Numerical Methods for Accurate Computation of Design Sensitivities for Two-Dimensional Navier-Stokes Problems

Gradient computations can be a limiting factor in algorithm efficiency and accuracy for optimization based design. In this paper, we present three parameterized flow problems and consider the evaluation of state sensitivities both theoretically and numerically. Existence and uniqueness results are given for the sensitivities of a specific group of two-dimensional Navier-Stokes problems. We then turn our attention to obtaining numerical approximations to state sensitivities. We show convergence of our numerical sensitivities using a problem having an exact solution. Next, two problems, flow around a cylinder and flow over a bump, are used to evaluate several computational schemes. In particular, a local projection scheme for improved state derivative approximations and the use of an adaptive finite element scheme are shown to be important techniques for obtaining accurate sensitivity approximations. Lastly, we evaluate the impact of these computational techniques on cost function and gradient calculation.     

非定常N-S方程的双重网格数值模拟

本文采用全离散双重网格算法（时间变量采用Eular全隐式格式离散,空间变量采用混合有限元离散）,对非定常Navier-Stokes（N-S）方程进行数值模拟.双重网格算法的基本思想是,首先在粗网格有限元空间X^H上求解一个非线性问题,然后在细网格有限元空间Xh（h＜＜H）上求解一个线性问题.数值实验结果表明：在保持几乎相同精度的前提下,双重网格算法比标准有限元算法节省近一半的计算时间,说明了新算法求解非定常N-S方程的可行性和高效性.     

Spectral Direction Splitting Schemes for the Incompressible Navier-Stokes Equations

We propose and analyze spectral direction splitting schemes for the incompressible Navier-Stokes equations. The schemes combine a Legendre-spectral method for the spatial discretization and a pressure-stabilization/direction splitting scheme for the temporal discretization, leading to a sequence of one-dimensional elliptic equations at each time step while preserving the same order of accuracy as the usual pressure-stabilization schemes. We prove that these schemes are unconditionally stable, and present numerical results which demonstrate the stability, accuracy, and efficiency of the proposed methods.     

Boundary element analysis of Navier-Stokes equations

As arrangements, the fundamental solutions of anisotropic convective diffusion equations of transient incompressible viscous fluid flow and boundary elements analysis of the diffusion equation are presented. Secondly, by considering that convective diffusion equations and Navier-Stokes equations are mathematical formulations of mass and momentum conservation law respectively, and that consequently, both physical contents and equation styles are analogous, boundary integral formulations for Navier-Stokes equations are proposed on the basis of formulation of diffusion equations.     

Some numerical schemes using curvilinear coordinate grids for incompressible and compressible Navier-Stokes equations

In this review paper some numerical schemes recently developed by the authors and their coworkers for analysing the cascade flows of turbomachinery are described. These schemes use the curvilinear coordinate grid and solve the momentum equations of contravariant velocities (volume flux). The compressible flow schemes are based on the delta-form approximate-factorization finite-difference scheme, and are improved by using the diagonalization, the flux difference splitting and thetvd schemes to save computational effort and to increase stability and resolvability. Furthermore, using higher-order compacttvd muscl schemes, we can capture not only shock waves but also contact surfaces very sharply. On the other hand, the incompressible flow schemes are based on the well-knownSMAC scheme, and are extended to the curvilinear coordinate grid and further to the implicit scheme to reduce computations. These schemes, like thesmac scheme, satisfy the continuity condition identically, and suppress the occurrence of spurious errors. In both the compressible and incompressible schemes, for the turbulent flow thek-ɛ turbulence model with the law of the wall or considering the low Reynolds number effects is employed, and for the unsteady flow the Crank-Nicholson method is employed and the solution at each time step is obtained by the Newton iteration. Use of the volume flux instead of the physical velocity is inevitable for theMAC type schemes, and makes it easy to impose boundary conditions. Finally, some calculated results using the present schemes are shown.     

