Accurate and efficient multiscale simulation of a heterogeneous elastic beam via computation on small sparse patches
Authors: Roberts Anthony John , Tran-Duc Thien , Bunder Judith E. , Kevrekidis Yannis 
Published: April 2024

Modern `smart' materials have complex microscale structure, often with unknown macroscale closure. The Equation-Free Patch Scheme empowers us to non-intrusively, efficiently, and accurately simulate over large scales through computations on only small well-separated patches of the microscale system. Here the microscale system is a solid beam of random heterogeneous elasticity. The continuing challenge is to compute the given physics on just the microscale patches, and couple the patches across un-simulated macroscale space, in order to establish efficiency, accuracy, consistency, and stability on the macroscale. Dynamical systems theory supports the scheme. This research program is to develop a systematic non-intrusive approach, both computationally and analytically proven, to model and compute accurately macroscale system levels of general complex physical and engineering systems.
References

R. A. Biezemans, C. Le Bris, F. Legoll, and A. Lozinski. Non-intrusive implementation of a wide variety of Multiscale Finite Element Methods. Comptes Rendus. Mécanique 351 (2023), pp. 1–46. doi: 10.5802/crmeca.178
M. P. Brenner and P. Koumoutsakos. Editorial: Machine learning and Physical Review Fluids: An editorial perspective. Phys. Rev. Fluids 6.7 (2021), p. 070001. doi: 10.1103/PhysRevFluids.6.070001
J. E. Bunder, I. G. Kevrekidis, and A. J. Roberts. Equation-free patch scheme for efficient computational homogenisation via self-adjoint coupling. Numer. Math. 149.2 (2021), pp. 229–272. doi: 10.1007/s00211-021-01232-5
J. E. Bunder, A. J. Roberts, and I. G. Kevrekidis. Good coupling for the multiscale patch scheme on systems with microscale heterogeneity. J. Comput. Phys. 337 (2017), pp. 154–174. doi: 10.1016/j.jcp.2017.02.004 References C175
M. Cao and A. J. Roberts. Multiscale modelling couples patches of nonlinear wave-like simulations. IMA J. Appl. Math. 81.2 (2016), pp. 228–254. doi: 10.1093/imamat/hxv034
J. Divahar, A. J. Roberts, T. W. Mattner, J. E. Bunder, and I. G. Kevrekidis. Two novel families of multiscale staggered patch schemes efficiently simulate large-scale, weakly damped, linear waves. Comput. Meth. Appl. Mech. Eng. 413 (2023), p. 116133. doi: 10.1016/j.cma.2023.116133. (Cit. on pp. C163, C165, C172).
S. Lucarini, M. V. Upadhyay, and J. Segurado. FFT based approaches in micromechanics: fundamentals, methods and applications. Model. Sim. Mat. Sci. Eng. 30.2 (2021), p. 023002. doi: 10.1088/1361-651X/ac34e1
J. Maclean, J. E. Bunder, and A. J. Roberts. A toolbox of Equation-Free functions in Matlab/Octave for efficient system level simulation. Numer. Alg. 87 (2021), pp. 1729–1748. doi: 10.1007/s11075-020-01027-z
J. Maclean, J. E. Bunder, I. G. Kevrekidis, and A. J. Roberts. An equation free algorithm accurately simulates macroscale shocks arising from heterogeneous microscale systems. IEEE J. Multiscale Multiphys. Comput. Tech. 6 (2021), pp. 8–15. doi: 10.1109/JMMCT.2021.3054012
A. J. Majda and I. Grooms. New perspectives on superparameterization for geophysical turbulence. J. Comput. Phys. Frontiers in Computational Physics 271 (2014), pp. 60–77. doi: 10.1016/j.jcp.2013.09.014
K. Matouš, M. G. D. Geers, V. G. Kouznetsova, and A. Gillman. A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials. J. Comput. Phys. 330 (2017), pp. 192–220. doi: 10.1016/j.jcp.2016.10.070
K. Raju, T.-E. Tay, and V. B. C. Tan. A review of the FE2 method for composites. Multiscale Multidisc. Model. Exp. Design 4 (2021), pp. 1–24. doi: 10.1007/s41939-020-00087-x
A. J. Roberts. Macroscale, slowly varying, models emerge from the microscale dynamics in long thin domains. IMA J. Appl. Math. 80.5 (2015), pp. 1492–1518. doi: 10.1093/imamat/hxv004
A. J. Roberts and I. G. Kevrekidis. General tooth boundary conditions for equation free modelling. SIAM J. Sci. Comput. 29.4 (2007), pp. 1495–1510. doi: 10.1137/060654554
A. J. Roberts, T. MacKenzie, and J. Bunder. A dynamical systems approach to simulating macroscale spatial dynamics in multiple dimensions. J. Eng. Math. 86.1 (2014), pp. 175–207. doi: 10.1007/s10665-013-9653-6
A. J. Roberts, J. Maclean, and J. E. Bunder. Equation-Free function toolbox for Matlab/Octave. Tech. rep. https://github.com/uoa1184615/EquationFreeGit, 2019–2024
G. Samaey, A. J. Roberts, and I. G. Kevrekidis. Equation-free computation: An overview of patch dynamics. Multiscale methods: bridging the scales in science and engineering. Ed. by J. Fish. Oxford University Press, 2010. Chap. 8, pp. 216–246. doi: 10.1093/acprof:oso/9780199233854.003.0008
J. Somnic and B. W. Jo. Status and challenges in homogenization methods for lattice materials. Materials 15.2 (2022), p. 605. doi: 10.3390/ma15020605
H. Whitney. Differentiable manifolds. Annal. Math. 37.3 (1936), pp. 645–680. doi: 10.2307/1968482