Interpolation-based reduced-order modelling for steady transonic flows via manifold learning

This paper presents a parametric reduced-order model (ROM) based on manifold learning (ML) for use in steady transonic aerodynamic applications. The main objective of this work is to derive an efficient ROM that exploits the low-dimensional nonlinear solution manifold to ensure an improved treatment of the nonlinearities involved in varying the inflow conditions to obtain an accurate prediction of shocks. The reduced-order representation of the data is derived using the Isomap ML method, which is applied to a set of sampled computational fluid dynamics (CFD) data. In order to develop a ROM that has the ability to predict approximate CFD solutions at untried parameter combinations, Isomap is coupled with an interpolation method to capture the variations in parameters like the angle of attack or the Mach number. Furthermore, an approximate local inverse mapping from the reduced-order representation to the full CFD solution space is introduced. The proposed ROM, called Isomap+I, is applied to the two-dimensional NACA 64A010 airfoil and to the 3D LANN wing. The results are compared to those obtained by proper orthogonal decomposition plus interpolation (POD+I) and to the full-order CFD model.     

POD-identification reduced order model of linear transport equations for control purposes

Intrusive reduced order modeling techniques require access to the solver's discretization and solution algorithm, which are not available for most computational fluid dynamics codes. Therefore, a nonintrusive reduction method that identifies the system matrix of linear fluid dynamical problems with a least-squares technique is presented. The methodology is applied to the linear scalar transport convection-diffusion equation for a 2D square cavity problem with a heated lid. The (time-dependent) boundary conditions are enforced in the obtained reduced order model (ROM) with a penalty method. The results are compared and the accuracy of the ROMs is assessed against the full order solutions and it is shown that the ROM can be used for sensitivity analysis by controlling the nonhomogeneous Dirichlet boundary conditions.     

跨音速极限环型颤振的高效数值分析方法

事先建立一个低阶的非线性、非定常气动力模型是开展非线性流场中气动弹性问题研究的一个捷径. 基于CFD方法, 通过计算结构在流场中自激振动的响应来获得系统的训练数据. 采用带输出反馈的循环RBF神经网络, 建立时域非线性气动力降阶模型.耦合结构运动方程和非线性气动力降阶模型, 采用杂交的线性多步方法计算结构在不同速度(动压)下的响应历程, 从而获得模型极限环随速度(动压)变化的特性. 两个典型的跨音速极限环型颤振算例表明, 基于气动力降阶模型方法的计算结果与直接CFD仿真结果吻合很好, 与后者相比其将计算效率提高了1~2个数量级.      

Development of temporal and spatial high‐order schemes for two‐fluid seven‐equation two‐pressure model and its applications in two‐phase flow benchmark problems

Current existing main nuclear thermal‐hydraulics (T‐H) system analysis codes, such as RALAP5, TRACE, and CATHARE, play a crucial role in the nuclear engineering field for the design and safety analysis of nuclear reactor systems. However, two‐fluid model used in these T‐H system analysis codes is ill posed, easily leading to numerical oscillations, and the classical first‐order methods for temporal and special discretization are widely employed for numerical simulations, yielding excessive numerical diffusion. Two‐fluid seven‐equation two‐pressure model is of particular interest due to the inherent well‐posed advantage. Moreover, high‐order accuracy schemes have also attracted great attention to overcome the challenge of serious numerical diffusion induced by low‐order time and space schemes for accurately simulating nuclear T‐H problems. In this paper, the semi‐implicit solution algorithm with high‐order accuracy in space and time is developed for this well‐posed two‐fluid model and the robustness and accuracy are verified and assessed against several important two‐phase flow benchmark tests in the nuclear engineering T‐H field, which include two linear advection problems, the oscillation problem of the liquid column, the Ransom water faucet problem, the reversed water faucet problem, and the two‐phase shock tube problem. The following conclusions are achieved. (1) The proposed semi‐implicit solution algorithm is robust in solving two‐phase flows, even for fast transients and discontinuous solutions. (2) High‐order schemes in both time and space could prevent excessive numerical diffusion effectively and the numerical simulation results are more accurate than those of first‐order time and space schemes, which demonstrates the advantage of using high‐order schemes.     

