This work proposes a numerically efficient implementation of sparse parallel vectors within the open-source finite element library deal.II. The main algorithmic ingredient is the matrix-free evaluation of the Hamiltonian operator by cell-wise quadrature. Based on an a-priori chosen support for each vector, we develop algorithms and data structures to perform (i) matrix-free sparse matrix multivector products (SpMM), (ii) the projection of an operator onto a sparse sub-space (inner products), and (iii) post-multiplication of a sparse multivector with a square matrix. The node-level performance is analyzed using a roofline model. Our matrix-free implementation of finite element operators with sparse multivectors achieves a performance of 157 GFlop/s on an Intel Cascade Lake processor with 20 cores. Strong and weak scaling results are reported for a representative benchmark problem using quadratic and quartic finite element base},
author = {Davydov, Denis and Kronbichler, Martin},
doi = {10.1145/3399736},
faupublication = {yes},
journal = {ACM Transactions on Parallel Computing},
pages = {1-30},
peerreviewed = {Yes},
title = {{Algorithms} and {Data} {Structures} for {Matrix}-{Free} {Finite} {Element} {Operators} with {MPI}-{Parallel} {Sparse} {Multi}-{Vectors}},
volume = {7},
year = {2020}
}
@article{faucris.236084656,
abstract = {This work investigates matrix-free algorithms for problems in quasi-static finite-strain hyperelasticity. Iterative solvers with matrix-free operator evaluation have emerged as an attractive alternative to sparse matrices in the fluid dynamics and wave propagation communities because they significantly reduce the memory traffic, the limiting factor in classical finite element solvers. Specifically, we study different matrix-free realizations of the finite element tangent operator and determine whether generalized methods of incorporating complex constitutive behavior might be feasible. In order to improve the convergence behavior of iterative solvers, we also propose a method by which to construct level tangent operators and employ them to define a geometric multigrid preconditioner. The performance of the matrix-free operator and the geometric multigrid preconditioner is compared to the matrix-based implementation with an algebraic multigrid (AMG) preconditioner on a single node for a representative numerical example of a heterogeneous hyperelastic material in two and three dimensions. We find that matrix-free methods for finite-strain solid mechanics are very promising, outperforming linear matrix-based schemes by two to five times, and that it is possible to develop numerically efficient implementations that are independent of the hyperelastic constitutive law.},
author = {Davydov, Denis and Pelteret, Jean-Paul and Arndt, Daniel and Kronbichler, Martin and Steinmann, Paul},
doi = {10.1002/nme.6336},
faupublication = {yes},
journal = {International Journal for Numerical Methods in Engineering},
keywords = {adaptive finite-element method; finite-strain; geometric multigrid; hyperelasticity; matrix-free},
note = {CRIS-Team Scopus Importer:2020-03-20},
peerreviewed = {Yes},
title = {{A} matrix-free approach for finite-strain hyperelastic problems using geometric multigrid},
year = {2020}
}
@article{faucris.113956304,
abstract = {In this contribution several staggered schemes used to couple continuum mechanics (CM) and molecular mechanics (MM) are proposed. The described approaches are based on the atomistic-to-continuum correspondence, obtained by spatial averaging in the spirit of Irving and Kirkwood, and Noll. Similarities between this and other concurrent coupling schemes are indicated, thus providing a broad overview of different approaches in the field. The schemes considered here are decomposed into the surface-type (displacement or traction boundary conditions) and the volume-type. The latter restricts the continuum displacement field (and possibly its gradient) in some sense to the atomistic (discrete) displacements using Lagrange multipliers. A large-strain CM formulation incorporating Lagrange multipliers and a strategy to solve the resulting coupled linear system using an iterative solver is presented. Finally, the described coupling methods are numerically examined using two examples: uniaxial deformation and a plate with a hole relaxed under surface tension. Accuracy and convergence rates of each method are reported. It was found that the displacement (surface) coupling scheme and the Lagrangian (volume) scheme based on either discrete displacements or the

