An Adaptive Time-Step Backward Differentiation Algorithm to Solve Stiff Ordinary Differential Equations: Application to Solve Activated Sludge Models

A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system’s stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters.

A varying time-step explicit numerical inte-gration algorithm for solving motion equa-tion

If a traditional explicit numerical integration algorithm is used to solve motion equation in the finite element simulation of wave motion, the time-step used by numerical integration is the smallest time-step restricted by the stability criterion in computational region. However, the excessively small time-step is usually unnecessary for a large portion of computational region. In this paper, a varying time-step explicit numerical integration algorithm is introduced, and its basic idea is to use different time-step restricted by the stability criterion in different computational region. Finally, the feasibility of the algorithm and its effect on calculating precision are verified by numerical test.

Finite-difference modeling with variable grid-size and adaptive time-step in porous media

Forward modeling of elastic wave propagation in porous media has great importance for understanding and interpreting the influences of rock properties on characteristics of seismic wavefield. However,the finite-difference forward-modeling method is usually implemented with global spatial grid-size and time-step; it consumes large amounts of computational cost when small-scaled oil/gas-bearing structures or large velocity-contrast exist underground. To overcome this handicap,combined with variable grid-size and time-step,this paper developed a staggered-grid finite-difference scheme for elastic wave modeling in porous media. Variable finite-difference coefficients and wavefield interpolation were used to realize the transition of wave propagation between regions of different grid-size. The accuracy and efficiency of the algorithm were shown by numerical examples. The proposed method is advanced with low computational cost in elastic wave simulation for heterogeneous oil/gas reservoirs.

On time-step in structural seismic response analysis under ground displacement/acceleration

There are two models in use today to analyze structural responses when subjected to earthquake ground motions, the Displacement Input Model （DIM） and the Acceleration Input Model （AIM）. The time steps used in direct integration methods for these models are analyzed to examine the suitability of DIM. Numerical results are presented and show that the time-step for DIM is about the same as for AIM, and achieves the same accuracy. This is contrary to previous research that reported that there are several sources of numerical errors associated with the direct application of earthquake displacement loading, and a very small time step is required to define the displacement record and to integrate the dynamic equilibrium equation. It is shown in this paper that DIM is as accurate and suitable as, if not more than, AIM for analyzing the response of a structure to uniformly distributed and spatially varying ground motions.

RETHINKING TO FINITE DIFFERENCE TIME-STEP INTEGRATIONS

The numerical time step integrations of PDEs are mainly carried out by the finitedifference method to date. However,when the time step becomes longer, it causes theproblem of numerical instability,. The explicit integration schemes derived by the singlepoint precise integration method given in this paper are proved unconditionally stable.Comparisons between the schemes derived by the finite difference method and theschemes by the method employed in the present paper are made for diffusion andconvective-diffusion equations. Nunierical examples show the superiority of the singlepoint integration method.

Interval of effective time-step size for the numerical computation of nonlinear ordinary differential equations

由于满足计算的不确定性原理,需适当选取时间步长以保证非线性常微分方程组数值解的可靠性,目前尚未见关于有效步长区间的理论结果。本文对于给定的误差限,将方法截断误差与机器舍入误差的相关曲线分别进行平移,从而得到一种确定有效步长近似区间的方法,并推导出近似区间相比于原区间的相对误差公式。另外,研究了有效步长区间随积分时间的变化规律,并对已有的数值结果给出解释。本文所得结论可为数值求解常微分方程组选取有效步长并得到可靠的数值解提供理论支持。

Assessment of corrected time-step method for nominal ionospheric gradient calculation: A comparative analysis with spatial approaches

