Computational Engineering
Optimisation workbench
Stage D's tools in one place, in the order you would actually use them: find out what matters, then explore, then search, then decide between things that cannot both be best.
This page gathers the interactives from Modules 14 to 17. It teaches nothing the modules do not — it exists because optimisation work is iterative, and moving between four lessons to run four tools is not how the work is done.
The order matters and it is the order below. A sensitivity study first, because optimising a variable the answer does not depend on is wasted effort. Then exploration, then a search method chosen for the landscape, and only then a decision between non-dominated options.
What optimisation cannot do for you
Every result on this page is the best design found, by that method, from that start, in that many evaluations. It is not a global optimum unless something has proved it is one, and nothing here proves it.
- A compression member also has to be stable, so its real volume is larger than the load path suggests. Compare like with like, or design the members.
- Connections are not in this number. Where connections drive member size — timber especially — the ranking can reverse.
- It assumes member area is proportional to force, which is true for ties and only approximately true for struts.
- The lightest structure is not automatically the cheapest, the lowest-carbon, the simplest or the most robust.
The landscapes
The search tools work on two shared test landscapes, so the methods can be compared on equal terms. Both are seeded, so a number read off a tool is the number the lessons quote.
Single minimum
One basin. Any downhill method reaches the bottom, and the only question is how many evaluations it spends.
Several minima
Several basins of different depth. Where a downhill method ends up depends entirely on where it started.
Where each tool is taught
Module 14 — Sensitivity and design-space exploration
Which inputs the answer actually depends on — and why a wide plausible range beats a large exponent.
Module 15 — Structural optimisation
Topology, layout, shape and size — and a measure that lets you compare layouts before a single member has been chosen.
Module 16 — Optimisation algorithms
How a search finds a good design, why the answer depends on where it started, and the result that says no method wins everywhere.
Module 17 — Multi-objective design and Pareto decisions
What to do when the objectives compete — and why the tool must present the trade-off rather than announce a winner.
Sensitivity explorer
Module 14Before optimising anything, find out what the answer actually depends on. A wide plausible range beats a large exponent.
Try it
Sensitivity explorer
A floor beam with five inputs, each with a range justified from the project rather than assumed. Read the elasticities against the exponents you already know.
Output
- Low bound
- 0.00055
- Base case
- 0.00182
- High bound
- 0.00490
- Spread high/low
- 8.97
- Bracket valid?
- yes
| Variable | Elasticity | Range | Swing | Monotonic? |
|---|---|---|---|---|
| Imposed load present in service | 0.51 | 0.40–2.50 kN/m² | 88.7 % | yes |
| Section depth | -2.87 | 400.00–533.00 mm | -78.7 % | yes |
| Span | 3.96 | 8.70–9.30 m | 26.4 % | yes |
| End restraint stiffness | -0.04 | 0.00–60000.00 kN·m/rad | -24.9 % | yes |
| Young's modulus | -0.96 | 205.00–215.00 GPa | -4.6 % | yes |
Every variable moved the output the same way across its whole range, so pushing them all to their worst ends really does bracket the answer.
Where each range comes from
- Span: Grid set by the architect; construction tolerance and a possible late grid shift of one column line.
- Section depth: The serial sizes available within the agreed floor zone.
- Young's modulus: Structural steel. One of the few genuinely tight inputs, and its narrow range is the point.
- Imposed load present in service: What is actually on the floor when it vibrates, not the ultimate design value. The widest range here, and usually the one that decides the answer.
- End restraint stiffness: A nominally pinned connection behaves as continuous at vibration strains and as a pin at ultimate. The range spans both.
Two things this study is doing at once
- Verifying: for deflection the elasticities must be +4 for span, −3 for depth and −1 for modulus, because those are the exponents in the formula. Anything else means the model is not doing what beam theory says.
- Exploring: the tornado ordering directs effort at the inputs with the largest swing, which are not the ones with the largest elasticity. Young's modulus has an elasticity of −1 and almost no swing, because it is known to ±2 %.
- And it cannot find interactions. Varying one input while holding the others at base values is silent about combinations, and no amount of finer sampling fixes that.
What this shows: Swing = elasticity × range, and the ranking follows swing — so a tight input with a big exponent can matter less than a loose one with a small exponent.
Design space explorer
Module 14How quickly a design space becomes too large to enumerate.
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
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
1312× the cost, and it is what finds interactions
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.
Ground structure explorer
Module 15Layout optimisation is a search, and the answer depends on where you started.
Try it
Ground structure explorer
Start with every member and remove the ones carrying least. The tool backs off when a removal would create a mechanism.
