Interpolants of Lattice Functions for the Analysis of Atomistic/Continuum Multiscale Methods

We introduce a general class of (quasi-)interpolants of functions defined on a Bravais lattice, and establish several technical results for these interpolants that are crucial ingredients in the analysis of atomistic models and atomistic/continuum multi-scale methods

Elinvar effect in β−\beta-Ti simulated by on-the-fly trained moment tensor potential

A combination of quantum mechanics calculations with machine learning (ML) techniques can lead to a paradigm shift in our ability to predict materials properties from first principles. Here we show that on-the-fly training of an interatomic potential described through moment tensors provides the same accuracy as state-of-the-art {\it ab inito} molecular dynamics in predicting high-temperature elastic properties of materials with two orders of magnitude less computational effort. Using the technique, we investigate high-temperature bcc phase of titanium and predict very weak, Elinvar, temperature dependence of its elastic moduli, similar to the behavior of the so-called GUM Ti-based alloys [T. Sato {\ it et al.}, Science {\bf 300}, 464 (2003)]. Given the fact that GUM alloys have complex chemical compositions and operate at room temperature, Elinvar properties of elemental bcc-Ti observed in the wide temperature interval 1100--1700 K is unique.Comment: 15 pages, 4 figure

Numerical Methods for Multilattices

Among the efficient numerical methods based on atomistic models, the quasicontinuum (QC) method has attracted growing interest in recent years. The QC method was first developed for crystalline materials with Bravais lattice and was later extended to multilattices (Tadmor et al, 1999). Another existing numerical approach to modeling multilattices is homogenization. In the present paper we review the existing numerical methods for multilattices and propose another concurrent macro-to-micro method in the numerical homogenization framework. We give a unified mathematical formulation of the new and the existing methods and show their equivalence. We then consider extensions of the proposed method to time-dependent problems and to random materials.Comment: 31 page

Constrained Density Functional Theory: A Potential-Based Self-Consistency Approach

Chemical reactions, charge transfer reactions, and magnetic materials are notoriously difficult to describe within Kohn−Sham density functional theory, which is strictly a groundstate technique. However, over the last few decades, an approximate method known as constrained density functional theory (cDFT) has been developed to model low-lying excitations linked to charge transfer or spin fluctuations. Nevertheless, despite becoming very popular due to its versatility, low computational cost, and availability in numerous software applications, none of the previous cDFT implementations is strictly similar to the corresponding ground-state self-consistent density functional theory: the target value of constraints (e.g., local magnetization) is not treated equivalently with atomic positions or lattice parameters. In the present work, by considering a potential-based formulation of the self-consistency problem, the cDFT is recast in the same framework as Kohn−Sham DFT: a new functional of the potential that includes the constraints is proposed, where the constraints, the atomic positions, or the lattice parameters are treated all alike, while all other ingredients of the usual potentialbased DFT algorithms are unchanged, thanks to the formulation of the adequate residual. Tests of this approach for the case of spin constraints (collinear and noncollinear) and charge constraints are performed. Expressions for the derivatives with respect to constraints (e.g., the spin torque) for the atomic forces and the stress tensor in cDFT are provided. The latter allows one to study striction effects as a function of the angle between spins. We apply this formalism to body-centered cubic iron and first reproduce the well-known magnetization amplitude as a function of the angle between local magnetizations. We also study stress as a function of such an angle. Then, the local collinear magnetization and the local atomic charge are varied together. Since the atomic spin magnetizations, local atomic charges, atomic positions, and lattice parameters are treated on an equal footing, this formalism is an ideal starting point for the generation of model Hamiltonians and machine-learning potentials, computation of second or third derivatives of the energy as delivered from density-functional perturbation theory, or for second-principles approaches

Continuum Surface Energy from a Lattice Model

We investigate connections between the continuum and atomistic descriptions of deformable crystals, using certain interesting results from number theory. The energy of a deformed crystal is calculated in the context of a lattice model with general binary interactions in two dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. The main contribution is an explicit formula for the surface energy density as a function of the deformation gradient and boundary normal. The result is valid for a large class of domains, including faceted (polygonal) shapes and regions with piecewise smooth boundaries.Comment: V. 1: 10 pages, no fig's. V 2: 23 pages, no figures. Misprints corrected. Section 3 added, (new results). Intro expanded, refs added.V 3: 26 pages. Abstract changed. Section 2 split into 2. Section (4) added material. V 4, 28 pages, Intro rewritten. Changes in Sec.5 (presentation only). Refs added.V 5,intro changed V.6 address reviewer's comment

Варианты метода коллокации и наименьших невязок для решения задач математической физики в выпуклых четырехугольных областях

