Continuum Approximation Research Articles

Locating Urban Consolidation Centers under Shipper Rationality

An urban consolidation center (UCC) is a logistics facility used to combine deliveries of multiple shippers for reducing freight traffic in dense urban areas. Despite their benefits, such as reducing less-than-truckload deliveries, former applications of UUCs have faced backlash from shippers that were forced to use them even though they did not experience any cost savings. Such failures are often due to an unsuitable location of the UCC and the additional costs of facility operation and maintenance that are unfairly divided between the shippers. This paper introduces shipper rationality to measure the willingness of shippers to use an optimally located UCC. We use a continuum approximation approach to derive the routing costs of the shippers as closed-form expressions. We present three objective functions depending on the preference of the regulatory agency that chooses the location of the UCC, allowing rational shippers to join only if they experience cost savings from the UCC. We discuss influential factors in the optimal location of a UCC, including the customer density of each shipper, the establishment cost of the UCC, and the location of each shipper’s fulfillment center.

A linear-time crystal-growth algorithm for discretization of continuum approximation

Continuous approximation is regarded as a scalable and insightful method for acquiring near-optimum solutions to various location problems. A continuous approximation solution is nonetheless only a continuous density function and a further discretization procedure is needed to obtain a discrete location solution for engineering practice. Inspired by the process of “crystal growth”, this paper proposes a constructive heuristic algorithm as an alternative to the classic meta-heuristic disk algorithm (Ouyang and Daganzo, 2006) in discretizing a continuous solution from a continuum approximation location model. The main idea of this algorithm is to rasterize the space into a set of small cells (either regular triangles or squares) and repeatedly grow a core cell into a full-service area according to a certain visiting sequence. Thus, it has only linear time complexity proportional to the number of cells in the space. Numerical examples are conducted to test the performance of the proposed algorithm. The results indicate that this algorithm can solve discrete facility locations more efficiently and exhibits robust performance compared with the disk algorithm.

A hybrid discrete–continuum framework for modelling filtration

Typical mathematical frameworks for modelling the blocking behaviour of a filter due to particle deposition fall into one of two categories: a continuum approximation, whereby particle deposition is assumed to occur in such a way that all pores in the material are in the same state of blocking at any given time; or a discrete model, where blocking is treated as individual events in both space and time. While the former is computationally inexpensive, the latter allows for variation from pore to pore. This pore-to-pore variation has been shown to provide a qualitative change in the observed filtration behaviour that is essential to reproduce experimental observations. We present a hybrid model that describes the location of particle depositions in a continuum manner while retaining a discrete, stochastic component to capture the time at which a blocking event occurs. The model is able to grade between the aforementioned extreme continuum and discrete cases through a parameter that controls the spatial extent of a blocking event. This enables us to uncover the way in which the nature of the blocking process changes between these two pre-existing models. The model also captures the key ingredients of a fully discrete stochastic model at a fraction of the computational cost, making it ready to use to describe other complex filtration scenarios.

Adsorption of Cisplatin Molecules in Nanoporous Inorganic Materials as Drug Delivery Vehicles

The use of nanoparticles as anticancer cargo systems for drug delivery is a promising modality, as they avoid the known toxicity of anticancer drugs on healthy cells by the delivery of multiple drugs to the target cells. Here, the adsorption behavior of cisplatin drug molecules in two different inorganic materials, silica and metallic gold, is investigated mathematically. The 6–12 Lennard-Jones potential, together with the continuum approximation, is adapted to calculate the molecular interatomic energies between molecules. For each material, the relation between the pore radius ℓ and the minimum energy is determined, and the results indicate that the minimum energy occurs when the radii are ℓ=5.3 and ℓ=4.7 Å for the silica and gold nanopores, respectively. The method is promising for applications in the design of novel nanocapsules for future targeted drug and gene delivery.

Design of heterogeneous flexible-route public transportation networks under low demand

This paper presents design methods for a flexible-route transit system, in which vehicles travel within predetermined areas to provide door-to-door service. The main advantage of this system is that passengers no longer have to access transit stations in order to gain service. This system is suitable for low and heterogeneous passenger demand distribution as it features a hybrid system layout that includes both a hub-and-spoke network in the peripheral region and a grid network in the central region, along with heterogeneous local routes that address local demand variations. Continuum approximation (C.A.) is used to reduce the computation burden by formulating the design problem with respect to a few decision variables. We compare the performance of the proposed transit system with (i) the typical fixed-route system, (ii) the homogeneous flexible-route grid system, and (iii) the flexible-route grid system with local routes, in hypothetical settings. It is found through our numerical examples that the integration of the three proposed features (i.e. flexible transit, local tubes, and hybrid structure), as compared to counterparts with only two or fewer features, yields lower combined agency and user costs under the assumed low heterogeneous demand distribution. We then apply the design framework to a more realistic case for the City of Changzhi, China. An implementable design for Changzhi is developed, and its performance verified with simulations, demonstrating accuracy and applicability of the proposed continuum approximation model.