1-continuous approximation of the motion. A novel micromorphic approach is presented to accommodate the resulting higher-order gradient contributions arising in this highly-nonlinear and coupled problem within a classical finite element setting. Our formulation accounts for all material and geometric nonlinearities, as well as the coupling between the mechanical, electrical and micromorphic fields. The highly-nonlinear system of governing equations is derived using the Dirichlet principle and approximately solved using the finite element method. A series of numerical examples serve to elucidate the theory and to provide insight into this intriguing effect that underpins or influences many important scientific and technical applications.},
author = {McBride, A. T. and Davydov, Denis and Steinmann, Paul},
doi = {10.1016/j.cma.2020.113320},
faupublication = {yes},
journal = {Computer Methods in Applied Mechanics and Engineering},
keywords = {Finite element method; Flexoelectricity; Micromorphic continua},
note = {CRIS-Team Scopus Importer:2020-08-28},
peerreviewed = {Yes},
title = {{Modelling} the flexoelectric effect in solids: {A} micromorphic approach},
volume = {371},
year = {2020}
}
@inproceedings{faucris.231305877,
address = {LEIDEN},
author = {Mcbride, A. and Davydov, Denis and Steinmann, Paul},
booktitle = {ADVANCES IN ENGINEERING MATERIALS, STRUCTURES AND SYSTEMS: INNOVATIONS, MECHANICS AND APPLICATIONS},
date = {2019-09-02/2019-09-04},
doi = {10.1201/9780429426506-47},
faupublication = {yes},
month = {Jan},
note = {CRIS-Team WoS Importer:2020-01-10},
pages = {278-279},
peerreviewed = {unknown},
publisher = {CRC PRESS-BALKEMA},
title = {{Modelling} the flexoelectric effect in solids},
venue = {Cape Town},
year = {2019}
}
@article{faucris.117533284,
abstract = {

Possible links between discrete and continuum formulations have been discussed in the scientific community since several decades. Different atomistic expressions for the continuum fields (most importantly stresses) were proposed. As an example, it is possible to link discrete models to a continuum formulation based on spatial averaging in the Eulerian configuration followed by probability (statistical) averaging. As an alternative to Eulerian averaging, Lagrangian averaging has been recently proposed. Both approaches allow calculation of the local continuum fields from atomistic simulations.

From the continuum formulation perspective, the behavior of solids at the nanoscale can be captured based on a surface-enhanced continuum (SEC) theory whereby the surface is equipped with its own constitutive structure. A distinct advantage of continuum models over their atomistic counterparts is the increased computational efficiency.

In this contribution we compare atomistic fields obtained from molecular statics (MS) simulations to their counterpart, obtained from numerical approximations to the SEC theory. Bulk elastic parameters for the continuum constitutive model are obtained directly from the atomistic model. A representative numerical simulation of face-centered-cubic (FCC) copper is used to compare the two approaches.

The ability of the continuum formulation enhanced with a surface energy to model size effects, as observed in the atomistic simulations, is shown. The local fields evaluated using both the continuum and discrete approach are in a good agreement. The dependence of the results obtained from the atomistic-to-continuum procedure on both the averaging radius and the proximity of the free surface is studied. Eulerian and Lagrangian averaging approaches are shown to give comparable results for the here considered FCC crystal.

