Continuum Approximation Research Articles

Constitutive modelling for understanding stress-stretch behaviour of Lennard-Jones non-crystalline molecular Solid

Abstract Non-crystalline molecular solid materials have many scientific and engineering applications. This study develops a constitutive equation for understanding stress-stretch behaviour of non-crystalline molecular solid using Lennard-Jones (LJ) intermolecular interaction. The strain energy derived from Lennard-Jones interactions between molecules. Based on the excluded volume (spherical volume occupied by the molecules maintaining centre to centre distance with a reference molecule) and density of the molecules, strain energy density is developed. In order to relate the molecular approach with continuum approximation, the excluded volume and density are expressed as a function of strain invariants of right Cauchy-Green deformation tensor. Finally, the constitutive equation in the form of Cauchy stress tensor is developed using the present strain energy density function. The present constitutive model is used to study finite deformations of the molecular solid like uniaxial extension. We compare our theoretical results with the experimental data of flexible polyurethane foams and obtain very good agreements. The current constitutive model can predict the deformation of micro/nano engineering system components.

Quantum vortices, M2-branes and black holes

We study the partition functions of BPS vortices and magnetic monopole operators, in gauge theories describing N M2-branes. In particular, we explore two closely related methods to study the Cardy limit of the index on S2 × ℝ. The first method uses the factorization of this index to vortex partition functions, while the second one uses a continuum approximation for the monopole charge sums. Monopole condensation confines most of the N2 degrees of freedom except N32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {N}^{\\frac{3}{2}} $$\\end{document} of them, even in the high temperature deconfined phase. The resulting large N free energy statistically accounts for the Bekenstein-Hawking entropy of large BPS black holes in AdS4 × S7. Our Cardy free energy also suggests a finite N version of the N32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {N}^{\\frac{3}{2}} $$\\end{document} degrees of freedom.

Soliton solutions of the TWPA-SNAIL transmission line circuit equation under continuum approximation via the Jacobi elliptic function expansion method

Soliton solutions of the TWPA-SNAIL transmission line circuit equation under continuum approximation via the Jacobi elliptic function expansion method

Feeder bus service design under spatially heterogeneous demand

In rapidly sprawling urban areas and booming intercity express rail networks, efficiently designed feeder bus systems are more essential than ever to transport passengers to and from trunk-line rail terminals. When the feeder service region is sufficiently large, the spatial heterogeneity in demand distribution must be considered. This paper develops continuum approximation models for optimizing a heterogeneous fixed-route feeder network in a rectangular service region next to a rail terminal. Our work enhances previous studies by: (i) optimizing heterogeneous stop spacings along with line spacings and headways; (ii) accounting for passenger boarding and alighting numbers on bus dwell times and patron transfer delays at the rail terminal; and (iii) examining the advantages of asymmetric coordination between trunk and feeder schedules in both service directions. To tackle the increased modeling complexity, we introduce a semi-analytical method that combines analytically derived properties of the optimal solution with an iterative search algorithm. Local transit agencies can readily utilize this approach to design a real fixed-route feeder system.This paper reveals many findings and insights not previously reported. For instance, integrating the heterogeneous stop spacing optimization further reduces the system cost (by 4% under specific operating conditions). The cost savings increase with demand heterogeneity but decrease with the demand rate and service region size. Choosing the layout of feeder lines where buses pick up and drop off passengers along the service region’s shorter side also significantly lowers the system cost (by 6% when the service region’s aspect ratio is 1 to 2). Furthermore, coordinating trunk and feeder schedules in both service directions yields an additional cost saving of up to 20%.