Preferential attachment network model with aging and initial attractiveness

In this paper, we generalize the growing network model with preferential attachment for new links to simultaneously include aging and initial attractiveness of nodes. The network evolves with the addition of a new node per unit time, and each new node has m new links that with probability Π i are connected to nodes i already present in the network. In our model, the preferential attachment probability Π i is proportional not only to k i + A, the sum of the old node i's degree k i and its initial attractiveness A, but also to the aging factor , where τ i is the age of the old node i. That is, . Based on the continuum approximation, we present a mean-field analysis that predicts the degree dynamics of the network structure. We show that depending on the aging parameter α two different network topologies can emerge. For α < 1, the network exhibits scaling behavior with a power-law degree distribution P(k) ∝ k −γ for large k where the scaling exponent γ increases with the aging parameter α and is linearly correlated with the ratio A/m. Moreover, the average degree k(t i , t) at time t for any node i that is added into the network at time t i scales as where 1/β is a linear function of A/m. For α > 1, such scaling behavior disappears and the degree distribution is exponential.

Description of the buckling of a chain of hard spheres in terms of Jacobi functions

A linear chain of hard spheres, confined by a transverse harmonic potential, buckles under compression between two hard walls. Jacobi functions provide exact analytic solutions of a differential equation (related to the Duffing equation) for the displacement profile of this chain, within a continuum approximation. Here we explore these solutions, describing their forms and the way in which they vary as system parameters are changed. This is illustrated by reference to two-dimensional diagrams in which each such solution is represented by a point and contour plots illustrate various characteristics of the solutions (period, compression, localization, etc.). Our findings enrich the study of the buckling instability for a linear chain of particles. The approach presented here has the advantage of being based on a simple chain of hard spheres and is straightforward in its interpretation. As such these results may provide insight into more complex experiments, such as those involving the buckling of ion chains.

Analytical soliton solutions for cold bosonic atoms (CBA) in a zigzag optical lattice model employing efficient methods

This research finds an equation in a continuous domain and a discrete equation governing the system of cold bosonic atoms (CBA) in a zigzag optical lattice using a continuum approximation. Many solutions to the equation were obtained using two distinct methods: the three-wave approach (multi-wave interaction, rational solutions, and rational solution interaction) and the extended sub-equation method. These analytical approaches are more effective, consistent, and comprehensive mathematical tools for obtaining various exact closed-form solutions for a wide range of fractional space-time nonlinear evolution equations encountered in optical physics, condensed matter physics, and plasma physics. The solutions generated are in the form of hyperbolic and trigonometric solutions, and other-form solutions are obtained. Three-dimensional graphics and contour plots are often used to depict the graphical representations of the combined soliton solutions. These findings will aid our understanding of the dynamics of the zigzag optical grids and many other structures formed by colder bosonic atoms. The applied approaches are more simple, efficient, and straightforward to obtain the closed-form solutions for various nonlinear evolution equations in the fields of nonlinear sciences and physical engineering.

Сплошносредная модель биоконвекции с центростремительной силой

The paper is devoted to mathematical modeling of the unusual collective behavior of emperor penguins in winter. Recent observations of flocks of penguins of this species in their natural habitat have shown that penguins use various forms of adaptive behavior that help them survive in theconditions of the Antarctic winter. Despite the fact that each penguin is well protected from cold, these sea-birds are periodically forced to gather in dense groups, within which the temperature necessary for survival is maintained. Observations have revealed the effect of a sudden liquefaction of a dense huddle of penguins, accompanied by a macroscopic circulation of birds from the edge of the group to its center and back. This paper is the first to assert that the observed phenomenon is bioconvection, which should be interpreted as a second-order phase transition. Moreover, the circulation of birds in a dense group can be explained by the mechanism of thermal convection, with the desire of penguins to move toward the temperature maximum playing the role of an effective force field. In the continuum approximation, we propose a mathematical model of the phenomenon, which is reduced to the equation of a self-gravitating porous disk saturated with an incompressible fluid that generates heat. We derive the governing equations in the Darcy-Boussinesq approximation and formulate a nonlinear boundary value problem. An exact solution of the linearized problem for infinitesimal perturbations of the base state is obtained, and the critical values of the control parameter for the onset of bioconvection are calculated. For finite-amplitude perturbations, the boundary value problem is solved using a finite-difference method. We show that in the case of axial symmetry, the most dangerous is the four-vortex motion. In the case of wind breaking the symmetry of the problem, the two-vortex circulation of penguins becomes the most dangerous. The numerical results obtained based on our theoretical developments are compared with observations of penguins in the Antarctic.

