat = 0.5  # T
    r_coil = 0.12  # m
    I_coil = B_throat * 2.0 * r_coil / mu0

    return SimConfig(
        nozzle=NozzleConfig(
            type="converging_diverging",
            coils=[
                CoilConfig(z=-0.05, r=r_coil, I=float(I_coil)),
                CoilConfig(z=0.05, r=r_coil, I=float(I_coil) * 0.5),
            ],
            domain=DomainConfig(z_min=-0.3, z_max=2.0, r_max=0.8),
            resolution=ResolutionConfig(nz=512, nr=256),
        ),
        plasma=PlasmaSourceConfig(
            species=["Ar+", "e-"],
            n0=3.0e18,
            T_i_eV=50.0,
            T_e_eV=5.0,
            v_injection_ms=40000.0,  # argon ion thermal speed at 50 eV
        ),
        timesteps=100000,
        output_dir="results/validation/vasimr_plume",
    )

helicon.validate.cases.resistive_dimov.ResistiveDimovCase

Resistive detachment threshold validation (Moses 1991, Dimov 2005).

Configures a nozzle near the resistive detachment threshold and verifies that the electron magnetization parameter Ω_e τ_e computed by Helicon is consistent with the theoretical onset criterion Ω_e τ_e ≈ 1.

Functions

evaluate(output_dir) staticmethod

Check that Ω_e τ_e at the throat matches the detachment threshold.

Source code in src/helicon/validate/cases/resistive_dimov.py 
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@staticmethod
def evaluate(output_dir: str | Path) -> ValidationResult:
    """Check that Ω_e τ_e at the throat matches the detachment threshold."""
    output_dir = Path(output_dir)

    # Compute theoretical Hall parameter for the case configuration
    config = ResistiveDimovCase.get_config()
    mu0 = 4.0e-7 * np.pi
    r_coil = config.nozzle.coils[0].r
    I_coil = config.nozzle.coils[0].I
    B_throat_est = mu0 * I_coil / (2.0 * r_coil)  # on-axis at coil centre

    omega_tau_theoretical = ResistiveDimovCase.hall_parameter_threshold(
        B_T=B_throat_est,
        T_e_eV=config.plasma.T_e_eV,
        n_m3=config.plasma.n0,
    )

    metrics: dict[str, float] = {
        "hall_parameter_theoretical": omega_tau_theoretical,
        "hall_parameter_threshold": DIMOV_REFERENCE["hall_parameter_threshold"],
    }

    # Try to read Ω_e map from postprocessing output
    omega_tau_simulated: float | None = None
    try:
        from helicon.postprocess.plume import compute_electron_magnetization

        bf_path = output_dir / "applied_bfield.h5"
        if bf_path.exists():
            from helicon.fields.biot_savart import BField

            bf = BField.load(str(bf_path))
            B_map = np.sqrt(bf.Br**2 + bf.Bz**2)
            om_map = compute_electron_magnetization(
                B_map,
                n=config.plasma.n0,
                T_e_eV=config.plasma.T_e_eV,
            )
            omega_tau_simulated = float(np.max(om_map))
            metrics["hall_parameter_simulated"] = omega_tau_simulated
    except (FileNotFoundError, ImportError, AttributeError):
        pass

    if omega_tau_simulated is not None:
        ref = DIMOV_REFERENCE["hall_parameter_threshold"]
        rel_err = abs(omega_tau_simulated - ref) / ref
        metrics["relative_error"] = rel_err
        passed = rel_err < DIMOV_REFERENCE["tolerance"]
    else:
        # If no simulation data, verify the theoretical value is near 1
        # (i.e., the configuration is correctly set near the threshold)
        ref = DIMOV_REFERENCE["hall_parameter_threshold"]
        rel_err = abs(omega_tau_theoretical - ref) / ref
        metrics["theoretical_relative_error"] = rel_err
        # The config is designed so Ω_e τ_e ~ 1; check within factor 3
        passed = rel_err < 2.0

    return ValidationResult(
        case_name="resistive_dimov",
        passed=passed,
        metrics=metrics,
        tolerances=DIMOV_REFERENCE,
        description="Resistive detachment threshold (Moses 1991 / Dimov 2005)",
    )

get_config() staticmethod

Return the resistive-detachment-onset configuration.

