| Abstract | Element-Free Galerkin Method (EFGM) is best applied to problems where intensity of
responses is not uniform, deformation is excessive, or cracks occur and propagate. This study
explores the feasibility and the advantages of varying the domain of influence (DOI) and the
integration cells in relation to the intensity of the responses to minimize errors and improve
efficiency of EFGM. Several strategies for varying the sizes of DOI and the integration cells
are investigated and tested. As a rule of thumb, DOis associated with any node must be large
enough to ensure that at least m nodes are visible from each integration point, where m is the
number of monomials in basis function. At the same time, integration cells must be fine
enough to accurately evaluate the contribution of all visible nodes.
In standard EFGM, all DOis are maintained uniform in size. Thus, they must be large
enough to cover sufficient number of nodes in the low-gradient areas in order to ensure a good
matrix condition in the Moving Least Square (MLS) approximation. However, large DOis will
cause a large array of nodes to be coupled; thus solution not only is expensive, but also cannot
reflect locality effects.
On the issue of numerical integration, the commonly used uniform background cells will
yield an inaccurate solution for high node-concentrated area or wastes excessive computing
time in the node-sparse area. In this study, for lD problems, integration over the entire
problem domain is subdivided into a series of integration sub-domains, each being defined as
interval between two nodes. For 2D problems, integration over the entire problem domain is
subdivided into a network of triangular sub-domain cells, each being defined as a triangle
formed over three nodes. By this procedure, the size of integration cells can be varied with the
density of nodes, while the number of integration points in each cell is constant.
The numerical tests in this study confirm that the proposed strategy for varying the sizes of
DOI and integration cells works effectively for all test examples. This should serve as the
initial step towards to a breakthrough in adaptive discretization of EFGM in the future
extension of this study. By establishing an algorithm to generate a network of triangular cells
based on any given distribution of nodes, the triangulation process can be automated and
transparent to users. Thus, with a good estimator of error norm, one can determine if more
nodes should be added to the mid-sides of any triangle. Then, all nodes including the new ones
will be considered in the new triangulation, and the new discretized domain can be reanalyzed. This process can be repeated until a situation can be achieved where error norms in
all triangles are controlled under a specified limit. |