Convergence study of isogeometric analysis based on Bézier extraction in electronic structure calculations

Abstract

Behavior of various, even hypothetical, materials can be predicted via ab-initio electronic structure calculations providing all the necessary information: the total energy of the system and its derivatives. In case of non-periodic structures, the existing well-established methods for electronic structure calculations either use special bases, predetermining and limiting the shapes of wave functions, or require artificial computationally expensive arrangements, like large supercells. We developed a new method for non-periodic electronic structures based on the density functional theory, environment-reflecting pseudopotentials and the isogeometric analysis with Bézier extraction, ensuring continuity for all quantities up to the second derivative. The approach is especially suitable for calculating the total energy derivatives and for molecular-dynamics simulations. Its main assets are the universal basis with the excellent convergence control and the capability to calculate precisely the non-periodic structures even lacking in charge neutrality. Within the present paper, convergence study for isogeometric analysis vs. standard finite-element approach is carried out and illustrated on sub-problems that appear in our electronic structure calculations method: the Poisson problem, the generalized eigenvalue problem and the density functional theory Kohn-Sham equations applied to a benchmark problem.

Multiscale finite element calculations in Python using SfePy

“…and K is the multiindex attaining 11,22,33,12,13,23 for a 3D problem because of the used CoefSymSym class. In a similar way, the coefficients P H,1 can be introduced as Here, the CoefSym class is employed due to the second-order coefficient which can be represented as a vector with dimension sym.…”

“…SfePy has been employed by our group for a range of topics in multiscale modelling in biomechanics and materials science, including multiscale models of biological tissues (bone, muscle tissue with blood perfusion) [9,38,10,34,41,40], a fish heart model with active contraction [27], computations of acoustic transmission coefficients across interfaces of arbitrary microstructure geometry [35], computations of phononic band gaps [39,37], the finite element formulation of the Schroedinger equation [47,12,11], and other applications.…”

Evaluating Hellmann–Feynman forces within non-local pseudopotentials

“…An efficient evaluation of Hellmann-Feynman forces is essential for geometry optimization tasks and for molecular dynamics calculations in computational material physics. In this paper we introduce the expression for Hellmann-Feynman forces (HF forces, HFF) used in our newly developed ab-initio realspace code for non-periodic electronic structure calculations, based on the density functional theory, finite element method (FEM) (or its variant -isogeometric analysis [1,2]) and non-local environment-reflecting pseudopotentials [3].…”

A Software for Generating and Optimizing Pseudopotentials

“…Our research in this field is motivated by the interest in a detailed modelling of cracks in iron, where we wish to compute potentials for molecular dynamics simulations around a crack tip that would be as precise as possible. The potentials will be constructed with help of our ab-initio electronic calculations code (Vackář et al, 2011;Cimrman et al, 2018), that relies on the pseudopotential approach. This contribution is devoted to describing a new implementation of the approach developed in (Vackář, 1992) for generating and optimizing the so called environment-reflecting all-electron pseudopotentials.…”