Perform mesh convergence studies, Richardson extrapolation, and Grid Convergence Index (GCI) calculations. Enter solutions from three mesh levels to estimate numerical discretization error and verify accuracy.
The core idea is that the numerical solution $f$ approaches the exact solution $f_{exact}$ as the characteristic mesh size $h$ goes to zero. For well-behaved simulations, the error reduces at a predictable rate.
$$ f = f_{exact}+ g h^p + \text{(higher order terms)}$$Here, $f$ is your computed result (e.g., drag force, max stress), $f_{exact}$ is the unknown true solution, $g$ is a constant, $h$ is the mesh size, and $p$ is the observed order of accuracy. The simulator uses results from different $h$ values to solve for $p$ and $f_{exact}$.
Richardson Extrapolation uses this error model. With solutions $f_1$, $f_2$, $f_3$ from meshes of size $h_1$, $h_2$, $h_3$ (where $h_1 > h_2 > h_3$), we can estimate the exact solution.
$$ f_{extrapolated}\approx f_1 + \frac{f_1 - f_2}{r^p - 1} $$where $r = h_2 / h_1$ is the grid refinement ratio. The Grid Convergence Index (GCI) is then calculated as a safety-factor multiplied error estimate: $GCI = F_s |\epsilon|$, where $\epsilon$ is the relative error between meshes and $F_s$ is a safety factor (typically 1.25 for three or more meshes). This gives a practical error band for your engineering result.
Aerodynamic Drag Prediction: Before building a full-scale car model for wind tunnel testing, engineers run CFD simulations. They use this tool to determine the mesh fineness needed to predict drag coefficient within a 2% error band, saving weeks of trial-and-error mesh tuning.
Pressure Vessel Stress Analysis: In FEA of a chemical tank, the maximum stress near a nozzle is critical. Convergence analysis confirms that the chosen mesh captures the stress concentration factor accurately, ensuring the design meets safety standards without being overly conservative.
Electronic Chip Cooling: Simulating heat sink performance requires resolving thin boundary layers. Using Richardson extrapolation on three different mesh densities, a thermal engineer can reliably predict the peak chip temperature without running an impossibly dense 50-million-cell simulation.
Turbine Blade Fatigue Life: Calculating oscillating forces on a wind turbine blade involves complex fluid-structure interaction. Reporting the GCI alongside the predicted fatigue cycles is now a best practice in credible simulation studies, providing transparency about numerical uncertainty.
First, understand that “refining the mesh does not always guarantee a closer approximation to the correct solution.” For instance, near geometric singularities in a model (such as sharp corners or contact points), stresses may continue to diverge and fail to “converge” even as the mesh is refined. In such cases, since the stress theoretically becomes infinite, you need to shift your interpretation—for example, changing the goal of the CAE from “identifying the maximum stress value” to “understanding the stress distribution around that area.”
Next, the three meshes should be refined “systematically.” This is a golden rule. For example, instead of simply setting global element sizes to 1mm, 0.7mm, and 0.5mm, maintain a constant refinement ratio $r$ (recommended to be 1.3 or greater), and set sizes like 1mm, 0.77mm (1/1.3), 0.59mm (1/1.3^2). Otherwise, the calculation of the apparent order of accuracy $p$ becomes unstable, and the extrapolation results lose reliability.
Also, do not blindly trust the GCI value alone. Even if GCI_fine is below 2%, the physical quantity you are focusing on might be converging only “by chance.” It is common, for example, for displacement to have converged while maximum stress has not. Always check the convergence individually for multiple important output variables.