Newton Iterative Parallel Finite Element Algorithm for the Steady Navier-Stokes Equations

A combination method of the Newton iteration and parallel finite element algorithm is applied for solving the steady Navier-Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the Newton iterations of m times for a nonlinear problem on a coarse grid in domain Ω and computing a linear problem on a fine grid in some subdomains Ω _j ⊂Ω with j=1,…,M in a parallel environment. Then, the error estimation of the Newton iterative parallel finite element solution to the solution of the steady Navier-Stokes equations is analyzed for the large m and small H and h≪H. Finally, some numerical tests are made to demonstrate the the effectiveness of this algorithm.     

A numerical flow simulation based on the rate form of the equation of state

A finite-difference numerical method of solution for the unsteady, incompressible Navier-Stokes equations using primitive variables is presented. The rate form of the equation of state is used for the calculation of pressure. This form of the equation of state is well-suited for use with the unsteady form of the conservation equations (mass, momentum and energy). An implicit algorithm is used for the time integration for greater numerical stability. This method is used to solve a known benchmark problem in steady-state natural convection as a test of steady-state accuracy. The results of the simulation are compared to the benchmark.     

The FON method for the steady two-dimensional Navier-Stokes equations

The Fourth-Order Newton (FON) method, employing a direct linear biharmonic solver, is used to solve the finite-difference formulation of the steady, incompressible Navier-Stokes equations in an iterative Newton scheme. Accurate solutions to a number of problems for small and large Reynols numbers are obtained in a few iterations, employing strained grids. Special attention is paid to the problem of separation caused by non-parallel entrance flow.     

Application of spectral methods to the solution of Navier-Stokes equations

In this paper a number of topics that arise in application of Chebyshev expansion methods to the solution of the Navier-Stokes equations are addressed. These include the equivalence of finite differences and finite elements approaches, a new velocity-pressure formulation that permits easy extension to three dimensions, evaluation of the importance of pressure boundary conditions and the virtues of collocation over the tau method for satisfying boundary conditions. Example results from the velocity-pressure formulation which eliminate a non-linear momentum equation in favor of the linear continuity equation are presented. The results are for a 2-D unsteady flow on a flat plate at large Reynolds numbers. The behavior of an unsteady disturbance in such a flow is examined and compared with previous stream-function vorticity results of Murdock.     

Steady and unsteady flow past a rotating circular cylinder at low Reynolds numbers

Results of calculations of the steady and unsteady flows past a circular cylinder which is rotating with constant angular velocity and translating with constant linear velocity are presented. The motion is assumed to be two-dimensional and to be governed by the Navier-Stokes equations for incompressible fluids. For the unsteady flow, the cylinder is started impulsively from rest and it is found that for low Reynolds numbers the flow approaches a steady state after a large enough time. Detailed results are given for the development of the flow with time for Reynolds numbers 5 and 20 based on the diameter of the cylinder. For comparison purposes the corresponding steady flow problem has been solved. The calculated values of the steady-state lift, drag and moment coefficients from the two methods are found to be in good agreement. Notable, however, are the discrepancies between these results and other recent numerical solutions to the steady-state Navier-Stokes equations. Some unsteady results are also given for the higher Reynolds numbers of 60, 100 and 200. In these cases the flow does not tend to be a steady state but develops a periodic pattern of vortex shedding.     