Alleviating bus bunching via modular vehicles

The notorious phenomenon of bus bunching prevailing in uncontrolled bus systems produces irregular headways and downgrades the level of service by increasing passengers’ expected waiting time. Modular autonomous vehicles (MAVs), due to their ability to split and merge en route, have the potential to help both late and early buses recover from schedule deviation while providing continuous service. In this paper, we propose a novel bus bunching alleviation strategy for MAV-aided transit systems. We first consider a soft vehicle capacity constraint and establish a continuum approximation (CA) model (Model I) to capture the system dynamics intertwined with the MAV splitting and merging operations, and then establish an infinite-horizon stochastic optimization model to determine the optimal splitting and merging strategy. To capture the reality that passengers may fail to board an overcrowded bus, we propose a second model (Model II) by extending Model I to accommodate a hard vehicle capacity constraint. Based on the characteristics of the problem, we develop a customized deep Q-network (DQN) algorithm with multiple relay buffers and a penalized ruin state applicable for both models to optimize the strategy for each MAV. Numerical results show that the strategy obtained via the DQN algorithm is an effective bunch-proof strategy and has a better performance than the myopic strategy for MAV-aided systems and the two-way-looking strategy for conventional bus systems. Sensitivity analyses are also conducted to examine the effectiveness and benefits of the proposed strategy across different operation scenarios.

Stress field on intermediate length scales during triaxial compression

Defining a continuum stress field in a granular medium involves an approximation, as the underlying contact force network between particles is inhomogeneous and random. The continuum approximation introduces some baseline noise or variability in stress, which becomes more significant as the length scale of interest approaches the typical size of the grains. In this study, the authors evaluate how the stress variability resulting from the homogenisation of intergranular forces changes as the homogenisation scale is increased. This is done by obtaining the probability distribution function of stresses from a large number of repeated discrete-element method-based simulations of the triaxial compression of granular samples. The numerical experiments share sample dimensions, generation protocols and imposed stresses, but involve randomly generated sets of granular packings made of different numbers of particles. The results from this study are used to propose and develop a consistent general framework to rigorously describe mesoscale stress fields and hence fill the gap between existing micro- and macromechanical approaches. A backbone curve is obtained expressing the change in stress variability as the mesoscale is traversed. The results deduced from the backbone curve are in good agreement with those of independent numerical and physical experiments on granular soils.

A Physical Insight into Computational Fluid Dynamics and Heat Transfer

Mathematical equations that describe all physical processes are valid only under certain assumptions. One of them is the minimum scales used for the given description. In fact, this prohibits the use of derivatives in the mathematical models of the physical processes. This article represents a derivative-free approach for the mathematical modelling. The proposed approach for CFD and numerical heat transfer is based on the conservation and phenomenological laws, and physical constraints on the minimum problem-dependent spatial and temporal scales (for example, on the average free path of molecules and the average time of their collisions for gases). This leads to the derivative-free governing equations (the discontinuum approximation) that are very convenient for numerical simulation. The theoretical analysis of governing equations describing the fundamental conservation laws in the continuum and discontinuum approximations is given. The article demonstrates the derivative-free approach based on the correctly defined macroparameters (pressure, temperature, density, etc.) for the mathematical description of physical and chemical processes. This eliminates the finite-difference, finite-volume, finite-element or other approximations of the governing equations from the computational algorithms.

On dynamic fundamental diagrams: Implications for automated vehicles

The traffic fundamental diagram (FD) describes the relationships among fundamental traffic variables of flow, density, and speed. FD represents fundamental properties of traffic streams, giving insights into traffic performance. This paper presents a theoretical investigation of dynamic FD properties, derived directly from vehicle car-following (control) models to model traffic hysteresis. Analytical derivation of dynamic FD is enabled by (i) frequency-domain representation of vehicle kinematics (acceleration, speed, and position) to derive vehicle trajectories based on transfer function and (ii) continuum approximation of density and flow, measured along the derived trajectories using Edie's generalized definitions. The formulation is generic: the derivation of dynamic FD is possible with any analytical car-following (control) laws for human-driven vehicles or automated vehicles (AVs). Numerical experiments shed light on the effects of the density-flow measurement region and car-following parameters on the dynamic FD properties for an AV platoon.

