FELIX-2.0: New version of the finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation

doi:10.1016/j.cpc.2017.12.007

Computer Physics Communications

Volume 225, April 2018, Pages 180-191

https://doi.org/10.1016/j.cpc.2017.12.007 Get rights and content

Abstract

The time-dependent generator coordinate method (TDGCM) is a powerful method to study the large amplitude collective motion of quantum many-body systems such as atomic nuclei. Under the Gaussian Overlap Approximation (GOA), the TDGCM leads to a local, time-dependent Schrödinger equation in a multi-dimensional collective space. In this paper, we present the version 2.0 of the code FELIX that solves the collective Schrödinger equation in a finite element basis. This new version features: (i) the ability to solve a generalized TDGCM+GOA equation with a metric term in the collective Hamiltonian, (ii) support for new kinds of finite elements and different types of quadrature to compute the discretized Hamiltonian and overlap matrices, (iii) the possibility to leverage the spectral element scheme, (iv) an explicit Krylov approximation of the time propagator for time integration instead of the implicit Crank–Nicolson method implemented in the first version, (v) an entirely redesigned workflow. We benchmark this release on an analytic problem as well as on realistic two-dimensional calculations of the low-energy fission of ²⁴⁰Pu and ²⁵⁶Fm. Low to moderate numerical precision calculations are most efficiently performed with simplex elements with a degree 2 polynomial basis. Higher precision calculations should instead use the spectral element method with a degree 4 polynomial basis. We emphasize that in a realistic calculation of fission mass distributions of ²⁴⁰Pu, FELIX-2.0 is about 20 times faster than its previous release (within a numerical precision of a few percents).

Program summary

Program Title: FELIX-2.0

Program Files doi: http://dx.doi.org/10.17632/t8b4h9g88r.1

Licensing provisions: GPLv3

Programming language: C++

Journal reference of previous version: Computer Physics Communications 200, 350–363 (2016)

Does the new version supersede the previous version?: Yes

Reasons for the new version: FELIX-2.0 extends the physics capabilities of the previous version, since it includes the option to have a metric term in the collective Schrödinger equation. The computational efficiency of the code is considerably increased thanks to the implementation of several new numerical methods. This version also provides a more flexible and robust workflow to perform calculation of fission fragment distributions more efficiently.

Summary of revisions: The new version includes the ability to use a metric term in the TDGCM+GOA equation. Numerical integration methods (both spatial and time integration) have been updated to more efficient schemes: matrix elements are now computed with Gauss–Legendre or Gauss–Legendre–Lobatto quadratures, the time propagation of the wave packet is computed with the Krylov approximation of the propagator instead of the previous Crank–Nicolson scheme. A new type of finite element based on n-dimensional orthotope is implemented, which enables a spectral element scheme for spatial discretization. Finally, the workflow and inputs/outputs of the code were entirely redesigned to provide more flexibility and be more user-friendly.

Nature of problem: The Gaussian overlap approximation to the time-dependent generator coordinate method [1,2] yields a local, time-dependent Schrödinger equation in a small, multi-dimensional collective space. Its solution provides the time-evolution of the collective probability amplitude. For applications to nuclear fission, scission configurations are defined by a hyper-surface in this collective space referred to as the frontier. Distributions of nuclear observables such as charge or mass distributions of the fission fragments are extracted from the probability for the system to pass through any given section of the frontier. This probability is computed by first solving the evolution equation up to times greater than $1 0^{- 20}$ s and then integrating the flux of probability through the frontier over the entire timerange.

Solution method: FELIX-2.0 solves the time-dependent Schrödinger equation by first discretizing the N-dimensional collective space with the continuous Galerkin Finite Element Method. This produces a large set of coupled, time-dependent Schrödinger equations characterized by the sparse overlap and Hamiltonian matrices. The solution is evolved in small time steps by applying an explicit and unitary propagator built as a Krylov approximation [3] of the exponential of the Hamiltonian.

Additional comments including Restrictions and Unusual features: Although the implementation of the program gives it the ability to solve the TDGCM+GOA equation in a generic N-dimensional collective space, it has only been tested on 1-, 2- and 3-dimensional meshes.

