Time Step Selection Research Articles

Adaptive time step selection for spectral deferred correction

AbstractSpectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.

High-pressure core meltdown and hydrogen generation during In-vessel and Ex-vessel phases of VVER-1000 nuclear reactor and effect of heat structures

Following unprecedented Fukushima Daiichi severe accident, which significantly amplify the production of hydrogen and the associated risk of explosions, particularly in accidents like station blackout (SBO) that jeopardize containment integrity and reactor safety, a thorough investigation and comprehension of the phenomenon known as high-pressure melt ejection (HPME) are necessary. This study examines how the SBO accident develops without operator intervention or access to emergency diesel generators through full nuclear power plant simulation, including containment, core, primary, and secondary circuits. Accurate time step selection is crucial for the simulation to produce reliable results for high-pressure melt ejection processes. While the process is faster and easier under Low Pressure Melt Ejection (LPME) conditions, time step tuning can significantly lengthen the (HPME) simulation (approximately one month for each scenario). After thorough examinations and analysis of various simulation findings, three cases were evaluated for the ex-vessel phase, illustrating the most likely scenarios of high-pressure molten material splashing into the cavity. The findings demonstrate that during the in-vessel phase, 578 kg of hydrogen is created. Additionally, during the ex-vessel phase, there is a significant increase in the containment’s temperature and pressure due to the high-temperature hydrogen and significant water and steam leakage. The temperature and pressure inside the containment rise well above the safety criteria due to the leakage of 1529 kg of hydrogen from the cavity and lead to hydrogen explosion due to MELCOR message. The calculated amount of hydrogen gas is within the explosion limit range of 18.3% to 59% of the containment atmospheric pressure volume. Moreover, the height and diameter of the cavity alter due to the direct heating and corrosion of the cavity walls, ultimately compromising the integrity of the plant in the alpha mode. Based on the results obtained, it can be seen that if a large portion of the molten material is deposited on the concrete surface (case III) instead of dispersed in the cavity atmosphere (case I), the rate of hydrogen production and therefore the total amount of hydrogen (and its related energy) produced will be increased and could lead to deflagration to detonation situation.

Algebraically stable SDIRK methods with controllable numerical dissipation for first/second-order time-dependent problems

In this paper, a family of four-stage singly diagonally implicit Runge-Kutta methods are proposed to solve first-/second-order time-dependent problems, exhibiting the following numerical properties: fourth-order accuracy in time, unconditional stability, controllable numerical dissipation, and adaptive time step selection. The BN-stability condition is employed as a constraint to optimize parameters in the Butcher table, having significant benefits, and hence is recommended for nonlinear dynamics problems in contrast to existing methods. Numerical examples involving both first- and second-order linear/nonlinear dynamics problems validate the proposed method, and numerical results reveal that the proposed methods are free from the order reduction phenomenon when applied to nonlinear dynamics problems. The performance of adaptive time-stepping using the embedded scheme is further illustrated by the phase-field modeling problem. Additionally, the advantages and disadvantages of three-stage third-order accurate algebraically stable methods are discussed. The proposed high-order time integration can be readily integrated into high-order spatial discretization methods, such as the high-order spectral element method employed in this paper, to obtain high-order discretization in space and time dimensions.

Nonlinear biomechanics of diseased carotid arteries

The aim of this article is to analyse the nonlinear biomechanics of diseased carotid arteries as a potential tool for predicting the onset of cerebral strokes. A two-way coupled three-dimensional (3D) hyperelastic fluid-structure interaction (FSI) analysis of a diseased carotid artery with abnormal luminal projections at the carotid bulb known as carotid web (CaW) is conducted on the geometry of a patient artery, built upon employing CT angiography images. The blood-flow model incorporates non-Newtonian pulsatile turbulent fluid, and the artery wall is considered hyperelastic subject to blood-induced motion. The hemodynamics of artery induced by transient boundary conditions is determined, specifically focusing on shifts in crucial hemodynamic parameters such as wall shear stress (WSS) and alterations in the blood velocity pattern. Structural assessment of the artery wall involves quantifying the von Mises (VM) stress and deformation field. The analysis demonstrates that different CaW models result in different flow patterns for a selection of time steps in a cardiac cycle. The findings reveal that the presence of the web, as the most common disease among younger adults, can significantly influence the hemodynamic parameters and potentially accelerate the formation of thrombus and atherosclerosis. Hemodynamic analysis can potentially predict specific sites prone to plaque formation and rupture within the carotid artery.