Navier-Stokes方程的集中质量非协调有限元法

本文通过所谓的速度-压力型公式讨论了Navier-Stokes方程的集中质量非协调有限元法(半离散情形)。首先给出了所讨论方程的集中质量非协调有限元逼近格式,其次对所讨论方程的真解与逼近格式的解之间的误差进行了分析,最后利用Navier-Stokes投影算子及其性质,得到了在确定模意义下的速度、压力误差估计,且某些误差估计能达到最优。     

Two-Grid Finite Element Methods for the Steady Navier-Stokes/Darcy Model

Two-grid finite element methods for the steady Navier-Stokes/Darcy model are considered. Stability and optimal error estimates in the $H^1$-norm for velocity and piezometric approximations and the $L^2$-norm for pressure are established under mesh sizes satisfying $h=H^2$. A modified decoupled and linearised two-grid algorithm is developed, together with some associated optimal error estimates. Our method and results extend and improve an earlier investigation, and some numerical computations illustrate the efficiency and effectiveness of the new algorithm.     

求解非定常N-S方程的稳定化分数步长法研究

本文研究了求解非定常Navier-Stokes方程的稳定化分数步长法.首先,通过一阶精度的算子分裂,将非线性项和不可压缩条件分裂到两个不同的子问题中,并对非线性项采用Oseen迭代.格式分为两步:第一步求解一个线性椭圆问题;第二步求解一个广义的Stokes问题.这两个子问题关于速度都满足齐次Dilichlet边界条件.同时,在格式的第二步添加了局部稳定化项,使用等阶序对来加强数值解的稳定性.通过能量估计方法,对速度与压力做了收敛性分析和误差估计.最后,数值实验验证了方法的有效性.     

非定常不可压缩Navier-Stokes方程的特征稳定化非协调有限元方法

数值求解非定常不可压缩Navier-Stokes方程的难点之一在于强烈的非线性容易引发非物理震荡,本文结合可以有效减弱此种震荡的特征线离散方法,基于局部Gauss积分之差的稳定化格式,采用最低等阶非协调混合有限元对NCP1-P1,构造出求解非定常不可压缩Navier-Stokes方程的特征稳定化非协调混合有限元方法。证明了该方法的全离散格式是无条件稳定的,并给出逼近解的相应误差估计。     

Interpolation functions in control volume finite element method

 The main contribution of this paper is the study of interpolation functions in control volume finite element method used in equal order and applied to an incompressible two-dimensional fluid flow. Especially, the exponential interpolation function expressed in the elemental local coordinate system is compared to the classic linear interpolation function expressed in the global coordinate system. A quantitative comparison is achieved by the application of these two schemes to four flows that we know the analytical solutions. These flows are classified in two groups: flows with privileged direction and flows without. The two interpolation functions are applied to a triangular element of the domain then; a direct comparison of the results given by each interpolation function to the exact value is easily realized. The two functions are also compared when used to solve the discretized equations over the entire domain. Stability of the numerical process and accuracy of solutions are compared. Received: 20 October 2002 / Accepted: 2 December 2002     

一种同时求多项式零点的加速迭代法

本文讨论了在无重根情况下,利用改进的Newton迭代法对一种同时求多项式零点的并行迭代法进行加速,得到了一种新的加速迭代法。首先证明了该方法是收敛的,并且理论证明出收敛阶至少是5阶;其次,分析了该方法的计算效率;最后通过实际的数值算例表明:计算收敛阶和定理结论是一致的,且本算法具有较高的计算效率。     

