My new paper on the connection between peridynamics and RKPM (and high-order peridynamics) is accepted without revision, and will be online shortly. Stay tuned!
Generalized Reproducing Kernel Peridynamics: Unification of Local and Non-Local Meshfree Methods, and an Arbitrary-order Accurate Generalized State-Based Peridynamic Formulation
State-based peridynamics is a non-local reformulation of solid mechanics that replaces the force density of the divergence of stress with an integral of the action of force states on bonds local to a given position, which precludes differentiation with the aim to model strong discontinuities effortlessly. A popular implementation is a meshfree formulation where the integral is discretized by quadrature points, which results in a series of unknowns at the points under the strong-form collocation framework. In this work, the meshfree discretization of state-based peridynamics under the correspondence principle is examined and compared to traditional meshfree methods based on the classical local formulation of solid mechanics. It is first shown that the way in which the peridynamic formulation approximates differentiation can be unified with the implicit gradient approximation, and is this is termed the generalized reproducing kernel peridynamic approximation. This allows the construction of non-local deformation gradients with arbitrary-order accuracy, as well as non-local approximations of higher-order derivatives. A high-order accurate non-local divergence of stress is then proposed to replace the force density in the original state-based peridynamics, in order to obtain global arbitrary-order accuracy in the numerical solution. These two operators used in conjunction with one another is termed the generalized reproducing kernel peridynamic method. The strong-form collocation version of the method is tested against benchmark solutions to examine and verify the high-order accuracy and convergence properties of the method.
A Non-ordinary State-based Godunov-peridynamics Formulation for Strong Shocks in Solids
The theory and meshfree implementation of peridynamics has been proposed to model problems involving transient strong discontinuities such as dynamic fracture and fragment-impact problems. For effective application of numerical methods to these events, essential shock physics and Gibbs instability should be addressed. The Godunov scheme for shock treatment has been shown an effective approach for tackling these two issues but has not been considered yet for peri-dynamics. This work introduces a physics-based shock modeling formulation for non-ordinary state-based peridynamics, in which the Godunov scheme is introduced by embedding the Rie-mann solution into the force state, resulting in a shock formulation free of tuneable parameters. Several benchmark problems are solved to demonstrate the effectiveness of the proposed formulation for modeling problems involving shocks in solids.
We cordially invite you to present your work in the thematic session “Recent Advances and Applications in Meshfree and Particle Methods” as part of the 2019 International Mechanical Engineering Congress & Exposition (IMECE2019) to be held November 8-14, 2019 at Calvin L. Rampton Salt Palace Convention Center, Salt Lake City, Utah.
The minisymposium is listed as 11-28 under Track 11: Mechanics of Solids, Structures and Fluids. Abstracts (presentation only) should be submitted by the deadline of July 22, 2019, at the IMECE 2019 website: https://event.asme.org/IMECE
The aim of this minisymposium is to bring together experts working on these methods, share research results and identify the emergent needs towards more rapid progress in advancing the important fields of meshfree and particle methods. Topics of interest for this minisymposium include, but are not limited to the following:
· Recent advances in meshfree, peridynamics, material point methods, and smoothed particle hydrodynamics
· Recent advances in formulations for extreme material distortion, fragmentation, contact and impact, and material instability
· Recent advances in numerical integration
· Characterization and stabilization of numerical instabilities
· Strong form collocation meshfree methods
· Simulations involving multiple and coupled physics
· Simulations involving multiple time and/or length scales
· Simulation of multi-phase interactions
· Simulation of structural responses to extreme loading
· Simulation of microstructural behavior and microstructure evolution
· Simulation of manufacturing processes
· Simulation of bio and nano mechanics and material system responses
· Applications of meshfree methods to simulation of natural disasters
· Nonlocal mechanics and computation
· New applications for which meshfree and particle methods are superior to conventional mesh-based methods
· Parallel-computing, scalable algorithms, and large-scale simulations
Sheng-Wei Chi, University of Illinois, <email@example.com>
J. S. Chen, University of California, San Diego, <firstname.lastname@example.org>
Mike Hillman, Pennsylvania State University, <email@example.com>
C. T. Wu, Livermore Software Technology Corporation, <firstname.lastname@example.org>