Parametric studies, DoE and optimisation

A single simulation tells you how one design performs. It does not tell you whether a different one would perform better, which dimensions matter, or how much the answer moves when a tolerance or an operating condition shifts. Those are the questions design work actually turns on, and all of them need many runs rather than one.

The methods below are ways of getting the most information from the fewest solver runs, because the solver run is the expensive thing.

The simplest approach: vary one parameter across a range, run each value, plot the result. Inlet velocity against pressure drop; fillet radius against peak stress.

Sweeps are easy to set up and easy to read, and their limitation is combinatorial. Two parameters at ten values each is a hundred runs; four is ten thousand. A full factorial sweep stops being affordable almost immediately, which is what the next method exists to solve.

Design of experiments chooses which combinations to run so that the influence of each parameter — and the interactions between them — can be extracted from far fewer runs than an exhaustive grid. Latin hypercube sampling, factorial and space-filling designs all serve this purpose.

The output that usually matters most is not the best design found, but the ranking: which two or three parameters actually drive the result. That is frequently the whole return on the exercise, because it tells you what to spend the rest of the project on and what to stop arguing about.

A DoE also supports fitting a surrogate model — a cheap approximation of the response — which can then be explored interactively without running the solver again.

Related but distinct. Sensitivity asks how much the output moves per unit change of an input. Uncertainty quantification asks something more useful: given that your inputs are not known exactly — manufacturing tolerance, material scatter, an operating condition with a range — what is the distribution of the output?

The distinction matters for decisions. A design that is optimal at nominal values but falls off a cliff within tolerance is worse than a slightly duller design that holds up across the range. Optimisation without this step reliably finds the cliff edge, because that is exactly where the best nominal number tends to sit.

Optimisation closes the loop: an algorithm proposes a design, the solver evaluates it, and the result decides what to propose next, repeating until it converges or the budget runs out. It needs three things stated precisely — the parameters it may change, the objective it is improving, and the constraints it must respect.

The constraints are where most of the engineering judgement lives. An optimiser given “minimise drag” and nothing else will happily return something that cannot be manufactured, does not fit, or has no structural integrity. It is doing exactly what it was told; the specification was incomplete.

• Gradient-based methods converge quickly but need derivatives and can settle into a local optimum.
• Derivative-free methods — genetic algorithms, pattern search — need only the objective value, cope with awkward and discontinuous responses, and cost many more evaluations.
• Surrogate-based optimisation runs the algorithm against a fitted model rather than the solver, checking back with real runs periodically. Usually the affordable option when a single evaluation takes hours.

Parametric optimisation is limited to the parameters you thought to expose. Shape optimisation goes further and lets the surface itself move — but a surface has thousands of degrees of freedom, and computing a gradient for each by finite difference would mean thousands of runs.

The adjoint method is what makes this tractable. By solving one additional adjoint problem alongside the primal solution, it produces the sensitivity of your objective to every surface point at roughly the cost of a single extra run, regardless of how many design variables there are. The output is a map over the geometry showing where moving the surface would help and by how much.

That map is worth having even if you never run an automatic optimisation loop — it tells a designer which region of a part is worth their attention.

Sweeps, DoE and the run matrix need no external tool — they are orchestration around a solver you already have. NLopt and pagmo provide derivative-free optimisation algorithms in-process. Dakota covers uncertainty quantification and calibration. For adjoint shape optimisation, SU2 has both discrete and continuous adjoint formulations, and OpenFOAM has adjointOptimisationFoam, less maturely.