Q: What refinement ratio r should I use for the three-mesh study?

A refinement ratio r between 1.5 and 2.0 is recommended. r is computed as (N₃/N₂)^(1/d) where d is the spatial dimension. In 3D this means multiplying element count by 8 (r = 2.0) or 3.375 (r = 1.5). When r < 1.1 the order of accuracy calculation loses reliability. Non-integer ratios are supported by the tool via the arithmetic-mean r approximation.

Q: What is the relationship between the Richardson extrapolated value φ_ext and GCI_fine?

The Richardson-extrapolated value φ_ext is the estimated true solution at infinite mesh refinement. GCI_fine is the estimated upper bound of relative error of the finest mesh result φ₃ relative to φ_ext: GCI_fine = Fs × |ε₃₂| / (r^p − 1), where Fs = 1.25 for well-converged solutions. Together they define a confidence band: the true solution is expected within φ₃ ± GCI_fine × φ₃ with high probability.

Question 1

What is an acceptable GCI (Grid Convergence Index) value?

Accepted Answer

GCI_fine below 2% is generally considered to confirm mesh convergence. Values below 5% are often acceptable in engineering practice, while ASME V&V 20 recommends 3% or less. This tool displays GCI < 2% in green, 2–5% in orange, and > 5% in red. The GCI is a normalized error band around the Richardson-extrapolated exact solution.

Question 2

Why does the observed order of accuracy p become negative or anomalous?

Accepted Answer

Non-monotonic convergence (where the sequence φ₁→φ₂→φ₃ is not monotone) causes the p formula to diverge because ε₂₁/ε₃₂ < 0. This occurs with oscillatory convergence or nonlinear solutions. In these cases, manually set p = 2 (or a theoretically expected value). The tool displays a warning and defaults to p = 2.0 when monotonic convergence cannot be confirmed.

Question 3

What refinement ratio r should I use for the three-mesh study?

Accepted Answer

A refinement ratio r between 1.5 and 2.0 is recommended. r is computed as (N₃/N₂)^(1/d) where d is the spatial dimension. In 3D this means multiplying element count by 8 (r = 2.0) or 3.375 (r = 1.5). When r < 1.1 the order of accuracy calculation loses reliability. Non-integer ratios are supported by the tool via the arithmetic-mean r approximation.

Question 4

What is the relationship between the Richardson extrapolated value φ_ext and GCI_fine?

Accepted Answer

The Richardson-extrapolated value φ_ext is the estimated true solution at infinite mesh refinement. GCI_fine is the estimated upper bound of relative error of the finest mesh result φ₃ relative to φ_ext: GCI_fine = Fs × |ε₃₂| / (r^p − 1), where Fs = 1.25 for well-converged solutions. Together they define a confidence band: the true solution is expected within φ₃ ± GCI_fine × φ₃ with high probability.

Mesh Convergence / Richardson Extrapolation / GCI

Mesh Settings

Mesh 1 (Coarse)

Mesh 2 (Medium)

Mesh 3 (Fine)

Accuracy Settings

φ vs Mesh Size h (log-log)

Convergence Rate Reference Lines

Error from Extrapolated Value |φᵢ − φ_ext| vs N (log-log)

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