Implementation of high‐order compact finite‐difference method to parabolized Navier–Stokes schemes

The numerical solution to the parabolized Navier–Stokes (PNS) and globally iterated PNS (IPNS) equations for accurate computation of hypersonic axisymmetric flowfields is obtained by using the fourth‐order compact finite‐difference method. The PNS and IPNS equations in the general curvilinear coordinates are solved by using the implicit finite‐difference algorithm of Beam and Warming type with a high‐order compact accuracy. A shock‐fitting procedure is utilized in both compact PNS and IPNS schemes to obtain accurate solutions in the vicinity of the shock. The main advantage of the present formulation is that the basic flow variables and their first and second derivatives are simultaneously computed with the fourth‐order accuracy. The computations are carried out for a benchmark case: hypersonic axisymmetric flow over a blunt cone at Mach 8. A sensitivity study is performed for the basic flowfield, including profiles and their derivatives obtained from the fourth‐order compact PNS and IPNS solutions, and the effects of grid size and numerical dissipation term used are discussed. The present results for the flowfield variables and also their derivatives are compared with those of other basic flow models to demonstrate the accuracy and efficiency of the proposed method. The present work represents the first known application of a high‐order compact finite‐difference method to the PNS schemes, which are computationally more efficient than Navier–Stokes solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

Explicit formulations and performance of LSFD method on Cartesian mesh

Performance of the LSFD method is compared with conventional FD schemes. Generally, 9-point stencils for 2D cases and 27-point stencils for 3D cases are used for the approximation of the first and second order derivatives obtained with conventional central difference schemes. When the same stencils are used, explicit LSFD formulations for approximation of the first and second order derivatives are presented. The LSFD formulations are actually a combination of conventional central difference schemes along relevant mesh lines. It has been found that LSFD formulations need much less iteration steps than the conventional FD schemes to converge, and the ratio of mesh spacing in the x and y directions is an important parameter in the LSFD application, with a great impact on stability of LSFD computation.     

On accurate boundary conditions for a shape sensitivity equation method

This paper studies the application of the continuous sensitivity equation method (CSEM) for the Navier–Stokes equations in the particular case of shape parameters. Boundary conditions for shape parameters involve flow derivatives at the boundary. Thus, accurate flow gradients are critical to the success of the CSEM. A new approach is presented to extract accurate flow derivatives at the boundary. High order Taylor series expansions are used on layered patches in conjunction with a constrained least‐squares procedure to evaluate accurate first and second derivatives of the flow variables at the boundary, required for Dirichlet and Neumann sensitivity boundary conditions. The flow and sensitivity fields are solved using an adaptive finite‐element method. The proposed methodology is first verified on a problem with a closed form solution obtained by the Method of Manufactured Solutions. The ability of the proposed method to provide accurate sensitivity fields for realistic problems is then demonstrated. The flow and sensitivity fields for a NACA 0012 airfoil are used for fast evaluation of the nearby flow over an airfoil of different thickness (NACA 0015). Copyright © 2005 John Wiley & Sons, Ltd.     

An optimal method for seismic drift design of concrete buildings using gradient and Hessian matrix calculations

This paper describes a novel seismic optimal design method for the reinforced concrete frame. First, an optimal mathematical model with time-dependent constraints, i.e., inter-story drift constraints, is established for achieving minimum weight design. Second, the inequality constraint problem with time-dependent constraints is converted into a sequence of appropriately formed unconstrained problems using the integral interior point penalty function method. Third, an efficient algorithm of the first and second derivatives of the inter-story drift with respect to design variables is formulated based on Newmark-β method. Gradient and Hessian matrix of the integral interior penalty function are also computed. Fourth, Marquardt’s method is employed to solve a sequence of unconstrained problems. Finally, the minimum weight design of a three-story, two-bay planar frame is demonstrated using the new optimization method and the augmented Lagrange multiplier method. The comparative results show the seismic optimal design method presented in this paper is more efficient than the augmented Lagrange multiplier method in terms of computational time. The proposed new method is an effective and efficient approach for minimum weight design of the reinforced concrete frames subjected to earthquake excitation.     

