求解带有间断系数泊松方程的修正有限体积

利用修正的有限体积方法求解带有间断系数的泊松方程,改进是对基于笛卡尔坐标系下的调和平均系数进行的。数值实验表明新格式二阶逐点收敛并且在界面处具有二阶精度,新方法较已有的求解不连续扩散系数的算术平均法和调和平均法,特别是在系数跳跃较大的情况下更具优势。     

Unsteady development of a deformable bubble rising in a quiescent liquid

An accurate finite-volume based numerical method for the simulation of an isothermal two-phase flow, consisting of a rising deformable bubble translating in a quiescent, unbounded liquid, is presented. This direct simulation method is built on a sharp interface concept and developed on an Eulerian, Cartesian fixed-grid with a cut-cell scheme and marker points to track the moving interface. The unsteady Navier–Stokes equations in both liquid and gas phases are solved separately. The mass continuity and momentum flux conditions are explicitly matched at the true surface phase boundary to determine the evolving interface shape and movement of the bubble. The highlights of this method are that it utilizes a combined Eulerian–Lagrangian approach, and is capable of treating the interface as a sharp discontinuity. A fixed underlying grid is used to represent the control volume. The interface, however, is denoted by a separate set of marker particles which move along with the interface. A quadratic curve fitting algorithm with marker points is used to yield smooth and accurate information of the interface curvatures. This numerical scheme can handle a wide range of density and viscosity ratios. The bubble is assumed to be spherical and at rest initially, but deforms as it rises through the liquid pool due to buoyancy. Additionally, the flow is assumed to be axisymmetric and incompressible. The bubble deformation and dynamic motion are characterized by the Reynolds number, the Weber number, the density ratio and the viscosity ratio. The effects of these parameters on the translational bubble dynamics and shape are given and the physical mechanisms are explained and discussed. Results for the shape, velocity profile and various forces acting on the bubble are presented here as a function of time until the bubble reaches terminal velocity. The range of Reynolds numbers investigated is 1 < Re < 100, and that of Weber number is 1 < We < 10.     

A direct numerical simulation of axisymmetric cryogenic two-phase flows in a pipe with phase change

An accurate finite-volume based numerical method for the simulation of a cryogenic two-phase flow with phase change heat transfer, consisting of a saturated liquid slug translating in its own superheated vapor in a circular pipe is presented. This method is built on a sharp interface concept and developed on an Eulerian Cartesian fixed-grid with a cut-cell scheme and marker points to track the moving interface. The unsteady, axisymmetric Navier–Stokes equations in both liquid and vapor phases are solved separately. The mass continuity, momentum flux conditions and conservation of energy are explicitly matched at the true boundary between the two phases to determine the interface shape and movement. A quadratic curve fitting algorithm with marker points is used to yield smooth and accurate information of the interface curvatures.It is uniquely demonstrated for the first time with the current method that conservation of mass is strictly enforced for continuous infusion of flow into the domain of computation. The method has been used to compute the velocity, pressure and temperature fields and the deformation of the liquid core. It is also shown that the current method is capable of producing accurate results for a wide range of Reynolds number, Re, Weber number, We, Jakob number, Ja and large property jumps at the interface.     

A conservative and monotone mixed-hybridized finite element approximation of transport problems in heterogeneous domains

In this article, we discuss the numerical approximation of transport phenomena occurring at material interfaces between physical subdomains with heterogenous properties. The model in each subdomain consists of a partial differential equation with diffusive, convective and reactive terms, the coupling between each subdomain being realized through an interface transmission condition of Robin type. The numerical approximation of the problem in the two-dimensional case is carried out through a dual mixed-hybridized finite element method with numerical quadrature of the mass flux matrix. The resulting method is a conservative finite volume scheme over triangular grids, for which a discrete maximum principle is proved under the assumption that the mesh is of Delaunay type in the interior of the domain and of weakly acute type along the domain external boundary and internal interface. The stability, accuracy and robustness of the proposed method are validated on several numerical examples motivated by applications in biology, electrophysiology and neuroelectronics.     

An augmented approach for Stokes equations with a discontinuous viscosity and singular forces

For Stokes equations with a discontinuous viscosity across an arbitrary interface or/and singular forces along the interface, it is known that the pressure is discontinuous and the velocity is non-smooth. It has been shown that these discontinuities are coupled together, which makes it difficult to obtain accurate numerical solutions. In this paper, a new numerical method that decouples the jump conditions of the fluid variables through two augmented variables has been developed. The GMRES iterative method is used to solve the Schur complement system for the augmented variables that are only defined on the interface. The augmented approach also rescales the Stokes equations in such a way that a fast Poisson solver can be used in each iteration. Numerical tests using examples that have analytic solutions show that the new method has average second order accuracy for the velocity in the infinity norm. An example of a moving interface problem is also presented.     