Development of robust finite elements for general purpose structural analysis

The finite element method emerged out of the old work and energy methods and matrix structural analysis to become a numerical procedure to solve practical stress analysis problems in solid and structural mechanics. With the impetus given by the rapid development of computer technology, it became the most overwhelmingly popular analysis and design computational tool for a very wide spectrum of engineering science, e.g. fluid mechanics, heat transfer and electro-magnetics. Today, there are very powerful general-purpose software codes that make analyses and design tasks that were once considered to be intractable, routinely simple. Many of these are closely held proprietary codes owned and used in-house by large engineering firms or sold or licensed and supported by specialist companies. (Recent estimates indicate that the market for these codes has reached a turnover of a billion dollars and that industries and institutions spend several tens of billions of dollars in running such codes.) These codes are rarely given out in source code. In order to have an in-house code that could be continuously up-graded and enhanced, NAL initiated some work to develop a medium-sized general purpose code (about 20,000 lines of FORTRAN code) for the analysis of laminated composite structures (FEPACS - finite element package for analysis of composite structures), recognising the importance that laminated composites were assuming in aerospace structural technology. Several key elements commonly found in general purpose packages (GPP) used by the aerospace, automobile and mechanical engineering industries were identified. These were re-designed incorporating anisotropic composite capabilities and validated. Many hurdles were faced during this task and required an examination of the basic issues at a paradigmatic level. Concepts such as consistency and variational correctness were introduced and studied critically. These guidelines played a critical role in developing robust versions of the elements and are briefly covered in this review. The paradigms also helped to identify procedures to performa priori error estimates for the quality of approximation and this allowed the elements being developed to be critically validated. The article concludes with a summary of what has been achieved and also suggests areas where the concepts can be applied fruitfully in the study of the displacement type finite element method.     

Meshless point collocation for the numerical solution of Navier-Stokes flow equations inside an evaporating sessile droplet

The Navier-Stokes flow inside an evaporating sessile droplet is studied in the present paper, using sophisticated meshfree numerical methods for the computation of the flow field. This problem relates to numerous modern technological applications, and has attracted several analytical and numerical investigations that expanded our knowledge on the internal microflow during droplet evaporation. Two meshless point collocation methods are applied here to this problem and used for flow computations and for comparison with analytical and more traditional numerical solutions. Particular emphasis is placed on the implementation of the velocity-correction method within the meshless procedure, ensuring the continuity equation with increased precision. The Moving Least Squares (MLS) and the Radial Basis Function (RBF) approximations are employed for the construction of the shape functions, in conjunction with the general framework of the Point Collocation Method (MPC). An augmented linear system for imposing the coupled boundary conditions that apply at the liquid-gas interface, especially the zero shear-stress boundary condition at the interface, is presented. Computations are obtained for regular, Type-I embedded nodal distributions, stressing the positivity conditions that make the matrix of the system stable and convergent. Low Reynolds number (Stokes regime), and elevated Reynolds number (Navier-Stokes regime) conditions have been studied and the solutions are compared to those of analytical and traditional CFD methods. The meshless implementation has shown a relative ease of application, compared to traditional mesh-based methods, and high convergence rate and accuracy.     

Theoretical and numerical results of a deterministic two-dimensional vortex method

The first part of the paper reviews results obtained in earlier work: (1) The outline of the derivation of an integral equation of Fredholm type with respect to vorticity from the Navier-Stokes equations, and (2) the analytical results for two deterministic vortex methods which are based on the corespreading model. The aim of the second part is to confirm the results of the previous analysis and to estimate the accuracy of these methods numerically. In the present paper, the model problem of a transient flow past an impulsively started circular cylinder is studied by several numerical methods. The numerical results show that (1) the Gaussian core-spreading methods are comparable with the random walk vortex method, (2) the numerical fluctuation of the deterministic methods is small, and (3) the number of panels is smaller than those in the random walk vortex method to obtain the flow with almost the same accuracy. The theoretical results in § 2–4 of the present paper are summarized from “Core-spreading vortex methods in two-dimensional viscous flows”,Computer methods in applied mechanics and engineering (in press), by Kida and Nakajima, with permission from Elsevier Science, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK. The experimental picture is transferred from “The early stage of development of the wake behind an impulsively started cylinder for 40<Re<⁴”, by R Bouard and M Coutanceau,J. Fluid Mech. (1980) 101: 583–607.     