基于特征正交分解的非定常气动力建模技术

采用特征正交分解(proper orthogonal decomposition, POD)方法, 建立了基于状态空间的非定常气动力降阶模型, 并耦合结构方程, 建立了降阶的气动弹性系统, 开展了颤振分析的初步研究, 计算效率提高了2~3个数量级. 具体过程是:首先获取全阶系统的频域快照构成关联矩阵, 通过对关联矩阵进行奇异值分解提取流场模态(或流场基), 对低能量模态截断形成降阶子空间, 并将其映射到全阶系统, 从而形成基于状态空间的降阶非定常气动力模型. 对气动弹性标模AGARD445.6进行算例验证, 证明了降阶方法正确, 可以提供高效、高精度的气动弹性分析.      

Efficient unsteady aerodynamic loads prediction based on nonlinear system identification and proper orthogonal decomposition

In the present work, an efficient surrogate-based framework is developed for the prediction of motion-induced surface pressure fluctuations and integral force and moment coefficients. The model construction is realized by performing forced-motion computational fluid dynamics (CFD) simulations, while the result is processed via the proper orthogonal decomposition (POD) to obtain the predominant flow modes. Subsequently, a nonlinear system identification is carried out with respect to the applied excitation and the resulting POD coefficients. For the input/output model identification task, a recurrent local linear neuro-fuzzy approach is employed in order to capture the linear and nonlinear characteristics of the dynamic system. Once the reduced-order model (ROM) is trained, it can substitute the flow solver within unsteady aerodynamic or aeroelastic simulation frameworks for a given configuration at fixed freestream conditions. For demonstration purposes, the ROM approach is applied to the LANN wing in high subsonic and transonic flow. Due to the characteristic lambda-shock system, the unsteady aerodynamic surface pressure distribution is dominated by nonlinear effects. Numerical investigations show a good correlation between the results obtained by the ROM methodology in comparison to the full-order CFD solution. In addition, the surrogate approach yields a significant speed-up regarding unsteady aerodynamic calculations, which is beneficial for multidisciplinary computations.     

Low-order modelling of shallow water equations for sensitivity analysis using proper orthogonal decomposition

This article presents a reduced-order model (ROM) of the shallow water equations (SWEs) for use in sensitivity analyses and Monte-Carlo type applications. Since, in the real world, some of the physical parameters and initial conditions embedded in free-surface flow problems are difficult to calibrate accurately in practice, the results from numerical hydraulic models are almost always corrupted with uncertainties. The main objective of this work is to derive a ROM that ensures appreciable accuracy and a considerable acceleration in the calculations so that it can be used as a surrogate model for stochastic and sensitivity analyses in real free-surface flow problems. The ROM is derived using the proper orthogonal decomposition (POD) method coupled with Galerkin projections of the SWEs, which are discretised through a finite-volume method. The main difficulty of deriving an efficient ROM is the treatment of the nonlinearities involved in SWEs. Suitable approximations that provide rapid online computations of the nonlinear terms are proposed. The proposed ROM is applied to the simulation of hypothetical flood flows in the Bordeaux breakwater, a portion of the ‘Rivière des Prairies' located near Laval (a suburb of Montreal, Quebec). A series of sensitivity analyses are performed by varying the Manning roughness coefficient and the inflow discharge. The results are satisfactorily compared to those obtained by the full-order finite volume model.     

Relaxation system based sub-grid scale modelling for large eddy simulation of Burgers' equation

An implicit sub-grid scale model for large eddy simulation is presented by utilising the concept of a relaxation system for one dimensional Burgers' equation in a novel way. The Burgers' equation is solved for three different unsteady flow situations by varying the ratio of relaxation parameter (ε) to time step. The coarse mesh results obtained with a relaxation scheme are compared with the filtered DNS solution of the same problem on a fine mesh using a fourth-order CWENO discretisation in space and third-order TVD Runge-Kutta discretisation in time. The numerical solutions obtained through the relaxation system have the same order of accuracy in space and time and they closely match with the filtered DNS solutions.     

