4. Global sensitivity analysis

Local sensitivity analysis answers a narrow question: if one input moves a little around a nominal point, how much does the output move? Global sensitivity analysis (GSA) asks a harder and more useful question: across the entire plausible range of every input, how much of the output variance is attributable to each input, and how much to interactions between inputs? For CAE outputs — peak stress, first natural frequency, drag coefficient, maximum temperature — this distinction matters because design spaces are wide, inputs are correlated in practice, and nonlinear interaction effects are common. Two families of methods dominate engineering practice: variance-based Sobol decomposition and the elementary effects (Morris) screening method.

Variance decomposition and Sobol indices

Sobol analysis rests on the ANOVA-HDMR decomposition. If the model output is Y = f(X₁, …, Xk) with independent inputs, the total variance can be split into contributions from each input alone, each pair, each triple, and so on:

Var(Y) = Σᵢ Vᵢ + Σᵢ<ⱼ Vᵢⱼ + … + V₁₂…ₖ

The first-order Sobol index is the fraction of variance explained by input i alone:

Sᵢ = Var(E[Y | Xᵢ]) / Var(Y)

The total-order index captures everything input i participates in, including all interactions:

Sᵢᵀ = 1 − Var(E[Y | X₋ᵢ]) / Var(Y)

The gap Sᵢᵀ − Sᵢ is the interaction share attributable to input i. When ΣSᵢ ≈ 1, the model is essentially additive and a first-order ranking is trustworthy. When ΣSᵢ is well below 1 — say 0.6 — interactions dominate, and any optimization or robustness study that ignores them will mislead. This single diagnostic is often the most valuable output of a Sobol run.

In practice, Sobol indices are estimated by Monte Carlo. The Saltelli scheme builds two independent sample matrices A and B of size N × k, then forms k hybrid matrices AB(i) in which column i of A is replaced by column i of B. With model evaluations f(A), f(B), and f(AB(i)), the Saltelli–Jansen estimator gives:

Sᵢ  ≈ (1/N) Σ f(B)ⱼ · [f(AB⁽ⁱ⁾)ⱼ − f(A)ⱼ] / Var(Y)
Sᵢᵀ ≈ (1/2N) Σ [f(A)ⱼ − f(AB⁽ⁱ⁾)ⱼ]² / Var(Y)

Total cost is N(k+2) model runs. For a CAE model with k = 8 inputs and N = 1024 base samples, that is 10,240 solves — feasible for a 2-minute steady-state thermal run, prohibitive for a 6-hour transient. This cost is precisely why surrogate models (Chapter 5 onward) are usually built first and Sobol indices computed on the response surface rather than the solver.

Sample size N must be large enough for the estimator to converge. A practical check: run the analysis at N and 2N; if the ranking of Sᵢᵀ changes or any index shifts by more than ~0.05, increase N. Report confidence intervals from bootstrap resampling rather than point estimates alone.

Elementary effects: cheap screening before expensive quantification

When k is large (15–40 geometric or material parameters is typical in CAE), full Sobol analysis is wasteful because many inputs will turn out irrelevant. The Morris elementary effects method screens first. Each input is perturbed one at a time along a random trajectory through the normalized design space, and the elementary effect for input i is:

EEᵢ = [f(x₁,…,xᵢ+Δ,…,xₖ) − f(x)] / Δ

Repeating over r trajectories yields three statistics per input: μ (mean effect, sensitive to sign cancellation), μ* (mean absolute effect, the robust importance measure), and σ (standard deviation of effects, a proxy for nonlinearity or interaction). A common screening rule is to plot μ* against σ: inputs in the upper-right quadrant are both influential and interaction-heavy and deserve full Sobol treatment; inputs near the origin can be frozen. Cost is r(k+1) runs — with r = 10 and k = 20, only 210 evaluations.

