Numerical Comparison of the Particle Finite Element Method Against an Eulerian Formulation
The main goal of this paper is to validate experimentally the principal conclusions previously published in [17 ]. Two manufactured test cases were considered with their respective analytic solutions. First, a scalar transport equation is considered writt
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Abstract The main goal of this paper is to validate experimentally the principal conclusions previously published in [17]. Two manufactured test cases were considered with their respective analytic solutions. First, a scalar transport equation is considered written in such a way that several parameters are included to stress the limiting situation where the Eulerian and the Lagrangian approaches behave better. The results show conditions to be fulfilled in order to choose between both formulations, according to the problem parameters. A brief discussion about the projection needed for PFEM-2 method is included, specially due to its impact on the error convergence rate. Lately, an extension to Navier-Stokes equations is introduced using also a manufactured case to verify again the same conclusions. This paper intends to establish the first steps towards a mathematical error analysis for the particle finite element method which supports the preliminary theoretical and experimental results presented here.
J.M. Gimenez • N. Nigro Centro de Investigación de Métodos Computacionales, CIMEC-UNL/CONICET, Predio CONICET Santa Fe, Colectora Ruta Nac 168, Km 472, Paraje El Pozo, 3000 Santa Fe, Argentina e-mail: [email protected] http://www.cimec.org.ar Facultad de Ingeniería y Ciencias Hídricas - Universidad Nacional del Litoral, Ciudad Universitaria, Paraje “El Pozo”, Santa Fe, Argentina http://www.fich.unl.edu.ar P. Morin Instituto de Matemática del Litoral, IMAL-UNL/CONICET, Predio CONICET Santa Fe, Colectora Ruta Nac 168, Km 472, Paraje El Pozo, 3000 Santa Fe, Argentina http://www.imal.santafe-conicet.gov.ar S. Idelsohn () International Center for Numerical Methods in Engineering. CIMNE Edificio C1, Campus Norte UPC, C/ Gran Capitán S/N, 08034 Barcelona, Spain http://www.cimne.com ICREA, Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain e-mail: [email protected] © Springer International Publishing Switzerland 2016 Y. Bazilevs, K. Takizawa (eds.), Advances in Computational Fluid-Structure Interaction and Flow Simulation, Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-3-319-40827-9_2
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1 Introduction Over the last decades, computer simulation of incompressible fluid flow has been mainly based on the Eulerian formulation of the fluid mechanics equations on fixed domains [3]. During this period, hardware has evolved considerably increasing the speed performance of computations and allowing better facilities for data entry and the display of results. However in these decades there have been no substantial improvements on the numerical methods used concerning the efficiency of the algorithm. In most practical engineering problems, very fine mesh and very small time-steps are needed to reach acceptable results. This handicap exceeds most time the efficiency of current powerful computers. More recently, particle-based methods in which each particle is followed in a Lagrangian manner have been used for fluid flow problems. Monaghan [2
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