Pub Date : 2017-09-09DOI: 10.4208/cicp.oa-2017-0192
B. Treeby, Elliott S. Wise, B. Cox
A class of nonstandard pseudospectral time domain (PSTD) schemes for solving time-dependent hyperbolic and parabolic partial differential equations (PDEs) is introduced. These schemes use the Fourier collocation spectral method to compute spatial gradients and a nonstandard finite difference scheme to integrate forwards in time. The modified denominator function that makes the finite difference time scheme exact is transformed into the spatial frequency domain or k-space using the dispersion relation for the governing PDE. This allows the correction factor to be applied in the spatial frequency domain as part of the spatial gradient calculation. The derived schemes can be formulated to be unconditionally stable, and apply to PDEs in any space dimension. Examples of the resulting nonstandard PSTD schemes for several PDEs are given, including the wave equation, diffusion equation, and convection-diffusion equation.
{"title":"Nonstandard Fourier Pseudospectral Time Domain (PSTD) Schemes for Partial Differential Equations","authors":"B. Treeby, Elliott S. Wise, B. Cox","doi":"10.4208/cicp.oa-2017-0192","DOIUrl":"https://doi.org/10.4208/cicp.oa-2017-0192","url":null,"abstract":"A class of nonstandard pseudospectral time domain (PSTD) schemes for solving time-dependent hyperbolic and parabolic partial differential equations (PDEs) is introduced. These schemes use the Fourier collocation spectral method to compute spatial gradients and a nonstandard finite difference scheme to integrate forwards in time. The modified denominator function that makes the finite difference time scheme exact is transformed into the spatial frequency domain or k-space using the dispersion relation for the governing PDE. This allows the correction factor to be applied in the spatial frequency domain as part of the spatial gradient calculation. The derived schemes can be formulated to be unconditionally stable, and apply to PDEs in any space dimension. Examples of the resulting nonstandard PSTD schemes for several PDEs are given, including the wave equation, diffusion equation, and convection-diffusion equation.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74247041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-09-08DOI: 10.1142/S0219876218501086
V. D. Camiola, V. Tozzini
The evaluation of collective modes is fundamental in the analysis of molecular dynamics simulations. Several methods are available to extract that information, i.e normal mode analysis, principal component and spectral analysis of trajectories, basically differing by the quantity considered as the nodal one (frequency, amplitude, or pattern of displacement) and leading to the definition of different kinds of collective excitations and physical spectral observables. Different views converge in the harmonic regime and/or for homo-atomic systems. However, for anharmonic and out of equilibrium dynamics different quantities bring different information and only their comparison can give a complete view of the system behavior. To allow such a comparative analysis, we review and compare the different approaches, applying them in different combination to two examples of physical relevance: graphene and fullerene C60.
{"title":"Collective mode mining from molecular dynamics simulations: a comparative approach","authors":"V. D. Camiola, V. Tozzini","doi":"10.1142/S0219876218501086","DOIUrl":"https://doi.org/10.1142/S0219876218501086","url":null,"abstract":"The evaluation of collective modes is fundamental in the analysis of molecular dynamics simulations. Several methods are available to extract that information, i.e normal mode analysis, principal component and spectral analysis of trajectories, basically differing by the quantity considered as the nodal one (frequency, amplitude, or pattern of displacement) and leading to the definition of different kinds of collective excitations and physical spectral observables. Different views converge in the harmonic regime and/or for homo-atomic systems. However, for anharmonic and out of equilibrium dynamics different quantities bring different information and only their comparison can give a complete view of the system behavior. To allow such a comparative analysis, we review and compare the different approaches, applying them in different combination to two examples of physical relevance: graphene and fullerene C60.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":"284 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91543133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article explains how to apply the computer algebra package GAP (this http URL) in the computation of the problems in quantum physics, in which the application of Lie algebra is necessary. The article contains several exemplary computations which readers would follow in the desktop PC: such as, the brief review of elementary ideas of Lie algebra, the angular momentum in quantum mechanics, the quark eight-fold way model, and the usage of Weyl character formula (in order to construct weight modules, and to count correctly the degeneracy). This short article is a complement to the article "Computer Algebra and Material Design" [arXiv:1612.02275] by one of the authors (A.K.). The latter article is also available in the arXiv.
