Keywords: meshfree, meshless, moving least squares, smooth particle hydrodynamics, element-free Galerkin, coupling with finite elements, incompressibility.

Meshfree methods have proven in recent years that they are an alternative where limitations of conventional computational methods, such as finite elements, finite volumes or finite difference methods, are apparent. For instance, smooth particle hydrodynamic (SPH) methods are widely used for fast-transient dynamic simulations such as explosions or impact problems, because of their low computational cost and the ability to handle severe distortions [6,3]. Other meshless methods, such as element-free Galerkin (EFG) [1] or reproducing kernel particle methods (RKPM) [5] can also deal with large distortions and go beyond finite element (FE) computations [4], in addition, for dealing with moving discontinuities. For instance, simulations of failure, where the simulation of the propagation of cracks with arbitrary and complex paths is needed. However, these later formulations have a higher computational cost due to the use of Gauss quadratures or specific techniques to accurately integrate the weak form.

Nevertheless, when remeshing is not an important issue, for instance in the absence of large distortions, FE computations are preferred by practitioners for two main reasons. Firstly, it should be noted that most users are familiar with FE methods, and mesh-free methods are still seen as research techniques. However, this will, of course, change with time. Secondly, FE computations in the absence of remeshing can be very efficient from a computational point of view, and thus less costly than mesh-free computations. However, these advantages disappear when intensive remeshing is required. This is the case with problems such as large distortions, impact and blast. In these cases meshfree methods go clearly beyond FE methods. In fact, as the range of phenomena that need to be simulated in engineering practice broadens, the limitations of conventional computational methods have become apparent. There are many problems of industrial and academic interest which cannot be easily treated using these classical mesh-based methods. In particular those where it is necessary to deal with extremely large deformations of the mesh (for example, simulation of manufacturing processes such as extrusion and molding).

The objective of meshfree methods is to eliminate at least part of this mesh dependence by constructing the approximation entirely in terms of nodes (usually called particles in the context of meshfree methods). Moving discontinuities or interfaces can usually be treated without remeshing with minor cost and accuracy degradation. Thus the range of problems that can be addressed by meshfree methods is much wider than mesh-based methods. Moreover, large deformations can be handled more robustly with meshfree methods because the approximation is not based on elements whose distortion may degrade the accuracy. This is useful in both fluid and solid computations. Several review papers and books have been published on meshfree methods. See, for instance, [2] for a large number of references.

In this paper, after a brief recollection of the major approximation issues, critical topics for the practical use of meshfree methods are discussed. In particular, meshfree methods must be (and in fact are) blended with finite elements in commercial codes. Thus an appropriate coupling between meshfree and finite elements methods is necessary. Moreover, the uncoupling between the support and the integration domain in particle methods gives extra freedom that can be used, for instance, to efficiently treat incompressible problems. There are however other issues that must be tackled in future research to completely introduce meshfree methods in engineering practice. For instance, efficient and sound a posteriori error estimators that will allow us to compute engineering output bounds and, most importantly, efficient and robust integration techniques.

1

Belytschko T, Lu YY and Gu L. Element free Galerkin methods. Int. J. Numer. Methods Eng., 37(2):229-256, 1994. doi:10.1002/nme.1620370205

2

Huerta, A, Belytschko, T, Fernandez-Mendez, S and Rabczuk, T. Meshfree methods. In Stein E, de Borst, R and Hughes TJR, editors, Encyclopedia of Computational Mechanics, volume 1 Fundamentals, chapter 10, pages 279-309. John Wiley & Sons, Ltd., Chichester, 2004.

3

Johnson GR and Beissel SR. Normalized smoothing functions for SPH impact computations. Comput. Methods Appl. Mech. Eng., 39(16):2725-2741, 1996.

4

Jun S, Liu WK and Belytschko T. Explicit Reproducing Kernel Particle Methods for large deformation problems. Int. J. Numer. Methods Eng., 41:137-166, 1998. doi:10.1002/(SICI)1097-0207(19980115)41:1<137::AID-NME280>3.0.CO;2-A

5

Liu WK, Jun S and Zhang YF. Reproducing kernel particle methods. Int. J. Numer. Methods Fluids, 20(8-9):1081-1106, 1995. doi:10.1002/fld.1650200824

6

Sweegle JW, Hicks DL and Attaway SW. Smoothed particle hydrodynamics stability analysis. J. Comput. Phys., 116(1):123-134, 1995. doi:10.1006/jcph.1995.1010

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