Proc. Computational Modelling of Concrete Structures (EURO-C),
Badgastein, Austria, March 31 - April 3, 1998,
ed. R. de Borst, N. Bicanic, H. Mang, and G. Meschke, Balkema, Rotterdam, 311-319.


Milan Jirásek
Swiss Federal Institute of Technology
LSC -DGC, EPFL, 1015 Lausanne, Switzerland


The paper discusses the applicability of the element-free Galerkin (EFG) method to problems with strain localization. It is explained why the EFG technique fails for a standard local continuum, even when correct energy dissipation is ensured by adjusting the softening modulus as a function of nodal spacing. The source of disastrous stress oscillations, leading to the occurrence of multiple softening bands, is analyzed and illustrated by a simple uniaxial example. It is demonstrated that the oscillations are substantially reduced when the model is reformulated as nonlocal, provided that the radius of influence in the EFG formulation is not too large compared to the internal length of the nonlocal continuum. It is also shown that for regularized localization problems the accuracy of the EFG solution can be superior to that obtained by the finite element method. Finally, the potential of the method is illustrated by failure analysis of a two-dimensional beam model.

Concluding Remarks

The present paper has addressed certain basic issues related to the performance of the EFG method in problems with localization due to strain softening. We have shown that if the theoretical solution exhibits discontinuities, standard EFG performs poorly compared to FEM because the smooth shape functions generated by the MLS technique cannot capture sudden jumps. Modification of the shape functions incorporating information on the discontinuity is routinely applied in studies that use the discrete approach to modeling of fracture. When the propagating crack cuts the link between a node and an integration point, the integration point is removed from the domain of influence of the node. This can be done relatively easily because the crack faces are assumed to be stress-free and no forces are transmitted across the crack. Furthermore, the modification is local because the shape functions and their derivatives have to be recomputed in each step only in a small region around the crack tip. A similar technique would become too complex and computationally expensive if the model uses a smeared description of cracking. Instead of a clean stress-free crack we would have to treat a zone of highly localized strain, in which the ``links'' between points are only gradually disconnected. So it seems that for standard strain-softening continua the effort invested into the development of a reliable EFG technique would not pay off.

The situation changes if the model is enriched by a localization limiter that regularizes the solution. Compared to FEM, EFG has a better ability to reproduce the highly localized but smooth strain profile. Another advantage is that adaptive refinement might be greatly facilitated by the ``meshless'' character of EFG. Nonlocal models are sometimes claimed to be too expensive because they require a mesh size that corresponds to the characteristic length of the continuum, which can be very small compared to the size of the structure. However, the mesh has to be very fine only in the regions of intense straining. Adaptive techniques can make the application of nonlocal models in real design problems more economic, and the implementation of a refinement procedure for an EFG computational grid should be easier than for a finite element mesh.

It would be interesting to explore the potential of other meshless techniques, especially of those that exhibit slower rates of convergence but are computationally less demanding. In practical applications of nonlinear analysis, the important property of a method is not its asymptotic rate of convergence but its accuracy for a reasonable number of degrees of freedom. It is therefore possible that some techniques that do not converge at an impressive rate for linear problems would be handy in nonlinear applications.

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EPFL / 13 January 1998 /