Localization of electromagnetic waves in a zigzag lattice of waveguides with competing cubic and quintic nonlinear responses

The zigzag waveguide array is considered, where even waveguides are composed of optically linear material, and their nearest neighbors are characterized by positive cubic nonlinearity and negative fifth-order nonlinearity. In the continuum (long-wave) approximation the solutions of the system of coupled wave equations are found, which describe waves localized in the lattice and harmonic propagating waves. Keywords: nonlinear waves, discrete photonics, waveguides, solitons.

Локализация электромагнитных волн в зигзагообразной решетке волноводов с конкурирующими нелинейными откликами третьего и пятого порядков

The zigzag waveguide array is considered, where even waveguides are composed of optically linear material, and their nearest neighbors are characterized by positive cubic nonlinearity and negative fifth-order nonlinearity. In the continuum (long-wave) approximation the solutions of the system of coupled wave equations are found, which describe waves localized in the lattice and harmonic propagating waves.

New Optical Solitons with Variational Principle and Collective Variable Strategy for Cold Bosons in Zig‐Zag Optical Lattices

The theme of this piece of research is to investigate the collective variable (CV) as well as semi‐inverse techniques to explore a significant model of cold bosonic atoms in a zig‐zag optical lattice. The system is reduced to an important equation by utilizing the continuum approximation and explains the soliton’s dynamics in sense of pulse variables. These parameters are amplitude, temporal position, chirp, width, frequency, and phase which are termed as collective variables (CVs). The proposed methods are more straightforward, succinct, accurate, and simple to calculate. Furthermore, to employ the computational counterfeit on the system of six ordinary differential equations that denote all the CVs incorporated in the supposed ansatz, a well‐established computational method which is the Runge–Kutta scheme of order four is applied. The CV method is exerted to resolve the evolution of pulse parameters with the propagation distance and graphical illustrations which are also given. Moreover, figures reveal the fascinating periodic oscillations of frequency, width, amplitude, and chirp of soliton. Solitons and their numerical behavior to interpret fluctuations in CVs are presented for several values of super‐Gaussian pulse parameters. Also, the results for the semi‐inverse method which are bright solitons are provided in the form of 2D, 3D, and density plots for the distinct values of the fractal parameter to understand their physical significance. This scheme is effective in finding variational principles of various nonlinear evolution equations. Some compelling characteristics pertaining to the current scrutiny are also deduced.

Design of a two-echelon last-mile delivery model

Due to high congestion in cities and growing demand for last-mile delivery services, several companies have been implementing two-echelon distribution strategies over the past few years. Notably, the installation of urban transshipment points has gained increasing attention, used by logistics operators to transfer goods from large freight trucks to smaller and more agile vehicles for last-mile delivery. Nevertheless, the main challenge is how to decide the number and location of these facilities under the presence of demand uncertainty. In this paper, we develop a two-stage stochastic program to design two-echelon last-mile delivery networks under demand uncertainty. This approach decomposes the problem into strategic decisions (facility location) and operational decisions (daily distribution of goods). To address large-scale instances, we solve the model through the sample average approximation (SAA) technique and estimate the optimal routing costs (of the SAA counterpart) using a continuous approximation method. Using a real-world case study with more than 1300 customers from New York City, our results provide several managerial insights regarding the mix of transportation modes, facility location, and the impact of allowing the outsourcing of customer demand. We provide extensive validation of the two-stage stochastic program results through a simulation-based approach and the calculation of the value of the stochastic solutions. • We develop a two-stage stochastic program to design two-echelon last-mile delivery networks under demand uncertainty. • We use sample average approximation and continuum approximation of route cost to address large-scale instances. • For a real-world case study, we provide managerial insights regarding the mix of transportation modes, facility locations, and the benefits of outsourcing deliveries. • We extensively validate our results using a simulation-based approach and assess the value of accounting for stochasticity.

