Coarse Time Step Research Articles

The Transition from Aerosol- to Updraft-Limited Susceptibility Regime in Large-Eddy Simulations with Bulk Microphysics

Large-eddy simulation (LES) is often used as a benchmark simulation in climate science and is suggested as a fundamental tool to examine, e.g., marine cloud brightening. Therefore, it is necessary to critically evaluate if these high-resolution models can skillfully simulate expected physical phenomena. This study focuses on the first indirect aerosol effect in warm stratocumulus clouds. We investigate if an LES code with explicit aerosol-cloud interactions and a widely used two-moment bulk microphysical scheme can reproduce well-known cloud droplet number susceptibility regimes previously identified by observations and supported by detailed parcel model simulations—the updraft-limited regime (typically occurring at high aerosol number concentrations and low updraft speeds) and the aerosol-limited regime (typically occurring at low aerosol number concentrations and high updraft speeds). Our simulations show that the LES in its default configuration cannot reproduce the two regimes if the initial droplet radius of newly activated droplets (rid) is estimated by integrating the wet aerosol size distribution. The main reason is related to the relatively coarse (but commonly used) time step in the model (Δt ≈ 1s), which is too long to resolve relevant microphysical processes adequately at high aerosol concentrations. A regime transition does occur if the timestep is decreased to Δt ≈ 0.1s and if a renormalization procedure is applied, which limits the number of activated droplets so that the water mass of the newly activated droplets cannot exceed the available amount of supersaturated water vapor. Another way to obtain a regime transition is to increase rid to values >1 µm. However, a clear recommendation for the choice of rid cannot be made upon physical arguments. An alternative solution could be to introduce a sub-time-stepping or adaptive time-stepping algorithm to calculate droplet formation and growth, particularly for updraft-limited conditions. Our study highlights the importance of critically evaluating LES results to guarantee that relevant physical processes are properly represented.

A high-order multi-time-step scheme for bond-based peridynamics

A high-order multi-time-step scheme (MTS) for the bond-based peridynamic (PD) model, an extension of classical continuous mechanics widely used for analyzing discontinuous problems like cracks, is proposed. The MTS scheme discretizes the spatial domain with a meshfree method and advances in time with a high-order Runge–Kutta method. To effectively handle discontinuities (cracks) that appear in a local subdomain in the solution, the scheme employs the Taylor expansion and Lagrange interpolation polynomials with a finer time step size, that is, coarse and fine time step sizes for smooth and discontinuous subdomains, respectively, to achieve accurate and efficient simulations. By eliminating unnecessary fine-scale resolution imposed on the entire domain, the MTS scheme outperforms the standard STS scheme for PD by significantly reducing computational costs, particularly for problems with discontinuous solutions, as demonstrated by comprehensive theoretical analysis and numerical experiments.

Single-component Multi-rate Real Time Hybrid Simulation Pilot Test on a Composite Structure

This paper represents a single component multi-rate Real-Time Hybrid Simulation (mrRTHS) strategy for structural assessment of a cantilever Glass Fiber Reinforced Polymer (GFRP) beam loaded at the tip by a sinusoidal point load. This emulated structure is implemented as a simplified wind turbine blade in terms of geometry, scale and load – here with special attention paid to the root and max-chord section. Thus, the experimental substructure comprises the clamped end of the GFRP beam while the free end makes up the numerical substructure. The partitioning between the numerical and experimental substructure – referred to here as the shared boundary – Includes a discrete point with 3 degrees-of-freedom (dof). The numerical substructure generates a displacement signal through a Taylor basis with a coarse time step to optimize computational resources. Using the previous displacement data points, a finer control signal is generated to ensure accurate actuator control in the transfer system. A DIC and inertia compensator is implemented to account for the compliance and dynamics imposed by the load train in the transfer system. The structural response is investigated by mrRTHS for an execution frequency in the range: 0.074Hz – 2.96Hz for the sinusoidal point load. The system performance is evaluated against an experimental test setup of the emulated structure – referred to here as the experimental reference. With a root-mean-square (RMS) error in the order of 8-20% between the mrRTHS and reference, the system proved successful in terms of stability and overall correlation at the shared boundary, which is considered an important milestone towards single component mrRTHS on a structure like e.g. a wind turbine blade, aircraft wing or similar cantilever-shaped large load carrying structure.

