1. CAE-driven design under uncertainty

In simulation-based engineering, a single deterministic CAE run answers a narrow question: given one fixed set of inputs, what is the response? Real products never operate at that single point. Material properties scatter between batches, sheet thickness varies within tolerance bands, loads fluctuate across duty cycles, and manufacturing processes introduce geometric deviation. Design decisions made on a single nominal run therefore carry hidden risk. CAE-driven design under uncertainty reframes the question: given the joint variability of inputs, what is the distribution of the response, and which inputs actually drive it?

Sensitivity analysis (SA) and response surface modeling (RSM) are the two computational instruments that make this reframing tractable. SA quantifies how output variability partitions across inputs; RSM builds a cheap surrogate that replaces expensive solver calls during probabilistic loops. Together they convert an expensive, opaque simulation model into an analyzable input–output relationship that supports robust design decisions.

Why the Nominal Run Is Not Enough

Consider a bracket whose peak von Mises stress under a nominal 8 kN load is 210 MPa against a 235 MPa yield limit. The nominal safety factor of 1.12 looks acceptable. But if yield strength scatters with a coefficient of variation (CoV) of 5% and the load with a CoV of 10%, the probability of exceeding yield may be far from negligible. A deterministic workflow cannot see this because it never propagates the scatter. The engineering consequence is a design that passes review but fails in service — or, equally costly, an over-conservative design that adds mass because the team could not quantify which tolerances mattered.

Uncertainty in CAE models falls into two broad classes. Aleatory uncertainty is irreducible statistical scatter — material batch variation, environmental load. Epistemic uncertainty stems from incomplete knowledge — an uncalibrated contact stiffness, a coarse mesh, an assumed boundary condition. The two demand different treatments: aleatory uncertainty is characterized by distributions and propagated; epistemic uncertainty is reduced by calibration, verification, or conservatism. Confusing them is a common source of misleading probabilistic results.

The Role of Sensitivity Analysis

Sensitivity analysis answers two distinct questions, and the distinction matters operationally.

• Local sensitivity — the partial derivative ∂Y/∂Xᵢ evaluated at the nominal point, usually normalized as (∂Y/∂Xᵢ)(Xᵢ/Y) to give a dimensionless elasticity. It is cheap (one baseline plus one perturbation per input) and directly interpretable: a normalized sensitivity of 0.6 means a 1% change in that input produces roughly a 0.6% change in the output. Its limitation is that it is valid only near the nominal point and says nothing about interaction or nonlinearity.
• Global sensitivity — variance-based measures such as Sobol indices, which decompose output variance into contributions from each input and their interactions. The first-order index Sᵢ is the fraction of output variance explained by Xᵢ alone; the total index Sᵢᵀ includes all interactions involving Xᵢ. Global SA is robust to nonlinearity and is the correct tool when inputs scatter over wide ranges.

The practical payoff is prioritization. If a screening study shows that thickness and yield strength together explain 85% of stress variance while fillet radius explains 3%, the tolerance budget and the test campaign should follow that ranking. Sensitivity results also feed tolerance allocation: tightening the tolerance on a high-sensitivity dimension buys far more reliability per unit cost than tightening a low-sensitivity one.

The Role of Response Surface Modeling

Monte Carlo simulation with a full FE model is often impractical: 10,000 samples × 20 minutes per solve is over 3,300 CPU-hours. RSM addresses this by fitting a surrogate — a polynomial, radial basis function, Kriging/Gaussian process, or polynomial chaos expansion — to a designed set of solver runs, then sampling the surrogate millions of times at negligible cost.

The workflow has a specific structure. A design of experiments (DoE) selects sample points: a Latin hypercube or Sobol sequence for space-filling coverage, or a central composite design when a quadratic polynomial is intended. The solver runs at those points, and the surrogate is fitted and validated against a hold-out set using metrics such as R² and, more importantly for engineering, the normalized maximum absolute error (NMAE) or root-mean-square error (RMSE) on the response of interest. Only a validated surrogate should be used for probability estimation.

How the Two Fit Together

The diagram below shows the integrated workflow: uncertain inputs enter a DoE, the CAE solver produces responses, a surrogate is fitted and validated, and the surrogate then serves both global sensitivity analysis and Monte Carlo reliability estimation. The feedback arrow is essential — if validation error is unacceptable, the DoE is refined (added points, higher-order basis) before any probabilistic conclusion is trusted.

A Concrete Example

Take the bracket above with four uncertain inputs: Young's modulus E (200 GPa, CoV 3%), yield strength σ_y (235 MPa, CoV 5%), flange thickness t (6.0 mm, ±0.3 mm uniform), and applied load F (8 kN, CoV 10%). The response of interest is peak von Mises stress. A 60-point Latin hypercube DoE is run through the nonlinear FE model, and a Kriging surrogate is fitted. Validation on 15 hold-out points gives R² = 0.97 and NMAE = 4.2% — acceptable for a screening-level reliability estimate.

Global SA on the surrogate returns total Sobol indices of approximately S_Fᵀ ≈ 0.52, S_tᵀ ≈ 0.31, S_σyᵀ ≈ 0.12, and S_Eᵀ ≈ 0.05. The interpretation is direct: load and thickness dominate stress variance, while elastic modulus is nearly irrelevant. A tolerance-tightening budget should therefore target thickness control and load specification, not material modulus. Monte Carlo with 200,000 surrogate evaluations then estimates P(stress > σ_y) ≈ 1.8 × 10⁻², a reliability index β ≈ 2.1 — below the typical target of β ≥ 3.0 for this class of structure. The design needs either a thickness increase or a load derating, and the sensitivity ranking tells the team which lever is more effective.

Where the Workflow Fits in Practice

In a mature simulation workflow, SA and RSM sit between detailed CAE verification and design release. The sequence is typically: verify the deterministic model against test or hand calculation; define input distributions from material databases, process capability data, and load spectra; screen inputs with a coarse DoE and local sensitivities; build and validate a surrogate; run global SA and reliability analysis; then feed the ranking back into tolerance allocation, test planning, and design iteration. The surrogate also becomes a reusable asset — a fast what-if model that designers can query without solver licenses or queue time.

The main failure modes are predictable. Distributions are often guessed rather than sourced, which corrupts every downstream probability. Surrogates are sometimes extrapolated beyond their DoE domain. Local sensitivities are occasionally reported as if they were global, hiding interaction effects. And epistemic uncertainty is sometimes buried inside aleatory distributions, producing probabilities that look precise but rest on unverified assumptions. Keeping the two uncertainty classes separate, validating the surrogate on independent points, and reporting sensitivity and probability with their error bands are the practices that keep CAE-driven design under uncertainty trustworthy.