A Comparison of Atomistic and Enhanced Continuum Approaches at Finite Temperature - Davydov, Denis; Javili, Ali; Steinmann, Paul; McBride, Andrew (Surface effects in solid mechanics, 2012)
UnivIS-Import:2015-04-16:Pub.2012.tech.FT.FT-TM.acompa_7

Algorithms and Data Structures for Matrix-Free Finite Element Operators with MPI-Parallel Sparse Multi-Vectors - Davydov, Denis; Kronbichler, Martin (ACM Transactions on Parallel Computing, 2020)
Traditional solution approaches for problems in quantum mechanics scale as O(M3), where M is the number of electrons. Various methods have been proposed to address this issue and obtain a linear scaling O(M). One promising formulation is the direct minimization of energy. Such methods take advantage of physical localization of the solution, allowing users to seek it in terms of non-orthogonal orbitals with local support. 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

A matrix-free approach for finite-strain hyperelastic problems using geometric multigrid - Davydov, Denis; Pelteret, Jean-Paul; Arndt, Daniel; Kronbichler, Martin; Steinmann, Paul (International Journal for Numerical Methods in Engineering, 2020)
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.
CRIS-Team Scopus Importer:2020-03-20
Keywords: adaptive finite-element method; finite-strain; geometric multigrid; hyperelasticity; matrix-free

Comparison of several staggered atomistic-to-continuum concurrent coupling strategies - Davydov, Denis; Pelteret, Jean-Paul; Steinmann, Paul (Computer Methods in Applied Mechanics and Engineering, 2014)
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 H1 norm derived from continuous displacement fields provide the best performance. © 2014 Elsevier B.V.
UnivIS-Import:2015-03-09:Pub.2014.tech.FT.FT-TM.compar
Keywords: Atomic-to-continuum coupling methods; Concurrent multiscale methods; Finite elements; Irving-Kirkwood-Noll procedure; Large strain; Molecular mechanics

Computational electro- and magneto-elasticity for quasi-incompressible media immersed in free space - Pelteret, Jean-Paul; Davydov, Denis; McBride, Andrew; Vu, Duc Khoi; Steinmann, Paul (International Journal for Numerical Methods in Engineering, 2016)
In this work, a mixed variational formulation to simulate quasi-incompressible electro-active or magneto-active polymers immersed in the surrounding free space is presented. A novel domain decomposition is used to disconnect the primary coupled problem and the arbitrary free-space mesh update problem. Exploiting this decomposition, we describe a block-iterative approach to solving the linearised multiphysics problem, and a physically and geometrically based, three-parameter method to update the free space mesh. Several application-driven example problems are implemented to demonstrate the robustness of the mixed formulation for both electro-elastic and magneto-elastic problems involving both finite deformations and quasi-incompressible media

Convergence study of the h-adaptive PUM and the hp-adaptive FEM applied to eigenvalue problems in quantum mechanics - Davydov, Denis; Gerasimov, Tymofiy; Pelteret, Jean-Paul; Steinmann, Paul (Advanced Modeling and Simulation in Engineering Sciences, 2017)
In this paper the h-adaptive partition-of-unity method and the h- and hp-adaptive finite element method are applied to eigenvalue problems arising in quantum mechanics, namely, the Schrödinger equation with Coulomb and harmonic potentials, and the all-electron Kohn–Sham density functional theory. The partition-of-unity method is equipped with an a posteriori error estimator, thus enabling implementation of error-controlled adaptive mesh refinement strategies. To that end, local interpolation error estimates are derived for the partition-of-unity method enriched with a class of exponential functions. The efficiency of the h-adaptive partition-of-unity method is compared to the h- and hp-adaptive finite element method. The latter is implemented by adopting the analyticity estimate from Legendre coefficients. An extension of this approach to multiple solution vectors is proposed. Numerical results confirm the theoretically predicted convergence rates and remarkable accuracy of the h-adaptive partition-of-unity approach. Implementational details of the partition-of-unity method related to enforcing continuity with hanging nodes are discussed.

