Tran-Duc, T.Bunder, J.E.Roberts, A.J.2025-07-222025-07-222025Computational Mechanics, 2025; 1-220178-76751432-0924https://hdl.handle.net/2440/146210OnlinePublWe develop a multiscale patch scheme for studying the system level characteristics of heterogeneous functional graded beams in 3D via accurate computational homogenisation. The algorithm is an extension of our previous work for 2D beams (Tran-Duc et al. in Int. J. Solids Struct. 292:112719, 2024) to explore out-of-plane dynamics of 3D beams of functional graded materials. The scheme computes the detailed microscale elastic equations only in sparsely spaced, small patches of the domain (akin to FE²), and via symmetry-preserving interpolation between these patches. We develop new applications of the scheme to two classes of functionally graded beams, namely cross-sectionally graded and axially graded. Our approach accurately and provably predicts the macroscale system-wide behaviour. Beam deflection and natural frequencies from the patch computations agree very well with both existing experimental data and the full-domain computations, which provides a new validation of the approach and a new characterisation of the interaction between bending and twisting in graduated beams. The scheme is stable and robust, with errors consistently small and controllable by varying 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. The scheme also accurately predicts the homogenised dynamics of periodic micro-structured materials, such as metamaterials, by simply ensuring patches are a multiple of the micro-period. Localised phenomena, such as material failures or cracks or boundary layers, may also be accurately encompassed by fully resolving them within a patch.en© The Author(s) 2025. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.multiscale patch algorithm; elasticity; functionally graded materials; beam; numerical methodsJournal article10.1007/s00466-025-02614-4733043Bunder, J.E. [0000-0001-5355-2288]Roberts, A.J. [0000-0001-8930-1552]