{"title":"Lie algebra in quantum physics by means of computer algebra","authors":"I. Kikuchi, Akihito Kikuchi","doi":"10.17605/OSF.IO/YRG7V","DOIUrl":"https://doi.org/10.17605/OSF.IO/YRG7V","url":null,"abstract":"This article explains how to apply the computer algebra package GAP (this http URL) in the computation of the problems in quantum physics, in which the application of Lie algebra is necessary. The article contains several exemplary computations which readers would follow in the desktop PC: such as, the brief review of elementary ideas of Lie algebra, the angular momentum in quantum mechanics, the quark eight-fold way model, and the usage of Weyl character formula (in order to construct weight modules, and to count correctly the degeneracy). \u0000This short article is a complement to the article \"Computer Algebra and Material Design\" [arXiv:1612.02275] by one of the authors (A.K.). The latter article is also available in the arXiv.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82611414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Abert, C. Huber, F. Bruckner, C. Vogler, G. Wautischer, D. Suess
We present a finite-difference method for the topology optimization of permanent magnets that is based on the FFT accelerated computation of the stray-field. The presented method employs the density approach for topology optimization and uses an adjoint method for the gradient computation. Comparsion to various state-of-the-art finite-element implementations shows a superior performance and accuracy. Moreover, the presented method is very flexible and easy to implement due to various preexisting FFT stray-field implementations that can be used.
{"title":"A fast finite-difference algorithm for topology optimization of permanent magnets","authors":"C. Abert, C. Huber, F. Bruckner, C. Vogler, G. Wautischer, D. Suess","doi":"10.1063/1.4998532","DOIUrl":"https://doi.org/10.1063/1.4998532","url":null,"abstract":"We present a finite-difference method for the topology optimization of permanent magnets that is based on the FFT accelerated computation of the stray-field. The presented method employs the density approach for topology optimization and uses an adjoint method for the gradient computation. Comparsion to various state-of-the-art finite-element implementations shows a superior performance and accuracy. Moreover, the presented method is very flexible and easy to implement due to various preexisting FFT stray-field implementations that can be used.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80162202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In many problems involving particle accelerators and relativistic plasmas, the accurate modeling of relativistic particle motion is essential for accurate physical predictions. Here, we extend a charge-conserving finite element time-domain (FETD) particle-in-cell (PIC) algorithm for the time-dependent Maxwell-Vlasov equations on irregular (unstructured) meshes to the relativistic regime by implementing and comparing three particle pushers: (relativistic) Boris, Vay, and Higuera-Cary. We illustrate the application of the proposed relativistic FETD-PIC algorithm for the analysis of particle cyclotron motion at relativistic speeds, harmonic particle oscillation in the Lorentz-boosted frame, and relativistic Bernstein modes in magnetized charge-neutral (pair) plasmas.
{"title":"Relativistic Extension of a Charge-Conservative Finite Element Solver for Time-Dependent Maxwell-Vlasov Equations","authors":"D. Na, H. Moon, Y. Omelchenko, F. Teixeira","doi":"10.1063/1.5004557","DOIUrl":"https://doi.org/10.1063/1.5004557","url":null,"abstract":"In many problems involving particle accelerators and relativistic plasmas, the accurate modeling of relativistic particle motion is essential for accurate physical predictions. Here, we extend a charge-conserving finite element time-domain (FETD) particle-in-cell (PIC) algorithm for the time-dependent Maxwell-Vlasov equations on irregular (unstructured) meshes to the relativistic regime by implementing and comparing three particle pushers: (relativistic) Boris, Vay, and Higuera-Cary. We illustrate the application of the proposed relativistic FETD-PIC algorithm for the analysis of particle cyclotron motion at relativistic speeds, harmonic particle oscillation in the Lorentz-boosted frame, and relativistic Bernstein modes in magnetized charge-neutral (pair) plasmas.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79563404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-07-09DOI: 10.4208/cicp.OA-2016-0179d
G. Khakimzyanov, D. Dutykh, O. Gusev
In the present manuscript, we consider the problem of dispersive wave simulation on a rotating globally spherical geometry. In this Part IV, we focus on numerical aspects while the model derivation was described in Part III. The algorithm we propose is based on the splitting approach. Namely, equations are decomposed on a uniformly elliptic equation for the dispersive pressure component and a hyperbolic part of shallow water equations (on a sphere) with source terms. This algorithm is implemented as a two-step predictor-corrector scheme. On every step, we solve separately elliptic and hyperbolic problems. Then, the performance of this algorithm is illustrated on model idealised situations with an even bottom, where we estimate the influence of sphericity and rotation effects on dispersive wave propagation. The dispersive effects are quantified depending on the propagation distance over the sphere and on the linear extent of generation region. Finally, the numerical method is applied to a couple of real-world events. Namely, we undertake simulations of the Bulgarian 2007 and Chilean 2010 tsunamis. Whenever the data is available, our computational results are confronted with real measurements.