Transient state analysis of separated flow around a sphere

Transient state solutions of the Navier-Stokes equations were obtained for incompressible flow around a sphere accelerating from zero initial velocity to its terminal free falling velocity. By assuming rotational symmetry about the axis in the direction of motion, the Navier-Stokes equations and the continuity equation were simplified in terms of vorticity and stream function. The instantaneous acceleration of the falling sphere was calculated by considering the difference between the gravitational force and the drag force in a transient state. A set of implicit finite difference equations was developed. In order to obtain accurate information around the body, an exponential transformation along the radial direction was used to provide finer meshes in the vicinity of the surface of the sphere. The vorticity equation was solved by an alternating direction implicit (ADI) method while the stream function equation was solved by a successive over-relaxation (SOR) method. Simultaneous solutions were obtained. Transient state solutions were compared with steady state solutions for Reynolds numbers up to 300. Separations first occurred at a Reynolds number 20 for steady state flows and at Reynolds numbers 22·46 and 28·24 for transient state flows with terminal Reynolds numbers of 100 and 300, respectively. Separation angles, sizes of separation regions, and drag coefficents were calculated for both steady and unsteady states. Good agreement was obtained with existing experimental data in the steady state.     

Approximation of stochastic partial differential equations by a kernel-based collocation method

In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations (SPDEs). Using an implicit time-stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on the numerical solution of the elliptic equations at each time step. The estimator of the solution of the elliptic equations is given as a linear combination of reproducing kernels derived from the differential and boundary operators of the SPDE centred at collocation points to be chosen by the user. The random expansion coefficients are computed by solving a random system of linear equations. Numerical experiments demonstrate the feasibility of the method.     

Implicit solutions of the unsteady Navier-Stokes equation for laminar flow through an orifice within a pipe

A technique is presented for the numerical solution of the unsteady Navier-Stokes equation for laminar flow through an orifice within a pipe. The solution is accomplished through the rearrangement of the equation of motion into a vorticity transport equation (VTE) and a definition-of-vorticity equation (DVE) which are solved by an implicyt numerical method. An initial series of studies was performed to analyze flow development at upstream and downstream infinity for the case of constantly increasing flow until a Reynolds number of 5 was reached followed by a period of constant flow until steady flow was approached. The solution during this series of studies never failed to produce convergent results, although a damped instability was observed when very large time increments were used. Results are presented for the unsteady development of flow far upstream of the orifice for non-dimentional flow accelerations of 1, 10, and 100. These demonstrate that the penetration of vorticity decreases with increasing acceleration, and that the fraction of volume flow adjacent to the wall increases with increasing acceleration. By allowing this solution to proceed during the period of constant flow, it was found that the stream function distribution approached a stedy conditon more rapidly than the vorticity distribution. Results are also presented for the asymptotic solution for steady flow through an orifice at a Reynolds number of 5 for an orifice-to-pipe dia ratio of 0·5. These results compared very favorably with those from a recent steady flow solution.     

Computation of unsteady viscous flow and aeroacoustic noise of cross flow fans

Time-accurate viscous flow solutions are sought for the prediction of unsteady flow characteristics and associated aeroacoustic blade tonal noise of a cross flow fan. The two-dimensional incompressible Navier-Stokes equations in a moving coordinate are time-accurately solved by an unstructured finite-volume method on triangular meshes, and a sliding mesh technique is utilized at the interface between the domain rotating with blades and the stationary one for allowing the unsteady interactions. An accuracy assessment of the present method is made by comparing the fan performances with experimental data for a rotational speed at 1000 rpm and the Reynolds number 5300 based on blade tip speed and chord length. With the computed unsteady viscous flow solutions, sound pressure is predicted using Curle’s equation and narrow-band noise characteristics of three impellers with a uniform and two random pitch (type-A and -B) blades are compared by their sound pressure level spectra. Also, the frequency modulations of the blade passing frequency noise by random pitch fans are discussed.     

Application of a posteriori error estimation to finite element simulation of incompressible Navier-Stokes flow

The main goal of this paper is to study adaptive mesh techniques, using a posteriori error estimates, for the finite element solution of the Navier-Stokes equations modeling steady and unsteady flows of an incompressible viscous fluid. Among existing operator splitting techniques, the θ-scheme is used for time integration of the Navier-Stokes equations. Then, a posteriori error estimates, based on the solution of a local system for each triangular element, are presented in the framework of the generalized incompressible Stokes problem, followed by its practical application to the case of incompressible Navier-Stokes problem. Hierarchical mesh adaptive techniques are developed in response to the a posteriori error estimation. Numerical simulations of viscous flows associated with selected geometries are performed and discussed to demonstrate the accuracy and efficiency of our methodology.     

