4. Global sensitivity analysis fundamentals

Local sensitivity methods—partial derivatives, one-at-a-time perturbation, or normalized gradient coefficients—answer a narrow question: how much does the output move when a single input is nudged, with everything else frozen? For CAE models this is often misleading. A stiffened panel may show a large local derivative with respect to skin thickness, yet once rib spacing and skin thickness vary together, the two effects can cancel or reinforce each other. Global sensitivity analysis (GSA) abandons the frozen-input assumption and apportions the entire output variance to individual inputs and to their interactions. This chapter develops the variance-based formulation, defines the Sobol indices and the total-effect measure, and shows how to compute and interpret them for a CAE output.

Variance as the quantity to be explained

Let the model be Y = f(X₁, X₂, …, X_d), where each input X_i is an independent random variable with a specified distribution (uniform, normal, lognormal, or an empirical range). Independence is the working assumption of the classical Sobol decomposition; correlated inputs require a different treatment and are addressed later in the course. The output Y is a scalar response of interest—peak von Mises stress, first natural frequency, maximum deflection, or a fatigue life estimate.

The total variance V = Var(Y) is the budget we decompose. The Sobol–Hoeffding decomposition writes the model as a sum of terms of increasing interaction order:

f(X) = f₀ + Σᵢ fᵢ(Xᵢ) + Σᵢ<ⱼ fᵢⱼ(Xᵢ, Xⱼ) + … + f₁₂…d(X₁, …, X_d)

where f₀ = E[Y], each first-order term is the conditional expectation minus the mean, fᵢ(Xᵢ) = E[Y | Xᵢ] − f₀, and each second-order term captures the residual interaction between Xᵢ and Xⱼ after removing the main effects. Because the terms are orthogonal under independence, the variances add exactly:

V = Σᵢ Vᵢ + Σᵢ<ⱼ Vᵢⱼ + … + V₁₂…d
Vᵢ = Var(E[Y | Xᵢ])

The quantity Vᵢ is the first-order variance contribution of input Xᵢ: the expected reduction in output variance if Xᵢ were fixed at its true value and all other inputs were left free. This is the precise sense in which GSA measures importance—not the slope of the response, but the share of output scatter that an input controls.

First-order and total-effect Sobol indices

Normalizing by the total variance gives the first-order Sobol index:

Sᵢ = Vᵢ / V = Var(E[Y | Xᵢ]) / Var(Y)

Sᵢ lies in [0, 1] and represents the fraction of output variance attributable to Xᵢ alone. The sum of all first-order indices is at most 1; the shortfall 1 − Σᵢ Sᵢ is the variance locked up in interactions.

The total-effect index S_Tᵢ answers the complementary question: how much variance remains when Xᵢ is the only input left free? Equivalently, it is the expected residual variance if every input except Xᵢ were fixed:

S_Tᵢ = 1 − Var(E[Y | X_~i]) / Var(Y) = E[Var(Y | X_~i)] / Var(Y)

where X_~i denotes all inputs other than Xᵢ. The total index includes Xᵢ's main effect plus every interaction it participates in. Two derived quantities are diagnostically valuable:

• Interaction share S_Tᵢ − Sᵢ: how much of Xᵢ's influence is mediated by coupling with other inputs. A large gap signals that the input cannot be screened out by a one-at-a-time study.
• Closure check Σᵢ S_Tᵢ ≥ 1, with equality only when the model is purely additive. If Σᵢ S_Tᵢ is close to 1, the model is nearly additive and first-order indices tell almost the whole story.

Estimating the indices from CAE runs

The conditional expectations above are integrals over the input distribution and are not available in closed form for a finite element model. The standard estimator is the Saltelli scheme, which builds two independent sample matrices A and B of size N × d and forms d hybrid matrices AB⁽ⁱ⁾ in which column i of A is replaced by column i of B. With model evaluations y_A, y_B, and y_AB⁽ⁱ⁾, the Jansen estimators are:

V̂ᵢ  = (1/N) Σₙ y_B[n] · (y_AB⁽ⁱ⁾[n] − y_A[n])
V̂_Tᵢ = (1/(2N)) Σₙ (y_A[n] − y_AB⁽ⁱ⁾[n])²
V̂    = Var(y_A ∪ y_B)

The total cost is N(d + 2) model evaluations. For a crash or transient thermal model this is the dominant expense, so the sample size N must be chosen against the convergence of S_Tᵢ, not against the mean. A practical workflow is to run a pilot with N = 256, inspect the bootstrap confidence intervals on S_Tᵢ, and increase N until the ranking of the top inputs is stable. Convergence of the total index is slower than that of the first-order index because it depends on the tail of the conditional variance.

Reading the indices on a CAE example

Consider a representative example (values are illustrative, not measured): a bracket model with five inputs—Young's modulus E, plate thickness t, fillet radius r, bolt preload P, and friction coefficient μ—and a scalar output equal to peak von Mises stress. A Saltelli run with N = 1024 yields the following index table.

The sum of total indices is 1.25, indicating meaningful interaction structure. Thickness alone explains 46% of stress variance, but the fillet radius and preload together contribute another 0.65 through coupling—fixing r at its nominal value while varying t would understate the stress scatter. Conversely, E and μ can be frozen for the subsequent response-surface study, cutting the input dimension from five to three and reducing the design size substantially.

Practical interpretation and pitfalls

Three habits separate a defensible GSA from a decorative one. First, define the output scalar before sampling. Peak stress is a maximum over the mesh and is sensitive to mesh density and to which element happens to be critical; a 99th-percentile or a fixed-location probe is more stable and more reproducible. Second, state the input distributions. Sobol indices are properties of the joint input distribution, not of the model alone; a uniform ±10% range and a normal distribution with the same bounds give different indices. Third, report uncertainty on the indices. Bootstrap resampling of the Saltelli sample gives confidence intervals that let a reader judge whether S_T for E (0.05) is genuinely smaller than for μ (0.03) or whether the difference is sampling noise.

A common failure mode is to interpret a small first-order index as "unimportant." The preload row above shows why this is wrong: S_P = 0.08 would suggest preload is a minor player, yet S_TP = 0.27 places it third in overall influence because it interacts strongly with thickness and fillet radius. Screening on Sᵢ alone would have discarded a variable that materially widens the stress distribution. The total-effect index is therefore the correct screening criterion, and the first-order index is the correct attribution criterion—they answer different questions and should be reported together.

Finally, note the cost implication. The variance-based approach is model-agnostic and handles nonlinearity and interaction rigorously, but it demands N(d + 2) runs. When the CAE model is expensive, the indices are often computed on a cheap surrogate—a polynomial response surface, Gaussian process, or polynomial chaos expansion—fitted to a modest design. That substitution is legitimate only if the surrogate's own predictive uncertainty is small relative to the index differences being interpreted; otherwise the GSA inherits the surrogate's error. The next chapters build exactly this bridge, using the sensitivity ranking to decide which inputs the response surface must resolve and which can be frozen.