Shallow Diffusion for Fast Speech Enhancement (Student Abstract)

Recently, the field of Speech Enhancement has witnessed the success of diffusion-based generative models. However, these diffusion-based methods used to take multiple iterations to generate high-quality samples, leading to high computational costs and inefficiency. In this paper, we propose SDFEN (Shallow Diffusion for Fast spEech eNhancement), a novel approach for addressing the inefficiency problem while enhancing the quality of generated samples by reducing the iterative steps in the reverse process of diffusion method. Specifically, we introduce the shallow diffusion strategy initiating the reverse process with an adaptive time step to accelerate inference. In addition, a dedicated noisy predictor is further proposed to guide the adaptive selection of time step. Experiment results demonstrate the superiority of the proposed SDFEN in effectiveness and efficiency.

A Lagrangian method for slurry flow modeling in hydraulic fractures

We developed a new numerical simulator of slurry flow in hydraulic fractures. It is based on a continuum model developed under the assumptions of lubrication theory. We consider slurry containing fluid and proppant components. Proppant particles settle under gravity and bridge in the flow domain. Fluid fluxes are calculated from an elliptic boundary value problem for given fracture geometry and sources/sinks distribution. We solve it using a conservative finite volume method and then apply empirical closure relations to get the velocity field for solids. The transport of slurry components is described by nonlinear hyperbolic equations with fluxes depending on proppant concentration and particle size/fracture width ratio. These equations admit solutions with concentration shock waves and rarefaction fans. They must be represented accurately in the computational algorithm. We utilize an Eulerian approach for the discretization of nonlinear proppant fluxes. After getting the fluxes on the interfaces between Eulerian grid cells we follow a Lagrangian approach to simulate the proppant transport along the given velocity field. In this study we discuss key implementation aspects: time step selection, regularization, fluxes discretization, seeding, reseeding, and movement of Lagrangian particles. We verify the simulator on benchmark solutions focused on advection, batch proppant settling, slurry slumping and fluid displacement by proppant.

Turbulence characterization of instantaneous airflow above passengers with different air distribution systems in a commercial airliner cabin

The environment inside airliner cabins has attracted increasing attention. Instantaneous airflow, as one of the fundamental parameters of airflow field, affects aspects of the airliner cabin environment such as occupant comfort and airborne disease transmission. This study evaluated the characteristics of instantaneous airflow in airliner cabins with different ventilation systems (mixing ventilation and displacement ventilation) and focused on the region above passengers' heads, where complex turbulence characteristics are found. This investigation used ultrasonic anemometers for measurements at ten selected points above passengers' heads. The energy ratio on the low-frequency scale with mixing ventilation was greater than that with displacement ventilation. The instantaneous airflow with mixing ventilation was close to that of natural wind, while the instantaneous airflow with displacement ventilation was close to that of mechanical wind. Compared with mixing ventilation, displacement ventilation produced a smaller vortex length and a shorter residence time. Moreover, there was no long-term vortex in the flow field under displacement ventilation, and therefore, pollutants would be discharged more quickly from the cabin. In addition, this paper distinguishes the spatial and temporal turbulence scales of mixing and displacement ventilation, thus providing a reference for the selection of grid size and time steps in computational fluid dynamics simulations.

