Flutter prediction by an Euler method on non-moving Cartesian grids with gridless boundary conditions

A method is presented for the prediction of transonic flutter by the Euler equations on a stationary Cartesian mesh. Local grid refinement is established through a series of embedded meshes, and a gridless method is implemented for the treatment of surface boundary conditions. For steady flows, the gridless method applies surface boundary conditions using a weighted average of the flow properties within a cloud of nodes in the vicinity of the surface. The weighting is established with shape functions derived using a least-squares fitting of the surrounding nodal cloud. For unsteady calculations, a perturbation of the shape functions is incorporated to account for a fluctuating surface normal direction. The nature of the method provides for efficient and accurate solution of transient flow problems in which surface deflections are small (i.e. flutter calculations) without the need for a deforming mesh. Although small deviations in angle of attack are considered, the mean angle of attack can be large. Results indicate good agreement with available experimental data for unsteady flow, and with computational results addressing flutter of the Isogai wing model obtained using traditional moving mesh algorithms.     

Ghost-cell method for analysis of inviscid three-dimensional flows on Cartesian-grids

The present paper deals with the implementation of non-penetration boundary conditions at solid walls for three-dimensional inviscid flow computations on Cartesian grids. The crux of the method is the curvature-corrected symmetry technique (CCST) developed by the present authors for body-fitted grids. The method introduces ghost cells near the boundaries whose values are developed from an assumed flow-field model in vicinity of the wall consisting of a vortex flow, with locally symmetric distribution of entropy and total enthalpy. In three dimensions this procedure is implemented in the so-called “osculating plane”. This method was shown to be substantially more accurate than traditional surface boundary condition approaches. This improved boundary condition is adapted to a Cartesian mesh formulation, which we have termed the “ghost-cell method”. In this approach, all cell centers exterior to the body are computed with fluxes at the six surrounding cell faces, without any cut cell. A multiple-valued point technique is used to compute sharp edges. The merits of the ghost-cell method for three-dimensional inviscid flow computations are established by computing compressible and transonic flows about a sphere, an oblate and a prolate spheroid, a cylindrical wing with an end-plate, the ONERA M6 wing and detailed comparison to body-fitted grid computations and to published data. The computed results show the surface non-penetration condition to be satisfied in the limit of vanishing cell size and the method to be second-order accurate in space. The comparison with body-fitted results proves that the accuracy is comparable to the accuracy of CCST computations on body-fitted grids and remarkably superior to body-fitted computations based on traditional pressure extrapolation, non-penetration boundary conditions. In addition, we prove that the results are independent of the position of the body with respect to the grid. Finally, we show that the ONERA M6 wing results compare very well with published data.     

大展弦比机翼μ控制

利用大展弦比机翼后缘不同位置上的操纵面进行颤振主动控制,通过将大展弦比机翼简化为包含弯曲和扭转两种模态的悬臂梁结构,根据片条理论,建立包含操纵面运动规律的大展弦比机翼气动弹性方程.由于简化的数值模型与实际模型之间存在一定的误差,通常模型的运动方程包含有不确定参量用来表示建模误差.鲁棒控制方法能够得到一个有效控制器,控制这种带有模型不确定参量的运动方程.文中论述利用鲁棒μ控制方法,研究有两个操纵面大展弦比机翼的鲁棒控制问题.仿真结果表明鲁棒μ控制可以有效地抑制大展弦比机翼的受扰振动,提高颤振临界速度,且两个操纵面共同控制效果比单操纵面显著.     

Flux-based level-set method for two-phase flows on unstructured grids

In this paper we deal with the application of the flux-based level set method to moving interface computations on unstructured grids. The focus lies on the overcoming of the known difficulties of level set methods, e.g. accurate computations of important geometric properties, reliable and precise reinitialization of the level set function and the adaption of standard discretization methods to the moving boundary case. The basic building block of our approach is the high-resolution flux-based level set method for general advection equation (Frolkovi? and Mikula in SIAM J Sci Comput 29(2):579–597, 2007, Frolkovi? and Wehner in Comput Vis Sci 12(6):626–650, 2009). We extend this method for the problem of reinitialization of the level set function on unstructured grids by using quadratic interpolation to compute distances for nodes close to the interface. To realize numerical simulation for some applications with moving boundaries, we adapt the approach of ghost fluid method (Gibou and Fedkiw in J Comput Phys 202:577–601, 2005) for unstructured grids. The idea is to describe the development of the moving boundary with a level set formulation while the computational grid remains fixed and the boundary conditions are enforced using some extrapolation. Our main motivation is the numerical solution of two-phase incompressible flow problems. Additionally to previously mentioned steps, we introduce further numerical schemes in the framework of finite volume discretization for the flow. Possible jumps of the pressure and the directional derivative of velocity at the interface are modeled directly within the method using the approach of extended approximation spaces. Besides that, an algorithm for the computations of curvature is considered that exhibits the second order accuracy for some examples. Numerical experiments are provided for the presented methods.     