A variable‐fidelity aerodynamic model using proper orthogonal decomposition

A variable‐fidelity aerodynamic model based on proper orthogonal decomposition (POD) of an ensemble of computational fluid dynamics (CFD) solutions at different parameters is presented in this article. The ensemble of CFD solutions consists of two subsets of numerical solutions or snapshots computed at two different nominal orders of accuracy or discretization. These two subsets are referred to as the low‐fidelity and high‐fidelity solutions or data, whereby the low fidelity corresponds with computations made at the lower nominal order of accuracy or coarser discretization. In this model, the relatively inexpensive low‐fidelity data and the more accurate but expensive high‐fidelity data are considered altogether to devise an efficient prediction methodology involving as few high‐fidelity analyses as possible, while obtaining the desired level of detail and accuracy. The POD of this set of variable‐fidelity data produces an optimal linear set of orthogonal basis vectors that best describe the ensemble of numerical solutions altogether. These solutions are projected onto this set of basis vectors to provide a finite set of scalar coefficients that represent either the low‐fidelity or high‐fidelity solutions. Subsequently, a global response surface is constructed through this set of projection coefficients for each basis vector, which allows predictions to be made at parameter combinations not in the original set of observations. This approach is used to predict supersonic flow over a slender configuration using Navier–Stokes solutions that are computed at two different levels of nominal accuracy as the low‐fidelity and high‐fidelity solutions. The numerical examples show that the proposed model is efficient and sufficiently accurate. Copyright © 2016 John Wiley & Sons, Ltd.     

Comparing analytical and numerical solution of nonlinear two and three‐dimensional hydrostatic flows

New test cases for frictionless, three‐dimensional hydrostatic flows have been derived from some known analytical solutions of the two‐dimensional shallow water equations. The flow domain is a paraboloid of revolution and the flow is determined by the initial conditions, the nonlinear advective terms, the Coriolis acceleration and by the hydrostatic pressure. Wetting and drying is also included. Some specific properties of the exact solutions are discussed under different hypothesis and relative importance of the forcing terms. These solutions are proposed for testing the stability, the accuracy and the efficiency of numerical models to be used for simulating environmental hydrostatic flows. The computed solutions obtained with a semi‐implicit finite difference—finite volume algorithm on unstructured grid are compared with the corresponding analytical solutions in both two and three space dimension. Excellent agreement are obtained for the velocity and for the resulting water surface elevation. Comparison of the computed inundation area also shows a good agreement with the analytical solution with degrading accuracy observed when the inundation area becomes relatively large and for long simulation time. Copyright © 2006 John Wiley & Sons, Ltd.     

基于SCI/ERA方法的高效气动力降阶模型

由正交Walsh函数构造Walsh-单信号-复合-输入,对其作用下的计算流体力学响应采用单信号-复合-输入/特征系统实现算法SCI/ERA(Single-Composite-Input/Eigensystem Realization Algorithm)辨识得到离散时间非定常气动力状态空间降阶模型。通过对Isogai机翼剖面气动弹性算例的计算证明该方法具有和非定常计算流体力学方法相当的精度同时模型维数降低2个数量级;在模型构造时间上,SCI/ERA方法比脉冲/ERA方法计算效率提高24%,同时内存占用减小34%;由理论分析可知当耦合结构模态数目增加时,SCI/ERA方法所需的计算开销增幅远小于脉冲/ERA方法;采用频域平衡特征正交分解BPOD(Balanced Proper Orthogonal Decomposition)方法可以准确地从降阶模型中提取出一个低频二次降阶模型,同时保持与原模型相当的精度。二次降阶后模型维数进一步减小88%。     

Solution of time-domain acoustic wave propagation problems using a RBF interpolation model with “a priori” estimation of the free parameter