Cutoff-Based Modeling of Coulomb Interactions for Atomistic-to-Continuum Multiscale Methods - Boddu, Vishal; Davydov, Denis; Eidel, Bernhard; Steinmann, Paul (Multiscale Science and Engineering, 2019)
Atomic interactions in a large class of functional materials, essentially—all dielectrics, polarizable solids and ionic solids, involve long-range Coulomb interactions. Yet atomistic-to-continuum multiscale methods, such as the quasicontinuum method, are currently applicable only in cases where the atomic interactions are short-ranged. This restriction on the nature of atomic-level interactions for multiscale methods excludes numerous materials that are central to various scientific and industrial applications. A number of studies have pointed out unphysical artifacts in molecular simulations upon using the direct cutoff-based truncated sum to evaluate Coulomb interactions in ionic solids. However, recently it is understood that the artifacts of the direct cutoff-based truncated sum can be significantly minimized if a suitable correction term is added. In this work we examine whether the Wolf summation method, one of the prominent cutoff-based methods, is suitable to carry out the accumulation of the Coulomb interactions within the context of atomistic-to-continuum multiscale methods. In this regard, we choose the quasicontinuum method from the existing collection of multiscale methods for demonstration. It is found that the Wolf summation method can be applied if the atomistic system has charge ordering; otherwise, the error in the accumulation of the Coulomb interactions could be large.<br

DFT-FE – A massively parallel adaptive finite-element code for large-scale density functional theory calculations - Motamarri, Phani; Das, Sambit; Rudraraju, Shiva; Ghosh, Krishnendu; Davydov, Denis; Gavini, Vikram (Computer Physics Communications, 2019)
We present an accurate, efficient and massively parallel finite-element code, DFT-FE, for large-scale ab-initio calculations (reaching ∼100,000 electrons) using Kohn–Sham density functional theory (DFT). DFT-FE is based on a local real-space variational formulation of the Kohn–Sham DFT energy functional that is discretized using a higher-order adaptive spectral finite-element (FE) basis, and treats pseudopotential and all-electron calculations in the same framework, while accommodating non-periodic, semi-periodic and periodic boundary conditions. We discuss the main aspects of the code, which include, the various strategies of adaptive FE basis generation, and the different approaches employed in the numerical implementation of the solution of the discrete Kohn–Sham problem that are focused on significantly reducing the floating point operations, communication costs and latency. We demonstrate the accuracy of DFT-FE by comparing the energies, ionic forces and periodic cell stresses on a wide range of problems with popularly used DFT codes. Further, we demonstrate that DFT-FE significantly outperforms widely used plane-wave codes—both in CPU-times and wall-times, and on both non-periodic and periodic systems—at systems sizes beyond a few thousand electrons, with over 5−10 fold speedups in systems with more than 10,000 electrons. The benchmark studies also highlight the excellent parallel scalability of DFT-FE, with strong scaling demonstrated up to 192,000 MPI tasks. Program summary: Program Title: DFT-FE Program Files doi: http://dx.doi.org/10.17632/tgdmgvmfft.1 Licensing provisions: LGPL v3 Programming language: C/C++ External routines/libraries: p4est (http://www.p4est.org/), deal.II (https://www.dealii.org/), BLAS (http://www.netlib.org/blas/), LAPACK (http://www.netlib.org/lapack/), ELPA (https://elpa.mpcdf.mpg.de/), ScaLAPACK (http://www.netlib.org/scalapack/), Spglib (https://atztogo.github.io/spglib/), ALGLIB (http://www.alglib.net/), LIBXC (http://www.tddft.org/programs/libxc/), PETSc (https://www.mcs.anl.gov/petsc), SLEPc (http://slepc.upv.es). Nature of problem: Density functional theory calculations. Solution method: We employ a local real-space variational formulation of Kohn–Sham density functional theory that is applicable for both pseudopotential and all-electron calculations with arbitrary boundary conditions. Higher-order adaptive spectral finite-element basis is used to discretize the Kohn–Sham equations. Chebyshev polynomial filtered subspace iteration procedure (ChFSI) is employed to solve the nonlinear Kohn–Sham eigenvalue problem self-consistently. ChFSI in DFT-FE employs Cholesky factorization based orthonormalization, and spectrum splitting based Rayleigh–Ritz procedure in conjunction with mixed precision arithmetic. Configurational force approach is used to compute ionic forces and periodic cell stresses for geometry optimization. Restrictions: Exchange–correlation functionals are restricted to Local Density Approximation (LDA) and Generalized Gradient Approximation (GGA), with and without spin. The pseudopotentials available are optimized norm conserving Vanderbilt (ONCV) pseudopotentials and Troullier–Martins (TM) pseudopotentials. Calculations are non-relativistic. Unusual features: DFT-FE handles all-electron and pseudopotential calculations in the same framework, while accommodating periodic, non-periodic and semi-periodic boundary conditions. Additional comments: DFT-FE github https://github.com/dftfeDevelopers/dftfe
CRIS-Team Scopus Importer:2019-10-01
Keywords: Ab-initio molecular dynamics; All-electron; Electronic structure; Ionic forces; Mixed-precision arithmetic; Pseudopotential; Real-space; Spectral finite-elements

