Efficient prediction of static and dynamical responses of functional graded beams using sparse multiscale patches

Abstract We 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 fe $$ ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mn>2</mml:mn> </mml:mmultiscripts> </mml:math> ), 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 $$17$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>17</mml:mn> </mml:mrow> </mml:math> when the patches cover $$27\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>27</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> 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.