A Scalable Multiple-Time-Step Efficient Numerical Algorithm for Wind–Road Vehicle–Train–Bridge Dynamic Simulations

During the past two decades, more and more rail-cum-road bridges have been built to meet the ever-increasing demand of highway and railway transportation in China. The rail-cum-road bridges are expected to withstand multiple types of loadings during normal operation, including wind, road vehicle, and train. Accurate quantifications of the coupling dynamics among bridge, road vehicle, and train are essential to ensure the safety and reliability of rail-cum-road bridges during their lifetime. The present study develops an analytical framework to simulate the complex wind–road vehicle–train–bridge (WRVTB) system. In addition, a scalable multiple-time-step (SMTS) algorithm is proposed to overcome the computational efficiency issue of traditional single-time-step (STS) algorithms for solving the WRVTB interaction. The SMTS algorithm utilizes domain decomposition techniques to enable a fine and a coarse time-step to be assigned for a road vehicle–train subsystem and bridge subsystem, which greatly improves the computational efficiency for solving the WRVTB interaction. A numerical simulation of coupled WRVTB analysis is conducted based on a prototype rail-cum-road bridge to demonstrate the computational efficiency and accuracy of the proposed method. The results indicate that both the MTS and the SMTS algorithms are able to achieve better accuracy and efficiency simultaneously in comparison with the STS algorithm. Although the SMTS algorithm does not exhibit obvious superiority over the MTS algorithm in terms of the overall accuracy and computational efficiency, the major advantage of the proposed SMTS algorithm over the MTS algorithm is the flexibility. When the SMTS algorithm is adopted, a refined time-step can be assigned to a subsystem of interest, while relatively coarse time-steps can be assigned to the remaining subsystems. As such, the desired accuracy of the subsystem of interest is guaranteed at a low computational cost.

Novel Girsanov correction based Milstein schemes for analysis of nonlinear multi-dimensional stochastic dynamical systems

This work proposes Girsanov corrected explicit, semi-implicit, and implicit Milstein approximations for the solution of nonlinear stochastic differential equations. The solution trajectories provided by the Milstein schemes are corrected by employing the change of measures, aimed at removing the error associated with the diffusion process incurred due to the transformation between two probability measures. The change of measures invoked in the Milstein schemes ensures that the solution from the mapping is measurable with respect to the filtration generated by the error process. The proposed scheme incorporates the error between the approximated mapping and the exact representation as an innovation that is accounted in the Milstein trajectories as an additive term. Numerical demonstrations using parametrically and non-parametrically excited stochastic oscillators, a practical problem involving ring type gyroscope, and a 7-degree-of-freedom nonlinear structural system subjected to both stationary and non-stationary excitation demonstrate the improvement in the solution accuracy for the proposed schemes with much coarser time steps when compared with the classical Milstein approximation with finer time steps.

Performance Analysis of Speculative Parallel Adaptive Local Timestepping for Conservation Laws

Stable simulation of conservation laws, such as those used to model fluid dynamics and plasma physics applications, requires the satisfaction of the so-called Courant-Friedrichs-Lewy condition. By allowing regions of the mesh to advance with different timesteps that locally satisfy this stability constraint, significant work reduction can be attained when compared to a time integration scheme using a single timestep size. However, parallelizing this algorithm presents considerable difficulty. Since the stability condition depends on the state of the system, dependencies become dynamic and potentially non-local. In this article, we present an adaptive local timestepping algorithm using an optimistic (Timewarp-based) parallel discrete event simulation. We introduce waiting heuristics to limit misspeculation and a semi-static load balancing scheme to eliminate load imbalance as parts of the mesh require finer or coarser timesteps. Last, we outline an interface for separating the physics of the specific conservation law from the temporal integration allowing for productive adoption of our proposed algorithm. We present a misspeculation study for three conservation laws, demonstrating both the productivity of the local timestepping API, for which 74% of the lines of code are reused across different conservation laws, and the robustness of the waiting heuristics—at most 1.5% of element updates are rolled back. Our performance studies demonstrate up to a 2.8× speedup versus a baseline unoptimized local timestepping approach, a 4x improvement in per-node throughput compared to an MPI parallelization of synchronous timestepping, and scalability up to 3,072 cores on NERSC’s Cori Haswell partition.

