Input-output reduced-order modeling of unsteady flow over an airfoil at a high angle of attack based on dynamic mode decomposition with control

Due to the damage caused by stall flutter, the investigation and modeling of the flow over a wind turbine airfoil at high angles of attack are essential. Dynamic mode decomposition (DMD) and dynamic mode decomposition with control (DMDc) are used to analyze unsteady flow and identify the intrinsic dynamics. The DMDc algorithm is found to have an identification problem when the spatial dimension of the training data is larger than the number of snapshots. IDMDc, a variant algorithm based on reduced dimension data, is introduced to identify the precise intrinsic dynamics. DMD, DMDc and IDMDc are all used to decompose the data for unsteady flow over the S809 airfoil that are obtained by numerical simulations. The DMD results show that the dominant feature of a static airfoil is the adjacent shedding vortices in the wake. For an oscillating airfoil, the DMDc results may fail to consider the effect of the input and have an identification problem. IDMDc can alleviate this problem. The dominant IDMDc modes show that the intrinsic flow for the oscillating case is similar to the unsteady flow over the static airfoil. Moreover, the input–output model identified by IDMDc can give better predictions for different oscillating cases than the identified DMDc model.     

Computational modeling of phase separation and coarsening in solder alloys

Solders represent highly versatile and useful materials. They provide a broad range of technical applications such as soldering in automotive processing, microelectromechanical systems (MEMS) and solar panels. Due to the fascinating variety of microstructural changes solder materials underlie, their micromorphological dynamics have been extensively studied in the past decades by experimental, analytical and numerical approaches. The evolved microstructure exerts a significant effect, in particular, in very small components such as solder joints in microelectronic packages. In order to capture the essence of the microstructural evolution in solder alloys with a diffusion theory of heterogeneous solid mixtures we employ an extended Cahn–Hilliard phase-field model. In our contribution we introduce different numerical schemes to treat Cahn–Hilliard equation. Here we focus on the innovative isogeometric finite element approach and outline its considerable benefits in comparison to the other methods. To this end we present numerical simulations of phase decomposition and coarsening controlled by diffusion for eutectic binary solders Sn–Pb and Ag–Cu illustrating the versatility of this approach. A concluding computational study of a three-dimensional phase separation event within a solder ball geometry will corroborate the quality of our model.     

Theoretical analysis and numerical simulation for the spill procedure of liquid fuel of fuel air explosive with shell

Addressing the problem of spill of liquid fuel of fuel air explosive (FAE) with shell, this paper deduces variation relationship of fracture radius and fracture velocity based on continuity and motion equations of shell using the Taylor fracture criteria of cylindrical shells, and analyzes the correctness of the approximating process in detail. Furthermore, analytical solutions and numerical simulations for the liquid fuel spill process were obtained using Adomian's decomposition method (ADM) and the Runge-Kutta numerical method. Results show that both the numerical solution and the analytical solution based on ADM are closer to the result from experiments and computations from commercial dynamics numerical software.     

Controlling torsional vibrations of drill strings via decomposition of traveling waves

Torsional vibrations in drill strings, especially stick-slip vibrations, are detrimental to the drilling process as they slow down the rate of penetration and may lead to failure of the drilling equipment. We present a method for controlling these vibrations by exactly decomposing the drill string dynamics into two traveling waves traveling in the direction of the top drive and in the direction of the drill bit. The decomposition is derived from the wave equation governing the string vibrations and is achieved with only two sensors that can be placed directly at the top drive and at a short distance below the top drive (e.g., 5 m). Therefore, downhole measurements along the string and at the bit are not necessary, which is a major advantage compared to other control concepts for drill string dynamics. The velocity of the top drive is then controlled in order to absorb the wave traveling in the direction of the top drive, thus achieving a reflection coefficient of zero for the frequency range of the undesired torsional vibrations. The proposed algorithm is implemented for both a numerical example and an experimental setup; results show that the control concept works very effectively.     

The method of dynamic mode decomposition in shallow water and a swirling flow problem

In this paper, a new vector‐filtering criterion for dynamic modes selection is proposed that is able to extract dynamically relevant flow features from dynamic mode decomposition of time‐resolved experimental or numerical data. We employ a novel modes selection criterion in parallel with the classic selection based on modes amplitudes, in order to analyze which of these procedures better highlight the coherent structures of the flow dynamics. Numerical tests are performed on two distinct problems. The efficiency of the proposed criterion is proved in retaining the most influential modes and reducing the size of the dynamic mode decomposition model. By applying the proposed filtering mode technique, the flow reconstruction error is shown to be significantly reduced. Copyright © 2016 John Wiley & Sons, Ltd.     

