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A Column Space Based Approach to Solve Systems of Multivariate Polynomial Equations ⁎ ⁎

IFAC-PapersOnLine 2021年第9期54卷 137-144页

作者： Christof Vermeersch Bart De Moor Center for Dynamical Systems Signal Processing and Data Analytics (STADIUS) Dept. of Electrical Engineering (ESAT) KU Leuven Kasteelpark Arenberg 10 3001 Leuven Belgium

We propose a novel approach to solve systems of multivariate polynomial equations, using the column space of the Macaulay matrix that is constructed from the coefficients of these polynomials. A multidimensional realization problem in the null space of the Macaulay matrix results in an eigenvalue problem, the eigenvalues and eigenvectors of which yield the common roots of the system. Since this null space based algorithm uses well-established numerical linear algebra tools, like the singular value and eigenvalue decomposition, it finds the solutions within machine precision. In this paper, on the other hand, we determine a complementary approach to solve systems of multivariate polynomial equations, which considers the column space of the Macaulay matrix instead of its null space. This approach works directly on the data in the Macaulay matrix, which is sparse and structured. We provide a numerical example to illustrate our new approach and to compare it with the existing null space based root-finding algorithm.

关键词： Macaulay matrix multivariate polynomials numerical algorithms realization theory matrix algebra

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A new method to improve the numerical stability of the hierarchical equations of motion for discrete harmonic oscillator modes

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JOURNAL OF CHEMICAL PHYSICS 2020年第20期153卷 204109页

作者： Yan, Yaming Xing, Tao Shi, Qiang Chinese Acad Sci Inst Chem Beijing Natl Lab Mol Sci State Key Lab Struct Chem Unstable & Stable Speci Beijing 100190 Peoples R China Univ Chinese Acad Sci Beijing 100049 Peoples R China Huairou Natl Comprehens Sci Ctr Phys Sci Lab Beijing 101407 Peoples R China

The hierarchical equations of motion (HEOMs) have developed into an important tool in simulating quantum dynamics in condensed phases. Yet, it has recently been found that the HEOM may become numerically unstable in simulations using discrete harmonic oscillator modes [I. S. Dunn, et al., J. Chem. Phys. 150, 184109 (2019)]. In this paper, a new set of equations of motion are obtained based on the equivalence between the HEOM for discrete harmonic oscillator modes and the mixed quantum-classical Liouville equation. The new set of equations can thus be regarded as the expansion of the same phase space partial differential equation using different basis sets. It is shown that they have similar structures as the original HEOM but are free from the problem of numerical instability. The new set of equations are also incorporated into the matrix product state method, where it is found that the trace of the reduced density operator is not well conserved during the propagation. A modified time-dependent variational principle is then proposed to achieve better trace conservation.

关键词： Excitons: Energetics and Spatio-temporal Dynamics Topics numerical algorithms Quantum dissipation Harmonic oscillator Matrix product state Excitation energy transfer

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A General Approach to Lifting-Line Theory, Applied to Wings with Sweep

A General Approach to Lifting-Line Theory, Applied to Wings ...

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AIAA Science and Technology Forum and Exposition (AIAA SciTech)

作者： Reid, Jackson T. Hunsaker, Douglas F. Utah State Univ Mech & Aerosp Engn 4130 Old Main Hill Logan UT 84321 USA

ISBN: (数字)9781624105951

ISBN: (纸本)9781624105951

Implementations of lifting-line theory model the flow over a finite wing using a sheet of semi-infinite vortices extending from a vortex filament placed along the locus of aerodynamic centers of the wing. Prandtl's classical implementation is restricted to straight wings in flows without sideslip. In this work, it is shown that these limitations can be overcome if, at the control points where induced velocity is calculated, the second derivative of the locus of aerodynamic centers is zero and the trailing vortices are perpendicular to the locus. Therefore, a general implementation of lifting-line theory is presented that conditionally forces the second derivative of the locus of aerodynamic centers to zero at each control point, and joints each trailing vortex such that there is a finite segment of the trailing vortex perpendicular to the locus of aerodynamic centers. Consideration is given to modeling the locus of aerodynamic centers of non-straight wings and the section aerodynamic properties of such wings. The resulting general formulation is analyzed to determine sensitivity, accuracy, and numerical convergence. The general implementation demonstrates second-order convergence when the control points are clustered at the root and tips of the wing, and produces results that closely match those of a higher-order panel method and experimental data.

