Shape Optimisation

What Shape Optimisation Changes

Shape optimisation modifies the boundaries of a structure without fundamentally changing its topology. The number of holes, the number of members, and the connectivity between them all stay the same; what changes is the geometry of the boundaries. Typical examples include fillet radius, hole location and shape, transition profile, bracket curvature, panel edge shape, aero-structural surface definition, and local reinforcement geometry. The structure is still the same structure — it just has smoother, better-placed curves.

Why Shape Matters: Stress Concentration

A geometric discontinuity forces load to flow around it, and the local stress rises above the nominal value. The ratio of that peak to the nominal stress is the stress concentration factor. A sharp transition, a small fillet, or an off-centre hole produces a high local stress even when the surrounding field is modest. The goal of shape optimisation is not to make every surface beautiful; it is to smooth the load path so the material is used more evenly.

Kt = sigma_max / sigma_nominal

where:
  Kt           = stress concentration factor (dimensionless)
  sigma_max    = peak local stress at the discontinuity
  sigma_nominal = nominal stress in the net section away from the feature

fillet-stress

Kt is a geometric property, not a material property. It depends on shape alone. Two identical geometries in different materials have the same Kt — but the consequences of that Kt differ: a brittle material will fracture at the peak, while a ductile material may yield locally and redistribute.

Two Approaches: Parameterised CAD and Mesh Morphing

There are two broad ways to drive a shape optimisation. In the parameterised CAD approach, design variables map to real geometric features — a fillet radius, a hole diameter, a spline control point. The optimiser moves these, the CAD regenerates, and the mesh is rebuilt. This gives a clean, manufacturable result but is limited to the parameters the engineer chose to expose. In the mesh morphing approach, the optimiser deforms the existing mesh directly, moving nodes on the surface and propagating the deformation into the interior. This allows much more general shape changes without re-meshing, but the resulting geometry must be reverse-engineered back into CAD, and mesh quality can degrade as deformation grows.

• Parameterised CAD: design variables are real dimensions — radius, length, angle, control-point coordinates. Clean output, limited freedom.
• Mesh morphing: design variables are node displacements on the boundary. Broad freedom, but mesh distortion limits step size and the final shape must be reconstructed.
• Hybrid approaches use free-form deformation boxes or radial basis functions to morph the mesh while keeping the underlying parameterisation tractable.

Sensitivities and Geometric Constraints

Shape optimisation lives or dies on the quality of the sensitivities — the derivative of the objective (typically compliance or peak stress) with respect to each shape variable. When sensitivities are accurate, gradient-based optimisers converge in a handful of iterations. When they are noisy — as they often are for peak-stress objectives, where the location of the maximum can jump between elements — the optimiser may chase noise rather than real improvement.

• Minimum fillet radii dictated by manufacturing — casting, machining, forging, or additive all impose different limits.
• Packaging constraints — the optimised shape must still fit within the surrounding envelope and not foul adjacent parts.
• Interface constraints — mating faces, bolt patterns, and seal surfaces usually cannot move and must be held fixed.
• Manufacturing constraints — draft angles for casting, tool access for machining, minimum wall thickness for additive.
• Mesh quality — large shape changes can distort elements to the point where stress results are no longer trustworthy.

Why Reducing a Local Peak Can Move the Problem Elsewhere

A shape optimisation that targets peak stress at one location will often lower it there — and raise it somewhere else. The total load has not changed; it has merely been redirected. Enlarging a fillet spreads the stress concentration over a longer transition, but if the transition ends in a thinner section, the stress in that thinner section rises. Moving a hole away from a high-stress region may reduce the concentration at the hole but increase bending in the ligament on the other side. This is why shape optimisation should be monitored globally, not just at the feature of interest.

Always track the global stress field during a shape optimisation, not just the objective location. The optimiser may be quietly pushing stress into a region you are not watching.

OPTIMISING A FILLET IS NOT THE SAME AS OPTIMISING A STRUCTURE

Shape optimisation operates on boundaries that already exist. It cannot add a rib where there is none, cannot remove a redundant member, and cannot relocate a joint. A well-optimised fillet on a poorly placed bracket is still a poorly placed bracket. Shape optimisation is a refinement tool, not a conceptual design tool. It is most powerful when the topology and layout have already been chosen well — through topology optimisation, engineering judgement, or experience — and the remaining gains are in the detail of the boundaries. Treating shape optimisation as a replacement for thinking about layout leads to locally excellent, globally mediocre structures.

Engineering judgement — what can change the conclusion

For Shape Optimisation, the harmonised review should concentrate on ensuring geometry changes remain represented by a stable parameterisation and do not exploit mesh artefacts, unmanufacturable features or load/boundary simplifications. The engineering value comes from identifying the assumptions that can move the governing margin or failure mode, then testing those assumptions deliberately rather than adding complexity indiscriminately. Where simplified and high-fidelity methods coexist, the simpler method should be used as an independent trend or magnitude check so that agreement is based on physics rather than shared modelling assumptions.

Key takeaways

• Shape optimisation changes boundaries, not topology — fillets, holes, and transitions improve, but the structure is fundamentally the same.
• Stress concentration factor Kt is purely geometric: Kt = sigma_max / sigma_nominal. The same shape produces the same Kt in any material.
• Parameterised CAD gives clean, manufacturable output but limited freedom; mesh morphing gives broad freedom but must be reverse-engineered back to CAD.
• A shape optimisation that lowers stress at one feature can raise it elsewhere — always monitor the global field, not just the point of interest.
• Manufacturing, packaging, interface, and mesh-quality constraints bind shape optimisation just as tightly as they bind every other method.
• Shape optimisation refines detail; it does not fix poor topology or poor layout. It is a finishing tool, not a conceptual design tool.