[1] J.J. Griffin, J. A. Wheeler, Collective Motions in Nuclei by the Method of Generator Coordinates, Phys. Rev. C 108, 311-327 (1975)

[2] P. G. Reinhard, K. Goeke, The Generator-Coordinate Method and Quantized Collective Motion in Nuclear Systems, Rep. Prog. Phys. 50, 1-64 (1987)

[3] Y. Saad, Iterative Methods for Sparse Linear Systems: Second Edition, SIAM, 2003

Introduction

A completely microscopic description of the fission process is a major challenge for nuclear theory. Fission is a time-dependent, non-equilibrium, quantum many-body problem where more than 200 nucleons interact over large time scales $τ$ (typically $τ > 1 0^{- 20}$ s) in a coherent way to break the system into two or more fragments. Of particular interest to applications of fission in either science (nucleosynthesis, superheavy science) or applications (energy production) are the properties of the fission fragments, in particular their charge and mass distributions. One of the most effective approaches to computing such observables in a quantum-mechanical framework is the Gaussian overlap approximation (GOA) of the time-dependent generator coordinate method (TDGCM) [[1], [2]]. This approach relies on the energy density functional (EDF) formalism and reduces the dynamics of the complete system to a local, collective Schrödinger-like equation involving only a few arbitrary degrees of freedom referred to as collective variables. This approach was originally introduced for the description of low-energy neutron-induced fission in the 1980s [[3], [4], [5]] but was not applied on a large scale because of computational limitations at the time. Thanks to progress in computing capabilities, the TDGCM+GOA approach to fission has been recently applied to predictions of fission fragment distributions in 2-dimensional collective space [[6], [7], [8], [9], [10]]. These studies, together with others [11], have highlighted the need to take into account additional degrees of freedom in order to increase the accuracy of calculations. The first version of FELIX was a first step towards the goal of providing the community with a numerical tool capable of solving the TDGCM+GOA equation for an arbitrary number of collective variables [12]. In this paper, we present a major upgrade of FELIX which contains both new physics features and much improved numerical performances.

In Section 2, we review the new and upgraded features of FELIX-2.0 compared to the previous version. The convergence properties of the different numerical methods implemented are benchmarked in Section 3. We also present in the same section a comparison of the performances between FELIX-2.0 and FELIX-1.0. The Section 4 is finally devoted to the practical installation and usage of the package.

Section snippets

The TDGCM+GOA with a metric

The time-dependent extension to the generator coordinate method provides an appropriate formalism to describe the slow and large amplitude motion of nuclei [[2], [13]]. In this approach, we assume that the many-body state $| Ψ (t) 〉$ of the fissioning system takes the generic form $| Ψ (t) 〉 = \int_{q} f (q, t) | Φ_{q} 〉 d q .$ The set ${| Φ_{q} 〉}_{q}$ is made of known many-body states parametrized by a vector of continuous variables $q \equiv (q_{1}, \dots, q_{N})$ . Each of these $q_{i}$ is a collective variable and must be chosen based on the physics of the

Benchmarks

In this section, we investigate the convergence properties and overall performance of the numerical methods implemented in FELIX-2.0. This analysis is performed based on the results of three benchmark runs. The first one simulates an oscillating system in a 2-dimensional harmonic oscillator and its exact solution is known analytically. The other two consist in realistic calculations of the low-energy fission yields of $^{256}$ Fm and $^{240}$ Pu in a $(Q_{20}, Q_{30})$ collective space.

Usage of FELIX-2.0

The package is composed of the following directories and files:

$•$
README.md, AUTHORS, LICENSE: contains the basic instructions about how to build, use and distribute this package;
$•$
Makefile: a standard GNU makefile to build the solver, the tools, the tests, and the documentation;
$•$
src/: C++ source files of the TDGCM solver and of the tools;
$•$
tools/: additional C++ and Python source files to handle the inputs and outputs of the TDGCM solver;
$•$
tests/: a unitary test suite for the package as

Acknowledgments

This software is based on pugixml library (http://pugixml.org). Pugixml is Copyright (C) 2006–2015 Arseny Kapoulkine. This work was partly performed under the auspices of the US Department of Energy by the Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Oak Ridge Leadership Computing Facility located in

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W. Younes, D. Gogny, Lawrence Livermore National Laboratory, Fragment Yields Calculated in a Time-Dependent Microscopic...

W. Younes, D. Gogny, Lawrence Livermore National Laboratory, Collective Dissipation from Saddle to Scission in a...

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^☆: This paper and its associated computer program are available via the Computer Physics Communication homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).

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FELIX-2.0: New version of the finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation☆

Abstract

Program summary

Introduction

Section snippets

The TDGCM+GOA with a metric

Benchmarks

Usage of FELIX-2.0

Acknowledgments

Nuclear Physics A

Comput. Phys. Comm.

Comput. Phys. Comm.

Comput. Phys. commun.

Nuclear Phys. A

Phys. Lett. B

Phys. Rev.

Rep. Progr. Phys.

Phys. Rev. C