Numerical solution of three‐dimensional velocity–vorticity Navier–Stokes equations by finite difference method

This paper describes the finite difference numerical procedure for solving velocity–vorticity form of the Navier–Stokes equations in three dimensions. The velocity Poisson equations are made parabolic using the false‐transient technique and are solved along with the vorticity transport equations. The parabolic velocity Poisson equations are advanced in time using the alternating direction implicit (ADI) procedure and are solved along with the continuity equation for velocities, thus ensuring a divergence‐free velocity field. The vorticity transport equations in conservative form are solved using the second‐order accurate Adams–Bashforth central difference scheme in order to assure divergence‐free vorticity field in three dimensions. The velocity and vorticity Cartesian components are discretized using a central difference scheme on a staggered grid for accuracy reasons. The application of the ADI procedure for the parabolic velocity Poisson equations along with the continuity equation results in diagonally dominant tri‐diagonal matrix equations. Thus the explicit method for the vorticity equations and the tri‐diagonal matrix algorithm for the Poisson equations combine to give a simplified numerical scheme for solving three‐dimensional problems, which otherwise requires enormous computational effort. For three‐dimensional‐driven cavity flow predictions, the present method is found to be efficient and accurate for the Reynolds number range 100?Re?2000. Copyright © 2004 John Wiley & Sons, Ltd.     

A method for the analysis of dynamic response of structure containing non-smooth contactable interfaces

A novel single-step method is proposed for the analysis of dynamic response of visco-elastic structures containing non-smooth contactable interfaces. In the method, a two-level algorithm is employed for dealing with a nonlinear boundary condition caused by the dynamic contact of interfaces. At the first level, and explicit method is adopted to calculate nodal displacements of global viscoelastic system without considering the effect of dynamic contact of interfaces and at the second level, by introducing contact conditions of interfaces, a group of equations of lower order is derived to calculate dynamic contact normal and shear forces on the interfaces. The method is convenient and efficient for the analysis of problems of dynamic contact. The accuracy of the method is of the second order and the numerical stability condition is wider than that of other explicit methods. The project supported by the National Natural Science Foundation of China (59578032) and the Key Project of the Ninth Five-Year Plan (96221030202)     

An efficient method for structural optimization

Based on the dual theory of nonlinear mathematical programming and the second order Taylor series expansions of functions, an efficient algorithm for structural optimum design has been developed. The main advantages of this method are the generality in use, the efficiency in computation and the capability in identifying automatically the set of active constraints. On the basis of the virtual work principle, formulas in terms of element stresses for the first and second order derivatives of nodal displacement and stress with respect to design variables are derived. By applying the Saint-Venant's principle, the computational efforts involved in the Hessian matrix associated with the iterative expression can be significantly reduced. This method is especially suitable for optimum design of large scale structures. Several typical examples have been optimized to test its uasefulness.     

一种基于面-面局部搜索的接触算法

接触-碰撞算法是影响动态接触问题数值分析精度的重要因素,发展健壮、精确的局部搜索算法对提高数值模拟精度具有重要意义。为解决点-面接触搜索算法存在的盲区问题,提出了一种基于面-面的局部搜索算法,并发展了相应的接触力计算方法。新的接触算法以面心坐标与特征长度表征接触片,进行预搜索快速排除不会发生接触的潜在接触对;利用主从接触片的投影与侵入深度关系确定接触片间的接触状态;利用等参逆变换解析给出接触点,避免了方程组的迭代求解。典型算例与工程实例计算表明,本文算法消除了接触搜索的盲区,具有很好的健壮性与计算精度。     

非结构重叠网格方法研究及应用

航空、航天和兵器技术等领域的研究中存在大量包含运动边界的流场。非结构重叠网格方法是一种高效的处理动边界问题的新方法。围绕相对运动的每个物体单独生成非结构网格,在网格重叠区域通过搜索和插值完成网格系之间的信息传递,提出了动态八叉树搜索算法,发展了绝对坐标系和相对坐标系相结合的流场求解方式,采用二阶精度Van Leer/Hanel格式和四阶Runge-Kutta法分别进行空间和时间离散,形成了一种新的非结构重叠网格算法。对三维Riemann问题的求解结果与精确解能很好吻合,证明了本文的重叠网格算法具有较好的时空离散精度和插值精度。对7.62mm步枪射击过程进行了数值模拟,描述了弹丸离开膛口后膛口流场的发展过程,与实验结果体现的发展过程较为吻合,验证了本文提出的非结构网格算法体系具有较好的计算性能,是研究含动边界复杂流场的一种有效手段。     

