Meshfree implementation for the real‐space electronic‐structure calculation of crystalline solids

Jun, Sukky

doi:10.1002/nme.943

Cited by 10 publications

(7 citation statements)

References 23 publications

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“…Modern partition of unity finite element (PUFE) methods provide an elegant and highly efficient solution to the quantum-mechanical problem. Jun [55] has reported non-self-consistent results using a meshfree basis to solve the Schrödinger equation in periodic solids, whereas Chen and coworkers [56] and Suryanarayana and coworkers [57] have employed meshfree partitionof-unity basis functions in full self-consistent calculations. Finite element basis functions as the partition-of-unity, however, offer significant advantages: locality is retained to the maximum extent possible, which facilitates parallel implementation, variational convergence is ensured by the minmax theorem [25], numerical integration errors can be strictly controlled, and Dirichlet as well as periodic boundary conditions are readily imposed.…”

Section: Introductionmentioning

confidence: 99%

Partition of unity finite element method for quantum mechanical materials calculations

Pask

Sukumar

2017

Extreme Mechanics Letters

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The current state of the art for large-scale quantum-mechanical simulations is the planewave (PW) pseudopotential method, as implemented in codes such as VASP, ABINIT, and many others. However, since the PW method uses a global Fourier basis, with strictly uniform resolution at all points in space, it suffers from substantial inefficiencies in calculations involving atoms with localized states, such as first-row and transition-metal atoms, and requires significant nonlocal communications, which limit parallel efficiency. Real-space methods such as finite-differences (FD) and finite-elements (FE) have partially addressed both resolution and parallel-communications issues but have been plagued by one key disadvantage relative to PW: excessive number of degrees of freedom (basis functions) needed to achieve the required accuracies. In this paper, we present a real-space partition of unity finite element (PUFE) method to solve the Kohn-Sham equations of density functional theory. In the PUFE method, we build the known atomic physics into the solution process using partition-of-unity enrichment techniques in finite element analysis. The method developed herein is completely general, applicable to metals and insulators alike, and particularly efficient for deep, localized potentials, as occur in calculations at extreme conditions of pressure and temperature. Full self-consistent Kohn-Sham calculations are presented for LiH, involving light atoms, and CeAl, involving heavy atoms with large numbers of atomic-orbital enrichments. We find that the new PUFE approach attains the required accuracies with substantially fewer degrees of freedom, typically by an order of magnitude or more, than the PW method. We compute the equation of state of LiH and show that the computed lattice constant and bulk modulus are in excellent agreement with reference PW results, while requiring an order of magnitude fewer degrees of freedom to obtain.

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Section: Introductionmentioning

confidence: 99%

Partition of unity finite element method for quantum mechanical materials calculations

Pask

Sukumar

2017

Extreme Mechanics Letters

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“…Now, for the trial and test functions to be admissible in the weak formulation (9), they must span a subspace of the Bloch-periodic space V, i.e., satisfy the Bloch-periodic boundary condition (3b). Referring to (16), this can be accomplished in two distinct ways:…”

Section: Schrödinger Equationmentioning

confidence: 99%

Classical and enriched finite element formulations for Bloch‐periodic boundary conditions

Sukumar

Pask

2008

Numerical Meth Engineering

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SUMMARYIn this paper, classical and enriched finite element formulations to impose Bloch-periodic boundary conditions are proposed. Bloch-periodic boundary conditions arise in the description of wave-like phenomena in periodic media. We consider the quantum-mechanical problem in a crystalline solid, and derive the weak formulation and matrix equations for the Schrödinger and Poisson equations in a parallelepiped unit cell under Bloch-periodic and periodic boundary conditions, respectively.For such second-order problems, these conditions consist of value-and derivative-periodic parts. The value-periodic part is enforced as an essential boundary condition by construction of a value-periodic basis, whereas the derivative-periodic part is enforced as a natural boundary condition in the weak formulation. We show that the resulting matrix equations can be obtained by suitably specifying the connectivity of element matrices in the assembly of the global matrices or by modifying the Neumann * Correspondence to: N. Sukumar, Department of Civil and Environmental Engineering, University of California, One Shields Avenue, Davis, CA 95616. E-mail: nsukumar@ucdavis.edu

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“…There have also been implementations of the method with £nite elements [127,177], and even meshless methods [79]. Finite element discretization methods may be successful in reducing the total number of variables involved, but they are far more dif£cult to implement.…”

Section: Finite Differences In Real Spacementioning

confidence: 99%

Numerical Methods for Electronic Structure Calculations of Materials

Saad¹,

Chelikowsky²,

Shontz³

2010

SIAM Rev.

265

166

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The goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems. The paper is intended for a diverse scienti£c computing audience. For this reason, we assume the reader does not have an extensive background in the related physics. Our overview focuses on the nature of the numerical problems to be solved, their origin, and on the methods used to solve the resulting linear algebra or nonlinear optimization problems. It is common knowledge that the behavior of matter at the nanoscale is, in principle, entirely determined by the Schrödinger equation. In practice, this equation in its original form is not tractable. Successful, but approximate, versions of this equation, which allow one to study nontrivial systems, took about £ve or six decades to develop. In particular, the last two decades saw a ¤urry of activity in developing effective software. One of the main practical variants of the Schrödinger equation is based on what is referred to as Density Functional Theory (DFT). The combination of DFT with pseudopotentials allows one to obtain in an ef£cient way the ground state con£guration for many materials. This article will emphasize pseudopotentialdensity functional theory, but other techniques will be discussed as well.

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Meshfree implementation for the real‐space electronic‐structure calculation of crystalline solids

Cited by 10 publications

References 23 publications

Partition of unity finite element method for quantum mechanical materials calculations

Partition of unity finite element method for quantum mechanical materials calculations

Classical and enriched finite element formulations for Bloch‐periodic boundary conditions

Numerical Methods for Electronic Structure Calculations of Materials

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