Multi-Scale Finite-Element Method (MS-FEM)

This projects investigates the applicability of the Multi-Scale Finite-Element Method (MS-FEM) of T. Hou to diffusion equations in perforated domains. One targetted application is pollutant dispersion in cities. Pollutant dispersion is extremely dependent of the geometry of the city but its full account leads to very time consuming simulations. The MS-FEM method is able to provide real-time responses to critical events, which is extremely useful in crisis management.

The MS-FEM method relies on the expansion of the solution on special basis functions which are pre-calculated by means of local simulations on a fine mesh and which represent a ‘model’ of the microstructure of the flow. By contrast to sub-grid modeling methodologies, the Multi-Scale basis functions are calculated from the actual geometry of the domain and do not depend on an (often arbitrary) analytical model of the microstructure. The extension of the MS-FEM to perforated domains relies on a penalization model inside the holes. A direct account of the geometry is possible but is costly and requires some adaptation of the MS-FEM methodology.

The picture below presents the solution of the diffusion equation in a geometry representative of an urban area. The top picture (1) is the reference solution, while the bottom left picture (a) represents the solution of the MS-FEM on a 15 X 8 mesh, while the right picture (b) represents the solution of the MS-FEM on a 60 X 32 mesh. The fine grid solution is undistinguishable from the reference solution but the coarse grid one is already quite satisfactory. The CPU speed-up is comprised between 4 and 6 orders of magnitude (excluding the time to construct the Multi-Scale Finite-Element basis).

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