A Study on CFL Conditions for the DG Solution of Conservation Laws on Adaptive Moving Meshes

The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin (DG) solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping. A commonly used selection of time step is a direct extension based on Courant-Friedrichs-Levy (CFL) conditions established for fixed and uniform meshes. In this work, we provide a mathematical justification for those time step selection strategies used in practical adaptive DG computations. A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws. Based on the analysis, a new selection strategy of the time step is proposed, which takes into consideration the coupling of the $\alpha$-function (that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity) and the heights of the mesh elements. The analysis also suggests several stable combinations of the choices of the $\alpha$-function in the numerical scheme and in the time step selection. Numerical results obtained with a moving mesh DG method for Burgers' and Euler equations are presented. For comparison purpose, numerical results obtained with an error-based time step-size selection strategy are also given.

An extended Vlasov–Fokker–Planck approach for kinetic simulations of laser plasmas

Vlasov–Fokker–Planck simulation codes occupy an important niche in modeling laser-produced plasmas, since they are well suited to studying the effect of collisions on electron kinetic phenomena, especially energy transport. One of the most important elements of energy transport is the absorption of laser light by the plasma; however, simulating this in detail requires resolving oscillations of the laser light, whose characteristic timescale is orders of magnitude shorter than the simulation time needed to study transport physics. For this reason, most Vlasov–Fokker–Planck codes used to study electron transport in laser plasmas rely on simplified models of the laser–plasma coupling. Their underlying assumptions nominally preclude their use for modeling laser light having short-scale structure in space or time, such as broadband lasers. In this work, we derive a more general computational framework suitable for arbitrarily structured laser fields. Our approach is based on an extended set of Vlasov–Fokker–Planck equations that separately solve for the low- and high-frequency plasma response. We implement these extended Vlasov–Fokker–Planck equations in the spherical harmonic code K2 and demonstrate the performance of the method on several laser absorption test problems, with particular attention to the judicious selection of time steps, time integrators, and spherical harmonic truncation, according to the intensity and spectrum of the laser light under consideration. Comparison with the widely used Langdon absorption operator shows the Langdon operator performs remarkably well for predicting laser heating in the simple cases considered here, even in situations that would seem to violate its underlying assumptions.

Monthly Runoff Forecasting Based on Interval Sliding Window and Ensemble Learning

In recent years, machine learning, a popular artificial intelligence technique, has been successfully applied to monthly runoff forecasting. Monthly runoff autoregressive forecasting using machine learning models generally uses a sliding window algorithm to construct the dataset, which requires the selection of the optimal time step to make the machine learning tool function as intended. Based on this, this study improved the sliding window algorithm and proposes an interval sliding window (ISW) algorithm based on correlation coefficients, while the least absolute shrinkage and selection operator (LASSO) method was used to combine three machine learning models, Random Forest (RF), LightGBM, and CatBoost, into an ensemble to overcome the preference problem of individual models. Example analyses were conducted using 46 years of monthly runoff data from Jiutiaoling and Zamusi stations in the Shiyang River Basin, China. The results show that the ISW algorithm can effectively handle monthly runoff data and that the ISW algorithm produced a better dataset than the sliding window algorithm in the machine learning models. The forecast performance of the ensemble model combined the advantages of the single models and achieved the best forecast accuracy.

On the role of the architecture for spring discharge prediction with deep learning approaches

AbstractUnderstanding karst spring flow is important to accommodate the increasing water demand caused by the population growth and manage the freshwater water resource effectively. However, due to the spatial and temporal heterogeneity and complex hydrological processes in karst systems, predicting karst spring discharge remains challenging. In this study, three deep learning‐based models, including long short‐term memory (LSTM), gated recurrent unit (GRU) and simple recurrent neural network (RNN), are framed with an encoder–decoder architecture to provide multiple‐step‐ahead spring discharge prediction. The encoder–decoder architecture includes an encoder that reads and encodes the input sequence into a vector and decoder that deciphers the vector and outputs the predicted sequence. Three hybrid models called LSTM‐ED, GRU‐ED and simple RNN‐ED are compared with single‐step models and multiple‐step models without the encoder–decoder architecture to investigate the role of the encoder–decoder architecture on multi‐step‐ahead prediction. The sensitivity of the selection of input time and lead time steps on the karst spring discharge prediction is evaluated. The predicted results are compared with the observed spring discharge. It implies that: (1) LSTM‐ED, GRU‐ED and RNN‐ED models obtain similar results on predicting karst spring discharge multiple time steps ahead; (2) three hybrid multiple‐step models outperform the single‐step models in making consistent and accurate spring discharge predictions; (3) the multiple‐step models framed with an encoder–decoder architecture obtain better spring discharge prediction results than the single‐step models and multiple‐step models without the encoder–decoder structure; (4) the LSTM‐ED, GRU‐ED and simple RNN‐ED models are sensitive to the selection of lead time and insensitive to the selection of input time step. A short lead time typically yields a more accurate spring discharge prediction.