}, author = {Davydov, Denis and Javili, Ali and Steinmann, Paul}, doi = {10.1016/j.commatsci.2012.11.053}, faupublication = {yes}, journal = {Computational Materials Science}, keywords = {Atomistic to continuum; Surface elasticity; Nano-materials; Surface-enhanced continuum theory}, note = {UnivIS-Import:2015-03-09:Pub.2012.tech.FT.FT-TM.onmole}, pages = {510-519}, peerreviewed = {Yes}, title = {{On} molecular static and surface-enhanced continuum modeling of nano-structures}, volume = {69}, year = {2013} } @article{faucris.121722744, abstract = {In this paper, details of an implementation of a numerical code for computing the Kohn--Sham equations are presented and discussed. A fully self-consistent method of solving the quantum many-body problem within the context of density functional theory using a real-space method based on finite element discretisation of realspace is considered. Various numerical issues are explored such as (i) initial mesh motion aimed at co-aligning ions and vertices; (ii) a priori and a posteriori optimization of the mesh based on Kelly's error estimate; (iii) the influence of the quadrature rule and variation of the polynomial degree of interpolation in the finite element discretisation on the resulting total energy. Additionally, (iv) explicit, implicit and Gaussian approaches to treat the ionic potential are compared. A quadrupole expansion is employed to provide boundary conditions for the Poisson problem. To exemplify the soundness of our method, accurate computations are performed for hydrogen, helium, lithium, carbon, oxygen, neon, the hydrogen molecule ion and the carbon-monoxide molecule. Our methods, algorithms and implementation are shown to be stable with respect to convergence of the total energy in a parallel computational environment.}, author = {Davydov, Denis and Young, Toby and Steinmann, Paul}, doi = {10.1002/nme.5140}, faupublication = {yes}, journal = {International Journal for Numerical Methods in Engineering}, pages = {n/a--n/a}, peerreviewed = {Yes}, title = {{On} the adaptive finite element analysis of the {Kohn}-{Sham} equations: {Methods}, algorithms, and implementation.}, year = {2015} } @article{faucris.216837534, abstract = {Although there are numerous formulae for atomic-level fluxes, they are expressed either in terms of a singlet density, resulting from Irving and Kirkwood's statistical mechanics formulation of hydrodynamical equations, or a pair density, proposed in kinetic theories of transport processes. Flux formulae using singlet density have been further developed and widely implemented in molecular dynamics (MD) simulations by either replacing the Dirac delta with a volumetric averaging function or performing a surface average of the flux operators. Pair density-based flux formulae have also been further developed by using spatial-averaging kernels; these formulae, however, have rarely been implemented or used in modern MD. In this work, distributional calculus is used to reformulate the fluxes in momentum and energy transport processes. The formulation results demonstrate that these two types of existing flux formulae are mathematically equivalent when expressed with the Dirac delta. The lasting confusion regarding these two different types of flux formulae from two different formalisms is thus resolved.}, author = {Diaz, Adrian and Davydov, Denis and Chen, Youping}, doi = {10.1098/rspa.2018.0688}, faupublication = {yes}, journal = {Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences}, keywords = {Atomistic flux; Dirac delta distribution; Irving-Kirkwood procedure; Local stress and heat flux; Statistical mechanics formalism; Transient transport fluxes}, note = {CRIS-Team Scopus Importer:2019-05-02}, peerreviewed = {Yes}, title = {{On} the equivalence of the two foundational formulations for atomistic flux in inhomogeneous transport processes}, volume = {475}, year = {2019} } @article{faucris.226684088, abstract = {The simulation of complex engineering structures built from magneto-rheological elastomers is a computationally challenging task. Using the FE2 method, which is based on computational homogenisation, leads to the repetitive solution of micro-scale FE problems, causing excessive computational effort. In this paper, the micro-scale FE problems are replaced by POD reduced models of comparable accuracy. As these models do not deliver the required reductions in computational effort, they are combined with hyper-reduction methods like the Discrete Empirical Interpolation Method (DEIM), Gappy POD, Gauss-Newton Approximated Tensors (GNAT), Empirical Cubature (EC) and Reduced Integration Domain (RID). The goal of this work is the comparison of the aforementioned hyper-reduction techniques focusing on accuracy and robustness. For the application in the FE2 framework, EC and RID are favourable due to their robustness, whereas Gappy POD rendered both the most accurate and efficient reduced models. The well-known DEIM is discarded for this application as it suffers from serious robustness deficiencies.}, author = {Brands, Benjamin and Davydov, Denis and Mergheim, Julia and Steinmann, Paul}, doi = {10.3390/mca24010020}, faupublication = {yes}, journal = {Mathematical and Computational Applications}, note = {CRIS-Team WoS Importer:2019-09-17}, peerreviewed = {Yes}, title = {{Reduced}-{Order} {Modelling} and {Homogenisation} in {Magneto}-{Mechanics}: {A} {Numerical} {Comparison} of {Established} {Hyper}-{Reduction} {Methods}}, volume = {24}, year = {2019} } @article{faucris.113486384, abstract = {This paper is the second part of a series dedicated to reviewing the fundamental link between discrete and continuum formulations, which is established by space averaging followed by probability density averaging. On obtaining the continuum balance laws of mass and linear momentum in the part I, here the balance laws of angular momentum and energy are re-derived from a discrete (atomistic/molecular) description. Different approaches (explicit and implicit) for the consideration of the potential energy are reviewed. Thereby for the explicit approach ambiguous possibilities for the localization of the potential energy are briefly discussed. Thereby we conclude that the explicit approach is preferable from the practical application point of view, however it becomes cumbersome when applied to multi-body interactions systems, whereas the implicit approach has no ambiguity in the localization of the potential energy to each particle and is easily applicable to any multi-body potential. Possible solutions for continuum fluxes (couple stresses, heat flux) are postponed until part III of the series.}, author = {Davydov, Denis and Steinmann, Paul}, doi = {10.1177/108128651349030}, faupublication = {yes}, journal = {Mathematics and Mechanics of Solids}, note = {UnivIS-Import:2015-03-09:Pub.2013.tech.FT.FT-TM.review}, pages = {-}, peerreviewed = {Yes}, title = {{Reviewing} the roots of continuum formulations in molecular systems. {Part} {II}: {Energy} and angular momentum balance equations}, volume = {-}, year = {2013} } @article{faucris.113966864, abstract = {This contribution is the third part in a series devoted to the fundamental link between discrete particle systems and continuum descriptions. The basis for such a link is the postulation of the primary continuum fields such as density and kinetic energy in terms of atomistic quantities using space and probability averaging.