基于S-R和分解定理的三维几何非线性无网格法

S-R(strain-rotation)和分解定理克服了经典有限变形理论的一些缺点, 使其可以为几何非线性数值分析提供可靠的理论基础. 对于大变形问题, 由于无网格法(element-free method)避免了对单元网格的依赖, 从而从根本上避免了有限单元法(finite element method, FEM)的单元畸变问题, 保证了求解精度. 因此, 将无网格法和S-R和分解定理结合起来势必能建立一套更加合理可靠的几何非线性数值计算方法. 目前基于S-R 定理的无网格数值方法研究较少并且只能用于二维平面问题的求解, 但实际上绝大多数问题都必须以三维模型来进行处理, 因此建立适用于三维情况的S-R无网格法是非常有必要的. 本文给出了适用于三维情况的S-R 无网格法: 采用由更新拖带坐标法和势能率原理推导出来的增量变分方程, 利用基于全局弱式的无网格Galerkin 法(EFG)得到了用于求解三维空间问题的离散格式. 利用MATLAB编制三维S-R 无网格法程序, 对受均布载荷的三维悬臂梁和四边简支矩形板结构的非线性弯曲问题进行了计算. 最后将所得的数值结果与已有文献进行了比较, 验证了本文的三维S-R无网格数值算法的合理性、有效性和准确性. 本文的三维S-R无网格数值算法可以作为一种可靠的三维几何非线性数值分析方法.      

Parallel data-driven decomposition algorithm for large-scale datasets: with application to transitional boundary layers

Many fluid flows of engineering interest, though very complex in appearance, can be approximated by low-order models governed by a few modes, able to capture the dominant behavior (dynamics) of the system. This feature has fueled the development of various methodologies aimed at extracting dominant coherent structures from the flow. Some of the more general techniques are based on data-driven decompositions, most of which rely on performing a singular value decomposition (SVD) on a formulated snapshot (data) matrix. The amount of experimentally or numerically generated data expands as more detailed experimental measurements and increased computational resources become readily available. Consequently, the data matrix to be processed will consist of far more rows than columns, resulting in a so-called tall-and-skinny (TS) matrix. Ultimately, the SVD of such a TS data matrix can no longer be performed on a single processor, and parallel algorithms are necessary. The present study employs the parallel TSQR algorithm of (Demmel et al. in SIAM J Sci Comput 34(1):206–239, 2012), which is further used as a basis of the underlying parallel SVD. This algorithm is shown to scale well on machines with a large number of processors and, therefore, allows the decomposition of very large datasets. In addition, the simplicity of its implementation and the minimum required communication makes it suitable for integration in existing numerical solvers and data decomposition techniques. Examples that demonstrate the capabilities of highly parallel data decomposition algorithms include transitional processes in compressible boundary layers without and with induced flow separation.     

DYNAMICAL APPROACH STUDY OF SPURIOUS STEADY-STATE NUMERICAL SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS II. GLOBAL ASYMPTOTIC BEHAVIOR OF TIME DISCRETIZATIONS ∗

SUMMARY

The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 × 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how “numerical” basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DCs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless ofthe stability ofthe spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.     

An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit

In this article, we propose a numerical scheme to solve the Kac model of the Boltzmann equation for multiscale rarefied gas dynamics. Formally, this scheme is shown to be uniformly stable with respect to the Knudsen number, consistent with the fluid-diffusion limit for small Knudsen numbers, and with the Kac equation in the kinetic regime. Our approach is based on the micro–macro decomposition which leads to an equivalent formulation of the Kac model that couples a kinetic equation with macroscopic ones. This method is validated with various test cases and compared to other standard methods.     

基于李群局部标架的多柔体系统动力学建模与计算

多柔体系统动力学主要研究由多个具有运动学约束、存在大范围相对运动的柔性部件构成的动力学系统的建模、计算和控制.多柔体系统不仅具有柔体大变形导致的几何非线性,更具有大范围刚体运动引起的几何非线性,其非线性程度远高于计算结构力学所研究的几何非线性问题.本文基于李群局部标架(local frame of Lie group, LFLG),讨论如何发展一套新的多柔体系统动力学建模和计算方法体系, 具体内容包括:基于局部标架的梁、板壳单元,适用于长时间历程计算的多柔体系统碰撞动力学积分算法,结合区域分解技术的大规模多柔体系统动力学并行求解器, 以及若干验证性算例.上述基于李群局部标架的方法体系可在计算中消除刚体运动带来的几何非线性问题,使柔体系统的广义惯性力、广义弹性力及其雅可比矩阵满足刚体运动的不变性,使多柔体系统动力学与大变形结构力学相互统一,有望推动新一代多柔体系统动力学建模和计算软件的发展.      

