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Статті в журналах з теми "Particle methods (Numerical analysis)"

Автор: Grafiati

Опубліковано: 4 червня 2021

Оновлено: 26 січня 2023

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1

Neunzert, Helmut, and Jens Struckmeier. "Particle Methods for the Boltzmann Equation." Acta Numerica 4 (January 1995): 417–57. http://dx.doi.org/10.1017/s0962492900002579.

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In the following chapters we will discuss particle methods for the numerical simulation of rarefied gas flows.We will mainly treat a billiard game, that is, our particles will be hard spheres. But we will also touch upon cases where particles have internal energies due to rotation or vibration, which they exchange in a collision, and we will talk about chemical reactions happening during a collision.Due to the limited size of this paper, we are only able to mention the principles of these real-gas effects. On the other hand, the general concepts of particle methods to be presented may be used for other kinds of kinetic equations, such as the semiconductor device simulation. We leave this part of the research to subsequent papers.

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2

KOSHIZUKA, Seiichi. "Numerical Analysis of Continuous Media Using Particle Methods." JOURNAL OF THE JAPAN WELDING SOCIETY 75, no. 2 (2006): 126–28. http://dx.doi.org/10.2207/jjws.75.126.

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3

Bagtzoglou, Amvrossios C., Andrew F. B. Tompson, and David E. Dougherty. "Projection functions for particle-grid methods." Numerical Methods for Partial Differential Equations 8, no. 4 (July 1992): 325–40. http://dx.doi.org/10.1002/num.1690080403.

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4

Havlak, Karl J., and Harold Dean Victory. "On Deterministic Particle Methods for Solving Vlasov--Poisson--Fokker--Planck Systems." SIAM Journal on Numerical Analysis 35, no. 4 (August 1998): 1473–519. http://dx.doi.org/10.1137/s0036142996302529.

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5

Wollman, Stephen. "On the Approximation of the Vlasov--Poisson System by Particle Methods." SIAM Journal on Numerical Analysis 37, no. 4 (January 2000): 1369–98. http://dx.doi.org/10.1137/s0036142999298528.

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6

Ganguly, Keshab, and H. D. Victory, Jr. "On the Convergence of Particle Methods for Multidimensional Vlasov–Poisson Systems." SIAM Journal on Numerical Analysis 26, no. 2 (April 1989): 249–88. http://dx.doi.org/10.1137/0726015.

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7

Draghicescu, C. I. "An Efficient Implementation of Particle Methods for the Incompressible Euler Equations." SIAM Journal on Numerical Analysis 31, no. 4 (August 1994): 1090–108. http://dx.doi.org/10.1137/0731057.

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8

Patterson, Robert I. A., and Wolfgang Wagner. "Cell Size Error in Stochastic Particle Methods for Coagulation Equations with Advection." SIAM Journal on Numerical Analysis 52, no. 1 (January 2014): 424–42. http://dx.doi.org/10.1137/130924743.

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9

Guo, Meizhai, Megan S. Lord, and Zhongxiao Peng. "Quantitative wear particle analysis for osteoarthritis assessment." Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine 231, no. 12 (October 5, 2017): 1116–26. http://dx.doi.org/10.1177/0954411917735081.

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Анотація:

Osteoarthritis is a degenerative joint disease that affects millions of people worldwide. The aims of this study were (1) to quantitatively characterise the boundary and surface features of wear particles present in the synovial fluid of patients, (2) to select key numerical parameters that describe distinctive particle features and enable osteoarthritis assessment and (3) to develop a model to assess osteoarthritis conditions using comprehensive wear debris information. Discriminant analysis was used to statistically group particles based on differences in their numerical parameters. The analysis methods agreed with the clinical osteoarthritis grades in 63%, 50% and 61% of particles for no osteoarthritis, mild osteoarthritis and severe osteoarthritis, respectively. This study has revealed particle features specific to different osteoarthritis grades and provided further understanding of the cartilage degradation process through wear particle analysis – the technique that has the potential to be developed as an objective and minimally invasive method for osteoarthritis diagnosis.

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10

Victory, Jr., H. D., Garry Tucker, and Keshab Ganguly. "The Convergence Analysis of Fully Discretized Particle Methods for Solving Vlasov–Poisson Systems." SIAM Journal on Numerical Analysis 28, no. 4 (August 1991): 955–89. http://dx.doi.org/10.1137/0728051.