In this part, solutions to the flux quantities (stress, couple stress, and heat flux), which arise in the balance laws of linear and angular momentum, and energy are discussed based on the Noll’s lemma. We show especially that the expression for the stress is not unique. Integrals of all the fluxes over space are derived. It is shown that the integral of both the microscopic Noll–Murdoch and Hardy couple stresses (more precisely their potential part) equates to zero. Space integrals of the Hardy and the Noll–Murdoch Cauchy stress are equal and symmetric even though the local Noll–Murdoch Cauchy stress is not symmetric. Integral expression for the linear momentum flux and the explicit heat flux are compared to the virial pressure and the Green–Kubo expression for the heat flux, respectively.

It is proven that in the case when the Dirac delta distribution is used as kernel for spatial averaging, the Hardy and the Noll–Murdoch solution for all fluxes coincide.

The heat fluxes resulting from both the so-called explicit and implicit approaches are obtained and compared for the localized case. We demonstrate that the spatial averaging of the localized heat flux obtained from the implicit approach does not equate to the expression obtained using a general averaging kernel. In contrast this happens to be true for the linear momentum flux, i.e. the Cauchy stress.

}, author = {Davydov, Denis and Steinmann, Paul}, doi = {10.1177/1081286513516480}, faupublication = {yes}, journal = {Mathematics and Mechanics of Solids}, note = {UnivIS-Import:2015-03-09:Pub.2014.tech.FT.FT-TM.review}, pages = {-}, peerreviewed = {Yes}, title = {{Reviewing} the roots of continuum formulations in molecular systems. {Part} {III}: {Stresses}, {Couple} {Stresses}, {Heat} {Fluxes}}, year = {2014} } @article{faucris.112862464, abstract = {The link between atomistic quantities and continuum fields has been the subject of research for at least half a century. Nevertheless, there are still many open questions and misleading discussions in the literature. Therefore, based on the fundamental principles of classical mechanics and statistical physics we construct the basic framework for the link between the atomistic and continuum worlds. In doing so, considerable attention is paid to the central force decomposition and multi-body potentials, balance of angular momentum for the system of particles and its relationship to the extended third Newton axiom and the difference between the theorem of change of kinetic energy and the energy balance law. A number of general theorems related to the convolution properties of statistically averaged quantities, as well as their rates are also proven. These theorems make the derivation of balance equations far simpler when compared to the approaches used by others. Such theorems also make the link between space–time averaging and space–probability averaging more transparent.

In this contribution the balance laws of mass and linear momentum are derived. The remaining balance laws of angular momentum and energy as well as the particular forms of fluxes, such as the stress, are discussed in the follow-up contributions of this series.