Variational principles of elastic-viscous dynamics in laplace transformation form,F. E. M. formulation and numerical method

The author gives variational principles of elastic-viscous dynamics in spectral resolving form^[1], it will be extended to Laplace transformation form in this paper, mixed variational principle of shell dynamics and variational principle of dynamics of elastic-viscous-porous media are concerned, for the latter, F. E. M. formulation has been worked out.Variational principles in Laplace transformation form have concise forms, for the sake of utilizing F. E. M. conveniently it is necessary to find values of preliminary time function at some instants, when values of Laplace transformation at some points are known, but there are no efficient methods till now. In this paper, a numerical method for finding discrete values of preliminary function is presented, from numerical example we see such a method is efficient.By combining both methods stated above, variational principles in Laplace transformation form and numerical method, a quite wide district of solid dynamic problems can be solved by ths aid of digital computers.     

Multiscale analysis of gas dynamic fields

On the basis of numerical data processing, it is shown that a joint analysis of singularities detected in an initial gas dynamic field and at the first level of its wavelet decomposition allows one to remove the most part of numerical artifacts caused by numerical errors in computations. The application of asymmetric wavelets for this decomposition leads to a displacement of discontinuities. It is also shown that the values of the field can be identified with the coefficients of its wavelet decomposition.     

A non-intrusive approach for the reconstruction of POD modal coefficients through active subspaces

Reduced order modeling (ROM) provides an efficient framework to compute solutions of parametric problems. Basically, it exploits a set of precomputed high-fidelity solutions—computed for properly chosen parameters, using a full-order model—in order to find the low dimensional space that contains the solution manifold. Using this space, an approximation of the numerical solution for new parameters can be computed in real-time response scenario, thanks to the reduced dimensionality of the problem. In a ROM framework, the most expensive part from the computational viewpoint is the calculation of the numerical solutions using the full-order model. Of course, the number of collected solutions is strictly related to the accuracy of the reduced order model. In this work, we aim at increasing the precision of the model also for few input solutions by coupling the proper orthogonal decomposition with interpolation (PODI)—a data-driven reduced order method—with the active subspace (AS) property, an emerging tool for reduction in parameter space. The enhanced ROM results in a reduced number of input solutions to reach the desired accuracy. In this contribution, we present the numerical results obtained by applying this method to a structural problem and in a fluid dynamics one.     

Advances in dynamics and control of tethered satellite systems

The concept of tethered satellite system （TSS） promises to revolutionize many aspects of space exploration and exploitation. It provides not only numerous possible and valuable applications, but also challenging and interesting problems related to their dynamics, control, and physical implementation. Over the past decades, this exciting topic has attracted significant attention from many researchers and gained a vast number of analytical, numerical and experimental achievements with a focus on the two essential aspects of both dynamics and control. This review article presents the historic background and recent hot topics for the space tethers, and introduces the dynamics and control of TSSs in a progressive manner, from basic operating principles to the state-of-the-art achievements.     

A high‐fidelity low‐cost aerodynamic model using proper orthogonal decomposition

In this paper a high‐fidelity aerodynamic model is presented for use in parametric studies of weapon aerodynamics. The method employs a reduced‐order model obtained from the proper orthogonal decomposition (POD) of an ensemble of computational fluid dynamics (CFD) solutions with varying parameters. This decomposition produces an optimal linear set of orthogonal basis functions that best describe the ensemble of numerical solutions. These solutions are then projected onto this set of basis functions to provide a finite set of scalar coefficients that represent the solutions. A pseudo‐continuous representation of these projection coefficients is constructed, which allows predictions to be made of parameter combinations not in the original set of observations. The paper explores the performance of a few design‐of‐experiment approaches for the generation of the initial ensemble of computational experiments. Response surface construction methods based on parametric and non‐parametric models for the pseudo‐continuous representation of the projection coefficients are also evaluated. The model has been applied to two‐flow problems related to high‐speed weapon aerodynamics, inviscid flow around a flare‐stabilized hypersonic projectile and supersonic turbulent flow around a fin‐stabilized projectile with drooping nose control. Comparisons of model predictions with high‐fidelity CFD simulations suggest that the POD provides a reliable and robust approach to the construction of reduced‐order models. The practicality of the model is shown to be sensitive to the technique used to generate the ensemble of observations from which the model is constructed, while the accuracy of the approach depends on the pseudo‐continuous representation of the projection coefficients. Copyright © 2009 John Wiley & Sons, Ltd.     