A direct-forcing immersed boundary method for the thermal lattice Boltzmann method

In this study, a direct-forcing immersed boundary method (IBM) for thermal lattice Boltzmann method (TLBM) is proposed to simulate the non-isothermal flows. The direct-forcing IBM formulas for thermal equations are derived based on two TLBM models: a double-population model with a simplified thermal lattice Boltzmann equation (Model 1) and a hybrid model with an advection–diffusion equation of temperature (Model 2). As an interface scheme, which is required due to a mismatch between boundary and computational grids in the IBM, the sharp interface scheme based on second-order bilinear and linear interpolations (instead of the diffuse interface scheme, which uses discrete delta functions) is adopted to obtain the more accurate results. The proposed methods are validated through convective heat transfer problems with not only stationary but also moving boundaries – the natural convection in a square cavity with an eccentrically located cylinder and a cold particle sedimentation in an infinite channel. In terms of accuracy, the results from the IBM based on both models are comparable and show a good agreement with those from other numerical methods. In contrast, the IBM based on Model 2 is more numerically efficient than the IBM based on Model 1.     

A moving-grid finite element algorithm for supersonic flow interaction between moving bodies

The present study describes a finite element scheme which provides for the computer simulation of the motion of one finite element mesh relative to another along a smooth interface through which flow occurs. The scheme is designed to simulate transient flow interaction phenomena encountered in situations in which two or more solid bodies move relative to one another through an arbitrary flow field. Particular applications include simulations of rotor-stator flow interactions in turbomachinery. Although the method is applicable to any type of flow interaction problem, the applications considered here focus on the modeling of supersonic flow interactions characterized by the time-dependent Euler equations, and on problems of pure advection of a scalar field, all in two-dimensional flow domains. Several numerical examples are given to demonstrate the robustness and the performance of the numerical scheme.     

High Order Accurate Finite Difference Modeling of Seismo-Acoustic Wave Propagation in a Moving Atmosphere and a Heterogeneous Earth Model Coupled Across a Realistic Topography

We develop a numerical method for simultaneously simulating acoustic waves in a realistic moving atmosphere and seismic waves in a heterogeneous earth model, where the motions are coupled across a realistic topography. We model acoustic wave propagation by solving the linearized Euler equations of compressible fluid mechanics. The seismic waves are modeled by the elastic wave equation in a heterogeneous anisotropic material. The motion is coupled by imposing continuity of normal velocity and normal stresses across the topographic interface. Realistic topography is resolved on a curvilinear grid that follows the interface. The governing equations are discretized using high order accurate finite difference methods that satisfy the principle of summation by parts. We apply the energy method to derive the discrete interface conditions and to show that the coupled discretization is stable. The implementation is verified by numerical experiments, and we demonstrate a simulation of coupled wave propagation in a windy atmosphere and a realistic earth model with non-planar topography.     

An ISPH scheme for numerical simulation of multiphase flows with complex interfaces and high density ratios

Multiphase problems with high density ratios and complex interfaces deal with numerical instabilities and require accurate considerations for capturing the multiphase interfaces. An Incompressible Smoothed Particle Hydrodynamics (ISPH) scheme is presented to simulate such problems. In order to keep the present scheme simple and stable, well-established formulations are used for discretizing the spatial derivatives and a repulsive force is applied at the multiphase interface between particles of different fluids to maintain the interface sharpness. Special considerations are included to overcome the difficulties to model severe physical discontinuities at the interface and surface tension effects are taken into account. Different particle shifting schemes are also tested for a range of problems. Several two phase flows are investigated and the presented scheme is validated against both analytical and numerical solutions. A detailed study is also carried out on the influence of the repulsive force in an ISPH scheme showing that this simple treatment efficiently enhances the interface capturing features. The comparisons indicate that the proposed scheme is robust and capable of simulating a wide range of multiphase problems with complex interfaces including low to high ratios for density and viscosity.     

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.     

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.     

Stratified smooth two-phase flow using the immersed interface method

The immersed interface method is used to derive a numerical method for solving fully developed, stratified smooth two-phase flow in pipes. This sharp interface technique makes the representation of the interface independent of the grid structure, and it allows for using an arbitrary shaped interface. The two-dimensional steady-state axial momentum equation is discretized and solved using a finite difference scheme on a composite, overlapping grid with local grid refinement near the interface and near the pipe wall. A low Reynolds number k-ε turbulence model is adopted to account for the effect of turbulence. A level set function is used to represent the interface. Numerical results are presented for laminar and turbulent flows. The numerical method compares well with analytical solution for laminar flow, and it shows acceptable agreement with experimental data for turbulent flow. A few examples are given to demonstrate the capability of the method to solve flow problems with a complex shaped interface.     

Accurate adaptive level set method and sharpening technique for three dimensional deforming interfaces

In this paper, we demonstrate improved accuracy of the level set method for resolving deforming interfaces by proposing two key elements: (1) accurate level set solutions on adapted Cartesian grids by judiciously choosing interpolation polynomials in regions of different grid levels and (2) enhanced re-initialization by an interface sharpening procedure. The level set equation is solved using a fifth order WENO scheme or a second order central differencing scheme depending on availability of uniform stencils at each grid point. Grid adaptation criteria are determined so that the Hamiltonian functions at nodes adjacent to interfaces are always calculated by the fifth order WENO scheme. This selective usage between the fifth order WENO and second order central differencing schemes is confirmed to give more accurate results compared to those in literature for standard test problems. In order to further improve accuracy especially near thin filaments, we suggest an artificial sharpening method, which is in a similar form with the conventional re-initialization method but utilizes sign of curvature instead of sign of the level set function. Consequently, volume loss due to numerical dissipation on thin filaments is remarkably reduced for the test problems.     