DETERMINATION OF PHYSICAL AND MECHANICAL PROPERTIES OF POLYETHYLENE AND POLYMER NANOCOMPOSITES USING THE METHODS OF MOLECULAR DYNAMICS

Today, molecular dynamics (MD) simulation become an effective method for investigating and predicting the physical and mechanical properties (PMP) of materials at the atomic scale. In this work, initial configurations of molecular systems of pure polyethylene (PE) and carbon nanotube-reinforced nanocomposites (PE-CNT) are created and described by the united atom model and Dreiding force field. Equilibration of the initial configurations of molecular models carried out using NVT and NPT ensembles in LAMMPS. Simulated complex of PMP of PE and PE-CNT includes elastic modulus, Poisson's ratio, bulk modulus, shear modulus, yield strength, mass isobaric heat capacity, and coefficient of linear thermal expansion (CLTE). The results verification of the PE molecular model showed the obtained MD simulation data either coincide with available literature data or are close to them. The results justify the use of the same algorithms to obtaine reliable data on the PMP of PE-CNT. Comparing the complex PMP of molecular models of pure PE and PE-CNT, it was established the elastic modulus of PE-CNT α=1.4 % increases by 10.4% at 300 K and by 29.3% at 320 K. The yield strength of PE-CNT α=1.4 % also increases by 33.2 % at a strain rate of 109 s−1 and a temperature of 300 K compared to PE. Meanwhile, mass isobaric heat capacity and CLTE decrease with increasing temperature: mass isobaric heat capacity decreases by 1.3 % at 300 K and by 4.4 % at 320 K; CLTE decreases by 20.2 % at 300 K and by 22.6 % at 320 K. Nonlinear two-parameter dependencies of the PMP of PE-CNT obtained in the temperature range (280–320) K and the volume fraction of fillers (0–1.5) %. These results allow to develop of new composite materials without conducting sufficiently complex and time-consuming numerical experiments based on MD simulation. The obtained data on the complex PMP are necessary for modeling the thermo-elastic-plastic state of products made of PE-CNT under operational conditions in a continuum approximation.

Numerical modelling of continuous casting of round billets with turbulence in the melt

The present work aims to solve the continuous casting benchmark problem in axisymmetry with turbulence in the melt by the meshless method. The physical model of the liquid-solid system that involves mass, momentum and energy conservation is formulated in the mixture continuum approximation. The melt is assumed Newtonian and incompressible. A k-epsilon Reynolds averaged Navier-Stokes turbulence model is implemented with Abe-Kondoh-Nagano closures. The mushy region is modelled as a Darcy porous media with the Kozeny-Carman permeability model. The solution of partial differential equations is implemented locally using collocation with radial basis functions and explicit time-stepping. The velocity pressure coupling is performed using the fractional step method. The validation of the model is assessed by comparing its outcomes with the results of the finite volume method. A sensitivity study of the varying casting speed and temperature on the velocity, temperature, and solid fraction fields is shown.

MCell4 with BioNetGen: A Monte Carlo simulator of rule-based reaction-diffusion systems with Python interface.

Biochemical signaling pathways in living cells are often highly organized into spatially segregated volumes, membranes, scaffolds, subcellular compartments, and organelles comprising small numbers of interacting molecules. At this level of granularity stochastic behavior dominates, well-mixed continuum approximations based on concentrations break down and a particle-based approach is more accurate and more efficient. We describe and validate a new version of the open-source MCell simulation program (MCell4), which supports generalized 3D Monte Carlo modeling of diffusion and chemical reaction of discrete molecules and macromolecular complexes in solution, on surfaces representing membranes, and combinations thereof. The main improvements in MCell4 compared to the previous versions, MCell3 and MCell3-R, include a Python interface and native BioNetGen reaction language (BNGL) support. MCell4's Python interface opens up completely new possibilities for interfacing with external simulators to allow creation of sophisticated event-driven multiscale/multiphysics simulations. The native BNGL support, implemented through a new open-source library libBNG (also introduced in this paper), provides the capability to run a given BNGL model spatially resolved in MCell4 and, with appropriate simplifying assumptions, also in the BioNetGen simulation environment, greatly accelerating and simplifying model validation and comparison.

