Data‐driven projection method in fluid simulation

Physically based fluid simulation requires much time in numerical calculation to solve Navier–Stokes equations. Especially in grid‐based fluid simulation, because of iterative computation, the projection step is much more time‐consuming than other steps. In this paper, we propose a novel data‐driven projection method using an artificial neural network to avoid iterative computation. Once the grid resolution is decided, our data‐driven method could obtain projection results in relatively constant time per grid cell, which is independent of scene complexity. Experimental results demonstrated that our data‐driven method drastically speeded up the computation in the projection step. With the growth of grid resolution, the speed‐up would increase strikingly. In addition, our method is still applicable in different fluid scenes with some alterations, when computational cost is more important than physical accuracy. Copyright © 2016 John Wiley & Sons, Ltd.     

Application of a fractional step method for the numerical solution of the shallow water waves in a rotating rectangular basin

A numerical method is applied to the problem of an incompressible fluid in a slowly rotating rectangular basin for the simulation of wave propagation in shallow water. The present work is a complete study of the wave motion through evaluation of the wave height and the velocity components. The results are found by the application of a fractional step method and illustrated graphically. The technique is applied by splitting the shallow water equations and successive integration in every direction along the characteristics using the Riemann invariants associated with cubic spline interpolation. It has the advantage of reducing the multidimensional matrix inversion problem into an equivalent one-dimensional problem. Numerical results are represented in three dimensions for the velocity components at different times. The distribution of temperature and concentration are also calculated and plotted.     

Adaptive space–time finite element methods for the wave equation on unbounded domains

Comprehensive adaptive procedures with efficient solution algorithms for the time-discontinuous Galerkin space–time finite element method (DGFEM) including high-order accurate nonreflecting boundary conditions (NRBC) for unbounded wave problems are developed. Sparse multi-level iterative schemes based on the Gauss–Seidel method are developed to solve the resulting fully-discrete system equations for the interior hyperbolic equations coupled with the first-order temporal equations associated with auxiliary functions in the NRBC. Due to the local nature of wave propagation, the iterative strategy requires only a few iterations per time step to resolve the solution to high accuracy. Further cost savings are obtained by diagonalizing the mass and boundary damping matrices. In this case the algebraic structure decouples the diagonal block matrices giving rise to an explicit multi-corrector method. An h-adaptive space–time strategy is employed based on the Zienkiewicz–Zhu spatial error estimate using the superconvergent patch recovery (SPR) technique, together with a temporal error estimate arising from the discontinuous jump between time steps of both the interior field solutions and auxiliary boundary functions. For accurate data transfer between meshes, a new enhanced interpolation (EI) method is developed and compared to standard interpolation and projection. Numerical studies of transient radiation and scattering demonstrate the accuracy, reliability and efficiency gained from the adaptive strategy.     

Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science

In this paper, we suggest a fractional functional for the variational iteration method to solve the linear and nonlinear fractional order partial differential equations with fractional order initial and boundary conditions by using the modified Riemann-Liouville fractional derivative proposed by G. Jumarie. Fractional order Lagrange multiplier has been considered. Solution has been plotted for different values of α.     

一种基于SMP的并行逐次超松弛迭代法

逐次超松弛迭代方法被广泛应用于油藏数值模拟中压力方程的求解.其并行实现是提高模拟速度的重要途径.传统并行方案大都只是在一次迭代内进行数据划分,而没有进一步将数据划分与迭代空间划分相结合,故针对SOR算法和SMP（symmetric multi-processors）系统的特点,以OpenMP为并行化实现工具,提出了基于SMP的并行逐次超松弛迭代方法（parallelSOR）.方法通过改变不同迭代步内数据点的更新次序,使不同区域内的数据点可以并行执行多次迭代.总结出针对三维油藏区域在数据空间划分和迭代空间合并上相对较优的策略,分析了迭代过程中网格块的生长形状.与传统的并行策略相比,该方法具有可减小同步开销、改进数据局部性、cache命中率高等优点.实验结果表明,该方法具有较高的加速比和效率.     

Computation of taylor vortex flow by a transient implicit method

A finite difference method is presented for the computation of steady axisymmetric solutions of Navier-Stokes equations using the time dependent stream function, vorticity, and tangential velocity formulation. The scheme involves implicit fractional steps and fast Fourier transforms. Upwind differencing for convective terms is used in order to increase the stability for high values of the Reynolds number. The method is applied to the flow in an annulus of rectangular cross section with rotating walls. Attention is focused upon the problem of centrifugal instabilities, non-uniqueness of the steady state solution, and selection of wavelengths in the supercritical range.     