Regional Power Planning Robust to Multiple Models: Meeting Mexico's 2050 Climate Goals

As countries set climate goals, they face questions on how these goals can be reached. Important studies have used a top-down approach, employing and comparing multiple Integrated Assessment Models (IAMs) to identify possible development pathways. These studies, however, are at very coarse time-steps; and do not include information on the geographic distribution of resources or infrastructure. Moreover, the multi-model approach, while useful, leaves questions as to how policy-makers and planners can use the divergent results. Using Mexico as a case study, we employ a bottom-up model of the electric power system to identify critical geographic areas of investment for installed capacity and transmission that are robust across a set of IAM-derived climate mitigation pathways. We find that, despite a lack of robustness in the location of installed capacity investments, investment in transmission expansion is fairly robust across pathways, as it is driven largely by the location of load rather than of generation.

Robust Convergence of Parareal Algorithms with Arbitrarily High-Order Fine Propagators

The aim of this paper is to analyze the robust convergence of a class of parareal algorithms for solving parabolic problems. The coarse propagator is fixed to the backward Euler method and the fine propagator is a high-order single step integrator. Under some conditions on the fine propagator, we show that there exists some critical $J_*$ such that the parareal solver converges linearly with a convergence rate near $0.3$, provided that the ratio between the coarse time step and fine time step named $J$ satisfies $J \ge J_*$. The convergence is robust even if the problem data is nonsmooth and incompatible with boundary conditions. The qualified methods include all absolutely stable single step methods, whose stability function satisfies $|r(-\infty)|<1$, and hence the fine propagator could be arbitrarily high-order. Moreover, we examine some popular high-order single step methods, e.g., two-, three- and four-stage Lobatto IIIC methods, and verify that the corresponding parareal algorithms converge linearly with a factor $0.31$ and the threshold for these cases is $J_* = 2$. Intensive numerical examples are presented to support and complete our theoretical predictions.

A new combined scheme of H1-Galerkin FEM and TGM for bacterial equations

In this paper, a new combined second order fully discrete scheme of an H1-Galerkin finite element method (H1-GFEM) with two grid method (TGM) is developed for the bacterial equations. Based on Q11/Q1,0×Q0,1 element pair, the existence and uniqueness of the approximation solution are proved, and the superclose and superconvergent estimates of order O(h2+H4+τ2) for the original variables in H1-norm and intermediate variables in L2-norm are derived, respectively. Some numerical results are provided to show that the proposed method has a very good performance and its computing cost is only about half of H1-GFEM for the considered problem. Here, the sizes of coarse mesh, fine mesh and time step are denoted by H,h,τ, respectively.

The Advanced Multilevel Predictor-Corrector Quasi-Static Method for Pin-Resolved Neutron Kinetics Simulation

The Advanced Multilevel Predictor-Corrector Quasi-static Method (AML-PCQM) is proposed in this work. The four computational levels, including transport, Multi-Group (MG) Coarse Mesh Finite Difference (CMFD), One-Group (1G) CMFD, and Exact Point-Kinetics Equation (EPKE), are coupled with a new dynamic iteration strategy. In each coupling algorithm, the original Transient Fixed Source Problem (TFSP) is solved in the predictor process using coarse time step, and then the flux distribution is factorized to the functions of amplitude and shape in the next corrector process. Finally, multiple fine time steps are used to adjust the predicted solution. Two heterogeneous single assembly problems with the prompt control rod withdrawal event are used to verify the AML-PCQM scheme’s accuracy and efficiency. The numerical results obtained by different cases are compared and analyzed. The final results indicate that the AML-PCQM performs the remarkable advantages of efficiency and accuracy with the reference cases.

Multi-rate Real Time Hybrid Simulation operated on a flexible LabVIEW real-time platform

