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The concept of multiscale modelling has emerged over the last few decades to describe procedures that seek to simulate continuum-scale behaviour using information gleaned from computational models of finer scales in the system, rather than resorting to empirical constitutive models. A large number of such methods have been developed, taking a range of approaches to bridging across multiple length and time scales. Here we introduce some of the key concepts of multiscale modelling and present a sampling of methods from across several categories of models, including techniques developed in recent years that integrate new fields such as machine learning and material design.

Your institute does not have access to this article. Nature Communications Open Access 07 July Nature Communications Open Access 28 June Curtin, W. Fish, J. Bridging the scales in nano engineering and science. Nanoparticle Res. Article Google Scholar. Practical Multiscaling Wiley, Ghosh, S. Ericksen, J. On the Cauchy—Born rule. Solids 13 , — Arroyo, M. Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy—Born rule. B 69 , Friesecke, G.

Validity and failure of the Cauchy—Born hypothesis in a two-dimensional mass—spring lattice. Nonlinear Sci. Voigt, W. Leipzig , — Reuss, A. Account of the liquid limit of mixed crystals on the basis of the plasticity condition for single crystal.

Dvorak, G. On transformation strains and uniform-fields in multiphase elastic media. A , — CAS Google Scholar. Oskay, C.

Eigendeformation-based reduced order homogenization for failure analysis of heterogeneous materials. Methods Appl. Yuan, Z. Multiple scale eigendeformation-based reduced order homogenization. Bakhvalov, N. Hill, R. Elastic properties of reinforced solids: some theoretical principles. Solids 11 , — Tolman, R. The Principles of Statistical Mechanics Clarendon, Dirac, P. The Principles of Quantum Mechanics 4th edn Clarendon, Sanchez-Palencia, E. Suquet, P. Elkhodary, K. Archetype-blending continuum theory.

Duarte, C. Generalized finite element methods for three-dimensional structural mechanics problems. Hou, T. A multiscale finite element method for elliptic problems in composite materials and porous media. Weinan, E. Heterogeneous multiscale method: a general methodology for multiscale modeling.

B 67 , Hughes, T. Multiscale enrichment based on partition of unity. Methods Eng. Chen, W. A generalized space—time mathematical homogenization theory for bridging atomistic and continuum scales. Generalized mathematical homogenization of atomistic media at finite temperatures in three dimensions.

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Oden, J. Multi-scale modeling of physical phenomena: adaptive control of models. Jones, R. Mallat, S. A theory for multiresolution signal decomposition—the wavelet representation. IEEE Trans. Pattern Anal. Beylkin, G. A multiresolution strategy for reduction of elliptic PDEs and eigenvalue problems.

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