Analysis and control of dynamic stall phenomenon using Navier-Stokes formulation involving vorticity,stream function and circulation

An unsteady Navier-Stokes (ns) analysis is developed for studying flow past a maneuvering body. The present inclusion ofcirculation in the earlierns analysis of the authors makes it feasible,for the first time, to accurately simulate the asymptotic far-field boundary condition. In the overall analysis, a clustered conformal mesh withC-grid topology is used, with the governing differential equations being solved using the implicitadi-bge technique. The effects of grid size and clustering on the flow solution, and the effect of grid stretching on the far-field solution, are studied using the flow configuration with Re=45,000 and constantrate pitch-up motion, with . The results obtained for this case compare very satisfactorily with available experimental data. Other numerical results are also used for carefully validating the analysis developed here. Anactive control strategy, consisting of modulated suction/injection at the airfoil surface, is studied, and provides satisfactory control of the unsteady separation process and, hence, of the dynamic stall vortex. This research is supported, in part, byafosr Grant Nos. 90-0249 and F49620-92-J-0292, and Ohio Supercomputer Center Grant No.pes080.     

A 3D implicit unstructured-grid finite volume method for structural dynamics

In this work, a new vertex-based finite volume method (FVM) using unstructured grids and cell-based data structure is proposed for computational analysis of two-and three-dimensional (2D/3D) general structural dynamic problems. The governing equations are spatially discretized by the FVM and an implicit dual time stepping scheme is employed to integrate the equations in time. The proposed method is applied to calculate deformations and dynamics of 2D and 3D cantilevers, as well as simply supported and clamped square plates. Computational results obtained are found to agree well with analytical solutions. It can be a viable alternative to the traditional finite element method (FEM) for structural dynamic calculations. And it can be seamlessly integrated into FVM-based Computational Fluid Dynamics (CFD) solver for simulating fluid-structure interaction (FSI).     

A finite volume procedure for fluid flow,heat transfer and solid-body stress analysis

A unified cell-centered unstructured mesh finite volume procedure is presented for fluid flow, heat transfer and solid-body stress analysis. An in-house procedure (A. W. Date, Solution of Transport Equations on Unstructured Meshes with Cell-Centered Colocated Variables. Part I: Discretization, International Journal of Heat and Mass Transfer, vol. 48 (6), 1117-1127, 2005) is extended to include the solid-body stress analysis. The transport terms for a cell-face are evaluated in a structured grid-like manner. The Cartesian gradients at the center of each cell-face are evaluated using the coordinate transformation relations. The accuracy of the procedure is demonstrated by solving several benchmark problems involving different boundary conditions, source terms, and types of loading.     

A parallelized Lattice-Gas solver for transient Navier–Stokes-flow: Implementation and simulation results

The last decade has seen the development of Lattice-Gas (LG) schemes as a complementary if not alternative method for the simulation of moderate Reynolds-Number Navier–Stokes flow. After a short theoretical introduction we present a detailed discussion of implementation features for a specific 2D-LG algorithm, which runs in parallel on a workstation-cluster, discuss simulation results and compare one of them to experimental studies. Finally, we attempt to point out present problems and perspectives of these methods.     

Finite element/finite volume approaches with adaptive time stepping strategies for transient thermal problems

Transient thermal analysis of engineering materials and structures by space discretization techniques such as the finite element method (FEM) or finite volume method (FVM) lead to a system of parabolic ordinary differential equations in time. These semidiscrete equations are traditionally solved using the generalized trapezoidal family of time integration algorithms which uses a constant single time step. This single time step is normally selected based on the stability and accuracy criteria of the time integration method employed. For long duration transient analysis and/or when severe time step restrictions as in nonlinear problems prohibit the use of taking a larger time step, a single time stepping strategy for the thermal analysis may not be optimal during the entire temporal analysis. As a consequence, an adaptive time stepping strategy which computes the time step based on the local truncation error with a good global error control may be used to obtain optimal time steps for use during the entire analysis. Such an adaptive time stepping approach is described here. Also proposed is an approach for employing combinedFEM/FVM mesh partitionings to achieve numerically improved physical representations. Adaptive time stepping is employed thoughout to practical linear/nonlinear transient engineering problems for studying their effectiveness in finite element and finite volume thermal analysis simulations. Additional support and computing times were furnished by Minnesota Supercomputer Institute at the University of Minnesota.