An improved reproducing kernel particle method for nearly incompressible finite elasticity Original Research Article

The previously developed reproducing kernel particle method (RKPM) employs a high-order quadrature rule for desired domain integration accuracy. This leads to an over-constrained condition in the limit of incompressibility, and volumetric locking and pressure oscillation were encountered. The employment of a large support size in the reproducing kernel shape function increases the dependency in the discrete constraint equations at quadrature points and thereby relieves locking. However, this approach consumes high CPU and it cannot effectively resolve pressure oscillation difficulty. In this paper, a pressure projection method is introduced by locally projecting the pressure onto a lower-order space to reduce the number of independent discrete constraint equations. This approach relieves the over-constrained condition and thus eliminates volumetric locking and pressure oscillation without the expense of employing large support size in RKPM. The method is developed in a general framework of nearly incompressible finite elasticity and therefore linear problems are also applicable. To further reduce the computational cost, a stabilized reduced integration method based on an assumed strain approach on the gradient matrix associated with the deformation gradient is also introduced. The resulting stiffness matrix and force vector of RKPM are obtained explicitly without numerical integration.