Algorithm 706: DCUTRI: an algorithm for adaptive cubature over a collection of triangles

Berntsen, Jarle; Espelid, Terje O.

doi:10.1145/131766.131772

Cited by 37 publications

(24 citation statements)

References 10 publications

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“…The numerical integration algorithm that we have developed constructs a quadrature rule over each finite element that satisfies a given error tolerance for all the above integrands. Adaptive integration schemes are normally recursive in nature and have a few common ingredients: a quadrature rule that can be applied to the integration domain to provide a local estimate of the integral; a procedure to estimate the local integration error; a strategy to partition the integration domain into smaller divisions of the same shape; and a stopping criterion [98][99][100]. Our quadrature construction algorithm is specifically designed for high accuracy on parallelepipeds.…”

Section: Adaptive Quadraturementioning

confidence: 99%

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

Pask¹,

Sukumar²,

Guney³

et al. 2011

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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...

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Section: Adaptive Quadraturementioning

confidence: 99%

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

Pask¹,

Sukumar²,

Guney³

et al. 2011

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“…Define fo(fo ) (1) lrz(f) = f(x, y)dy dx = f(x, y)dxdy. 2 With f equal to the function given in the first 3 examples (with a, b > 0 in the cases 1.…”

Section: ((Ax + By)/(cx +mentioning

confidence: 99%

On integrating vertex singularities using extrapolation

Espelid

1994

BIT

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Abstract.A new approach to the integration of vertex singularities is described. This approach is based on a non-uniform subdivision of the region of integration and the technique fits well to the subdivision strategy used in many adaptive algorithms. A nice feature with this approach is that it can be used in any dimension and on any region of integration which can be subdivided into subregions of the same form. The strategy can be applied both to vertex singularities and internal point singularities. In the latter case this can be done without an initial subdivision of the region in order to put the singular point in a vertex. It turns out that the technique has excellent numerical stability properties.Subject Classification: AMS(MOS) 65D30, 65B05.

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“…We will discuss how to compute numerical estimates to integration problems of type (1), where the function involved, f , is a product of a homogeneous function f α (x 1 , x 2 , . .…”

Section: Homogeneous Functions: Basic Error Expansionmentioning

confidence: 99%

Integrating Singularities using Non-uniform Subdivision and Extrapolation

Espelid

1993

Numerical Integration IV

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A new approach to the computation of approximations to multidimensional integrals over an n-dimensional hyper-rectangular region, when the integrand is singular, is described. This approach is based on a non-uniform subdivision of the region of integration and the technique fits well to the subdivision strategy used in many adaptive algorithms. The strategy can be applied to vertex singularities, line singularities and more general subregion singularities. The technique turns out to have good numerical stability properties.

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Algorithm 706: DCUTRI: an algorithm for adaptive cubature over a collection of triangles

Cited by 37 publications

References 10 publications

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

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

On integrating vertex singularities using extrapolation

Integrating Singularities using Non-uniform Subdivision and Extrapolation

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