Vol.:(0123456789) 1 3 Computational Mechanics (2019) 64:563–583 https://doi.org/10.1007/s00466-019-01670-x ORIGINAL PAPER A convected‑particle tetrahedron interpolation technique in the material‑point method for the mesoscale modeling of ceramics R. B. Leavy 1,2  · J. E. Guilkey 1  · B. R. Phung 1  · A. D. Spear 1  · R. M. Brannon 1 Received: 14 June 2018 / Accepted: 12 January 2019 / Published online: 4 February 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract Convected particle domain interpolation, which is known to boost the accuracy of the material-point method, is applied in a form called convected-particle tetrahedron interpolation (CPTI). CPTI exploits the efciency of tetrahedral tessellations to represent complex structural geometries, while still solving feld equations on a rectilinear background grid. Advantages include anti-locking and an ability to handle extremely large deformations without sufering typical Eulerian advection errors. CPTI is demonstrated to resolve long-standing errors caused by spuriously ragged (stair-stepped) surfaces, and it is also shown to accommodate mathematically rigorous evaluation of surface integrals in models for contact and friction. Benefts of this work are illustrated in mesoscale simulations of an aluminum oxynitride ceramic. Keywords Material point method · Convected particle domain interpolation · Ceramics · Crystal plasticity · Aluminum oxynitride · Verifcation 1 Introduction This paper addresses some challenges in mesoscale mod- eling of polycrystalline materials that arise when using conventional fnite-element methods. Specifcally, the Uin- tah material-point method (MPM) framework [1] is herein used to demonstrate that a convected-particle tetrahedron interpolation (CPTI) method advances the state of the art by allowing conformal representations of material boundaries (e.g., grain boundaries) while retaining other advantages of MPM over the fnite-element method (FEM). The scheme is verifed and shown to represent an advance over predecessor integration schemes for two inhomogeneous deformations (involving extremely large axial and shear stretches). Addi- tionally, a crystal elastic-viscoplastic model is implemented in Uintah to simulate deformation of randomly oriented anisotropic grains in a statistical volume element (SVE) of aluminum oxynitride (AlON). To set nomenclature, only terse reminders of well-estab- lished equations are provided so that attention can focus on the newer aspects of this research. 2 Numerical methods 2.1 Material‑point method The material-point method is an updated-Lagrangian method capable of robustly handling extraordinarily large deforma- tions [2–4]. The MPM saves discrete continuum-feld data such as displacement, velocity, and stress at Lagrangian material points known as particles. The term “particle” is unfortunate, because particles in the MPM are better regarded as fnite Lagrangian domains that tesselate the body. As summarized below, various versions of the MPM can be categorized by diferences in how felds are averaged over these particle domains [5]. A crucial concept is that an average of a feld over an arbitrarily deformed particle domain can be approximated by a weighted average over a geometrically simpler “characteristic” domain. The weak form of the momentum equation is solved on an updated- Lagrangian background grid that is typically chosen (for convenience and efciency) to be rectilinear at the begin- ning of a time step. Spatially varying felds are taken to be * R. B. Leavy Brian.Leavy@utah.edu 1 Mechanical Engineering, University of Utah, Salt Lake City, UT, USA 2 Impact Physics Branch, Army Research Lab, Aberdeen Proving Ground, MD, USA