This paper presents a real-time hybrid simulation (RTHS) strategy where the numerical and experimental substructures are executed at two different rates to optimize computational resources while maintaining an effective actuator control. The concept is referred to here as multi-rate real-time hybrid simulation (mrRTHS), and this approach is intended to enable low-cost RTHS by facilitating testing in the case of limited computational resources. Operated on a Laboratory Virtual Engineering Workshop (LabVIEW) real-time target, the mrRTHS concept is demonstrated through both a single- and multipledegree-of-freedom (SDOF) and (MDOF) mass-spring-damper system. The numerical substructure generates a displacement signal with a coarse time step of Δt. Using the current and three previous displacement data points, a finer control signal is defined with a time step of δt, using a third-order polynomial algorithm–referred to here as the polynomial fitting extrapolator. Both the numerical substructure and polynomial fitting extrapolator is executed with a sampling rate of Δt by an on-board single-core processor–referred to here as the digital signal processor (DSP). Through a field-programmable gate array (FPGA) the control signal is compensated and transmitted to the transfer system through an I/O module with a sampling rate of 1 kHz (i.e. δt = 0.001 sec). The ratio between Δt and δt are an integer–referred to here as the execution ratio. For an execution ratio of 1:5 and 1:10 the system performance is evaluated against a numerical model of the emulated structure–referred to here as the reference structure. For both the SDOF and MDOF system, a good correlation between the mrRTHS and reference is achieved with execution ratios of 1:5 and 1:10. When changing the execution ratio from 1:5 to 1:10, approximately 50% reduction of the required computational resources on the DSP is achieved.

An Efficient Non-Iterative Hybrid Method for Analyzing Train–Rail–Bridge Interaction Problems

The dynamic interaction problem for the train–rail–bridge (TRB) systems presents a computational challenge, especially for the analysis of large-size TRB coupling systems. To address this issue, an efficient non-iterative hybrid method (NHM) is proposed. With this method, the integrated TRB system is divided into three subsystems, i.e. the train subsystem, the rail subsystem, and the bridge subsystem. Based on the individual subsystems, a multi-step[Formula: see text] technique is adopted in which a fine time step is used to analyze the high-frequency coupling vibration for the train and rail subsystems, and a coarse time step is adopted to calculate the low-frequency coupling vibration for the rail and bridge subsystem. Additionally, Zhais explicit integral method is used to predict the displacement of the wheelsets and the rail at the current time step before using the Newmark method. The proposed method incorporates the advantages of Zhai’s explicit method and the MS technique to avoid the iteration that may be required for the train–rail coupled analysis. The simulation fidelity and computational efficiency of the proposed method are demonstrated in the analysis of two examples of typical high-speed railway bridges. It was demonstrated that the proposed method can significantly enhance the computational efficiency, while maintaining a higher precision with a larger time step in comparison with other existing methods.

Modelling of vehicle-track related dynamics: a development of multi-finite-element coupling method and multi-time-step solution method

ABSTRACT In this paper, modelling methods are developed to achieve dynamics exploration for the vehicle–track system and its substructures. First, a coupled model, with the satisfaction of wheel–rail displacement compatibility and force equilibrium within each time step, is constructed for the characterisation of vehicle–track interaction, where the vehicle is modelled as a multi-rigid-body system, the ballasted tracks are modelled as a combination of elasticity, damping and mass structures by finite-element method, and time-varying wheel–rail contacts in geometry, creep and force are depicted at the three-dimensional space. On the foundation of vehicle–track interactions, a multi-finite-element coupling method originated from the energy variation principle is presented to integrate multiple substructures, such as bridge and soil subgrade, into the vehicle–track coupled system. Due to the variation of dominant frequencies of the vehicle–track system and its substructures, a multi-time-step solution method is proposed to achieve a hybrid solution of fine and coarse time step size for the vehicle–track system and substructure system, respectively. Numerical examples are presented to show the accuracy and practicality of the proposed model and methods.

Consistent Transport Transient Solvers of the High-Fidelity Transport Code PROTEUS-MOC

This paper presents the three new pin-resolved transient solvers of PROTEUS-MOC developed in a consistent way to the latest steady-state solver. A new transient fixed source problem (TFSP) solver was developed without relying on the isotropic approximation of the angular flux time derivative. A moving axial mesh scheme was also implemented to model the control rod movement accurately with coarse axial meshes. In addition, in order to reduce the computational time further, an improved quasi-static method (IQM) solver and a predictor-corrector quasi-static method (PCQM) solver were developed in a consistent way to the TFSP solver. Initial verification tests were performed using the C5G7-TD benchmark problems. The results of the direct TFSP solver agreed very well with the MPACT and NECP-X solutions within ~2.5%. Additional analyses suggested that the observed differences are mainly due to the coarse time steps used in the MAPCT and NECP-X calculations. These results indicate that the direct TFSP solver of PROTEUS-MOC was correctly implemented and the moving axial mesh scheme is working properly. Numerical tests of IQM and PCQM against the direct TFSP solver showed that the IQM and PCQM solvers can reduce the computational time about 10 to 100 times without any significant loss of accuracy by allowing larger time steps. The PCQM calculation with the quadratic interpolation of kinetics parameters (KPs) showed the best performance among the four combinations of the IQM and PCQM solvers and the linear and quadratic interpolation schemes of KPs. This study also showed that the different delayed neutron precursor models of six and eight families can cause larger power differences than the different high-fidelity transient codes and that the adjoint scalar flux weighting can cause significant errors in KPs and subsequently in power evolution. In addition, the transient analyses of a modified C5G7 benchmark problem containing a void channel similar to the hodoscope channel of the Transient Reactor Test (TREAT) facility showed that the isotropic approximation of the angular flux time derivative can cause nonnegligible errors in the time-dependent power distribution. This study also demonstrated that PROTEUS-MOC can be used for transient analyses of reactors with internal void regions.

Urban-scale variation in pollen concentrations: A single station is insufficient to characterize daily exposure.

Epidemiological analyses of airborne allergenic pollen often use concentration measurements from a single station to represent exposure across a city, but this approach does not account for the spatial variation of concentrations within the city. Because there are few descriptions of urban-scale variation, the resulting exposure measurement error is unknown but potentially important for epidemiological studies. This study examines urban scale variation in pollen concentrations by measuring pollen concentrations of 13 taxa over 24-hr periods twice weekly at 25 sites in two seasons in Detroit, Michigan. Spatio-temporal variation is described using cumulative distribution functions and regression models. Daily pollen concentrations across the 25 stations varied considerably, and the average quartile coefficient of dispersion was 0.63. Measurements at a single site explained 3-85% of the variation at other sites, depending on the taxon, and 95% prediction intervals of pollen concentrations generally spanned one to two orders of magnitude. These results demonstrate considerable heterogeneity of pollen levels at the urban scale, and suggest that the use of a single monitoring site will not reflect pollen exposure over an urban area and can lead to sizable measurement error in epidemiological studies, particularly when a daily time-step is used. These errors might be reduced by using predictive daily pollen levels in models that combine vegetation maps, pollen production estimates, phenology models and dispersion processes, or by using coarser time-steps in the epidemiological analysis.

A Novel Three-Dimensional FDTD Subgridding Method for the Coupling Analysis of Shielded Cavity Excited by Ambient Wave

The finite-difference time-domain (FDTD) method used for the coupling analysis of the shielded cavity usually needs amount of elements, which would seriously affect the computation cost. To relieve this problem, a novel three-dimensional FDTD subgridding method combined with an automesh generator technique is presented. First, the automesh generator technique is applied to read the triangular element data from the stereolithography format file of the shielded cavity to fastly mesh the structure of the cavity by the coarse and fine grids of FDTD. Then, the subgridding interfaces are separated to temporal and spatial parts to implement temporal and spatial subgridding algorithms, respectively, to obtain the electric and magnetic fields on the interfaces. To ensure the stability and continuity of the subgridding interfaces, the missing subgrid and fine grid magnetic fields nearest temporal and spatial subgridding interfaces are obtained by the Taylor expansions and linear interpolation schemes of the coarse grid magnetic fields, respectively, at the end of each fine time step. And the coarse grid electric fields in the subgrid and fine grid nearest temporal and spatial subgridding interfaces should be updated via some interpolation schemes of the subgrid and fine grid electric fields at the end of each coarse time step, respectively. The accuracy, efficiency, and longer time stability of the presented method have been verified by comparing with the finite-integration time-domain method via the simulation of a cavity with one slot excited by the plane wave. Finally, the suitability of this presented method applied for the coupling analysis of the arbitrary shielded cavity excited by the ambient wave is confirmed.

Conservative multirate multiscale simulation of multiphase flow in heterogeneous porous media

Accurate and efficient simulation of multiphase flow in heterogeneous porous media motivates the development of space-time multiscale strategies for the coupled nonlinear flow (pressure) and saturation transport equations. The flow equation entails heterogeneous high-resolution (fine-scale) coefficients and is global (elliptic or parabolic). The time-dependent saturation profile, on the other hand, may exhibit sharp local gradients or discontinuities (fronts) where the solution accuracy is highly sensitive to the time-step size. Therefore, accurate flow solvers need to address the multiscale spatial scales, while advanced transport solvers need to also tackle multiple time scales. This paper presents the first multirate multiscale method for space-time conservative multiscale simulation of sequentially coupled flow and transport equations. The method computes the pressure equation at the coarse spatial scale with a multiscale finite volume technique, while the transport equation is solved by taking variable time-step sizes at different locations of the domain. At each coarse time step, the developed local time-stepping technique employs an adaptive recursive time step refinement to capture the fronts accurately. The applicability (accuracy and efficiency) of the method is investigated for a wide range of two-phase flow simulations in heterogeneous porous media. For the studied cases, the proposed method is found to provide 3 to 4 times faster simulations. Therefore, it provides a promising strategy to minimize the tradeoff between accuracy and efficiency for field-scale applications.

