二维欧拉程序中一种新的分界面跟踪技术

１．引言 二维流体力学计算方法按其采用Ｌａｇｒａｎｇｅ坐标系还是Ｅｕｌｅｒ坐标系,分为Ｌａｓｒａｎｇｅ方法和 Ｅｕｌｅｒ方法两大类． Ｌａｇｒａｎｇｅ坐标是随材料质团运动的,由于每个有限网格点总是与初始时同一部分流体相联系,所以拉氏程序能够清晰显示解域内部多种物质的分界向和自由界向,而且描述材料分界面的运动是自动完成的．然而在二维计算中,当材料发生大变形时,拉氏网格发生扭曲,甚至使相邻网格彼此相交和重叠,使计算过程无法进行下去．因此,Ｌａｇｒａｎｇｅ方法在二维计算中有其局限性,只能用了变形较小的模…     

A volume-of-fluid method for simulation of compressible axisymmetric multi-material flow

A two-dimensional Eulerian hydrodynamic method for the numerical simulation of inviscid compressible axisymmetric multi-material flow in external force fields for the situation of pure fluids separated by macroscopic interfaces is presented. The method combines an implicit Lagrangian step with an explicit Eulerian advection step. Individual materials obey separate energy equations, fulfill general equations of state, and may possess different temperatures. Material volume is tracked using a piecewise linear volume-of-fluid method. An overshoot-free logically simple and economic material advection algorithm for cylinder coordinates is derived, in an algebraic formulation. New aspects arising in the case of more than two materials such as the material ordering strategy during transport are presented. One- and two-dimensional numerical examples are given.     

The search for stable finite element methods for simple relativistic systems

Three finite element methods are applied to the describing static, spherically symmetric fluid bodies in general relativity. The usual Ritz method with linear splines is unconditionally unstable, although the instability is mild. As most Ritz and Galerkin methods will produce similar unstable methods a weighted residual method with cubic Lagrange splines is tried. This again proves unstable, and it seems that this approximation is inherently inappropriate for hyperbolic equations. However, the cubic Hermite spline approximation, with some modification, proves to be both A-stable and S-Stable, and is accurate to order h³.     

Eulerian Based Interpolation Schemes for Flow Map Construction and Line Integral Computation with Applications to Lagrangian Coherent Structures Extraction

We propose and analyze a new class of Eulerian methods for constructing both the forward and the backward flow maps of sufficiently smooth dynamical systems. These methods improve previous Eulerian approaches so that the computations of the forward flow map can be done on the fly as one imports or measures the velocity field forward in time. Similar to typical Lagrangian or semi-Lagrangian methods, the proposed methods require an interpolation at each step. Having said that, the Eulerian method interpolates d components of the flow maps in the d dimensional space but does not require any \((d+1)\)-dimensional spatial-temporal interpolation as in the Lagrangian approaches. We will also extend these Eulerian methods to compute line integrals along any Lagrangian particle. The paper gives a computational complexity analysis and an error estimate of these Eulerian methods. The method can be applied to a wide range of applications for flow map constructions including the finite time Lyapunov exponent computations, the coherent ergodic partition, and high frequency wave propagations using geometric optic.     

The finite mass mesh method

The finite mass method is a purely Lagrangian scheme for the spatial discretisation of the macroscopic phenomenological laws that govern the flow of compressible fluids. In this article we investigate how to take into account long range gravitational forces in the framework of the finite mass method. This is achieved by incorporating an extra discrete potential energy of the gravitational field into the Lagrangian that underlies the finite mass method. The discretisation of the potential is done in an Eulerian fashion and employs an adaptive tensor product mesh fixed in space, hence the name finite mass mesh method for the new scheme. The transfer of information between the mass packets of the finite mass method and the discrete potential equation relies on numerical quadrature, for which different strategies will be proposed. The performance of the extended finite mass method for the simulation of two-dimensional gas pillars under self-gravity will be reported. Communicated by: G. Wittum     

