ENRICHED FINITE ELEMENT METHODS IN QUANTUMMECHANICAL (AB INITIO)
MATERIALS CALCULATIONS
Doublewell potential and enrichment functions for
pseudoatomic wavefunctions
(Researchwork
supported by DOE, NSFDMS and UC Lab Fees Research Program, 2007–2011)
First principles (ab initio) quantum mechanical simulations based on
KohnSham
density functional theory (DFT) are a vital component of modern
materials research. The parameter free, quantum mechanical nature of the
theory facilitates both fundamental understanding and robust predictions across
the gamut of materials systems, from metallic actinides to insulating organics.
However, the solution of the equations of DFT (coupled Schrodinger and Poisson
equations) is a formidable task and this has severely limited the range of
materials systems that can be investigated by such rigorous, quantum mechanical
means. Current stateoftheart approaches for DFT calculations extend to more
complex problems by adding more grid points (finitedifference methods) or
basis functions (planewave and finiteelement methods) without regard to the
nature of the complexity, leading to substantial inefficiencies in the
treatment of highly inhomogeneous systems such as those involving firstrow,
transitionmetal or actinide atoms. This work, initiated with
John
Pask at LLNL in 2006, attempts to
overcome
this basic limitation of current approaches by employing partitionofunity
enrichment techniques in finiteelement analysis to build the known atomic physics into
the solution process, thus substantially reducing the degrees of freedom
required and increasing the size of problems that can be addressed.
We are collaborating with
Zhaojun Bai (project) on the development of
efficient and scalable finite element solvers for generalized eigenproblems
in quantum mechanics.
Reports and Publications
 C. Albrecht, C. Klaar, J. E. Pask,
M. A. Schweitzer, N. Sukumar and
A. Ziegenhagel (2018), "Orbitalenriched Flattop
Partition of Unity Method for the Schrödinger Eigenproblem,"
Computer Methods in Applied Mechanic and Engineering,
Vol. 342, pp. 224–239.
Available at
arXiv:1801.09596
 Y. Cai, Z. Bai, J. E. Pask and N. Sukumar (2018),
"Convergence Analysis of a Locally Accelerated Preconditioned Steepest
Descent Method for HermitianDefinite Generalized Eigenvalue Problems,"
Journal of Computational Mathematics,
Vol. 36, Number 5, pp. 739–760.
 J. E. Pask and N. Sukumar
(2017), "Partition of Unity Finite Element Method for Quantum
Mechanical Materials Calculations,"
Extreme Mechanics Letters,
Vol. 11, pp. 8–17.
Available at
arXiv:1611.00731
 Y. Cai, Z. Bai, J. E. Pask and N. Sukumar
(2013), "Hybrid Preconditioning for Iterative Diagonalization of IllConditioned Generalized Eigenvalue Problems in Electronic Structure
Calculations," Journal of Computational Physics,
Vol. 255, pp. 16–30.
Available at arXiv:1308.2445
 S. E. Mousavi, J. E. Pask and N. Sukumar
(2012), "Efficient Adaptive Integration of Functions with Sharp
Gradients and Cusps in nDimensional Parallelepipeds,"
International Journal for Numerical Methods in Engineering,
Vol. 91, Number 4, pp. 343–357.
Available at
arXiv:1202.5341
 J. E. Pask, N. Sukumar and S. E. Mousavi
(2012), "Linear Scaling Solution of the AllElectron Coulomb Problem
in Solids,"
International Journal for Multiscale Computational Engineering,
Vol. 10, Number 1, pp. 83–99.
Available at
arXiv:1004.1765
 J. E. Pask, N. Sukumar, M. Guney and W. Hu
(March 2011), "PartitionOfUnity FiniteElement Method for Large Scale
Quantum Molecular Dynamics on Massively Parallel Computational Platforms,"
Department of Energy LDRD Grant 08ERD052, Report No.
LLNLTR470692.
 N. Sukumar and J. E. Pask
(2009), "Classical and Enriched Finite Element Formulations for
BlochPeriodic Boundary Conditions,"
International Journal for Numerical Methods in Engineering,
Vol. 77, Number 8, pp. 1121–1138.
[Errata: a = 5.7 bohr for the harmonic oscillator
problem in Sec. 4.2.1; a = 4 bohr and σ = 1.5
for the periodic Gaussian problem in Sec. 4.2.2]
 N. Sukumar (February 2009),
"Finite Element Methods in Quantum Mechanics"
in iMechanica Web Blog: Journal Club Theme of February 2009.
Presentations
 "Finite Element and Enriched PartitionofUnity
Methods for Wave Problems," Invited Lecture,
Department of Civil and
Environmental Engineering,
Duke University,
Durham, NC, April 2018.
 "OrbitalEnriched Partition of Unity Finite
Element Method for Ab Initio Density Functional
Calculations," Invited Seminar,
Department of Mechanical Engineering,
UM  Shanghai Jiao Tong
University Joint Institute,
Shanghai, China, August 7, 2017.
 "Computational Methods for Nonlinear Eigenvalue
Problems (with Zhaojun Bai),"
International
Summer School on Scientific Computing,
State Key
Laboratory of Scientific and Engineering Computing,
Chinese Academy of Sciences, Beijing, China, July 24  August 5, 2017.
 "OrbitalEnriched Partition of Unity Finite Element Method for Ab Initio Density Functional Calculations," Invited Seminar,
PRISM/CEE Seminar,
Princeton University,
Princeton, NJ, April 2017.