In the last decade, Meshless Methods have found widespread application in different fields of engineering and science. Beyond novelty, their mathematical simplicity and numerical accuracy have been the key of their rapid dissemination. Among meshless techniques, RBF (Radial Basis Functions) based methods can be simple and general to solve the problems related to multiple areas of applied physics and engineering. In the specific field of acoustics, there are usually two possible approaches for solving a problem: time- and frequency-domain. In this paper, the authors propose a local time-domain approach to establish an efficient methodology for the solution of large-scale acoustic wave propagation problems. For this purpose, a local interpolation scheme, based on the reproduction of the local wave field using RBFs (MultiQuadric and Gaussian), is implemented and its accuracy is verified against known closed-form solutions. An explicit time-domain marching procedure is adopted, and the quality of the numerical results is also compared with that obtained using standard space-time Finite-Difference schemes. Additionally, the RBF interpolation model is used to simulate the propagation of a Ricker pulse in two simple test cases, and applied to simulate a more complex configuration, corresponding to an underwater sound propagation problem. In this frame, the results are also compared with those computed using a fourth-order in space and second-order in time Finite-Difference scheme.     

Efficiency of randomised dynamic mode decomposition for reduced order modelling

ABSTRACT

The purpose of this paper is the identification of a reduced order model (ROM) from numerical code output by non-intrusive techniques (i.e. not requiring projecting of the governing equations onto the reduced basis modes). In this paper, we perform a comparison between two methods of model order reduction based on dynamic mode decomposition (DMD). The first method is a deterministic (classic) DMD technique endowed with a dynamic filtering criterion of selection of modes used in the ROM model. The second method is an adaptive randomised DMD algorithm (ARDMD) based on a randomised singular value decomposition. This produced an accelerating algorithm, which is endowed with a few additional advantages. In addition, the reduced order model is guaranteed to satisfy the boundary conditions of the full model, which is crucial for surrogate modelling. For numerical illustration, we use the shallow water equations model.     

Principal interval decomposition framework for POD reduced‐order modeling of convective Boussinesq flows

A principal interval decomposition (PID) approach is presented for the reduced‐order modeling of unsteady Boussinesq equations. The PID method optimizes the lengths of the time windows over which proper orthogonal decomposition (POD) is performed and can be highly effective in building reduced‐order models for convective problems. The performance of these POD models with and without using the PID approach is investigated by applying these methods to the unsteady lock‐exchange flow problem. This benchmark problem exhibits a strong shear flow induced by a temperature jump and results in the Kelvin–Helmholtz instability. This problem is considered a challenging benchmark problem for the development of reduced‐order models. The reference solutions are obtained by direct numerical simulations of the vorticity and temperature transport equations using a compact fourth‐order‐accurate scheme. We compare the accuracy of reduced‐order models developed with different numbers of POD basis functions and different numbers of principal intervals. A linear interpolation model is constructed to obtain basis functions when varying physical parameters. The predictive performance of our models is then analyzed over a wide range of Reynolds numbers. It is shown that the PID approach provides a significant improvement in accuracy over the standard Galerkin POD reduced‐order model. This numerical assessment of the PID shows that it may represent a reliable model reduction tool for convection‐dominated, unsteady‐flow problems. Copyright © 2015 John Wiley & Sons, Ltd.     

A numerical scheme for solving a periodically forced Reynolds equation

A modified Reynolds equation is used to model the air‐film in a high‐speed squeeze‐film bearing. The axial position of the bearing stator is prescribed as a finite amplitude periodic oscillation. A numerical approach is considered for solving the uncoupled and coupled periodic problems associated with this model. The uncoupled problem requires the computation of the squeeze‐film dynamics when the rotor is held at a fixed axial position and the coupled problem incorporates the additional air–rotor interaction since the rotor position is unknown and modelled as a spring‐mass‐damper system. The details of a Fourier spectral collocation scheme are provided for the reduction of the modified Reynolds equation to a system of non‐linear, first‐order ordinary differential equations in space. Using the Matlab boundary value problem solver bvp4c this system of equations is solved to give the periodic pressure distributions and rotor heights. The high degree of accuracy in the spectral collocation scheme is demonstrated through comparison with an appropriate analytical solution. Further analysis indicates that the direct periodic solver is at least 10 times faster than the equivalent Crank–Nicholson finite‐difference scheme. For changing values of a selected physical parameter the method of arc‐length continuation is employed to track branches of solutions computed using the spectral collocation scheme. A selection of results is presented to demonstrate the range of accessible solutions and the robust nature of the numerical scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

