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## Marrying continuum and molecular physics: the Andersen-Parrinello-Rahman method revised into a scale bridging device

#### CECAM-HQ-EPFL, Lausanne, Switzerland

#### Organisers

Multi-scale modelling has become a hot topic in computational physics, to the point of turning into a successful catchphrase. In terms of solid results, however, its success has not been overwhelming. It is regrettable, in particular, that concurrent multi-scale methods coupling molecular dynamics with continuum mechanics are even less developed and less well understood that methods linking Newtonian molecular dynamics with some viable approximation of the underlying quantum structure [1].

In partitioned-domain methods [2–4], the atomistic and continuum models are completely and independently defined in different space regions, partially overlapping. The coupling between molecular and continuum degrees of freedom takes place only within interface belts. Sophisticated smoothing and filtering techniques may painfully mitigate, but not eradicate, severe interface artefacts. Moreover, typical numerical experiments feature a finite element mesh size as small as 5–10 atomic spacings, i.e., of nanometre order.

In hierarchical multi-scale methods [5–7], the whole domain is covered by a macroscopic grid on which a continuum model is computed. The molecular model enters as a local refinement for obtaining the information purposely left out from the continuum formulation (typically, the stress- strain relation). In our opinion, this approach holds a far better promise than the previous one. Its current implementations, however, suffer from the feeble coupling between molecular and continuum degrees of freedom, operated by manipulating the periodic boundary conditions on each MD cell à la Lees-Edwards, and hence affecting directly only the relatively few molecules that cross the cell boundary.

In our opinion, a definite boost to hierarchical multi-scale methods would be given by the adoption of an Andersen-Parrinello-Rahman formulation of MD [8–10], revised with a view to concurrent molecular-continuum simulations [11,12]. This method is characterised by the decomposition of particle velocity into the sum of a spatially tidy entrainment velocity, parameterised by the cell deformation rate, and a disordered streaming velocity. Independently of the intentions of the originators, the introduction of collective degrees of freedom affecting simultaneously all the particles within one computational cell lends itself naturally to a seamless coupling with continuum mechanics, hinging on the idea of identifying each MD cell with an infinitesimally small piece of a continuous medium.

The fields entering the continuum description are adequately sampled on an array of positions (think of Gauss points in a finite element model), whose typical spacing H is enormously larger than the average intermolecular distance d. The reference size h of the MD cell associated with each of these macroscopic sampling positions is large enough with respect to d in order to allow for a decent molecular sampling, and still much smaller than H. With this approach (where full advantage may be taken of the fact that, typically, H/h >> h/d ) the molecules in each cell interact directly with each other, while being indirectly affected by those in the other cells via their collective degrees of freedom, governed by the force balance and compatibility equations of the continuum field theory. In turn, the stress-strain relation characterising the response of the medium arises as an emergent property of molecular dynamics.

With this workshop, we aim to attenuate the two interconnected obstacles that hinder substantial progress in the application of multi-scale modelling to engineering problems, involving the close-coupling of molecular simulations with simulations at truly macroscopic scales (this is one of the topics of interest in CECAM’s 2015 call for workshop proposals):

1) Dearth of direct interaction with specialists in continuum physics, traditionally foreign to the CECAM community;

2) Lack of an agreed upon conceptual framework, encompassing both molecular and continuum physics, on which sound and effective scale-bridging techniques may flourish.

To fulfil this goal, we plan to convene a small, well-assorted group of computationally-oriented scientists, eager to combine complementary expertises in molecular and continuum physics, and let them discuss in depth what we deem the most promising avenue to further progress in coupled atomistic-continuum modelling.

We will investigate in detail if, as we believe, the Andersen-Parrinello-Rahman method, revised and reconsidered in a multi-scale perspective, may provide a major breakthrough in this direction. This method, despite having been introduced 35 years ago in an ad hoc way, may today be justified from first (Newtonian) principles. Accordingly, it should be regarded more as a subtle upscaling strategy than as a mere computational trick. Properly reinterpreted, it provides the best and most general identification of the fundamental constructs of continuum mechanics in terms of molecular quantities. We propose therefore to

1) Present and discuss in detail the APR-based hierarchical multi-scale method;

2) Consolidate the results of the discussion by (i) identifying the critical issues that need priority attention in order to develop and implement the method; (ii) defining a first set of test problems and benchmarks (to be augmented after the workshop); (iii) organising a web page (if appropriate within the ‘Multi-Scale’ section of the E-CAM website) to collect this material and to host other theoretical and computational information.

Correspondingly, the format of this meeting will have to be very different from that of a traditional workshop. First of all, its size will be smaller (a dozen participants, organisers included). Secondly, only two speakers (one of the organisers and one participant) will introduce the method and highlight its features in five half-day sessions out of six, distributed in four days (see Program). These presentations will be done in a lecture format, leaving plenty of space for in-depth discussions, intended to be the key element of each session. As such, this is not a school, but an opportunity for experts in the field to assess the viability of our work hypothesis and identify directions for its implementation. If funding allows, however, we would like to invite also one or two junior researchers with experience in the field. The last session of the workshop will focus the previous work by:

1) Reviewing the critical issues diagnosed in the previous five sessions and identifying the work groups who will concentrate on each of them within collaborations initiated at the workshop;

2) Structuring the web page dedicated to the APR-based hierarchical multi-scale method and its applications.

## References

**Italy**

Giovanni Ciccotti ( University "La Sapienza" ) - Organiser

Antonio DiCarlo ( Università Roma Tre ) - Organiser & speaker