Jitney-lite: a flexible-route feeder service for developing countries

The paper develops a novel strategy for delivering feeder service in support of trunk-line transit. The strategy is well suited to developing countries, where costs of emergent communication technologies often preclude their use. The strategy, termed Jitney-lite, is a form of collective transportation that provides a degree of flexibility. Patrons who board an outbound Jitney-lite vehicle at a transit station are delivered to their doorsteps. On the return trip to the station, the vehicle boards new patrons in the manner of traditional, fixed-route, fixed-stop feeder-bus service. Continuum approximation models are formulated, both for Jitney-lite and traditional services. The models are used to determine the conditions for which one service form imparts lower generalized costs than the other. A case study of the Bangkok Metropolitan Region reveals that Jitney-lite tends to generate lower costs in the city's periphery, where travel demands are relatively low. The service can be especially cost-effective where sidewalks and other pedestrian infrastructure are underdeveloped, and where residents earn middle-to-high incomes.

Integrating COVID-19 health risks into crowding costs for transit schedule planning

The public transport sector worldwide experienced the worst impact in recent history, in terms of ridership loss, due to the COVID-19 pandemic. The pandemic negatively affected passengers' perceptions of public transport and is likely to make a lasting impact on ridership, trip patterns, and modal share. Without any supportive changes to transit operations, ridership is likely to decline. This study explores the setting of frequencies in transit lines and proposes a two-part methodology that addresses the changing perceptions of users, especially in a health-related context. The first part develops a mathematical model that expresses the pre-COVID-19 cost of passenger crowding as an integral part of user costs to determine the optimal headway that considers the trade-offs between user and operator costs. A continuum approximation for the demand of the bus line has been used in the derivation. The second part extends the developed model to include both the costs of the health risks associated with the COVID-19 pandemic and crowding. The developed models will help transit planners and operators to plan and adapt operations to changing health risks during the pandemic and post-pandemic. Several numerical examples are provided to describe the uses and applications of the analytical models using information obtained from the literature.

Effective Continuum Approximations for Permeability in Brown-Coal and Other Large-Scale Fractured Media

The stability of open-pit brown-coal mines is affected by the manner in which water is transmitted or retained within their slopes. This in turn is a function of the in-situ fracture network at those mines. Fracture networks in real mines exhibit significant degrees of heterogeneity; encompassing a wide range of apertures, inter-fracture separations, and orientations. While each of these factors plays a role in determining fluid movement, over the scale of a mine it is often impractical to precisely measure, let alone simulate, the behaviour of each fracture. Accordingly, effective continuum models capable of representing the bulk effects of the fracture network are needed to understand the movement of fluid within these slopes. This article presents an analysis of the fracture distribution within the slopes of a brown coal mine and outlines a model to capture the effects on the bulk permeability. A stress-dependent effective-fracture-permeability model is introduced that captures the effects of the fracture apertures, spacing, and orientation. We discuss how this model captures the fracture heterogeneity and the effects of changing stress conditions on fluid flow. The fracture network data and the results from the effective permeability model demonstrate that in many cases slope permeability is dominated by highly permeable but low-probability fractures. These results highlight the need for models capable of capturing the effects of heterogeneity and uncertainty on the slope behaviour.

Influence of the next-nearest neighbor and the boson–boson interactions on U-shaped, W-shaped profile and modulation instability gain spectra in a zig–zag optical lattice

In this paper, we employ the auxiliary equation method (AEM) to show the leverage of the nearest neighbors hopping and the boson–boson interaction in a zig zag optical lattice on the stressed optical solitons (os) and the Modulation Instability (MI) gain of cold bosonic atoms. The pedestal of the work is set on the continuum approximation of a cold bosonic atoms in a zig zag optical lattice which has demonstrated in detailed in refs. [Tala-Tebue E, Rezazadeh H, Djoufack ZI, et al. Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44; Djoufack ZI, TalaTebue E, FotsaNgafo F, et al. Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving nextnearest neighbor interactions. Optik 164(2018):575–589]. The results collected have enclosed bright, dark optical solitons and the combined bright-dark and bright-bright optical solitons. Compared the acquired results with Tala-Tebue et al. [Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44], further the os including U-shaped and W-shaped profile have been drop in and the powerful of the nearest neighbors hopping and the boson–boson interaction has been exposed across the spatiotemporal and plot evolution of the bright and dark os. Therewith, the MI bands have been stressed and it is observed some new attitude of the MI gain spectra under the induction of the model parameters. The collected results are new in literature in our knowledge and will improve the previous results acquired in refs. [Tala-Tebue E, Rezazadeh H, Djoufack ZI, et al. Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44; Djoufack ZI, TalaTebue E, FotsaNgafo F, et al. Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving nextnearest neighbor interactions. Optik 164(2018):575–589; Zhao XH, Tian B, Liu DY, et al. Dark solitons interaction for a (2+1)-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain. Superlattices Microstruct 100(2016):587–595; Uddin MF, Hafez MG, Hammouch Z, et al. Periodic and rogue waves for Heisenberg models of ferromagnetic spin chains with fractional beta derivative evolution and obliqueness. Waves Random Complex Media 2020. https://doi.org/10.1080/17455030.2020.1722331; Inc M, Aliyu AI, Yusuf A. Optical solitons to the nonlinear Shrdingers equation with spatio-temporal dispersion using complex amplitude ansatz. J Mod Opt 2017;64(21):2273–2280].

