Discretization Number Research Articles

Biased versus unbiased numerical methods for stochastic simulations

Approximate numerical methods are one of the most used strategies to extract information from many-interacting-agents systems. In particular, numerical approximations are of extended use to deal with epidemic, ecological and biological models, since unbiased methods like the Gillespie algorithm can become unpractical due to high CPU time usage required. However, the use of approximations has been debated and there is no clear consensus about whether unbiased methods or biased approach is the best option. In this work, we derive scaling relations for the errors in approximations based on binomial extractions. This finding allows us to build rules to compute the optimal values of both the discretization time and number of realizations needed to compute averages with the biased method with a target precision and minimum CPU-time usage. Furthermore, we also present another rule to discern whether the unbiased method or biased approach is more efficient. Ultimately, we will show that the choice of the method should depend on the desired precision for the estimation of averages.

A Dantzig-Wolfe decomposition-based algorithm for capacitated passenger assignment problem with time-varying demand in high-speed railway networks

This paper proposes an algorithm for solving the schedule-based passenger assignment problem, with explicit consideration of continuous time-varying demand and tight capacity constraints for large-scale high-speed railway (HSR) networks. We construct the space–time travel network and formulate the assignment model by destination-based arc flow variables, then design a Dantzig-Wolfe (D-W) decomposition-based algorithm to solve the large-scale model of this problem with less memory requirement and computational complexity. The sub-problem in each iteration can be solved as a single-source (destination) minimum cost flow problem without capacity constraint by the “rooftops” method, which provides an effective technique to tackle both continuous and discrete time-varying demand in the assignment problem. It is further demonstrated that the assignment solutions for discrete time-varying demand can converge to that for continuous time-varying demand with the increase of demand discretization number. Three real case studies with different HSR network scales show that the proposed algorithm is efficient in memory and computational complexity, especially when highly accurate solutions for passenger assignment on large-scale HSR networks are needed.

STATIC BENDING STRENGTH OF SANDWICH COMPOSITE PLATES WITH TETRACHIRAL HONEYCOMBS

The article presents a numerical strength analysis of sandwich plates with solid face layers and tetrachiral honeycomb core layer under static bending conditions. An aluminum alloy was chosen as the material of plates. For honeycomb core layers, the discretization (number of unit cells) and the relative density were varied with a constant thickness. Calculations were performed for the case of bending with rigidly clamped ends and three-point bending within the framework of the theory of elasticity by the finite element method. The strength analysis enables one to determine the load values, at which the maximal stresses according to the von Mises criterion were equal to the conventional yield stress of the material. The aim of this work is to study the effect of discretization and relative density of honeycomb core layers of tetrachiral type on the str ength of sandwich plates.

Pressure-Transient Performances of Fractured Horizontal Wells in the Compartmentalized Heterogeneous Unconventional Reservoirs

In order to investigate pressure performance of multiple fractured horizontal wells (MFHWs) penetrating heterogeneous unconventional reservoir and avoid the high computational cost of numerical simulation, a semi-analytical model for MFHWs combining Green function solution and boundary element method has been obtained, where the reservoir is divided into different homogeneous substructures and coupled at interface boundaries by plane source function in a closed rectangular parallelepiped. Hydraulic fractures are assumed uniform flux and dual porosity model is used for natural fractures system. Then the model is validated by compared with analytical solution of MFHWs in a homogeneous reservoir and trilinear flow model, which shows that this model can achieve high accuracy even with a small interface discretization number, and it can consider the radial flow around each hydraulic fractures. Finally, the pressure responses with heterogeneous parameters of reservoirs are discussed including heterogeneous permeability, non-uniform block-length and fracture half-length distribution as well as dual porosity parameters like elastic storage ratio and crossflow ratio.

On the discretization of the ice thickness distribution in the NEMO3.6-LIM3 global ocean–sea ice model

