📚 Course ContentsChapter 2 Design of Experiments fundamentals

2. Design of Experiments fundamentals

Before a single simulation job is submitted, a Design of Experiments (DOE) study must answer a deceptively simple question: which runs should I execute, and why those and not others? In a CAE workflow, every run costs solver time, license tokens, and engineering attention, so the experimental plan is an economic decision as much as a statistical one. This section establishes the vocabulary and mechanics that every later sampling strategy, surrogate model, and sensitivity index depends on: factors, levels, responses, randomization, blocking, and replication.

Factors and Levels

A factor is an input variable whose influence on the system you intend to study. In CAE, factors are the parameters you can legitimately vary in the model: sheet thickness, material yield strength, mesh density, friction coefficient, inlet velocity, spring stiffness, or a geometric offset. Factors come in two flavors. A continuous factor takes values on a real interval, such as thickness from 0.8 mm to 2.0 mm. A categorical factor takes discrete, unordered settings, such as material grade (DP600, DP780, HSLA350) or solver formulation (implicit vs. explicit).

A level is a specific value or setting assigned to a factor. A two-level continuous factor might use 0.8 mm and 2.0 mm; a three-level factor adds a midpoint at 1.4 mm. The number of levels determines the resolution at which you can detect curvature in the response. Two levels can only fit a linear effect; three levels allow a quadratic term. This is why a full factorial with k factors at two levels requires 2^k runs — 5 factors already demand 32 simulations, and 10 factors demand 1024, which is usually infeasible.

Two practical rules govern factor selection. First, screen before you optimize: if you have 15 candidate parameters, run a screening design (e.g., a Plackett–Burman or fractional factorial) to identify the 4–6 that matter, then study those intensively. Second, define factor ranges from physics, not convenience. A thickness range that crosses a buckling threshold or a material range that crosses a strain-rate transition will produce a response surface with a discontinuity that no polynomial surrogate can capture.

Responses

A response is the measured output of a run. In CAE, responses are extracted from solver output: peak von Mises stress, maximum intrusion, first natural frequency, mass, peak acceleration, or a scalar objective such as mass-normalized energy absorption. Responses must be unambiguous and extractable by script; a response that requires an engineer to eyeball a contour plot is not a response, it is an opinion.

Response selection deserves the same rigor as factor selection. A common mistake is to track a single scalar (e.g., peak stress) when the design intent is multi-objective (stiffness and mass and cost). Each additional response increases the data you must extract per run but does not increase the number of runs — so extract generously, then analyze selectively. Also record run metadata (solver version, wall-clock time, convergence status) alongside responses; a non-converged run is not a data point, it is a missing value, and treating it as a valid response silently corrupts the sensitivity analysis.

Randomization

Randomization is the deliberate shuffling of run order so that nuisance variables — machine drift, license contention, ambient temperature, a solver patch applied mid-study — do not align with a factor. If you run all low-thickness cases on Monday and all high-thickness cases on Tuesday, and the cluster was throttled on Tuesday, thickness and throttling are confounded: you cannot separate their effects.

In CAE, randomization is often skipped because "the solver is deterministic." That is precisely the trap. The solver may be deterministic, but the environment is not: queue priority, node allocation, and floating-point reduction order across MPI ranks can all shift results at the 10^-4 level. Randomizing run order costs nothing and protects against systematic bias. In practice, generate the design matrix, then apply a random permutation to the row order before submission, and log the permutation.

Blocking

Blocking groups runs that share a known nuisance condition so that the nuisance effect is removed from the factor-effect estimate. If your study must span two hardware queues with different core counts, treat the queue as a block: run a balanced subset of the design on each queue. The block effect is then estimated and subtracted, rather than being mistaken for a factor effect.

Blocking differs from randomization in intent. Randomization spreads unknown nuisance variation; blocking isolates known nuisance variation. Use blocking when you can name the nuisance (queue, operator, mesh family, solver version) and randomization for everything you cannot name.

Replication

Replication is running the same factor combination more than once to estimate pure error. It is the only way to distinguish a real factor effect from numerical noise. In CAE, replication is frequently confused with repetition: re-running the same input file on the same mesh with the same solver produces bit-identical results and estimates nothing. True replication perturbs the noise sources you care about — remesh with a different seed, re-run with a different number of MPI ranks, or use a slightly different but equally valid modeling choice.

A practical compromise: replicate the center point 3–5 times. The variance of center-point responses gives a cheap, defensible estimate of pure error, which in turn sets the denominator for every significance test in the analysis.

Worked example — a 3-factor, 2-level screening study. Factors: sheet thickness t (0.8 / 2.0 mm), yield strength σy (350 / 780 MPa), and friction coefficient μ (0.05 / 0.20). Response: peak von Mises stress σmax extracted from the solver's .dat output. A full 23 factorial gives 8 runs; add 3 center-point replicates at (1.4 mm, 565 MPa, 0.125) for a total of 11. Randomize the 11 run orders, block by queue if two queues are used, and record σmax for each. The center-point variance then tells you whether the 0.8→2.0 mm thickness effect is larger than solver noise.

How the Pieces Fit Together

The six concepts are not independent checkboxes; they form a chain of decisions. Factors and levels define the design space. Responses define what you measure. Randomization and blocking protect the validity of the comparison. Replication quantifies the uncertainty of that comparison. Skip any link and the sensitivity indices you compute later will look precise while being wrong.

Common Pitfalls in CAE DOE

Confounding factors with mesh. If you refine the mesh only for high-thickness cases, thickness and mesh density are confounded. Hold mesh quality constant across the design, or treat mesh family as a blocking variable.
Treating non-converged runs as data. A run that hits the iteration limit is a missing value. Log it, exclude it, and if many runs fail, the factor range is too aggressive.
Ignoring solver noise. Before trusting a 2% effect, run a center-point replicate set and check whether 2% exceeds the noise band. If it does not, the effect is not resolvable at your current solver settings.
Over-blocking. Blocking on too many nuisance variables fragments the design and inflates variance. Block only on nuisances you can name and that you expect to matter.

With these six concepts in place — factors, levels, responses, randomization, blocking, and replication — the next step is to choose a specific design geometry (full factorial, fractional factorial, central composite, Latin hypercube) that fits your run budget while preserving the effects you care about. That choice is where the statistics of DOE meets the economics of CAE.