Model reduction for supersonic cavity flow using proper orthogonal decomposition(POD)and Galerkin projection

The reduced-order model （ROM） for the two-dimensional supersonic cavity flow based on proper orthogonal decomposition （POD） and Galerkin projection is investigated. Presently, popular ROMs in cavity flows are based on an isentropic assumption, valid only for flows at low or moderate Mach numbers. A new ROM is constructed involving primitive variables of the fully compressible Navier-Stokes （N-S） equations, which is suitable for flows at high Mach numbers. Compared with the direct numerical simulation （DNS） results, the proposed model predicts flow dynamics （e.g., dominant frequency and amplitude） accurately for supersonic cavity flows, and is robust. The comparison between the present transient flow fields and those of the DNS shows that the proposed ROM can capture self-sustained oscillations of a shear layer. In addition, the present model reduction method can be easily extended to other supersonic flows.

The proper orthogonal decomposition method for the Navier-Stokes equations

The proper orthogonal decomposition (POD) method for the instationary Navier-Stokes equations is considered. Several numerical approaches to evaluating the POD eigenfunctions are presented. The POD eigenfunctions are applied as a basis for a Galerkin projection of the instationary Navier-Stokes equations. And a low-dimensional ordinary differential models for fluid flows governed by the instationary Navier-Stokes equations are constructed. The numerical examples show that the method is feasible and efficient for optimal control of fluids.

Proper orthogonal decomposition analysis of coherent motions in a turbulent annular jet

A three-dimensional incompressible annular jet is simulated by the large eddy simulation(LES)method at a Reynolds number Re=8500.The time-averaged velocity field shows an asymmetric wake behind the central bluff-body although the flow geometry is symmetric.The proper orthogonal decomposition(POD)analysis of the velocity fluctuation vectors is conducted to study the flow dynamics of the wake flow.The distribution of turbulent kinetic energy across the three-dimensional POD modes shows that the first four eigenmodes each capture more than 1%of the turbulent kinetic energy,and hence their impact on the wake dynamics is studied.The results demonstrate that the asymmetric mean flow in the near-field of the annular jet is related to the first two POD modes which correspond to a radial shift of the stagnation point.The modes 3 and 4 involve the stretching or squeezing effects of the recirculation region in the radial direction.In addition,the spatial structure of these four POD eigenmodes also shows the counter-rotating vortices in the streamwise direction downstream of the flow reversal region.

基于POD和Kriging的水滴收集量快速预测方法

数值模拟手段预测翼面处水滴撞击特性通常较为耗时,为快速准确计算结冰条件下的翼面处水滴收集量,提出了基于本征正交分解和代理模型的水滴收集量快速预测方法。首先对FAR 25部附录C中连续最大结冰条件进行优化拉丁超立方采样,通过数值模拟手段获得各采样点在翼面处水滴收集量分布,从而构造样本空间。在此基础上,利用本征正交分解(Proper orthogonal decompostion,POD)方法找到表达和重构水滴收集量的本征模态以及相应的拟合系数。最后,利用Kriging模型建立样本空间中各采样点与拟合系数间的代理模型,实现翼面处水滴收集量分布的快速预测。经多组工况验证表明:该方法可较为准确地预测翼面处水滴收集量分布,其计算成本较数值模拟方法大幅降低,能够为无人机防除冰设计提供有益参考。

基于POD-αATS的油浸变压器瞬态温升降阶自适应变步长计算方法

针对油浸式电力变压器瞬态温升计算效率过低的问题,该文提出本征正交分解-αATS(proper orthogonal decomposition-adaptive time stepping based onαfactor,POD-αATS)降阶自适应变步长瞬态计算方法。首先,推导变压器绕组瞬态温升计算的有限元离散方程;其次,采用POD降阶算法改善传统瞬态计算中存在的条件数过大及方程阶数过高的问题;同时对于瞬态计算中的时间步长选择问题,提出适用于非线性问题的αATS变步长策略;然后,为验证方法的有效性,基于110 kV油浸式电力变压器绕组的基本结构建立二维八分区数值计算模型,同时将计算结果与基于110 kV绕组的温升实验结果进行对比。数值计算及实验结果表明,所提算法与全阶定步长算法在流场和温度场中的精度几乎相同,且流场计算效率提升约45倍,温度场计算效率提升约38倍,计算速度得到显著提高。这一点在温升实验中同样得到验证,说明该文所提算法的准确性、高效性及一定的工程实用性。

