Design of Experiments for Engineering Analysis

DoE applies to tests and to simulation campaigns alike

Design of experiments is the discipline of choosing where to gather data so that the resulting dataset contains real information about the system being studied. It applies equally to physical experiments — a wind-tunnel campaign, a component test matrix — and to simulation campaigns, where each "experiment" is an expensive high-fidelity run and the question is which runs to spend the budget on. The principles are the same: the factors that may affect the response are identified, the levels at which each is studied are chosen, and the combinations to be run are selected so that the main effects and the interactions among factors can be separated. A well-designed campaign extracts more information per run; a poorly designed one spends the same budget and leaves the important questions unanswered.

Factors, responses, levels and interactions

A factor is an input that is deliberately varied — a load magnitude, a material property, a geometry parameter, a boundary condition. A response is an output that is measured or computed — a peak stress, a displacement, a natural frequency, a fatigue life. The levels of a factor are the values at which it is studied. An interaction is the situation in which the effect of one factor depends on the level of another: a thickness increase may reduce stress at one load level and increase it at another, or a material change may help under one boundary condition and hurt under another. Interactions are not exotic; they are common in structural and mechanical systems, and they are exactly what unstructured data collection tends to miss.

• Factor — input deliberately varied (load, material, geometry, boundary condition).
• Response — output measured or computed (stress, displacement, frequency, life).
• Level — value at which a factor is studied.
• Interaction — the effect of one factor depends on the level of another.
• Screening — identifying which factors matter before detailed study.
• Factorial design — combinations of factor levels run together so interactions can be estimated.
• Latin hypercube sampling — each factor sampled across its range with controlled spread.
• Space-filling designs — points spread to cover the design space with few gaps.

Why one factor at a time can hide interactions

The intuitive way to study a system is to vary one factor at a time while holding the others fixed. This is simple and interpretable, and it is adequate when factors act independently. But when factors interact — when the effect of a thickness change depends on the load level, or the effect of a joint stiffness depends on the boundary condition — a one-factor-at-a-time study can miss the interaction entirely, because it never runs the combinations in which the interaction appears. The study reports a main effect that is only valid at the fixed values of the other factors, and the engineer may not realise that the response would change qualitatively elsewhere in the space. A factorial or space-filling design runs combinations specifically so that interactions can be seen.

ONE-FACTOR-AT-A-TIME STUDIES CAN HIDE PARAMETER INTERACTIONS.

Coverage: one factor at a time versus a space-filling design

The diagram contrasts the coverage of a two-factor space under a one-factor-at-a-time study and a space-filling design. The one-factor-at-a-time study samples along the axes through a central point and leaves most of the space unvisited; any behaviour that lives in the corners or off the axes is invisible. A space-filling design spreads points across the whole space, so interactions and off-axis behaviour are sampled. The diagram also shows, schematically, that two factors together can alter the response in a way that varying them in isolation would not reveal.

DESIGN-SPACE COVERAGE: ONE FACTOR AT A TIME vs SPACE-FILLING

  Parameter B
  │
  │     ●           (one-factor-at-a-time:
  │     │            only the axes through
  │     │            the centre are sampled;
  │     │            corners and off-axis
  │  ●──┼──●         regions are unvisited)
  │     │
  │     │
  │     ●
  │
  └──────────────────► Parameter A

  Parameter B
  │
  │   ●    ●        (space-filling design:
  │     ●      ●     points spread across
  │  ●     ●    ●    the whole space;
  │     ●      ●     interactions and
  │   ●    ●         off-axis behaviour
  │                   are sampled)
  └──────────────────► Parameter A

  INTERACTION EXAMPLE
  ┌─────────────────────────────────────────────┐
  │  Parameter A alone: small effect             │
  │  Parameter B alone: small effect             │
  │  A and B together: response changes sharply  │
  │  → visible only when both are varied jointly  │
  └─────────────────────────────────────────────┘

Design-space coverage and expensive simulations

When each run is a cheap measurement, a full factorial design — every combination of every level of every factor — is affordable and gives complete information about main effects and interactions within the chosen levels. When each run is an expensive high-fidelity simulation, a full factorial quickly becomes unaffordable as the number of factors grows, and the design must extract the most information from the fewest runs. This is the regime where space-filling designs — Latin hypercube sampling, optimal designs, maximin or minimax distance designs — are used. The goal is to cover the design space with few large gaps, so that a surrogate built on the data is well supported everywhere in the region of interest and not just along a few axes. The link between DoE and surrogate modelling is direct: a surrogate is only as good as the data behind it, and that data is only as informative as the design that produced it.

DoE methods and their applications

The table summarises the common design types. Screening designs are used early, to identify which of many candidate factors actually matter before spending budget on a detailed study. Factorial designs estimate main effects and interactions but scale steeply with factor count. Space-filling designs are preferred for surrogate construction because they avoid large unsampled regions. Optimal designs are tailored to a chosen model form and criterion. The choice depends on the number of factors, the cost per run, whether interactions are expected and what the data will be used for.

The classic misuse: ad-hoc point selection

The most common failure is selecting simulation points by convenience — the values already in a model, the cases that are easy to set up, the points a colleague suggested — rather than through a structured design. A dataset built this way can look substantial and still systematically miss the regions that control the response. A surrogate built on it will be well informed where the points happen to cluster and uninformed exactly where the decision is sensitive. The cost of a structured design is small compared with the cost of running an expensive campaign and discovering afterwards that the data does not answer the question.

SELECTING SIMULATION POINTS AD HOC OR BY CONVENIENCE RATHER THAN THROUGH A STRUCTURED DESIGN CAN PRODUCE A DATASET THAT SYSTEMATICALLY MISSES THE REGIONS THAT CONTROL THE ENGINEERING RESPONSE.