A. Prakash

K. B. Nakshatrala, A. Prakash, K. D. Hjelmstad

This paper addresses some numerical and theoretical aspects of dual Schur domain decomposition methods for linear first-order transient partial differential equations. In this work, we consider the trapezoidal family of schemes for integrating the ordinary differential equations (ODEs) for each subdomain and present four different coupling methods, corresponding to different algebraic constraints, for enforcing kinematic continuity on the interface between the subdomains. Method 1 (d-continuity) is based on the conventional approach using continuity of the primary variable and we show that this method is unstable for a lot of commonly used time integrators including the mid-point rule. To alleviate this difficulty, we propose a new Method 2 (Modified d-continuity) and prove its stability for coupling all time integrators in the trapezoidal family (except the forward Euler). Method 3 (v-continuity) is based on enforcing the continuity of the time derivative of the primary variable. However, this constraint introduces a drift in the primary variable on the interface. We present Method 4 (Baumgarte stabilized) which uses Baumgarte stabilization to limit this drift and we derive bounds for the stabilization parameter to ensure stability. Our stability analysis is based on the ``energy'' method, and one of the main contributions of this paper is the extension of the energy method (which was previously introduced in the context of numerical methods for ODEs) to assess the stability of numerical formulations for index-2 differential-algebraic equations (DAEs).

M. Vimal, S. Sandfeld, A. Prakash

Atomistic simulations have now established themselves as an indispensable tool in understanding deformation mechanisms of materials at the atomic scale. Large scale simulations are regularly used to study the behavior of polycrystalline materials at the nanoscale. In this work, we propose a method for grain segmentation of an atomistic configuration using an unsupervised machine learning algorithm that clusters atoms into individual grains based on their orientation. The proposed method, called the Orisodata algorithm, is based on the iterative self-organizing data analysis technique and is modified to work in the orientation space. The working of the algorithm is demonstrated on a 122 grain nanocrystalline thin film sample in both undeformed and deformed states. The Orisodata algorithm is also compared with two other grain segmentation algorithms available in the open-source visualization tool Ovito. The results show that the Orisodata algorithm is able to correctly identify deformation twins as well as regions separated by low angle grain boundaries. The model parameters have intuitive physical meaning and relate to similar thresholds used in experiments, which not only helps obtain optimal values but also facilitates easy interpretation and validation of results.