An extended Galerkin weak form and a point interpolation method with continuous strain field and superconvergence using triangular mesh

Abstract

A point interpolation method (PIM) with continuous strain field (PIM-CS) is developed for mechanics problems using triangular background mesh, in which PIM shape functions are used to construct both displacement and strain fields. The strain field constructed is continuous in the entire problem domain, which is achieved by simple linear interpolations using locally smoothed strains around the nodes and points required for the interpolation. A general parameterized functional with a real adjustable parameter α are then proposed for establishing PIM-CS models of special property. We prove theoretically that the PIM-CS has an excellent bound property: strain energy obtained using PIM-CS lies in between those of the compatible FEM and NS-PIM models of the same mesh. Techniques and procedures are then presented to compute the upper and lower bound solutions using the PIM-CS. It is discovered that an extended Galerkin (x-Galerkin) model, as special case resulted from the extended parameterized functional with α = 1, is outstan-

Smoothed Particle Hydrodynamics (SPH): an Overview and Recent Developments

“…The W2 formulation can create various models with special properties, such upper bound property [20][21][22][23], ultra-accurate and supper convergent solutions [22,[24][25][26][27][28][29], and even nearly exact solutions [30,31]. These W2 formulations have a theoretical foundation on the novel G space theory [16][17][18][19].…”

“…Compared to the SPH formulations on density change in (28), this summation density approach conserves mass exactly, but suffers from serious boundaries deficiency due to the particle inconsistency. A frequently used way to remediate the boundaries deficiency is the following normalization form by the summation of the smoothing function itself [52,53] …”

“…Similarly substituting the cohesive pressure into the energy (see (28)), it is possible to get its analogous contribution to energy…”

“…The second part in (113) describes the cohesion between particles. Using SPH particle approximation for pressure (see (28)), and considering this cohesive pressure part separately, we can get…”

Strain Field Construction

“…We usually choose all the field nodes first, and then may add in the middle points of cell edges and the centroidals of the triangular cells as the interpolation points. Such selection of strain interpolation points are used in PIM-CS [14,31], SαFEM [38], and SC-PIM [29,30].…”