{"title":"Dispersive shallow water wave modelling. Part IV: Numerical simulation on a globally spherical geometry","authors":"G. Khakimzyanov, D. Dutykh, O. Gusev","doi":"10.4208/cicp.OA-2016-0179d","DOIUrl":"https://doi.org/10.4208/cicp.OA-2016-0179d","url":null,"abstract":"In the present manuscript, we consider the problem of dispersive wave simulation on a rotating globally spherical geometry. In this Part IV, we focus on numerical aspects while the model derivation was described in Part III. The algorithm we propose is based on the splitting approach. Namely, equations are decomposed on a uniformly elliptic equation for the dispersive pressure component and a hyperbolic part of shallow water equations (on a sphere) with source terms. This algorithm is implemented as a two-step predictor-corrector scheme. On every step, we solve separately elliptic and hyperbolic problems. Then, the performance of this algorithm is illustrated on model idealised situations with an even bottom, where we estimate the influence of sphericity and rotation effects on dispersive wave propagation. The dispersive effects are quantified depending on the propagation distance over the sphere and on the linear extent of generation region. Finally, the numerical method is applied to a couple of real-world events. Namely, we undertake simulations of the Bulgarian 2007 and Chilean 2010 tsunamis. Whenever the data is available, our computational results are confronted with real measurements.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83767133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-07-05DOI: 10.4208/CICP.OA-2017-0213
Jiequn Han, Linfeng Zhang, R. Car, E. Weinan
We present a simple, yet general, end-to-end deep neural network representation of the potential energy surface for atomic and molecular systems. This methodology, which we call Deep Potential, is "first-principle" based, in the sense that no ad hoc approximations or empirical fitting functions are required. The neural network structure naturally respects the underlying symmetries of the systems. When tested on a wide variety of examples, Deep Potential is able to reproduce the original model, whether empirical or quantum mechanics based, within chemical accuracy. The computational cost of this new model is not substantially larger than that of empirical force fields. In addition, the method has promising scalability properties. This brings us one step closer to being able to carry out molecular simulations with accuracy comparable to that of quantum mechanics models and computational cost comparable to that of empirical potentials.
{"title":"Deep Potential: a general representation of a many-body potential energy surface","authors":"Jiequn Han, Linfeng Zhang, R. Car, E. Weinan","doi":"10.4208/CICP.OA-2017-0213","DOIUrl":"https://doi.org/10.4208/CICP.OA-2017-0213","url":null,"abstract":"We present a simple, yet general, end-to-end deep neural network representation of the potential energy surface for atomic and molecular systems. This methodology, which we call Deep Potential, is \"first-principle\" based, in the sense that no ad hoc approximations or empirical fitting functions are required. The neural network structure naturally respects the underlying symmetries of the systems. When tested on a wide variety of examples, Deep Potential is able to reproduce the original model, whether empirical or quantum mechanics based, within chemical accuracy. The computational cost of this new model is not substantially larger than that of empirical force fields. In addition, the method has promising scalability properties. This brings us one step closer to being able to carry out molecular simulations with accuracy comparable to that of quantum mechanics models and computational cost comparable to that of empirical potentials.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76326998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-06-28DOI: 10.1103/PhysRevMaterials.1.043603
Chi Chen, Z. Deng, Richard Tran, Hanmei Tang, I. Chu, S. Ong
In this work, we present a highly accurate spectral neighbor analysis potential (SNAP) model for molybdenum (Mo) developed through the rigorous application of machine learning techniques on large materials data sets. Despite Mo's importance as a structural metal, existing force fields for Mo based on the embedded atom and modified embedded atom methods still do not provide satisfactory accuracy on many properties. We will show that by fitting to the energies, forces and stress tensors of a large density functional theory (DFT)-computed dataset on a diverse set of Mo structures, a Mo SNAP model can be developed that achieves close to DFT accuracy in the prediction of a broad range of properties, including energies, forces, stresses, elastic constants, melting point, phonon spectra, surface energies, grain boundary energies, etc. We will outline a systematic model development process, which includes a rigorous approach to structural selection based on principal component analysis, as well as a differential evolution algorithm for optimizing the hyperparameters in the model fitting so that both the model error and the property prediction error can be simultaneously lowered. We expect that this newly developed Mo SNAP model will find broad applications in large-scale, long-time scale simulations.
