Penalty-finite-element analysis of 3-D Navier-Stokes equations

The steady-state Navier-Stokes equations in three dimensions are solved by a penalty-finite-element method for the problems of wall-driven cavity flow in a cubical box and natural convection in a cubical cavity subjected to differential side heating. The present solutions are compared with recent finite-difference solutions of the wall-driven cavity problem for Reynolds numbersR_E = 100 and 400. The agreement is found to be very good.     

A modified full multigrid algorithm for the Navier-Stokes equations

A modified full multigrid (FMG) method for the solution of the Navier-Stokes equations is presented. The method proposed is based on a V-cycle omitting the restriction procedure for dependent variables but retaining it for the residuals. This modification avoids possible mismatches between the mass fluxes and the restricted velocities as well as the turbulent viscosity and the turbulence quantities on the coarse grid. In addition, the pressure on the coarse grid can be constructed in the same way as the velocities. These features simplify the multigrid strategy and corresponding programming efforts. This algorithm is applied to accelerate the convergence of the solution of the Navier-Stokes equations for both laminar and high-Reynolds number turbulent flows. Numerical simulations of academic and practical engineering problems show that the modified algorithm is much more efficient than the FMG-FAS (Full Approximation Storage) method.     

An explicit Chebyshev pseudospectral multigrid method for incompressible Navier-Stokes equations

The two-dimensional steady incompressible Navier-Stokes equations in the form of primitive variables have been solved by Chebyshev pseudospectral method. The pressure and velocities are coupled by artificial compressibility method and the NS equations are solved by pseudotime method with an explicit four-step Runge-Kutta integrator. In order to reduce the computational time cost, we propose the spectral multigrid algorithm in full approximation storage (FAS) scheme and implement it through V-cycle multigrid and full multigrid (FMG) strategies. Four iterative methods are designed including the single grid method; the full single grid method; the V-cycle multigrid method and the FMG method. The accuracy and efficiency of the numerical methods are validated by three test problems: the modified one-dimensional Burgers equation; the Taylor vortices and the two-dimensional lid driven cavity flow. The computational results fit well with the exact or benchmark solutions. The spectral accuracy can be maintained by the single grid method as well as the multigrid ones, while the time cost is greatly reduced by the latter. For the lid driven cavity flow problem, the FMG is proved to be the most efficient one among the four iterative methods. A speedup of nearly two orders of magnitude can be achieved by the three-level multigrid method and at least one order of magnitude by the two-level multigrid method.     

An unfactored implicit scheme with multigrid acceleration for the solution of the Navier-Stokes equations

An unfactored implicit difference scheme for the steady state solution of the multidimensional Navier-Stokes equations of a compressible fluid is presented. The hyperbolic part is approximated by a high resolution scheme based on flux-vector splitting and upwind-biased differences to avoid the necessity of artificial dissipation terms and to construct a diagonal dominant solution matrix. Consequently, an iterative inversion of the solution matrix can be performed without any time step restriction. The rate of convergence is improved by using the indirect multigrid concept in form of the FAS scheme. The method is formulated for a body-fitted, curvilinear coordinate system. The computational results for laminar subsonic, transonic and supersonic steady-state flows which are compared with analytical and other numerical results as well as with experimental data illustrate the efficiency and the accuracy of the algorithm.     

Parallel multigrid solution of the Navier-Stokes equations on general 2D domains

Multigrid methods are distinguished by their optimal (sequential) efficiency and by the fact that all their algorithmical components are fully parallelizable. For this reason, this class of numerical methods is especially attractive for use on parallel (MIMD, local memory) computers. In this paper, we describe a parallel multigrid solver for steady-state incompressible Navier-Stokes equations on general domains which is currently being developed at the GMD. Due to the geometrical generality of the problem, our approach is based on a non-staggered (nodal-point) finite volume scheme on multi-block boundary fitted grids. The typical instability of non-staggered schemes is overcome by suitably modifying the discrete continuity equation without affecting the overall order of consistency.