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11

Victory, Jr., H. D., and Edward J. Allen. "The Convergence Theory of Particle-In-Cell Methods for Multidimensional Vlasov–Poisson Systems." SIAM Journal on Numerical Analysis 28, no. 5 (October 1991): 1207–41. http://dx.doi.org/10.1137/0728065.

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12

Vuollekoski, H., S. L. Sihto, V. M. Kerminen, M. Kulmala, and K. E. J. Lehtinen. "A numerical comparison of different methods for determining the particle formation rate." Atmospheric Chemistry and Physics 12, no. 5 (March 1, 2012): 2289–95. http://dx.doi.org/10.5194/acp-12-2289-2012.

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Abstract. Different methods of determining formation rates of 3 nm particles are compared, basing on analysis of simulated data, but the results are valid for analyses of experimental particle size distribution data as well, at least within the accuracy of the applied model. The study shows that the method of determining formation rates indirectly from measured number concentration data of 3–6 nm particles is generally in good agreement with the theoretical calculation with a systematic error of 0–20%. While this accuracy is often enough, a simple modification to the approximative equation for the formation rate is recommended. A brief study on real atmospheric data implied that in some cases the accuracy gain may be significant.

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13

Belytschko, Ted, Yong Guo, Wing Kam Liu, and Shao Ping Xiao. "A unified stability analysis of meshless particle methods." International Journal for Numerical Methods in Engineering 48, no. 9 (2000): 1359–400. http://dx.doi.org/10.1002/1097-0207(20000730)48:9<1359::aid-nme829>3.0.co;2-u.

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14

Vuollekoski, H., S. L. Sihto, V. M. Kerminen, M. Kulmala, and K. E. J. Lehtinen. "A numerical comparison of different methods for determining the particle formation rate." Atmospheric Chemistry and Physics Discussions 10, no. 8 (August 10, 2010): 18781–805. http://dx.doi.org/10.5194/acpd-10-18781-2010.

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Анотація:

Abstract. Different methods of determining formation rates of 3 nm particles are compared, basing on analysis of simulated data, but the results are valid for analyses of experimental particle size distribution data as well. The study shows that the method of determining formation rates indirectly from measured number concentration data of 3–6 nm particles is generally in good agreement with the theoretical calculation with a systematic error of 0–20%. While this is often accurate enough, a simple modification to the approximative equation for the formation rate is recommended. Additionally, the temporal connection between the concentration of the nucleating vapour and the formation rate is studied. It is concluded that the often used power-law connecting these two is inaccurate.

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15

Chertock, Alina, and Alexander Kurganov. "On a practical implementation of particle methods." Applied Numerical Mathematics 56, no. 10-11 (October 2006): 1418–31. http://dx.doi.org/10.1016/j.apnum.2006.03.024.

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16

Belytschko, T., Y. Krongauz, J. Dolbow, and C. Gerlach. "On the completeness of meshfree particle methods." International Journal for Numerical Methods in Engineering 43, no. 5 (November 15, 1998): 785–819. http://dx.doi.org/10.1002/(sici)1097-0207(19981115)43:5<785::aid-nme420>3.0.co;2-9.

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17

Liu, Wing Kam, Sukky Jun, Shaofan Li, Jonathan Adee, and Ted Belytschko. "Reproducing kernel particle methods for structural dynamics." International Journal for Numerical Methods in Engineering 38, no. 10 (May 30, 1995): 1655–79. http://dx.doi.org/10.1002/nme.1620381005.

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18

Mroczka, Janusz, Mariusz Woźniak, and Fabrice R. A. Onofri. "Algorithms and methods for analysis of the optical structure factor of fractal aggregates." Metrology and Measurement Systems 19, no. 3 (October 1, 2012): 459–70. http://dx.doi.org/10.2478/v10178-012-0039-2.

