液体通过滑阀的阶跃响应的仿真研究

精确地了解滑阀的动态特性对液压控制系统的合理设计是很重要的。可是，滑阀的瞬变流体结构由于其非稳定性的复杂所以有关的信息较少。该文对流体通过滑阀的阶跃响应进行了数值解析计算。该研究所用的数值解析方法是先用等间隔的交错矩形网格。基于有限体积法将基础方程式离散化，然后用类似于SIMPLER法的迭代算法解离散化所得的差分方程组。压力差阶跃变化的情况下，做出了流动场随时间变化的流线图和流量及再附着点距离的时间变化图。数值计算结果说明了流体通过滑阀的阶跃响应可用具有不同的时间常数的指数函数来模拟。该文证实了理论模型和数值计算结果一致。     

通过液压元件的瞬流特性的仿真研究

液流通过节流孔和滑阀的过渡过程进行了数值解析。压力差为阶跃变化的情况下做出了节流孔和滑阀的流动场随时间变化的流线图。数值计算的结果说明了滑阀内流量的时间响应和管节流孔一样可用具有两个不同的时间常数指数函数来表示。为了迅速地对流量的流动特性进行仿真，本文提出了简单易算的时间常数公式，证实了理论模型和数值计算结果一致。     

Numerical Modeling of Degenerate Equations in Porous Media Flow

In this paper is introduced a new numerical formulation for solving degenerate nonlinear coupled convection dominated parabolic systems in problems of flow and transport in porous media by means of a mixed finite element and an operator splitting technique, which, in turn, is capable of simulating the flow of a distinct number of fluid phases in different porous media regions. This situation naturally occurs in practical applications, such as those in petroleum reservoir engineering and groundwater transport. To illustrate the modelling problem at hand, we consider a nonlinear three-phase porous media flow model in one- and two-space dimensions, which may lead to the existence of a simultaneous one-, two- and three-phase flow regions and therefore to a degenerate convection dominated parabolic system. Our numerical formulation can also be extended for the case of three space dimensions. As a consequence of the standard mixed finite element approach for this flow problem the resulting linear algebraic system is singular. By using an operator splitting combined with mixed finite element, and a decomposition of the domain into different flow regions, compatibility conditions are obtained to bypass the degeneracy in order to the degenerate convection dominated parabolic system of equations be numerically tractable without any mathematical trick to remove the singularity, i.e., no use of a parabolic regularization. Thus, by using this procedure, we were able to write the full nonlinear system in an appropriate way in order to obtain a nonsingular system for its numerical solution. The robustness of the proposed method is verified through a large set of high-resolution numerical experiments of nonlinear transport flow problems with degenerating diffusion conditions and by means of a numerical convergence study.     

A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes

A Lagrangian numerical scheme for solving nonlinear degenerate Fokker–Planck equations in space dimensions \(d\ge 2\) is presented. It applies to a large class of nonlinear diffusion equations, whose dynamics are driven by internal energies and given external potentials, e.g. the porous medium equation and the fast diffusion equation. The key ingredient in our approach is the gradient flow structure of the dynamics. For discretization of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. Thanks to that time discretisation, the fully discrete solution inherits energy estimates from the original gradient flow, and these lead to weak compactness of the trajectories in the continuous limit. Consistency is analyzed in the planar situation, \(d=2\). A variety of numerical experiments for the porous medium equation indicates that the scheme is well-adapted to track the growth of the solution’s support.     

Diagonalization based Parallel-in-Time method for a class of fourth order time dependent PDEs

In this paper, we design, analyze and implement efficient time parallel methods for a class of fourth order time-dependent partial differential equations (PDEs), namely the biharmonic heat equation, the linearized Cahn–Hilliard (CH) equation and the nonlinear CH equation. We use a diagonalization technique on all-at-once system to develop efficient iterative time parallel methods for investigating the solution behaviour of the said equations. We present the convergence analysis of Parallel-in-Time (PinT) algorithms. We verify our findings by presenting numerical results.     

Numerical inverse isoparametric mapping in remeshing and nodal quantity contouring

In finite element analysis, isoparametric mapping defined as [(ξ, η) → (x, y): X = N_iξ_i] is widely used. It is a one-to-one mapping and its construction is especially elegant for elements of a variable number of nodes showing its versatile applicability to model curved boundaries. In certain analyses, such as remeshing in dynamic analyses or contouring and others, the inverse of this mapping is inevitably valuable, but its determination is not so straightforward. To avoid solving a system of nonlinear equations, generally an iterative technique of order N² in a two-dimensional mesh is often resorted to. In the paper, this technique is improved by systematically bisecting along a predefined line that reduces the iterations to only order N. Its applications in remeshing and nodal quantity contouring are demonstrated and a possible extension for stress contouring is also discussed. The FORTRAN subroutines for the technique proposed are also given.     

A triparametric family of three-step optimal eighth-order methods for solving nonlinear equations

A new triparametric family of three-step optimal eighth-order iterative methods free from second derivatives are proposed in this paper, to find a simple root of nonlinear equations. Convergence analysis as well as numerical experiments confirms the eighth-order convergence and asymptotic error constants.     

