Accelerated convergence of the numerical simulation of incompressible flow in general curvilinear co‐ordinates by discretizations on the double‐staggered grids

The convergence rate of a methodology for solving incompressible flow in general curvilinear co‐ordinates is analyzed. Double‐staggered grids (DSGs), each defined by the same boundaries as the physical domain, are used for discretization. Both grids are MAC quadrilateral meshes with scalar variables (pressure, temperature, etc.) arranged at the center and the Cartesian velocity components at the middle of the sides of the mesh cells. The problem was checked against benchmark solutions of natural convection in a squeezed cavity, heat transfer in concentric horizontal cylindrical annuli, and a hot cylinder in a duct. Poisson's pressure‐correction equations that arise from the SIMPLE‐like procedure are solved by several methods: successive overrelaxation, symmetric overrelaxation, modified incomplete factorization preconditioner, conjugate gradient (CG), and CG with preconditioner. A genetic algorithm was developed to solve problems of numerical optimization of SIMPLE‐like calculation time in a space of iteration numbers and relaxation parameters. The application provides a means of making an unbiased comparison between the DSGs method and the widely used interpolation method. Furthermore, the convergence rate was demonstrated by application to the calculation of natural convection heat transfer in concentric horizontal cylindrical annuli. Calculation times when DSGs were used were 2–10 times shorter than those achieved by interpolation. With the DSGs method, calculation time increases slightly with increasing non‐orthogonality of the grids, whereas an interpolation method calls for very small iteration parameters that lead to unacceptable calculation times. Copyright © 2007 John Wiley & Sons, Ltd.     

极坐标平面问题的四阶差分及应用

推导出了极坐标系下双调和方程的差分公式，用逐次超松弛迭代法求出圆板平面应力问题的差分解，并和解析解作比较，验证了差分公式的正确性，为解决圆域及其类似区域的平面问题提供了新方法．     

Nonstationary motion of a viscous fluid in a long tube

A number of problems are solved for the nonstationary motion of a viscous compressible fluid in a tube with elastic walls. It is assumed that the tube is semi-infinite, its axis horizontal, and that at one of its ends the flow rate of the fluid can change. The solution of each of the problems is reduced to the finding a generalized solution to a nonlinear system of partial differential equations for two functions — the mean values of the velocity and pressure in the tube section — with certain constant or null initial conditions and with a boundary condition specifying the time dependence of some function of the velocity and the pressure at the end of the tube. It is noted that the same problems can be solved by successive approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 35–43, November–December, 1980.     

Parallel Multisplitting AOR method for solving a class of system of nonlinear algebraic equations

ＰＡＲＡＬＬＥＬＭＵＬＴＩＳＰＬＩＴＴＩＮＧＡＯＲＭＥＴＨＯＤＦＯＲＳＯＬＶＩＮＧＡＣＬＡＳＳＯＦＳＹＳＴＥＭＯＦＮＯＮＬＩＮＥＡＲＡＬＧＥＢＲＡＩＣＥＱＵＡＴＩＯＮＳＢａｉＺｈｏｎｇｚｈｉ（白中治）（ＩｎｓｔｉｔｕｉｅｏｆＭａｔｈｅｍａｔｉｃｓＦｕ...     

A component framework for the parallel solution of the incompressible Navier–Stokes equations with Radau‐IIA methods

An efficient solution strategy for the simulation of incompressible fluids needs adequate and accurate space and time discretization schemes. In this paper, for the space discretization, we use an inf–sup stable finite element method and for the time discretization, Radau‐IIA methods of higher order, which have the advantage that the pressure component has convergence order s in time, where s is the number of internal stages. The disadvantage of this approach is that we have a high computational amount of work, because large nonlinear systems of equations have to solved. In this paper, we use a transformation of the coefficient matrix and the simplified Newton method. This approach has the effect that our large nonlinear systems split into smaller ones, which can now also be solved in parallel. For the parallelization of the code we use the software component technology and the Component Template Library. Numerical examples show that high order in the pressure component can be achieved and that the proposed solution technique is very effective. Copyright © 2015 John Wiley & Sons, Ltd.     

