Isogeometric analysis in electronic structure calculations

Abstract

In electronic structure calculations, various material properties can be obtained by means of computing the total energy of a system as well as derivatives of the total energy w.r.t. atomic positions. The derivatives, also known as HellmanFeynman forces, require, because of practical computational reasons, the discretized charge density and wave functions having continuous second derivatives in the whole solution domain. We describe an application of isogeometric analysis (IGA), a spline modification of finite element method (FEM), to achieve the required continuity. The novelty of our approach is in employing the technique of Bézier extraction to add the IGA capabilities to our FEM based code for ab-initio calculations of electronic states of non-periodic systems within the density-functional framework, built upon the open source finite element package SfePy. We compare FEM and IGA in benchmark problems and several numerical results are presented.

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.…”

Mixing algorithms for fixed-point iterations in self-consistent electronic structure calculations

“…with the charge density ρ(r) = N i n i |ψ i (r)| 2 on the right-hand side. See (Cimrman et al, 2018b) for details. The problem (1), (2) is nonlinear and needs to be solved iteratively: we use the stadard DFT loop algorithm outlined in Fig.…”

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

“…In this paper, we study the convergence properties of the electronic structure calculation method presented in [13] based on…”

“…The description of the essential characteristics of the real space code for electronic structure calculations can be found in [51] . Our electronic structure calculation method has been implemented in the open source finite element package SfePy [12] (Simple Finite Elements in Python, http://sfepy.org ), which is a general package for solving (systems of) partial differential equations (PDEs) by the finite element method (FEM), cf. [45] .…”