New cubic reference table based image steganography

Tradition reference table based (RTB) methods employ two pixels to conceal a secret digit according to a two-dimensional reference table. In this paper, x-dimensional reference table framework is defined, which can be regarded as generalization of the prior works. The available pixels to be embedded are limited in two-dimensional space, so the proposed methods extend the dimensional space of reference table. Theoretical analyses are given to justify the effectiveness of the proposed construction. Two novel RTB methods named CRT (cubic reference table) and CRT-PVD (cubic reference table and pixel value differencing) are presented. The former is independent of image contents while embedding, and the latter is depended on the discriminated image smoothness. Experimental results show that two proposed methods can achieve better performance compared with the prior works.

     

An Algebraic Multigrid Method for Higher-order Finite Element Discretizations

In this paper, we will design and analyze a class of new algebraic multigrid methods for algebraic systems arising from the discretization of second order elliptic boundary value problems by high-order finite element methods. For a given sparse stiffness matrix from a quadratic or cubic Lagrangian finite element discretization, an algebraic approach is carefully designed to recover the stiffness matrix associated with the linear finite element disretization on the same underlying (but nevertheless unknown to the user) finite element grid. With any given classical algebraic multigrid solver for linear finite element stiffness matrix, a corresponding algebraic multigrid method can then be designed for the quadratic or higher order finite element stiffness matrix by combining with a standard smoother for the original system. This method is designed under the assumption that the sparse matrix to be solved is associated with a specific higher order, quadratic for example, finite element discretization on a finite element grid but the geometric data for the underlying grid is unknown. The resulting new algebraic multigrid method is shown, by numerical experiments, to be much more efficient than the classical algebraic multigrid method which is directly applied to the high-order finite element matrix. Some theoretical analysis is also provided for the convergence of the new method.     

Lagrange–Galerkin method for unsteady free surface water waves

We present a new numerical technique to approximate solutions to unsteady free surface flows modelled by the two-dimensional shallow water equations. The method we propose in this paper consists of an Eulerian–Lagrangian splitting of the equations along the characteristic curves. The Lagrangian stage of the splitting is treated by a non-oscillatory modified method of characteristics, while the Eulerian stage is approximated by an implicit time integration scheme using finite element method for spatial discretization. The combined two stages lead to a Lagrange–Galerkin method which is robust, second order accurate, and simple to implement for problems on complex geometry. Numerical results are shown for several test problems with different ranges of difficulty.     

Response of structures to planar blast loads – A finite element engineering approach

Design and validation of structures against blast loads are important for modern society in order to protect and secure its citizen. Since it is a challenge to validate and optimise protective structures against blast loads using full-scale experimental tests, we have to turn our attention towards advanced numerical tools like the finite element method. Several different finite element techniques can be used to describe the response of structures due to blast loads. Some of these are: (1) a pure Lagrangian formulation, (2) an initial Eulerian simulation (to determine the load) followed by a Lagrangian simulation (for the structural response) and (3) a hybrid technique that combines the advantages of Eulerian and Lagrangian methods to have a full coupling between the blast waves and the deformation of the structure. Ideally, all blast simulations should be carried out using the fully coupled Eulerian–Lagrangian approach, but this may not be practical as the computational time increases considerably when going from a pure Lagrangian to a fully coupled Eulerian–Lagrangian simulation. A major goal in this study is to investigate if a pure Lagrangian formulation can be applied to determine the structural response in a specified blast load problem or if more advanced approaches such as the fully coupled Eulerian–Lagrangian approach is required for reliable results. This is done by conducting numerical simulations of an unprotected 20 ft ISO container exposed to a blast load of 4000 kg TNT at 120 m standoff distance using the three different approaches presented above. To validate and discuss the results, the simulated response of the container is compared to available data from a full-scale blast test under such conditions.     

