Structural Optimisation

The Broader Discipline

Structural optimisation is the mathematical discipline of finding the best structural design according to a defined objective, subject to constraints. It is not a single method but a family of methods that differ in what they are allowed to change. Sizing optimisation adjusts thicknesses and cross-sections. Shape optimisation moves boundaries. Topology optimisation decides where material exists at all. Material and composite optimisation choose the material system and its architecture. Multidisciplinary optimisation couples structural performance with aerodynamics, thermal, manufacture and cost. The common thread is the mathematical formulation.

optimisation-hierarchy

Mathematical Formulation

Every structural optimisation problem can be written in the same general form. The objective function f(x) is what you want to minimise (mass, compliance, cost). The design variables x are the parameters you are allowed to change. The inequality constraints g_i(x) ≤ 0 and equality constraints h_j(x) = 0 represent the limits the design must satisfy — stress, displacement, buckling, frequency, manufacturability.

    minimise    f(x)
    subject to  g_i(x)  ≤  0      i = 1, ..., m      (inequality constraints)
                h_j(x)  ｰ  0      j = 1, ..., p      (equality constraints)
                x_L  ≤  x  ≤  x_U                    (design variable bounds)

    where  x   = vector of design variables
           f   = objective function (e.g. mass)
           g_i = inequality constraints (e.g. stress ≤ allowable)
           h_j = equality constraints (e.g. fixed volume)

Objective Functions, Constraints and Design Variables

The art of structural optimisation is in choosing the right objective, the right constraints, and the right variables. A mass objective with a stiffness constraint is the most common formulation, but others are equally valid: minimise deflection subject to a mass constraint, minimise cost subject to stress constraints, or maximise the first natural frequency subject to a mass constraint. The design variables must be ones you can actually change in the real design — a variable that cannot be manufactured is not a design variable, it is a wish.

• Objective: what are you minimising (or maximising)? Mass, compliance, cost, first frequency, buckling load
• Constraints: what must the design satisfy? Stress, deflection, buckling, frequency, manufacturing limits
• Design variables: what can you change? Thickness, section, shape, topology, material, lay-up
• Bounds: what are the limits on each variable? Minimum gauge, maximum stock size, standard increments

Sensitivities

Gradient-based optimisers need the sensitivity of the objective and each constraint to every design variable — the derivative df/dx and dg_i/dx. These sensitivities tell the optimiser which direction to move. For FEA-based optimisation, sensitivities are computed efficiently using the adjoint method, which gives all sensitivities in a single solve regardless of the number of design variables. Without sensitivities, the optimiser must use finite differences, which is prohibitively expensive for more than a handful of variables.

The adjoint method makes large-scale gradient-based optimisation affordable. It computes sensitivities for thousands of variables at the cost of one additional linear solve per load case.

Gradient-Based vs Gradient-Free Methods

The choice of optimisation algorithm depends on the smoothness of the problem and the availability of sensitivities.

Local vs Global Optima

Gradient-based methods find the nearest local optimum — the best design in the neighbourhood of the starting point. They do not guarantee finding the global optimum, the best design anywhere in the design space. For problems with a single dominant optimum (most well-posed structural problems), this is rarely a problem. For multi-modal problems with several competing designs, a global method or a multi-start gradient approach is needed. In practice, most structural optimisation uses gradient-based methods because they scale to thousands of variables, and the engineering problem is posed to be unimodal.

Computational Cost and Surrogate Models

Each optimisation iteration requires at least one FEA solve, and often several (for multiple load cases or sensitivity computation). For large models or nonlinear analysis, this cost can make a full optimisation impractical. Surrogate models — cheap mathematical approximations of the expensive analysis — reduce cost by fitting a response surface to a small number of FEA runs and optimising on the surrogate instead. Design of experiments (DoE) techniques choose the FEA runs to make the surrogate as accurate as possible with the fewest samples.

• Surrogate models: polynomial response surface, Kriging, radial basis functions — fit to DoE samples
• Design of experiments: Latin hypercube, full factorial, space-filling — choose samples efficiently
• Adaptive sampling: add FEA runs where the surrogate is uncertain to improve accuracy iteratively

Comparison of Optimisation Levels

The different levels of structural optimisation vary widely in design freedom, computational cost, manufacturing impact and the stage of design at which they are applied.

THE BEST TIME TO OPTIMISE

Design freedom is highest at the concept stage and falls sharply as a programme matures. A topology decision made at the concept stage can save tens of kilograms; the same decision attempted at the detailed stage may be impossible because tooling has been committed, interfaces frozen and certification started. The cheapest optimisation is the one performed earliest.

1. Concept stage: topology and material selection — maximum freedom, maximum impact, lowest cost of change
2. Preliminary design: shape and composite lay-up optimisation — still significant freedom, tooling not yet committed
3. Detailed design: sizing optimisation — limited to thickness and section changes, low cost but low impact
4. Certification stage: no optimisation — design is frozen, only verification and justification remain

design-freedom

By the detailed design stage, the only optimisation left is sizing. If you have not considered topology and load paths by then, you are optimising a poor concept — and no amount of sizing will fix that.

Multidisciplinary Optimisation

Real structures rarely serve a single discipline. An aircraft wing is a structural, aerodynamic, aeroelastic and manufacturing problem simultaneously. Multidisciplinary optimisation (MDO) couples the analyses of several disciplines and optimises across all of them. The challenge is that each discipline has its own solver, its own mesh and its own sensitivities, and coupling them is computationally and organisationally expensive. In practice, MDO is applied at the concept and preliminary stages where the coupling is strongest and the design freedom is greatest.

Key takeaways

• Structural optimisation is the umbrella discipline encompassing sizing, shape, topology, material, composite and multidisciplinary optimisation.
• All structural optimisation can be expressed as: minimise f(x) subject to constraints, where x are the design variables.
• The choice of optimisation level — sizing, shape or topology — determines how much design freedom you have and how late in the programme it can still be applied.
• Gradient-based methods suit smooth problems with many variables; gradient-free methods suit discontinuous or highly nonlinear problems.
• Design freedom falls sharply as a programme matures — the cheapest optimisation is the one performed early.