2. Design space definition for iterative CAE studies

An iterative CAE study is only as good as the design space it explores. Before a single solver run is launched, the engineer must decide what can change, how much it may change, what must never be violated, and what "better" actually means. These four decisions — parameterization, variables, constraints, and response metrics — form the design space definition. Get them wrong and the optimizer will either converge to an infeasible or unmanufacturable design, or waste hundreds of solver hours exploring regions no engineer would ever accept. This chapter treats design space definition as a deliberate engineering artifact, not a by-product of the CAD model.

Parameterization: Turning Geometry into Controllable Inputs

Parameterization is the mapping from a set of scalar inputs to a fully defined CAE model. A good parameterization is complete (every input needed to rebuild the model is exposed), minimal (no redundant inputs that fight each other), and robust (small input changes produce small, valid geometry changes). In practice this means driving the CAD through a named parameter table rather than editing sketches by hand.

Consider a bracket stiffened by a triangular rib. A workable parameter set might be:

plate_length      L  = 180 mm   (driven, fixed)
plate_width       W  = 90 mm    (driven, fixed)
rib_height        h  = 35 mm    (variable)
rib_thickness     t  = 4 mm     (variable)
fillet_radius     r  = 6 mm     (variable)
mount_hole_dia    d  = 10 mm    (driven, fixed)

Each variable must be tied to a regeneration-safe feature. If r exceeds h/2, the fillet may fail to rebuild — a classic parameterization defect that surfaces only when the optimizer pushes the variable to its bound. The remedy is either a constraint (see below) or a reparameterization using a normalized ratio such as r = k·h with 0.1 ≤ k ≤ 0.4, which keeps the geometry valid by construction.

Variables: Choosing the Right Degrees of Freedom

Not every dimension should become a design variable. A useful filter is to ask three questions of each candidate: Does it materially affect a response metric? Is it free to change in production? Can it be varied independently of the others? Dimensions that fail any of these should be frozen as constants, because every added variable multiplies the number of solver evaluations needed for a given sampling density.

For the bracket, freezing L, W, and d is justified: they are set by the mating interface and the fastener standard. The three remaining variables — h, t, r — are the genuine design freedom. A three-variable space with a Latin Hypercube sample of 30 points is tractable; a ten-variable space with the same density would need thousands of runs.

Variables also need types. Continuous variables (h, t) suit gradient-based or response-surface methods. Discrete variables — sheet thicknesses from a standard gauge list, for example — require enumeration or mixed-integer handling. If t must be a stocked gauge, declare it discrete with the candidate set {3, 4, 5, 6} mm rather than a continuous range, otherwise the optimizer will return 4.37 mm and the result is unmanufacturable.

Constraints: Encoding Physics and Manufacturability

Constraints fall into three families, and each must be declared explicitly in the study setup:

• Bound constraints — the box limits on each variable, e.g. 20 ≤ h ≤ 50 mm, 3 ≤ t ≤ 8 mm. These define the sampling domain and must reflect real manufacturing limits, not arbitrary round numbers.
• Geometric constraints — relationships that keep the model valid, e.g. r ≤ 0.4·h or t ≤ 0.25·W. These prevent rebuild failures and are best enforced in the parameterization itself where possible.
• Response constraints — performance limits evaluated by the solver, e.g. σ_vm,max ≤ 180 MPa (yield 250 MPa with a 1.4 safety factor) and δ_tip ≤ 0.5 mm. These are the constraints that make the study an engineering optimization rather than a pure shape search.

Response constraints deserve special care because they are expensive: each one requires a full solve. A common mistake is to overload the study with a dozen response constraints, most of which are never active. Start with the two or three that genuinely govern the design, verify after the first iteration batch which constraints are active, and prune the rest.

Response Metrics: Defining "Better" Precisely

A response metric is a scalar extracted from the CAE result that the study will optimize or report. It must be unambiguous (a single number, not a field), reproducible (same mesh, same extraction rule every run), and aligned with the objective. For the bracket, the natural objective is mass, with the responses being peak von Mises stress, tip deflection, and first natural frequency.

The extraction rule matters as much as the metric. "Peak stress" taken over a coarse mesh will differ from the same quantity on a refined mesh, so the mesh must be held constant across all iterations, or the metric must be sampled at a fixed probe location. Document the extraction rule in the study definition so that a colleague can reproduce the number exactly.

Putting the Design Space Together

The figure below shows how the four elements combine into a single study definition. The parameter table feeds the CAD regeneration; the variables and bounds define the sampling domain; the constraints filter candidate designs; and the response metrics score the survivors for ranking.

Worked Example: First Iteration Batch

With the design space defined as above, the first batch is a 30-point Latin Hypercube sample over h ∈ [20, 50], t ∈ {3,4,5,6}, and k ∈ [0.1, 0.4]. Each point regenerates the CAD, meshes with a fixed element size of 2 mm, and runs a linear static solve under LC1 (a 2 kN downward load at the free end) plus a free-free modal solve. The extraction script writes one row per design:

design_id, h_mm, t_mm, k, mass_g, sigma_max_MPa, delta_mm, f1_Hz, feasible
D001,      32.4, 4,    0.22, 214.6, 168.3, 0.41, 134.2, 1
D002,      47.1, 3,    0.35, 198.2, 241.7, 0.63, 118.5, 0
D003,      25.8, 5,    0.18, 236.9, 142.0, 0.33, 151.8, 1
...

After the batch, inspect the feasible column. If fewer than roughly 20% of samples are feasible, the bounds are too aggressive and should be tightened before spending more solver time. If nearly all samples are feasible, the constraints are not binding and the objective can be pushed harder. This diagnostic — the feasible fraction — is the single most useful early check on a design space definition, and it is exactly what the feedback loop in the figure is for.

Common Design Space Defects

Three failure modes recur in practice. Over-parameterization exposes dimensions that the optimizer exploits in unphysical ways — for example, thinning a wall until stress concentrates at a fillet the mesh cannot resolve. Under-constraint omits a manufacturability or stiffness limit, producing a design that is optimal on paper but fails in service. Metric drift changes the extraction rule mid-study, so early and late designs are scored on different scales. Each is prevented by the same discipline: write the design space definition down as a versioned artifact, review it before the first solve, and treat any change to it as a new study rather than a tweak.

The design space definition is therefore not paperwork — it is the contract between the engineer and the optimizer. Parameterization guarantees the model can be rebuilt, variables and bounds define what is legitimately free, constraints encode what must hold, and response metrics define what "best" means. Once these four are fixed and documented, the iterative CAE comparison in later chapters has a stable foundation on which rankings can be trusted.