Meshfree methods for solid mechanics have been in development for nearly 30 years. Motivation for their development include alleviation of the burden of conformal mesh creation and the desire to overcome the limitations of traditional mesh-based discretizations in extreme-deformation applications.  A meshfree method requires the definition of both meshfree shape functions and a consistent and stable quadrature scheme.  A major remaining impediment to the adoption of Galerkin meshfree methods for nonlinear solid mechanics is the development of an efficient stable meshfree quadrature scheme for integrating the weak form of the governing equations.  The quadrature scheme should ideally be sufficient to avoid the necessity for artificial stabilization and associated parameters.

Here, a reproducing-kernel Bubnov-Galerkin meshfree method is presented that uses a stable and consistent quadrature scheme constructed using a finer discretization of reproducing kernels.  The quadrature weights are obtained by integrating the shape functions of a refined cloud of meshfree points.  Verification examples are presented demonstrating the expected rates of convergence for the quadrature rules of various orders.  For solution of boundary value problems, the derivatives of the shape functions of the original meshfree discretization are corrected to satisfy the discrete form of the divergence theorem as required for consistency.  This correction methodology was originally developed for use in general polyhedral finite elements.  In this methodology, an L2 minimization problem is formulated that minimizes the difference between the original shape function derivatives and the projected values while also satisfying the discrete divergence constraints.  This minimization process is local to the support of each meshfree point or node.  The resulting corrected derivatives are then consistent with the discrete divergence constraint.  The spectrum of the stiffness matrix using various ratios of quadrature points and degree-of-freedom meshfree points is used to study the stability of the overall Bubnov-Galerkin meshfree method.  

Several 2D verification problems and examples for solid mechanics are presented. These include the engineering patch test, beam bending with shear loading, and plate with hole.   Convergence rates are studied for each example along with sensitivity to irregularity in meshfree point placement within a domain.  Several 3D example problems are presented including the large deformation of a hyperelastic beam and the linear elastic stress analysis of a geometrically complex machine part.  

Further research is needed in the nearly incompressible regime of material behavior.  One simple approach to handle near incompressibility, e.g. plasticity, is to use traditional Fbar methodologies with a coarser discretization of meshfree points representing the dilational response, while adhering to standard heuristics for ratios in degrees of freedom for shear and dilational deformation fields. 


A Consistent and Stable Quadrature Scheme for Galerkin Meshfree Methods for Applications in Nonlinear Solid Mechanics

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Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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