Modeling of heat and mass transfer in the discontinuum approximation

The article presents a theoretical analysis of the governing equations expressing the fundamental conservation laws in the continuum and discontinuum approximations, and methods for solving problems of hydrodynamics as one of the most important subfields of continuum mechanics. This article is an attempt to more accurately describe physicochemical macro-processes. It is shown that the most suitable equations for computer modeling are the conservation laws under natural constraints on the minimum spatial and time scales, i.e., equations without partial derivatives and constraints on the solution smoothness. Using the continuity and thermal conductivity equations, a phenomenological method for constructing and numerically solving the governing equations is presented, and comparison with the traditional approach is given.

Multi-agent deployment in 3-D via reaction–diffusion system with radially-varying reaction

This paper considers the problem of the deployment of a set of agents distributed on a disk-shaped grid onto three-dimensional (3-D) profiles, by using a continuum approximation (valid in the limit for a large number of agents) and then a control methodology for partial differential equations (PDEs). The agents’ collective behavior is modeled by a pair of radially-varying diffusion–reaction PDEs in polar coordinates, whose state determines the agents’ position. Having a radially-varying reaction coefficient not only increases the challenge of kernel equations becoming singular in radius, but also brings more potential deployment manifolds. To stabilize and increase the convergence of the deployment, a boundary controller and a boundary observer are designed by combining an infinite-dimensional backstepping approach with a Fourier series decomposition technique, thus driving all agents to the desired profile. A key feature of the presented result is that the desired profile only needs to be known by the leaders, with the followers only needing to follow a simple control strategy which requires only the measurement of its current position and communication with its neighbors as defined by the multi-agent system topology. The method provides closed-loop exponential stability with any prescribed convergence rate in the L2 norm. Simulation tests are shown to prove the effectiveness of the proposed algorithm.

Простейшая аппроксимация интегральных коэффициентов в уравнениях корреляционной магнитодинамики для ферромагнетиков

The equations of correlational magnetodynamics (CMD) describe a magnet in the continuum approximation. The main problem in constructing CMD is the calculation of integral coefficients, in particular, the coefficient describing the production of short-range order, depending on the three-particle distribution functions and the structure of the crystal lattice. The work provides the simplest approximations for the integral coefficients of CMD based on the value of pair correlations at the phase transition point. To ensure an equilibrium solution, the coefficients are additionally determined in the upper part of the phase plane according to the assumption of a helical magnetization structure. The resulting approximation provides qualitative agreement with the simulation results within the framework of the original atomistic model of the magnet, and at the same time it turns out to be simple enough for further analysis.

Scattering of water waves by multiple rows of vertical thin barriers

The reflection and transmission of waves by a periodic array of thin fixed identical vertical barriers extending uniformly through the depth of the fluid is considered. The water wave problem, which also has analogues in acoustics and electromagnetics, involves scattering by a large finite number of equally spaced rows, each consisting of an array of barriers with a linear periodic arrangement extending to infinity. The main purpose of the work is to compare the results of two approximate methods of solution based on homogenisation techniques with exact results allowing us to determine when the results can be expected to be in good agreement. When no approximation is made, a method of solution (Discrete Model) describing the propagation of incident waves through the exact barrier arrangement is derived using Floquet–Bloch eigensolutions for the corresponding infinite periodic array. It is shown how the solution can be reduced to a simple pair of integral equations which results only from matching the region containing barrier arrays to the two exterior domains. Numerical solutions are, nevertheless, relatively complicated and two approximate models are developed which considerably simplify the numerical effort required. In one approximation (Continuum Model I) it is assumed only that the spacing between two adjacent rows of barriers is small and is complementary to the commonly-used wide-spacing approximation. The complication of the geometry is removed by replacing the governing equations and boundary conditions with an effective continuous medium and a simplified pair of integral equations now govern the solutions. In a second approximation (Continuum Model II), a low-frequency homogenisation approach based on a long-wave assumption is invoked and matching now leads to closed-form expressions for the reflection and transmission coefficients. Results concentrate on the comparison between different methods which allows us to assess the validity of each continuum model. This understanding will be important in developing approximate models that describe the operation of wave energy farms consisting of large arrays of oscillating flaps designed to absorb energy.