关键词： Aerodynamic Properties Horseshoe Vortex Side Slip Angle Aerodynamic Angle High Aspect Ratio Rectangular Wing Low Aspect Ratio Wings Airfoil Geometry Symmetric Airfoil numerical algorithms

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Finite-discrete element modelling of sea ice sheet elasticity, sea ice sheet fracture, and ice-structure interaction - A three-dimensional, lattice-based approach

Finite-discrete element modelling of sea ice sheet elasticit...

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作者： Lilja, Ville-Pekka Aalto University

学位级别：博士

In this thesis, the elastic and inelastic properties of an ice sheet modelled by a new hybrid, three-dimensional finite-discrete element (FE-DE) method were examined. Ice-structure interaction between an ice sheet and a conical offshore structure was studied as well. By this new method, an ice sheet is modelled with undeformable, i.e. rigid, discrete elements. The mass centroids of the discrete elements connect then via an in-plane beam lattice of co-rotational, viscously damped, de-cohesive Timoshenko beam finite elements. A centroidal-Voronoi-tessellation-based iterative scheme (CVT) was applied in creating the studied FE-DE meshes, i.e. the modelled ice sheets. Due to the internally damped, de-cohesive, lattice-based construction, the mechanical response of a modelled ice sheet turns out to be both strain rate- and size-dependent (dependent on both the absolute and relative sizes), the investigation of which formed an integral part of the present study. A general objective of this thesis was to study the applicability of the new, hybrid FE-DE method in modelling the elasticity and fracture of sea ice sheets. In order to understand the effects of scale and to demonstrate the feasibility of the approach in studying ice mechanics applications in general, i.e. the ice-structure interaction, several conceptually simple constitutive tests with square FE-DE sheet samples of varying side lengths, thicknesses, and discrete element sizes were performed. The results presented gave a partial guideline for choosing the microscale material parameters of a CVT-tessellated, lattice-based FE-DE model of an ice sheet in order to achieve a desired macroscale response, both elastic and inelastic. Furthermore, the results provided substantial insight into the functional dependencies each studied physical quantity has. In addition, the elastic bending tests showed that the model is able to emulate a free Kirchhoff-Love plate in bending on a Winkler-type foundation with a good accura

关键词： ice ice-structure interaction plates beam lattice networks fracture mechanics size effect numerical algorithms centroidal Voronoi tessellation jää jää-rakenne-vuorovaikutus laatat palkkiverkkorakenteet murtumismekaniikka kokoefekti yhdistetty diskreetti-elementtimenetelmä numeeriset algoritmit

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Comparison of analytical and numerical estimations of laser beam spreading in a turbulent atmosphere 25

Comparison of analytical and numerical estimations of laser ...

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25th International Symposium On Atmospheric and Ocean Optics - Atmospheric Physics

作者： Konyaev, Petr Kanev, Feodor VE Zuev Inst Atmospher Opt Tomsk Russia

ISBN: (数字)9781510631694

ISBN: (纸本)9781510631694

Two possible methods to solve scalar parabolic wave equation describing beam propagation in a randomly inhomogeneous medium have been compared in the article. The analytical approach was based on additive approximation, while numerically the equation was solved by the split-step Fourier algorithm. As was shown calculation of the beam radius with the use of both methods provides approximately the same results in the interval of weak fluctuations of the index of refraction. Increase of distortion intensity leads to divergence of data obtained analytically and in computer simulations. This disagreement can be explained by intrinsic limitations of additive approximation evolving in the regime of strong amplitude-phase fluctuations.