Matrix-Free Locally Adaptive Finite Element Solution of Density-Functional Theory With Nonorthogonal Orbitals and Multigrid Preconditioning - Davydov, Denis; Heister, Timo; Kronbichler, Martin; Steinmann, Paul (physica status solidi (b), 2018)
In this paper, we propose a new numerical method to find the ground state energy of a given physical system within the Kohn–Sham density functional theory. The h-adaptive finite element method is adopted for spatial discretization and implemented with matrix-free operator evaluation. The ground state energy is found by performing unconstrained minimization with non-orthogonal orbitals using the limited memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. A geometric multigrid preconditioner is applied to improve the convergence. The clear advantage of the proposed approach is demonstrated on selected examples by comparing the performance to other methods such as preconditioned steepest descent minimization. The proposed method provides a solid framework toward (Formula presented.) (N) complexity for the locally adaptive real-space solution of density functional theory with finite elements.
CRIS-Team Scopus Importer:2022-08-05
Keywords: adaptive finite element method; density functional theory; geometric multigrids; matrix-free; quasi-Newton method

Modelling the flexoelectric effect in solids: A micromorphic approach - McBride, A. T.; Davydov, Denis; Steinmann, Paul (Computer Methods in Applied Mechanics and Engineering, 2020)
Flexoelectricity is characterised by the coupling of the second gradient of the motion and the electrical field in a dielectric material. The presence of the second gradient is a significant obstacle to obtaining the approximate solution using conventional numerical methods, such as the finite element method, that typically require a C1-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.
CRIS-Team Scopus Importer:2020-08-28
Keywords: Finite element method; Flexoelectricity; Micromorphic continua

Modelling the flexoelectric effect in solids - Mcbride, A.; Davydov, Denis; Steinmann, Paul (ADVANCES IN ENGINEERING MATERIALS, STRUCTURES AND SYSTEMS: INNOVATIONS, MECHANICS AND APPLICATIONS, 2019)
CRIS-Team WoS Importer:2020-01-10

On molecular static and surface-enhanced continuum modeling of nano-structures - Davydov, Denis; Javili, Ali; Steinmann, Paul (Computational Materials Science, 2013)
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.
UnivIS-Import:2015-03-09:Pub.2012.tech.FT.FT-TM.onmole
Keywords: Atomistic to continuum; Surface elasticity; Nano-materials; Surface-enhanced continuum theory

On the adaptive finite element analysis of the Kohn-Sham equations: Methods, algorithms, and implementation. - Davydov, Denis; Young, Toby; Steinmann, Paul (International Journal for Numerical Methods in Engineering, 2015)
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.

