Partition-of-unity finite-element method for large scale quantum molecular dynamics on massively parallel computational platforms

Abstract

Over the course of the past two decades, quantum mechanical calculations have emerged as a key component of modern materials research. However, the solution of the required quantum mechanical equations is a formidable task and this has severely limited the range of materials systems which can be investigated by such accurate, quantum mechanical means. The current state of the art for large-scale quantum simulations is the planewave (PW) method, as implemented in now ubiquitous VASP, ABINIT, and QBox codes, among many others. However, since the PW method uses a global Fourier basis, with strictly uniform resolution at all points in space, and in which every basis function overlaps every other at every point, it suffers from substantial inefficiencies in calculations involving atoms with localized states, such as first-row and transition-metal atoms, and requires substantial nonlocal communications in parallel implementations, placing critical limits on scalability. In recent years, real-space methods such as finite-differences (FD) and finite-elements (FE) have been developed to address these deficiencies by reformulating the required quantum mechanical equations in a strictly local representation. However, while addressing both resolution and parallel-communications problems, such local real-space approaches have been plagued by one key disadvantage relative to planewaves: excessive degrees of freedom (grid points, basis functions) needed to achieve the required accuracies. And so, despite critical limitations, the PW method remains the standard today.In this work, we show for the first time that this key remaining disadvantage of real-space methods can in fact be overcome: by building known atomic physics into the solution process using modern partition-of-unity (PU) techniques in finite element analysis. Indeed, our results show order-of-magnitude reductions in basis size relative to state-of-the-art planewave based methods. The method developed here is completely general, applicable to any crystal symmetry and to both metals and insulators alike. We have developed and implemented a full self-consistent Kohn-Sham method, including both total energies and forces for molecular dynamics, and developed a full MPI parallel implementation for large-scale calculations. We have applied the method to the gamut of physical systems, from simple insulating systems with light atoms to complex d-and f-electron systems, requiring large numbers of atomic-orbital enrichments. In every case, the new PU FE method attained the required accuracies with substantially fewer degrees of freedom, typically by an order of magnitude or more, than the current state-of-the-art PW method. Finally, our initial MPI implementation has shown excellent parallel scaling of the most time-critical parts of the code up to 1728 processors, with clear indications of what will be required to achieve comparable scaling for the rest.Having shown that the key remaining disadvantage of real-space methods can in fact be overcome, the work has attrac...

Real time time-dependent density functional theory using higher order finite-element methods

“…However, this can be alleviated by augmenting the finite-element basis with numerical atom-centered basis. This idea has been successfully used for ground-state DFT 57,[80][81][82] , and its extension to RT-TDDFT is currently being investigated. Further, assessing the transferability of pseudopotentials for electron dynamics, enabled by the unified treatment of all-electron and pseudoptential calculations, is another interesting direction for future investigation.…”

“…VII A 1). Moreover, one can mitigate the need of a refined mesh for the all-electron calculation by using an enriched finite-element basis, wherein the standard (classical) finite-element basis are augmented with numerical atom-centered basis 57,[80][81][82] . This idea has successfully attained 100 − 300x speedup over the standard (classical) finite-elements for ground-state DFT calculations 57 , and can be extended to RT-TDDFT to further the capabilities of finite-elements.…”

Orbital-enriched flat-top partition of unity method for the Schrödinger eigenproblem

“…The second drawback of the PUFEM identified in [5] is related to the fact that the Galerkin discretization of (4) yields a Hermitian generalized eigenproblem (9) where H and S are in general sparse but not diagonal matrices, which renders the solution of (9) more challenging since the computation of S −1 is implicitly required, i.e., one effectively needs to solve the standard eigenproblem S −1 Hũ = εũ.…”

Convergence study of the h-adaptive PUM and the hp-adaptive FEM applied to eigenvalue problems in quantum mechanics

“…Application of the PUM to the above problems holds a significant promise to improve on accuracy of a standard (nonenriched) FE approximation. The corresponding numerical evidence can be found in [49,53], where convergence studies for PUM solutions obtained on uniformly refined meshes are performed.…”