AN EFFICIENT SOLUTION FOR 2D RAYLEIGH-BÉNARD CONVECTION USING FFT

研究提高二维方腔瑞利-贝纳德对流 直接数值模拟求解方法的计算效率问题.对于非定常湍流热对流, 压力泊松方程的求解是影响整个计算效率的关键. 利用快速傅里叶变换(fast Fourier transform,FFT)解耦并结合追赶法, 可实现压力泊松方程的直接求解.通过与跳点超松弛迭代法在求解精度和计算速度对比, 可以看到, 利用FFT压力泊松方程直接方法计算热对流问题是高效的.还给出了典型状态的热对流初始羽流和大尺度环流温度场, 以及系列瑞利数(Ra)计算结果的宏观传热努塞数(Nu)变化.     

A new numerical technique for the analysis of parametrically excited nonlinear systems

A new computational scheme using Chebyshev polynomials is proposed for the numerical solution of parametrically excited nonlinear systems. The state vector and the periodic coefficients are expanded in Chebyshev polynomials and an integral equation suitable for a Picard-type iteration is formulated. A Chebyshev collocation is applied to the integral with the nonlinearities reducing the problem to the solution of a set of linear algebraic equations in each iteration. The method is equally applicable for nonlinear systems which are represented in state-space form or by a set of second-order differential equations. The proposed technique is found to duplicate the periodic, multi-periodic and chaotic solutions of a parametrically excited system obtained previously using the conventional numerical integration schemes with comparable CPU times. The technique does not require the inversion of the mass matrix in the case of multi degree-of-freedom systems. The present method is also shown to offer significant computational conveniences over the conventional numerical integration routines when used in a scheme for the direct determination of periodic solutions. Of course, the technique is also applicable to non-parametrically excited nonlinear systems as well.     

Buckling experiments on stiffened cast-epoxy conical shells

This paper describes a casting technique for fabricating high-quality plastic structural models and presents results on the use of such specimens to parametrically study the effect of base-ring stiffness on the critical buckling pressure of a ring-stiffened conical shell. The fabrication technique involves machining a metal mold to the desired configuration and vacuum drawing the plastic material into the mold. A room-temperature-curing translucent thermoset epoxy was the casting material selected. The casting technique allows many high-quality specimens to be produced and each specimen is capable of being repeatedly tested without failure. The conical shell was modified for successive tests by machining the epoxy base-ring configuration to reduce its stiffness. A shell-of-revolution computer program which uses a nonlinear axisymmetric prebuckling strain field to obtain a bifurcation-buckling solution was used to guide the selection of configurations tested. The shell experimentally exhibited asymmetric collapse behavior and the ultimate load was considerably higher than the analyticalbifurcation prediction. The asymmetric buckling-mode shape, however, initially appeared at a pressure near the analysis-bifurcation solution. Comparison of experimental and analytical prebuckling strains at pressure magnitudes below the initiation of asymmetric collapse showed good agreement.     

基于Wilson-θ和Newmark-β法的非线性动力学方程改进算法

Wilson-θ法和Newmark-β法是非线性动力学方程求解的常用方法。它们的一个基本步骤是,将方程改写为增量平衡的形式,在每一个积分步长内用状态参量修正平衡方程的系数矩阵,其本质是在单个步长内对系统的非线性环节进行了线性化处理。本文基于增量思想分别改进了Wilson-θ法和Newmark-β法,根据即时解给出下一步的猜测解,然后对猜测解进行迭代校正,最终得到收敛的近似解。算例表明,改进算法的精度更高,且收敛准则简单。更为重要的是,本文方法无须对非线性项进行线性化处理,因而计算效率更高,适应范围更广。     

Mathematical model of two-phase fluid nonlinear flow in low-permeability porous media with applications

IntroductionItisasuccessfulexampleinadevelopmentstoryofscienceandtechnologyformechanicsoffluidsinporousmediatocombinewithengineeringtechnology .Fieldsinfluencedbythemechanicsinvolveddevelopmentofoil_gasandgroundwaterresources,controlonseawaterintrusionandsubsidenceandgeologichazards,geotechnicalengineeringandbioengineering ,andairlineindustry[1～ 7].Aproblemonnonlinearflowinlow_permeabilityporousmediaisbutonlyabasiconeindifferentkindsofengineeringfields,butalsooneoffrontlineresearchfieldsofmod…     

