Sizing Optimisation

What Sizing Optimisation Changes

Sizing optimisation is the most constrained level of structural optimisation: the geometry and topology are fixed, and only the dimensions of the existing members are adjusted. It does not move boundaries or remove material from the design space; it changes how thick each wall is, how deep each beam is, or how large each tube diameter is. Because the changes are parametric and smooth, sizing optimisation is computationally inexpensive, numerically robust, and applicable at any stage of design — including detailed design where topology and shape are already frozen.

• Shell thickness: skin gauges, web thicknesses, wall thicknesses of thin-wall sections
• Plate thickness: flange and base-plate gauges
• Beam sections: cross-section depth, width, flange thickness, web thickness
• Tube diameter and wall thickness: round and square tubes
• Stiffener dimensions: stiffener height, flange width, thickness
• Sandwich core thickness and face sheet thickness
• Composite laminate thickness (via ply count) and individual ply thickness

Continuous vs Discrete Variables

Sizing variables can be continuous or discrete. A continuous thickness variable can take any value within its bounds — 2.13 mm, 2.47 mm, 3.01 mm. A discrete variable is restricted to a set of available values — standard plate gauges, stock tube sizes, or integer ply counts. Continuous optimisation is smooth and efficient; discrete optimisation is harder and may require gradient-free methods or rounding strategies. In practice, many sizing optimisations run with continuous variables and then round to the nearest standard gauge in a post-processing step — but this rounding can violate constraints and must be verified.

Rounding a continuous optimisation result to standard gauges can push a marginally satisfactory design over its constraint. Always re-run verification on the rounded, buildable design — not on the continuous optimum.

Standard Formulation

The standard sizing optimisation minimises mass subject to the full set of structural constraints. The design variables are the thicknesses (or sections) of each design region. The objective is total mass; the constraints ensure the design satisfies all failure modes.

    minimise    M(t)  =  Σ ρ_e · A_e(t_e) · L_e
    subject to  σ_e(t)   ≤  σ_allow          (stress)
                δ(t)     ≤  δ_max            (deflection)
                λ_1(t)   ≥  λ_req            (buckling factor)
                f_1(t)   ≥  f_req            (first frequency)
                t_min    ≤  t_e  ≤  t_max     (manufacturing bounds)

    where  t_e  = thickness (or section) of element e
           ρ_e  = density,  A_e = area,  L_e = length
           λ_1  = first buckling eigenvalue
           f_1  = first natural frequency

Sensitivities and Gradient Methods

Sizing optimisation is well suited to gradient-based methods. The sensitivities of mass and stress with respect to thickness are smooth and inexpensive to compute — mass is directly proportional to thickness, and stress sensitivities follow from the element stiffness derivative. Buckling and frequency sensitivities are more involved but still tractable through the adjoint or direct differentiation methods. Because the variable count in a sizing problem is moderate (tens to hundreds of design regions, not thousands of elements), gradient-based optimisers converge reliably and quickly.

Sizing optimisation is the natural home of gradient-based methods. The problem is smooth, the sensitivities are cheap, and the variable count is manageable. Reserve gradient-free methods for genuinely discrete or discontinuous problems.

Design Constraints

Sizing optimisation is not just about mass and stress. Real designs must satisfy a bundle of constraints, many of which come from manufacturing rather than analysis.

• Minimum gauges: the thinnest material that can be procured, handled or manufactured
• Standard stock sizes: plate, sheet and tube are available in discrete gauges — not arbitrary thicknesses
• Manufacturing increments: casting, machining and additive processes have minimum step sizes
• Symmetry requirements: symmetric structures must have symmetric thickness distributions
• Buckling constraints: thin walls buckle before yielding — sizing must maintain stability
• Frequency constraints: avoid resonance with known excitation frequencies
• Damage tolerance: sufficient thickness to tolerate assumed damage for the inspection interval

Local vs Global Sizing

Sizing optimisation can be local or global. Local sizing allows every element to have its own thickness, maximising design freedom but producing a result with hundreds of distinct gauges. Global sizing (or zoning) groups elements into design regions that share a single thickness, reducing the number of distinct gauges at the cost of some optimality. In practice, a purely local optimisation is a starting point — the result must be zoned and rationalised before it becomes a buildable design.