}, author = {Davydov, Denis and Steinmann, Paul}, faupublication = {yes}, journal = {Mathematics and Mechanics of Solids}, note = {UnivIS-Import:2015-03-09:Pub.2012.tech.FT.FT-TM.review}, pages = {-}, peerreviewed = {Yes}, title = {{Reviewing} the roots of continuum formulations in molecular systems. {Part} {I}: {Particle} dynamics, statistical physics, mass and linear momentum balance equations}, year = {2012} } @article{faucris.119174924, abstract = {Size effects in a system composed of a polymer matrix with a single silica nanoparticle are studied using molecular dynamics and surface-enhanced continuum approaches. The dependence of the composites mechanical properties on the nanoparticles radius was examined. Mean values of the elastic moduli obtained using molecular dynamics were found to be lower than those of the polystyrene matrix alone. The surface-enhanced continuum theory produced a satisfactory fit of macroscopic stresses developing during relaxation due to the interface tension and uniaxial deformation. Neither analytical nor finite-element solutions correlated well with the size-effect in elastic moduli predicted by the molecular dynamics simulations.}, author = {Davydov, Denis and Voyiatzis, E and Chatzigeorgiou, Georgios and Liu, S and Steinmann, Paul and Böhm, M and Müller-Plathe, F.}, doi = {10.1080/1539445X.2014.959597}, faupublication = {yes}, journal = {Soft Materials}, keywords = {Molecular dynamics; Nanocomposite; Size effect; Surface energy; Surface-enhanced continuum}, pages = {S142-S151}, peerreviewed = {Yes}, title = {{Size} effects in a silica-polystyrene nanocomposite: {Molecular} dynamics and surface-enhanced continuum approaches}, volume = {12}, year = {2014} } @article{faucris.222695631, abstract = {This paper provides an overview of the new features of the finite element library deal.II, version 9.1.}, author = {Arndt, Daniel and Bangerth, Wolfgang and Clevenger, Thomas C. and Davydov, Denis and Fehling, Marc and Garcia-Sanchez, Daniel and Harper, Graham and Heister, Timo and Heltai, Luca and Kronbichler, Martin and Maguire Kynch, Ross and Maier, Matthias and Pelteret, Jean-Paul and Turcksin, Bruno and Wells, David}, doi = {10.1515/jnma-2019-0064}, faupublication = {yes}, journal = {Journal of Numerical Mathematics}, note = {CRIS-Team Scopus Importer:2019-07-19}, peerreviewed = {Yes}, title = {{The} deal.{II} {Library}, {Version} 9.1}, year = {2019} } @article{faucris.108859344, author = {Bangerth, Wolfgang and Davydov, Denis and Heister, Timo and Heltai, Luca and Kanschat, Guido and Kronbichler, Martin and Maier, Matthias and Turcksin, Bruno and Wells, David}, doi = {10.1515/jnma-2016-1045}, faupublication = {yes}, journal = {Journal of Numerical Mathematics}, pages = {135-141}, peerreviewed = {Yes}, title = {{The} \texttt{deal.{II} {Library}, {Version} 8.4}, volume = {24}, year = {2016} } @article{faucris.119685544, author = {Arndt, Daniel and Bangerth, Wolfgang and Davydov, Denis and Heister, Timo and Heltai, Luca and Kronbichler, Martin and Maier, Matthias and Pelteret, Jean-Paul and Turcksin, Bruno and Wells, David}, doi = {10.1515/jnma-2017-0058}, faupublication = {yes}, journal = {Journal of Numerical Mathematics}, pages = {137--145}, peerreviewed = {Yes}, title = {{The} \texttt{deal.{II} {Library}, {Version} 8.5}, volume = {25}, year = {2017} } @inproceedings{faucris.120169984, author = {Pelteret, Jean-Paul and Saxena, Prashant and Walter, Bastian and Davydov, Denis and Steinmann, Paul}, booktitle = {85th Annual Meeting of the International Association of Applied Mathematics and Mechanics}, date = {2014-03-10/2014-03-14}, faupublication = {yes}, peerreviewed = {unknown}, title = {{Towards} a computational model of magneto-sensitive polymers}, venue = {Erlangen}, year = {2014} } @inproceedings{faucris.109958904, author = {Pelteret, Jean-Paul and Davydov, Denis and Javili, Ali and Walter, Bastian and Steinmann, Paul}, booktitle = {5th International Conference on Computational Methods for Coupled Problems in Science and Engineering}, date = {2013-06-17/2013-06-19}, faupublication = {yes}, peerreviewed = {unknown}, title = {{Towards} a computational model of thermal- and magneto-sensitive polymers}, venue = {Ibiza}, year = {2013} }