A nonasymptotic analysis for re‐solving heuristic in online matching

We investigate an online edge‐weighted bipartite matching problem with general capacity constraints. In this problem, the resources are offline and nonreplenishable with different capacities. Demands arrive online and each requests a certain amount of resources. The goal is to maximize the reward generated by successful matches. We model the offline optimization problem as a deterministic linear program (LP) and present multiple randomized online algorithms based on the solution to the offline LP. We analyze the performance guarantee of each algorithm in terms of its competitive ratio (CR). Importantly, we introduce a re‐solving heuristic that periodically recomputes the offline LP and uses the updated offline solution to guide the online algorithm decisions. We find that the algorithm's CR can be significantly improved when re‐solving at carefully selected time steps. Finally, we investigate the value of the demand distribution in further improving the algorithm efficiency. We conduct extensive numerical studies to demonstrate the efficiency of the proposed algorithms. The effect of market conditions on the algorithm performance is also investigated.

Stable timestep formulas for high-order advection–diffusion and Navier–Stokes solvers

We derive three-dimensional stable timestep formulas for high-order explicit time integration of the advection–diffusion equation. The proposed formulas cover explicit first, second, third, and fourth-order Runge–Kutta integrators in time as well as upwind, central, second-order, high-order upwind (k-schemes), and flux-limiters for the advection term along with central discretization on the diffusion term. The stability rules are then compared to numerical computations of stable timesteps of the advection–diffusion equation and show excellent agreement. Finally, the formulas are implemented in a finite volume Navier–Stokes code and used to guide stable timestep selection to assess their suitability for Navier–Stokes solvers. All cases that were tested showed stable non-diverging solutions and significant improvement over common CFL-type stability specification. Despite its basis in the linear advection–diffusion equation, this work could be useful in providing a starting point for stable timestep selection for nonlinear evolutionary-type PDEs such as the Navier–Stokes equations.

Simulation of Impulse Corona on HVDC Overhead Transmission Lines With Background DC Space Charges by Method of Characteristics

This paper proposes a time-domain method of characteristics to simulate the impulse corona with background space charge, and derives the ordinary differential equations of the space charge generated by the impulse corona along the electric field line. The corresponding boundary conditions, the selection of time step and space step, and the update strategy of space charge are given. The calculation result of the impulse corona onset voltage is consistent with the test result, proving that the background space charge has almost no effect on the impulse corona onset electric field which is obtained by Kaptzov's assumption. The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q-u</i> curve obtained by simulation is compared with the test results to verify the effectiveness of the simulation method. The simulation method is used to obtain the development process of space charge during the application of the opposite polarity impulse voltage. By analyzing the space charge distribution during the process of impulse corona, and the corresponding microscopic charge distribution and macroscopic charge amount at different corona states, the characteristics of impulse corona are explained.

