  We develop a multiscale patch scheme for studying the system level
characteristics of heterogeneous functional graded beams. The algorithm
computes the detailed beam dynamics on the microscale, but only in small
patches of the beam domain, and then applies symmetry-preserving interpolation
to these patches to accurately predict the macroscale behaviour. To validate
the algorithm, two examples of functionally graded beams are investigated,
namely cross-sectionally graded and axially graded. Gradient patterns are
defined via volume fractions of aluminium and silicon carbine either over the
beam's cross section or along its axial direction. In these examples the
multiscale patch scheme only computes over a fraction of the beam's
full-domain. Beam deflection and natural frequencies from the patch
computations agree very well with existing experimental data and the
full-domain computations. The algorithm is stable and robust, with errors
consistently small and reliably reducible by increasing the number of patches.
The reduction in the spatial domain of computation substantially improves the
computational efficiency, with the computational time reducing by a factor of
up to 17 when the patches cover 27% of the beam.
