First principles (ab initio) quantum mechanical simulations based on Kohn-Sham 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 state-of-the-art approaches for DFT calculations extend to more complex problems by adding more grid points (finite-difference methods) or basis functions (planewave and finite-element methods) without regard to the nature of the complexity, leading to substantial inefficiencies in the treatment of highly inhomogeneous systems such as those involving first-row, transition-metal or actinide atoms. This work, initiated with John Pask at LLNL in 2006, attempts to overcome this basic limitation of current approaches by employing partition-of-unity enrichment techniques in finite-element 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

• O. Čertík, J. E. Pask, I. Fernando, R. Goswami, N. Sukumar, L. A. Collins, G. Manzini and J. Vackář (2023), "High-Order Finite Element Method for Atomic Structure Calculations," Computer Physics Communications, Vol. 297, Article 109051.
• C. Albrecht, C. Klaar, J. E. Pask, M. A. Schweitzer, N. Sukumar and A. Ziegenhagel (2018), "Orbital-enriched Flat-top 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 Hermitian-Definite 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 Ill-Conditioned 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 n-Dimensional 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 All-Electron 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), "Partition-Of-Unity Finite-Element Method for Large Scale Quantum Molecular Dynamics on Massively Parallel Computational Platforms," Department of Energy LDRD Grant 08-ERD-052, Report No. LLNL-TR-470692.
• N. Sukumar and J. E. Pask (2009), "Classical and Enriched Finite Element Formulations for Bloch-Periodic 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 Partition-of-Unity Methods for Wave Problems," Invited Lecture, Department of Civil and Environmental Engineering, Duke University, Durham, NC, April 2018.

• "Orbital-Enriched 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.

• "Orbital-Enriched 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 Kohn-Sham 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.

• "Partition-of-Unity Enriched Finite Elements to Solve the Hartree-Fock 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 All-Electron Coulomb Problem in Solids (with J. E. Pask and S. E. Mousavi)," Poster Presentation, CECAM - Jülich Summer School 2011 on Fast Methods for Long-Range Interactions in Complex Systems, Jülich, Germany, September 2011.

• "A New Real-Space Formulation for the All-Electron 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.

• "Partition-of-Unity Finite Elements for Large, Accurate Quantum-Mechanical 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 Quantum-Mechanical 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 All-Electron 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 Kohn-Sham 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 Electronic-Structure 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 Real-Space Finite Element Method to Solve the Kohn-Sham 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 Electronic-Structure 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 Large-Scale 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 Real-Space 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), "Real-Space Mesh Techniques in Density-Functional 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.

Electronic-Structure Methods

• FEM for Large-Scale Ab Initio Electronic-Structure Calculations (LLNL)

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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