The new versions of the collocations and least residuals (CLR) method of high-order accuracy are proposed and implemented for the numerical solution of the boundary value problems for PDE in the convex quadrangular domains. Their implementation and numerical experiments are performed by the examples of solving the biharmonic and Poisson equations. The solution of the biharmonic equation is used for simulation of the stress-strain state of an isotropic plate under the action of the transverse load. Differential problems are projected into the space of fourth-degree polynomials by the CLR method. The boundary conditions for the approximate solution are put down exactly on the boundary of the computational domain. The versions of the CLR method are implemented on the grids, which are constructed by two different ways. In the first version, a “quasiregular” grid is constructed in the domain, the extreme lines of this grid coincide with the boundaries of the domain. In the second version, the domain is initially covered by a regular grid with rectangular cells. Herewith, the collocation and matching points that are situated outside the domain are used for approximation of the differential equations in the boundary cells that had been crossed by the boundary. In addition the “small” irregular triangular cells that had been cut off by the domain boundary from rectangular cells of the initial regular grid are joined to adjacent quadrangular cells. This technique allowed to essentially reduce the conditionality of the system of linear algebraic equations of the approximate problem in comparison with the case when small irregular cells together with other cells were used as independent ones for constructing an approximate solution of the problem. It is shown that the approximate solution of problems converges with high order and matches with high accuracy with the analytical solution of the test problems in the case of the known solution in numerical experiments on the convergence of the solution of various problems on a sequence of grids. Предложены и реализованы новые варианты метода коллокации и наименьших невязок (КНН) для численного решения краевых задач для уравнений с частными производными в выпуклых четырехугольных областях. Их реализация и численные эксперименты выполнены на примерах решения уравнений Пуассона и бигармонического. Решение второго уравнения использовано для моделирования напряженно–деформированного состояния изотропной пластины, находящейся под действием поперечной нагрузки. Дифференциальные задачи методом КНН проектировались в пространство полиномов четвертой степени. Граничные условия для приближенного решения задач выписывались точно на границе расчетной области. Реализованы варианты метода КНН на сетках, построенных двумя различными способами. В первом варианте в области строится некоторая “квазирегулярная” сетка, крайние линии которой совпадают с границами области. Во втором — область сначала накрывается регулярной сеткой с прямоугольными ячейками. При этом в граничных ячейках, которые пересекла граница, для аппроксимации дифференциальных уравнений использованы “законтурные” (расположенные вне расчетной области) точки коллокации и точки согласования решения задачи. Кроме этого, “малые” нерегулярные треугольные ячейки, отсеченные границей области от прямоугольных ячеек начальной регулярной сетки, присоединялись к соседним четырехугольным ячейкам. Этот прием позволил существенно уменьшить обусловленность системы линейных алгебраических уравнений приближенной задачи по сравнению со случаем, когда малые ячейки наряду с другими ячейками использовались как самостоятельные для построения приближенного решения задачи. В численных экспериментах по сходимости приближенного решения различных задач на последовательности сеток установлено, что оно сходится с повышенным порядком и с высокой точностью совпадает с аналитическим решением задачи в случае, когда оно известно.

Atomic-scale representation and statistical learning of tensorial properties

This chapter discusses the importance of incorporating three-dimensional symmetries in the context of statistical learning models geared towards the interpolation of the tensorial properties of atomic-scale structures. We focus on Gaussian process regression, and in particular on the construction of structural representations, and the associated kernel functions, that are endowed with the geometric covariance properties compatible with those of the learning targets. We summarize the general formulation of such a symmetry-adapted Gaussian process regression model, and how it can be implemented based on a scheme that generalizes the popular smooth overlap of atomic positions representation. We give examples of the performance of this framework when learning the polarizability and the ground-state electron density of a molecule

Machine-learning Driven Synthesis of TiZrNbHfTaC5 High-Entropy Carbide

Synthesis of high-entropy carbides (HEC) requires high temperatures that can be provided by electric arc plasma method. However, the formation temperature of a single-phase sample remains unknown. Moreover, under some temperatures multi-phase structures can emerge. In this work we developed an approach for a controllable synthesis of HEC TiZrNbHfTaC5 based on theoretical and experimental techniques. We used canonical Monte Carlo (CMC) simulations with the machine learning interatomic potentials to determine the temperature conditions for the formation of single-phase and multi-phase samples. In full agreement with the theory, the single-phase sample, produced with electric arc discharge, was observed at 2000 K. Below 1200 K the sample decomposed into (Ti-Nb-Ta)C and a mixture of (Zr-Hf-Ta)C, (Zr-Nb-Hf)C, (Zr-Nb)C, and (Zr-Ta)C. Our results demonstrate the conditions for the formation of HEC and we anticipate that our approach can pave the way towards targeted synthesis of multicomponent materials.Comment: 16 pages, 8 figure