CONICET | Buscador de Institutos y Recursos Humanos

One of the main drawbacks of the explicit integration using Eulerian formulationsis the restricted stability of the solution with the time steps and with the spatial discretization.For the case of the Navier-Stokes equations, it is well known that the time step to be used inthe solution is stable only for time step smaller than two critical values: the Courant-Friedrichs-Lewy (CFL) number and the Fourier number. The first one is concerning with theconvective terms and the second one with the diffusive ones. Both numbers must be less thanone to have stable algorithms. For convection dominant problems like high Reynolds numberflows, the condition CFL<1 becomes crucial and limit the use of explicit method oroutdistance it to be efficient. On the other hand, implicit solutions using Eulerianformulations is restricted in the time step size due to the lack of convergence of the convectivenon-linear terms. Both time integrations, explicit or implicit are, in most cases, limited to CFLno much larger than one. The possibility to perform parallel processing and the recentupcoming of new processors like GPU and GPGPU increase the possibilities of the explicitintegration in time due to the facility to parallelize explicit methods having results with speed-up closed to one. Although the incompressible condition cannot be solved explicitly, thesolution of the momentum conservation equations with an explicit integration of theconvective terms together with a parallel processing reduces considerably the computing timeto solve the whole problem provided that a large time-step may be preserved independently tothe discretization in space. Only to remember the new Particle Finite Element Method, calledPFEM 2nd generation (PFEM-2) uses a Lagrangian formulation with an explicit timeintegrator without the CFL<1 restriction for the convective terms. This allows large time-steps, independent of the spatial discretization, having equal or better precision that animplicit integration. Moreover, PFEM-2 has two versions, one for moving mesh withpermanent remeshing and one for fixed mesh [1]. In this lecture we will present some recentadvances in the Particle Finite Element Method (PFEM) to solve the incompressible Navier-Stokes equations coupled with another fields like in multiphysics exploiting some nice featuresfound in the fixed version. On the other hand we will also present the moving mesh versionapplied to multifluids using a parallel remeshing that makes this efficient in terms of cpu time.This updated proposal will be tested numerically and compared in terms of accuracy as incomputing cpu time with other more standard Eulerian formulations.

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