Conjugate gradient solution of linear equations

The conjugate gradient method is an ingenious method for iterative solution of sparse linear equations. It is now a standard benchmark for parallel scientific computing. In the author's opinion, the apparent mystery of this method is largely due to the inadequate way in which it is presented in textbooks. This tutorial explains conjugate gradients by deriving the computational steps from elementary mathematical concepts. The computation is illustrated by a numerical example and an algorithmic outline. © 1998 John Wiley & Sons, Ltd.     

A cascadic geometric filtering approach to subdivision

A new approach to subdivision based on the evolution of surfaces under curvature motion is presented. Such an evolution can be understood as a natural geometric filter process where time corresponds to the filter width. Thus, subdivision can be interpreted as the application of a geometric filter on an initial surface. The concrete scheme is a model of such a filtering based on a successively improved spatial approximation starting with some initial coarse mesh and leading to a smooth limit surface.

In every subdivision step the underlying grid is refined by some regular refinement rule and a linear finite element problem is either solved exactly or, especially on fine grid levels, one confines to a small number of smoothing steps within the corresponding iterative linear solver. The approach closely connects subdivision to surface fairing concerning the geometric smoothing and to cascadic multigrid methods with respect to the actual numerical procedure. The derived method does not distinguish between different valences of nodes nor between different mesh refinement types. Furthermore, the method comes along with a new approach for the theoretical treatment of subdivision.     

A Lagrangian meshless finite element method applied to fluid–structure interaction problems

A method is presented for the solution of the incompressible fluid flow equations using a Lagrangian formulation. The interpolation functions are those used in the meshless finite element method and the time integration is introduced in a semi-implicit way by a fractional step method. Classical stabilization terms used in the momentum equations are unnecessary due to the lack of convective terms in the Lagrangian formulation. Furthermore, the Lagrangian formulation simplifies the connections with fixed or moving solid structures, thus providing a very easy way to solve fluid–structure interaction problems.     

基于机载激光雷达点云的断裂线自动提取方法

基于机载激光雷达（LIDAR）点云生产高精度的数字高程模型（DTM）需要进行断裂线的存储与表达,在分析现有断裂线提取方法不足的基础上,提出一种从LIDAR点云自动提取断裂线的方法。该方法利用离散的点云构建三角网,建立点云之间的拓扑关系,根据三角网面片之间的法向差异提取候选断裂线点,采用“方向优先”追踪策略实现断裂线的追踪处理,并利用“线性迭代法”实现断裂线的光滑输出。实验结果表明,该方法可以快速从LIDAR点云中自动提取断裂线信息,具有一定的应用价值。     

Study on a grid interface algorithm for solutions of incompressible Navier–Stokes equations

Grid interface treatment is a crucial issue in solving unsteady, three-dimensional, incompressible Navier–Stokes equations by domain decomposition methods. Recently, a mass flux based interpolation (MFBI) interface algorithm was proposed for Chimera grids [Tang HS, Jones SC, Sotiropoulos F. An overset grid method for 3D unsteady incompressible flows. J Comput Phys, 2003;191:567–600] and it has been successfully applied to a variety of flows. MFBI determines velocity and pressure at grid interfaces by mass conservation and interpolation, and it is easy to implement. Compared with the commonly used standard interpolation, which directly interpolates velocity as well as pressure, the proposed interface algorithm gives fewer solution oscillations and faster convergence rates. This paper makes a study on MFBI. Starting with discussions about grid connectivity, it is shown that MFBI is second-order accurate for mass flux across grid interface. It is also derived that the scheme provides second-order accuracy for momentum flux. In addition, another version of MFBI is presented. At last, numerical examples are presented to demonstrate that MFBI honors mass flux balance at grid interfaces and it leads to second-order accurate solutions.     

Starting algorithms for Gauss Runge-Kutta methods for Hamiltonian systems

Among the symplectic integrators for the numerical solution of general Hamiltonian systems, implicit Runge-Kutta methods of Gauss type (RKG) play an important role. To improve the efficiency of the algorithms to be used in the solution of the nonlinear equations of stages, accurate starting values for the iterative process are required. In this paper, a class of starting algorithms, which are based on numerical information computed in two previous steps, is studied. For two- and three-stages RKG methods, explicit starting algorithms for the stage equations with orders three and four are derived. Finally, some numerical experiments comparing the behaviour of the new starting algorithms with the standard first iterant based on Lagrange interpolation of stages in the previous step are presented.     

Spectral method of decoupling the vorticity and stream function for the incompressible fluid flows

A spectral method is proposed for the vorticity-stream function equations of the incompressible fluid flows. It is effective to overcome the lack of vorticity boundary condition. This method decouples the vorticity and stream function. At each time step, first, the vorticity is explicitly solved and the stream function is evaluated by a Poisson-like equation; then the vorticity is determined by a Poisson-like equation again. The numerical experiments show that this method is of efficiency and high accuracy.     