Streaming Approach to In Situ Selection of Key Time Steps for Time‐Varying Volume Data

AbstractKey time steps selection, i.e., selecting a subset of most representative time steps, is essential for effective and efficient scientific visualization of large time‐varying volume data. In particular, as computer simulations continue to grow in size and complexity, they often generate output that exceeds both the available storage capacity and bandwidth for transferring results to storage, making it indispensable to save only a subset of time steps. At the same time, this subset must be chosen so that it is highly representative, to facilitate post‐processing and reconstruction with high fidelity. The key time steps selection problem is especially challenging in the in situ setting, where we can only process data in one pass in an online streaming fashion, using a small amount of main memory and fast computation. In this paper, we formulate the problem as that of optimal piece‐wise linear interpolation. We first apply a method from numerical linear algebra to compute linear interpolation solutions and their errors in an online streaming fashion. Using that method as a building block, we can obtain a global optimal solution for the piece‐wise linear interpolation problem via a standard dynamic programming (DP) algorithm. However, this approach needs to process the time steps in multiple passes and is too slow for the in situ setting. To address this issue, we introduce a novel approximation algorithm, which processes time steps in one pass in an online streaming fashion, with very efficient computing time and main memory space both in theory and in practice. The algorithm is suitable for the in situ setting. Moreover, we prove that our algorithm, which is based on a greedy update rule, has strong theoretical guarantees on the approximation quality and the number of time steps stored. To the best of our knowledge, this is the first algorithm suitable for in situ key time steps selection with such theoretical guarantees, and is the main contribution of this paper. Experiments demonstrate the efficacy of our new techniques.

On the application of the GS4-1 framework for fluid dynamics and adaptive time-stepping via a universal A-posteriori error estimator

PurposeThe purpose of this paper is to describe a novel universal error estimator and the adaptive time-stepping process in the generalized single-step single-solve (GS4-1) computational framework, applied for the fluid dynamics with illustrations to incompressible Navier–Stokes equations.Design/methodology/approachThe proposed error estimator is universal and versatile that it works for the entire subsets of the GS4-1 framework, encompassing the nondissipative Crank–Nicolson method, the most dissipative backward differential formula and anything in between. It is new and novel that the cumbersome design work of error estimation for specific time integration algorithms can be avoided. Regarding the numerical implementation, the local error estimation has a compact representation that it is determined by the time derivative variables at four successive time levels and only involves vector operations, which is simple for numerical implementation. Additionally, the adaptive time-stepping is further illustrated by the proposed error estimator and is used to solve the benchmark problems of lid-driven cavity and flow past a cylinder.FindingsThe proposed computational procedure is capable of eliminating the nonphysical oscillations in GS4-1(1,1)/Crank–Nicolson method; being CPU-efficient in both dissipative and nondissipative schemes with better solution accuracy; and detecting the complex physics and hence selecting a suitable time step according to the user-defined error threshold.Originality/valueTo the best of the authors’ knowledge, for the first time, this study applies the general purpose GS4-1 family of time integration algorithms for transient simulations of incompressible Navier–Stokes equations in fluid dynamics with constant and adaptive time steps via a novel and universal error estimator. The proposed computational framework is simple for numerical implementation and the time step selection based on the proposed error estimation is efficient, benefiting to the computational expense for transient simulations.

Difference-based Equivalent Static Load Method with adaptive time selection and local stiffness adaption

Structural optimization of crash-related problems usually involves nonlinearities in geometry, material, and contact. The Equivalent Static Load (ESL) method provides a method to solve such problems. It has previously been extended to employ an individual Finite Element model describing the deformed geometry at each considered time step under the name Difference-based Equivalent Static Load (DiESL) method. This paper demonstrates how an appropriate selection of the time steps in each cycle can further improve the convergence behavior of the DiESL method. It is shown that the adaptive selection of time steps leads to better objective values and more reliable convergence to the presumed global optimum. Furthermore, the DiESL extension enables the adaption of path-dependent structural properties of the original nonlinear problem like material stiffness in each linear auxiliary load case. In this paper, an adaption of the Young’s modulus on element level in the linear auxiliary problem corresponding to the local plasticization in the nonlinear dynamic problem is successfully implemented. Here, the test examples indicate that an observable improvement can only be obtained if neither the elements in the elastic nor in the plastic range are dominating the structure’s behavior.

