## Final Activity Report Summary - NanoSim (Towards a general approach to multiscale simulation and its application to nanotechnology)

Many science fields demand the simulation of physical phenomena with an increasing level of complexity. This requires elaborated numerical techniques and efficient implementation that will allow solving these complex problems in an accurate way. In a multi-scale problem, the unknown variables have a very wide range of scales, from micro-scales to macro-scales, and are strongly coupled. This means that we cannot capture the macro-scale component of the unknowns without a correct approximation of the effect of micro-scales over the macro-scales. Let us put two examples which are illustrative of this situation.

The Navier-Stokes partial differential equations govern the movement of fluids. For high enough velocities, the fluid dynamics become chaotic, i.e. turbulent. There is a wide range of scales interacting in this phenomenon, from the micro-vortexes that cannot be seen by the human eye to the big vortexes that we can easily observe. The numerical approximation of partial differential equations is attained as follows; the continuum problem is discretised, meaning that we look for a solution (velocities and pressures in this case) in a finite dimensional space of functions, instead of the infinite dimensional one to which these unknowns belong. This discretisation is based on a mesh or grid. Roughly speaking, instead of solving the problem at the infinite points of the fluid, we restrict ourselves to evaluate the unknowns in a set of vertices or nodes of the mesh. By doing that, it becomes obvious that we can only solve for those scales that can be captured by the mesh. All the sub-scales that cannot be captured, because the mesh is too coarse to represent them, are lost during this discretisation process. The loss of sub-scales means that the effect of the sub-scale over the large scales is lost and the numerical approximation of the large scales becomes meaningless. Thus, this is a multi-scale problem and its complication is clear. A straightforward approximation of the large scales is inaccurate and the approximation of all scales is unaffordable, since it requires an extremely fine mesh and becomes too expensive to be solved. The only affordable way to solve the abovementioned problem is by modelling the sub-scales effect over the large scales. This is the origin of turbulence models in computational physics and, more recently, variational multiscale methods in the finite element community.

Nanotechnology aims to create devices of such a small size that their structural analysis can be out of the range of applicability of the classical continuum theory for solid mechanics. For example, fracture and dislocation cannot be represented by the continuum theory but are basic in these analyses. The simulation of these problems requires the coupling of atomistic models, which are very accurate but extremely expensive, in singular regions, where the continuum theory is not valid, and continuum models in the rest of the domain. This is the only way to capture the micro-scales that are important at the macroscopic level while maintaining affordable computational cost. The reason for which the micro-scale behaviour is needed in some regions is very natural. A crack in a device originates from an atomistic level as well as its propagation in the tip; however the size of the crack can become ‘macro’ at some point. This is another multi-scale problem of key interest nowadays.

In the frame of this project, we developed and analyzed numerical algorithms for the approximation of the type of problems commented above. In a first stage, AtC coupling was tackled. We presented a set of efficient methods, based on interface coupling of atomistic and continuum models. These methods were applied to some nano-scale problems, like nano-indentation. On the other hand, we actively worked on the development of the variational multi-scale method for CtC coupling. We proved new theoretical results for these methods, especially for the Navier-Stokes equations. Furthermore, we used these techniques for different problems in cardiovascular engineering. We were recently applying these techniques to magnetohydrodynamics in fusion reaction technology and liquid crystal modelling for liquid crystal display (LCD) devices.

The Navier-Stokes partial differential equations govern the movement of fluids. For high enough velocities, the fluid dynamics become chaotic, i.e. turbulent. There is a wide range of scales interacting in this phenomenon, from the micro-vortexes that cannot be seen by the human eye to the big vortexes that we can easily observe. The numerical approximation of partial differential equations is attained as follows; the continuum problem is discretised, meaning that we look for a solution (velocities and pressures in this case) in a finite dimensional space of functions, instead of the infinite dimensional one to which these unknowns belong. This discretisation is based on a mesh or grid. Roughly speaking, instead of solving the problem at the infinite points of the fluid, we restrict ourselves to evaluate the unknowns in a set of vertices or nodes of the mesh. By doing that, it becomes obvious that we can only solve for those scales that can be captured by the mesh. All the sub-scales that cannot be captured, because the mesh is too coarse to represent them, are lost during this discretisation process. The loss of sub-scales means that the effect of the sub-scale over the large scales is lost and the numerical approximation of the large scales becomes meaningless. Thus, this is a multi-scale problem and its complication is clear. A straightforward approximation of the large scales is inaccurate and the approximation of all scales is unaffordable, since it requires an extremely fine mesh and becomes too expensive to be solved. The only affordable way to solve the abovementioned problem is by modelling the sub-scales effect over the large scales. This is the origin of turbulence models in computational physics and, more recently, variational multiscale methods in the finite element community.

Nanotechnology aims to create devices of such a small size that their structural analysis can be out of the range of applicability of the classical continuum theory for solid mechanics. For example, fracture and dislocation cannot be represented by the continuum theory but are basic in these analyses. The simulation of these problems requires the coupling of atomistic models, which are very accurate but extremely expensive, in singular regions, where the continuum theory is not valid, and continuum models in the rest of the domain. This is the only way to capture the micro-scales that are important at the macroscopic level while maintaining affordable computational cost. The reason for which the micro-scale behaviour is needed in some regions is very natural. A crack in a device originates from an atomistic level as well as its propagation in the tip; however the size of the crack can become ‘macro’ at some point. This is another multi-scale problem of key interest nowadays.

In the frame of this project, we developed and analyzed numerical algorithms for the approximation of the type of problems commented above. In a first stage, AtC coupling was tackled. We presented a set of efficient methods, based on interface coupling of atomistic and continuum models. These methods were applied to some nano-scale problems, like nano-indentation. On the other hand, we actively worked on the development of the variational multi-scale method for CtC coupling. We proved new theoretical results for these methods, especially for the Navier-Stokes equations. Furthermore, we used these techniques for different problems in cardiovascular engineering. We were recently applying these techniques to magnetohydrodynamics in fusion reaction technology and liquid crystal modelling for liquid crystal display (LCD) devices.