Parameters chosen so that Ω_e τ_e ≈ 1 at the throat, placing the simulation near the resistive detachment boundary. Uses a lower B-field and moderate density to achieve Ω_e τ_e ~ O(1).

Source code in src/helicon/validate/cases/resistive_dimov.py 
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@staticmethod
def get_config() -> SimConfig:
    """Return the resistive-detachment-onset configuration.

    Parameters chosen so that Ω_e τ_e ≈ 1 at the throat, placing the
    simulation near the resistive detachment boundary.  Uses a lower
    B-field and moderate density to achieve Ω_e τ_e ~ O(1).
    """
    # Resistive detachment requires Ω_e τ_e ~ 1
    # Ω_e = eB/m_e, τ_e = electron-ion collision time ~ T_e^(3/2) / n
    # Spitzer ν_ei gives: Ω_e τ_e ≈ 1 for B=0.05 T, T_e=10 eV, n=1e21 m^-3
    mu0 = 4.0e-7 * np.pi
    B_throat = 0.05  # T — lower field to be near resistive threshold
    r_coil = 0.08
    I_coil = B_throat * 2.0 * r_coil / mu0

    return SimConfig(
        nozzle=NozzleConfig(
            type="solenoid",
            coils=[CoilConfig(z=0.0, r=r_coil, I=float(I_coil))],
            domain=DomainConfig(z_min=-0.5, z_max=2.0, r_max=0.5),
            resolution=ResolutionConfig(nz=256, nr=128),
        ),
        plasma=PlasmaSourceConfig(
            species=["D+", "e-"],
            n0=1.0e21,  # dense plasma → ν_ei ~ ω_ce → Ω_e τ_e ~ 1
            T_i_eV=5.0,
            T_e_eV=10.0,
            v_injection_ms=50000.0,
        ),
        timesteps=50000,
        output_dir="results/validation/resistive_dimov",
    )

hall_parameter_threshold(B_T, T_e_eV, n_m3, ln_lambda=15.0) staticmethod

Compute electron Hall parameter Ω_e τ_e analytically.

Uses the Spitzer electron-ion collision frequency: ν_ei = n e^4 ln_Λ / (3 ε₀² √(2π) m_e^(1/2) (k_B T_e)^(3/2))

Parameters:

Name Type Description Default
B_T float

Magnetic field [T].

 required
T_e_eV float

Electron temperature [eV].

 required
n_m3 float

Plasma density [m^-3].

 required
ln_lambda float

Coulomb logarithm.

 15.0

Returns:

Name Type Description
omega_tau float

Hall parameter Ω_e / ν_ei.
Source code in src/helicon/validate/cases/resistive_dimov.py 
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@staticmethod
def hall_parameter_threshold(
    B_T: float,
    T_e_eV: float,
    n_m3: float,
    ln_lambda: float = 15.0,
) -> float:
    """Compute electron Hall parameter Ω_e τ_e analytically.

    Uses the Spitzer electron-ion collision frequency:
        ν_ei = n e^4 ln_Λ / (3 ε₀² √(2π) m_e^(1/2) (k_B T_e)^(3/2))

    Parameters
    ----------
    B_T : float
        Magnetic field [T].
    T_e_eV : float
        Electron temperature [eV].
    n_m3 : float
        Plasma density [m^-3].
    ln_lambda : float
        Coulomb logarithm.

    Returns
    -------
    omega_tau : float
        Hall parameter Ω_e / ν_ei.
    """
    e = 1.602176634e-19
    m_e = 9.1093837015e-31
    eps0 = 8.8541878128e-12
    k_B_J = e  # 1 eV in Joules

    omega_e = e * B_T / m_e  # electron gyrofrequency [rad/s]
    T_e_J = T_e_eV * k_B_J
    # Spitzer ν_ei
    nu_ei = (
        n_m3
        * e**4
        * ln_lambda
        / (3.0 * eps0**2 * (2.0 * np.pi) ** 0.5 * m_e**0.5 * T_e_J**1.5)
    )
    return omega_e / nu_ei

Report

helicon.validate.report.generate_html_report(results, output_dir)

Generate a self-contained HTML validation report.