DNS of flows with helical symmetry

Some flows such as the wakes of rotating devices often display helical symmetry. We present an original DNS code for the dynamics of such helically symmetric systems. We show that, by enforcing helical symmetry, the three-dimensional Navier–Stokes equations can be reduced to a two-dimensional unsteady problem. The numerical method is a generalisation of the vorticity/streamfunction formulation in a circular domain, with finite differences in the radial direction and spectral decomposition along the azimuth. When compared to a standard three-dimensional code, this allows us to reach larger Reynolds numbers and to compute quasi-steady patterns. We illustrate the importance of helical pitch by some physical cases: the dynamics of several helical vortices and a quasi-steady vortex flow. We also study the linear dynamics and nonlinear saturation in the Batchelor vortex basic flow, a paradigmatic example of trailing vortex instability. We retrieve the behaviour of inviscid modes and present new results concerning the saturation of viscous centre modes.     

A shock-capturing scheme for body-fitted meshes

A finite difference scheme based on flux difference splitting is presented for the solution of the two-dimensional Euler equations of gas dynamics in a generalized co-ordinate system. The scheme is based on numerical characteristic decomposition and solves locally linearized Riemann problems using upwind differencing. The decomposition is for a generalized co-ordinate system and a convex equation of state. This ensures good shock-capturing properties when incorporated with operator splitting and the advantage of using body-fitted co-ordinates. The resulting scheme is applied to supersonic flow of real air' past a circular cylinder.     

Impulsive control induced effects on dynamics of single and coupled ODE systems

The dynamics of differential system can be changed very obviously after inputting impulse signals. Previous studies show that the single chaotic system can be controlled to periodic motions using impulsive control method. It was well known that the dynamics of hyper-chaotic and coupled systems are very important and more complex than those of a single system. In this paper, particular impulsive control of the hyper-chaotic Lü system was proposed, which is with outer impulsive signals. It can be seen that such impulsive strategy can generate chaos from periodic orbit or control chaos to periodic orbit etc. For the first time, impulsive control induced effects on dynamics of coupled systems are considered in this paper, where the impulse effect has outer input signals. Many interesting and useful results are obtained. The coupled system can realize synchronization and its synchronization manifold can be changed with such impulsive control signals. Strict theories are given, and numerical simulations confirm the correctness of theoretical results.     

燃料大幅晃动等几何分析仿真及航天器耦合动力学研究

现代航天器肩负许多周期长且复杂的航天任务,通常需要携带大量的液体燃料.贮箱中液体燃料大幅晃动会严重影响航天器的姿态稳定性和控制精度,是现代航天器耦合动力学建模和精确控制研究的重要问题.本文提出了一种新的液体大幅晃动数值仿真方法,采用等几何分析方法对贮箱内气体和液体整体进行建模和空间离散,采用压力修正的分步法对控制方程进行时间离散,结合水平集方法划分气体和液体区域并且实时追踪液体晃动自由面.提出了一种质量修正方法以消除水平集函数演化产生的液体质量误差.基于燃料大幅晃动等几何分析仿真方法,对携带太阳能帆板的充液航天器进行动力学建模和耦合运动数值仿真.对于太阳能帆板的振动问题则采用Kirchhoff-Love板理论建模和模态分析法数值求解.通过将数值仿真结果与解析解对比,证明了本文给出方法的正确性.本文还对燃料大幅晃动下的航天器刚-液-柔耦合运动进行了数值仿真,发现液体晃动对航天器的姿态变化和结构振动的幅值和频率具有不可忽视的影响.     

A control volume finite element method for three-dimensional three-phase flows

A novel control volume finite element method with adaptive anisotropic unstructured meshes is presented for three-dimensional three-phase flows with interfacial tension. The numerical framework consists of a mixed control volume and finite element formulation with a new P₁DG-P₂ elements (linear discontinuous velocity between elements and quadratic continuous pressure between elements). A “volume of fluid” type method is used for the interface capturing, which is based on compressive control volume advection and second-order finite element methods. A force-balanced continuum surface force model is employed for the interfacial tension on unstructured meshes. The interfacial tension coefficient decomposition method is also used to deal with interfacial tension pairings between different phases. Numerical examples of benchmark tests and the dynamics of three-dimensional three-phase rising bubble, and droplet impact are presented. The results are compared with the analytical solutions and previously published experimental data, demonstrating the capability of the present method.