Adaptive-Time-Step High-Frame-Rate Particle Image Velocimetry

A frequent configuration in computational fluid mechanics combines an explicit time advancing scheme for accuracy purposes and a computational grid with a very small portion of much smaller elements than in the remaining mesh. Examples of such situations are the traveling of a discontinuity followed by a moving mesh, and the large eddy simulation of high Reynolds number flows around bluff bodies where together very thin boundary layers and vortices of much more important size need to be captured. For such configurations, explicit time advancing schemes with global time stepping are very costly. In order to overcome this problem, the multirate time stepping approach represents an interesting improvement. The objective of such schemes, which allow to use different time steps in the computational domain, is to avoid penalizing the computational cost of the time advancement of unsteady solutions which can become large due to the use of small global time steps imposed by the smallest elements such as those constituting the boundary layers. In the present work, a new multirate scheme based on control volume agglomeration is proposed for the solution of the compressible Navier-Stokes equations possibly equipped with turbulence models. The method relies on a prediction step where large time steps are performed with an evaluation of the fluxes on macro-cells for the smaller elements for stability purpose, and on a correction step in which small time steps are employed only for the smaller elements. The efficiency of the proposed method is evaluated on several benchmarks flows: the problem of a moving contact discontinuity (inviscid flow), the computation with hybrid turbulence model of flows around bluff bodies like a tandem cylinders at Reynolds number $1.66 × 10 5$ , a circular cylinder at Reynolds number $8.4 × 10 6$ , and a flow around a space probe model at Reynolds number $1 × 10 6$ .

SUMMARYThe Eulerian–Lagrangian Localized Adjoint Method (ELLAM) is well adapted for advection‐dominated transport problems. The method can use large time steps because the advection term is accurately approximated without any CFL restriction.A new formulation of ELLAM was developed by Younes et al. (Adv. Water Resour. 2006; 29:1056–1074) for unstructured triangular meshes. This formulation uses only strategic numerical integration points and thus requires a very limited number of integration points (usually 1 per element). To avoid numerical dispersion due to interpolation when several time steps are used, the method continuously tracks characteristics, and only changes due to dispersion are interpolated at each time step.In this work, we study the applicability and efficiency of this formulation for highly heterogeneous domains. The results show that (i) this formulation remains efficient even for highly heterogeneous domains, (ii) special care must be taken for the approximation of the dispersive term when large time steps are used, (iii) interpolation at intermediate times can be used to reduce memory requirement for problems with a large number of elements and small time steps, and (iv) the method can be used successfully for heterogeneous problems including injection and pumping wells, and remains much more efficient than the discontinuous Galerkin finite element method even when small time steps are required. Copyright © 2011 John Wiley & Sons, Ltd.

Detailed switching models (DSM) of ac–dc converters are available in many offline and real-time electromagnetic transient (EMT) simulation programs. However, such discrete models typically require small simulation time steps for accurately handling the switching events without interpolation. This paper proposes a new parametric modeling approach for line-commutated and 2-level voltage-source ac–dc converters. The proposed methodology is based on parametric functions that relate ac and dc variables in the instantaneous sense, which allows the reconstruction of the waveforms of voltages and currents (with the same of details as the switching models of converters) without topological changes in the converter circuit. Thus, it can operate at larger time steps without requiring interpolation, thereby enhancing simulation efficiency. The computational advantages of the proposed models over the conventional switching models are demonstrated in offline PSCAD and real-time RTDS NovaCor simulators in terms of maximum possible time step size and accuracy. It is shown that the proposed models can accurately run with much larger time steps (up to ∼200 μs for LCRs, and up to ∼50 μs for VSCs) compared to detailed switching models, which are limited to relatively small time steps of ∼10 μs.

Currently, the mechanical dynamics of soft tissue deformation is achieved by numerical time integrations such as the explicit or implicit integration; however, the explicit integration is stable only under a small time step, whereas the implicit integration is computationally expensive in spite of the accommodation of a large time step. This paper presents a cellular neural network method for stable simulation of soft tissue deformation dynamics. The non-rigid motion equation is formulated as a cellular neural network with local connectivity of cells, and thus the dynamics of soft tissue deformation is transformed into the neural dynamics of the cellular neural network. Results show that the proposed method can achieve good accuracy at a small time step. It still remains stable at a large time step, while maintaining the computational efficiency of the explicit integration. The proposed method can achieve stable soft tissue deformation with efficiency of explicit integration for surgical simulation.

