Computational Engineering / Sensitivity and design-space exploration

Bracketing and its limits

Three runs that bound an answer, the assumption they rest on, and what happens when it fails.

Lesson review: expert verification remains open

This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Checks needed for project use

A computational engineering reviewer must check algorithms, convergence claims, statistical assumptions and numerical examples beyond the cases explicitly recorded.
Validate the sensitivity lab’s numerical derivatives, units and range conventions independently.
Joint parameter ranges and model interactions need project-specific engineering review.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

Read the lesson register and outstanding review notes

Why this matters

A full factorial study of five inputs at eight values uses 32768 runs. Simpler studies can reduce the cost, but a three-run bracket is justified only when the response has consistent monotonic directions over the whole feasible range. Check that assumption before treating the endpoints as bounds.

By the end of this lesson you should be able to

Construct a bracketing set of runs
State the monotonicity assumption that bracketing rests on
Recognise a non-monotonic response and say what it costs
Explain why one-at-a-time studies cannot find interactions

Bracketing

If the output is monotonic in each variable throughout the feasible domain, with a consistent direction for each, its extremes lie at the corresponding domain corners. Two extreme runs and a base case can then be useful. Individual sweeps at the base point do not establish that condition throughout a multidimensional range.

Resonance and changing load paths can produce interior extremes. Nonlinearity alone does not defeat bracketing: the scalar second-order amplifier is nonlinear but monotonic over its precritical range. Discrete load patterns and correlated inputs require a feasible-domain assessment.

Interactions

A one-at-a-time study shows response along the sampled lines. It can reveal a turning point on one of those lines, but does not establish the absence of other turning points or interactions. Joint sampling, a suitable experimental design or a justified analytical argument is needed to assess combined effects.

Report the parameter ranges, response sampling, derivative assumptions, any justified bounds and the interactions or interior regions not assessed.

Try it

Design space explorer

Sweep two variables together and see the surface. Then compare the cost of doing that for all five.

Horizontal axis

Vertical axis

Values per variable

Darker blue is the smallest deflection (0.97 mm); orange is the largest (2.90 mm). The shading is a secondary cue — the range is stated in the label so it does not depend on colour.

Cells sampled: 81
Smallest deflection: 0.974 mm
Largest deflection: 2.898 mm
Ratio across the space: 2.98
One-at-a-time runs, all 5 variables: 45
Full factorial, all 5 variables: 59,049

What the surface shows that two sweeps cannot

If the contours were straight parallel bands, the two variables would act independently and two separate sweeps would tell you everything.
Where the contours curve or fan out, the effect of one variable depends on the value of the other — an interaction, and one-at-a-time sampling is blind to it.
Five variables at nine values each is 59 049 runs for a full factorial against 45 for one-at-a-time. That gap is why Module 16 exists.

What this shows: A two-variable sweep shows the interaction that two one-variable sweeps cannot — and a full factorial over five variables is unaffordable.

Worked example

A bracket that does not contain the worst case

Given

A floor whose footfall response factor is being assessed
The mass present in service is uncertain between 0.4 and 2.5 kN/m² of imposed load
At 0.4 the response factor is 6.2; at 2.5 it is 5.8
At the base case of 1.2 it is 11.4

Find

What a three-run bracket would have concluded, and what is actually true

Predict first

A one-at-a-time sensitivity study finds that no single input moves the output by more than 15 %. What can be concluded about the combined effect of all the uncertainties?

Practice

A study varies 4 inputs at 6 values each in a full factorial design. How many runs does that require?

Practice

Three independent optional changes produce +12%, −9% and +15% relative to a base output. Assuming exact additivity, and allowing each change to be either applied or omitted, what is the largest increase?

Check yourself

When does taking the corner combinations fail to bracket the answer?

Summary

Three runs bracket an answer if every response is monotonic
Resonance and changing load paths may create interior extrema; a nonlinear response can still be monotonic
A one-at-a-time study can detect non-monotonicity, which is why it comes first
It cannot detect interactions, and no amount of finer sampling fixes that
Report that interactions were not explored — it is what makes the rest usable