Nonlinear prediction of subsonic aerodynamic loads on wings and bodies at high angles of attack

This paper predicts numerically the nonlinear aerodynamic loads on wings and bodies at high angles of attack in subsonic flow separating along certain known lines. It is assumed that the separation vortices are symmetric and unbursting. The wing mean surfaces and body surfaces are simulated by bound vortex lattices, and the separation vortex surfaces by free vortex lines. The vortex system satisfies boundary conditions on wing and body surfaces, separation conditions, and the condition that free vortex lines are tangent to local velocity. The velocity induced by a vortex is computed by the Biot-Savart law with Göthert transformation for subsonic small perturbation flows. The vortex strengths and the free vortex locations are solved by relaxation method. Then the aerodynamic loads on wings and bodies are computed. They agree well with experimental tests.     

Bezier曲面的凸性分析及保凸拼接

通过分析Bezier曲面局部凸的充要条件与控制顶点网格形状的关系,在几何上将网格的形状与曲面的凸性之间建立联系,导出Bezier曲面局部凸的几个充要条件。将凸性分析的结论应用到Bezier曲面拼接中,得到Bezier曲面保凸拼接的条件,并且证明带有公共边界线的两个局部凸Bezier曲面的Cn(n≥1)拼接后,仍然是凸的且凸性相同。最后,给出几个三次Bezier曲面保凸拼接实例。     

On structural optimization with aeroelasticity constraints

The optimal design of a cantilever wing in incompressible flow is considered. The wing is modelled as a full depth sandwich wing using finite element analysis. A doublet lattice panel method is used for computation of the unsteady aerodynamic loads. The weight of the wing is minimized using the thicknesses of the composite face sheets as design variables subject to constraints on flutter and divergence speed. Imperfection sensitivity of the final design is analysed and general aspects of imperfection sensitivity in optimization subject to aeroelasticity constraints are discussed in some detail.     

Spatial direct numerical simulation of transitional boundary layer over compliant surfaces

An iterative implicit fractional step method is developed and employed for the simulation of transitional boundary layer over compliant surfaces. The three-dimensional perturbation Navier-Stokes equations are discretized by curvilinear finite volumes on a collocated grid system. A multigrid procedure is used for computations associated with the pressure-Poisson equation, while simulation is carried out by MPI-based parallel computation with domain decomposition. Results in the literature for oblique linear waves and the non-linear breakdown of a wave triad over a rigid wall are repeated to check the accuracy of the codes that had been developed. Oblique linear TS (Tollmien-Schlichting) waves over finite-length compliant membranes are generally found to coexist with CIFI (compliance induced flow instability) or FISI (flow induced surface instabilities) waves. The latter waves usually possess longer wavelengths and thus propagate at a larger oblique wave angles than the TS waves. Simulation reveals that compliant surfaces may slow down the development of secondary instabilities during the early stages of laminar-turbulent transition. However, during the later stages of fundamental wave breakdown, interactions with CIFI and edge-generated waves may increase the amplitudes of the original 2D and 3D TS waves, leading to an earlier breakdown on compliant surfaces. Linear interaction between the flow and compliant membranes has been assumed.     

Numerical simulation of 3-D wing flutter with fully coupled fluid-structural interaction

A numerical methodology coupling Navier-Stokes equations and structural modal equations for predicting 3-D transonic wing flutter is developed in this paper. A dual-time step implicit unfactored Gauss-Seidel iteration with the Roe scheme is employed for the flow solver. A modal approach is used for the structural response. The flow and structural solvers are fully coupled via successive iterations within each physical time step. The mesh-deformation strategy is described. The accuracy of the modal approach is validated with ANSYS. The results indicate that the first five modes are sufficient to accurately model the wing-structure response for the studied case of this paper. The computed flutter boundary of AGARD wing 445.6 at free stream Mach numbers ranging from 0.499 to 1.141 agrees well with the experiment.     

