Publications

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We propose here new projection methods for treating near-incompressibility in small and large deformation elasticity and plasticity within the framework of particle and meshfree methods. Using the $\bar{B}$ and $\bar{F}$ techniques as our point of departure, we develop projection methods for the conforming reproducing kernel method and the immersed-particle or material point-like methods. The methods are based on the projection of the dilatational part of the appropriate measure of deformation onto lower-dimensional approximation spaces, according to the traditional $\bar{B}$ and $\bar{F}$ approaches, but tailored to meshfree and particle methods. The presented numerical examples exhibit reduced stress oscillations and are free of volumetric locking and hourglassing phenomena.

Window functions provide a base for the construction of approximation functions in many meshfree methods. They control the smoothness and extent of the approximation functions and are commonly defined using Euclidean distances which helps eliminate the need for a meshed discretization, simplifying model development for some classes of problems. However, for problems with complicated geometries such as nonconvex or multi-body domains, poor solution accuracy and convergence can occur unless the extents of the window functions, and thus approximation functions, are carefully controlled, often a time consuming or intractable task. In this paper, we present a method to provide more control in window function design, allowing efficient and systematic handling of complex geometries. “Conforming” window functions are constructed using Bernstein–Bézier splines defined on local triangulations with constraints imposed to control smoothness. Graph distances are used in conjunction with Euclidean metrics to provide adequate information for shaping the window functions. The conforming window functions are demonstrated using the Reproducing Kernel Particle Method showing improved accuracy and convergence rates for problems with challenging geometries. Conforming window functions are also demonstrated as a means to simplify the imposition of essential boundary conditions.

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The Reproducing Kernel Particle Method (RKPM), a meshfree method, has been implemented in Sandia's Sierra/SolidMechanics in a collaboration between Sandia and the University of California San Diego's Center for Extreme Events Research (UCSD/CEER). Meshfree methods, like RKPM, have an advantage over mesh-based methods, like the Finite Element Method (FEM), in applications where achieving or maintaining a quality mesh becomes burdensome or impractical. For example, using FEM for problems with very large deformations will result in poor element Jacobians which causes problems with the parametric mapping. RKPM constructs the approximation functions in the physical domain, circumventing the parametric mapping issue. Also, reconstructing the approximation functions at very large deformations is straight-forward. RKPM has an advantage over traditional meshfree methods such as Smoothed-Particle Hydrodynamics (SPH) due to its ability to reproduce linear or higher-order functions exactly. This removes the tensile instabilities that are present in SPH and allows preservation of angular momentum. The point of this memo is to explore the capabilities and limitations of the current implementation by testing it on three different applications: 1) a quasi-static ductile shearing problem 2) a dynamic concrete panel perforation problem and 3) a set of dynamic metal panel perforation problems. In summary, areas where RKPM appears to be a promising alternative to current methods have been identified. Also, outstanding inefficiencies and issues (bugs) with code are noted, ways to improve the capabilities using material from literature are mentioned and areas deserving of new research are highlighted.

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