Parallel-in-Time Simulation Algorithm for Power Electronics: MMC-HVdc System

The complexity in simulating power electronics, such as modular multilevel converters (MMCs), requires simulation algorithms to speed up the process. Existing simulation algorithms exploit spatial parallelism to speed up the simulation. With the rise in the complexity of power electronics and the presence of an increased number of states within them, there are limits in the speedup using spatial parallelism. In this article, a temporal parallelism algorithm based on parallel-in-time methods is developed for simulation of power electronics systems. The temporal parallelism algorithm is based on the computation of power electronics states on coarse and fine time steps using different models. The models of power electronics systems used in coarse and fine time steps are average-value and detailed models, respectively. The updates to states on coarse time step are computed serially and are used to initialize the states on the fine time step. The updates on fine time step are computed in parallel. A translation method is proposed in this article to update the states on fine time step from the simulations in the coarse time step, and vice versa. The proposed algorithm is applied to simulate MMCs and is validated with respect to a detailed reference MMC model. The proposed algorithm recorded a speedup of up to $3.47 \times $ with five cores.

Splitting-based domain decomposition methods for two-phase flow with different rock types

In this paper, we are concerned with the global pressure formulation of immiscible incompressible two-phase flow between different rock types. We develop for this problem two robust schemes based on domain decomposition (DD) methods and operator-splitting techniques. The first scheme follows a sequential procedure in which the (global) pressure, the saturation-advection and the saturation-diffusion problems are fully decoupled. In this scheme, each problem is treated individually using various DD approaches and specialized numerical methods. The coupling between the different problems is explicit and the time-marching is with no iterations. To adapt to different time scales of problem components and different rock types, the novel scheme uses a multirate time stepping strategy, by taking multiple finer time steps for saturation-advection within one coarse time step for saturation-diffusion and pressure, and permits independent time steps for the advection step in the different rocks. In the second scheme, we review the classical Implicit Pressure–Explicit Saturation (IMPES) method (by decoupling only pressure and saturation) in the context of multirate coupling schemes and nonconforming-in-time DD approaches. For the discretization, the saturation-advection problem is approximated with the explicit Euler method in time, and in space with the cell-centered finite volume method of first order of Godunov type. The saturation-diffusion problem is approximated in time with the implicit Euler method and in space with the mixed finite element method, as in the pressure problem. Finally, in a series of numerical experiments, we investigate the practicality of the proposed schemes, the accuracy-in-time of the multirate and nonconforming time strategies, and compare the convergence of various DD methods within each approach.

Rate optimization of fractional flow reservoir model based on the continuous adjoint method

In this paper, continuous adjoint method is applied to research the rate optimization of fractional flow reservoir model. The continuous adjoint equations are derived by the variational theory and the stable numerical schemes of adjoint equations are obtained through numerical methods. Firstly, the continuous adjoint equations are linear equations and evolve from the end simulation time. The convection velocity direction of the continuous adjoint saturation equation is opposite to the convection direction of the saturation equation. Secondly, the total simulation time has significant impact on the adjoint variables and the adjoint gradient. After water breakthrough in production well, the adjoint variables exhibit spatial variety that gradually changes from production well to injection well. With this observation, we find that this variety is related to the fractional flow function and its derivative value. The sharp change of fractional flow function and its derivative at the end of the simulation time can lead to large fluctuation of the adjoint gradient along the time step. Thirdly, the influence of time interval on optimization results is studied. Increasing the number of time steps will increase the optimal value, but the value of the difference between optimal and initial value will not increase significantly. When the variable number of time steps is adopted, this difference decreases as the number of time steps increases, which indicates that the coarse time step optimization can provide a better initial control value for the fine time step optimization. An adaptive time step from coarse step to fine step is thus recommended for optimization.