An unconstrained sizing optimisation that gives every element its own thickness is mathematically optimal and practically useless. The engineering work begins after the optimiser stops.

Why Unconstrained Optima Are Impractical

A sizing optimisation with no manufacturing constraints will produce a thickness distribution that is theoretically optimal but unbuildable. Adjacent elements may differ by 0.01 mm — a distinction no factory can maintain. A skin may taper continuously from 4 mm to 0.8 mm, requiring a material that does not exist. A beam may need a section that is not a standard size. The raw optimisation result is a mathematical optimum, not an engineering design. The gap between the two is bridged by zoning and rationalisation.

robust-design

ENGINEERING THE RESULT

Zoning and rationalisation turn a raw sizing result into a practical set of gauges. This is not post-processing; it is engineering. The analyst examines the optimised thickness distribution, identifies regions of similar thickness, groups them into zones with a single gauge each, and selects standard sizes that satisfy all constraints. The zoned design is then re-analysed to confirm that the rationalisation has not violated any margin. The process is iterative: zone, check, adjust, re-check — until the design is both buildable and satisfactory.

1. Run the sizing optimisation with continuous variables and manufacturing bounds
2. Examine the optimised thickness distribution — identify natural groupings and transitions
3. Define zones: groups of elements sharing a single gauge, with boundaries at structural transitions
4. Select standard gauges for each zone — the nearest available size that satisfies all constraints
5. Re-run the full verification on the zoned, standardised design — confirm no constraint is violated
6. Iterate: adjust zone boundaries or gauges where margins are negative or excessively positive
7. Document the final gauge map and the justification for each zone

The number of distinct gauges in the final design is a design decision, not an optimisation output. A structure with six gauges is easier to procure, inspect and certify than one with twenty — and the mass penalty of rationalisation is usually small.

Zoning in Practice

A good zoning strategy respects both the structural requirements and the manufacturing reality. Zones should follow structural features — panels between stiffeners, regions between frames, areas of similar load — rather than arbitrary element boundaries. Transitions between zones should occur where stress gradients are low, so the thickness step does not create a stress concentration. Ply drops in composite zones should follow the same logic: drop plies where stresses are low, not where they are high.

• Zone boundaries at structural transitions: frames, stiffeners, joints — not mid-panel
• Thickness transitions where stress gradients are low — avoid steps at stress concentrations
• Minimum zone size: a zone too small to justify a distinct gauge should be merged with its neighbour
• Composite ply drops: drop plies in low-stress regions, taper gradually, avoid stacking drops

Verification of the Rationalised Design

The final step in any sizing optimisation is to verify the rationalised, zoned, standard-gauge design. The continuous optimum is not the design that will be built; the zoned design is. It must be re-analysed with the actual gauges, the actual material properties and the full set of load cases. Margins that were positive in the continuous optimum may be negative in the zoned design, particularly at zone boundaries where the thickness steps down. The verification closes the loop and confirms that the engineering, not just the mathematics, is sound.

Report margins on the buildable, zoned design — not on the continuous optimum. A margin that exists only in the mathematical solution is not a margin at all.

Key takeaways

• Sizing optimisation adjusts thickness and cross-section while keeping the geometry and topology fixed — it is the lowest-freedom, lowest-cost optimisation level.
• The standard formulation minimises mass subject to stress, deflection, buckling and frequency constraints, with thickness or section as design variables.
• Unconstrained or loosely constrained sizing produces impractical thickness changes — dozens of distinct gauges that cannot be procured or built.
• Zoning and rationalisation turn a raw sizing result into a practical set of gauges — this is engineering, not post-processing.
• Gradient-based methods are efficient for sizing because sensitivities are smooth and the variable count is moderate.