基于FPGA的高速高质量图像旋转

为了进行高质量、高速的图像旋转变换 ,通过对传统图像旋转矩阵的分解 ,将图像在二维空间中的旋转运算分解成为三次一维空间内的平移运算 ,从而将用于图像旋转运算的二维插值运算简化为在一维空间中进行的一维插值运算。为了保证图像旋转后的质量 ,采用 3阶 B-样条对每次平移后像素点的灰度值进行插值运算 ,并提出了一种基于 IIR和 FIR数字滤波器的 3阶 B-样条插值法的高速实现方案 ;最后针对 2 5 6灰度级 ,2 5 6× 2 5 6像素的图像设计出一种基于 FPGA的高速、高质量的硬件图像旋转及显示系统     

A new operator splitting method for the numerical solution of partial differential equations II

We extend the applications of a new method for splitting operators in partial differential equations introduced by us (A. Rouhi and J. Wright, A new operator splitting method for the numerical solution of partial differential equations, Comput. Phys. Commun. 85 (1995) 18–28, and Spectral implementation of a new operator splitting method for solving partial differential equations, Comput. Phys. (1995), to be published.) to equations in two spatial dimensions, and show how the method allows the use of explicit time stepping methods in some instances when other methods require implicit time stepping. This odd-even splitting method also enables one to increase the order of accuracy of time stepping in a straightforward manner. Our main examples will be the two-dimensional Navier-Stokes equations and the shallow water equations. In the first example we show how the pressure term can be dealt with in simple geometries. We will then discuss the treatment of the diffusion term. Next we will discuss how fast waves can be treated by explicit methods using the odd-even splitting, while retaining all stability and accuracy advantages of usual implicit methods. Our example here will be the shallow water equations in two dimensions.     

Cubic Bézier approximation of a digitized curve

In this paper we present an efficient technique for piecewise cubic Bézier approximation of digitized curve. An adaptive breakpoint detection method divides a digital curve into a number of segments and each segment is approximated by a cubic Bézier curve so that the approximation error is minimized. Initial approximated Bézier control points for each of the segments are obtained by interpolation technique i.e. by the reverse recursion of De Castaljau's algorithm. Two methods, two-dimensional logarithmic search algorithm (TDLSA) and an evolutionary search algorithm (ESA), are introduced to find the best-fit Bézier control points from the approximate interpolated control points. ESA based refinement is proved to be better experimentally. Experimental results show that Bézier approximation of a digitized curve is much more accurate and uses less number of points compared to other approximation techniques.     

Implementing surfactant mass balance in 2D FEM�CALE models

Practical aspects of implementing surfactant mass balance computation in finite elements models, where the model geometry shape change is captured by utilizing the arbitrary Lagrange–Eulerian method are discussed briefly. The discussion and the reported simulations are carried out in two-dimensional Cartesian coordinates. Two alternative approaches to formulating the governing equation of surfactant mass balance for solving it computationally are presented and discussed. One of the approaches is based on computing the boundary curvature and boundary tangential velocity, as well as their differentials on the boundary, directly. The other approach is based on reformulating the governing equation in order to track the proportional rate of change of local surface area. As a conclusion, it is found that though both of the presented approaches can be configured to perform adequately in terms of surfactant mass conservation, surface differentials that are necessary to compute the surface curvature and surface tangential velocity in the first one of the methods evoke numerical oscillations near those points of the boundary where it is not smooth. The text is accompanied by example simulations and figures.     

An efficient motion magnification system for real-time applications

The human eye cannot see subtle motion signals that fall outside human visual limits, due to either limited resolution of intensity variations or lack of sensitivity to lower spatial and temporal frequencies. Yet, these invisible signals can be highly informative when amplified to be observable by a human operator or an automatic machine vision system. Many video magnification techniques have recently been proposed to magnify and reveal these signals in videos and image sequences. Limitations, including noise level, video quality and long execution time, are associated with the existing video magnification techniques. Therefore, there is value in developing a new magnification method where these issues are the main consideration. This study presents a new magnification method that outperforms other magnification techniques in terms of noise removal, video quality at large magnification factor and execution time. The proposed method is compared with four methods, including Eulerian video magnification, phase-based video magnification, Riesz pyramid for fast phase-based video magnification and enhanced Eulerian video magnification. The experimental results demonstrate the superior performance of the proposed magnification method regarding all video quality metrics used. Our method is also 60–70% faster than Eulerian video magnification, whereas other competing methods take longer to execute than Eulerian video magnification.     

