  We present a quasi-conforming embedded reproducing kernel particle method
(QCE-RKPM) for modeling heterogeneous materials that makes use of techniques
not available to mesh-based methods such as the finite element method (FEM) and
avoids many of the drawbacks in current embedded and immersed formulations
which are based on meshed methods. The different material domains are
discretized independently thus avoiding time-consuming, conformal meshing. In
this approach, the superposition of foreground (inclusion) and background
(matrix) domain integration smoothing cells are corrected by a quasi-conforming
quadtree subdivision on the background integration smoothing cells. Due to the
non-conforming nature of the background integration smoothing cells near the
material interfaces, a variationally consistent (VC) correction for domain
integration is introduced to restore integration constraints and thus optimal
convergence rates at a minor computational cost. Additional interface
integration smoothing cells with area (volume) correction, while
non-conforming, can be easily introduced to further enhance the accuracy and
stability of the Galerkin solution using VC integration on non-conforming
cells. To properly approximate the weak discontinuity across the material
interface by a penalty-free Nitsche's method with enhanced coercivity, the
interface nodes on the surface of the foreground discretization are also shared
with the background discretization. As such, there are no tunable parameters,
such as those involved in the penalty type method, to enforce interface
compatibility in this approach. The advantage of this meshfree formulation is
that it avoids many of the instabilities in mesh-based immersed and embedded
methods. The effectiveness of QCE-RKPM is illustrated with several examples.