Predicting solutions of the stochastic fractional order dynamical system using machine learning

The solution of fractional-order systems has been a complex problem for our research. Traditional methods like the predictor-corrector method and other solution steps are complicated and cumbersome to derive, which makes it more difficult for our solution efficiency. The development of machine learning and nonlinear dynamics has provided us with new ideas to solve some complex problems. Therefore, this study considers how to improve the accuracy and efficiency of the solution based on traditional methods. Finally, we propose an efficient and accurate nonlinear auto-regressive neural network for the fractional order dynamic system prediction model (FODS-NAR). First, we demonstrate by example that the FODS-NAR algorithm can predict the solution of a stochastic fractional order system. Second, we compare the FODS-NAR algorithm with the famous and good reservoir computing (RC) algorithms. We find that FODS-NAR gives more accurate predictions than the traditional RC algorithm with the same system parameters, and the residuals of the FODS-NAR algorithm are closer to 0. Consequently, we conclude that the FODS-NAR algorithm is a method with higher accuracy and prediction results closer to the state of fractional-order stochastic systems. In addition, we analyze the effects of the number of neurons and the order of delays in the FODS-NAR algorithm on the prediction results and derive a range of their optimal values.     

Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 1: methodology and numerical procedure

A methodology for computing three‐dimensional interaction between waves and fixed bodies is developed based on a fully non‐linear potential flow theory. The associated boundary value problem is solved using a finite element method (FEM). A recovery technique has been implemented to improve the FEM solution. The velocity is calculated by a numerical differentiation technique. The corresponding algebraic equations are solved by the conjugate gradient method with a symmetric successive overrelaxation (SSOR) preconditioner. The radiation condition at a truncated boundary is imposed based on the combination of a damping zone and the Sommerfeld condition. This paper (Part 1) focuses on the technical procedure, while Part 2 [Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 2. Numerical results and validation. International Journal for Numerical Methods in Fluids 2001] gives detailed numerical results, including validation, for the cases of steep waves interacting with one or two vertical cylinders. Copyright © 2001 John Wiley & Sons, Ltd.     

A direct analysis of nonlinear systems with external periodic excitations via normal forms

This study presents a direct methodology for a quantitative analysis of nonlinear dynamic systems with external periodic forcing via an application of the theory of normal forms. Rather than introducing a new state variable to reduce the problem to a homogeneous one, a set of time-dependant near-identity transformations is applied to construct the normal forms. In the process, the total response of the system is expressed as superposition of a steady state solution and a transient solution. A steady state solution of the system is obtained by the method of harmonic balance and the transient solution is obtained by solving a set of time periodic homological equations. The proposed method can be applied to time-invariant as well as time varying systems. After discussing the time-invariant case, the methodology is extended to systems with time-periodic coefficients. The case of time periodic systems is handled through an application of the Lyapunov–Floquet (L–F) transformation. Application of the L–F transformation produces a dynamically equivalent system in which the linear part of the system is time-invariant, making the system amenable to near-identity transformations. An example for each type of system, namely, constant coefficients and time-varying coefficients, is included to demonstrate effectiveness of the method. Various resonance conditions are discussed. It is observed that the linear parametric excitation term need not be small as generally assumed in perturbation and averaging techniques. Results obtained by proposed methods are compared with numerical solutions. Close agreements are found in some typical applications.     

DYNAMICAL APPROACH STUDY OF SPURIOUS STEADY-STATE NUMERICAL SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS II. GLOBAL ASYMPTOTIC BEHAVIOR OF TIME DISCRETIZATIONS ∗

SUMMARY

The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 × 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how “numerical” basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DCs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless ofthe stability ofthe spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.     