Performance assessment of OpenFOAM and FLOW-3D in the numerical modeling of a low Reynolds number hydraulic jump

A comparative performance analysis of the CFD platforms OpenFOAM and FLOW-3D is presented, focusing on a 3D swirling turbulent flow: a steady hydraulic jump at low Reynolds number. Turbulence is treated using RANS approach RNG k-ε. A Volume Of Fluid (VOF) method is used to track the air–water interface, consequently aeration is modeled using an Eulerian–Eulerian approach. Structured meshes of cubic elements are used to discretize the channel geometry. The numerical model accuracy is assessed comparing representative hydraulic jump variables (sequent depth ratio, roller length, mean velocity profiles, velocity decay or free surface profile) to experimental data. The model results are also compared to previous studies to broaden the result validation. Both codes reproduced the phenomenon under study concurring with experimental data, although special care must be taken when swirling flows occur. Both models can be used to reproduce the hydraulic performance of energy dissipation structures at low Reynolds numbers.     

Collocation and galerkin finite element methods for viscoelastic fluid flow—I. : Description of method and problems with fixed geometry

Collocation and Galerkin finite element methods are developed for viscoelastic fluid flow in a fixed geometry. The collocation methods use Hermite cubic polynomials with a global coordinate transformation to permit irregular geometry. The Galerkin method uses isoparametric elements (transformed element by element) with bilinear polynomials for pressure and quadratic polynomials for velocity. Both methods are applied to two-dimensional flow in planar geometry and the Galerkin method is applied to axisymmetric cylindrical geometries as well. The fluid model is a nonlinear Maxwell model but is limited to small elastic components.

The two methods are applied to several test problems. Entry-length problems test the ability to model pressure singularities are velocity discontinuities. Stick-slip problems test the ability to model pressure singularities and stress discontinuities. Both test problems have analytic or accurate numerical solutions for Newtonian fluids so that the accuracy of the two methods is compared.     

三次样条构造的伪逆解法

为了提高三次样条构造的可行性, 基于矩阵的伪逆方法, 提出一种不依赖额外约束条件的三次样条构造的伪逆解法。该解法通过求解出三次样条二阶导数的最小范数解, 从而较好地构造出三次样条函数。理论分析及数值实验结果表明该三次样条构造的伪逆解法具有简单、有效等特点。综合分析各种构造解法的性质, 对各种三次样条构造解法进行归类比较, 为在实际工程计算应用中选择合适的三次样条构造解法提供了指导方向。     

Toward design optimization of a Pelton turbine runner

The objective of the present paper is to propose a strategy to optimize the performance of a Pelton runner based on a parametric model of the bucket geometry, massive particle based numerical simulations and advanced optimization strategies to reduce the dimension of the design problem. The parametric model of the Pelton bucket is based on four bicubic Bézier patches and the number of free parameters is reduced to 21. The numerical simulations are performed using the finite volume particle method, which benefits from a conservative, consistent, arbitrary Lagrangian Eulerian formulation. The resulting design problem is of High-dimension with Expensive Black-box (HEB) performance function. In order to tackle the HEB problem, a preliminary exploration is performed using 2’000 sampled runners geometry provided by a Halton sequence. A cubic multivariate adaptive regression spline surrogate model is built according to the simulated performance of these runners. Moreover, an original clustering approach is proposed to decompose the design problem into four sub-problems of smaller dimensions that can be addressed with more conventional optimization techniques.     

A Roe scheme for the Bi-temperature model of magnetohydrodynamics

In this paper, a Roe scheme for the bi-temperature magnetohydrodynamics (MHD) model is set up. A Roe matrix is obtained for the one-dimensional system in Eulerian coordinates. These results are extended to the two-dimensional case. One- and two-dimensional numerical examples are provided, showing Roe solver efficiency.