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Abstract We introduce numerical methods and algorithms to estimate the main parameters of fractal-like particle aggregates from their optical structure factor (i.e. light scattering diagrams). The first algorithm is based on a direct and simple method, but its applicability is limited to aggregates with large size parameter and intermediate fractal dimension. The second algorithm requires to build calibration curves based on accurate particle agglomeration and particle light scattering models. It allows analyzing the optical structure factor of much smaller aggregates, regardless of their fractal dimension and the size of the single particles. Therefore, this algorithm as well as the introduction of a criterial curve to detect the different scattering regimes, are thought to be powerful tools to perform reliable and reproducible analysis.

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19

Oyinbo, Sunday Temitope, and Tien-Chien Jen. "Feasibility of numerical simulation methods on the Cold Gas Dynamic Spray (CGDS) Deposition process for ductile materials." Manufacturing Review 7 (2020): 24. http://dx.doi.org/10.1051/mfreview/2020023.

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Анотація:

The techniques of cold gas dynamic spray (CGDS) coating involve the deposition of solid, high speed micron to nano particles onto a substrate. In contrast to a thermal spray, CGDS does not melt particles to retain their physico-chemical properties. There have been many advantages in developing microscopic analysis of deformation mechanisms with numerical simulation methods. Therefore, this study focuses on four cardinal numerical methods of analysis which are: Lagrangian, Smoothed Particles Hydrodynamics (SPH), Arbitrary Lagrangian-Eulerian (ALE), and Coupled Eulerian-Lagrangian (CEL) to examine the Cold Gas Dynamic Spray (CGDS) deposition system by simulating and analyzing the contact/impact problem at deformation zone using ductile materials. The details of these four numerical approaches are explained with some aspects of analysis procedure, model description, material model, boundary conditions, contact algorithm and mesh refinement. It can be observed that the material of the particle greatly influences the deposition and the deformation than the material of the substrate. Concerning the particle, a higher-density material such as Cu has a higher initial kinetic energy, which leads to a larger contact area, a longer contact time and, therefore, better bonding between the particle and the substrate. All the numerical methods studied, however, can be used to analyze the contact/impact problem at deformation zone during cold gas dynamic spray process.

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20

Havlak, Karl J., and Harold Dean Victory, Jr. "The Numerical Analysis of Random Particle Methods Applied to Vlasov–Poisson Fokker-Planck Kinetic Equations." SIAM Journal on Numerical Analysis 33, no. 1 (February 1996): 291–317. http://dx.doi.org/10.1137/0733016.

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21

Mazumdar, D., and R. I. L. Guthrie. "An analysis of numerical methods for solving the particle trajectory equation." Applied Mathematical Modelling 12, no. 4 (August 1988): 398–402. http://dx.doi.org/10.1016/0307-904x(88)90069-8.

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22

Brenier, Y., and G. H. Cottet. "Convergence of Particle Methods with Random Rezoning for the Two-Dimensional Euler and Navier–Stokes Equations." SIAM Journal on Numerical Analysis 32, no. 4 (August 1995): 1080–97. http://dx.doi.org/10.1137/0732049.

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23

Cui, Shumo, Alexander Kurganov, and Alexei Medovikov. "Particle methods for PDEs arising in financial modeling." Applied Numerical Mathematics 93 (July 2015): 123–39. http://dx.doi.org/10.1016/j.apnum.2014.04.005.

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24

Huang, H., C. T. Dyka, and S. Saigal. "Hybrid particle methods in frictionless impact-contact problems." International Journal for Numerical Methods in Engineering 61, no. 13 (2004): 2250–72. http://dx.doi.org/10.1002/nme.1146.

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25

Galindo-Torres, Sergio Andres, Dorival Pedroso, David Williams, and Hans Mühlhaus. "An Analysis of the Strength of Anisotropic Granular Assemblies via Discrete Methods." Applied Mechanics and Materials 553 (May 2014): 525–30. http://dx.doi.org/10.4028/www.scientific.net/amm.553.525.

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This paper presents a study on the macroscopic strength characteristics of granular assemblies with three-dimensional complex-shaped particles. Different assemblies are considered, with both isotropic and anisotropic particle geometries. The study is conducted using the Discrete Element Method (DEM), with so-called sphero-polyhedral particles, and simulations of mechanical true triaxial tests for a range of Lode angles and confining pressures. The observed mathematical failure envelopes are investigated in the Haigh-Westergaard stress space, as well as on the deviatoric-mean pressure plane. It is verified that the DEM with non-spherical particles produces results that are qualitatively similar to experimental data and previous numerical results obtained with spherical elements. The simulations reproduce quite well the shear strength of assemblies of granular media, such as higher strength during compression than during extension. In contrast, by introducing anisotropy at the particle level, the shear strength parameters are greatly affected, and an isotropic failure criterion is no longer valid. It is observed that the strength of the anisotropic assembly depends on the direction of loading, as observed for real soils.