{"title":"Accurate Force Field for Molybdenum by Machine Learning Large Materials Data","authors":"Chi Chen, Z. Deng, Richard Tran, Hanmei Tang, I. Chu, S. Ong","doi":"10.1103/PhysRevMaterials.1.043603","DOIUrl":"https://doi.org/10.1103/PhysRevMaterials.1.043603","url":null,"abstract":"In this work, we present a highly accurate spectral neighbor analysis potential (SNAP) model for molybdenum (Mo) developed through the rigorous application of machine learning techniques on large materials data sets. Despite Mo's importance as a structural metal, existing force fields for Mo based on the embedded atom and modified embedded atom methods still do not provide satisfactory accuracy on many properties. We will show that by fitting to the energies, forces and stress tensors of a large density functional theory (DFT)-computed dataset on a diverse set of Mo structures, a Mo SNAP model can be developed that achieves close to DFT accuracy in the prediction of a broad range of properties, including energies, forces, stresses, elastic constants, melting point, phonon spectra, surface energies, grain boundary energies, etc. We will outline a systematic model development process, which includes a rigorous approach to structural selection based on principal component analysis, as well as a differential evolution algorithm for optimizing the hyperparameters in the model fitting so that both the model error and the property prediction error can be simultaneously lowered. We expect that this newly developed Mo SNAP model will find broad applications in large-scale, long-time scale simulations.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85987632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-06-16DOI: 10.1007/978-3-319-72374-7_26
C. Mendl, S. Palpacelli, A. Kamenev, S. Succi
{"title":"Quantum Lattice Boltzmann Study of Random-Mass Dirac Fermions in One Dimension","authors":"C. Mendl, S. Palpacelli, A. Kamenev, S. Succi","doi":"10.1007/978-3-319-72374-7_26","DOIUrl":"https://doi.org/10.1007/978-3-319-72374-7_26","url":null,"abstract":"","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":"51 1","pages":"321-330"},"PeriodicalIF":0.0,"publicationDate":"2017-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91039501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The finite difference time domain (FDTD) method has been successfully applied to obtain energies and wave functions for two electrons in a quantum dot modeled by a three dimensional harmonic potential. The FDTD method uses the time-dependent Schr"odinger equation (TDSE) in imaginary time. The TDSE is numerically solved with an initial random wave function and after enough simulation time, the wave function converges to the ground state wave function. The excited states are determined by using the same procedure for the ground state with additional constraints that the wave function must be orthogonal with all lower energy wave functions. The numerical results for energies and wave functions for different parameters of confinement potentials are given and compared with published results using other numerical methods. It is shown that the FDTD method gives accurate energies and wave functions.
{"title":"The Finite Difference Time Domain (FDTD) Method to Determine Energies and Wave Functions of Two-Electron Quantum Dot","authors":"I Wayan Sudiarta, L. M. Angraini","doi":"10.1063/1.5064196","DOIUrl":"https://doi.org/10.1063/1.5064196","url":null,"abstract":"The finite difference time domain (FDTD) method has been successfully applied to obtain energies and wave functions for two electrons in a quantum dot modeled by a three dimensional harmonic potential. The FDTD method uses the time-dependent Schr\"odinger equation (TDSE) in imaginary time. The TDSE is numerically solved with an initial random wave function and after enough simulation time, the wave function converges to the ground state wave function. The excited states are determined by using the same procedure for the ground state with additional constraints that the wave function must be orthogonal with all lower energy wave functions. The numerical results for energies and wave functions for different parameters of confinement potentials are given and compared with published results using other numerical methods. It is shown that the FDTD method gives accurate energies and wave functions.","PeriodicalId":8424,"journal":{"name":"arXiv: Computational Physics","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72671280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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