Accurate and efficient multiscale simulation of a heterogeneous elastic beam via computation on small sparse patches




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.


  • 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/ 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/
  • 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/
  • 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., 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

Author Biography

Anthony John Roberts, University of Adelaide, South Australia

Four books and a toolbox:

* (2020) Linear Algebra for the 21st Century
Oxford University Press. isbn: 978-0-19-885640-5, 978-0-19-885639-9

* (2015) Model emergent dynamics in complex systems
SIAM, Philadelphia. isbn: 9781611973556.

* (2009) Elementary calculus of financial mathematics
SIAM, Philadelphia. isbn: 978-0-898716-67-2.

* (1994) A one-dimensional introduction to continuum mechanics
World Sci. isbn: 978-981-02-1913-0.

* (2020) with John Maclean, and J. E. Bunder; Equation-Free
function toolbox for Matlab/Octave.





Proceedings Computational Techniques and Applications Conference