# 15.1. Overview

The Conforming Reproducing Kernel (CRK) method uses ideas from meshfree methods, such as the reproducing kernel particle method (RKPM), but uses a mesh to provide boundary information and a structure for domain integration [[1], [2]]. The goal of CRK is to provide the advantages of meshfree methods (robustness in large deformation, greatly reduced sensitivity to discretization quality) while overcoming some of the drawbacks (solution quality near convex boundaries or material interfaces, enforcement of essential boundary conditions, computational efficiency, etc.). CRK uses robust integration techniques for nearly incompressible materials [[3]] and is being used to explore bond-based material failure methodologies [[2]].

The domain of the CRK kernels are defined by “stars” of elements surrounding a given node. Kernel functions are defined on the stars and then subject to the reproducing conditions to form a basis for the CRK formulation. Examples kernel supports include one-star, two-star and spherical-star. The one-star kernel includes all the elements attached to a node. For tetrahedral elements, the one-star kernel recovers linear finite element shape functions after enforcing linear reproducing conditions. The two-star includes the one-star plus an additional ring of elements (all the elements attached to the one-star). The spherical-star uses all the elements fully or partially contained in a ball about the node, providing the lowest discretization sensitivity of these kernel shapes. All kernel shapes are modified to conform to essential boundaries and material interfaces. Kernel values, and thus shape functions are forced to zero at these locations for nodes that are not on the boundary (i.e., the functions “conform” to the boundary). This results in simple essential boundary condition enforcement and greatly improved solution quality near these locations when compared to RKPM or other common meshfree methods where kernel and shape functions overlap boundaries and interfaces.

Domain integration is accomplished using strain-smoothing techniques [[4]]. Derivatives are consistently approximated using integration-cell boundary integrals of the field as opposed to direct derivative evaluation at Gaussian quadrature points, as is done in most finite element methods. The strain-smoothing method has proven to be robust for large deformation and an efficient way to integrate consistently. However, as with finite elements and Gaussian quadrature, nearly incompressible materials require additional considerations to avoid displacement and pressure instabilities. CRK uses a sub-cell strain-smoothing approach in an \(\mathbf{\overline{F}}\) framework to handle the near-incompressibility constraints [[3]]. An integration cell is represented as a collection of sub-cells. Each sub-cell provides a unique deviatoric portion of the deformation gradient but the volumetric portion is the volume weighted average of the all the sub-cells in the cell. Appropriate balancing of the number of cells, sub-cells and nodal degrees-of-freedom are used to provide solutions that are stable in both pressure and displacement.

CRK supports both hexahedral and tetrahedral meshes in general. Tetrahedral meshes are typically preferred due to the relative ease of meshing. Many features are tailored to tetrahedral inputs.

[1]

Jacob J. Koester and Jiun-Shyan Chen. Conforming window functions for meshfree methods. Computer Methods in Applied Mechanics and Engineering, 347:588 – 621, 2019. doi:https://doi.org/10.1016/j.cma.2018.12.042.

[2] (1,2)

Jacob Koester, Michael Tupek, and Scott Mitchell. An agile design-to-simulation workflow using a new conforming moving least squares method. Technical Report SAND2019-11851, Sandia National Laboratories, Albuquerque, NM, 2019.

[3] (1,2)

Georgios Moutsanidis, Jacob Koester, Michael Tupek, Yuri Bazilevs, and Jiun-Shyan Chen. Treatment of near-incompressibility in meshfree and immersed-particle methods. Computational Particle Mechanics, 2019.

[4]

Jiun-Shyan Chen, Cheng-Tang Wu, Sangpil Yoon, and Yang You. A stabilized conforming nodal integration for galerkin mesh-free methods. International Journal for Numerical Methods in Engineering, 50(2):435–466, 2001.