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26

Guo, Aixia, Tsorng-Whay Pan, Jiwen He, and Roland Glowinski. "Numerical Methods for Simulating the Motion of Porous Balls in Simple 3D Shear Flows Under Creeping Conditions." Computational Methods in Applied Mathematics 17, no. 3 (July 1, 2017): 397–412. http://dx.doi.org/10.1515/cmam-2017-0012.

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AbstractIn this article, two novel numerical methods have been developed for simulating fluid/porous particle interactions in three-dimensional (3D) Stokes flow. The Brinkman–Debye–Bueche model is adopted for the fluid flow inside the porous particle, being coupled with the Stokes equations for the fluid flow outside the particle. The rotating motion of a porous ball and the interaction of two porous balls in bounded shear flows have been studied by these two new methods. The numerical results show that the porous particle permeability has a strong effect on the interaction of two porous balls.

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27

Filbet, Francis, and Luis Miguel Rodrigues. "Asymptotically Stable Particle-In-Cell Methods for the Vlasov--Poisson System with a Strong External Magnetic Field." SIAM Journal on Numerical Analysis 54, no. 2 (January 2016): 1120–46. http://dx.doi.org/10.1137/15m104952x.

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28

Koshizuka, Seiichi, and Mikio Sakai. "Numerical Analysis of Continuum Mechanics and Multi-phase Flow Using Particle Methods." Journal of the Society of Powder Technology, Japan 46, no. 2 (February 10, 2009): 106–13. http://dx.doi.org/10.4164/sptj.46.106.

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29

Jun, Sukky, Wing Kam Liu, and Ted Belytschko. "Explicit Reproducing Kernel Particle Methods for large deformation problems." International Journal for Numerical Methods in Engineering 41, no. 1 (January 15, 1998): 137–66. http://dx.doi.org/10.1002/(sici)1097-0207(19980115)41:1<137::aid-nme280>3.0.co;2-a.

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30

Joldes, Grand Roman, Adam Wittek, and Karol Miller. "Stable time step estimates for mesh-free particle methods." International Journal for Numerical Methods in Engineering 91, no. 4 (June 1, 2012): 450–56. http://dx.doi.org/10.1002/nme.4290.

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31

Cottet, G. H., J. M. Etancelin, F. Perignon, and C. Picard. "High order semi-Lagrangian particle methods for transport equations: numerical analysis and implementation issues." ESAIM: Mathematical Modelling and Numerical Analysis 48, no. 4 (June 30, 2014): 1029–60. http://dx.doi.org/10.1051/m2an/2014009.

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32

Li, Ying, Jin Yun Pu, and Shuo Zhang. "Numerical Analysis of Water Mist Drop Survival Time." Applied Mechanics and Materials 178-181 (May 2012): 958–61. http://dx.doi.org/10.4028/www.scientific.net/amm.178-181.958.

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Анотація:

in order to study the survival time of the water mist, the quality, energy and motion models were established by using mathematical methods, and then the three models were integrated by fourth-order Runge-Kutta method; at last, by analysis of simulation data, the conclusion were obtained that the survival time of the droplet had been related the particle size and ambient temperature, and their running distance had been related to the particle size and initial velocity.

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33

Rabczuk, T., and T. Belytschko. "Adaptivity for structured meshfree particle methods in 2D and 3D." International Journal for Numerical Methods in Engineering 63, no. 11 (2005): 1559–82. http://dx.doi.org/10.1002/nme.1326.

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34

Liu, Wing Kam, and Sukky Jun. "Multiple-scale reproducing kernel particle methods for large deformation problems." International Journal for Numerical Methods in Engineering 41, no. 7 (April 15, 1998): 1339–62. http://dx.doi.org/10.1002/(sici)1097-0207(19980415)41:7<1339::aid-nme343>3.0.co;2-9.

