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Construct accurate multi-continuum micromorphic homogenisations in multi-D space-time with computer algebra

A. J. Roberts

TL;DR

The paper presents a phase-shift embedding and invariant-manifold framework for multi-continuum micromorphic homogenisation that works at finite scale separation and does not rely on averaging. By coupling phase-shift ensembles with rigorous dynamical-systems theory, it constructs M-mode macroscale models (including bi- and tri-continuum forms) that faithfully encode microscale physics, with provable error control and applicability to nonlinear, non-autonomous, and multi-D problems. The approach is demonstrated in 1-D diffusion, 2-D elasticity, and high-contrast laminates, revealing nonlocal regularisation and higher-order gradient effects that extend the predictive reach beyond classical homogenisation. Computer algebra plays a central role, enabling high-order calculations and flexible parameter choices, including functionally graded materials, while providing systematic guidelines for mode selection and boundary-condition treatment. The methodology offers a unified, physically grounded route to design and analyze enriched continua with adjustable spatial-temporal resolution, directly tied to microscopic mechanisms and phase shifts.

Abstract

Homogenisation empowers the efficient macroscale system level prediction of physical scenarios with intricate microscale structures. Here we develop an innovative powerful, rigorous and flexible framework for asymptotic homogenisation of dynamics at the \emph{finite} scale separation of real physics, with proven results underpinned by modern dynamical systems theory. The novel systematic approach removes most of the usual assumptions, whether implicit or explicit, of other methodologies. By no longer assuming averages the methodology constructs so-called multi-continuum or micromorphic homogenisations systematically informed by the microscale physics. The developed framework and approach enables a user to straightforwardly choose and create such homogenisations with clear physical and theoretical support, and of highly controllable accuracy and fidelity.

Construct accurate multi-continuum micromorphic homogenisations in multi-D space-time with computer algebra

TL;DR

The paper presents a phase-shift embedding and invariant-manifold framework for multi-continuum micromorphic homogenisation that works at finite scale separation and does not rely on averaging. By coupling phase-shift ensembles with rigorous dynamical-systems theory, it constructs M-mode macroscale models (including bi- and tri-continuum forms) that faithfully encode microscale physics, with provable error control and applicability to nonlinear, non-autonomous, and multi-D problems. The approach is demonstrated in 1-D diffusion, 2-D elasticity, and high-contrast laminates, revealing nonlocal regularisation and higher-order gradient effects that extend the predictive reach beyond classical homogenisation. Computer algebra plays a central role, enabling high-order calculations and flexible parameter choices, including functionally graded materials, while providing systematic guidelines for mode selection and boundary-condition treatment. The methodology offers a unified, physically grounded route to design and analyze enriched continua with adjustable spatial-temporal resolution, directly tied to microscopic mechanisms and phase shifts.

Abstract

Homogenisation empowers the efficient macroscale system level prediction of physical scenarios with intricate microscale structures. Here we develop an innovative powerful, rigorous and flexible framework for asymptotic homogenisation of dynamics at the \emph{finite} scale separation of real physics, with proven results underpinned by modern dynamical systems theory. The novel systematic approach removes most of the usual assumptions, whether implicit or explicit, of other methodologies. By no longer assuming averages the methodology constructs so-called multi-continuum or micromorphic homogenisations systematically informed by the microscale physics. The developed framework and approach enables a user to straightforwardly choose and create such homogenisations with clear physical and theoretical support, and of highly controllable accuracy and fidelity.
Paper Structure (63 sections, 11 theorems, 83 equations, 7 figures)

This paper contains 63 sections, 11 theorems, 83 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Overview of the article's structure
  3. Introduce the novel methodology
  4. Strong theoretical support
  5. Quantifying errors and uncertainty
  6. Clarify and guide multi-continua choices
  7. Functionally graded materials
  8. Computer algebra handles complicated details
  9. Systematic multi-continuum homogenisation for 1-D spatial systems
  10. Phase-shift embedding
  11. Invariant manifolds of multi-modal any-order homogenisation
  12. Equilibria
  13. Spectrum at each equilibrium
  14. Multi-modal, multi-continuum, homogenisations exist
  15. Multi-modal, multi-continuum, homogenisations are relevant
  16. ...and 48 more sections

Key Result

lemma 1

For every solution $\fu(t,{\RaisedName{\x}{\mathcal{ X}}},\theta)\in\HH^N_{\mathbb D}$ of the embedding Eemdifpde, and for every phase $\phi$, the defined field $u_\phi(t,x):=\fu(t,x,x+\phi)$ satisfies the heterogeneous diffusion Eshdifpde. Specifically, for the case of phase $\phi=0$, the field

Figures (7)

  • Figure 1: 4% compression of a material with circular inclusions, seen near either end, may develop a nontrivial microscale structure, seen near the middle TGuo2024. Macroscale homogenisation of such microscale structures may best be via a multi-continuum model.
  • Figure 2: cylindrical domain of the embedding \ref{['Eemdifpde']} for field $\fu(t,{\RaisedName{\x}{\mathcal{ X}}},\theta)$: the background colour represents the microscale variation in $\kappa(\theta)$. Obtain solutions of the heterogeneous \ref{['Eshdifpde']} on the blue line as $u_\phi(t,x) := \fu(t,x,x+\phi)$ for any constant phase $\phi$.
  • Figure 3: Mercer--Roberts plot for the series in heterogeneity $a$ of the coefficient of $U_1$ in a high-order extension of \ref{['EE3Mevol1']} for $\partial_t^\alpha U_{1}$. The extrapolated intercepts to $1/n=0$ predict the location of convergence limiting singularities in the complex $a$-plane.
  • Figure 4: the sub-cell structure of the bi-continuum, im field \ref{['EhcegManifold2u']} in the high-contrast laminate problem. Specifically, the non-dimensional case of $\ell=2\pi$, $\kappa_1=1$, and a thin insulating layer $\eta/\ell=0.06$ located at $\theta=\pm\pi$ of insulation parameter $\chi=1$ so $\kappa_0=0.06$.
  • Figure 5: schematic domain of the multiscale embedding \ref{['Eempde']} for a field $\fu(t,{\RaisedName{\x}{\mathcal{ X}}},\thetav)$, for ${\RaisedName{\x}{\mathcal{ X}}}\in\XX\subset\RR$ and for $\thetav\in\Theta:=[0,1]^2$. Here the periodicities $\ell_1=1.62$ and $\ell_2=0.72$ so $\cE=1.620.72$. We obtain solutions of the heterogeneous \ref{['Egenpde']} on such blue lines as $u_\phiv(t,x):=\fu(t,x,\phiv+\cE^+x)$ for every constant phase $\phiv\in\RR^2$, here $\phiv=(0.82,0.32)$, and where the third argument of has components modulo $1$.
  • ...and 2 more figures

Theorems & Definitions (25)

  • definition 1: multi-continuum models
  • lemma 1
  • proof
  • lemma 2: converse
  • definition 2: invariant manifold (im)
  • corollary 1: first approximation
  • proof
  • lemma 3
  • proof
  • definition 3: asymptotic order
  • ...and 15 more