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

Sukumar, N.; Pask, John E.

doi:10.1002/nme.2457

Cited by 87 publications

(90 citation statements)

References 23 publications

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“…The term 1/r α containing the singularity is used as the weight function in these quadrature rules similar to the quadratures presented by Haegemans [60]. Even though this study targeted integrands with vertex singularities in PUFE methods, the stringent demands on accuracy in non-singular PUFE applications such as acoustics [61] and Schrödinger and Poisson solutions in quantum mechanics [62] reinforces the need and importance of developing accurate and efficient quadrature rules for enriched finite element methods.…”

Section: Discussionmentioning

confidence: 99%

Generalized Duffy transformation for integrating vertex singularities

Mousavi

Sukumar

2009

Comput Mech

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For an integrand with a 1/r vertex singularity, the Duffy transformation from a triangle (pyramid) to a square (cube) provides an accurate and efficient technique to evaluate the integral. In this paper, we generalize the Duffy transformation to power singularities of the form p(x)/r α , where p is a trivariate polynomial and α > 0 is the strength of the singularity. We use the map (u, v, w) → (x, y, z) : x = u β , y = xv, z = xw, and judiciously choose β to accurately estimate the integral. For α = 1, the Duffy transformation (β = 1) is optimal, whereas if α = 1, we show that there are other values of β that prove to be substantially better. Numerical tests in two and three dimensions are presented that reveal the improved accuracy of the new transformation. Higher-order partition of unity finite element solutions for the Laplace equation with a derivative singularity at a re-entrant corner are presented to demonstrate the benefits of using the generalized Duffy transformation.

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Section: Discussionmentioning

confidence: 99%

Generalized Duffy transformation for integrating vertex singularities

Mousavi

Sukumar

2009

Comput Mech

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“…Finally, we briefly compare the proposed enriched finite element method with the other existing methods which in a similar spirit seek to augment the classical finite element basis with other basis functions that efficiently capture the known physics in regions of interest. One such approach is that of partition-of-unity finite element method (PUFEM) 80,81 , wherein a typical discretization can be defined as 82,83 Another such approach is that of gaussian finite element mixed basis 84 , wherein a given choice of gaussian basis is used to the augment the classical finite element basis instead of atomic solutions to the Kohn-Sham problem, as used in the present work. We note that compared to the gaussian basis the atomic solutions provide a more natural choice for augmenting functions and also provide for better control over the conditioning of the basis through the use of smooth cutoff functions on the radial part of the atomic orbitals.…”

Section: Inverse Of Overlap Matrixmentioning

confidence: 99%

Large-scale all-electron density functional theory calculations using an enriched finite-element basis

Kanungo

Gavini

2017

Phys. Rev. B

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We present a computationally efficient approach to perform large-scale all-electron density functional theory calculations by enriching the classical finite element basis with compactly supported atom-centered numerical basis functions that are constructed from the solution of the Kohn-Sham (KS) problem for single atoms. We term these numerical basis functions as enrichment functions, and the resultant basis as the enriched finite element basis. The compact support for the enrichment functions is obtained by using smooth cutoff functions, which enhances the conditioning and maintains the locality of the enriched finite element basis. The integrals involved in the evaluation of the discrete KS Hamiltonian and overlap matrix in the enriched finite element basis are computed using an adaptive quadrature grid that is constructed based on the characteristics of enrichment functions. Further, we propose an efficient scheme to invert the overlap matrix by using a block-wise matrix inversion in conjunction with special reduced-order quadrature rules, which is required to transform the discrete Kohn-Sham problem to a standard eigenvalue problem. Finally, we solve the resulting standard eigenvalue problem, in each self-consistent field iteration, by using a Chebyshev polynomial based filtering technique to compute the relevant eigenspectrum. We demonstrate the accuracy, efficiency and parallel scalability of the proposed method on semiconducting and heavymetallic systems of various sizes, with the largest system containing 8694 electrons. We obtain accuracies in the ground-state energies that are ∼ 1 mHa with reference ground-state energies employing classical finite element as well as gaussian basis sets. Using the proposed formulation based on enriched finite element basis, for accuracies commensurate with chemical accuracy, we observe a staggering 50−300 fold reduction in the overall computational time when compared to classical finite element basis. Further, we find a significant outperformance by the enriched finite element basis when compared to the gaussian basis for the modest system sizes where we obtained convergence with gaussian basis. We also observe good parallel scalability of the numerical implementation up to 384 processors for a representative benchmark system comprising of 280-atom silicon nano-cluster.

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“…Such kind of periodic boundary conditions are referred to as Bloch periodic boundary conditions [10]. The expressions of Bloch's periodic boundary conditions for electric and magnetic field components take following form:…”

Section: Set Up Periodic Boundary Conditionsmentioning

confidence: 99%