广义有限差分法在含阻抗边界空腔声学分析中的应用

声学分析在噪声控制、室内隔音等工程计算中有着重要的作用. 由于现实生活中的声学模型往往伴随着吸声材料, 因此分析含阻抗边界条件的声学问题显得十分必要. 广义有限差分法是一种新型区域型无网格数值离散方法, 该方法基于多元函数泰勒级数展开式和加权最小二乘拟合, 将控制方程中未知参量的各阶偏导数表示为相邻节点函数值的线性组合. 本文首次将广义有限差分法应用于含阻抗边界条件空腔声学问题的分析中, 建立了空腔声场问题的广义有限差分法数值离散格式. 与传统算法相比, 所建立的数值模型具有无需网格剖分和数值积分、计算精度高、适用于大规模声学分析等优点. 通过具有解析解的经典算例, 研究了总节点数目和局部支撑点数目对数值结果的影响, 得到了最大计算频率与节点间距之间关系的经验公式. 此外, 将广义有限差分法应用于无解析解的二维和三维复杂声学模型, 并与COMSOL Multiphysics软件所得的有限元结果进行了比较分析. 数值实验表明, 该算法是一种高效、精确、稳定、收敛的数值模拟方法, 在含阻抗边界空腔声学分析中具有广阔的应用前景.      

波动方程的精细逐步积分法

应用现有的波动方程求解方法解决工程实际问题尚存在一定的局限性。本文在结构动力方程精细逐步积分的基础上,提出了波动方程初边值问题的精细逐步积分法,并分别给出了不同边界条件下的精细逐步积分格式。此数值方法虽然是显式积分方法,却是无条件稳定的。分别用精细逐步积分法和其它已有的方法对两个算例进行了计算,一个是有解析解的例子,该例验证了此方法的准确性,另一个例子是求解由波动方程及初始条件和边界条件组成的有杆抽油系统预测模型,此例验证了精细逐步积分法的高效性。     

Fast, Robust and Accurate Digital Image Correlation Calculation Without Redundant Computations

High-efficiency and high-accuracy deformation analysis using digital image correlation (DIC) has become increasingly important in recent years, considering the ongoing trend of using higher resolution digital cameras and common requirement of processing a large sequence of images recorded in a dynamic testing. In this work, to eliminate the redundant computations involved in conventional DIC method using forward additive matching strategy and classic Newton–Raphson (FA-NR) algorithm without sacrificing its sub-pixel registration accuracy, we proposed an equivalent but more efficient DIC method by combining inverse compositional matching strategy and Gauss-Newton (IC-GN) algorithm for fast, robust and accurate full-field displacement measurement. To this purpose, first, an efficient IC-GN algorithm, without the need of re-evaluating and inverting Hessian matrix in each iteration, is introduced to optimize the robust zero-mean normalized sum of squared difference (ZNSSD) criterion to determine the desired deformation parameters of each interrogated subset. Then, an improved reliability-guided displacement tracking strategy is employed to achieve further speed advantage by automatically providing accurate and complete initial guess of deformation for the IC-GN algorithm implemented on each calculation point. Finally, an easy-to-implement interpolation coefficient look-up table approach is employed to avoid the repeated calculation of bicubic interpolation at sub-pixel locations. With the above improvements, redundant calculations involved in various procedures (i.e. initial guess of deformation, sub-pixel displacement registration and sub-pixel intensity interpolation) of conventional DIC method are entirely eliminated. The registration accuracy and computational efficiency of the proposed DIC method are carefully tested using numerical experiments and real experimental images. Experimental results verify that the proposed DIC method using IC-GN algorithm and the existing DIC method using classic FA-NR algorithm generate similar results, but the former is about three to five times faster. The proposed reliability-guided IC-GN algorithm is expected to be a new standard full-field displacement tracking algorithm in DIC.     