Abstract. The ice thickness distribution (ITD) is one of the core constituents of modern sea ice models. The ITD accounts for the unresolved spatial variability of sea ice thickness within each model grid cell. While there is a general consensus on the added physical realism brought by the ITD, how to discretize it remains an open question. Here, we use the ocean–sea ice general circulation model, Nucleus for European Modelling of the Ocean (NEMO) version 3.6 and Louvain-la-Neuve sea Ice Model (LIM) version 3 (NEMO3.6-LIM3), forced by atmospheric reanalyses to test how the ITD discretization (number of ice thickness categories, positions of the category boundaries) impacts the simulated mean Arctic and Antarctic sea ice states. We find that winter ice volumes in both hemispheres increase with the number of categories and attribute that increase to a net enhancement of basal ice growth rates. The range of simulated mean winter volumes in the various experiments amounts to ∼30 % and ∼10 % of the reference values (run with five categories) in the Arctic and Antarctic, respectively. This suggests that the way the ITD is discretized has a significant influence on the model mean state, all other things being equal. We also find that the existence of a thick category with lower bounds at ∼4 and ∼2 m for the Arctic and Antarctic, respectively, is a prerequisite for allowing the storage of deformed ice and therefore for fostering thermodynamic growth in thinner categories. Our analysis finally suggests that increasing the resolution of the ITD without changing the lower limit of the upper category results in small but not negligible variations of ice volume and extent. Our study proposes for the first time a bi-polar process-based explanation of the origin of mean sea ice state changes when the ITD discretization is modified. The sensitivity experiments conducted in this study, based on one model, emphasize that the choice of category positions, especially of thickest categories, has a primary influence on the simulated mean sea ice states while the number of categories and resolution have only a secondary influence. It is also found that the current default discretization of the NEMO3.6-LIM3 model is sufficient for large-scale present-day climate applications. In all cases, the role of the ITD discretization on the simulated mean sea ice state has to be appreciated relative to other influences (parameter uncertainty, forcing uncertainty, internal climate variability).

Effect of discrete number of velocity directions of phonon on thermal conductivity prediction in the cross-plane direction of superlattices

In this paper, we analyze the effect of the discrete number of velocity directions of the phonon on thermal conductivity prediction in the cross-plane direction of superlattices based on solving the phonon Boltzmann transport equation (BTE) numerically. The simulation shows that the accuracy of the calculation is mainly affected by discretization number of polar angle (Nθ). To improve the efficiency in the two-dimensional situation, choosing a proper discrete number of velocity directions of the phonon can be used. Besides, the relationship between the control angle (M) and Knudsen (Kn) number is concluded when the relative error is less than 5%.

A clique covering MIP model for the irregular strip packing problem

The irregular strip packing problem consists in the cutting of a set of two-dimensional pieces from an object of fixed width using the minimum possible length. Despite its economic importance for many industries, few exact studies have addressed this problem. Recently, a mixed integer programming model in which pieces are placed on a grid has been proposed. Although the model has proved the optimality for some large instances, it has a large number of non-overlap constraints, which grows quickly according to the discretization resolution and number of distinct pieces. This paper proposes a clique covering model to reduce the number of constraints and improve the linear relaxation. The model has outperformed the previous model in most evaluated instances and obtained an optimal solution for instances with up to 25 pieces (22 distinct pieces) subject to grid discretization.

Comparative analysis of discretization methods in Bayesian networks

A key step in implementing Bayesian networks (BNs) is the discretization of continuous variables. There are several mathematical methods for constructing discrete distributions, the implications of which on the resulting model has not been discussed in literature. Discretization invariably results in loss of information, and both the discretization method and the number of intervals determines the level of such loss. We designed an experiment to evaluate the impact of commonly used discretization methods and number of intervals on the developed BNs. The conditional probability tables, model predictions, and management recommendations were compared and shown to be different among models. However, none of the models did uniformly well in all comparison criteria. As we cannot justify using one discretization method against others, we recommend caution when discretization is used, and a verification process that includes evaluating alternative methods to ensure that the conclusions are not an artifact of the discretization approach.

Micromixing simulation of novel large-double-blade impeller

The novel large-double-blade impeller spans a wide range of industrial application areas owing to its obvious advantages in the rapid reactions. So it is indeed necessary to have a profound understanding of micromixing characteristics about the novel large-double-blade impeller. This paper establishes a model to analyze the three main factors’ effect on the segregation index through the numerical simulation which is used to research the performance of novel large-double-blade impeller. The three main factors are separately feed discretization number, feeding location and agitation speed. Through the analysis above, the turbulent energy dissipation rate and concentration distribution of H+, I3− would be deduced. Meanwhile, it compared with the Fullzone (FZ) impeller and double ribbon (DHR) impeller. The results show that (1) when the feed discretization number is over 35, the segregation index would tend to be the fixed specific value. And only in that situation, the effect of macro mixture could be ignored. (2) The optimized feeding location is in the blade area with a vast, turbulent energy dissipation rate to achieve a better micromixing performance, which is demonstrate by the consumption curve of H+ and the generation contour of I3−. (3) The main spot of the consumption of H+ and the generation of I3− is on the one side of the feeding location. Besides, compared with FZ impeller and DHR impeller in the same scale and operating condition, novel large-double-blade impeller has a better micromixing performance.

Solving the discrete S-model kinetic equations with arbitrary order polynomial approximations

A numerical method for solving the model Boltzmann equation using arbitrary order polynomials is presented. The S-model is solved by a discrete ordinate method with velocity space discretized with a truncated Hermite polynomial expansion. Physical space is discretized according to the Conservative Flux Approximation scheme with extension to allow non-uniform grid spacing. This approach, which utilizes Legendre polynomials, allows the spatial representation and flux calculation to be of arbitrary order. High order boundary conditions are implemented. Various results are shown to demonstrate the utility and limitations of the method with comparison to solutions of the Euler and Navier–Stokes equations and from the Direct Simulation Monte Carlo and Unified Gas Kinetic schemes. The effect of both the velocity space discretization and Knudsen number on the convergence properties of the scheme are also investigated.

