INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids (2010) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.2327 Novel finite particle formulations based on projection methodologies D. Asprone 1 , F. Auricchio 2 and A. Reali 2, ∗, † 1 Department of Structural Engineering, University of Naples ‘Federico II’, Italy 2 Department of Structural Mechanics, University of Pavia, Italy SUMMARY Particle methods are among the most studied meshless approaches, with applications ranging from solid mechanics to fluid-dynamics and thermo-dynamics. The objective of the present paper is to analyze the behavior of finite particle formulations based on projection methodologies, investigating in particular how these approaches behave in approximating 1D and 2D second-order problems. Moreover, the issue of choosing suitable projection functions is discussed and 1D and 2D numer- ical tests, showing the second-order accuracy of the methods under investigation, are performed. Copyright 2010 John Wiley & Sons, Ltd. Received 27 January 2010; Accepted 10 February 2010 KEY WORDS: meshless methods; particle methods; projection functions; second-order accuracy; smoothed particle hydrodynamics; 2D Poisson problem 1. INTRODUCTION A large number of numerical methods has been recently proposed in the literature to address advanced mechanical problems, such as those involving rapid deformations, high-intensity forces, large displacement fields. In many of these cases, in fact, classical finite element methods (FEM) suffer from mesh distortion, spurious numerical errors and, above all, mesh sensitivity. Hence, to overcome such issues, a variety of numerical methods, belonging to the family of the so- called meshless techniques, has been widely investigated and applied. The objective of employing these methods is to avoid the introduction of a mesh for the continuum, preferring a particle discretization, with the goal of obtaining an easier treatment of large and rapid displacements. Thus, meshless methods have been widely applied, mainly to fluid dynamics problems, where particle approaches appear to be more feasible. However, recently, a number of researchers have tried to extend meshless methods also to solid mechanics problems. Among the several meshless numerical methods proposed, particle methods, and in partic- ular smoothed particle hydrodynamics (SPH), have been widely implemented and investigated. Historically, SPH was introduced by Lucy [1] and Gingold and Monagan [2] to treat astrophysics problems, and, then, a variety of formulations has been proposed to apply its principles to different problems, such as incompressible flows [3], elasticity [4], fracture of solids [5, 6], heat conduction [7]. Furthermore, in order to address a number of criticalities and issues, several improvements have been proposed: for instance, Swegle et al. [8] highlighted ‘tension instability’, fixing it through ∗ Correspondence to: A. Reali, Department of Structural Mechanics, University of Pavia, via Ferrata 1, 27100, Pavia, Italy. † E-mail: alessandro.reali@unipv.it Copyright 2010 John Wiley & Sons, Ltd.