2. Defining factors and responses

A factorial study is only as trustworthy as the variables you choose to manipulate and the metrics you choose to observe. Before any DOE matrix is generated or any solver run is launched, the analyst must translate an engineering question into a precise, testable set of factors (controllable CAE inputs) and responses (quantitative outputs extracted from simulation results). This chapter focuses on that translation: how to pick factors that are physically meaningful and numerically well-behaved, how to define responses that are stable and comparable across runs, and how to assign measurement scales so that the subsequent ANOVA and regression analysis is valid.

From engineering question to factor list

A factor is a CAE input parameter that the analyst can set independently and hold at a specified level during a run. Good factors share three properties: they are controllable (set directly in the input deck or GUI), independent (not derived from another factor in a way that creates collinearity), and physically relevant (the response is expected to change with them). Typical CAE factor families include geometry dimensions (fillet radius, wall thickness, rib spacing), material properties (Young's modulus, yield stress, damping ratio), boundary and loading conditions (bolt preload, pressure magnitude, friction coefficient), and mesh or solver settings when they are part of the study scope (element order, contact stiffness scale factor).

Two practical rules prevent most downstream problems. First, avoid derived factors: if you vary both plate thickness t and cross-sectional area A = w·t, the two factors are functionally linked and the design matrix becomes rank-deficient. Choose the independent primitives (w and t) and let the derived quantity be computed. Second, fix nuisance variables: mesh density, solver tolerance, and contact algorithm should be locked at validated settings so that run-to-run variation reflects only the factors under study. If mesh sensitivity is itself a question, treat it as a separate factor with a clearly documented level set, not as an uncontrolled setting.

Choosing levels and the measurement scale

Each factor is studied at two or more levels. For a two-level factorial, the low and high levels should bracket the realistic operating range, not the extreme limits of the material or geometry. A common mistake is to set levels so wide that the response becomes nonlinear or the solver fails at the high corner; a common opposite mistake is to set levels so narrow that the effect is buried in numerical noise. A defensible starting point is ±10–20% around the nominal design value for geometry and material factors, and the validated operating envelope for load factors.

The measurement scale of a factor determines how its effect is coded in the model. Continuous factors (thickness, modulus, pressure) are coded as −1 and +1 at the low and high levels, with the center point at 0. Categorical factors (material grade A vs. B, element type linear vs. quadratic) are coded as indicator variables and cannot be interpolated between levels. Mixing the two in one design is legitimate, but the analysis must treat them differently: continuous factors yield a slope, categorical factors yield a level contrast. Record the coding explicitly, because the regression coefficients are only interpretable relative to the coded scale.

Defining responses that survive the solver

A response is a scalar extracted from simulation output that answers part of the engineering question. The critical requirement is stability: the same input must produce the same response value within a tight tolerance, or the DOE will attribute numerical scatter to factor effects. Responses that are sensitive to mesh, time-step, or convergence path are poor candidates unless those sensitivities are controlled.

Well-behaved CAE responses typically include: peak von Mises stress at a named location, maximum displacement at a probe node, first natural frequency from a modal solve, mass or volume of a component, peak reaction force, and fatigue life at a critical location. Each must be tied to a fixed extraction location or query — a named node set, a defined path, or a result probe — so that every run reports the same physical quantity. A stress read at "the maximum anywhere in the model" is unstable because the location can jump between runs; a stress read at a fixed gauge node set is stable and comparable.

Where the engineering question concerns a field rather than a point, define the response as a reduced scalar: area-averaged pressure, 95th-percentile stress over a surface, or total strain energy. Document the reduction rule precisely, because two analysts reducing the same field differently will produce non-comparable response columns.

Units, sign conventions, and transformation

State units and sign conventions once, at the top of the response definition, and apply them consistently. For displacement, define the positive direction relative to a named coordinate system; for stress, state whether tension is positive; for reaction force, state the sign relative to the applied load. Inconsistent sign conventions across runs silently corrupt interaction terms.