In this paper we re-examine the idea that implicit integrators with large time steps offer the best stability/performance trade-off for stiff systems. We make the surprising observation that performing a single large time step with n constraint solver iterations is less effective than computing n smaller time steps, each with a single constraint solver iteration. Based on this observation, our approach is to split every visual time step into n substeps of length Δt/n and to perform a single iteration of extended position-based dynamics (XPBD) in each such substep. When compared to a traditional implicit integrator with large time steps we find constraint error and damping are significantly reduced. When compared to an explicit integrator we find that our method is more stable and robust for a wider range of stiffness parameters. This result holds even when compared against more sophisticated implicit solvers based on Krylov methods. Our method is straightforward to implement, and is not sensitive to matrix conditioning nor is it to overconstrained problems.

We have studied two numerical strategies for solving a coupled system of dictinct nonlinear equations governing sediment transport in Lake Okeechobee. Using high-resolution bathymetry data of Lake Okeechobee, Florida, we study the numerical properties of the two strategies, from 1 core to 512 cores. The fully-explicit scheme is straightforward to implement and requires a relatively small amount of computation per time step. However, this simple numerical strategy has to use small time steps to ensure stability. These small time steps may render the explicit solver impractical for long-term and high-resolution basin simulations. As a comparison, we have implemented a semi-implicit scheme, where the two partial differential equations at each time step are solved implicitly in sequence. Numerical experiments show that this semi-implicit scheme is numerically stable even for very large time steps. Using a multicore-based cluster, we have carried out parallel simulations of sediment transport along a river chanel and into Lake Okeechobee.

PCB design aids

High-performance direct gravitational N-body simulations on graphics processing units

This paper presents a multiple time-step solver based on a time-domain explicit integration algorithm for improving the computational speed of high-voltage transmission line electromagnetic transient (EMT) simulation. For weakly rigid EMT models of high-voltage transmission lines, the previous precise Runge-Kutta integration method with a small time step is adopted; for strongly rigid nonlinear EMT models of high-voltage transmission lines, a an improved precise integration method using a large time step is used for the solver of EMT simulation. Practical simulations for overvoltages of high-voltage transmission line show that multiple time-step EMT solver can be applied to different cases of EMT simulation for transmission line.

Within the nuclear-electronic orbital (NEO) framework, the real-time NEO time-dependent density functional theory (RT-NEO-TDDFT) approach enables the simulation of coupled electronic-nuclear dynamics. In this approach, the electrons and quantum nuclei are propagated in time on the same footing. A relatively small time step is required to propagate the much faster electronic dynamics, thereby prohibiting the simulation of long-time nuclear quantum dynamics. Herein, the electronic Born-Oppenheimer (BO) approximation within the NEO framework is presented. In this approach, the electronic density is quenched to the ground state at each time step, and the real-time nuclear quantum dynamics is propagated on an instantaneous electronic ground state defined by both the classical nuclear geometry and the nonequilibrium quantum nuclear density. Because the electronic dynamics is no longer propagated, this approximation enables the use of an order-of-magnitude larger time step, thus greatly reducing the computational cost. Moreover, invoking the electronic BO approximation also fixes the unphysical asymmetric Rabi splitting observed in previous semiclassical RT-NEO-TDDFT simulations of vibrational polaritons even for small Rabi splitting, instead yielding a stable, symmetric Rabi splitting. For the intramolecular proton transfer in malonaldehyde, both RT-NEO-Ehrenfest dynamics and its BO counterpart can describe proton delocalization during the real-time nuclear quantum dynamics. Thus, the BO RT-NEO approach provides the foundation for a wide range of chemical and biological applications.

A simple direct-forcing immersed boundary projection method with prediction-correction for fluid-solid interaction problems

The explicit finite-difference scheme is widely used in seismic wave simulation. Using large time steps can significantly reduce the number of iterations and thus improve computational efficiency, but this approach faces two main challenges: reduced accuracy caused by temporal dispersion and restrictions imposed by the Courant-Friedrichs-Lewy (CFL) stability condition. Based on the wavefield iteration equation in matrix form, an explicit finite-difference method with time step n-tupling and an extended CFL limit is developed for acoustic wave simulation. By combining n successive iteration operators into a single-step operator, two types of n-tupling algorithms are constructed, effectively expanding the CFL stability limit by a factor of n and enabling time steps well beyond conventional thresholds. Both theoretical analysis and numerical experiments demonstrate that simulations with time step n-tupling achieve accuracy equivalent to conventional single-step methods while reducing computation time to approximately 1/n, resulting in a corresponding n-fold increase in overall computational efficiency. Time dispersion is suppressed using two complementary strategies: for relatively small base time steps, n-tupling inherently attains the accuracy of smaller steps; for relatively large base steps, a time-dispersion transform is applied to eliminate errors and maintain high numerical accuracy throughout the simulation.

