Proc. Computational Modelling of Concrete
Badgastein, Austria, March 31 - April 3, 1998,
ed. R. de Borst, N. Bicanic, H. Mang, and G. Meschke, Balkema, Rotterdam, 311-319.
ELEMENT-FREE GALERKIN METHOD
APPLIED TO STRAIN-SOFTENING MATERIALS
Swiss Federal Institute of Technology
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.
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
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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13 January 1998 /