Mesh Refinement Ratio and GCI Convergence Verification Method

A minimum of three levels. If you only compare two meshes, you can only say "the results are close, so it's probably okay." With three, you can estimate the convergence rate (order p) and predict the limit solution as mesh size → 0 via extrapolation. This is Richardson extrapolation, and quantifying its error band is the GCI (Grid Convergence Index).

Exactly. The three-mesh method is also specified as the standard procedure in ASME V&V 20 (the Verification & Validation guideline for computational fluid and solid mechanics). To state "mesh dependency is negligible" in papers or project reports, providing GCI values has become the international common language.

For structured grids, $r = 2$ is indeed simple, but the element count becomes 8 times larger in 3D. The computational cost explodes. That's why ASME V&V 20 recommends $r = \sqrt{2} \approx 1.41$. The element count only increases by about 2.8 times. Conversely, if $r < 1.3$, the difference between meshes becomes too small and gets buried in rounding error, making the estimation of the observed order $p$ unstable.

The most standard method is to back-calculate an equivalent length from the average volume of all elements. $d$ is the spatial dimension number; use area for 2D, volume for 3D.

Yes. For isotropic refinement, the relationship between the element count ratio and $r$ is as follows.

The discretized solution $f_h$ can be expressed relative to the true solution $f_{\text{exact}}$ by the following Taylor expansion.

Exactly. However, since higher-order terms are truncated, the extrapolated value is an "estimate," not the true value. That's precisely why we quantify the uncertainty with GCI.

Good question. In that case, it cannot be solved explicitly, so the following nonlinear equation is solved by iterative methods (fixed-point iteration or Newton's method).

$F_s$ is a "fudge factor" to increase confidence. Roache's original paper recommends $F_s = 3$ (conservative) for the two-mesh method and $F_s = 1.25$ for the three-mesh method. With the three-mesh method, the consistency of extrapolation accuracy has been verified using the observed order $\hat{p}$, so the safety factor can be smaller.

Let $f_1 = 248.3$ (fine), $f_2 = 251.6$ (medium), $f_3 = 260.1$ (coarse), and $r = \sqrt{2}$.

$\varepsilon_{21} = 251.6 - 248.3 = 3.3$, $\varepsilon_{32} = 260.1 - 251.6 = 8.5$

$\hat{p} = \ln(8.5/3.3) / \ln(\sqrt{2}) = \ln(2.576)/0.3466 \approx 2.73$

$\varepsilon = (251.6 - 248.3)/248.3 = 0.01329$

$\text{GCI}_{21}^{\text{fine}} = 1.25 \times 0.01329 / (1.414^{2.73} - 1) = 0.01661 / 1.634 \approx 0.0102 = 1.02\%$

This means that relative to the fine mesh result of 248.3 MPa, there is a discretization uncertainty of $\pm 1.0\%$ that can be reported.

There are three options. (1) Add an even finer mesh and recalculate, (2) Move the evaluation point away from the singularity (e.g., use section average stress instead of stress concentration point), (3) Use higher-order elements to reach the asymptotic range sooner. In practice, (2) often has the best cost-performance.

Indeed, for unstructured meshes, maintaining exactly the same topology is difficult. Therefore, the practical approach is to "uniformly scale the global mesh size by a factor of $1/r$." Apply local refinement to the same regions and with the same ratios for all three levels, unifying the "recipe" for mesh generation.

Report all of them is the correct answer. GCI is calculated independently for each quantity of interest; you cannot say "the model has a GCI of 1% overall." Local quantities (stress at a concentration point) and global quantities (total reaction force, maximum deflection) have completely different convergence behaviors. It's normal for local quantities to converge more slowly. Best practice is to evaluate at least three quantities directly relevant to design judgment and report their respective GCIs.

The top three causes for excessively large GCI are these.

• Evaluation at a singularity — At stress concentration points (notch tip, load point), the theoretical solution diverges to infinity, so it never converges no matter how fine the mesh. Countermeasure: Evaluate at a location away from the singularity or use section average stress.
• Refinement ratio too small — If $r < 1.1$, rounding error and discretization error cannot be distinguished. Countermeasure: Ensure $r \geq 1.3$.
• Analysis not converged — In nonlinear analysis, if the solver's convergence criteria are too loose, the solutions themselves for each mesh become inaccurate. Countermeasure: Set residuals sufficiently small.

Yes. A common pitfall for CAE beginners is performing "convergence study using maximum stress." Especially in models with sharp edges without fillets, the maximum stress monotonically increases as the mesh is refined and does not converge. The choice of evaluation quantity determines the success or failure of the mesh convergence study.