We present an efficient library-based numerical method for simulating the Hodgkin-Huxley (HH) neuronal networks. The key components in our numerical method involve (i) a pre-computed high resolution data library which contains typical neuronal trajectories (i.e., the time-courses of membrane potential and gating variables) during the interval of an action potential (spike), thus allowing us to avoid resolving the spikes in detail and to use large numerical time steps for evolving the HH neuron equations; (ii) an algorithm of spike-spike corrections within the groups of strongly coupled neurons to account for spike-spike interactions in a single large time step. By using the library method, we can evolve the HH networks using time steps one order of magnitude larger than the typical time steps used for resolving the trajectories without the library, while achieving comparable resolution in statistical quantifications of the network activity, such as average firing rate, interspike interval distribution, power spectra of voltage traces. Moreover, our large time steps using the library method can break the stability requirement of standard methods (such as Runge-Kutta (RK) methods) for the original dynamics. We compare our library-based method with RK methods, and find that our method can capture very well phase-locked, synchronous, and chaotic dynamics of HH neuronal networks. It is important to point out that, in essence, our library-based HH neuron solver can be viewed as a numerical reduction of the HH neuron to an integrate-and-fire (I&F) neuronal representation that does not sacrifice the gating dynamics (as normally done in the analytical reduction to an I&F neuron).

One of the most basic struggles in simulations of statistical and dynamical systems is how to appropriately balance the desire for simulation accuracy, obtained in the small time step limit, with simulation efficiency, obtained for large time steps. Obviously, we would like to have the benefits of both limits of accuracy and efficiency. Thus, understanding the influence of discrete time on the behavior of equations of motion is crucial for the understanding and optimization of numerical simulations in physical science and engineering. There are two different types of crucial discretization errors. One is that the simulated, discrete-time trajectory increasingly deviates from the true, continuous-time trajectory as the time step is increased. Another is that the simulated discrete-time velocity increasingly deviates from the expected velocity of the corresponding simulated trajectory as the time step is increased. While the former error is a signature that the simulated trajectory is not exactly the intended one, the simulation may still be representative of the true system. In contrast, the latter error is a signature of a fundamental internal inconsistency in the simulation, which may greatly complicate physical interpretation of the simulation results, especially when both configurational and kinetic measures are required. It is therefore crucially important to understand the relationship between a discrete-time coordinate and its corresponding velocity.It has previously been shown that, unlike in continuous-time, discrete-time configurational dynamics does not need a discrete-time velocity variable. This realization allowed for the derivation of the complete configurational GJ set of stochastic Verlet-type algorithms for simulating the Langevin equation. This set has the key feature of simultaneously giving exact, time-step independent statistical sampling of damped, noisy harmonic oscillators, and giving exact spatial transport values, drift and Einstein diffusion, on flat surfaces. The apparent fact that a discrete-time velocity is not required to simulate a coordinate trajectory implies that discrete-time velocity is an ambiguous concept, which, in turn, implies that multiple velocity definitions can be entertained in order to avoid internal simulation inconsistencies between configurational and kinetic measures. From the derivation of the stochastic GJ methods it was shown that a statistically correct velocity obtained at the same time as the coordinate (an on-site velocity) cannot be obtained. In contrast, a velocity at the half-step between consecutive configurational time steps (a half-step velocity) was identified to produce exact statistical kinetic response for harmonic oscillators regardless of the applied time step. However, only for a single GJ method does this velocity also give the correct drift value.We here derive new, practical velocity definitions that are optimized for statistical kinetic measures, such that, e.g., measures of kinetic energy and correlations with the configurational coordinate are exact for the noisy harmonic oscillator, while also producing the correct kinetic measure of transport on flat surfaces. We identify and analyze three half-step velocity definitions that fulfill these criteria, and we outline simple and practical Verlet-type algorithms that incorporate these velocities within the statistically desirable GJ format that provides the optimized configurational statistics. The application of the methods are demonstrated through characteristic Molecular Dynamics simulations of both simple and complex systems, illustrating how the resulting methods are able to efficiently and simultaneously simulate configurational and kinetic statistical ensembles in different situations using very large time steps.

We report modifications of the ph-AFQMC algorithm that allow the use of large time steps and reliable time step extrapolation. Our modified algorithm eliminates size-consistency errors present in the standard algorithm when large time steps are employed. We investigate various methods to approximate the exponential of the one-body operator within the AFQMC framework, distinctly demonstrating the superiority of Krylov methods over the conventional Taylor expansion. We assess various propagators within AFQMC and demonstrate that the Split-2 propagator is the optimal method, exhibiting the smallest time-step errors. For the HEAT set molecules, the time-step extrapolated energies deviate on average by only 0.19 kcal/mol from the accurate small time-step energies. For small water clusters, we obtain accurate complete basis-set binding energies using time-step extrapolation with a mean absolute error of 0.07 kcal/mol compared to CCSD(T). Using large time-step ph-AFQMC for the N2 dimer, we show that accurate bond lengths can be obtained while reducing CPU time by an order of magnitude.