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35

Mishnaevsky, Leon. "Computational Analysis of the Effects of Microstructures on Damage and Fracture in Heterogeneous Materials." Key Engineering Materials 306-308 (March 2006): 489–94. http://dx.doi.org/10.4028/www.scientific.net/kem.306-308.489.

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3D FE (finite element) simulations of the deformation and damage evolution of particle reinforced composites are carried out for different microstructures of the composites. Several new methods and programs for the automatic reconstruction of 3D microstructures of composites on the basis of the geometrical description of microstructures as well as on the basis of the voxel array data have been developed and tested. Different methods of reconstruction and generation of finite element models of 3D microstructures of composite materials (geometry-based and voxel array based) are discussed and compared. It was shown that FE analyses of the elasto-plastic deformation and damage of composite materials using the microstructural models of materials generated with these methods yield very close results. Numerical testing of composites with random, regular, clustered and gradient arrangements of spherical particles is carried out. The fraction of failed particles and the tensile stress-strain curves were determined numerically for each of the microstructures. It was found that the rate of damage growth as well as the critical applied strain, at which the damage growth in particles begins, depend on the particle arrangement, and increase in the following order: gradient < random < regular < clustered microstructure.

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36

Stachowiak, G. W., G. B. Stachowiak, and P. Campbell. "Application of numerical descriptors to the characterization of wear particles obtained from joint replacements." Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine 211, no. 1 (January 1, 1997): 1–10. http://dx.doi.org/10.1243/0954411961534620.

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The application of image analysis techniques to the characterization of wear particles from failed joint replacements has been described. Wear particles were extracted from periprosthetic tissues collected during revision surgery. Chemical digestive methods were used to separate the wear particles from the biological soft material. The particles isolated were examined by optical and scanning electron microscopy. Digitized particle images were analysed on a Macintosh computer by a specially developed software and by the image analysis program ‘Prism’. The following numerical descriptors were used to characterize the particles: particle size, boundary fractal dimension and shape parameters such as form factor, roundness, convexity and aspect ratio. Elemental composition of the particles was determined by energy dispersive X-ray spectroscopy. Three selected types of wear particles were analysed and compared: titanium (Ti)-based and calcium (Ca)-based particles from a hip prosthesis and ultra-high molecular weight polyethylene (UHMWPE) particles from a knee prosthesis. The particles exhibited significantly different sizes and their shape numerical descriptors were also different. From the results obtained it appears that computer image analysis of wear particle morphology can be employed to determine the wear processes occurring in the joints. In some cases, the condition of the joint can also be assessed based on this analysis.

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37

Guo, L., and K. Morita. "Numerical simulation of 3D sloshing in a liquid-solid mixture using particle methods." International Journal for Numerical Methods in Engineering 95, no. 9 (July 1, 2013): 771–90. http://dx.doi.org/10.1002/nme.4520.

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38

LAI, T. C., Y. S. MORSI, S. DAS, and A. OWIDA. "NUMERICAL ANALYSIS OF PARTICLE DEPOSITION IN ASYMMETRICAL HUMAN UPPER AIRWAYS UNDER DIFFERENT INHALATION CYCLES." Journal of Mechanics in Medicine and Biology 13, no. 04 (July 7, 2013): 1350068. http://dx.doi.org/10.1142/s0219519413500681.

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Анотація:

It is recognised that knowledge of air flow characteristics in the tracheo-bronchial tree is essential to the understanding of airway resistance, intrapulmonary gas mixing and deposition of airborne particles. Numerical and mathematical methods have previously been used extensively to obtain particle deposition patterns inside a single airway and various regions of the human lung for a range of physiological conditions. However, detailed analysis of particle deposition, in asymmetrical human upper airways, under transient conditions, has not been uncovered in published literature at the time this research commenced. In this research study, a commercial CFD package, called "CFX Workbench 11" was deployed to analyse flow fields, transient flow and particle deposition. This research work was an extension to earlier research published in 2008 by the present authors. The airway geometry applied in this current research was created by closely following the values published by another researcher (Horsfield), where the transient flows for three different breathing cycles were used as the input boundary conditions. The findings of the modelling presented herein indicated that the release position did not vary significantly at different time steps or with changes in particle size, but varied significantly with breathing patterns. Moreover, the rate of particle deposition at the wall was found to increase with the rising of the branching angles.