A linearly semi-implicit compact scheme for the Burgers–Huxley equation

In this paper, a linearly semi-implicit compact scheme is developed for the Burgers–Huxley equation. The equation is decomposed into two subproblems, i.e. a Burgers equation and a nonlinear ODE, by the operator splitting technique. The Burgers equation is solved by a linearly self-starting compact scheme which is fourth-order accurate in space and second-order accurate in time. The nonlinear ODE is discretized by a third-order semi-implicit Runge–Kutta method, which possesses good numerical stability with low computational cost. The numerical experiments show that the scheme provides the expected convergence order. Finally, several experiments are conducted to simulate the solutions of the Burgers–Huxley equation to validate our numerical method.     

Second-Order Accurate Computation of Curvatures in a Level Set Framework Using Novel High-Order Reinitialization Schemes

We present a high-order accurate scheme for the reinitialization equation of Sussman et al.(J. Comput. Phys. 114:146–159, [1994]) that guarantees accurate computation of the interface’s curvatures in the context of level set methods. This scheme is an extension of the work of Russo and Smereka (J. Comput. Phys. 163:51–67, [2000]). We present numerical results in two and three spatial dimensions to demonstrate fourth-order accuracy for the reinitialized level set function, third-order accuracy for the normals and second-order accuracy for the interface’s mean curvature in the L ¹- and L ^∞-norms. We also exploit the work of Min and Gibou (UCLA CAM Report (06-22), [2006]) to show second-order accurate scheme for the computation of the mean curvature on non-graded adaptive grids.     

Process monitoring in stamping operations through tooling integrated sensing

This paper introduces a new sensing method for stamping process monitoring based on the measurement of contact pressure distribution across the sheet metal-tooling interface, by means of an array of tooling-integrated force sensors. The role of numeric surface methods in estimating the contact pressure distribution on the sheet metal-tooling interface has been studied through finite element analysis and experiments. A finite element model is set up to model the contact interactions, based on the geometry of a customized stamping test fixture. Discrete samples of contact pressure taken from the FE model have been used to recreate continuous-pressure surfaces based on the Thin Plate Splines (TPS) and Bezier surface algorithms. It is shown that the temporal–spatial contact pressure distribution across the sheet metal-tooling interface can be effectively reconstructed through interpolation using spatially discrete sensor data. Comparison of surface-based pressure estimates with the FEA field solution indicates that the TPS-based method is more accurate than the Bezier method. The effectiveness of the surface modeling scheme is also evaluated experimentally by comparing the net press force calculated from numerical integration of the TPS surfaces with the experimentally measured value under different press speeds. The effectiveness of the new sensing method is further demonstrated by detection of slide misparallelism through analysis of the tooling interface pressure distribution. The study presents a new approach to enhancing process observability in manufacturing operations.     

The quadratic B-spline finite-element method for the coupled Schrödinger–Boussinesq equations

In this paper, a quadratic B-spline finite-element method is proposed for solving the coupled Schrödinger–Boussinesq equations numerically. A semi-discrete finite-element scheme is constructed for this system. The existence and uniqueness of the numerical solutions and the convergence of the discrete scheme are discussed. Numerical results indicate that the proposed method is accurate and efficient.     

填埋场典型复合衬垫系统单剪试验数值模拟

为研究填埋场典型复合衬垫系统接触面间的滑动特性,建立典型复合衬垫系统单剪试验的有限元模型,用ANSYS对其进行数值仿真分析.研究复合衬垫不同界面在不同法向力下的滑动特征;绘制衬垫系统不同界面节点的应力 位移关系曲线,得到不同法向力作用下各个界面不同位置点沿剪切位移方向和垂直剪切位移方向的切应力变化规律;分析接触面内节点的剪切位移随载荷步的变化特征.结果表明：随法向力的增加,滑动界面将发生转移;在低法向力（56 kPa）条件下,土工网 土工膜界面发生滑动;在高法向压力条件下（压力超过560 kPa）,土工膜 黏土界面发生滑动.     

A new fourth-order difference scheme for solving an N-carrier system with Neumann boundary conditions

In this paper, a numerical method is developed to solve an N-carrier system with Neumann boundary conditions. First, we apply the compact finite difference scheme of fourth order for discretizing spatial derivatives at the interior points. Then, we develop a new combined compact finite difference scheme for the boundary, which also has fourth-order accuracy. Lastly, by using a Padé approximation method for the resulting linear system of ordinary differential equations, a new compact finite difference scheme is obtained. The present scheme has second-order accuracy in time direction and fourth-order accuracy in space direction. It is shown that the scheme is unconditionally stable. The present scheme is tested by two numerical examples, which show that the convergence rate with respect to the spatial variable from the new scheme is higher and the solution is much more accurate when compared with those obtained by using other previous methods.