On the equivalence of the two foundational formulations for atomistic flux in inhomogeneous transport processes - Diaz, Adrian; Davydov, Denis; Chen, Youping (Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 2019)
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.
CRIS-Team Scopus Importer:2019-05-02
Keywords: Atomistic flux; Dirac delta distribution; Irving-Kirkwood procedure; Local stress and heat flux; Statistical mechanics formalism; Transient transport fluxes

Reduced-Order Modelling and Homogenisation in Magneto-Mechanics: A Numerical Comparison of Established Hyper-Reduction Methods - Brands, Benjamin; Davydov, Denis; Mergheim, Julia; Steinmann, Paul (Mathematical and Computational Applications, 2019)
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.
CRIS-Team WoS Importer:2019-09-17

Reviewing the roots of continuum formulations in molecular systems. Part II: Energy and angular momentum balance equations - Davydov, Denis; Steinmann, Paul (Mathematics and Mechanics of Solids, 2013)
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.
UnivIS-Import:2015-03-09:Pub.2013.tech.FT.FT-TM.review

Reviewing the roots of continuum formulations in molecular systems. Part III: Stresses, Couple Stresses, Heat Fluxes - Davydov, Denis; Steinmann, Paul (Mathematics and Mechanics of Solids, 2014)
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.
UnivIS-Import:2015-03-09:Pub.2014.tech.FT.FT-TM.review

Reviewing the roots of continuum formulations in molecular systems. Part I: Particle dynamics, statistical physics, mass and linear momentum balance equations - Davydov, Denis; Steinmann, Paul (Mathematics and Mechanics of Solids, 2012)
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.
UnivIS-Import:2015-03-09:Pub.2012.tech.FT.FT-TM.review

Size effects in a silica-polystyrene nanocomposite: Molecular dynamics and surface-enhanced continuum approaches - Davydov, Denis; Voyiatzis, E; Chatzigeorgiou, Georgios; Liu, S; Steinmann, Paul; Böhm, M; Müller-Plathe, F. (Soft Materials, 2014)
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.
Keywords: Molecular dynamics; Nanocomposite; Size effect; Surface energy; Surface-enhanced continuum

The deal.II Library, Version 9.1 - Arndt, Daniel; Bangerth, Wolfgang; Clevenger, Thomas C.; Davydov, Denis; Fehling, Marc; Garcia-Sanchez, Daniel; Harper, Graham; Heister, Timo; Heltai, Luca; Kronbichler, Martin; Maguire Kynch, Ross; Maier, Matthias; et al. (Journal of Numerical Mathematics, 2019)
This paper provides an overview of the new features of the finite element library deal.II, version 9.1.
CRIS-Team Scopus Importer:2019-07-19

The deal.II Library, Version 8.4 - Bangerth, Wolfgang; Davydov, Denis; Heister, Timo; Heltai, Luca; Kanschat, Guido; Kronbichler, Martin; Maier, Matthias; Turcksin, Bruno; Wells, David (Journal of Numerical Mathematics, 2016)

The deal.II Library, Version 8.5 - Arndt, Daniel; Bangerth, Wolfgang; Davydov, Denis; Heister, Timo; Heltai, Luca; Kronbichler, Martin; Maier, Matthias; Pelteret, Jean-Paul; Turcksin, Bruno; Wells, David (Journal of Numerical Mathematics, 2017)

Towards a computational model of magneto-sensitive polymers - Pelteret, Jean-Paul; Saxena, Prashant; Walter, Bastian; Davydov, Denis; Steinmann, Paul (85th Annual Meeting of the International Association of Applied Mathematics and Mechanics, 2014)

Towards a computational model of thermal- and magneto-sensitive polymers - Pelteret, Jean-Paul; Davydov, Denis; Javili, Ali; Walter, Bastian; Steinmann, Paul (5th International Conference on Computational Methods for Coupled Problems in Science and Engineering, 2013)