Remove members carrying less than this fraction of the largest force.
- Members at the start
- 23
- Members kept
- 23
- Members removed
- 0
- Structural volume at the start
- 10587 kN·m
- Structural volume now
- 10587 kN·m
- Still standing?
- yes
- Iterations
- 1
Nothing was removed at this threshold: every member is carrying more than the cut-off. Raise it and watch the lightly-loaded members go.
The same idea, applied to four bracing layouts
| Layout | Volume (kN·m) | Topology constant |
|---|---|---|
| K-bracing | 1433 | 0.000 |
| Knee brace (asymmetric) | 1600 | 0.000 |
| Single diagonal | 1733 | 0.000 |
| X-bracing | 1733 | 0.000 |
The topology constant is identical for all four — zero, here, because every external force has zero work on its own position. That is the check that the comparison is fair. If it drifted, something about the loads or the supports would have changed and the volumes would not be comparable.
What the load path does not include
- A compression member also has to be stable, so its real volume is larger than the load path suggests. Compare like with like, or design the members.
- Connections are not in this number. Where connections drive member size — timber especially — the ranking can reverse.
- It assumes member area is proportional to force, which is true for ties and only approximately true for struts.
- The lightest structure is not automatically the cheapest, the lowest-carbon, the simplest or the most robust.
What this shows: Layout optimisation is a search, not a formula — the answer depends on where you started and on the removal rule you chose.
Truss shape optimiser
Module 15There is a best depth and it is an interior minimum — and least material and least deflection are not the same depth.
Try it
Truss shape optimiser
Sweep the depth. Both material and stiffness have interior optima — at different depths.
2 m
- Least-material depth
- 5 m
- Volume there
- 13024 kN·m
- Inspecting depth
- 2 m
- Volume there
- 19120 kN·m
- Deflection there
- 38.90 mm
47 % above the minimum
least at about 8 m, deeper than the volume optimum
The volume curve has an interior minimum: shallow costs chord force, deep costs diagonal length. So does the deflection curve, at a different depth: at constant member area a very deep truss softens again, because its diagonals have become long. Two objectives, two different depths, from the same structure.
Reading it
- Chord force is moment over depth, so the shallow end is dominated by chord material.
- Diagonals lengthen in proportion to depth, so the deep end is dominated by web material — and by web flexibility, which is why the deflection turns round too.
- An interior optimum means a genuine trade-off, and a design pushed to either extreme is worse than one in the middle.
- If the truss is deflection-controlled, the material optimum is not the answer — and no single-objective optimiser will mention that. 'Deeper is always stiffer' is only true if you re-size the members as you go.
What this shows: Two objectives give two different answers from the same sweep — and which one governs depends on whether the design is strength-controlled or deflection-controlled.
Size optimiser
Module 15Sizing changes the forces, so one pass is never the answer.
Try it
Size optimiser
A fully-stressed design, iterated. Add a deflection limit or a discrete catalogue and watch what each costs.
Structure
The indeterminate one has a duplicated diagonal and both supports held horizontally.
Sections available
Zero means no deflection constraint — stress alone governs.
- Converged?
- yes
- Iterations
- 2
- Governing constraint
- stress
- Total volume
- 8200 cm³
- Mass
- 0.064 t
- Mean utilisation
- 1.000
- Deflection
- —
exactly critical everywhere — no reserve anywhere
| Member | Area (cm²) | Utilisation |
|---|---|---|
| AC | 5.00 | 1.000 |
| BC | 5.00 | 1.000 |
| AB | 4.00 | 1.000 |
Every member is at 100 % utilisation. The optimiser did exactly what it was told — and the result has no reserve anywhere, so a single member failing has nothing to redistribute into. Robustness is not an emergent property of minimising weight; if it matters it has to be a constraint.
Try this
- Determinate, stress only: converges in one or two passes with every member fully stressed. The forces do not depend on the sizes.
- Switch to indeterminate: sizing changes the forces, so the loop has to run several times. That is the snowball.
- Add a deflection limit: the governing constraint changes and the volume rises. Stress-only optimisation can violate serviceability without saying so.
- Switch to the catalogue: utilisations scatter, volume rises, and fabrication gets simpler.
What this shows: Sizing changes the forces, so one pass is never the answer in an indeterminate structure — and a fully-stressed design is exactly critical everywhere, which is not the same as robust.
Form-finding explorer
Module 15The shape is an output, not something you drew.
Try it
Form-finding explorer
Set a force density for each cable and the equilibrium shape falls out of a linear solve. The shape is an output.
Fixing this ratio is what turns a nonlinear problem into a linear one.
Raise the force density on the left half only and watch the shape warp.