关键词： Atmospheric turbulence numerical algorithms split-step Fourier method wave optics laser beams

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An atomistically informed kinetic Monte Carlo model for predicting solid solution strengthening of body-centered cubic alloys

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INTERNATIONAL JOURNAL OF PLASTICITY 2019年 122卷 319-337页

作者： Shinzato, Shuhei Wakeda, Masato Ogata, Shigenobu Osaka Univ Grad Sch Engn Sci Dept Mech Sci & Bioengn 1-3 Machikaneyama Toyonaka Osaka 5608531 Japan NIMS 1-2-1 Sengen Tsukuba Ibaraki 3050047 Japan Kyoto Univ Ctr Element Strategy Initiat Struct Mat ESISM Sakyo Ku Kyoto 6068501 Japan

In order to predict solid solution strengthening in body-centered cubic dilute substitutional alloys, we developed an atomistically informed kinetic Monte Carlo (kMC) model for screw dislocation motion, which is a major determinant of the yield strength of the BCC alloys. The kMC model only requires parameters obtainable using atomic simulations. The parameters in the developed kMC model were actually determined for Fe Si dilute alloys using atomistically derived activation energies of kink-pair nucleation and kink migration as well as their stress dependencies. The activation energies were computed using the nudged elastic band method with developed interatomic potentials based on first-principles density functional theory. Eventually, the critical resolved shear stress (CRSS), activation volume of the dislocation glide, and their temperature and solute concentration dependencies were directly obtained by two dimensional kMC dislocation glide simulations. Our kMC model qualitatively reproduced the trends of experimentally observed temperature and concentration dependencies of CRSS, and thus it can naturally describe solid solution strengthening, and softening without any empirical information. In addition, the limitations of the two dimensional model, including single slip and lack of non-Schmid effect, are discussed, which results in quantitative difference with the experimental CRSS.

关键词： Dislocations Metallic material numerical algorithms Strengthening mechanisms

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Advanced algorithms for studying the impact of climate changes on ozone levels in the atmosphere

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INTERNATIONAL JOURNAL OF ENVIRONMENT AND POLLUTION 2019年第1-3期66卷 212-238页

作者： Zlatev, Zahari Dimov, Ivan Farago, Istvan Georgiev, Krassimir Havasi, Agnes Aarhus Univ Dept Environm Sci Roskilde Denmark Bulgarian Acad Sci Inst Informat & Commun Technol Sofia Bulgaria MTA ELIE Numer Anal & Large Networks Res Grp H-1117 Budapest Hungary Budapest Univ Technol & Econ Dept Differential Equat Muegyet Rkp 3 H-1111 Budapest Hungary Eotvos Lorand Univ Dept Appl Anal & Computat Math Budapest Hungary

The impact of the increased temperature on some pollution levels is studied by developing climate scenarios and by running them on a period of 16 consecutive years. The mathematical model used in the experiments is a system of nonlinear partial differential equations (PDEs) and the space domain contains the whole of Europe together with some of its surroundings. Difficult computational problems arise after the direct discretisation of the spatial derivatives in the equations. The enormous computational problems can be treated successfully only by resolving five important tasks: 1) applying accurate and efficient numerical algorithms;2) selecting an appropriate splitting procedure;3) developing parallel codes;4) using high-speed computers;5) exploiting well the cache memories of the parallel computers. numerical results will be presented in order to illustrate both the efficiency of the code and the fact that the future climate changes will result in consistent increase of some pollution levels.

