📊 Grid Convergence Index (GCI)
Quantify CFD discretization error numerical uncertainty across 3 grid levels using Roache GCI and ASME V&V 20 standard.
⚡ Fortran 90 Engine
Double Precision (IEEE 754)
✓ ISO / ASME Validated
Cfd
📊 Solver Telemetry
● ACTIVE
📝 Three-Grid Verification Setup
ASME V&V 20 Formulations:
$$r_{21} = \frac{h_2}{h_1}, \quad r_{32} = \frac{h_3}{h_2}, \quad \epsilon_{21} = \phi_2 - \phi_1$$ $$p = \frac{1}{\ln(r_{21})} \left| \ln\left|\frac{\epsilon_{32}}{\epsilon_{21}}\right| + q(p) \right|$$ $$\phi_{ext}^{21} = \frac{r_{21}^p \phi_1 - \phi_2}{r_{21}^p - 1}$$ $$GCI_{fine}^{21} = \frac{F_s \cdot e_a^{21}}{r_{21}^p - 1}$$
$$r_{21} = \frac{h_2}{h_1}, \quad r_{32} = \frac{h_3}{h_2}, \quad \epsilon_{21} = \phi_2 - \phi_1$$ $$p = \frac{1}{\ln(r_{21})} \left| \ln\left|\frac{\epsilon_{32}}{\epsilon_{21}}\right| + q(p) \right|$$ $$\phi_{ext}^{21} = \frac{r_{21}^p \phi_1 - \phi_2}{r_{21}^p - 1}$$ $$GCI_{fine}^{21} = \frac{F_s \cdot e_a^{21}}{r_{21}^p - 1}$$
📊 Verification Uncertainty Report
Verification Summary
Richardson Extrapolated Value ($\phi_{ext}^{21}$ at $h \to 0$)
12851.2
Fine Grid $GCI_{fine} = \mathbf{\pm 4.028\%}$ | Apparent Order $p = \mathbf{1.32}$
Asymptotic Range: EXCELLENT: Grid within asymptotic range (0.95 <= AR <= 1.05). (Ratio $AR = \mathbf{1.02}$)
Apparent Order ($p$)1.32
Fine $GCI_{21}$4.03 %
Medium $GCI_{32}$6.67 %
Rel Error ($e_a$)2.01 %
Ratio $r_{21}$1.44
Ratio $r_{32}$1.44
📈 Solution Convergence vs Normalized Grid Spacing $h/h_1$
📋 Grid Convergence Analysis Breakdown
| Parameter | Value | Description |
|---|---|---|
| Convergence Type | Monotonic Convergence | Monotonic / Oscillatory behavior |
| Apparent Order ($p$) | 1.323 | ASME V&V 20 solution |
| Richardson Extrap ($\phi_{ext}$) | 12851.2 | Continuum solution at $h \to 0$ |
| Fine Grid GCI | 4.028 % | Numerical uncertainty on fine mesh |
| Medium Grid GCI | 6.672 % | Numerical uncertainty on medium mesh |
| Asymptotic Ratio ($AR$) | 1.02 | Target $AR \approx 1.0$ |
Fortran Solver Raw Output:
=============================================================== THERMOFLUIDCALC — ASME V&V 20 GRID CONVERGENCE INDEX (GCI) =============================================================== Dimension = 3 Safety Factor Fs = 1.2500 Fine Size h1 = 7.469008E-03 Fine Solution phi1 = 1.245000E+04 Medium Size h2 = 1.077217E-02 Medium Solution phi2= 1.220000E+04 Coarse Size h3 = 1.547196E-02 Coarse Solution phi3= 1.180000E+04 --------------------------------------------------------------- Ratio r21 (h2/h1) = 1.4422 Ratio r32 (h3/h2) = 1.4363 Convergence Type = Monotonic Convergence Apparent Order p = 1.3226 Richardson Extrap = 1.285123E+04 Approx Error ea21 = 2.008032E-02 Approx Error ea21 % = 2.0080% Extrap Error e_ext = 3.122079E-02 Extrap Error e_ext %= 3.1221% GCI Fine (GCI_21) = 4.028367E-02 GCI Fine (GCI_21) % = 4.0284% GCI Medium (GCI_32) = 6.672393E-02 GCI Medium (GCI_32)%= 6.6724% Asymptotic Ratio AR = 1.0205 --------------------------------------------------------------- Asymptotic Check = EXCELLENT: Grid within asymptotic range (0.95 <= AR <= 1.05). =============================================================== --- GCI CONVERGENCE PROFILE --- Grid_Index h_norm phi_val GCI_pct 0 0.000000E+00 1.285123E+04 0.000000E+00 1 1.000000E+00 1.245000E+04 4.028367E+00 2 1.442250E+00 1.220000E+04 6.672393E+00 3 2.071488E+00 1.180000E+04 6.672393E+00 --- END GCI PROFILE ---
📘 Calculation Methodology: Grid Convergence Index (GCI) & Richardson Extrapolation
Mathematical Model & Theory
The GCI method (ASME V&V 20 / Roache) calculates numerical discretization error and confidence bounds by comparing results across three geometrically scaled grids:
$$p = \frac{1}{\ln(r_{21})} \left| \ln\left|\frac{f_3 - f_2}{f_2 - f_1}\right| \right|, \quad GCI_{fine} = \frac{1.25 |(f_1 - f_2)/f_1|}{r_{21}^p - 1}$$
$$f_{extrapolated} = f_1 + \frac{f_1 - f_2}{r_{21}^p - 1}$$
Assumptions
- Monotonic asymptotic grid convergence.
- Uniform refinement ratio $r \ge 1.3$.
Academic References
- Roache, P. J. (1998): Verification and Validation in Computational Science and Engineering.
- ASME V&V 20-2009 Standard.
Worked Engineering Example
Problem Statement:
Three grids with $r = 1.414$ yield $C_D = [0.0345, 0.0328, 0.0322]$. Calculate apparent order $p$ and $GCI_{fine}$.
Step-by-step Solution:
1. $p = \ln((0.0345-0.0328)/(0.0328-0.0322)) / \ln(1.414) \approx 3.01$.
2. $GCI_{21} = 1.25 \times (0.0006/0.0322) / (1.414^{3.01} - 1) \approx 1.27\%$.
Final Result:
Fine grid uncertainty is $C_D = 0.0322 \pm \mathbf{1.27\%}$.
Three grids with $r = 1.414$ yield $C_D = [0.0345, 0.0328, 0.0322]$. Calculate apparent order $p$ and $GCI_{fine}$.
Step-by-step Solution:
1. $p = \ln((0.0345-0.0328)/(0.0328-0.0322)) / \ln(1.414) \approx 3.01$.
2. $GCI_{21} = 1.25 \times (0.0006/0.0322) / (1.414^{3.01} - 1) \approx 1.27\%$.
Final Result:
Fine grid uncertainty is $C_D = 0.0322 \pm \mathbf{1.27\%}$.