Disparate Time Scales Research Articles

Efficient Fluid-Thermal-Structural Time Marching with Computational Fluid Dynamics

This study focuses on the development of time-marching procedures for efficient and accurate fluid-thermal-structural analysis with time-accurate computational fluid dynamics. The developed procedures are based on a loosely coupled, partitioned framework for the fluid, thermal, and structural solvers—each with different second-order time integrators. The procedures also implement subcycling, where disparate time scales between the solvers are leveraged to minimize communication between solvers. The first scheme uses a second-order predictor of fluid loads, along with second-order interpolations of structural and thermal solutions. The second scheme is an extension that adds a corrector step to the structural and thermal solvers. The two schemes are benchmarked against both a strongly coupled approach with subiterations and a basic loosely coupled approach that omits the use of predictors/correctors. The schemes are examined in the context of the aerothermoelastic response of a panel in high supersonic flow, and are found to maintain second-order time accuracy with and without subcycling. Furthermore the approaches compare favorably against a strongly coupled approach at significantly reduced computational times, with the predictor-corrector approach yielding speedups of 2–4 times. In comparison, a basic scheme can yield characteristically different behavior.

Kinetic analysis of the ionization ratio effects on a three‐species plasma

In partially ionized plasmas, the energy transferred to electrically charged species by the electromagnetic field can be partly channelized to the population of neutrals, due to interspecies collisional processes. Depending on the relative density of neutrals, these effects may govern the collective plasma dynamics by drastically modifying particle dynamics and energy‐transport processes with respect to the fully ionized plasma‐approximation models. In this work, the influence of the ionization ratio ri on a partially ionized plasma is analysed by means of a three‐species one‐dimensional kinetic model to compute transient and steady state velocity‐dependent distribution functions. The conservative collision operators accounting for charge–charge and charge–neutral interactions allow studying several plasma scenarios with the same entire number of particles per unit of volume but for an increasing ri parameter, in the presence of a modulated signal‐like electric field. For a sequence of plasma scenarios of fixed ri, ranging from typical weakly ionized to highly ionized plasma values ri ∼ 10−7–10−4, the mass species flows are examined. These flows behave linearly with respect to ri up to a value ri ≃ 10−5 from which the quasi‐linear dependence is critically altered. The convection–diffusion equations are solved with the semianalytical Propagator Integral Method, which behaves well to deal with conservative operators, density, and field discontinuities, allowing for the use of collision terms of disparate time and spatial characteristic scales. The results can be relevant to a wide class of plasma systems and to analyse the ionization ratio effects on transport coefficients.

A fast and robust numerical scheme for solving models of charge carrier transport and ion vacancy motion in perovskite solar cells

Drift-diffusion models that account for the motion of both electronic and ionic charges are important tools for explaining the hysteretic behaviour and guiding the development of metal halide perovskite solar cells. Furnishing numerical solutions to such models for realistic operating conditions is challenging owing to the extreme values of some of the parameters. In particular, those characterising (i) the short Debye lengths (giving rise to rapid changes in the solutions across narrow layers), (ii) the relatively large potential differences across devices and (iii) the disparity in timescales between the motion of the electronic and ionic species give rise to significant stiffness. We present a finite difference scheme with an adaptive time step that is posed on a non-uniform staggered grid that provides second order accuracy in the mesh spacing. The method is able to cope with the stiffness of the system for realistic parameters values whilst providing high accuracy and maintaining modest computational costs. For example, a transient sweep of a current-voltage curve can be computed in only a few minutes on a standard desktop computer.

Kinetic Simulation of Collisional Magnetized Plasmas with Semi-implicit Time Integration

Plasmas with varying collisionalities occur in many applications, such as tokamak edge regions, where the flows are characterized by significant variations in density and temperature. While a kinetic model is necessary for weakly-collisional high-temperature plasmas, high collisionality in colder regions render the equations numerically stiff due to disparate time scales. In this paper, we propose an implicit–explicit algorithm for such cases, where the collisional term is integrated implicitly in time, while the advective term is integrated explicitly in time, thus allowing time step sizes that are comparable to the advective time scales. This partitioning results in a more efficient algorithm than those using explicit time integrators, where the time step sizes are constrained by the stiff collisional time scales. We implement semi-implicit additive Runge–Kutta methods in COGENT, a high-order finite-volume gyrokinetic code and test the accuracy, convergence, and computational cost of these semi-implicit methods for test cases with highly-collisional plasmas.

(Invited) Bridging Long Temporal Scales: Durability Analysis of Electrochemical Systems

Fundamental understanding of the mechanisms of degradation of electrochemical systems for energy storage and conversion is receiving intense investigation. A vast number of energy storage and conversion devices that share common features, i.e., a reliance on nano-scaled materials that are not thermodynamically stable, and whose costs are too high compared to existing technologies. Models have been developed with detailed physics for many degradation phenomena, for example, formation of the solid electrolyte interphase in lithium-ion batteries, dissolution and ripening of precious metal catalyst in proton exchange membrane fuel cells, and corrosion of catalyst supports. Increasingly these models deal with ever smaller time and length scales. At the same time, to allow for rapid advances in these electrochemical technologies, we must not only understand the degradation phenomena but also have some predictive capability that a physical model provides. Design procedures used for batteries for satellite applications illustrate this point. The aerospace industry requires that cells be tested for 70 % of life to become qualified for space. Therefore, a satellite battery for a 10-year mission would require a full 7 years of testing in the lab using the final design. As a result, these applications cannot take advantage of the latest technology. Similarly, fuel cells and energy storage devices for automobiles require useable lifetimes of more than 10 years. The development of thorough fundamental understanding of the mechanisms of failure combined with physics-based models is essential. The timescale for these detailed physical models are much shorter than the timescale for analysis of lifetime (years) behavior. Therefore, a computationally efficient means of bridging across these time scales is needed. Here our purpose was to develop the framework for analysis of systems that combine detailed performance and degradation. This framework has the potential to transform the way that these systems are designed and operated. The computational expense of these meticulous degradation models does not permit their practical use in higher-level systems analysis. It is important to include detailed physics, but computationally expensive. One answer is to use response surface methodology. Indeed, we have used this to incorporate electrochemical degradation into a hybrid-system design for a vehicle. Briefly this approach runs the detailed simulation many times to map the anticipated design space. Then surrogate models are developed to represent the output of the detailed model. One obstacle with this method is that each time the detailed model is modified with new physics or even just a parameter change, the process must be repeated from the beginning. An alternative approach is developed that follows the methodology of Kevrekidis et al. The detailed model is run with small time increments for a short period of time, long enough so the time derivative can be evaluated accurately. Then using a forward Euler scheme, for example, a large step in time is taken. The temporal variations of the dependent variables are well-behaved, as illustrated with particle size. The metric for success is straightforward—savings in computational time over running the full detailed model. These steps can be adjusted, much like adaptive time stepping, as needed to achieve the desired accuracy. In addition to the computational efficiency, the advantage over RSM is that if the underlying detailed model is changed, no changes are needed in the framework. This methodology is applied to the dissolution of platinum in proton exchange fuel cells investigated for long-term automotive application. gPROMS is used to solve the differential equations and provides a framework to bridge the disparate time scales.

Beyond Mean-Field Microkinetics: Toward Accurate and Efficient Theoretical Modeling in Heterogeneous Catalysis

Kinetics as the link between atomic scale properties and macroscopic functionalities is indispensable in describing surface chemical reactions and computation-based rational design of catalysts. Kinetic Monte Carlo (KMC) on the explicit lattice can resolve events taking place on the catalytic surfaces at the atomic level. It can explicitly account for spatial correlations due to lateral interactions among adsorbates, which have been proved to significantly affect the surface chemical reactions. However, the disparity in time scales of various processes (e.g., adsorption/desorption, diffusion, and reaction) usually makes brute force KMC simulations impractical. Here, we propose a method, namely XPK, to extend the phenomenological kinetics (PK) for the accurate and efficient microkinetic modeling of heterogeneous catalysis. XPK is achieved through a hybrid between the diffusion-only KMC on the explicit lattice to evaluate the reaction propensities and later an implicit lattice KMC in the PK form to evolve t...

Direct observation of ultrafast large-scale dynamics of an enzyme under turnover conditions

The functional cycle of many proteins involves large-scale motions of domains and subunits. The relation between conformational dynamics and the chemical steps of enzymes remains under debate. Here we show that in the presence of substrates, domain motions of an enzyme can take place on the microsecond time scale, yet exert influence on the much-slower chemical step. We study the domain closure reaction of the enzyme adenylate kinase from Escherichia coli while in action (i.e., under turnover conditions), using single-molecule FRET spectroscopy. We find that substrate binding increases dramatically domain closing and opening times, making them as short as ∼15 and ∼45 µs, respectively. These large-scale conformational dynamics are likely the fastest measured to date, and are ∼100-200 times faster than the enzymatic turnover rate. Some active-site mutants are shown to fully or partially prevent the substrate-induced increase in domain closure times, while at the same time they also reduce enzymatic activity, establishing a clear connection between the two phenomena, despite their disparate time scales. Based on these surprising observations, we propose a paradigm for the mode of action of enzymes, in which numerous cycles of conformational rearrangement are required to find a mutual orientation of substrates that is optimal for the chemical reaction.

A third-order class-D amplifier with and without ripple compensation

We analyse the nonlinear behaviour of a third-order class-D amplifier, and demonstrate the remarkable effectiveness of the recently introduced ripple compensation (RC) technique in reducing the audio distortion of the device. The amplifier converts an input audio signal to a high-frequency train of rectangular pulses, whose widths are modulated according to the input signal (pulse-width modulation) and employs negative feedback. After determining the steady-state operating point for constant input and calculating its stability, we derive a small-signal model (SSM), which yields in closed form the transfer function relating (infinitesimal) input and output disturbances. This SSM shows how the RC technique is able to linearise the small-signal response of the device. We extend this SSM through a fully nonlinear perturbation calculation of the dynamics of the amplifier, based on the disparity in time scales between the pulse train and the audio signal. We obtain the nonlinear response of the amplifier to a general audio signal, avoiding the linearisation inherent in the SSM; we thereby more precisely quantify the reduction in distortion achieved through RC. Finally, simulations corroborate our theoretical predictions and illustrate the dramatic deterioration in performance that occurs when the amplifier is operated in an unstable regime. The perturbation calculation is rather general, and may be adapted to quantify the way in which other nonlinear negative-feedback pulse-modulated devices track a time-varying input signal that slowly modulates the system parameters.

Multiple Pathways and Time Scales for Conformational Transitions in apo-Adenylate Kinase.

The open/close transition in adenylate kinase (AK) is regarded as a representative example for large-scale conformational transition in proteins, yet its mechanism remains unclear despite numerous experimental and computational studies. Using extensive (∼50 μs) explicit solvent atomistic simulations and Markov state analysis, we shed new lights on the mechanism of this transition in the apo form of AK. The closed basin of apo AK features an open NMP domain while the LID domain closes and rotates toward it. Therefore, although the computed structural properties of the closed ensemble are consistent with previously reported FRET and PRE measurements, our simulations suggest that NMP closure is likely to follow AMP binding, in contrast to the previous interpretation of FRET and PRE data that the apo state was able to sample the fully closed conformation for "ligand selection". The closed state ensemble is found to be kinetically heterogeneous; multiple pathways and time scales are associated with the open/close transition, providing new clues to the disparate time scales observed in different experiments. Besides interdomain interactions, a novel mutual information analysis identifies specific intradomain interactions that correlate strongly to transition kinetics, supporting observations from previous chimera experiments. While our results underscore the role of internal domain properties in determining the kinetics of open/close transition in apo AK, no evidence is observed for any significant degree of local unfolding during the transition. These observations about AK have general implications to our view of conformational states, transition pathways, and time scales of conformational changes in proteins. The key features and time scales of observed transition pathways are robust and similar from simulations using two popular fixed charge force fields.

Natural extension of fast-slow decomposition for dynamical systems.

Modeling and parameter estimation to capture the dynamics of physical systems are often challenging because many parameters can range over orders of magnitude and are difficult to measure experimentally. Moreover, selecting a suitable model complexity requires a sufficient understanding of the model's potential use, such as highlighting essential mechanisms underlying qualitative behavior or precisely quantifying realistic dynamics. We present an approach that can guide model development and tuning to achieve desired qualitative and quantitative solution properties. It relies on the presence of disparate time scales and employs techniques of separating the dynamics of fast and slow variables, which are well known in the analysis of qualitative solution features. We build on these methods to show how it is also possible to obtain quantitative solution features by imposing designed dynamics for the slow variables in the form of specified two-dimensional paths in a bifurcation-parameter landscape.

Analysis of Nonprocessive Molecular Motor Transport Using Renewal Reward Theory

We propose and analyze a mathematical model of cargo transport by non-processive molecular motors. In our model, the motors change states by random discrete events (corresponding to stepping and binding/unbinding), while the cargo position follows a stochastic differential equation (SDE) that depends on the discrete states of the motors. The resulting system for the cargo position is consequently an SDE that randomly switches according to a Markov jump process governing motor dynamics. To study this system we (1) cast the cargo position in a renewal theory framework and generalize the renewal reward theorem and (2) decompose the continuous and discrete sources of stochasticity and exploit a resulting pair of disparate timescales. With these mathematical tools, we obtain explicit formulas for experimentally measurable quantities, such as cargo velocity and run length. Analyzing these formulas then yields some predictions regarding so-called non-processive clustering, the phenomenon that a single motor cannot transport cargo, but that two or more motors can. We find that having motor stepping, binding, and unbinding rates depend on the number of bound motors, due to geometric effects, is necessary and sufficient to explain recent experimental data on non-processive motors.

Numerical simulation of optical breakdown in a liquid droplet induced by a laser pulse

Modelling and simulation of interaction of laser pulse with individual liquid droplets are of crucial importance in terms of development and code coupling for simulation of laser propulsion and detonation in gas-droplet systems. These applications are challenging and complex for many reasons, one among them is the disparate time and length scales that are required to resolve for an accurate physical representation of the problem. The injection of liquid droplets with low evaporation temperature causes optical breakdown on the individual droplet and leads to a drop of the minimum pulse energy of detonation in the gas-droplet mixture. The mathematical models of various stages of the optical breakdown on individual droplet and numerical methodology for computer modelling of the laser-induced breakdown are developed. Sub-models of optical breakdown on liquid droplet include droplet heating to boiling temperature, its evaporation and formation of vapor aureole around the droplet, ionization of vapor aureole and development of electron avalanche, appearance of micro-plasma spots and their expansion, propagation of shock wave inside the droplet and in the surrounding gas. The threshold intensity of optical breakdown on individual water droplet and its dependence on droplet radius, location of droplet, total energy and radius of laser pulse are studied.

Statistical Measures to Quantify Similarity between Molecular Dynamics Simulation Trajectories.

Molecular dynamics simulation is commonly employed to explore protein dynamics. Despite the disparate timescales between functional mechanisms and molecular dynamics (MD) trajectories, functional differences are often inferred from differences in conformational ensembles between two proteins in structure-function studies that investigate the effect of mutations. A common measure to quantify differences in dynamics is the root mean square fluctuation (RMSF) about the average position of residues defined by Cα-atoms. Using six MD trajectories describing three native/mutant pairs of beta-lactamase, we make comparisons with additional measures that include Jensen-Shannon, modifications of Kullback-Leibler divergence, and local p-values from 1-sample Kolmogorov-Smirnov tests. These additional measures require knowing a probability density function, which we estimate by using a nonparametric maximum entropy method that quantifies rare events well. The same measures are applied to distance fluctuations between Cα-atom pairs. Results from several implementations for quantitative comparison of a pair of MD trajectories are made based on fluctuations for on-residue and residue-residue local dynamics. We conclude that there is almost always a statistically significant difference between pairs of 100 ns all-atom simulations on moderate-sized proteins as evident from extraordinarily low p-values.

Full conjugate heat transfer modelling for steam turbines in transient operations

Steam turbines are increasingly required to operate more flexibly, leading to a need for more detailed calculations and design considerations of the thermal conditions during transient operations. A major challenge in transient thermal designs is the time scale disparity. For the natural cooling of a turbine, the physical process is typically in hours or tens of hours, but on the other hand, the time step sizes typically usable tend to be very small, in seconds or sub-seconds, due to the numerical stability requirement for natural convection as often encountered.The present paper reports a validation and demonstration study of a new loosely coupled conjugate heat transfer (CHT) approach developed for efficient and accurate simulations of steam turbine transient operations (Shut-down/Natural cooling). The verification and validation have been carried out for a complex 3D geometry configuration with attributes of realistic but geometrically simplified full-scale steam turbines and compared with a well-established simplified working method backed up by experimental data. Further comparisons are made against a direct baseline CHT method. The present results demonstrate a generally good agreement with the simplified working method. The new loosely coupled method is shown to give a marked speed-up (by a factor of 240) compared to the baseline directly coupled method with the same accuracy. It thus offers the potential for significant improvement in predicting the long-duration transient natural convection conjugate heat transfer problems of steam turbines in terms of computational efficiency and accuracy.

Simulation of thin-film battery response to periodic loading by a transition matrix approximation using boundary and nonlinearity error analysis

Simulating repeating loading events on dynamic systems can be challenging when large timescale disparities exist coupled with aperiodic effects. Batteries driving switched/pulsed loads represent one such situation. Large timescale disparity can be experienced by solid-state batteries driving switching microactuators or microelectronics, due to extremely short transient response times of microscale systems relative to some of the battery’s own dynamics. Projecting state changes over a long series of fast-timescale loading events using a transition matrix approach was shown previously to significantly reduce numerical expense of simulation compared to full modeling. Here we develop an approach for further accelerated simulation of a battery driving a microelectromechanical system (MEMS) actuator that quantifies errors and addresses overhead expenses in projecting battery states across multiple fast events. This is done with a definition of system states that allows efficient transition matrix generation, and an analysis of key errors associated with projection. This error analysis enables targeted modification to the transition matrix during projection. A case study explores these modeling approaches in a capacitively loaded, battery usage scenario of a piezoelectrically-driven microrobot where the proposed improvements reduce the numerical cost (function calls) by over 44x from the prior approach. Conditions for further simplified modeling are discussed.

Assessment of Unsteadiness Modeling for Transient Natural Convection

Turbine flexible operations with faster startups/shutdowns are required to accommodate emerging renewable power generations. A major challenge in transient thermal design and analysis is the time scale disparity. For natural cooling, the physical process is typically in hours, but on the other hand, the time-step sizes typically usable tend to be very small (subseconds) due to the numerical stability requirement for natural convection as often observed. An issue of interest is: What time-step sizes can and should be used in terms of stability as well as accuracy? In this work, the impact of flow temporal gradient and its modeling is examined in relation to numerical stability and modeling accuracy for transient natural convection. A source term-based dual-timing formulation is adopted, which is shown to be numerically stable for very large time-steps. Furthermore, a loosely coupled procedure is developed to combine this enhanced flow solver with a solid conduction solver for solving unsteady conjugate heat transfer (CHT) problems for transient natural convection. This allows very large computational time-steps to be used without any stability issues, and thus enables to assess the impact of using different time-step sizes entirely in terms of a temporal accuracy requirement. Computational case studies demonstrate that the present method can be run stably with a markedly shortened computational time compared to the baseline solver. The method is also shown to be more accurate than the commonly adopted quasi-steady flow model when unsteady effects are non-negligible.

On Large Eddy Simulation Based Conjugate Heat Transfer Procedure for Transient Natural Convection

Natural convection is an important heat transfer mode for flexible operations of gas turbines and steam turbines. Its prediction presents considerable challenges. The strong interdependence between fluid and solid parts points to the need for coupled fluid–solid conjugate heat transfer (CHT) methods. The fundamental fluid–solid time scale disparity is further compounded by the long-time scales of practical turbine flexible operations. In addition, if a high-fidelity flow model (e.g., large eddy simulation (LES)) is adopted to resolve turbulence fluctuations, extra mesh dependency on solid domain mesh may arise. In this work, understanding of the extra solid mesh dependency in a directly coupled LES based CHT procedure is gained by an interface response analysis, leading to a simple and clear characterization of erroneously predicted unsteady interface temperatures and heat fluxes. A loosely coupled unsteady CHT procedure based on a multiscale methodology for solving problems with large time scale disparity is subsequently developed. A particular emphasis of this work is on efficient and accurate transient CHT solutions in conjunction with the turbulence eddy resolved modeling (LES) for natural convection. A two-scale flow decomposition associated with a corresponding time-step split is adopted. The resultant triple-timing formation of the flow equations can be solved efficiently for the fluid–solid coupled system with disparate time scales. The problem statement, analysis, and the solution methods will be presented with case studies to underline the issues of interest and to demonstrate the validity and effectiveness of the proposed methodology and implemented procedure.

Multiscale Methods for Fracture: A Review $$^\bigstar $$ ★

The global response of a system is often governed by the material behaviour at smaller length scales. Investigating the system mechanics at the smallest scale does not always provide the complete picture. Therefore, in the ambitious objective to derive the overall full-scale global response using a bottom-up approach, multiscale methods coupling disparate length and time scales have been evolved in the past two decades. The major objective of the multiscale methods is to reduce the computational costs by coupling the inexpensive coarse-scale/continuum based models with expensive fine-scale models. The fine-scale region is employed in the critical areas, such as crack tips or core of the dislocation. To improve the efficiency the fine-scale domain is adaptively adjusted as the defects propagate. As a result, the accuracy of the fine-scale model is combined with the efficiency of the coarse-scale model, arriving at a computationally efficient and accurate multiscale model. Currently, multiscale methods are applied to study problems in numerous fields, involving multiphysics. In this article, we present an overview of the multiscale methods for fracture applications. We discussed the techniques to model the coarse- and fine-scale domains, details of the coupling methods, adaptivity, and efficient coarse-graining techniques. The article is concluded with comments on recent trends and future scope.

Dynamic-template-directed multiscale assembly for large-area coating of highly-aligned conjugated polymer thin films

Solution processable semiconducting polymers have been under intense investigations due to their diverse applications from printed electronics to biomedical devices. However, controlling the macromolecular assembly across length scales during solution coating remains a key challenge, largely due to the disparity in timescales of polymer assembly and high-throughput printing/coating. Herein we propose the concept of dynamic templating to expedite polymer nucleation and the ensuing assembly process, inspired by biomineralization templates capable of surface reconfiguration. Molecular dynamic simulations reveal that surface reconfigurability is key to promoting template–polymer interactions, thereby lowering polymer nucleation barrier. Employing ionic-liquid-based dynamic template during meniscus-guided coating results in highly aligned, highly crystalline donor–acceptor polymer thin films over large area (>1 cm2) and promoted charge transport along both the polymer backbone and the π–π stacking direction in field-effect transistors. We further demonstrate that the charge transport anisotropy can be reversed by tuning the degree of polymer backbone alignment.

Inductively coupled plasmas at low driving frequencies

We discuss the peculiarities of inductively coupled plasma (ICP) at low driving frequencies. The ratio of electric to magnetic field, decreases with decreasing frequency according to Faraday’s law—higher magnetic fields are required to induce the same electric field at lower frequencies. We point out that the ratio of can be non-uniform in space depending on primary coil configuration and the presence of ferromagnetic materials. In this paper, we consider examples of low-frequency ICPs with negligibly small magnetic fields in plasma. The disparity of time scales for ion transport and the electron energy relaxation results in nonlinear plasma dynamics at low frequencies. Numerical simulations demonstrate that at low frequencies, the presence of plasma has very little effect on spatial distributions of the electric and magnetic fields, which are determined solely by the coil geometry and by the presence of ferromagnetic cores. Simulations of plasma dynamics in ICP over a wide range of driving frequencies and gas pressures illustrate high-frequency, quasi-static and dynamic regimes of discharge operation and explain some trends observed in experiments.