A higher-order eulerian scheme for coupled advection-diffusion transport

A new accurate high-order numerical method is presented for the coupled transport of a passive scalar (concentration) by advection and diffusion. Following the method of characteristics, the pure advection problem is first investigated. Interpolation of the concentration and its first derivative at the foot of the characteristic is carried out with a fifth-degree polynomial. The latter is constructed by using as information the concentration and its first and second derivatives at computational points on current time level t in Eulerian co-ordinates. The first derivative involved in the polynomial is transported by advection along the characteristic towards time level t + Δt in the same way as is the concentration itself. Second derivatives are obtained at the new time level t + Δt by solving a system of linear equations defined only by the concentrations and their derivatives at grid nodes, with the assumption that the third-order derivatives are continuous. The approximation of the method is of sixth order. The results are extended to coupled transport by advection and diffusion. Diffusion of the concentration takes place in parallel with advection along the characteristic. The applicability and precision of the method are demonstrated for the case of a Gaussian initial distribution of concentrations as well as for the case of a steep advancing concentration front. The results of the simulations are compared with analytical solutions and some existing methods.     

Second-order accurate particle image velocimetry

 An adaptive, second-order accurate particle image velocimetry (PIV) technique is presented. The technique uses two singly exposed images that are interrogated using a modified cross-correlation algorithm. Consequently, any of the equipment commonly available for conventional PIV (such as dual head Nd: YAG lasers, interline transfer CCD cameras, etc.) can be used with this more accurate algorithm. At the heart of the algorithm is a central difference approximation to the flow velocity (accurate to order Δt ²) versus the forward difference approximation (accurate to order Δt) common in PIV. An adaptive interrogation region-shifting algorithm is used to implement the central difference approximation. Adaptive shifting algorithms have been gaining popularity in recent years because they allow the spatial resolution of the PIV technique to be maximized. Adaptive shifting algorithms also have the virtue of helping to eliminate velocity bias errors. The second- order accuracy resulting from the central difference approximation can be obtained with relatively little additional computational effort compared to that required for a standard first-order accurate forward difference approximation. The adaptive central difference interrogation (CDI) algorithm has two main advantages over adaptive forward difference interrogation (FDI) algorithms: it is more accurate, especially at large time delays between camera exposures; and it provides a temporally symmetric view of the flow. By comparing measurements of flow around a single red blood cell made using both algorithms, the CDI technique is shown to perform better than conventional FDI-PIV interrogation algorithms near flow boundaries. Cylindrical Taylor–Couette flow images, both experimental and simulated, are used to demonstrate that the CDI algorithm is significantly more accurate than conventional PIV algorithms, especially as the time delay between exposures is increased. The results of the interrogations are shown to agree quite well with analytical predictions and confirm that the CDI algorithm is indeed second-order accurate while the conventional FDI algorithm is only first-order accurate. Received: 15 June 2000/Accepted: 2 February 2001     

非线性随机结构动力可靠度的密度演化方法

建议了一类新的非线性随机结构动力可靠度分析方法。基于非线性随机结构反应分析的概率密度演化方法,根据首次超越破坏准则对概率密度演化方程施加相应的边界条件,求解带有初、边值条件的概率密度演化方程,可以给出非线性随机结构的动力可靠度。研究了数值计算技术,建议了具有自适应功能的TVD差分格式。以具有双线型恢复力性质的8层框架结构为例进行了地震作用下的动力可靠度分析,与随机模拟结果的比较表明,所建议的方法具有较高的精度和效率。     