- Solved?
- yes
- Maximum sag
- 2.2500 m
- Sag / span
- 0.2250
- Symmetric?
- yes
Uniform force densities and uniform loads give a symmetric hanging shape. Double q and the sag halves exactly — the relationship is linear, which is the whole trick.
Why the shape is not an aesthetic choice
- Fixing q = F/L for every cable makes the equilibrium equations linear in the coordinates, so the shape comes from a single linear solve.
- The result is funicular: it carries its load in pure axial tension, with no bending anywhere.
- Invert it and you have the funicular arch for that loading — which is why hanging models were used to design masonry vaults long before anyone could solve the equations.
- Set the load to zero and the cable becomes straight: with no transverse load there is nothing for it to hang under.
What this shows: In modelling you draw a shape and ask what it does. In form-finding you state what you want it to do and it tells you the shape.
Search algorithm comparison
Module 16No method wins both landscapes, and the evaluation count is the price.
Try it
Search algorithm comparison
Six methods on the same landscape, with the evaluation count for each. Everything is seeded, so the numbers here are the numbers the lesson quotes.
Landscape
One basin. Any downhill method reaches the bottom, and the only question is how many evaluations it spends.
Show the path of
Dark blue is the lowest objective value (1.015); orange the highest (14.348). The shading is a secondary cue — every value is in the table below.
| Method | Best found | Gap | Evaluations | Trapped? |
|---|---|---|---|---|
| Grid search | 1.0150 | 0.0150 | 441 | no |
| Gradient descent | 1.0000 | 0.0000 | 79 | no |
| Random search | 1.0432 | 0.0432 | 441 | no |
| Simulated annealing | 1.0019 | 0.0019 | 441 | no |
| Genetic algorithm | 1.0000 | 0.0000 | 401 | no |
| Particle swarm | 1.0001 | 0.0001 | 441 | no |
Gradient descent reached within 0.0000 of the global optimum in 79 evaluations.
What this method is doing
- Follows the slope downhill from where it started. It cannot leave the basin it began in, and it does not know there is anything outside it.
Things worth trying
- On the smooth landscape, compare gradient descent's evaluation count with everything else. It wins outright — right answer, a fifth of the budget.
- Switch to the rugged landscape without changing anything. The same method is now trapped.
- Move the start to (6, 3) on the rugged landscape and gradient descent finds the global optimum. Nothing about the method changed.
- Grid search always evaluates 441 times regardless. In five variables at the same resolution it would be 4 084 101.
What this shows: No method wins on every landscape — and on a rugged one the answer a downhill method gives is decided by where it started.
Pareto explorer
Module 17A front is a property of the objectives you chose. Change them and designs move on and off it.
Try it
Pareto explorer
Four scheme options for the same building, on any two objectives. The tool builds the front, marks what is dominated, and refuses to name a winner.
Option set
The second is three designs contrived so one sits in a dip.
Horizontal axis
Vertical axis
| Option | Embodied carbon | Structural depth | Status | Crowding |
|---|---|---|---|---|
| CLT and glulam hybrid | 447 | 465 | on the front | ∞ |
| Concrete band beam and slab | 666 | 374 | on the front | 2.00 |
| Composite steel frame | 782 | 480 | dominated (rank 1) | — |
| Reinforced concrete flat slab | 866 | 318 | on the front | ∞ |
- On the front
- 3 of 4
- Dominated
- 1
- Infeasible
- 0
- Reachable by a weighted sum
- 3 of 3
- Unreachable at any weighting
- none
Every front member is reachable by some weighting here, because this front happens to be convex. Switch to the concave demonstration and one design becomes unreachable at every weighting.
Knee: Concrete band beam and slab
- The knee is where the trade-off curve bends most sharply. It is a good place to start a conversation and a bad place to end one: it depends on the axes you chose and their scaling, and it knows nothing about who is paying or what they value.
Your decision
The tool will not choose. Pick an option and say why — a rejection or a selection with no reason cannot be reviewed and cannot be revisited.
Still missing: chosen, reason, decidedBy. Six months from now the choice will be a fact and the reason will be gone unless it is written down.
Try this
- On the four real schemes, plot carbon against depth: three are on the front and the composite steel frame is dominated by the concrete band beam on both axes.
- Now switch the vertical axis to mass. The front collapses to a single design — the CLT hybrid dominates everything. Nothing about the schemes changed; the question did.
- Switch to the concave demonstration. All three designs are non-dominated, and the balanced middle one wins at none of the 41 weightings.
What this shows: Everything on the front is a legitimate answer — and a weighted sum cannot reach a design that sits in a concave dip.