Scattering of a Rayleigh wave by an elastic three-quarter space

The steady state problem of the scattering of an incident Rayleigh wave by an elastic three-quarter space (270° wedge) is considered. The problem is reduced to the numerical solution of a pair of Fredholm integral equations of the second kind. The kernels of these equations consist of elementary functions. The reflection and transmission coefficients can be computed from two independent equations, which provides a check on both the accuracy and correctness of the results. The method used can be extended to other wedge angles.     

A novel numerical algorithm based on Galerkin–Petrov time-discretization method for solving chaotic nonlinear dynamical systems

Nonlinear chaotic systems yield many interesting features related to different physical phenomena and practical applications. These systems are very sensitive to initial conditions at each time-iteration level in a numerical algorithm. In this article, we study the behavior of some nonlinear chaotic systems by a new numerical approach based on the concept of Galerkin–Petrov time-discretization formulation. Computational algorithms are derived to calculate dynamical behavior of nonlinear chaotic systems. Dynamical systems representing weather prediction model and finance model are chosen as test cases for simulation using the derived algorithms. The obtained results are compared with classical RK-4 and RK-5 methods, and an excellent agreement is achieved. The accuracy and convergence of the method are shown by comparing numerically computed results with the exact solution for two test problems derived from another nonlinear dynamical system in two-dimensional space. It is shown that the derived numerical algorithms have a great potential in dealing with the solution of nonlinear chaotic systems and thus can be utilized to delineate different features and characteristics of their solutions.     

Reduced‐order modeling based on POD of a parabolized Navier–Stokes equation model I: forward model

A proper orthogonal decomposition (POD)‐based reduced‐order model of the parabolized Navier–Stokes (PNS) equations is derived in this article. A space‐marching finite difference method with time relaxation is used to obtain the solution of this problem, from which snapshots are obtained to generate the POD basis functions used to construct the reduced‐order model. In order to improve the accuracy and the stability of the reduced‐order model in the presence of a high Reynolds number, we applied a Sobolev H₁ norm calibration to the POD construction process. Finally, some numerical tests with a high‐fidelity model as well as the POD reduced‐order model were carried out to demonstrate the efficiency and the accuracy of the reduced‐order model for solving the PNS equations compared with the full PNS model. Different inflow conditions and different selections of snapshots were experimented to test the POD reduction technique. The efficiency of the H₁ norm POD calibration is illustrated for the PNS model with increasingly higher Reynolds numbers, along with the optimal dissipation coefficient derivation, yielding the best root mean square error and correlation coefficient between the full and reduced‐order PNS models. Copyright © 2011 John Wiley & Sons, Ltd.     

二维拉氏辐射流体力学人为解构造方法

人为构造解方法是复杂多物理过程耦合程序正确性验证的重要方法之一,适用于二维拉氏大变形网格的流体、辐射耦合人为解模型较为少见。针对拉氏辐射流体力学程序正确性验证的需要,从二维拉氏辐射流体力学方程组出发,基于坐标变换技术,给出了拉氏空间到欧氏空间的物理变量导数关系式,开展了辐射流体耦合的人为解构造方法研究,构造了一类质量方程无源项的二维人为解模型,并应用于非结构拉氏程序LAD2D辐射流体力学计算的正确性考核,为流体运动网格上的辐射扩散计算提供了一种有效手段。数值结果显示观测到的数值模拟收敛阶与理论分析一致。