Two practical cautions apply. First, μ* is scale-dependent, so normalize inputs to [0,1] and, where outputs span orders of magnitude, analyze log(Y). Second, Morris σ flags interaction but does not quantify it; a high σ is a prompt to run Sobol on that subset, not a substitute for it.

Two-stage global sensitivity workflow with Morris screening and Sobol quantification Flow from CAE inputs through Morris elementary effects screening to a reduced Sobol set, with interpretation quadrants for first-order and total-order indices. CAE input set k = 20 parameters plate thickness, E, ν, yield stress, load angle, fillet radius, … ranges from design spec Morris screening r = 10 trajectories r(k+1) = 210 runs outputs: μ*, σ per input cheap, ranks importance Reduced set k' = 6 active 14 frozen at nominal values documented, not silent Sobol on k' N = 1024 N(k'+2) = 8192 runs Sᵢ, Sᵢᵀ, ΣSᵢ quantifies variance share Interpreting the Sobol result 0 0.5 1.0 thickness S=0.38 Sᵀ=0.52 E S=0.30 Sᵀ=0.44 load angle S=0.15 Sᵀ=0.24 first-order Sᵢ total-order Sᵢᵀ ΣSᵢ = 0.83 → mostly additive, but thickness carries a 0.14 interaction share with E. Illustrative values, not measured data.

Reading the indices for CAE decisions

The numbers above are illustrative, but the interpretation pattern is standard. Thickness has S = 0.38 and Sᵀ = 0.52, so roughly 0.14 of the output variance comes from its interaction with other parameters — most likely with Young's modulus through bending stiffness. A designer who tunes thickness alone, holding E at nominal, will under-predict the spread in peak stress. Conversely, load angle has S = 0.15 and Sᵀ = 0.24; its interaction share is smaller, and a first-order ranking is adequate for it.

Three interpretation rules are worth committing to memory. First, rank by Sᵀ, not S, when deciding what to control tightly in manufacturing or what to include in a robustness study — total-order indices never understate an input's influence. Second, use ΣSᵢ as a model-structure diagnostic: if it is below about 0.7, invest in interaction-aware surrogate terms (quadratic or Kriging with interaction kernels) rather than a purely linear regression. Third, treat indices as estimates with uncertainty. Bootstrap the Saltelli estimator and report 95% intervals; an input whose interval overlaps another's should not be ranked above it with confidence.

A reproducible screening-then-quantify procedure

The following sequence is a defensible default for a CAE study with 10–30 inputs and a solver too slow for direct Monte Carlo:

  1. Define input ranges from design specification or tolerance stack-up, not from convenience. Record units and distribution (uniform for screening, then the actual distribution for Sobol if it is non-uniform).
  2. Normalize all inputs to [0,1] and, if the output spans more than one order of magnitude, transform to log(Y).
  3. Run Morris with r = 10 and Δ = 0.1 on a 4-level grid. Compute μ* and σ per input; plot μ* vs σ.
  4. Freeze inputs with μ* below roughly 5% of the maximum μ*, but record the freeze decision and re-check it if the design later changes.
  5. Run Saltelli–Sobol on the surviving set with N = 512, then N = 1024. Confirm convergence by comparing rankings and index shifts.
  6. Report Sᵢ, Sᵢᵀ, ΣSᵢ, and bootstrap intervals. State the frozen inputs and the variance they were assumed not to contribute.
A common failure mode: running Sobol directly on all 20 inputs with N = 128 to save time. The estimator is biased and noisy at that sample size, and the resulting ranking can invert the true order. If the budget cannot support N ≥ 512 on the full set, screen first — do not shrink N.

The practical payoff of this two-stage approach is that a study which would have required tens of thousands of solver calls collapses to a few hundred screening runs plus a few thousand Sobol runs on a reduced set. The indices then feed directly into the surrogate construction and optimization stages that follow, where knowing which inputs carry interaction variance determines whether a quadratic response surface is adequate or a more flexible model is required.

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