面向数字孪生模型应用的油浸式变压器绕组温度POD-RBFLP降阶计算方法

了解油浸式电力变压器绕组的温度情况是保证其运行稳定性的关键,也是当前针对油浸式变压器数字孪生分析的必然需求。为了快速地获得变压器绕组的稳态温度,该文提出一种基于本征正交分解(proper orthogonal decomposition,POD)和包含线性多项式的径向基函数响应面法(radial basis function response surface method including linear polynomial,RBFLP)的降阶计算模型。首先,讨论POD方法的降阶特性,并设计一种基于留一法交叉验证的自适应获得快照矩阵方法,以提高计算精度及效率;其次,采用响应面方法建立POD模态系数与绕组工况的相关关系,旨在实现通过绕组工况快速获得POD模态系数,从而跳过对降阶模型的复杂非线性计算,进而高效重构绕组温度场。相关算例表明,该方法具有较好的计算精度和效率,在50组测试工况下,与全阶计算相比,误差不超过2.5 K,且总计算时间仅为1.45 s;最后,基于110 kV变压器绕组搭建温升试验平台,试验结果表明,降阶计算结果相较于试验结果,平均计算误差不超过2 K,且单步计算时间仅为0.03 s,相较于同等规模的全阶计算,计算效率有较大幅度地提升。

包含快照选择策略的POD-QDEIM油浸式变压器绕组瞬态温升降阶算法研究及其实验验证

为了提高采用有限元方法对油浸式电力变压器绕组瞬态温升进行仿真的计算速度,该文提出一种包含快照选择策略的本征正交分解(proper orthogonal decomposition,POD)-基于列主元QR分解的离散经验插值(QR factorization-discrete empirical interpolation method,Q-DEIM)(POD-QDEIM)算法。首先,文章基于最小二乘有限元法与迎风有限元法推导绕组流热耦合计算方程;其次,通过引入POD方法,降低有限元方程阶数提高求解效率;同时提出POD自适应快照选择策略,在保证降阶模型精度的前提下,减小快照矩阵规模;随后,为了改善POD方法求解非线性问题时效率不高的劣势,文章结合基于Q-DEIM方法,通过插值的方法形成方程中的非线性项,从而提高每一时步非线性项的形成效率;最后,基于110 kV油浸式电力变压器绕组基本结构,建立了八分区分匝绕组传热模型以及绕组温升试验装置对所提算法进行验证,仿真及实验结果表明POD-QDEIM降阶算法的计算精度在工程上属于可接受范围,其计算效率较全阶计算提升了近11.72倍,说明该算法在油浸式电力变压器绕组瞬态温升快速仿真领域的工程应用价值。

Numerical investigation of the effect of wedge-type cavitating-bubble generator on attached unsteady cavitating flows using proper orthogonal decomposition method

The objective of this work is to reveal the effect of a passive control method called wedge-type cavitating-bubble generator(WCG)on the cloud cavitation dynamics of National Advisory Committee for Aeronautics(NACA)66 hydrofoil.The simulations are performed using the Partially-averaged Navier-Stokes(PANS)method coupled with the Zwart cavitation model.The proper orthogonal decomposition(POD)method is applied to extract the dominant flow structures.The results show that the WCG can induce the attached cavity to occur just behind the WCG instead of the hydrofoil leading edge.During the periodical time-evolution process of the unsteady cavity,it is found that the attached cavity with a larger scale around the hydrofoil with WCG has a rougher surface,accompanied with more shedding behaviors of small cavities.This is further illustrated by the POD modes,that is,the mode 1 and modes 2–4 present the large and small cavity vortex structures respectively.Meanwhile,the dominant frequencies of 50 Hz,47.5 Hz are given by the POD method respectively for the hydrofoils without and with WCG,which is in good agreement with that of FFT analysis.In addition,the correlation distribution of POD modal coefficients shows that the WCG can strengthen the vortex energy as well as the turbulence intensity.

风力机翼型S809绕流流动特性的POD和DMD对比分析

失速时的流动分离现象对风力机叶片的气动性能有重要影响,S809作为典型水平轴风力机翼型,在临界失速攻角下气动性能会大幅降低。基于流动特征提取的非定常流场降阶模型(reduced-order model,ROM)是进一步深入了解非定常流动的重要手段。本文通过计算流体力学方法得到轻、深失速攻角下翼型的流动特征,对时变速度场进行本征正交分解(proper orthogonal decomposition,POD)和动态模态分解(dynamic mode decomposition,DMD)分析,得到轻、深失速下翼型的非定常流场信息(能量占比、模态频率等)。通过两种方法的对比,结果表明,POD和DMD方法能够准确捕捉流动过程中的非定常结构和升力主频相同的典型模态,但是POD方法由于基于能量特征,在捕捉模态时会忽略与升力主频相近但能量较小的流动结构,而基于频率特征的DMD方法能够准确获得场的演化信息(增长率、频率等)。本文研究有利于针对主频结构发展相应的流动控制方法,从而改善翼型流场情况,提高气动性能。

