Curve Fitting — CAE Glossary

The most common application is material parameter identification. From stress-strain data obtained in tensile or compression tests, we determine the parameters of a constitutive relation (material model). For example, when analyzing a rubber seal component, we fit the Mooney-Rivlin model parameters $C_{10}$ and $C_{01}$ to the experimental data. CAE cannot exist without test data, so curve fitting is like the entrance to analysis.

Basically three steps: (1) choose a mathematical model (e.g., Mooney-Rivlin, Ogden, Arruda-Boyce, etc.), (2) optimize the unknown parameters of that model to minimize the error between it and the test data, and (3) validate the fitting results. The part where we "minimize the error" uses the least squares method.

The method squares the differences (residuals) between measured values $y_i$ and model predictions $f(x_i; \boldsymbol{\theta})$, sums them across all data points, and adjusts parameters $\boldsymbol{\theta}$ to minimize this sum. Mathematically:

Linear models like $f = a + bx$ can be solved in one step using the normal equations. However, nearly all material models used in CAE are nonlinear, so we use nonlinear least squares methods. The most common is the Levenberg-Marquardt algorithm, which combines gradient descent and the Gauss-Newton method. Abaqus's material calibration tool and Ansys's "Curve Fitting" feature often use this internally.

The strain energy density function for the Mooney-Rivlin model is:

That's the most common trap. Fitting only to uniaxial tensile data often produces poor predictions for equibiaxial tension or pure shear behavior. Since real components experience complex multiaxial stress states, the standard practice is to fit at least two types of data (uniaxial tension + equibiaxial or pure shear) simultaneously. I've seen cases where a model built on uniaxial data alone gave 30% error in stiffness predictions for an engine mount analysis.

Roughly, they're distinguished by strain range. Mooney-Rivlin works well for moderate strains (up to ~150%). For large strains (300% and above), the Ogden model is stronger. If you want to capture molecular chain locking behavior, use the Arruda-Boyce model. In practice, fit all candidate models and select the one with the highest $R^2$ and the least systematic bias in residuals. Built-in calibration tools in solvers let you compare multiple models at once.

The coefficient of determination $R^2$ being close to 1 is basic, but it's not enough. Three things matter: First, examine the residual plot—are residuals randomly scattered with no systematic bias (no U-shaped pattern)? Second, check extrapolation validity—do stresses become negative or diverge outside the fitted range? For hyperelastic models, check if the Drucker stability condition is satisfied. Third, verify physical reasonableness of parameters—if $C_{10}$ becomes negative, that's physically nonsensical and the fit cannot be used.

Fitting noisy data directly to a high-order model causes overfitting. The curve passes perfectly through the data points but oscillates wildly between them, becoming physically meaningless. Remedies include: (1) smoothing the data (moving average or Savitzky-Golay filter), (2) reducing model order, (3) adding regularization terms. But the real solution is improving the test data quality itself—better specimen preparation and measurement environment are the fundamental fix.

Exactly. The Johnson-Cook model is: