Time Adaptive Scheme Research Articles

A linear adaptive second‐order backward differentiation formulation scheme for the phase field crystal equation

AbstractIn this article, we present and analyze a linear fully discrete second order scheme with variable time steps for the phase field crystal equation. More precisely, we construct a linear adaptive time stepping scheme based on the second order backward differentiation formulation (BDF2) and use the Fourier spectral method for the spatial discretization. The scalar auxiliary variable approach is employed to deal with the nonlinear term, in which we only adopt a first order method to approximate the auxiliary variable. This treatment is extremely important in the derivation of the unconditional energy stability of the proposed adaptive BDF2 scheme. However, we find for the first time that this strategy will not affect the second order accuracy of the unknown phase function by setting the positive constant large enough such that The energy stability of the adaptive BDF2 scheme is established with a mild constraint on the adjacent time step radio . Furthermore, a rigorous error estimate of the second order accuracy of is derived for the proposed scheme on the nonuniform mesh by using the uniform bound of the numerical solutions. Finally, some numerical experiments are carried out to validate the theoretical results and the efficiency of the proposed scheme combined with the time adaptive strategy.

A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility

In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with a general mobility. The proposed fully-discrete scheme is carefully constructed based on the combination of first and second-order backward differentiation formulas with nonuniform time steps for temporal approximation and the central finite difference for spatial discretization. The discrete maximum bound principle is proved of the proposed scheme by using the kernel recombination technique under certain mild constraints on the time steps and the ratios of adjacent time step sizes. Furthermore, we rigorously derive the discrete H 1 H^{1} error estimate and energy stability for the classic constant mobility case and the L ∞ L^{\infty } error estimate for the general mobility case. Various numerical experiments are also presented to validate the theoretical results and demonstrate the performance of the proposed method with a time adaptive strategy.

A Time-Adaptive Order Reduction Technique for the Doyle-Fuller-Newman Electrochemical Model of Lithium-Ion Batteries

A time-adaptive reduced order model (ROM) is developed for the electrochemical model for lithium-ion cells derived by Doyle, Fuller, and Newman (DFN) [M. Doyle, T. F. Fuller and J. Newman, J. Electrochem. Soc., 140 1526 (1993)]. The main advantage of a time-adaptive strategy is that it does not require a set of full order model simulations to be generated beforehand and, thus, it is the most cost-effective alternative when no databases are available. However, the reduction of this electrochemical problem exhibits special features that require ad hoc solutions, preventing the application of generic strategies. This complexity is carefully analysed, focusing on mode selection, treatment of non-linearities and error estimation. Despite of all this analysis being done for a pseudo-two-dimensional DFN model, we show that such complexity is intrinsic to the physics of the electrochemical problem, making the analysis applicable to a pseudo-four-dimensional DFN model, where results prove that the benefits of a reduction in the number of degrees of freedom are more self-evident. The efficiency, robustness and accuracy of our method are remarkable, as shown by the macroscopic (cell voltage) and internal (variable distributions) results obtained from the simulation of two different electrochemical cells under several charge/discharge C-rates.

An a posteriori strategy for adaptive schemes in time for one-dimensional advection-diffusion transport equations

Stability condition is a more restrictive constraint that leads to unnecessary small-time steps with respect to the accuracy and results in computational time wastage. We propose a node by node adaptive time scheme to relax the stability constraint enabling a larger global time step for all the nodes. A nonlinear procedure for optimising both the schemes in time and space is proposed in view of increasing the numerical efficiency and reducing the computational time. The method is based on a four-parameter family of schemes we shall tune in function of the physical data (velocity, diffusion), the characteristic size in time and space, and the local regularity of the function leading to a nonlinear procedure. The a posteriori strategy we adopt consists in, given the solution at time tn, computing a candidate solution with the highest accurate schemes in time and space for all the nodes. Then, for the nodes that present some instabilities, both the schemes in time and space are modified and adapted in order to preserve the stability with a large time step. The updated solution is computed with node-dependent schemes both in time and space. For the sake of simplicity, only convection-diffusion problems are addressed as a prototype with a two-parameters five-points finite difference method for the spatial discretisation together with an explicit time two-parameters four-stages Runge–Kutta method. We prove that we manage to obtain an optimal time-step algorithm that produces accurate numerical approximations exempt of non-physical oscillations.

The worm-LBM, an algorithm for a high number of propagation directions on a lattice Boltzmann grid: The case of phonon transport

The lattice Boltzmann method (LBM) is a numerical approach to tackle problems described by a Boltzmann type-equation, where time, space, and velocities are discretized to describe scattering and advection. Even though the LBM executes advection along a lattice direction without numerical error, its usage in the high Knudsen number regime (ballistic) has been hindered by the ray effect problem (for dimensions greater than 1D). This problem has its origin in the low number of available propagation directions on standard LBM lattices. Here, to overcome this limitation, we propose the worm-lattice Boltzmann method (worm-LBM), which allows a high number of lattice directions by alternating in time the basic directions described within the next neighbor schemes. Additionally, to overcome the velocity anisotropy issue, which otherwise clearly manifests itself in the ballistic regime (e.g. the 2 higher grid velocity of the D2Q8 scheme along the diagonal direction compared to the axial one), the time-adaptive scheme (TAS) is proposed. The TAS method makes use of pausing advection on the grid, allowing to impose not only isotropic propagation but also arbitrary direction-dependent grid velocity. Last but not least, we propose a grid-mean free path (grid-MFP) correction to correctly handle the aforementioned velocity issue in the diffusive limit, without affecting the ballistic one. We provide a detailed description of the TAS method and the worm-LBM algorithm, and verify their numerical accuracy by using several transient diffusive-ballistic phonon transport cases, including different initial and boundary conditions. We demonstrate the accuracy of the new worm-LBM to describe problems where a high angular resolution (i.e. a high number of propagation directions) is required, as the in-plane thermal transport problem under adiabatic-diffusive boundary conditions. In this particular case, we show that schemes with a low number of propagation directions (D2Q8) result in an overestimation of the analytical Fuchs-Sondheimer solution for intermediate and high Knudsen numbers, and that schemes with a higher number of propagation directions are required to correctly describe the problem. Overall, the new, very accurate, and efficient worm-LBM algorithm, free of numerical smearing and false scattering, has the potential to be at the forefront of the numerical solvers to tackle the advective part of different equations in a wide field of applications.

Algorithms for most reliable routes on stochastic and time-dependent networks

This study presents algorithms to determine the most reliable routes on stochastic and time-dependent networks. The measure of reliability adopted is the probability of on-time arrival at the destination, given a threshold arrival-time. We propose two distinct algorithms to determine optimal time-adaptive strategy and optimal apriori path on stochastic and time-dependent networks. First, a decreasing order-of-time algorithm is proposed to determine the optimal strategy to the sink from all node and departure-time combinations. Second, a label-correcting, network pruning algorithm is proposed to determine the optimal path between the source and the sink for a given departure-time. The correctness of both the proposed algorithms is proved and their computational complexity expressions are derived. The efficacy of the proposed procedures is demonstrated on large-scale transportation networks. This work has the potential to facilitate wider application of stochastic and time-dependent networks in reliability-based modeling and analysis.

Simulating the temporal change of the active response of arteries by finite elements with high-order time-integrators

A new numerical method to model the active response of arteries is proposed. Vasoconstrictors and vasodilators in the bloodstream diffuse from the lumen into the arterial wall through the intima and cause the smooth muscle cells, mostly in the media, to contract. We combine the diffusion process with the mechanical model in Yosibash and Priel (Comput Methods Appl Mech Eng 237–240:51–66, 2012). Finite element computations of the fully coupled field problem using time-adaptive, high-order time-integration methods based on diagonally-implicit Runge–Kutta methods are investigated with respect to their convergence behavior for linear and non-linear loading paths. Since the blood pressure is periodic, highly non-linear external loading path, the step-size estimation has to be adapted to minimize step-size rejections. An example of an artery analysis that illustrates the advantage of the proposed time-adaptive scheme is provided.

Study on the restart algorithm for a buried hot oil pipeline based on wavelet collocation method

The influence of soil temperature field on the restart process of crude oil pipeline is not considered in previous studies. However, it cannot be ignored for the long-time restart of buried hot oil pipeline. Considering the long computation time for the solution of soil temperature field, the study on the hydraulic-thermal coupled acceleration algorithm for the restart process of buried hot oil pipeline is conducted. Firstly, the mathematical models considering the influence of soil temperature field are introduced in detail for the restart problem. Then based on wavelet collocation method, the adaptive gird is generated to reduce the computational cost of soil temperature field. Furthermore, the hydraulic-thermal coupled acceleration algorithm is proposed, which can realize the fast coupled solution of the restart process. Finally, the optimal values of threshold and coarsest level of resolution are investigated for the restart problem, and the time-adaptive strategy and no-time-adaptive strategy are compared. And the conclusions are drawn from the comparison and analysis of numerical results, which can provide beneficial guidance to the choices of threshold, coarsest level of resolution and adaptive strategy in future studies.

Finding the Most Reliable Strategy on Stochastic and Time-Dependent Transportation Networks: A Hypergraph Based Formulation

This study addresses the problem of determining the most reliable time-adaptive strategy on a stochastic and time-dependent transportation network. The reliability is measured as a conic combination of the mean and standard-deviation of travel time and is termed robust-cost. The stochastic time-dependent network is represented as a directed acyclic hypergraph, where the time-adaptive strategies correspond to the hyperpaths. This representation transforms the problem to that of determining the hyperpath with the least robust-cost on the constructed hypergraph. The minimum robust-cost strategy problem is difficult to solve because of the non-linear objective function. Consequently, the solution procedures commonly adopted in the literature —that are based on substrategy optimality and substrategy non-dominance —are not applicable to this problem. In this light, we propose a novel bounds-based iterative algorithm that determines the minimum robust-cost strategy on the stochastic and time-dependent networks. This algorithm needs to determine the least and K-best strategies in the second moment of travel time, for which an efficient procedure is also proposed. The algorithm is shown to be exact and exhibit parameterically polynomial behavior; computational tests were performed to demonstrate its efficiency. Further, tests showed that the minimum robust-cost strategy compromises little in terms of the mean travel time (0.2%–2.9%) —compared to least expected travel time strategy— with significant reduction in travel time variability (6.2%–29.8%).

Adaptive Q-aware Scheduling Algorithm for Multi-Service Flows in 802.16j Networks

The IEEE 802.16j standard defines Mobile Multi-hop Relay Wireless Cellular Networks. Introduction of Relays in WiMAX networks pose new challenges along with the advantages of High Throughput and Improved Coverage Area. Thus Scheduling plays a crucial role in Mobile Multihop Relay(MMR) Wireless Networks. The Queue aware scheduling algorithm [1] maximizes the throughput by considering Queues at Base Station(BS) and Relay Station(RS) as well along with Concurrent Transmission Technique. In Queue aware scheduling, as all the arrived packets are queued up in FCFS manner, the high priority packets are obviously delayed. In this paper, QOS guarantee is incorporated by considering multiple Queues for different service flows at Base Station for Queue aware scheduling algorithm. Real Time Adaptive Scheme (RTAS) [2] is modified to improve throughput and used for picking the packets from service class sub queues(SCSQ) to include them in BS input queue. The results show that high priority packets are given proper importance while considering low priority packets as well. Throughput is also increased because of Modified RTAS and Concurrent Transmission Technique of Queue aware scheduling. General Terms WiMAX, Scheduling, Algorithm, Throughput, Queue, Packet

A monolithic finite element approach using high-order schemes in time and space applied to finite strain thermo-viscoelasticity

This article addresses a thermo-mechanically coupled problem of thermo-viscoelasticity at finite strains using a monolithic approach. The underlying equations are based on the non-linear transient heat equation, the local equilibrium conditions and the evolution equations of the internal variables. The latter describe the hardening behavior of the material. If the method of vertical lines is applied, its first step–namely the spatial discretization–yields a system of differential-algebraic equations (DAE-system). Here, we employ the p-version of the finite element method based on integrated Legendre polynomials. This can lead to very precise solutions in the spatial domain. In order to be accurate in the time-domain as well, stiffly accurate, diagonally-implicit Runge–Kutta methods are applied to solve the DAE-system yielding a coupled system of non-linear algebraic equations. In this article, the system is solved monolithically by employing the Multilevel-Newton algorithm. Accordingly, a high-order result is obtained in the space and the time domain. The numerical concept is applied to a constitutive model of finite strain thermo-viscoelasticity. Several examples are applied to demonstrate the efficiency and applicability of the numerical scheme. It is especially the transient problems that call for time-adaptive schemes which are naturally embedded in the concept.

Implementation of HHT algorithm for numerical integration of multibody dynamics with holonomic constraints

The Hilber–Hughes–Taylor (HHT) algorithm is implemented in this paper to solve the dynamic problem of flexible multibody system with holonomic constraints. The incremental formulas of governing equations of motion, a set of differential algebraic equations of index 3, are presented and integrated direct by current approach. The parameterization of rotation has been paid a special attention, which makes the formula of HHT algorithm much more complicated because of the nonlinearity of rotations. The associated nonlinear algebraic equations are solved with the application of simplified Newton’s iteration to simplify the linearization of finite rotation. Two options are available for solving the incremental linear equations, one is the direct method based on LU decomposition and the other GMRES-based matrix-free method. Based on the estimated truncation error, the time adaptive scheme is designed to enhance the integrating speed. Numerical simulations prove the modified HHT algorithm behavior as the second-order accuracy with numerical damping. The application of conformal rotation vector improves the robustness of the current approach which makes it possible to integral rotation of arbitrary magnitude.

Ranking paths in stochastic time-dependent networks

In this paper we address optimal routing problems in networks where travel times are both stochastic and time-dependent. In these networks, the best route choice is not necessarily a path, but rather a time-adaptive strategy that assigns successors to nodes as a function of time. Nevertheless, in some particular cases an origin–destination path must be chosen a priori, since time-adaptive choices are not allowed. Unfortunately, finding the a priori shortest path is an NP-hard problem.In this paper, we propose a solution method for the a priori shortest path problem, and we show that it can be easily extended to the ranking of the first K shortest paths. Our method exploits the solution of the time-adaptive routing problem as a relaxation of the a priori problem. Computational results are presented showing that, under realistic distributions of travel times and costs, our solution methods are effective and robust.

Response analysis of nonlinear multi-degree-of-freedom systems with uncertain properties to non-Gaussian random excitations

The investigation reported in this paper is concerned with the development of an approach for response analysis of multi-degree-of-freedom (mdof) nonlinear systems with uncertain properties of large variations and under non-Gaussian nonstationary random excitations. The developed approach makes use of the stochastic central difference (SCD) method, time co-ordinate transformation (TCT), and adaptive time schemes (ATS). It is applicable to geometrically complicated systems idealized by the finite element method (FEM). For demonstration of its use and availability of results for direct comparison, a two-degree-of-freedom (tdof) nonlinear asymmetric system with uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitations is considered. Computed results obtained for the system with and without uncertain natural frequencies, and under Gaussian and non-Gaussian nonstationary random excitations are presented. It is concluded that the approach is relatively simple, accurate and efficient to apply.

Large Nonlinear Responses of Spatially Nonhomogeneous Stochastic Shell Structures Under Nonstationary Random Excitations

A novel approach for determining large nonlinear responses of spatially homogeneous and nonhomogeneous stochastic shell structures under intensive transient excitations is presented. The intensive transient excitations are modeled as combinations of deterministic and nonstationary random excitations. The emphases are on (i) spatially nonhomogeneous and homogeneous stochastic shell structures with large spatial variations, (ii) large nonlinear responses with finite strains and finite rotations, (iii) intensive deterministic and nonstationary random disturbances, and (iv) the large responses of a specific spherical cap under intensive apex nonstationary random disturbance. The shell structures are approximated by the lower order mixed or hybrid strain based triangular shell finite elements developed earlier by the author and his associate. The novel approach consists of the stochastic central difference method, time coordinate transformation, and modified adaptive time schemes. Computed results of a temporally and spatially stochastic shell structure are presented. Computationally, the procedure is very efficient compared with those entirely or partially based on the Monte Carlo simulation, and it is free from the limitations associated with those employing the perturbation approximation techniques, such as the so-called stochastic finite element or probabilistic finite element method. The computed results obtained and those presented demonstrate that the approach is simple and easy to apply.

Bollène-2002 Experiment: Radar Quantitative Precipitation Estimation in the Cévennes–Vivarais Region, France

Abstract The Bollène-2002 Experiment was aimed at developing the use of a radar volume-scanning strategy for conducting radar rainfall estimations in the mountainous regions of France. A developmental radar processing system, called Traitements Régionalisés et Adaptatifs de Données Radar pour l’Hydrologie (Regionalized and Adaptive Radar Data Processing for Hydrological Applications), has been built and several algorithms were specifically produced as part of this project. These algorithms include 1) a clutter identification technique based on the pulse-to-pulse variability of reflectivity Z for noncoherent radar, 2) a coupled procedure for determining a rain partition between convective and widespread rainfall R and the associated normalized vertical profiles of reflectivity, and 3) a method for calculating reflectivity at ground level from reflectivities measured aloft. Several radar processing strategies, including nonadaptive, time-adaptive, and space–time-adaptive variants, have been implemented to assess the performance of these new algorithms. Reference rainfall data were derived from a careful analysis of rain gauge datasets furnished by the Cévennes–Vivarais Mediterranean Hydrometeorological Observatory. The assessment criteria for five intense and long-lasting Mediterranean rain events have proven that good quantitative precipitation estimates can be obtained from radar data alone within 100-km range by using well-sited, well-maintained radar systems and sophisticated, physically based data-processing systems. The basic requirements entail performing accurate electronic calibration and stability verification, determining the radar detection domain, achieving efficient clutter elimination, and capturing the vertical structure(s) of reflectivity for the target event. Radar performance was shown to depend on type of rainfall, with better results obtained with deep convective rain systems (Nash coefficients of roughly 0.90 for point radar–rain gauge comparisons at the event time step), as opposed to shallow convective and frontal rain systems (Nash coefficients in the 0.6–0.8 range). In comparison with time-adaptive strategies, the space–time-adaptive strategy yields a very significant reduction in the radar–rain gauge bias while the level of scatter remains basically unchanged. Because the Z–R relationships have not been optimized in this study, results are attributed to an improved processing of spatial variations in the vertical profile of reflectivity. The two main recommendations for future work consist of adapting the rain separation method for radar network operations and documenting Z–R relationships conditional on rainfall type.

