A Particle Method with Second Order Approximation in Space-Time

IDELSOHN, SERGIO R.; NIGRO, NORBERTO M.; GIMENEZ, JUAN M.; AGUERRE, HORACIO J.

New York

Congreso; 13th World Congress on Computational Mechanics / 2nd Pan American Congress on Computational Mechanics; 2018

Lagrangian-based methods using particles are established methodologies which rely on the goodness of particles for solving the convective term of transport equations. Despite having shown impressive results for some particular applications and presenting promissory potential for parallel architectures, the efficiency of particle-based methods solving the incompressible Navier-Stokes equations currently does not justify its massive use in replace of classical Eulerian methodologies. Strategies such as PFEM-2 [1] have decreased the huge computational cost of particle methods, but mainly because of the low order schemes employed. This work presents the first proposal of a second order accurate particle method in both time and space for solving incompressible flows equations within a Lagrangian formulation. This methodology consists on reinterpreting the convective term of any transport equation as a source term estimated with particles. An updated version of the eXplicit Integration Following the Acceleration and Streamlines (X-IVAS) method [2] is used and a second order projection scheme is employed to transfer data from the particles to the mesh. In the case of incompressible flows, the modified momentum equation is used as velocity predictor without modifying the original rate of convergence of the Eulerian solution but improving the numerical approximation of the convective term.With the aim of presenting the methodology and particle-mesh interaction, firstly the solution of scalar transport equations is shown. Then, incompressible flow problems are solved where the rate of convergence of the method is assessed experimentally and the need of iterating the X-IVAS during velocity-pressure coupling is revealed. The implementation on the open source platform OpenFOAM allows employing general polyhedral meshes and obtaining  reliable computing times comparisons. Results reveals that, for the analyzed cases, the current method is able to obtain the same or lower level of error than a fast Eulerian alternative, but saving considerably total computing time.[1] ?Lagrangian versus Eulerian integration errors?; Sergio Idelsohn, Eugenio Oñate,  Norberto Nigro, Pablo Becker, Juan Gimenez; Comput. Methods Appl. Mech. Engrg. 293, 15,191?206.[2] ?Analysis of multi-fluid flows with large time steps using the Particle Finite Element Method?; Sergio Idelsohn, Julio Marti, Pablo Becker, Eugenio Oñate. Int. Journal for Num. Methods in Fluids, 75, 621-644.