An efficient nodal integration with quadratic exactness for three-dimensional meshfree Galerkin methods

Abstract

Quadratically consistent nodal integration (QCNI) for three-dimensional meshfree Galerkin methods with second order approximation is presented. The number of integration points is dramatically reduced since the weak form is evaluated only at approximation nodes. The stabilization for such reduced integration stems from the correction of nodal derivatives. Such correction is based on the orthogonality condition between the stress and strain difference in the framework of Hu-Washizu three-field variational principle. Taylor series expansion is employed such that a linear strain field in each background integration cell can be exactly reproduced. Three-dimensional quadratic patch test is exactly passed by QCNI and thus it possesses quadratic exactness. In contrast, the stabilized conforming nodal integration (SCNI) which is so far the most successful nodal integration technique can only reproduce a constant strain field in each integration cell and fails to pass the quadratic patch test. The comprehensive superiorities of the proposed QCNI over the existing SCNI in accuracy, convergence, efficiency and smoothness of the resulting stress fields are further demonstrated by several three-dimensional numerical examples. Especially, it is shown in some example that the accuracy of QCNI is surprisingly four order higher than that of SCNI.