First passage time of nonlinear ship rolling in narrow band non-stationary random seas

The generalized extended stochastic central difference (GESCD) method is applied to study the response statistics and first passage time of nonlinear ship rolling in narrow band stationary and non-stationary random seas. The GESCD method is based on a combination of the extended stochastic central difference method with a statistical linearization technique, modified adaptive time scheme, and time coordinate transformation. The extended stochastic central difference method is, however, an extension of the stochastic central difference method for the determination of the recursive mean square or covariance of responses of systems under narrow band stationary and non-stationary random disturbances. Approximate first passage probabilities of nonlinear systems based on the modified mean rate of various crossings proposed earlier by the first author were determined. It is concluded that the GESCD method is very accurate, simple and efficient to apply compared with Monte Carlo simulation. The proposed method is applicable to cases with large nonlinearities and intensive random excitations. The approximate first passage probabilities of the nonlinear system determined by the proposed approach are very accurate as they are in excellent agreement with those evaluated by the Monte Carlo simulation. It is believed that the model considered in this paper is a closer representation to reality than those reported earlier in the literature.

On space–time adaptive schemes for the numerical solution of PDEs

A fully adaptive numerical scheme for solving PDEs based on a finite volume discretization with explicit time discretization is presented. The local grid refinement is triggered by a multiresolution strategy which allows to control the approximation error in space. The costly fluxes are evaluated on the adaptive grid only. For automatic time step control a Runge-Kutta-Fehlberg method is used. A dynamic tree data structure allows memory compression and CPU time reduction. For validation different classical test problems are computed. The gain in memory and CPU time with respect to the finite volume scheme on a regular grid is reported and demonstrates the efficiency of the new method. Resume .N ous prici une methode numerique entierement adaptative pour les EDP, basee sur une discretisation spatiale en volumes finis et une integration temporelle explicite de type Runge-Kutta. Une strategie de type multi-resolution permet d'adapter localement le maillage tout en controlant l'erreur d'approximation en espace. Les flux sontes sur la grille adaptative uniquement. Une methode de type Runge-Kutta-Fehlberg est employee afin de choisir automatiquement le pas de temps tout en controlant l'erreur d'approximation. Nous proposons en outre une methode oul e pas de temps depend de l'´ echelle, afin d'´ eviter d'utiliser sur tous les niveaux le pas de temps qui garantit la stabilite numerique sur le niveau de grille le plus fin. La structure de donnees est organisee en arbre graduel, ce qui permet de reduire significativement la place memoire et le temps de calcul necessaires. Nous validons ce nouveau schema numeriquel'aide de differents cas-tests classiques. Nous estimons le gain en place memoire et en temps de calcul par rapport au meme calcul en volumes finis sur la grille la plus fine, afin de montrer l'efficacited e la m´

A Time Adaptive Scheme for the Solution of the Advection Equation in the Presence of a Transient Flow Velocity

A Time Adaptive Scheme for the Solution of the Advection Equation in the Presence of a Transient Flow Velocity

Responses of discretized systems under narrow band nonstationary random excitations. Part 2: nonlinear problems

In a companion paper, the extended stochastic central difference (ESCD) method was presented for computation of responses of linear multi-degrees-of-freedom (mdof) systems under narrow band stationary and nonstationary random excitations. The present paper is concerned with the generalization of the ESCD method for computation of nonlinear mdof systems. The generalization is based on a combination of the ESCD method with statistical linearization (SL) technique, modified adaptive time scheme (ATS), and time coordinate transformation (TCT). Unlike the conventional SL technique, the generalized ESCD method is applicable to mdof systems with large nonlinearities. Comparison is made of the computed results applying the generalized ESCD method to those obtained by the Monte Carlo simulation (MCS). Excellent agreements were obtained. It was observed that the proposed method is very efficient compared with the MCS.