Starting from the most simple Cartesian case, we discuss several possible multigrid approaches to the general 2D-problem. This motivates the basic design decisions of our multigrid solver in regard to both the discretization and the choice of multigrid components (smoothing schemes). Furthermore, the principal technique of parallelization (grid partitioning) is described as well as some fundamental aspects of the implementation (communication library).     

Development of a second order approximation for the Navier-Stokes equations

The purpose of this paper is the development of a 2nd order finite difference approximation to the steady state Navier-Stokes equations governing flow of an incompressible fluid in a closed cavity. The approximation leads to a system of equations that has proved to be very stable. In fact, numerical convergence was obtained for Reynolds numbers up to 20,000. However, it is shown that extremely small mesh sizes are needed for excellent accuracy with a Reynolds number of this magnitude. The method uses a nine point finite difference approximation to the convection term of the vorticity equation. At the same time it is capable of avoiding values at corner nodes where discontinuities in the boundary conditions occur. Figures include level curves of the stream and vorticity functions for an assortment of grid sizes and Reynolds numbers.     

Solution of 2-D Euler equations with a parallel code

A parallel program for the simulation of inviscid fluid flow around airfoils, based on the solution of unsteady Euler equations is presented. The concept of domain splitting and its connection with the selected parallelisation strategy is outlined. Furthermore, the implementation of the algorithm within the SUPRENUM environment as well as the treatment given to inter-process communication is also discussed. Finally, some topics about code development, use of simulation systems and preliminary validation test are quoted.     

A general finite-difference formulation with application to Navier-Stokes equations

A general finite-difference formulation is presented here for deriving an accurate and stable finite-difference scheme. The method introduces a new concept of “decay function” which is determined analytically at each grid point. An additional concept of “locally one-dimensional” is used in multi-dimensional equations. Two types of decay functions are derived, spatial decay functions and time-wise decay functions. In both cases, the properties of decay functions are discussed and the relation of the method to other difference schemes, such as central difference, upwind difference, explicit difference and implicit difference, are studied. Applications to one-, two- and three-dimensional vorticity equations are illustrated.     

Evaluation of multigrid acceleration for preconditioned time-accurate Navier-Stokes algorithms

The development of a two-dimensional time-accurate dual time step Navier-Stokes flow solver with time-derivative preconditioning and multigrid acceleration is described. The governing equations are integrated in time with both an explicit Runge-Kutta scheme and an implicit lower-upper symmetric-Gauss-Seidel scheme in a finite volume framework, yielding second-order accuracy in space and time. Issues concerning the implementation of multigrid for preconditioned, dual time step algorithms are discussed. Steady and unsteady computations were made of lid driven cavity flow, thermally driven cavity flow and pulsatile channel flow for a variety of conditions to validate the schemes and evaluate the effectiveness of multigrid for time-accurate simulations. Significant speedups were observed for steady and unsteady simulations. The speedups for unsteady simulations were problem dependent, a function of how rapidly the flow varied in time and the size of the allowable time step.     

Analysis and application of a parallel spectral element method for the solution of the Navier-Stokes equations

We present an analysis of the parallel spectral element method for solution of the unsteady incompressible Navier-Stokes equations in general three-dimensional geometries. The approach combines high-order spatial discretizations with iterative solution techniques in a way which exploits with high efficiency the currently available medium-grained distributed-memory parallel computers. Measured performance analysis on the Intel vector hypercubes and example Navier-Stokes calculations demonstrate that parallel processing can now be considered an effective fluid mechanics analysis tool.     

CDM based finite element code for concrete in 3-D

This paper presents a three dimensional finite element code DAMAG3D for nonlinear analysis of concrete type materials modeled as elastic-damage. The CDM model adopted is the one as proposed by SUARIS W, OUYANG C, FERNANDO V. M. Damage model for cyclic loading of concrete. J Engng Mech, American Society of Civil Engineers 1990; 116(5): 1020-35. for monotonic and cyclic loading of concrete structures. Code DAMAG3D is applied to simulate response of concrete under monotonically increasing load paths of uniaxial compression, Brazilian test, strip loading and patch loading, with reasonable correlation established with experimental results and results from other nonlinear constitutive models.     