Adaptive methods of lines for one-dimensional reaction-diffusion equations

Adaptive and non-adaptive finite difference methods are used to study one-dimensional reaction-diffusion equations whose solutions are characterized by the presence of steep, fast-moving flame fronts. Three non-adaptive techniques based on the methods of lines are described. The first technique uses a finite volume method and yields a system of non-linear, first-order, ordinary differential equations in time. The second technique uses time linearization, discretizes the time derivatives and yields a linear, second-order, ordinary differential equation in space, which is solved by means of three-point, fourth-order accurate, compact differences. The third technique takes advantage of the disparity in the time scales of the reaction and diffusion processes, splits the reaction--diffusion operator into a sequence of reaction and diffusion operators and solves the diffusion operator by means of either a finite volume method or a three-point, fourth-order accurate compact difference expression. The non-adaptive methods of lines presented in this paper may use equaliy or non-equally spaced fixed grids and require a large number of grid points to solve accurately one-dimensional problems characterized by the presence of steep, fast-moving fronts. Three adaptive methods for the solution of reaction-diffusion equations are considered. The first adaptive technique is static and uses a subequidistribution principle to determine the grid points, avoid mesh tangling and node overtaking and obtain smooth grids. The second adaptive technique is dynamic, uses an equidistribution principle with spatial and temporal smoothing and yields a system of first-order, non-linear, ordinary differential equations for the grid point motion. The third adaptive technique is hybrid, combines some features of static and dynamic methods, and uses a predictor-corrector strategy to predict the grid and solve for the dependent variables, respectively. The three adaptive techniques presented in this paper use physical co-ordinates and may employ finite volume or three-point, compact methods. The adaptive and non-adaptive finite difference methods presented in the paper are used to study a decomposition chemical reaction characterized by a scalar, one-dimensional reaction-diffusion equation, the propagation of a one-dimensional, confined, laminar flame in Cartesian co-ordinates and the Dwyer-Sanders model of one-dimensional flame propagation. It is shown that the adaptive moving method presented in this paper requires a smaller number of grid points than adaptive static, adaptive hybrid and non-adaptive methods. The adaptive hybrid method requires a smaller time step than adaptive static techniques, due to the lag between the grid prediction and the solution of the dependent variables. Non-adaptive methods of lines may yield temperature oscillations in front of and behind the flame front if Crank-Nicolson techniques are used to evaluate the time derivatives. Fourth-order accurate methods of lines in space yield larger temperature oscillations than second-order accurate methods of lines, and the magnitude of these oscillations decreases as the time step is decreased. It is also shown that three-point, fourth-order accurate discretizations of the spatial derivatives require the same number of grid points as second-order accurate, finite volume methods, in order to resolve accurately the structure of steep, fast-moving flame fronts.     

Immersed boundary conditions method for heat conduction problems in slots with time‐dependent geometry

A grid‐less, fully implicit, spectrally accurate algorithm for solving three‐dimensional, both stationary and time‐dependent, heat conduction problems in slots formed by either fixed or time‐dependent boundaries has been developed. The algorithm is based on the concept of immersed boundary conditions (IBC), where the physical domain is immersed within the computational domain and the boundary conditions take the form of internal constraints. The IBC method avoids the need to construct adaptive, time‐dependent grids resulting in the reduction of the required computational resources and, at the same time, maintaining accurate information about the location of the boundaries. The algorithm is spectrally accurate in space and capable of delivering first‐, second‐, third‐ and fourth‐order accuracy in time. Given a potentially large size of the resultant linear algebraic system, various methods that take advantage of the special structure of the coefficient matrix have been explored in search for an efficient solver, including a specialized direct solver as well as serial and parallel iterative solvers. The specialized direct solver has been found to be the most efficient from the viewpoints of the speed of the computations and the memory requirements. Copyright © 2010 John Wiley & Sons, Ltd.     

An efficient method for increasing the accuracy of mobility maps for ground vehicles

This paper presents an efficient method for increasing the accuracy of one key step regarding the process of determining a mobility map. That is, the interpolation of the original Digital Elevation Model (DEM) to a finer resolution before running multi-body-dynamics simulations. Specifically, this paper explores the use of fractal dimension and elevation range metrics for increasing the accuracy and reducing the computation time associated with the spatial interpolation ordinary kriging method. The first goal is to ensure the stationary variogram requirement. The second goal is to reduce kriging error or variance in the new predicted values. A novel segmentation-based approach has been proposed to divide the regions of interest into segments where stationarity is ensured. Empirical investigation based on real DEMs indicates the generality of the segmentation approach when natural and man-made terrains are considered. The proposed method leads to a more efficient computation burden and to more accurate results than the traditional approach.