 "Partition of Unity Finite Element Method for Quantum Mechanical Materials Calculations," Invited Seminar,
Department of Mechanical
Engineering and Materials Science,
Duke University,
Durham, NC, February 2017.
 "Partition of Unity Finite Element Method for Quantum Mechanical Materials Calculations," Invited Seminar,
Department of Mechanical
Engineering, UC Merced,
Merced, CA, October 2016.
 "Partition of Unity Finite Element Method for KohnSham Density Functional Calculations (with J. E. Pask),"
Invited Lecture,
USACM Workshop on Recent Advances in Computational Methods for Nanoscale Phenomena,
Ann Arbor, MI, August 2016.
 "Efficient Numerical Integration Schemes for Singularities, Sharp Gradients and Cusps: Applications in Fracture and Quantum Mechanics
(with
J. E. Pask and S. E. Mousavi),"
Symposium on Meshfree and Extended/Generalized Finite Element Methods,
PACAM XII,
Port of Spain, Trinidad, January 2012.
 "PartitionofUnity Enriched Finite Elements to Solve the HartreeFock Equations
(with
T. J. Martínez),"
Symposium on Current Multiscale Computations and Algorithms,
48th Annual Technical Conference of Society of
Engineering Sciences,
Evanston, IL, October 2011.
 "Enriched Finite Element Solution for the AllElectron Coulomb
Problem in Solids (with
J. E. Pask and S. E. Mousavi),"
Poster Presentation,
CECAM 
Jülich
Summer School 2011 on Fast Methods for LongRange Interactions in
Complex Systems, Jülich, Germany, September 2011.
 "A New RealSpace Formulation for the AllElectron Coulomb Problem in Crystalline
Solids (with
J. E. Pask and S. E. Mousavi),"
Symposium on Recent Developments in Nanoscale Modeling of Materials,
11th U.S. National Congress on Computational Mechanics, Minneapolis, MN, July 2011.
 "PartitionofUnity Finite Elements for Large, Accurate QuantumMechanical Materials
Calculations," Plenary Lecture,
ECCOMAS Thematic Conference on the Extended Finite Element Method (XFEM 2011),
Cardiff, UK, July 1, 2011.
 "Partition of Unity Finite Elements for QuantumMechanical
Calculations in Condensed Matter: Have Planewaves Finally Met Their
Match,"
Invited Lecture,
Department of Mechanical Engineering,
Mechanics of Computation Seminar,
Stanford University,
January 2010.
 "Linear Scaling Finite Element Solution for the AllElectron
Coulomb Problem in Solids,"
Invited Lecture,
Institute for Numerical Simulation, University of Bonn,
Bonn, Germany, October 1, 2009.
 "Partition of Unity Finite Element Method to Solve the KohnSham
Equations of Density Functional Theory (with
J. E. Pask),"
Keynote Lecture, Symposium on Advances in Multiscale and Multiphysics
Methods: From Quantum to Continuum,
8th
World Congress on Computational Mechanics WCCM8, Venice, Italy, July 2008.
 "Linear Scaling Enriched Finite Element Solution of the Coulomb
Problem in Solids,"
Invited Lecture, Structural Mechanics Seminar,
Georgia Institute of
Technology, Atlanta, GA, March 2008.
 "Partition of Unity Finite Elements for ElectronicStructure
Calculations in Molecules and Crystalline Solids,"
Invited Lecture,
Department of
Mechanical Engineering and Applied Mechanics,
University of
Pennsylvania, Philadelphia, PA, December 7, 2007.
 "A New RealSpace Finite Element Method to Solve the KohnSham
Equations of Density Functional Theory,"
Invited Lecture, Numerical Analysis and Scientific Computing
Seminar,
Courant Institute of
Mathematical Sciences,
New York University,
New York, NY, November 16, 2007.
 "Partition of Unity Finite Elements for ElectronicStructure
Calculations in Molecules and Crystalline Solids,"
Invited Lecture,
Joint Mechanical Engineering/Civil Engineering Seminar,
Columbia
University, New York, NY, November 9, 2007.
 "Partition of Unity Finite Elements in Quantum Mechanical
Materials Calculations,"
Invited Lecture, Seminar in Computational Mechanics Organized by
Laboratori de Calcul Numeric
and CIMNE,
Universitat Politecnica De Catalunya, Barcelona, Spain, September 15, 2006.
 "Partition of Unity Finite Element Method for LargeScale
Electronic Structure Calculations (with
J. E. Pask)," Seventh World Congress on Computational Mechanics,
Symposium on Enriched Finite Element Methods for Evolving Discontinuities
and Interfaces, Los Angeles, CA, July 2006.
Review Articles
 T. Torsti et al. (2006), "Three RealSpace Discretization Techniques
in Electronic Structure Calculations,"Physica Status Solidi. B, Basic
Research,
Vol. 243, Number 5, pp. 1016–1053.
 J. E. Pask and P. A. Sterne (2005), "Finite Element Methods in
Ab Initio Electronic Structure Calculations,"
Modelling and Simulation in Materials Science and Engineering,
Vol. 13, pp. R71–R96.
 T. L. Beck (2000), "RealSpace Mesh Techniques in DensityFunctional
Theory,"
Reviews of Modern Physics,
Vol. 72, Number 4, pp. 1041–1080.
 T. A. Arias (1999), "Multiresolution Analysis of Electronic Structure:
Semicardinal and Wavelet Bases,"
Reviews of Modern Physics,
Vol. 71, Number 1, pp. 267–311.
ElectronicStructure Methods
Disclaimer: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation
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