Rate of Convergence Analysis of Discretization and Smoothing Algorithms for Semiinfinite Minimax Problems

Discretization algorithms for semiinfinite minimax problems replace the original problem, containing an infinite number of functions, by an approximation involving a finite number, and then solve the resulting approximate problem. The approximation gives rise to a discretization error, and suboptimal solution of the approximate problem gives rise to an optimization error. Accounting for both discretization and optimization errors, we determine the rate of convergence of discretization algorithms, as a computing budget tends to infinity. We find that the rate of convergence depends on the class of optimization algorithms used to solve the approximate problem as well as the policy for selecting discretization level and number of optimization iterations. We construct optimal policies that achieve the best possible rate of convergence and find that, under certain circumstances, the better rate is obtained by inexpensive gradient methods.

Transient elution of particulate matter from hydrodynamic unit operations as a function of computational parameters and runoff hydrograph unsteadiness

While computational fluid dynamics (CFD) is utilized to simulate particulate matter (PM) separation and particle size distributions (PSDs) from unit operations, the role of computational parameters and runoff hydrograph unsteadiness to simulate intra-event elution of PM has not been examined. An Euler–Lagrangian CFD model is utilized to simulate discrete (Type I) separation of PM by a common hydrodynamic unit operation subject to unsteady runoff events and a hetero-disperse PM gradation. Utilizing a baffled hydrodynamic separator (HS) this study illustrates the potential of a CFD model to predict PM elution as a function of runoff hydrograph unsteadiness. The study hypothesizes that accurate simulation of unit behavior as a function of unsteadiness is dependent on mesh size (MS), time step (TS) and PSD discretization number (DN). Full-scale physical model and CFD results are compared. Intra-event results demonstrate that MS, TS and DN parameters significantly influence prediction of transient PM elution, PSDs and computational effort. Results demonstrate that each parameter generates error for transient PM elution as influenced by the unsteadiness level. In contrast, TS, MS and DN selection each have a statistically significantly smaller influence on event-based PM mass balances and PSDs.

AEROELASTIC ANALYSIS OF ROTOR BLADES USING CFD/CSD COUPLING IN HOVER MODE

A computational fluid dynamics (CFD) is coupled with a computational structural dynamics (CSD) to simulate the unsteady rotor flow with aeroelasticity effects. An unstructured upwind Navier-Stokes solver was developed for this simulation, with 2nd order time-accurate dual-time stepping method for temporal discretization and low Mach number preconditioning method. For turbulent flows, both the Spalart-Allmaras and Menter's SST model are available. Mesh deformation is achieved through a fast dynamic grid method called Delaunay graph map method for unsteady flow simulation. The rotor blades are modeled as Hodges & Dowell's nonlinear beams coupled flap-lag-torsion. The rotorcraft computational structural dynamics code employs the 15-dof beam finite element formulation for modeling. The structure code was validated by comparing the natural frequencies of a rotor model with UMARC. The flow and structure codes are coupled tightly with information exchange several times at every time step. A rotor blade model's unsteady flow field in the hover mode is simulated using the coupling method. Effect of blade elasticity with aerodynamic loads was compared with rigid blade.

Finite temperature infrared spectroscopy of polycyclic aromatic hydrocarbon molecules: Path-integral molecular dynamics

The vibrational spectra of the naphthalene, pyrene, and coronene molecules have been computed in the 0-3500 cm(-1) infrared range using classical and quantum molecular dynamics simulations based on a dedicated tight-binding potential energy surface. The ring-polymer molecular dynamics (RPMD) and partially adiabatic centroid molecular dynamics (CMD) methods have been employed to account for quantum nuclear effects. The contributions of quantum delocalization to the line shift and broadening are significant in the entire spectral range and of comparable magnitude as pure thermal effects. While the two methods generally produce similar results, the CMD method may converge slower at low temperature with increasing Trotter discretization number. However, and contrary to the CMD method, the RPMD approach suffers from serious resonance problems at high frequencies and low temperatures.