The effect of ionospheric delay on the ground-based augmentation system under normal conditions can be mitigated by determining the value of the nominal ionospheric gradient(σvig).The nominal ionospheric gradient is generally obtained from Continuously Operating Reference Stations data by using the spatial single-difference method(mixed-pair,station-pair,or satellite-pair)or the temporal single-difference method(time-step).The time-step method uses only a single receiver,but it still contains ionospheric temporal variations.We introduce a corrected time-step method using a fixed-ionospheric pierce point from the geostationary equatorial orbit satellite and test it through simulations based on the global ionospheric model.We also investigate the effect of satellite paths on the corrected time-step method in the region of the equator,which tends to be in a more north–south direction and to have less coverage for the east–west ionospheric gradient.This study also addresses the limitations of temporal variation correction coverage and recommends using only the correction from self-observations.All processes are developed under simulations because observational data are still difficult to obtain.Our findings demonstrate that the corrected time-step method yieldsσvig values consistent with other approaches.

基于定子电流法的绕线转子无刷双馈发电机偏心故障研究

绕线转子无刷双馈电机在船舶轴带发电以及风力发电等领域中具有良好的应用前景。转子偏心是一种常见的机械故障,威胁着现代化工业体系的稳定运转,因此实现其状态检测和故障诊断非常关键。由于无刷双馈电机具有特殊的定、转子绕组结构以及复杂的气隙磁场,因此传统的故障检测方法无法直接用于该类电机。该文通过定子电流法提取了功率绕组电流的特征频率分量作为故障指标,实现对静偏心、动偏心以及混合偏心的无创诊断。建立故障电机的时步有限元模型,并通过详细的故障机理分析,得出不同类型的偏心故障可能导致的特征频率信号。最后,以一台2/4对极无刷双馈电机为例,并通过有限元仿真、样机试验验证解析法的结论,为绕线转子无刷双馈电机偏心故障的精准检测提供了理论依据。

基于子循环自适应串行交错时间匹配算法的油浸式变压器绕组瞬态温升计算

针对传统算法存在的步长调整及耦合点选择问题,该文提出基于子循环自适应串行交错(SCAS)的时间匹配算法。首先,在步长调整方面,通过加入自适应(ATS)-启发式(HTS)混合变步长算法,解决步长无法实时调整的问题;然后,针对耦合点选择问题,该文在传统常规串行交错(CSS)时间匹配算法及现有改进方案子循环常规串行交错(SCSS)的基础上,提出SCAS时间匹配算法,采用自适应方案确定计算时间窗及预定耦合点,并引入指数平滑法对其精度进行保证;最后,该文基于110 kV油浸式电力变压器建立二维瞬态单分区分匝绕组模型,并将所提算法与其他传统方法进行对比分析。结果表明:在精度方面,与传统CSS算法相比SCAS算法的流场及温度场最大平均相对误差均不超过0.45%,在可接受误差范围内;在效率方面,SCAS算法总体计算效率较CSS算法提高了近20倍。同时,为了说明所提算法的工程价值,该文搭建基于110 kV变压器绕组模型的温升实验平台,将SCAS时间匹配算法应用于该模型中,实验结果表明:在精度方面,SCAS算法与实验达到稳态时最大绝对误差为2.7 K;在计算效率上,本文所提算法的计算效率较传统算法提升约46倍,计算步数约为传统的1/47,减少了计算冗余。

结构动力问题的高阶精确时步群积分方法

高阶精确时间积分方法可为与时间相关的高频复杂动力行为提供高精度的预测结果,但既有高阶精确时间积分方法普遍存在计算工作量偏大的问题,难以满足实际结构线性和非线性动力分析日益增长的计算需求。该文提出了一种基于时步群的高阶精确时间积分方法,将p(p≥2)个相邻的未知时步组成待求解的时步群,以结构动力方程积分解为基础构建逐时步群求解结构动态响应的时间积分方案。在对每个时步群进行积分的过程中,无需联立求解方程,仅通过矩阵乘法运算即可一次性地计算得到时步群内全部p个时步的动态响应。数值特性分析以及线性与非线性算例试验均表明,该文算法精度高、稳定性好、数值耗散可控,在选择较大的时间步距情形下依然能够稳定地获得高精度的计算结果。相较传统二阶精度时间积分方法,该文算法的计算效率亦有较大幅度地提高。