Space-time adaptive simulations for unsteady Navier-Stokes problems

In this paper we apply to the unsteady Navier-Stokes problems some results concerning a posteriori error estimates and adaptive algorithms known for steady Navier-Stokes, unsteady heat and reaction-convection-diffusion equations and unsteady Stokes problems. Our target is to investigate the real viability of a fully combined space and time adaptivity for engineering problems. The comparison between our numerical simulations and the literature results demonstrates the accuracy and efficiency of this adaptive strategy.     

A reduced spectral function approach for the stochastic finite element analysis

The stochastic finite element analysis of elliptic type partial differential equations with non-Gaussian random fields are considered. A novel approach by projecting the solution of the discretized equation into a reduced finite dimensional orthonormal vector basis is investigated. It is shown that the solution can be obtained using a finite series comprising functions of random variables and orthonormal vectors. These functions, called as the spectral functions, can be expressed in terms of the spectral properties of the deterministic coefficient matrices arising due to the discretization of the governing partial differential equation. Based on the projection in a reduced orthonormal vector basis, a Galerkin error minimization approach is proposed. The constants appearing in the Galerkin method are solved from a system of linear equations which has much smaller dimension compared to the original discretized equation. A hybrid analytical and simulation based computational approach is proposed to obtain the moments and probability density function of the solution. The method is illustrated using the stochastic nanomechanics of a zinc oxide (ZnO) nanowire deflected under the atomic force microscope (AFM) tip. The results are compared with the results obtained using direct Monte Carlo simulation, classical Neumann expansion and polynomial chaos approach for different correlation lengths and strengths of randomness.     

Optimal bang-bang control of linear stochastic systems with a small noise parameter

This study considers the problem of determining optimal feedback control laws for linear stochastic systems with amplitude-constrained control inputs. Two basic performance indices are considered, average time and average integral quadratic form. The optimization interval is random and defined as the first time a trajectory reaches the terminal regionR. The plant is modeled as a stochastic differential equation with an additive Wiener noise disturbance. The variance parameter of the Wiener noise process is assumed to be suitably small. A singular perturbation technique is presented for the solution of the stochastic optimization equations (second-order partial differential equation). A method for generating switching curves for the resulting optimal bang-bang control system is then developed. The results are applied to various problems associated with a second-order purely inertial system with additive noise at the control input. This problem is typical of satellite attitude control problems.     

Iterative algorithm to compute the maximal and stabilising solutions of a general class of discrete-time Riccati-type equations

In this article an iterative method to compute the maximal solution and the stabilising solution, respectively, of a wide class of discrete-time nonlinear equations on the linear space of symmetric matrices is proposed. The class of discrete-time nonlinear equations under consideration contains, as special cases, different types of discrete-time Riccati equations involved in various control problems for discrete-time stochastic systems. This article may be viewed as an addendum of the work of Dragan and Morozan (Dragan, V. and Morozan, T. (2009), ‘A Class of Discrete Time Generalized Riccati Equations’, Journal of Difference Equations and Applications, first published on 11 December 2009 (iFirst), doi: 10.1080/10236190802389381) where necessary and sufficient conditions for the existence of the maximal solution and stabilising solution of this kind of discrete-time nonlinear equations are given. The aim of this article is to provide a procedure for numerical computation of the maximal solution and the stabilising solution, respectively, simpler than the method based on the Newton–Kantorovich algorithm.     

带有随机跳跃干扰的线性二次随机最优控制问题

给出一类布朗运动和泊松过程混合驱动的正倒向随机微分方程解的存在唯一性结果, 应用这一结果研究带有随机跳跃干扰的线性二次随机最优控制问题,并得到最优控制的显式形 式,可以证明最优控制是唯一的.然后,引入和研究一类推广的黎卡提方程系统,讨论该方程系统 的可解性并由该方程的解得到带有随机跳跃干扰的线性二次随机最优控制问题最优的线性反馈.     