Parallel computational methods for 3D simulation of a parafoil with prescribed shape changes

In this paper we describe parallel computational methods for 3D simulation of the dynamics and fluid dynamics of a parafoil with prescribed, time-dependent shape changes. The mathematical model is based on the time-dependent, 3D Navier-Stokes equations governing the incompressible flow around the parafoil and Newton's law of motion governing the dynamics of the parafoil, with the aerodynamic forces acting on the parafoil calculated from the flow field. The computational methods developed for these 3D simulations include a stabilized space-time finite element formulation to accommodate for the shape changes, special mesh generation and mesh moving strategies developed for this purpose, iterative solution techniques for the large, coupled nonlinear equation systems involved, and parallel implementation of all these methods on scalable computing systems such as the Thinking Machines CM-5. As an example, we report 3D simulation of a flare maneuver in which the parafoil velocity is reduced by pulling down the flaps. This simulation requires solution of over 3.6 million coupled, nonlinear equations at every time step of the simulation.     

A projection method for solving nonsymmetric linear systems on multiprocessors

We consider the iterative solution of large sparse linear systems of equations arising from elliptic and parabolic partial differential equations in two or three space dimensions. Specifically, we focus our attention on nonsymmetric systems of equations whose eigenvalues lie on both sides of the imaginary axis, or whose symmetric part is not positive definite. This system of equation is solved using a block Kaczmarz projection method with conjugate gradient acceleration. The algorithm has been designed with special emphasis on its suitability for multiprocessors. In the first part of the paper, we study the numerical properties of the algorithm and compare its performance with other algorithms such as the conjugate gradient method on the normal equations, and conjugate gradient-like schemes such as ORTHOMIN(k), GCR(k) and GMRES(k). We also study the effect of using various preconditioners with these methods. In the second part of the paper, we describe the implementation of our algorithm on the CRAY X-MP/48 multiprocessor, and study its behavior as the number of processors is increased.     

A moving mesh approach to an ice sheet model

A moving mesh approach to the numerical modelling of problems governed by nonlinear time-dependent partial differential equations (PDEs) is applied to the numerical modelling of glaciers driven by ice diffusion and accumulation/ablation. The primary focus of the paper is to demonstrate the numerics of the moving mesh approach applied to a standard parabolic PDE model in reproducing the main features of glacier flow, including tracking the moving boundary (snout). A secondary aim is to investigate waiting time conditions under which the snout moves.     

Parallel solution of a traffic flow simulation problem

Computational Fluid Dynamics (CFD) methods for solving traffic flow continuum models have been studied and efficiently implemented in traffic simulation codes in the past. This is the first time that such methods are studied from the point of view of parallel computing. We studied and implemented an implicit numerical method for solving the high-order flow conservation traffic model on parallel computers. Implicit methods allow much larger time-step than explicit methods, for the same accuracy. However, at each time-step a nonlinear system must be solved. We used the Newton method coupled with a linear iterative method (Orthomin). We accelerated the convergence of Orthomin with parallel incomplete LU factorization preconditionings. We ran simulation tests with real traffic data from an 12-mile freeway section (in Minnesota) on the nCUBE2 parallel computer. These tests gave the same accuracy as past tests, which were performed on one-processor computers, and the overall execution time was significantly reduced.     

A Decoupled Preconditioning Technique for a Mixed Stokes–Darcy Model

We propose an efficient iterative method to solve the mixed Stokes–Darcy model for coupling fluid and porous media flow. The weak formulation of this problem leads to a coupled, indefinite, ill-conditioned and symmetric linear system of equations. We apply a decoupled preconditioning technique requiring only good solvers for the local mixed-Darcy and Stokes subproblems. We prove that the method is asymptotically optimal and confirm, with numerical experiments, that the performance of the preconditioners does not deteriorate on arbitrarily fine meshes.     

Numerical analysis of sandwich beams

This paper describes the development of a numerical model for the physical nonlinear analysis of simply supported sandwich beams, specifically with foamed-concrete cores and concrete faces. The long-term behaviour is included in view of creep and shrinkage of both faces and core. The structural behaviour of sandwich beams is described by a fourth-order differential equation in the deformation w and a second-order differential equation in the shear deformation of the core γ_k. The flexural stiffness of the core is taken into account. The general-solution procedure is based on the finite-difference method, together with a successive-substitution algorithm using the secant flexural moduli of the core and faces and the secant shear modulus of the core. The option of tension stiffening is incorporated to represent the nonlinear behaviour of reinforced concrete in tension. The tension stiffening is numerically calculated from a distributed tensile load instead of a load acting on both ends of the reinforced bar. Creep and shrinkage are calculated separately from the differential equations with an algorithm based on increments of time. With the presented model, the time-dependent deflection along the axis of the beam and the state of stress in every fibre can be calculated.     