Supratransmission phenomenon in a Fermi-Pasta-Ulam diatomic lattice.

The nonlinear supratransmission phenomenon in a Fermi-Pasta-Ulam (FPU) diatomic lattice with two forbidden bands is investigated. Using a decoupling ansatz for the motion of the two different sublattices combined with the continuum (quasidiscrete) approximation, we derived analytically the threshold amplitudes of supratransmission occurrence when a sinusoidal driving with frequency in the upper forbidden band (lower forbidden band gap between acoustic and optical modes) is applied at one end. The resulting estimate of the threshold of a lattice with a first heavy particle is different to the one obtained from a lattice with a first light particle, showing the influence of the driven particle and giving also the possibility to have two thresholds on each forbidden gap of a diatomic lattice by switching the order of light (m) and heavy (M) masses. In the lower forbidden band, the dependence of the supratransmission threshold on the mass ratio (μ=m/M) has been evidenced and it appears that for large (small) values of μ, that is μ>60%, the coupling between the two modes must (must not) be considered. Numerical explorations were subsequently performed with an emphasis on the dependence of the threshold on the driving frequency and also on the mass of the driven particle (light or heavy). A good agreement is found between the numerical and analytical thresholds. For the limit case where all the masses are identical, the results of the monoatomic FPU previously found in the literature are recovered.

The three-sided market of on-demand delivery

On-demand delivery services are three-sided markets that enable interactions between customers and suppliers with the help of crowdsourced drivers. Customers and suppliers may pay a commission to gain access to the service and drivers are granted a wage for providing their delivery service. This study characterizes the properties of three-side on-demand delivery markets, and proposes pricing strategies that enable the platform to manipulate the market towards profit or social welfare maximizing outcomes. We consider earning-sensitive independent drivers, price-and-time-sensitive customers, and price-sensitive suppliers. By assuming that all players are heterogeneous in their valuation of the service, we model their numbers as endogenously dependent on the price, wage, and commission of the platform. We also use the continuum approximation of the vehicle routing problem to derive customer waiting times. Our analyses show that suppliers internalize a portion of the driver wage, and the customers internalize the supplier commission. However, the platform requests lower commissions and offers a higher wage in social welfare maximization compared to profit maximization, which yields a negative profit for the platform in social welfare maximization. Moreover, we validate the results of the analytical model with numerical experiments on more realistic cases.

Multiscale Asymptotic Analysis Reveals How Cell Growth and Subcellular Compartments Affect Tissue-Scale Hormone Transport