考虑记忆时间的LSTM模型在赣江流域径流预报中的应用

在气候变化条件下,准确的径流预测对水资源的规划与管理十分重要。本文基于长短时记忆神经网络(LSTM)模型,采用赣江流域外洲、峡江以及栋背水文站的逐日流量以及CN05.1日降水数据构建3个不同面积流域的径流预测模型,并通过设置不同情景分析:模型的有效预见期与不同流域平均产汇流时间之间的关系,有效预见期内LSTM径流预测模型精度与记忆时间之间的关系,不同长度的预见期与模型最佳记忆时间之间的关系,同时探讨LSTM径流预测所需的记忆时间与流域面积的关系。结果表明:(1)综合考虑降水和前期径流情景下的径流预测效果最好,当预见期为1 d时,外洲、峡江、栋背站的纳什效率系数(NSE)分别可达0.98、0.96以及0.90;且其有效预见期与仅考虑降水信息的有效预见期相同,均与流域平均产汇流时间相近。(2)随着预见期的延长,不同情景下的预测精度均有不同程度的下降,其中仅考虑前期径流情景的下降率最大,说明降水信息较前期径流对径流预测效果的提升更重要。同时,随着流域面积的增加,相同预见期内径流预测精度均有所提升。(3)当预见期相同时,随记忆时间的延长,不同径流预测模型的预测精度均先上升至最高,接着具有下降趋势,最后逐渐趋于稳定。且在有效预见期内,随着预见期的延长,最佳记忆时间均有增大趋势,当达到最长的有效预见期时,对应的最佳记忆时间均为14 d。此外,在赣江流域的模拟结果表明,随着流域面积的增大,LSTM的最佳记忆时间减小。研究结果可为赣江流域的径流预报提供参考,同时有助于推求其他流域采用机器学习进行径流预测所需的最佳记忆时间。

一种可调延时超窄脉冲触发序列产生技术

为了实现高频信号在欠采样条件下的波形重构,递进延时超窄触发信号的产生成为顺序等效采样技术用于触发取样系统进行高频信号采样的关键。为此,设计了一种可编程延时触发序列产生及调理电路,在FPGA数字电路的控制下,通过计数器与延时模块产生可调延时触发序列,利用阶跃恢复二极管特性对产生的触发序列进行调理,产生一种延时步进可调、边沿极窄的脉冲信号。通过对该电路进行测试,结果表明,输出脉冲信号步进范围0~-2.4 ns可调,分辨率可达1 ps,且边沿跳变时间可以达到120 ps内,幅度可达到8 V左右。该延迟脉冲和调理电路可应用于通信、雷达等信号探测设备中,对于高频信号的获取与分析具有重要意义。

一种滑动检测算法下的滑坡位移时序分解方法

针对“阶跃式”滑坡位移时序分解模型力学解释性不强的缺陷,根据西原蠕变本构模型与自适应改进遗传算法模型,提出滑动R_(nl)阶跃点检测方法与改进加权移动平均修正阶跃项位移方法,并将该方法应用于白水河滑坡位移时序分解。将滑动R_(nl)阶跃点检测结果与MK检验结果、滑动t检验结果以及Bayes检测结果作对比。结果表明,滑动R_(nl)阶跃点检测结果更加准确与适用;同时将新型滑坡位移时序分解结果与二次移动平均时序分解结果、三次指数平滑时序分解结果以及VMD时序分解结果作对比。结果表明,新型滑坡位移时序分解方法解决了滑坡趋势项位移无规律、无力学解释性的问题,且在时序分解加法模式中单独引入滑坡位移预测中最重要的阶跃项位移,分析预测更具有针对性。因此,新型时序分解模型有一定的工程价值与时序预测借鉴价值。

An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon.The numerical simulation of the Cahn-Hilliardmodel needs very long time to reach the steady state,and therefore large time-stepping methods become useful.The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations.The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time.The proposed scheme is proved to be unconditionally energy stable and mass-conservative.An error estimate for the numerical solution is also obtained with second order in both space and time.By using this energy stable scheme,an adaptive time-stepping strategy is proposed,which selects time steps adaptively based on the variation of the free energy against time.The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

A High Order Adaptive Time-Stepping Strategy and Local Discontinuous Galerkin Method for the Modified Phase Field Crystal Equation

In this paper,we will develop a first order and a second order convex splitting,and a first order linear energy stable fully discrete local discontinuous Galerkin(LDG)methods for the modified phase field crystal(MPFC)equation.In which,the first order linear scheme is based on the invariant energy quadratization approach.The MPFC equation is a damped wave equation,and to preserve an energy stability,it is necessary to introduce a pseudo energy,which all increase the difficulty of constructing numerical methods comparing with the phase field crystal(PFC)equation.Due to the severe time step restriction of explicit timemarchingmethods,we introduce the first order and second order semi-implicit schemes,which are proved to be unconditionally energy stable.In order to improve the temporal accuracy,the semi-implicit spectral deferred correction(SDC)method combining with the first order convex splitting scheme is employed.Numerical simulations of the MPFC equation always need long time to reach steady state,and then adaptive time-stepping method is necessary and of paramount importance.The schemes at the implicit time level are linear or nonlinear and we solve them by multigrid solver.Numerical experiments of the accuracy and long time simulations are presented demonstrating the capability and efficiency of the proposed methods,and the effectiveness of the adaptive time-stepping strategy.

多工况下的直齿轮接触疲劳寿命仿真研究

基于直齿轮静力学分析的应力特征和瞬态动力学的应力时间子步历程,结合弹流润滑理论下的齿面摩擦因数,使用Workbench建立直齿轮接触疲劳寿命模型。通过弹性力学赫兹接触理论对静力学有限元仿真结果进行校核,确定应力历程输入中的放缩系数,估计材料的S-N曲线,并在Palmgren-Miner线性损伤理论框架下,利用nCode计算直齿轮接触疲劳寿命,得到寿命云图和危险节点位置。结果表明:齿轮接触疲劳的危险节点多出现在齿面沿齿宽方向线接触的两端位置,输入转速、负载转矩与接触疲劳寿命呈负相关,摩擦因数与接触疲劳寿命呈正相关,负载转矩对接触疲劳寿命的影响幅度最大。

计及季度趋势变时间步长风速仿真模型及应用

为提高风电并网系统充裕度评估精度,提出计及季度趋势的变时间步长风速仿真模型。该模型应用奇异谱分析从风速样本中提取季度趋势和随机波动分量,应用线性插值和奥恩斯坦-乌伦贝克过程对两分量建模,产生任意时间步长的仿真风速样本。将该模型嵌入蒙特卡罗模拟法,提出风电并网系统的季度充裕度评估方法。应用该模型仿真某观测站风速样本,评估风电并网IEEE-RTS发电系统的季度充裕度,结果表明,该模型可产生与实测风速样本统计指标等特征接近的任意时间步长仿真风速样本,且季度充裕度评估精度较高。

电离层层析辅助等离子体泡的TEC空间梯度研究

在严重的电离层活动时,发生的极端电离层扰动会对GNSS地基增强系统(Ground-based Augmentation Systems, GBAS)的用户构成完好性威胁,例如中国低纬地区出现的等离子体泡现象。本文针对等离子体泡造成的TEC空间梯度展开研究,提出了一种电离层层析辅助研究等离子体泡造成的TEC空间梯度的方法。首先,通过多分辨率电离层层析成像技术重构中国及周边中低纬地区在2017-09-08T12:00—17:00UT扰动最强烈时段的电离层层析图像,发现该时段内中国低纬地区出现电子密度耗竭的等离子体泡现象;其次,利用短基线双站法和时间步长法两种方法计算等离子体泡附近区域TEC空间梯度;最后,结合两者进行比较研究,验证了较大的TEC空间梯度往往存在于等离子体泡的侧壁周围,如本文在14:06UT出现在香港空域附近的最大TEC空间梯度为133.16 mm/km。同时还可以看出:层析反演的电离层图像不仅能直观地分析等离子体泡的三维电子密度分布,还可以帮助发现TEC空间大梯度产生的原因。

改进LSTM在金沙江流域中长期径流预报研究

径流非平稳性及超参数率定容易陷入“维数灾”的问题限制了神经网络(Long Short-Term Memory,LSTM)在中长期径流预报中的应用,针对上述问题,采取“分解-预测-重构”的思路,在数据预处理阶段采用变分模态分解(Variational Mode Decomposition,VMD)进行分解重构,耦合PSO(Particle Swarm Optimization)、SSA(Sparrow Search Algorithm)优化算法长短期记忆神经网络(LSTM),构建VMD-PSO-LSTM、VMD-SSA-LSTM、VMD-LSTM和LSTM四类模型进行对比试验研究,并探讨时滞步长对预报精度的影响。将模型应用于金沙江流域石鼓水文站月平均径流预报:(1)VMD-PSO-LSTM预报精度略高于VMD-SSA-LSTM,其次是VMD-LSTM,最后是LSTM。VMD-PSO-LSTM预报成果的MAE、MAPE、NSE、QR、RMSE分别为160.4、0.135、0.949、0.811、252.8;VMD-PSO-LSTM预报成果评价指标MAE、MAPE和RMSE分别比LSTM模型减小53.92%、50.67%、55.86%;NSE、QR分别比LSTM模型提升28.34%、84.48%;(2)在一定范围内,时滞步长越大,MAE、MAPE和RMSE增大,NSE、QR下降。【结论】对径流时间序列预处理,能够降低时间序列的非平稳性;加入优化算法能够在超参数空间中寻找到较优超参数组合,使得LSTM拟合能力较强,同时保持一定的泛化能力,从而提高预报精度。此外,时滞步长是影响预报精度的重要因素,加入优化算法能降低模型对时滞步长的敏感性。

基于头峰的多步分解反应过程热失控特征参数计算方法研究

目前微通道反应器在含能材料合成领域广泛应用,极大提高了合成过程的安全性,但是仍然需要关注物料的热稳定性。绝热条件下最大反应速率到达时间(TMR_(ad))与TMR_(ad)为24 h所对应的引发温度(T_(D24))是两个表征危险化学品及含能材料热分解危险性的重要特征参数,这两个参数的传统计算方法为单步N级法和数值计算法,存在分析过程费时费力的缺点。为此,根据差示扫描量热仪动态升温测试曲线,提出了基于头峰(即多峰曲线分峰后的第一个峰)的热分解失控特征参数计算方法,采用穷举法比较了该方法与模型计算法的T_(D24)偏差,进行了数值模拟验证,并基于文献实验计算了1,8-二硝基蒽醌、改性硝基胍(M-NQ)、1,5-二硝基蒽醌和3,4-二硝基呋咱基氧化呋咱(DNTF)4种物质的热失控特征参数。数值模拟结果表明,对于两步和三步连续反应,T_(D24)的最大偏差百分比分别为2.88%和6.9%,最大偏差为6.41℃;对于三步连续反应,T_(D24)最大偏差为5.39℃。结果表明,4种含能材料的T_(D24)计算偏差分别为-4.55,0.71,3.16℃和-0.84℃,与模型计算法得到的T_(D24)相比,偏差百分比的绝对值均小于2%,证实了T_(D24)计算方法的有效性,计算T_(D24)时偏差较小,计算简便,能够较为准确地获得其热分解失控特征参数。