Parameter identification of fractional-order Wiener system based on FF-ESG and GI algorithms

Fractional-order calculus has broad application scenarios in engineering and physics. Unlike integer-order calculus, fractional-order calculus has the ability to analyze nonclassical phenomena in science and engineering. For industrial processes with strong nonlinear characteristics, nonlinear models such as the Wiener model have become research hotspots. This paper studies the parameter identification of the fractional-order Wiener system. In this paper, the forgetting factor extended stochastic gradient (FF-ESG) algorithm and the gradient iterative (GI) algorithm are proposed to identify the parameters of the fractional-order Wiener system. Then, the convergence of the FF-ESG algorithm for the fractional-order Wiener system is analyzed. Both proposed algorithms can obtain exact parameter estimates, which are verified by a numerical example and a case study of a fluid control valve.     

Hierarchical gradient based iterative parameter estimation algorithm for multivariable output error moving average systems

According to the hierarchical identification principle, a hierarchical gradient based iterative estimation algorithm is derived for multivariable output error moving average systems (i.e., multivariable OEMA-like models) which is different from multivariable CARMA-like models. As there exist unmeasurable noise-free outputs and unknown noise terms in the information vector/matrix of the corresponding identification model, this paper is, by means of the auxiliary model identification idea, to replace the unmeasurable variables in the information vector/matrix with the estimated residuals and the outputs of the auxiliary model. A numerical example is provided.     

Wavelet strategy for flow and heat transfer in CNT-water based fluid with asymmetric variable rectangular porous channel

In the present work, the characteristics of physical model unsteady nanofluid flow and heat transfer in an asymmetric porous channel are analyzed numerically using wavelet collocation method. Using similarity transformation, unsteady two-dimensional flow model of nanofluid in a porous channel through expanding or contracting walls has been transformed into a system of nonlinear ordinary differential equations (ODEs). Then, the obtained nonlinear system of ODEs is solved via wavelet collocation method. The effect of various emerging parameters, such as nanoparticle volume fraction, Reynolds number (Re), and expansion ratio have been analyzed on velocity and temperature profiles. Numerical results have been presented in form of figures and tables. For some special cases, the obtained numerical results are compared with exact one and found that the results are good in agreement with exact solutions.

     

A finite element first-order equation formulation for the small-disturbance transonic flow problem

The nonlinear, mixed elliptic hyperbolic equation describing a steady transonic flow is considered. The original equation is replaced by a system of first-order equations that are hyperbolic in time and defined in terms of velocity components. Parabolic regularization terms are added to capture shock wave solutions and to damp iterative solution algorithms. A finite element Galerkin method in space and a Crank-Nicolson finite difference method in iterative time are used to reduce the problem to the solution of a system of algebraic equations. Stability and convergence characteristics of the iterative method are discussed. The numerical implementation of the method is explained, and numerical results are presented.     

Shape design optimization of stationary fluid-structure interaction problems with large displacements and turbulence

This paper presents general and efficient methods for analysis and gradient based shape optimization of systems characterized as strongly coupled stationary fluid-structure interaction (FSI) problems. The incompressible fluid flow can be laminar or turbulent and is described using the Reynolds-averaged Navier-Stokes equations (RANS) together with the algebraic Baldwin–Lomax turbulence model. The structure may exhibit large displacements due to the interaction with the fluid domain, resulting in geometrically nonlinear structural behaviour and nonlinear interface coupling conditions. The problem is discretized using Galerkin and Streamline-Upwind/Petrov–Galerkin finite element methods, and the resulting nonlinear equations are solved using Newtons method. Due to the large displacements of the structure, an efficient update algorithm for the fluid mesh must be applied, leading to the use of an approximate Jacobian matrix in the solution routine. Expressions for Design Sensitivity Analysis (DSA) are derived using the direct differentiation approach, and the use of an inexact Jacobian matrix in the analysis leads to an iterative but very efficient scheme for DSA. The potential of gradient based shape optimization of fluid flow and FSI problems is illustrated by several examples.     

A restarted iterative homotopy analysis method for two nonlinear models from image processing

Total variation (TV) minimization-based nonlinear models have been proven to be very useful and successful in image processing. A lot of effort has been devoted to overcome the nonlinearity of the model and at the same time to obtain fast numerical schemes. In this paper, we propose a restarted iterative homotopy analysis method (HAM) to improve the computational efficiency for the TV models and will show by experiments that this method demonstrates great potential for recovering the noise and with great speed in both image denoising and image segmentation models. The method modifies the existing HAM and makes it suitable to potentially solve other nonlinear partial differential equations arising from image processing models. In our examples, we will demonstrate the validity of a restarted HAM and that this method is efficient and robust even for images with large ratios of noise and with much less CPU time than other methods.     

The Hopf bifurcation theorem for quasilinear differential-algebraic equations

We prove a generalization of the Hopf bifurcation theorem for quasilinear differential equations (DAEs), i.e. equations of the form A(μ, χ)χ = G(μ, χ) where the matrix A(χ, μ) has constant but not full rank and hence the system cannot be made into an explicit ODE. The paper includes an appendix by J. Ernsthausen addressing the numerical calculation of the Hopf points in the DAE setting.