Efficient exponential compact higher order difference scheme for convection dominated problems

Present work is the development of a finite difference scheme based on Richardson extrapolation technique. It gives an exponential compact higher order scheme (ECHOS) for two-dimensional linear convection-diffusion equations (CDE). It uses a compact nine point stencil, over which the governing equations are discretized for both fine and coarse grids. The resulting algebraic systems are solved using a line iterative approach with alternate direction implicit (ADI) procedure. Combining the solutions over fine and coarse grids, initially a sixth order solution over coarse grid points is obtained. The resultant solution is then extended to finer grid by interpolation derived from the difference operator. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be monotone. The higher order accuracy and better rate of convergence of the developed algorithm have been demonstrated by solving numerous model problems.     

Numerical solution of incompressible Navier–Stokes equations using a fractional-step approach

A fractional step method for the solution of steady and unsteady incompressible Navier–Stokes equations is outlined. The method is based on a finite-volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (third and fifth) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when fifth-order upwind differencing and a modified production term in the Baldwin–Barth one-equation turbulence model are used with adequate grid resolution.     

A Multiquadric Interpolation Method for Solving Initial Value Problems

In this paper, an interpolation method for solving linear differential equations was developed using multiquadric scheme. Unlike most iterative formula, this method provides a global interpolation formula for the solution. Numerical examples show that this method offers a higher degree of accuracy than Runge-Kutta formula and the iterative multistep methods developed by Hyman (1978).     

An adaptive multi-grid algorithm for the numerical solution of quasilinear potential equations

This paper describes an iterative method for the numerical solution of a class of quasilinear potential equations using an adaptive multi-grid algorithm (MG-algorithm). The method of solution has been illustrated using one iteration step of MG-cycle. The prolongation and restriction operators, which need coarse-to-fine as well as fine-to-coarse grid transfer, have been chosen of very simple linear structure. A simple error estimation has been carried out to show that the correction equation suggested in [2] has to be modified to get an efficient MG-algorithm. Another simple approach has been suggested which is based on a two-level version and uses a linear correction equation only on the coarser grid. We also present computational results of several numerical experiments applied on a specific example of the minimal surface problem. A comparison between our methods and other methods applied on the example of the minimal surface problem has been presented.     

Numerical simulation of steady compressible flows using a zonal formulation. Part I: inviscid flows

Simulation of inviscid flows are usually based on the Euler equations, namely, conservation laws of mass, momentum, and energy written in terms of conservative variables. We present an alternative formulation where the velocity components are calculated from a generalized form of the Cauchy–Riemann equations with a non-homogeneous term representing vorticity. The vorticity is obtained in terms of the gradients of entropy and total enthalpy via Crocco's relation (an expression of the normal momentum equation). The entropy and the total enthalpy are calculated from the tangential momentum and energy equations. For external aerodynamics, most of the field is isentropic and irrotational, and the formulation reduces to potential flow. Hence, two zones are easily identifiable. An inner zone where vorticity could be important, and an outer zone where vorticity is negligible. Preliminary results for two and three dimensional transonic flows over airfoils and wings are presented and compared with standard Euler calculations.     

A high-order accurate numerical algorithm for three-dimensional transport prediction

The numerical solution of the three-dimensional pollutant transport equation is obtained with the method of fractional steps; advection is solved by the method of moments and diffusion by cubic splines. Topography and variable mesh spacing are accounted for with coordinate transformations. First estimate wind fields are obtained by interpolation to grid points surrounding specific data locations. Mass consistency is ensured by readjusting the three-dimensional wind field with a Sasaki variational technique. Numerical results agree with results obtained from analytical Gaussian plume relations for ideal conditions. The numerical model is used to simulate the transport of tritium released from the Savannah River Plant on 2 May, 1974. Predicted ground level air concentration 56 km from the release point is within 38% of the experimentally measured value.     

Fractional Jacobi Galerkin spectral schemes for multi-dimensional time fractional advection–diffusion–reaction equations

One of the ongoing issues with time fractional diffusion models is the design of efficient high-order numerical schemes for the solutions of limited regularity. We construct in this paper two efficient Galerkin spectral algorithms for solving multi-dimensional time fractional advection–diffusion–reaction equations with constant and variable coefficients. The model solution is discretized in time with a spectral expansion of fractional-order Jacobi orthogonal functions. For the space discretization, the proposed schemes accommodate high-order Jacobi Galerkin spectral discretization. The numerical schemes do not require imposition of artificial smoothness assumptions in time direction as is required for most methods based on polynomial interpolation. We illustrate the flexibility of the algorithms by comparing the standard Jacobi and the fractional Jacobi spectral methods for three numerical examples. The numerical results indicate that the global character of the fractional Jacobi functions makes them well-suited to time fractional diffusion equations because they naturally take the irregular behavior of the solution into account and thus preserve the singularity of the solution.