Determining how cell-scale processes lead to tissue-scale patterns is key to understanding how hormones and morphogens are distributed within biological tissues and control developmental processes. In this article, we use multiscale asymptotic analysis to derive a continuum approximation for hormone transport in a long file of cells to determine how subcellular compartments and cell growth and division affect tissue-scale hormone transport. Focusing our study on plant tissues, we begin by presenting a discrete multicellular ODE model tracking the hormone concentration in each cell’s cytoplasm, subcellular vacuole, and surrounding apoplast, represented by separate compartments in the cell-file geometry. We allow the cells to grow at a rate that can depend both on space and time, accounting for both cytoplasmic and vacuolar expansion. Multiscale asymptotic analysis enables us to systematically derive the corresponding continuum model, obtaining an effective reaction–advection–diffusion equation and revealing how the effective diffusivity, effective advective velocity, and the effective sink term depend on the parameters in the cell-scale model. The continuum approximation reveals how subcellular compartments, such as vacuoles, can act as storage vessels, that significantly alter the effective properties of hormone transport, such as the effective diffusivity and the induced effective velocity. Furthermore, we show how cell growth and spatial variance across cell lengths affect the effective diffusivity and the induced effective velocity, and how these affect the tissue-scale hormone distribution. In particular, we find that cell growth naturally induces an effective velocity in the direction of growth, whereas spatial variance across cell lengths induces effective velocity due to the presence of an extra compartment, such as the apoplast and the vacuole, and variations in the relative sizes between the compartments across the file of cells. It is revealed that hormone transport is faster across cells of decreasing lengths than cells with increasing lengths. We also investigate the effect of cell division on transport dynamics, assuming that each cell divides as soon as it doubles in size, and find that increasing the time between successive cell divisions decreases the growth rate, which enhances the effect of cell division in slowing hormone transport. Motivated by recent experimental discoveries, we discuss particular applications for transport of gibberellic acid (GA), an important growth hormone, within the Arabidopsis root. The model reveals precisely how membrane proteins that mediate facilitated GA transport affect the effective tissue-scale transport. However, the results are general enough to be relevant to other plant hormones, or other substances that are transported in a similar way in any type of cells.

Vlasov–Leont’ev Foundation Model for the Dynamic Analysis of a Laterally Loaded Pile in Homogeneous Viscoelastic Soil

The Vlasov–Leont’ev foundation model describes the soil displacement field as a product of separable functions and utilizes the calculus of variation to obtain the governing differential equations describing the motion of the structure/substructure and the soil. In this paper, the model is employed to solve a lateral pile dynamics (in a homogeneous linear viscoelastic soil) problem. The step-by-step formulation is demonstrated, and the results obtained are compared with existing formulations (Winkler-, cone-, approximate continuum–, and rigorous continuum–based formulations) having varying degrees of rigor and accuracy in the literature. The comparison is performed to evaluate the applicability of the present formulation for two class of problems (theories): half-space and soil stratum on a rigid base. It is found that the present formulation is applicable to problems in which the pile is embedded in a soil stratum on a rigid base (relevant to practice). Further, from the comparative results, discussions on the accuracy, computational efficiency, and ability to capture the mechanics of a laterally loaded pile–soil interaction problem are also provided. The advantages and limitations of the formulations are also included in the paper.

Voltage Controlled Ab-Initio Simulations of Metal-Insulator Interfaces Immersed in Electrolyte for Corrosion and Battery Applications Via the Quantum Continuum Approximation (QCA)

While Density Functional Theory (DFT) has been an invaluable tool for exploring atomic interaction of defects and interfaces, it has been difficult to include information about the voltage of the larger electrochemical system into these calculations. This is because the electric field at any given simulation region is dependent on the broader interface, which is usually 10s of nm long. Fully simulating this interface with DFT is computationally prohibitive, and most studies simulate smaller portions. This reduction eliminates the information about the surrounding electric field, however, and thus the interaction with the global potential is unaccounted for. In this talk, we introduce the Quantum Continuum Approximation (QCA) as a methodology for including interface voltage information into limited-size DFT calculations by self-consistently coupling DFT calculations of defect/interface regions to Poisson-Boltzmann descriptions of the surrounding insulating and electrolyte regions. By reducing the number of atoms needed to incorporate voltage information into DFT calculations by an order of magnitude, QCA makes the voltage-dependent first-principles study of full electrochemical interfaces viable. We demonstrate this methodology to examine Cl- ion insertion into aluminum/aluminum oxide interfaces as a function of voltage. We find good agreement between the voltage threshold for stability predicted by QCA and those found experimentally. We then demonstrate QCA as a tool for measuring Li+ ion transport barriers through different solid-electrolyte interface (SEI) regions in battery chemistry as a function of voltage. Our work demonstrates the ability to attach a potentiostat to DFT calculations of interfaces relevant to simulations of electrochemical applications at reasonable computational cost. SNL is managed and operated by NTESS under DOE NNSA contract DE-NA0003525