Finite difference procedure for solution of poisson equation over complex domains with Neumann boundary conditions

A generalized finite difference scheme for solving Poisson equation over multiply connected domain bounded by irregular boundaries at which Neumann boundary conditions are specified, is presented in this paper. The method used to treat the Neumann condition is a six-point gradient approximation method given by Greenspan[6]. The method is generalized to treat all types of grid intersections with the boundary. An efficient computational procedure is devised by eliminating the calculations at the boundary during the interations.The scheme is applied to the problem of forced convection heat transfer in a fully developed laminar flow through seven and nineteen rod-cluster assemblies. Fluid properties are assumed to be uniform. In arriving at the fast converging and efficient method from computational point of view, different iterative techniques, overrelaxation methods and boundary treatments were tried. The results of computations and the computer times are reported in the present paper.     

轴向运动功能梯度粘弹性梁横向振动的稳定性分析

基于Euler梁理论研究了轴向运动功能梯度粘弹性梁横向振动的稳定性问题.基于问题的数学模型和控制方程,利用微分求积法求得了轴向匀速运动功能梯度粘弹性梁亚临界区域内横向振动的复频率,分析其随着轴向运动速度、材料梯度指数等参数的变化情况,探讨上述参数对超临界区域失稳形式的影响.然后应用多尺度法结合边界条件分析了轴向速度带有周期扰动成分的变速运动功能梯度粘弹性梁的失稳问题,重点讨论了当速度扰动频率为固有频率二倍或者为两固有频率之和/差时所发生的次谐波共振及组合共振所导致的失稳.数值算例表明,随着梯度指数的增大,匀速运动功能梯度粘弹性梁的临界发散速度、耦合速度以及变速运动功能梯度粘弹性梁的稳定区域减小,且粘弹性系数的影响逐渐变弱,同等条件下,轴向运动功能梯度粘弹性固支梁比简支梁更为稳定.     

Surface orientation from a projected grid

Two simple methods are given for obtaining the surface shape using a projected grid. After the camera is calibrated to the 3-D workspace, the only input date needed for the computation of surface normals are grid intersect points in a single 2-D image. The first method performs nonlinear computations based on the distortion of the lengths of the grid edges and does not require a full calibration matrix. The second method requires that a full parallel projection model of the imaging is available, which enables it to compute 3-D normals using simple linear computations. The linear method performed better overall in the experiments, but both methods produced normals within 4-8° of known 3-D directions. These methods appear to be superior to methods based on shape-from-shading because the results are comparable, yet the equipment setup is simpler and the processing is not very sensitive to object reflectance     

Procedural CAD Model Edge Tolerance Negotiation for Surface Meshing

Geometrical models output from CAD software often require modification before they may be used for analysis-quality mesh generation. This is due primarily to the inconsistencies in tolerances used by the CAD operator and the tolerances required for analysis. This paper presents a method for construction of watertight surface meshes directly on imperfect non-modified CAD models. The method is based on a hierarchical grid topology structure that defines a surface mesh by a grid and a collection of curves defining the boundary. Curve boundaries on component surfaces are iteratively split and merged according to user-set tolerances, allowing adjacent surface meshes to become computationally watertight via their shared edge curves. The collection of watertight surface meshes may then be made model-inde-pendent through interactive agglomeration of the surface meshes, followed by refinement and decimation sweeps to remove artifacts of original surface edges. Interactive procedures used for difficult cases are also explained, as are ongoing efforts for further automation.     

Developments in Cartesian cut cell methods

This paper describes the Cartesian cut cell method, which provides a flexible and efficient alternative to traditional boundary fitted grid methods. The Cartesian cut cell approach uses a background Cartesian grid for the majority of the flow domain with special treatments being applied to cells which are cut by solid bodies, thus retaining a boundary conforming grid. The development of the method is described with applications to problems involving both moving bodies and moving material interfaces.     

Generalized Stochastic Sampling Method for Visualization and Investigation of Implicit Surfaces

Recently we proposed the stochastic sampling method (SSM), which can numerically generate sample points on complicated implicit surfaces quickly and uniformly. In this paper we generalize the method in two aspects: (1) We introduce two kinds of boundary conditions, so that we can sample a finite part of an open surface spreading infinitely. (2) We generalize the stochastic differential equation used in the SSM, so that its solutions can satisfy plural constraint conditions simultaneously. The first generalization enables us to visualize cut views of open surfaces. The second generalization enables us to visualize intersections of static and moving implicit surfaces, which leads to detailed investigation of intersections and other interesting applications such as visualization of contour maps.     

