**Multi-resolution flow simulations by smoothed particle hydrodynamics via
domain decomposition**

X Bian and Z Li and GE Karniadakis, JOURNAL OF COMPUTATIONAL PHYSICS, 297, 132-155 (2015).

DOI: 10.1016/j.jcp.2015.04.044

We present a methodology to concurrently couple particle-based methods via a domain decomposition (DD) technique for simulating viscous flows. In particular, we select two resolutions of the smoothed particle hydrodynamics (SPH) method as demonstration. Within the DD framework, a simulation domain is decomposed into two (or more) overlapping sub- domains, each of which has an individual particle scale determined by the local flow physics. Consistency of the two sub-domains is achieved in the overlap region by matching the two independent simulations based on Lagrangian interpolation of state variables and fluxes. The domain decomposition based SPH method (DD-SPH) employs different spatial and temporal resolutions, and hence, each sub-domain has its own smoothing length and time step. As a consequence, particle refinement and derefinement are performed asynchronously according to individual time advancement of each sub-domain. The proposed strategy avoids SPH force interactions between different resolutions on purpose, so that coupling, in principle, can go beyond SPH-SPH, and may allow SPH to be coupled with other mesoscopic or microscopic particle methods. The DD-SPH method is validated first for a transient Couette flow, where simulation results based on proper coupling of spatial-temporal scales agree well with analytical solutions. In particular, we find that the size of the overlap region should be at least r(c,1) + 2r(c,2), where r(c,1) and r(c,2) are cut off radii in the two sub-domains with r(c,1) = r(c,2). Subsequently, a perturbation wave is considered traveling either parallel or perpendicular to the hybrid interface. Compressibility is significant if transient behavior at short sonic-time-scale is relevant, while the fluid can be treated as quasi-incompressible at sufficiently long time scale. To this end, we propose a coupling of density fields from the two sub-domains. Finally, a steady Wannier flow is simulated, where a rotating cylinder is placed next to a wall. Lubrication effects are prominent in the gap between the cylinder and the bottom wall, rendering a high resolution necessary, whereas in the rest of the domain the flow can be simulated at much lower resolution. DD-SPH simulation results with both spatial and temporal resolution ratios up to 16 agree well with the results of a single high resolution simulation, but with the former two-orders of magnitude faster in the region away from the cylinder. (C) 2015 Elsevier Inc. All rights reserved.

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