Formulation and Solution of Dynamic Problems of Elastic Rod Systems Subjected to Boundary Conditions Described by Multivalued Relations

The dynamic behavior of rod systems under the action of external force factors described by multivalued (subdifferential) relations is studied. The mathematical formulation of the problem is given in the form of a dynamic quasivariational inequality. With the use of the Newmark difference scheme, successive approximations, and finite-element discretization, the problem is reduced to minimization of a convex nonsmooth finite-dimensional functional with respect to velocities at each time step. Introduction of auxiliary variables by the method of a modified Lagrangian reduces the problem of minimization of this functional to a sequence of smooth problems of nonlinear programming. The algorithm is verified using the numerical solution for a problem with one degree of freedom. The algorithm proposed is used to calculate the rods of deep-well pumps.     

Nongrey radiant heat transfer corrections to thermal conductivity measurements

A numerical investigation is presented to determine the influence of radiative transfer on experimental thermal conductivity measurements. The analysis is simplified by approximating the thermal conductivity test cell configuration by a parallel plate model. The problem is rigorously formulated in terms of a nonlinear integrodifferential equation, and the solution is obtained by the method of successive approximations. Both grey and nongrey results are presented. For water vapor at atmospheric pressure and 700 ?K, it is shown that radiative transfer between black surfaces may affect thermal conductivity measurements by 35 percent.     

Invariant solutions of the Westervelt model of nonlinear hydroacoustics without dissipation

We study three-dimensional Westervelt model of a nonlinear hydroacoustics without dissipation. We received all of its invariant submodels. We studied all invariant submodels described by the invariant solutions of rank 0 and 1. All invariant solutions of rank 0 and 1 are found either explicitly, or their search is reduced to the solution of the nonlinear integral equations. With a help of these invariant solutions we researched: (1) a propagation of the intensive acoustic waves (self-similar, axisymmetric, planar and one-dimensional) for which the acoustic pressure and a speed of its change, or the acoustic pressure and its derivative in the direction of one of the axes are specified at the initial moment of the time at a fixed point , (2) a spherically symmetric ultrasonic field for which the acoustic pressure and a speed of its change, or the acoustic pressure and its radial derivative are specified at the initial moment of the time at a fixed point. Solving of the boundary value problems describing these processes is reduced to the solving of nonlinear integral equations. We are established the existence and uniqueness of solutions of these boundary value problems under some additional conditions. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters. We found all the conservation laws of the first order for the Westerveld equation written in dimensionless variables.     

A Study of the Behavior of Implicit Pressure Explicit Saturation (IMPES) Schedules for Two-phase Flow in Dynamic Pore Network Models

We focus on the numerical difficulties that typify implicit pressure explicit saturation (IMPES) schedules in dynamic “ball-and-stick” pore network models for two-phase flow. We show that a time stepping procedure based on a prescribed maximum variation of the local capillary pressure rather than on a (usual) maximum variation of the local saturation along with the addition in the solution algorithm of suitable “flow constraints” (in Koplik and Lasseter, Soc. Pet. Eng. J. 25(1):89–100, 1985) provide more stability and a significant run time speed up. In particular, the slow convergence and the oscillatory behavior that typify IMPES schemes at low Ca values due to capillary pinning are efficiently suppressed.     

Exact solutions for nonlinear transient flow model including a quadratic gradient term

The models of the nonlinear radial flow for the infinite and finite reservoirs including a quadratic gradient term were presented. The exact solution was given in real space for flow equation including quadratic gradiet term for both constant-rate and constant pressure production cases in an infinite system by using generalized Weber transform.Analytical solutions for flow equation including quadratic gradient term were also obtained by using the Hankel transform for a finite circular reservoir case. Both closed and constant pressure outer boundary conditions are considered. Moreover, both constant rate and constant pressure inner boundary conditions are considered. The difference between the nonlinear pressure solution and linear pressure solution is analyzed. The difference may be reached about 8% in the long time. The effect of the quadratic gradient term in the large time well test is considered.     

On the existence of general solutions of the initial-value problem for the nonlinear Boltzmann equation with a cut-off

We consider a Boltzmann gas which fills all of space and is under the influence of a field of conservative external force whose potential is bounded from below.Assuming the intermolecular force has a cut-off, we prove existence and uniqueness for the general solution of the nonlinear Maxwell-Boltzmann equation at least in a finite interval of time. The solution can be constructed by the method of successive approximations in corresponding complete spaces. Definitions of these spaces are connected with an exponential function of the total energy of the molecule.Some indications of future generalizations and investigations are given.