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39

Fleissner, Florian, and Peter Eberhard. "Parallel load-balanced simulation for short-range interaction particle methods with hierarchical particle grouping based on orthogonal recursive bisection." International Journal for Numerical Methods in Engineering 74, no. 4 (2008): 531–53. http://dx.doi.org/10.1002/nme.2184.

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40

Johnson, Scott M., John R. Williams, and Benjamin K. Cook. "Quaternion-based rigid body rotation integration algorithms for use in particle methods." International Journal for Numerical Methods in Engineering 74, no. 8 (2008): 1303–13. http://dx.doi.org/10.1002/nme.2210.

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41

CARMONA, RENÉ, and STÉPHANE CRÉPEY. "PARTICLE METHODS FOR THE ESTIMATION OF CREDIT PORTFOLIO LOSS DISTRIBUTIONS." International Journal of Theoretical and Applied Finance 13, no. 04 (June 2010): 577–602. http://dx.doi.org/10.1142/s0219024910005905.

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Анотація:

The goal of the paper is the numerical analysis of the performance of Monte Carlo simulation based methods for the computation of credit-portfolio loss-distributions in the context of Markovian intensity models of credit risk. We concentrate on two of the most frequently touted methods of variance reduction in the case of stochastic processes: importance sampling (IS) and interacting particle systems (IPS) based algorithms. Because the subtle differences between these methods are often misunderstood, as IPS is often regarded as a mere particular case of IP, we describe in detail the two kinds of algorithms, and we highlight their fundamental differences. We then proceed to a detailed comparative case study based on benchmark numerical experiments chosen for their popularity in the quantitative finance circles.

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42

Takekawa, Junichi, Hitoshi Mikada, and Tada-nori Goto. "An accuracy analysis of a Hamiltonian particle method with the staggered particles for seismic-wave modeling including surface topography." GEOPHYSICS 79, no. 4 (July 1, 2014): T189—T197. http://dx.doi.org/10.1190/geo2014-0012.1.

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A Hamiltonian particle method (HPM), which is one of the mesh-free methods, can simulate seismic wavefields for models including surface topography in a simple manner. Numerical error caused by a curved free surface or by particles not aligned with the surface is not obvious in HPM. In general, the accommodation of irregular free surfaces requires more grids or particles in a minimum wavelength for achieving sufficient accuracy in the simulation. We tested the accuracy of HPM with staggered particles for simulating seismic-wave propagation including the surface topography, and we established the relationship between desired accuracy and spatial resolution. We conducted numerical simulations for models with a planar free surface aligned with the regular particle alignment and a dipping free surface. Our accuracy tests revealed that the numerical error strongly depends on the dipping angle of the slope. We concluded that about 25 particles in a minimum wavelength are required to calculate Rayleigh waves propagating along the irregular topography with good accuracy. Finally, we simulated Rayleigh wave propagation along irregular topography using a layered model with a hill. HPM can reproduce not only surface-wave propagation but also the reflected and refracted waves. Our numerical results were in good agreement with those from a finite-element method. Our investigations indicated that HPM could be a solution to simulate Rayleigh waves in the presence of complex surface topography.

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43

Ganguly, Keshab, J. Todd Lee, and H. D. Victory, Jr. "On Simulation Methods for Vlasov–Poisson Systems with Particles Initially Asymptotically Distributed." SIAM Journal on Numerical Analysis 28, no. 6 (December 1991): 1574–609. http://dx.doi.org/10.1137/0728080.

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44

Co’, Giampaolo, and Stefano De Leo. "Analytical and numerical analysis of the complete Lipkin–Meshkov–Glick Hamiltonian." International Journal of Modern Physics E 27, no. 05 (May 2018): 1850039. http://dx.doi.org/10.1142/s0218301318500398.