Efficient monolithic simulation techniques for the stationary Lattice Boltzmann equation on general meshes

In this paper, we present special discretization and solution techniques for the numerical simulation of the Lattice Boltzmann equation (LBE). In Hübner and Turek (Computing, 81:281–296, 2007), the concept of the generalized mean intensity had been proposed for radiative transfer equations which we adapt here to the LBE, treating it as an analogous (semi-discretized) integro-differential equation with constant characteristics. Thus, we combine an efficient finite difference-like discretization based on short-characteristic upwinding techniques on unstructured, locally adapted grids with fast iterative solvers. The fully implicit treatment of the LBE leads to nonlinear systems which can be efficiently solved with the Newton method, even for a direct solution of the stationary LBE. With special exact preconditioning by the transport part due to the short-characteristic upwinding, we obtain an efficient linear solver for transport dominated configurations (macroscopic Stokes regime), while collision dominated cases (Navier-Stokes regime for larger Re numbers) are treated with a special block-diagonal preconditioning. Due to the new generalized equilibrium formulation (GEF) we can combine the advantages of both preconditioners, i.e. independence of the number of unknowns for convection-dominated cases with robustness for stiff configurations. We further improve the GEF approach by using hierarchical multigrid algorithms to obtain grid-independent convergence rates for a wide range of problem parameters, and provide representative results for various benchmark problems. Finally, we present quantitative comparisons between a highly optimized CFD-solver based on the Navier-Stokes equation (FeatFlow) and our new LBE solver (FeatLBE).     

Manifold method coupled velocity and pressure for Navier-Stokes equations and direct numerical solution of unsteady incompressible viscous flow

Numerical manifold method (NMM) application to direct numerical solution for unsteady incompressible viscous flow Navier-Stokes (N-S) equations was discussed in this paper, and numerical manifold schemes for N-S equations were derived based on Galerkin weighted residuals method as well. Mixed covers with linear polynomial function for velocity and constant function for pressure was employed in finite element cover system. The patch test demonstrated that mixed covers manifold elements meet the stability conditions and can be applied to solve N-S equations coupled velocity and pressure variables directly. The numerical schemes with mixed covers have also been proved to be unconditionally stable. As applications, mixed cover 4-node rectangular manifold element has been used to simulate the unsteady incompressible viscous flow in typical driven cavity and flow around a square cylinder in a horizontal channel. High accurate results obtained from much less calculational variables and very large time steps are in very good agreement with the compact finite difference solutions from very fine element meshes and very less time steps in references. Numerical tests illustrate that NMM is an effective and high order accurate numerical method for unsteady incompressible viscous flow N-S equations.     

Determination of invariant tori bifurcation points

Two numerical methods are presented for the determination of invariant tori bifurcation points on a branch of periodic solutions of ordinary differential equations (lumped parameter systems). The methods are based on the shooting technique and the Newton method. The coefficients of characteristic polynomial of the monodromy matrix have to be evaluated in each iteration, however, a modification is suggested where it is not necessary. The convergence of the methods is very sensitive to an initial guess. A good guess can be obtained from results of continuation of the periodic solutions on a parameter. Applications to a hydrodynamic problem (a seven-mode model truncated Navier-Stokes equations for a two-dimensional flow of an incompressible fluid) and a problem from chemical reactor theory (two well mixed reaction cells with linear diffusion coupling and the Brusselator reaction kinetic scheme) demonstrate the effectiveness of the methods.     

广义系统稳态Kalman估值器

用现代时间序列分析方法,提出了广义离散线性随机系统稳态Kalman滤波、平滑 和预报的一种统一格式,给出了稳态Kalman估值器增益新算法,避免了求解Riccati方程.为 保证估值器的渐近稳定性,给出了选择初始估值的公式.仿真例子说明了所提出的结果的有 效性.