Three real-space discretization techniques in electronic structure calculations

A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the main approaches include finite-difference methods, various types of finite-elements and wavelets. This paper reports on the results of several code development projects that approach problems related to the electronic structure using these three different discretization methods. We review the ideas behind these methods, give examples of their applications, and discuss their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status solidi (b) - basic solid state physics" devoted to the CECAM workshop "State of the art developments and perspectives of real-space electronic structure techniques in condensed matter and molecular physics". v2: Minor stylistic and typographical changes, partly inspired by referee comment

Rigorous Simulation of 3D Masks

We perform 3D lithography simulations by using a finite-element solver. To proof applicability to real 3D problems we investigate DUV light propagation through a structure of size 9 microns times 4 microns times 65 nm. On this relatively large computational domain we perform rigorous computations (No Hopkins) taking into account a grid of 11 times 21 source points with two polarization directions each. We obtain well converged results with an accuracy of the diffraction orders of about one percent. The results compare well to experimental aerial imaging results. We further investigate the convergence of 3D solutions towards quasi-exact results obtained with different methods.Comment: 8 pages, 5 figures (see original publication for images with a better resolution

ELSI: A Unified Software Interface for Kohn-Sham Electronic Structure Solvers

Solving the electronic structure from a generalized or standard eigenproblem is often the bottleneck in large scale calculations based on Kohn-Sham density-functional theory. This problem must be addressed by essentially all current electronic structure codes, based on similar matrix expressions, and by high-performance computation. We here present a unified software interface, ELSI, to access different strategies that address the Kohn-Sham eigenvalue problem. Currently supported algorithms include the dense generalized eigensolver library ELPA, the orbital minimization method implemented in libOMM, and the pole expansion and selected inversion (PEXSI) approach with lower computational complexity for semilocal density functionals. The ELSI interface aims to simplify the implementation and optimal use of the different strategies, by offering (a) a unified software framework designed for the electronic structure solvers in Kohn-Sham density-functional theory; (b) reasonable default parameters for a chosen solver; (c) automatic conversion between input and internal working matrix formats, and in the future (d) recommendation of the optimal solver depending on the specific problem. Comparative benchmarks are shown for system sizes up to 11,520 atoms (172,800 basis functions) on distributed memory supercomputing architectures.Comment: 55 pages, 14 figures, 2 table

Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States

We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, led to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in $O(n\log n)$ complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D $n\times n\times n$ Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D ``density fitting`` scheme. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excited states, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is related to the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for finite lattice-structured systems, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a $L\times L\times L$ lattice manifests the linear in $L$ computational work, $O(L)$, instead of the usual $O(L^3 \log L)$ scaling by the Ewald-type approaches

Explicit Solution of the Time Domain Volume Integral Equation Using a Stable Predictor-Corrector Scheme

An explicit marching-on-in-time (MOT) scheme for solving the time domain volume integral equation is presented. The proposed method achieves its stability by employing, at each time step, a corrector scheme, which updates/corrects fields computed by the explicit predictor scheme. The proposedmethod is computationally more efficient when compared to the existing filtering techniques used for the stabilization of explicit MOT schemes. Numerical results presented in this paper demonstrate that the proposed method maintains its stability even when applied to the analysis of electromagnetic wave interactions with electrically large structures meshed using approximately half a million discretization elements

Thermodynamic Conditions in Quenching Chamber of Low Voltage Circuit Breaker

Práce se zabývá studiem procesů probíhajících při zhášení silnoproudého oblouku ve zhášecí komoře jističe. Je zaměřena na výpočet dynamiky tekutin a teplotního pole v okolí elektrického oblouku. V práci je dále popsán vliv vzdálenosti plechů v komoře a vliv tvarů plechů z hlediska aerodynamických podmínek uvnitř komory. Dalším cílem dosaženým touto prací je poskytnutí informací o vlivu polohy elektrického oblouku na termodynamické vlastnosti uvnitř komory. Toto je důležité, zejména pokud je oblouk do komory vtahován jinými silami, např. elektromagnetickými a během tohoto vtahovacího procesu mění svůj tvar i polohu. Za účelem co nejjednoduššího, ale zároveň co nejefektivnějšího řešení úkolu, byl vyvinut software určen speciálně pro výpočet dynamiky tekutin numerickou metodou konečných objemů (FVM). Tato metoda je, v porovnání s rozšířenější metodou konečných prvků (FEM), vhodnější pro výpočet dynamiky tekutin (CFD) zejména proto, že režie na výpočet jedné iterace jsou menší v porovnání s ostatními numerickými metodami. Další výhodou tohoto softwarového řešení je jeho modularita a rozšiřitelnost. Cely koncept softwaru je postaven na tzv. zásuvných modulech. Díky tomuto řešení můžeme využít výpočtové jádro pro další numerické analýzy, např. strukturální, elektromagnetickou apod. Jediná potřeba pro úspěšné používání těchto analýz je napsáni solveru pro konečné prvky (FEM). Jelikož je software koncipován jako multi–thread aplikace, využívá výkon současných vícejádrových procesorů naplno. Tato vlastnost se ještě více projeví, pokud se výpočet přesune z CPU na GPU. Jelikož současné grafické karty vyšších tříd mají několik desítek až stovek výpočetních jader a pracují s mnohem rychlejšími pamětmi, než CPU, je výpočetní výkon několikanásobně vyšší.Work deals with the study of processes that attend the electric arc extinction inside the quenching chamber of a circuit breaker. It is focused on several areas. The first one is concerned to fluid dynamics calculations (CFD) and the second one is aimed at thermal field calculations. In this work effects of metal plates distance together with metal plates shapes are described from aerodynamical point of view. Another objective solved by this work is to give information about influence of an electric arc position in a quenching chamber, which changed its shape due to forces acting on it during extinction process. For purpose of this work a new software solution for CFD was developed. Whole software concept is based on plug-ins. Due to this solution, the software§s calculation core can be used for other numerical analyses, like structural, electromagnetic, etc. The only requirement is to write a plug-in for these analyses. Because the software is designed as multi-threaded application, it can use the fully performance of current multi-core processors. Above mentioned property can be especially shown off, when a calculation is moved from CPU to GPU (Graphics Processing Units). Current high-end graphic cards have tens to hundreds cores and work with faster memories than CPU. Due to this fact, the simulation performance can raised manifold.