Discrete Phase Model Representation of Particulate Matter (PM) for Simulating PM Separation by Hydrodynamic Unit Operations

Modeling the separation of dilute particulate matter (PM) has been a topic of interest since the introduction of unit operations for clarification of rainfall-runoff. One consistent yet controversial issue is the representation of PM and PM separation mechanisms for treatment. While Newton's Law and surface overflow rate were utilized, many historical models represented PM as a lumped gravimetric index largely out of economy and lack of particle analysis methods. As a result such models did not provide information about particle fate in or through a unit operation. In this study, PM discrete phase modeling (DPM) and computational fluid dynamics (CFD) are applied to model PM fate as a function of particle size and flow rate in two common types of hydrodynamic separator (HS) units. The study examines the discretization requirements (as a discretization number, DN) and errors for particle size distributions (PSDs) that range from the common heterodisperse to a monodisperse PSD. PSDs are categorized based on granulometric indices. Results focus on ensuring modeling accuracy while examining the role of size dispersivity and overall PM fineness on DN requirements. The fate of common heterodisperse PSDs is accurately predicted for a DN of 16, whereas a single particle size index, commonly the d(50m), is limited to monodisperse PSDs in order to achieve similar accuracy.

Roles of saltation, sandblasting, and wind speed variability on mineral dust aerosol size distribution during the Puerto Rican Dust Experiment (PRIDE)

Recent field observations demonstrate that a significant discrepancy exists between models and measurements of large dust aerosol particles at remote sites. We assess the fraction of this bias explained by assumptions involving four different dust production processes. These include dust source size distribution (constant or dynamically changing according to saltation and sandblasting theory), wind speed distributions (using mean wind or a probability density function (PDF)), parent soil aggregate size distribution, and the discretization (number of bins) in the dust size distribution. The Dust Entrainment and Deposition global model is used to simulate the measurements from the Puerto Rican Dust Experiment (PRIDE) (2000). Using wind speed PDFs from observed National Centers for Environmental Prediction winds results in small changes in downwind size distribution for the production which neglects sandblasting, but it results in significant changes when production includes sandblasting. Saltation‐sandblasting generally produces more large dust particles than schemes which neglect sandblasting. Parent soil aggregate size distribution is an important factor when calculating size‐distributed dust emissions. Changing from a soil with large grains to a soil with smaller grains increases by 50% the fraction of large aerosols (D >5 μm) modeled at Puerto Rico. Assuming that the coarse medium sand typical of West Africa dominates all source regions produces the best agreement with PRIDE observations.

Dependence of the stability of semi-implicit atmospheric models on reference temperature profile

The linear stability of three-time-level finite-difference semi-implicit schemes is studied for hydrostatic atmospheric models with viscosity terms. The main objective is the investigation of stability dependence on choice of reference temperature profile. Characteristic equation for amplification factor is derived in the case of simplified baroclinic atmospheric numerical model, keeping the implicit approximation for the basic temperature profile and explicit approximation for its deviations. Analytical solutions for amplification factor are found in some special cases and computational experiments are performed for different values of model parameters, such as vertical discretization (number of levels and its distribution), temperature profile (basic and actual), and time step. The influence of the viscosity terms on scheme stability is also studied. Performed analysis confirms two main results obtained numerically in other models: instability generated by inappropriate determination of temperature profile is absolute and computational stability can be recovered if the basic reference profile is chosen to be warmer than the actual one.

Optimal techniques for class-dependent attribute discretization

Preprocessing of raw data has been shown to improve performance of knowledge discovery processes. Discretization of quantitative attributes is a key component of preprocessing and has the potential to greatly impact the efficiency of the process and the quality of its outcomes. In attribute discretization, the value domain of an attribute is partitioned into a finite set of intervals so that the attribute can be described using a small number of discrete representations. Discretization therefore involves two decisions, on the number of intervals and the placement of interval boundaries. Previous approaches for quantitative attribute discretization have used heuristic algorithms to identify partitions of the attribute value domain. Therefore, these approaches cannot be guaranteed to provide the optimal solution for the given discretization criterion and number of intervals. In this paper, we use linear programming (LP) methods to formulate the attribute discretization problem. The LP formulation allows the discretization criterion and the number of intervals to be integral considerations of the problem. We conduct experiments and identify optimal solutions for various discretization criteria and numbers of intervals.

The interpretation of the shear locking in beam elements

A new concept of explanation of the shear locking phenomenon occurring in the Timoshenko beam is presented. Instead of element matrix analysis the global expression of the equilibrium equation for a whole beam is considered. Based on the element stiffness matrix the equilibrium conditions for a regular discretized beam are derived in the form of one difference equation that converges to the analytical differential formulation for the continuous beam. It is clearly shown that application of the linear shape functions leads to an overly stiff element in the Mindlin elements because this approximation converges to the solution of a different equilibrium equation. From the stiffness matrices of exactly and reduced integrated linear and quadratic elements the difference equilibrium conditions are derived and they are compared with the exact equations of the bending beam problem. For each type of element the improved one is elaborated that assures the exact solution for beams irrespective of beam thickness and discretization (number of elements). The numerical analyses for various boundary conditions, beam load, discretization and thickness ratios are given.