2. Statistical foundations for CAE

CAE results are rarely single deterministic numbers. A crash simulation with varying material properties, a thermal model with uncertain convective coefficients, or a CFD case with mesh-dependent turbulence parameters all produce distributions of outcomes rather than one fixed value. Before any capability index (Cpk, Ppk) or control chart can be trusted, the analyst must understand the statistical machinery that turns a set of simulation runs into a defensible statement about process behavior. This section builds that foundation: which distributions arise in CAE, how sampling works when each "sample" is an expensive solver run, how to quantify uncertainty with confidence intervals, and how to test hypotheses about simulation outputs without fooling yourself.

Distributions encountered in CAE workflows

The choice of distribution is not cosmetic — it determines which capability formulas and control limits are valid. In simulation contexts, four families dominate:

• Normal (Gaussian): Appropriate for continuous responses driven by many small, additive, independent perturbations — e.g., displacement at a node under random manufacturing tolerance stack-up. Capability indices such as Cpk assume normality; applying them to skewed data silently misstates the defect rate.
• Lognormal: Arises when the response is a product of positive factors — fatigue life, time-to-failure, or stress concentration factors. Fatigue life data in particular is almost never normal; it is right-skewed and best modeled lognormally.
• Weibull: The standard for strength and life data, especially when a weakest-link failure mechanism governs (brittle fracture, bearing life). The shape parameter β encodes whether the failure rate is decreasing (β<1, infant mortality), constant (β=1, random), or increasing (β>1, wear-out).
• Uniform / bounded: Used to represent tolerances with hard limits (e.g., a dimension specified as 10.00 ± 0.05 mm with no tail beyond the limits). Treating a bounded tolerance as normal assigns nonzero probability to physically impossible values.

A practical first step in any CAE statistical study is to plot the histogram of the response and overlay candidate distributions. If the response is a maximum stress over a large model, the extreme-value (Gumbel) distribution may be more appropriate than the normal, because maxima are governed by tail behavior, not central tendency.

Sampling when each sample is a solver run

Classical statistics assumes cheap, independent samples. In CAE, each sample may cost hours of compute, so sampling strategy matters as much as sample size. Three approaches are common:

• Monte Carlo (random) sampling: Draw input parameters from their distributions and run the solver for each draw. Unbiased but converges slowly (error ∝ 1/√n), so it needs many runs for tail estimates.
• Latin Hypercube Sampling (LHS): Stratifies each input dimension so the full range is covered with fewer runs. For the same n, LHS typically gives lower variance in the estimated mean than plain Monte Carlo — valuable when n is capped by compute budget.
• Design of Experiments (DOE): Factorial or response-surface designs (e.g., a central composite design) fit a surrogate model, which is then sampled cheaply. This shifts the statistical burden from the solver to the metamodel, but introduces surrogate error that must itself be quantified.

Independence is a frequent trap. If consecutive runs reuse the same mesh or the same random seed for turbulence, the outputs are correlated, and the effective sample size is smaller than the nominal count. Always record seeds and check autocorrelation before computing standard errors.

Confidence intervals for simulation outputs

A confidence interval (CI) expresses the uncertainty in an estimated statistic, not the spread of individual results. For a mean response from n independent runs with sample standard deviation s, the 95% CI is x̄ ± t₀.₉₇₅,ₙ₋₁ · s/√n. The t-distribution (rather than z) is correct because s is estimated from the same small sample — and CAE samples are almost always small.

For proportions — e.g., the fraction of runs exceeding a stress limit — use the Wilson interval rather than the naive normal approximation, which behaves badly near 0 or 1. For capability indices, bootstrap the CI: resample the n outputs with replacement, recompute Cpk 2,000 times, and take the 2.5th and 97.5th percentiles. This avoids relying on the normality assumption that Cpk's closed-form CI requires.

Interpretation discipline matters. A 95% CI does not mean "95% probability the true mean lies in this interval" under frequentist semantics; it means that 95% of intervals constructed this way would contain the true mean. In practice, report the interval and its width, and state the sample size and distributional assumption alongside it.

Hypothesis testing in a simulation context

Hypothesis tests answer questions like: "Did the design change actually reduce mean stress, or is the difference within run-to-run noise?" The mechanics are standard — state H₀ and H₁, choose a test statistic, compare its p-value to α — but CAE adds specific hazards.

Multiple comparisons. Comparing five design variants pairwise means ten tests; at α=0.05, the chance of at least one false positive is roughly 40%. Apply a Bonferroni correction (α/m) or use Tukey's HSD when comparing all pairs.

Practical vs. statistical significance. With enough runs, a 0.1 MPa difference becomes "significant" but is engineering-irrelevant. Always pair the p-value with an effect size and a CI on the difference.

Paired designs. When comparing two solver configurations on the same set of input samples, use a paired t-test — it removes the between-sample variance and dramatically increases power.

The SVG below maps the decision path from a CAE response dataset to the correct test and interval, including the distribution check that gates the choice.

Worked example: comparing two solver configurations

Suppose a team runs 12 LHS samples through two turbulence-model configurations and records peak wall temperature. The paired differences (Config B minus Config A) have mean d̄ = −4.2 K and standard deviation s_d = 3.1 K. The paired t-statistic is t = d̄/(s_d/√12) = −4.2/(3.1/3.464) ≈ −4.69, with 11 degrees of freedom. The two-sided p-value is well below 0.01, so the reduction is statistically significant. The 95% CI on the difference is −4.2 ± 2.201·(3.1/3.464) ≈ [−6.2, −2.2] K. Because the entire interval lies below zero and the effect exceeds the 2 K engineering threshold, the change is both statistically and practically meaningful. Had the CI straddled zero, the correct conclusion would be "insufficient evidence," not "no effect."

These foundations — correct distributional assumptions, honest sampling, interval-based uncertainty, and disciplined hypothesis testing — are the prerequisites for everything that follows in capability analysis and statistical process control. Without them, control limits and capability indices become arithmetic performed on unjustified assumptions.