Continuous meshless approximations for nonconvex bodies by diffraction and transparency

Enrichment and coupling of the finite element and meshless methods

“…In reproducing kernel particle methods [1; 2] (RKPM) the interpolation functions are obtained in the framework of the MLS interpolation. The element-free Galerkin method [3][4][5][6][7][8] (EFG) can be viewed as a particular case of the previous formulation [16].…”

“…Such a situation may be of interest if a computation with ÿnite elements of degree p needs to be reÿned in a region without remeshing. The nodes of the original ÿnite element mesh are removed in but as many particles as needed are added in that region (see the crack propagation examples in the papers by Belytschko and co-workers [4][5][6]8]). …”

“…Meshless or particle methods such as reproducing kernel particle methods (RKPM) [1; 2] elementfree Galerkin (EFG) [3][4][5][6][7][8] or smooth particle hydrodynamics (SPH) [9; 10] among others (see [11; 12] for a general presentation), have nowadays proven their applicability in computational mechanics. They do not require to generate a mesh (a connectivity matrix) and thus, they are specially suited for certain problems, for instance adaptive reÿnement computations or discontinuous ÿeld problems (i.e.…”

An error estimate in the EFG method

“…In this approach, the boundaries of the body and any interior lines of discontinuity are considered opaque when constructing the weight functions. Another method for construction of approximations around the tip of a discontinuity is the diffraction method (Belytschko et al 1996a andOrgan et al 1996). This method treats the line of discontinuity as opaque but bases the weight function parameter on the length of the path which passes around the corner of the discontinuity.…”

Partition of unity-based thermomechanical meshfree method for two-dimensional crack problems

“…Though the visibility method introduces artificial discontinuities at the crack tip, it remains the most frequently applied method. The diffraction and transparency method [2,3,27] removes the artificial discontinuities of the visibility method but on cost of higher computational effort. These methods are difficult to extend in a three-dimensional setting and often require only approximate calculation of the derivatives of the shape functions.…”