关键词： large-scale environmental models systems of nonlinear PDEs numerical algorithms splitting techniques parallel codes high-speed computers cache memory climate changes ozone levels damaging effects

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Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrodinger equation

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COMMUNICATIONS IN NONLINEAR SCIENCE AND numerical SIMULATION 2019年 68卷 347-371页

作者： Vasylchenkova, A. Prilepsky, J. E. Shepelsky, D. Chattopadhyay, A. Aston Univ Aston Inst Photon Technol Birmingham B4 7ET W Midlands England Aston Univ Syst Analyt Res Inst Birmingham B4 7ET W Midlands England B Verkin Inst Low Temp Phys & Engn UA-61103 Kharkov Ukraine Kharkov Natl Univ UA-61022 Kharkov Ukraine Aston Univ Math Birmingham B4 7ET W Midlands England

Starting from a comparison of some established numerical algorithms for the computation of the eigenvalues (discrete or solitonic spectrum) of the non-Hermitian version of the Zakharov-Shabat spectral problem, this article delivers new algorithms that combine the best features of the existing ones and thereby allays their relative weaknesses. Our algorithm is modelled within the remit of the so-called direct nonlinear Fourier transform (NFT) associated with the focusing nonlinear Schrodinger equation. First, we present the data for the calibration of existing methods comparing the relative errors associated with the computation of the continuous NF spectrum. Then each method is paired with different numerical algorithms for finding zeros of a complex-valued function to obtain the eigenvalues. Next we describe a new class of methods based on the contour integrals evaluation for the efficient search of eigenvalues. After that we introduce a new hybrid method, one of our main results: the method combines the advances of contour integral approach and makes use of the iterative algorithms at its second stage for the refined eigenvalues search. The veracity of our new hybrid algorithm is established by estimating the convergence speed and accuracy across three independent test profiles. Along with the development of a new approach for the computation of the eigenvalues, our study also addresses the problem of computation of the so-called norming constants associated with the eigenvalues. We show that our formalism effectively amounts to accurate and fast enough computation of residues of the reflection coefficient in the upper complex half-plane of the spectral parameter. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Nonlinear Schrodinger equation Inverse scattering method numerical algorithms Signal processing

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Predicting elastic anisotropy of dual-phase steels based on crystal mechanics and microstructure

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INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES 2019年 151卷 639-649页

作者： Cantara, Aaron M. Zecevic, Milovan Eghtesad, Adnan Poulin, Camille M. Knezevic, Marko Univ New Hampshire Dept Mech Engn 33 Acad WayKingsbury HallW119 Durham NH 03824 USA

This work utilizes mean-field self-consistent and full-field fast Fourier transform-based homogenizations to study the effective elastic behavior of several steels: three dual-phase (DP), DP 590, DP 980, and DP 1180, and one martensitic (MS), MS 1700. Crystallographic textures and phase fractions of these steels are characterized using electron microscopy along with electron-backscattered diffraction to initialize the models. A comprehensive set of Young's modulus and Poisson's ratio data, measured at the ambient temperature as a function of orientation with respect to the rolling direction for each steel sheet, is used to calibrate and validate the models. The calibration of the models enabled us to estimate the single crystal elastic constants for both the martensitic phase and ferrite, while calculating the orientation dependent effective properties. Half of the data was used in the calibration. Subsequent predictions of the orientation dependent effective elastic properties for the remaining data verified that the estimated single crystal properties are reliable. As the steels exhibit a different level of anisotropy in their effective behavior, good predictions allowed us to discuss the role of texture, grain structure, phase fraction and distribution on the effective properties. The results of this work represent a significant incentive to introduce elastic anisotropy in numerical tools for simulating metal forming processes of dual-phase steels, in particular those processes involving springback, using the texture informed crystal mechanics-based models to more accurately estimate the effective elastic properties required by such simulations.

关键词： Microstructures Elastic material Anisotropic material numerical algorithms Dual-phase steel

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Reachable Domain of Spacecraft After a Gravity-Assist Flyby

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JOURNAL OF GUIDANCE CONTROL AND DYNAMICS 2019年第4期42卷 931-940页

作者： Chen, Qi Qiao, Dong Wen, Changxuan Beijing Inst Technol Sch Aerosp Engn Beijing 100081 Peoples R China Chinese Acad Sci Key Lab Space Utilizat Technol & Engn Ctr Space Utilizat Beijing 100094 Peoples R China

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