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

The paper explicitly claims that 5-point (order-4) inter-patch interpolation yields macroscale coupling errors of order O(H^4) when interpolating from patches I−2,…,I+2 to the edges of patch I, consistent with the candidate’s Lagrange-remainder derivation that shows edge field errors O(H^5) and hence O(H^4) at the macro level after division by H (for gradients) . The paper also invokes Whitney’s embedding idea to argue that the exchanged face data has dimension 216 for (ny,nz)=(6,6), and since 216>2·14>2·8 it almost surely parametrizes the slow manifold of dimension m in both basic (m=8) and micropolar (m=14) models; the candidate uses the same dimension count and the generic linear-projection corollary of Whitney, matching the paper’s logic . Finally, the paper’s numerical evidence that increasing the number of patches (smaller H) improves accuracy aligns with the candidate’s conclusion that the O(H^4) error decreases monotonically with refinement .