Topology optimisation asks how a fixed amount of material should be distributed within a given volume so that a structural measure is as good as possible. The field began with the homogenisation method of Bendsoe and Kikuchi (1988). The form used in most software is the density approach, in which each finite element carries a density variable between zero and one and intermediate values are penalised so that the solution tends toward solid and void (Bendsoe, 1989). The standard reference is the monograph of Bendsoe and Sigmund (2003), and the compact teaching code published by Sigmund (2001) made the algorithm widely understood.

Several numerical matters have to be handled for the result to mean anything. Without regularisation the solution depends on the mesh and shows checkerboard patterns; filtering of sensitivities or densities imposes a length scale and removes both (Sigmund and Petersson, 1998). The design variables are usually updated by an optimality criteria scheme or by the method of moving asymptotes (Svanberg, 1987). Stiffness problems are posed as minimum compliance under a volume constraint. Dynamic problems maximise a natural frequency, which is harder, because modes switch order during the iterations and low-density regions produce spurious local modes (Pedersen, 2000). A comparative account of the density approach and its alternatives is given by Sigmund and Maute (2013).

The raw output of an optimisation is seldom manufacturable. Restrictions that tie the result to a process, such as a minimum member size, symmetry planes, a constant section along an extrusion direction, or a draw direction for casting and for milling from one side, can be imposed during the optimisation, and their formulations are surveyed by Liu and Ma (2016).

The principles of stiff machine design are long established: short, direct force paths, closed sections for torsion, material placed far from the neutral axis, and attention to joints, which often contribute more compliance than the members they connect (Slocum, 1992). Structural behaviour is predicted today with finite element models coupled to models of drives and control, the approach described as the virtual machine tool (Altintas et al., 2005), and the dynamic compliance at the tool point is known to vary with axis position (Law et al., 2013). That last observation matters for a gantry beam, which is loaded at a different point for every position of the carriage.

The design space is the largest envelope the beam may occupy: the span between the side members, the height available below the top of the machine enclosure, and the depth that still leaves the carriage and spindle within the footprint. Regions that must remain solid are marked as non-design: the mounting faces and bolt bosses for the two linear guide rails, the seats of the ball screw or belt drive, and the end faces that bolt to the side members. Regions that must stay empty, such as cable routes and the travel envelope of the carriage, are excluded. A generous design space gives the optimiser room to find deep sections; the temptation to start from the existing beam should be resisted.

A beam optimised for one load is poor under any other. The load cases should therefore span the conditions of use, and Table 1 lists a minimal set. Cutting forces are applied at the tool tip and carried to the beam through a rigid or simplified carriage, so that the offset of the tool produces the correct bending and torsion. They are applied in each horizontal direction and vertically, with the carriage at mid-span and near one end. Inertia loads from acceleration of the beam axis and gravity acting on the carriage complete the set. Magnitudes come from the rated spindle power and the cutting parameters the machine is sold for, estimated with a standard cutting force model.

The simplest robust formulation minimises a weighted sum of the compliances of the static load cases subject to an upper limit on mass. The weights express how much each direction matters; equal weights on the two horizontal cutting directions and a smaller weight on gravity are a reasonable starting point. An alternative is to minimise mass subject to limits on tool point displacement in each case, which matches the way a specification is written but converges less readily. A lower bound on the first natural frequency, evaluated with the carriage mass attached, may be added as a constraint. Treating frequency as a constraint and not as the objective avoids most of the mode-switching difficulties described by Pedersen (2000).

The manufacturing route should be chosen before the run, since it determines the restrictions. For a beam milled from aluminium plate on a three-axis machine, a draw direction and a minimum member size no smaller than a practical rib thickness are appropriate. For an extruded profile, the density is constrained to be constant along the span, which reduces the problem to the design of a cross-section. For a sand casting in iron or aluminium, draw directions and a minimum wall are set by the foundry. Symmetry about the mid-span plane is normally imposed. The mesh should be fine enough for several elements across the smallest member, and the filter radius is set from the minimum member size. Running the optimisation at two or three mass limits shows how stiffness trades against weight and where additional material stops paying.

The density field is a suggestion, not a drawing. The designer reads it for its structural message: where the optimiser has formed a closed torsion box, where it has placed diagonal webs, how it has tied the rail seats to the end supports. This is then redrawn as clean geometry with constant wall thicknesses, fillets and tool access. Thin intermediate-density regions are either made solid or removed. For a small beam the interpretation often resolves into a recognisable form, such as a closed box with diagonal internal ribs and lightening pockets on the faces that carry little load, and this is acceptable: the value of the optimisation lies in showing which of the many possible ribbed boxes is the efficient one.

The rebuilt beam is analysed again, this time in the assembly. The linear guide carriages are represented by springs with the stiffness values given in the manufacturer's catalogue, the bolted end joints by contact or by spring elements, and the side members and base are included. The load cases of Table 1 are repeated and the tool point displacements compared with the targets and with the original beam analysed in the same model. A modal analysis gives the lowest natural frequencies and shows whether they are beam modes or rigid-body motion on the guides. Where the model reveals that joints dominate the compliance, further work on the beam is wasted, and the result should be reported as such.

A design procedure is complete only when its prediction has been checked. Two measurements are proposed for the first beam built. Static stiffness at the tool point is measured by loading the spindle nose against the table through a load cell and reading the relative displacement with an indicator, in the directions of the load cases. Dynamic behaviour is measured by impact testing, with an instrumented hammer and an accelerometer, to obtain the frequency response at the tool point and the lowest mode shapes (Ewins, 2000). Geometric accuracy of the assembled axis is checked by the methods of the machine tool test code (ISO, 2012). The same measurements on the original beam provide the baseline.