Reduced-order method for nuclear reactor primary circuit calculation

Accurate real-time simulations of nuclear reactor circuit systems are particularly important for system safety analysis and design.To effectively improve computational efficiency without reducing accuracy,this study establishes a thermal-hydraulics reduced-order model(ROM)for nuclear reactor circuit systems.The full-order circuit system calculation model is first established and verified and then used to calculate the thermal-hydraulic properties of the circuit system under different states as snapshots.The proper orthogonal decomposition method is used to extract the basis functions from snapshots,and the ROM is constructed using the least-squares method,effectively reducing the difficulty in constructing the ROM.A comparison between the full-order simulation and ROM prediction results of the AP1000 circuit system shows that the proposed ROM can improve computational efficiency by 1500 times while achieving a maximum relative error of 0.223%.This research develops a new direction and perspective for the digital twin modeling of nuclear reactor system circuits.

基于POD-ATS-HTS混合变步长方法的油浸式电力变压器绕组瞬态温升计算方法研究

针对当前油浸式电力变压器绕组瞬态温升仿真中,采用固定时间步长效率低的问题,提出一种混合变步长方法。首先,采用初始解优化算法,有效减少计算过程中方程的迭代次数;其次,结合本征正交分解算法(properorthogonal decomposition,POD),改善大规模有限元方程组存在的阶数过高、条件数过大的问题,提高了方程的求解效率和数值稳定性;第三,提出自适应(adaptivetimestepping,ATS)-启发式(heuristic time stepping,HTS)混合变步长算法,通过对时间步长的自适应与启发式调整,有效解决瞬态计算中计算效率与计算精度的对立问题;最后,建立油浸式电力变压器绕组二维单分区分匝的流热耦合仿真模型,以验证所提算法的正确性与高效性。数值计算结果表明:在流场中,与固定步长的计算结果相比,混合变步长算法的误差小于0.46%,计算效率提升了18.45倍;在温度场中,与固定步长的计算结果相比,所提算法的误差小于0.04%,计算效率提升了6倍。同时,通过与传统变步长算法的计算结果对比,说明所提混合变步长算法在计算精度、计算效率及变步长效果方面均具有一定优势。此外,还探讨混合变步长计算中,不同的参数设置对瞬态计算结果及状态变化过程的影响,为其工程应用奠定了一定基础。

A REDUCED-ORDER MFE FORMULATION BASED ON POD METHOD FOR PARABOLIC EQUATIONS

In this paper, we extend the applications of proper orthogonal decomposition （POD） method, i.e., apply POD method to a mixed finite element （MFE） formulation naturally satisfied Brezz-Babu^ka for parabolic equations, establish a reduced-order MFE formulation with lower dimensions and sufficiently high accuracy, and provide the error estimates between the reduced-order POD MFE solutions and the classical MFE solutions and the implementation of algorithm for solving reduced-order MFE formulation. Some numerical examples illustrate the fact that the results of numerical computation are consis- tent with theoretical conclusions. Moreover, it is shown that the new reduced-order MFE formulation based on POD method is feasible and efficient for solving MFE formulation for parabolic equations.

A NEW REDUCED-ORDER FVE ALGORITHM BASED ON POD METHOD FOR VISCOELASTIC EQUATIONS

A proper orthogonal decomposition （POD） technique is used to reduce the finite volume element （FVE） method for two-dimensional （2D） viscoelastic equations. A reduced-order fully discrete FVE algorithm with fewer degrees of freedom and sufficiently high accuracy based on POD method is established. The error estimates of the reduced- order fully discrete FVE solutions and the implementation for solving the reduced-order fully discrete FVE algorithm are provided. Some numerical examples are used to illus- trate that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order fully discrete FVE algorithm is one of the most effective numerical methods by comparing with corresponding numerical results of finite element formulation and finite difference scheme and that the reduced-order fully discrete FVE algorithm based on POD method is feasible and efficient for solving 2D viscoelastic equations.

Reduced-order finite element method based on POD for fractional Tricomi-type equation

The reduced-order finite element method （FEM） based on a proper orthogo- nal decomposition （POD） theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save mem- ory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element （FE） scheme is shown to be un- conditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations （FDEs）.

POD降阶方法在含局部非线性悬臂梁中的适应性分析

在非线性结构的振动控制设计中结构模型的阶数不宜过高,为此,本研究以含局部非线性的悬臂梁为研究对象,开展影响POD降阶方法所得低阶模型精度的研究。着重分析了非线性强弱、降阶模型的阶数、POD模态获取源信号的激振类型、响应信号的采样频率和响应信号采样时长等因素对降阶模型响应预测精度的影响。结果表明:对于强非线性的局部非线性悬臂梁系统,POD方法同样适用;在选取源信号的激振类型时,应避免选取脉冲激励信号;响应的采样频率与时长不一定要选取过大。最后,提出了一种针对含有噪声信号应用POD方法的解决方案,可为工程应用提供有益的参考。

An Extraction of the Dominant Rotor-Stator Interaction Modes by the Use of Proper Orthogonal Decomposition (POD)

The Proper Orthogonal Decomposition method is applied to the instantaneous velocity field within the rotor-stator inter-row region of a high-speed high-pressure centrifugal compressor. The processed data come from experiments and numerical simulations. In comparison with a Fourier transform, the POD gives the best modal approximation for both initial fields, in terms of the energy expressed on any given number of modes to be taken into account： to reach 98% of the total energy of the velocity field, the required number of POD modes is around nine times smaller than the number of Fourier harmonics. The individual POD modes are given and show that the unsteady rotor-stator interaction is already present in the very first modes.

基于双加权POD的建筑物风压场重构

本征正交分解法(Proper Orthogonal Decomposition,POD)是一种基于2阶统计量的降阶方法,通过寻找一组正交单位基使得随机场在新坐标下能有更简洁的描述。本文提出了面积和均方根双加权POD,将其应用于建筑物风压场重构。从均方值角度对POD进行推导,证明POD是均方值意义上的最佳展开方式;在新的推导框架下对POD进行双加权优化,使之能较好地捕捉风压场中能量较低的相干结构;对5∶1矩形风压场进行重构,初步验证了双加权POD的可行性。结果表明:双加权POD可以较好地重构5∶1矩形风压场,重构风压场各空间点的重构精度一致,且能够基本还原所有空间点的时程和功率谱密度。与面积加权POD相比,双加权POD能够显著提高风压场低能量区域的降阶模型重构精度。

基于POD有限元法的粘滞声波方程震源波场重构

叠前逆时偏移是地震勘探中一种流行的地下结构成像方法,其成像条件需要同时刻的震源波场值与检波器波场值.这在实际计算中就需要把正演模拟的所有时刻的震源波场数据全部存储下来,存储需求大.虽然震源波场重构技术可以降低对于波场数据的存储需求,但会引入额外的计算复杂度.为解决这个问题,本文提出了POD有限元法,并将其应用于粘滞震源波场重构.这里的本征正交分解(Proper Orthogonal Decomposition,POD)方法是一种降维方法,能够在降低数据量的同时提供足够的计算精度.数值算例显示,该方法比传统的有限元法更节省存储空间,能够加快重构速度.

基于POD-DNN代理模型的闸墩锚索有效预应力反演

为掌握锚索的有效预应力,确保混凝土预应力闸墩的安全可靠,建立了结合PSO算法的POD-DNN高效反演方法。使用LHS方法在设计空间中采样,并通过有限元计算相应的响应获得数据样本集;利用本征正交分解对数据集进行降维,建立POD-DNN代理模型;结合PSO算法,对混凝土预应力闸墩锚索有效预应力进行了反演。结果表明,该方法能够准确快速地反演锚索的有效预应力,适用于大体积混凝土水工建筑物的反演问题。

基于Twin Builder的110kV油浸式变压器3维磁场降阶模型及损耗分析

为解决3维变压器磁场有限元仿真计算效率低的问题,提出基于本征正交分解法的磁场降阶模型与基于二次多项式的损耗响应面模型。以1台110 kV变压器为算例,首先建立了考虑绕组涡流损耗的变压器磁场仿真模型,通过仿真分析了其在额定工况下的磁场及损耗分布;其次,通过仿真试验设计获取了多组工况下的磁场与损耗数据,在Twin Builder中训练得到变压器磁场的各降阶模块;最后,搭建面向数字孪生应用的变压器3维磁场降阶模型,实现了变压器磁场损耗的快速计算与云图分布的可视化。与计算单数据相比,仿真结果考虑了涡流效应的绕组损耗在靠近端部时有明显升高,最大值分别出现在高、低压绕组两端,约为绕组中部损耗的1.3~1.65倍。基于该磁场结果建立的降阶模型在计算效率方面较全阶模型有显著提升,将实际仿真时间缩短至0.76s,可满足数字孪生应用场景下的磁场快速计算需求。