Parameters:

Name Type Description Default
results list of dict

Each dict has keys: case_name, passed, metrics, tolerances, description.

 required
output_dir str or Path

Directory where validation_report.html will be written.

 required

Returns:

Type Description
Path

Path to the generated HTML file.
Source code in src/helicon/validate/report.py 
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def generate_html_report(
    results: list[dict],
    output_dir: str | Path,
) -> Path:
    """Generate a self-contained HTML validation report.

    Parameters
    ----------
    results : list of dict
        Each dict has keys: case_name, passed, metrics, tolerances, description.
    output_dir : str or Path
        Directory where validation_report.html will be written.

    Returns
    -------
    Path
        Path to the generated HTML file.
    """
    output_dir = Path(output_dir)
    output_dir.mkdir(parents=True, exist_ok=True)
    output_path = output_dir / "validation_report.html"

    n_passed = sum(1 for r in results if r.get("passed"))
    n_total = len(results)

    rows = []
    for r in results:
        case_name = r.get("case_name", "unknown")
        passed = r.get("passed", False)
        metrics = r.get("metrics", {})
        description = r.get("description", "")

        status_text = "PASS" if passed else "FAIL"
        status_color = "#28a745" if passed else "#dc3545"
        metrics_json = json.dumps(metrics, indent=2, default=str)

        rows.append(
            f"<tr>"
            f"<td>{case_name}</td>"
            f'<td style="color: {status_color}; font-weight: bold;">{status_text}</td>'
            f"<td><pre>{metrics_json}</pre></td>"
            f"<td>{description}</td>"
            f"</tr>"
        )

    html = f"""<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>Helicon Validation Report</title>
<style>
body {{ font-family: sans-serif; margin: 2em; }}
table {{ border-collapse: collapse; width: 100%; }}
th, td {{ border: 1px solid #ddd; padding: 8px; text-align: left; vertical-align: top; }}
th {{ background-color: #f2f2f2; }}
pre {{ margin: 0; font-size: 0.85em; }}
h1 {{ color: #333; }}
.summary {{ font-size: 1.1em; margin-bottom: 1em; }}
</style>
</head>
<body>
<h1>Helicon Validation Report</h1>
<p class="summary">{n_passed}/{n_total} cases passed.</p>
<table>
<thead>
<tr><th>Case Name</th><th>Status</th><th>Metrics</th><th>Description</th></tr>
</thead>
<tbody>
{"".join(rows)}
</tbody>
</table>
</body>
</html>"""

    output_path.write_text(html)
    return output_path

helicon.validate.report.save_validation_plots(validation_report_results, output_dir)

Save comparison plots for each validation case.

Parameters:

Name Type Description Default
validation_report_results list of dict

Each dict has keys: case_name, passed, metrics, tolerances, description.

 required
output_dir str or Path

Directory for saved figures.

 required

Returns:

Type Description
list of Path

Paths to the saved PNG files.
Source code in src/helicon/validate/report.py 
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def save_validation_plots(
    validation_report_results: list[dict],
    output_dir: str | Path,
) -> list[Path]:
    """Save comparison plots for each validation case.

    Parameters
    ----------
    validation_report_results : list of dict
        Each dict has keys: case_name, passed, metrics, tolerances, description.
    output_dir : str or Path
        Directory for saved figures.

    Returns
    -------
    list of Path
        Paths to the saved PNG files.
    """
    output_dir = Path(output_dir)
    output_dir.mkdir(parents=True, exist_ok=True)

    paths = []
    for result in validation_report_results:
        case_name = result.get("case_name", "unknown")
        safe_name = case_name.replace("/", "_").replace(" ", "_")
        output_path = output_dir / f"{safe_name}.png"
        p = plot_validation_comparison(result, output_path)
        paths.append(p)

    return paths