Some responses benefit from transformation before analysis. A response that spans orders of magnitude (fatigue life, clearance) is often analyzed on a log scale so that the factorial model captures multiplicative rather than additive effects. A response bounded below by zero (mass, life) may need a log or Box–Cox transform to satisfy the constant-variance assumption of ANOVA. Apply the transform to the response column, not to the factor levels, and report the transform in the results table.

Worked example: bracket stiffness study

Consider a steel mounting bracket whose tip deflection under a 500 N load must be reduced. The engineering question is: which of four inputs most strongly controls tip deflection, and are there interactions? The factor and response definitions below are illustrative and use assumed values for teaching purposes.

Note that the categorical factor has no center level; the design is a 2³×2 mixed-level factorial with center points only on the continuous factors. The response extraction locations are fixed node and surface sets, so every run reports the same physical quantity. The tip deflection is expected to be positive in +Z under the downward load convention defined in the model; that convention is recorded in the response definition, not left implicit.

Common pitfalls and how to catch them

• Response location drift: if the peak stress location moves between runs, the response is not comparable. Fix the extraction set and verify by re-running the nominal case twice.
• Factor collinearity: derived factors (area from width and thickness) create rank deficiency. Check the design matrix rank before running the solver.
• Level-induced solver failure: the high corner of a factor combination may not converge. Run the corner cases first as a screening step; if a corner fails, narrow the level range rather than dropping the run.
• Unit mismatch: mixing mm and m in the input deck produces responses off by 10³. Enforce a single unit system in the run table.
• Transform forgotten: analyzing a log-normal response on a linear scale inflates the residual variance and weakens the F-test. Record the transform in the response definition.

The diagram above makes the definition chain explicit: continuous and categorical factors feed the solver, which produces field results that are reduced to fixed-location scalar responses. The nuisance settings are held constant and shown separately so they are not mistaken for factors. The transform check sits on the response side because it applies to the response column, not to the factor levels.

Deliverable: factor–response definition sheet

The following sheet is a reusable template for documenting a study before runs begin. Fill one row per factor and one row per response; the coded and uncoded columns must both be populated, and the extraction rule must be specific enough that a second analyst could reproduce the response from the same result files.

STUDY: bracket_stiffness_v1
OBJECTIVE: rank factor effects on tip deflection; detect interactions
DESIGN: 2^3 x 2 mixed-level factorial + 3 center points (continuous only)

FACTORS
name            type        low(-1)  center(0)  high(+1)  units  coding
web_thickness   continuous  3.0      4.0        5.0       mm     coded = (x-4.0)/1.0
fillet_radius   continuous  2.0      4.0        6.0       mm     coded = (x-4.0)/2.0
bolt_preload    continuous  8.0      12.0       16.0      kN     coded = (x-12.0)/4.0
material_grade  categorical Q345     -          Q460      -      indicator: Q460=+1

FIXED (nuisance, not factors)
mesh_size = 2.0 mm; solver_tol = 1e-4; contact = bonded; load = 500 N, -Z

RESPONSES
name        extraction_rule                  units  scale    transform
tip_def     probe node set TIP, +Z           mm     linear   none
peak_stress max over surface set FILLET      MPa    linear   none
first_mode  modal solve, free-free, mode 1   Hz     linear   none

SIGN CONVENTIONS
displacement: +Z upward; load applied -Z; stress: tension positive

Two verification steps should precede the full run set. First, run the nominal (center) case twice and confirm that every response reproduces within a stated tolerance — for linear static solves, 0.1% is a reasonable target; for contact or nonlinear solves, 1–2% may be realistic. Second, run the four extreme corners of the continuous factor space and confirm convergence; if a corner fails, narrow the level range and update the sheet before generating the design matrix. These two checks catch the majority of response-definition and level-selection errors before they contaminate the ANOVA.

With factors, levels, responses, and scales documented in this form, the design matrix generation and the subsequent effect analysis become mechanical. The engineering judgment lives in the definitions — and that is exactly where the analyst's expertise should be concentrated.