A Continuous Model for Designing Corridor Systems with Modular Autonomous Vehicles Enabling Station-wise Docking

The “asymmetry” between spatiotemporally varying passenger demand and fixed-capacity transportation supply has been a long-standing problem in urban mass transportation (UMT) systems around the world. The emerging modular autonomous vehicle (MAV) technology offers us an opportunity to close the substantial gap between passenger demand and vehicle capacity through station-wise docking and undocking operations. However, there still lacks an appropriate approach that can solve the operational design problem for UMT corridor systems with MAVs efficiently. To bridge this methodological gap, this paper proposes a continuum approximation (CA) model that can offer near-optimal solutions to the operational design for MAV-based transit corridors very efficiently. We investigate the theoretical properties of the optimal solutions to the investigated problem in a certain (yet not uncommon) case. These theoretical properties allow us to estimate the seat demand of each time neighborhood with the arrival demand curves, which recover the “local impact” property of the investigated problem. With the property, a CA model is properly formulated to decompose the original problem into a finite number of subproblems that can be analytically solved. A discretization heuristic is then proposed to convert the analytical solution from the CA model to feasible solutions to the original problem. With two sets of numerical experiments, we show that the proposed CA model can achieve near-optimal solutions (with gaps less than 4% for most cases) to the investigated problem in almost no time (less than 10 ms) for large-scale instances with a wide range of parameter settings (a commercial solver may even not obtain a feasible solution in several hours). The theoretical properties are verified, and managerial insights regarding how input parameters affect system performance are provided through these numerical results. Additionally, results also reveal that, although the CA model does not incorporate vehicle repositioning decisions, the timetabling decisions obtained by solving the CA model can be easily applied to obtain near-optimal repositioning decisions (with gaps less than 5% in most instances) very efficiently (within 10 ms). Thus, the proposed CA model provides a foundation for developing solution approaches for other problems (e.g., MAV repositioning) with more complex system operation constraints whose exact optimal solution can hardly be found with discrete modeling methods.

Towards an agent-based model to simulate osseointegration in powder-bed 3D printed implant-like structures

The orthopedic industry is still searching for an efficient way to replace bone loss due to surgical procedures such as arthroplasty and limb-sparing surgery. Additive manufacturing (AM) presents an opportunity to manufacture affordable patient-specific implants. Optimization of the implant–bone interface to maximize osseointegration (bone ingrowth) has not been appropriately addressed. Mechanobiological models, suited to predict mechanical adaptation of bone, cannot be used to predict osseointegration inside implants as the implant is not exposed to any mechanical loading until it is fully accepted by the host body. Biological models relying on partial differential equations based on continuum approximation are not well-suited to predict the discrete phenomenon of osseointegration. This study proposes an agent-based modeling (ABM) approach for representing the osseointegration process for orthopedic implants produced by powder-bed additive manufacturing processes. Agent-Based Modeling (ABM) is a cellular automata based discrete computing technique that uses rule-based mathematics derived from experimental studies to simulate evolutionary phenomena. In this paper, osseointegration inside a hexagonal closed packing of AM powder particles is modeled using ABM. Cellular agents such as pre-osteoblasts and osteoblasts are realistically modeled as cubic cells. The proposed model underpredicts osseointegration at early stages but predicts osseointegration at around 21 days with sufficient accuracy when compared to the in vitro test conducted by Xue et al. in 2007.

External potential modifies memory of solute particles: A particle-viscous bath model

The response to external forces is one of the important properties of stochastic processes. In the present paper, we derive the generalized Langevin equation (GLE) for the description of the non-Markovian Brownian motion (BM) of a particle, e.g. a molecular solute in a solvent, in a harmonic potential. It is assumed that not only the solute particle but also the surrounding molecules respond to the external field. This effect is taken into account by a modification of the Caldeira-Legget particle-bath model. It is also assumed that the bath particles experience friction, modeled by the Stokes force. We show that under the influence of the external force the memory function and the random thermal force are affected by the external potential. Explicit expressions for these functions and the elastic force acting on the Brownian particle are obtained in the discrete variant of the model and its continuum approximation. In the latter case, analytical formulas for relevant time correlation functions describing the dynamics of the solute are obtained coming from the memory function that describes its free BM.