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The Lipkin–Meshkov–Glick is a simple, but not trivial, model of a quantum many-body system which allows us to solve the many-body Schrödinger equation without making any approximation. The model, which in its unperturbed case is composed only by two energy levels, includes two interacting terms. A first one, the [Formula: see text] interaction, which promotes or degrades pairs of particles, and a second one, the [Formula: see text] interaction, which scatters one particle in the upper and another in the lower energy level. In comparing this model with other approximation methods, the [Formula: see text] term interaction is often set to zero. In this paper, we show how the presence of this interaction changes the global structure of the system, generates degeneracies between the various eigenstates and modifies the energy eigenvalues structure. We present analytical solutions for systems of two and three particles and, for some specific cases, also for four, six and eight particles. The solutions for systems with more than eight particles are only numerical but their behavior can be well understood by considering the extrapolations of the analytical results. Of particular interest is the study of how the [Formula: see text] interaction affects the energy gap between the ground state and the first-excited state.

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Shang, Xiang-Yu, Ke Duan, Lian-Fei Kuang, and Qi-Yin Zhu. "Relation of EDL Forces between Clay Particles Calculated by Different Methods." Applied Sciences 12, no. 11 (May 31, 2022): 5591. http://dx.doi.org/10.3390/app12115591.

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Calculation of the electrostatic double layer force (EDL force) between clay particles is relevant as it is closely related to important macroscopic mechanical behaviors of clays. The popular method to calculate the EDL force is to integrate the electric potential and Maxwell stress along the boundary enclosing a simply connected domain within which a clay particle resides. The EDL force has also been calculated by the integration of the electrostatic force density over the preceding domain. However, the subtle relation of the EDL forces calculated by the different existing methods has not yet been investigated. By means of theoretical analysis and finite element simulation, it was shown that the force calculated by the integration of Maxwell stress along the complete boundary enclosing a multiply connected domain in which the clay particle is excluded, and that along the partial boundary enclosing the preceding simply connected domain represents the electrical attractive force and osmotic repulsive force, respectively, while the integration of the potential along both the same complete and partial boundary denotes the osmotic force. Numerical results showed that the calculated EDL force deviates from its actual value significantly with the decrease in distance between the chosen integral boundary and particle surface, and the deviation varies with surface potential and angle between particles. Moreover, the recommended minimum distance was proposed to be 10 times the thickness of the particle based on the present simulation results.

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46

Shinde, Suhas M., Dayanand M. Kawadekar, Pradeep A. Patil, and Virendra K. Bhojwani. "Analysis of micro and nano particle erosion by analytical, numerical and experimental methods: A review." Journal of Mechanical Science and Technology 33, no. 5 (May 2019): 2319–29. http://dx.doi.org/10.1007/s12206-019-0431-x.

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47

Bonet, J., and S. Kulasegaram. "Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations." International Journal for Numerical Methods in Engineering 47, no. 6 (February 28, 2000): 1189–214. http://dx.doi.org/10.1002/(sici)1097-0207(20000228)47:6<1189::aid-nme830>3.0.co;2-i.

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48

Bonet, Javier, and Sivakumar Kulasegaram. "Remarks on tension instability of Eulerian and Lagrangian corrected smooth particle hydrodynamics (CSPH) methods." International Journal for Numerical Methods in Engineering 52, no. 11 (December 20, 2001): 1203–20. http://dx.doi.org/10.1002/nme.242.

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49

Fu, Yao, Tong Wang, and Chuangang Gu. "Experimental and numerical analyses of gas–solid-multiphase jet in cross-flow." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 227, no. 1 (January 9, 2012): 61–79. http://dx.doi.org/10.1177/0954410011429420.

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In this article, jet influence on a gas–solid-multiphase channel flow was experimentally and numerically studied. The jet flow was found to have a diameter-selective controlling effect on the particles’ distribution. Jet flow formed a gas barrier in the channel for particles. While tiny particles could travel around and large particles could travel through, only particles on the 10 -µm scale were obviously affected. Three different calculation methods, Reynolds averaged Navier–Stokes, unsteady Reynolds averaged Navier–Stokes, and detached eddy simulation, were used to simulate this multiphase flow. By comparing the calculation results to the experimental results, it is found that all the three calculation methods could capture the basic phenomenon in the mean flow field. Nevertheless, there exist great differences in the transient flow field and particle distribution.

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50

Klar, A. "Computation of nonlinear functionals in particle methods." Computing 55, no. 3 (September 1995): 207–21. http://dx.doi.org/10.1007/bf02238432.

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