Flight force production by flapping insect wings in inclined stroke plane kinematics

The unsteady flow structure and the time-varying aerodynamic forces acting on a 2D dragonfly model wing are studied by numerically solving the Navier-Stokes equations. The incompressible Navier-Stokes equations are discretized and solved on a non-body confirming Cartesian grid; the concept of immersed boundary method is made use of to impose the no-slip boundary condition on the surface of the wing. The objective of the present study is to investigate the influence of the following kinematic parameters on the flight performance of inclined stroke plane hovering: Reynolds number (Re), stroke amplitude, wing rotational timing and rotational duration. While the effects of the above mentioned parameters on the stroke averaged force coefficients are the same in both horizontal and inclined stroke plane motions, the spatiotemporal dynamics of vorticity which produce the effects are entirely different. Our results also indicate that the drag mechanism proposed for tiny insects does not seem to augment the vertical force generation in inclined stroke plane motion.     

三次B样条插值的网格拼接和融合

目的 网格模型的拼接和融合是3维模型编辑的一个重要方面。为了提高3维模型之间拼接曲面的精度和效率,提出一种基于三次均匀B样条曲线曲面的网格融合方法。方法 首先,利用协变分析和数据驱动方法在目标模型上选定融合区域、确定要融合模型的大小及方向;其次,根据选定的3维网格模型,确定待拼接区域的边界,识别并记录边界点集,利用三次B样条插值边界点集;然后,对边界曲线进行双三次B样条曲面插值得到拼接区域连续曲面,并以此作为两模型拼接时的过渡面;最后,对拼接区域重采样,并对其三角化,以实现网格模型的无缝光滑拼接和融合。结果 为了验证本文方法对3维模型拼接的有效性,选取4组不同的模型,分别对其使用本文提出的融合拼接方法进行实验,对前两组模型的拼接效果进行了对比试验,实验结果表明,本文方法可以达到很好的拼接效果,对于融合区域以外的部分能够保持源模型的细节特征,拼接部分的过渡区域光顺平滑,拼接后的模型完整性佳。在运行时间相差0.05 s内,与数据驱动的建模方法相比,本文方法可以处理的节点数至少多2 000个,面片数至少多5 000个。结论 本文方法能够适用于具有任何边界的模型,在选取模型时,对于模型的形状、大小、拓扑结构等的要求较低,适用于新模型的快速建造,因此,该算法可应用于医学、商业广告、动画娱乐以及几何建模和制造等较为广阔的应用领域。     

Reliability based optimization in aeroelastic stability problems using polynomial chaos based metamodels

In this work, reliability based design optimization (RBDO) of two aeroelastic stability problems is addressed: (i) divergence, which arises in static aeroelasticity, and (ii) flutter, which arises in dynamic aeroelasticity. A set of design variables is considered as random variables, and the mean mass is minimized for a given set of constraints — including the probability of failure by divergence or flutter. The optimization process requires repeated evaluation of reliability, which is a major contributor to the total computational cost. To reduce this cost, a polynomial chaos expansion (PCE)-based metamodel is created over a grid in the parameter space. These precomputed PCEs are then interpolated for reliability calculation at intermediate points in the parameter space, as demanded by the optimization algorithm. Two new modifications are made to this method in this work. First, the Gauss quadrature rule is used — instead of statistical simulation — to estimate the chaos coefficients for higher computational speed. Second, to increase this computational gain further, a non-uniform grid is chosen instead of a uniform one, based on relative importance of the design parameters. This relative importance is found from a global sensitivity analysis. This new modified method is applied on a rectangular unswept cantilever wing model. For both optimization problems, it is observed that the proposed method yields accurate results with a considerable computational cost reduction, when compared to simulation based methods. The effect of grid spacing is also explored to achieve the best computational efficiency.     

基于CFD系统辨识的气弹分析及GPU并行算法初探

通过求解Euler方程获得运动翼段的非定常流场,并用CUDA语言对流场求解器进行GPU并行计算.使用ARMA(auto-regressive-moving-average)模型对非定常气动力进行辨识,由系统辨识模型得到的结果与全阶CFD计算结果十分吻合.基于降阶气动模型与结构的耦合,计算了具有S型颤振边界的气动弹性标准算例-Isogai Wing的跨音速颤振.本文给出的方法可以在保证气动弹性计算精度的前提下大幅提高计算效率.