Enhanced Meshfree Methods for Numerical Solution of Local and Nonlocal Theories of Solid Mechanics
Achieving good accuracy while keeping a low computational cost in numerical simulations
of problems involving large deformations, material fragmentation and crack propagations still
remains a challenge in computational mechanics. For these classes of problems, meshfree
discretizations of local and nonlocal approaches, have been shown to be effective as they avoid
some of the common issues associated with mesh-based techniques, such as the need for re-
meshing due to excessive mesh distortion. Nonetheless, other issues remain.
In the framework of local mechanics, the semi-Lagrangian reproducing kernel particle
method (RKPM) has been proved to be particularly effectively for material damage and frag-
mentation, as by reconstructing the field approximations in the current configuration it does
not require the deformation gradient to be positive definite. This, however, results in a high
Furthermore, for crack propagation problems, the use of classical local mechanics presents
many challenges, such as the need of accurately representing the singular stress field at crack tips. The peridynamic nonlocal theory circumvents these issues by reformulating solid mechanics in terms of integral equations. In engineering applications, a simple node-based discretization of peridynamics is typically employed. This approach is limited to first order convergence and often lacks the symmetry of interaction of the continuous form. The latter can be recovered through the use of the peridynamic weak form, which however involves costly double integration.
First, we first propose, in the context of local mechanics, a blending-based spatial coupling
scheme to transition from the computationally cheaper Lagrangian RKPM to the semi-Lagrangian RKPM. Next, we introduce an RK approximation to the field variables in strong form peridynam-ics to increase the order of convergence of peridynamic numerical solutions. Then, we develop an efficient n-th order symmetrical variationally consistent nodal integration scheme for RK enhanced weak form peridynamics.
Lastly, we propose a Waveform Relaxation Newmark algorithm for time integration of
the semi-discrete systems arising from meshfree discretizations of local and nonlocal dynamics
problems. This scheme retains the unconditional stability of the implicit Newmark scheme with
the advantage of the lower computational cost of explicit time integration schemes.
Numerical examples demonstrate the effectiveness of the proposed approaches.