Numerical Study of the Characteristics of Shock and Rarefaction Waves for Nonlinear Wave Equation

In everyday life human faces shock waves and rarefaction largely in their surroundings. Hence it’s necessary to know the behavior of these waves to protect destructive effects. The aim of this work to observe the propagation of shock and rarefaction waves in various dynamics due to solve non-linear hyperbolic inviscid Burgers’ equation numerically. The models adopted here two numerical schemes which enable us to solve non-linear hyperbolic Burgers’ equation numerically. The first order explicit upwind scheme (EUDS) and second order Lax-Wendroff schemes are used to solve this equation to improve our understanding of the numerical diffusion (smearing) and oscillations that can be present when using such schemes. In order to understand the behavior of the solution we use method of characteristics to find the exact solution of inviscid Burgers’ equation. Numerical solutions are studied for different initial conditions and the shock and rarefaction waves are investigated for Riemann problem. We present stability analysis of the schemes and establish stability condition which leads to determine time step selection in terms of spatial step size with maximum initial value. Numerical result for these schemes are compared with an exact solution of inviscid Burgers’ equation in terms of accuracy by error estimation. The numerical features of the rate of convergence are presented graphically. This analysis helps us to understand a wide range of physical phenomenon of the properties of wave as well as saves in several aspects in real life.

Simulation for influence of Y-shape fin on phase change of paraffin inside triplex pipe with using Al2O3 nanoparticles

Solidification phenomena within a chamber with different arrangement of fins were investigated in this article. The homogeneous mixture of paraffin (RT82) and Al2O3 were assumed as NEPCM (nanoparticle enhanced phase change material). Different cases have been simulated to show the impact of concentration of alumina, arrangement of fins and their height and thickness. Numerical technique with structure grid was utilized for simulation of this transient modeling. For selection of time step and grid size, different cases were analyzed and finally the optimum values were selected to reduce the computing price. For verification examination, empirical data was tested and nice accommodations were reported. The cases with outer arrangement of fins have better efficiency in view of solidification. After 900 s, the liquid fraction of case 8 reaches to 0.023 while this value for case 3 is 0.12. The third case has longest time and all NEPCM solidifies after 1582.2 s. The case 8 has minimum time of freezing and after 1072s, the process finishes. Changing NEPCM instead of pure RT82, causes the required time to decline about 13.71%, 11.98%, 13.75% and 11.71% for cases 1, 2, 3 and 4, respectively. With replacement of fins with b = 0.38in, t = 0.02in to outer wall, solidification time decreases about 14.45%.

Modelling of Soil Profile Pollutant Yield on Slaughterhouse Wastes Deposited Land

Aim: The study aimed at modeling the concentration of pollutants along soil profile using finite element method.&#x0D; Study Design: Data was generated from the laboratory on the concentrations of selected heavy metals at varying depths of land discharged slaughterhouses. This was used to estimate the level of nutrient build-up in the soil within these environs, hence, used to verify and validate the finite element analysis. The model upon validation was used to predict the rate of pollutant build-up in the soil within the slaughterhouses discharge areas.&#x0D; Methodology: A total of twelve composite samples were collected from three different land discharged locations. The three composite samples each were collected from the sampling locations at a depth of 0 to 10cm, 10 to 20cm, 20 to 30cm and 30 to 40cm. Four composite samples each were collected for analysis from the three sampling locations on specified sampling dates. The samples were then placed in sterile polythene bags and transported to the laboratory for processing. The laboratory results obtained for heavy metals were used for the generated model verification and validation, hence predictions for pollutants accumulation was done on a time step.&#x0D; Results: Model verification showed a good fit of a nonlinear polynomial curve for both the measured and predicted values with R² values of 0.9978 to 0.9985 for zinc and 0.9978 to 0.9984 for lead at a selected time step of 15years. It was observed however, that there was an increasing tendency to uniformity of concentration as the time step increased; this was due to parameters build-up with time in the soil.&#x0D; Conclusion: Finite element results revealed a high build-up in the concentration of pollutants (Zinc and Lead) in the land discharged slaughterhouses.