A new approach to code motion and its application to hoisting

An approach to code motion and hoisting, a program optimization technique, is discussed. The safety and profitability of optimization in general and hoisting in particular are analyzed. By restricting the analysis to a spanning tree imposed on the program graph, a linear algorithm is developed that provides sufficient but not necessary conditions for hoisting.     

指数函数方法及其在非线性发展方程中的应用

在求解非线性发展方程时,指数函数方法是一种非常简洁有效的方法。用此方法求解了2+1维Burgers方程和2+1维KP方程,并且得到了一些新的精确解。     

A comparative study of the parabolized Navier-Stokes code using various grid-generation techniques

The parabolized Navier-Stokes (PNS) equations are used to calculate the flow-field characteristics about the hypersonic research aircraft X-24C. A comparison of the results obtained using elliptic, hyperbolic and algebraic grid generators is presented. The outer bow shock is treated as a sharp discontinuity, and the discontinuities within the shock layer are captured. Surface pressures and heat-transfer results at angles of attack of 6° and 20°, obtained using the three grid generators, are compared. The PNS equations are marched downstream over the body in both Cartesian and cylindrical base coordinate systems, and the results are compared. A robust marching procedure is demonstrated by successfully using large marching-step size with the implicit shock fitting procedure. A correlation is found between the marching-step size. Reynolds number and the angle of attack at fixed values of smoothing and stability coefficients for the marching scheme.     

Translation of ATL to AGT and application to a code generator for Simulink

Software and Systems Modeling - Analysing and reasoning on model transformations has become very relevant for various applications such as ensuring the correctness of transformations. ATL is a...     

Iterative methods for stationary solutions to the steady-state compressible Navier-Stokes equations

Numerical experiments are presented for the solution of the steady-state compressible Navier-Stokes equations. One test problem is fixed supersonic flow past a double ellipse, and the various solution methods studied. The problem is discretized using Osher's scheme, first- and second-order accurate. The fastest convergence to steady state is obtained using Newton's method. Simplifications of Newton's method based on domain decomposition are shown to perform well, whereas line relaxation methods meet with difficulties.     

The derivation of a chiral substituent code for secondary alcohols and its application to the prediction of enantioselectivity

A chiral substituent code was proposed based on the features of secondary alcohols, in which a chiral center is attached to two substituents in addition to OH and H substituents. The new chirality code, which was generated by predefining positional information of four substituents attached to stereocenter, was applied to two datasets composed of secondary alcohols as the enantioselective products of asymmetric reactions. In the first dataset, the chemical reaction was catalyzed by a biocatalyst, lipase from Candida rugosa. The catalyst for the second dataset was (−)-diisopinocampheylchloroborane. The structure–enantioselectivity relationship models were constructed using random forests with the chiral substituent code as the input. The resulting models were assessed both in terms of single enantiomers and pairs of enantiomers. Satisfactory results were obtained for both datasets. Although the chiral substituent code was specifically developed for secondary alcohols, it can easily be extended to represent chiral compounds possessing a specific chiral center bonded to two variable substituents.     

A parallel multigrid method for constrained minimization problems and its application to friction,contact, and obstacle problems

The parallel solution of constrained minimization problems requires special care to be taken with respect to the information transfer between the different subproblems. Here, we present a nonlinear decomposition approach which employs an additional nonlinear correction step along the processor interfaces. Our approach is generic in the sense that it can be applied to a wide class of minimization problems with strongly local nonlinearities, including even nonsmooth minimization problems. We also describe the implementation of our nonlinear decomposition method in the object oriented library ObsLib \(++\). The flexibility of our approach and its implementation is presented along different problem classes as obstacle problems, frictional contact problems and biomechanical applications. For the same examples, number of iterations, computation time, and parallelization speedup are measured, and the results demonstrate that the implementation scales reasonably well up to 4096 processors.