A finite element method for diffusion dominated unsteady viscous flows

A general conforming finite element scheme for computing viscous flows is presented which is of second-order accuracy in space and time. Viscous terms are treated implicitly and advection terms are treated explicitly in the time marching segment of the algorithm. A method for solving the algebraic equations at each time step is given. The method is demonstrated on two test problems, one of them being a plane vortex flow for which asymptotic methods are used to obtain suitable numerical boundary conditions at each time step.     

Eulerian Vlasov codes

Four different Eulerian grid-based Vlasov solvers are discussed, namely a second order method and a fourth order method (symplectic integrator) using cubic splines for interpolation, the CIP (cubic interpolated propagation) method, and an Euler-Lagrange method applying two-dimensional cubic interpolants. The four methods will be presented by outlining their algorithm. The performance of the numerical methods will be compared by numerically solving the Vlasov-Poisson system for the distribution function on a fixed Eulerian grid, for the problem of a two-stream instability in a two-dimensional phase-space.     

Numerical solutions of fractional advection–diffusion equations with a kind of new generalized fractional derivative

In the current paper, the numerical solutions for a class of fractional advection–diffusion equations with a kind of new generalized time-fractional derivative proposed last year are discussed in a bounded domain. The fractional derivative is defined in the Caputo type. The numerical solutions are obtained by using the finite difference method. The stability of numerical scheme is also investigated. Numerical examples are solved with different fractional orders and step sizes, which illustrate that the numerical scheme is stable, simple and effective for solving the generalized advection–diffusion equations. The order of convergence of the numerical scheme is evaluated numerically, and the first-order convergence rate has been observed.     

Numerical solution of the shallow water equations with a fractional step method

A fractional step technique for the numerical solution of the shallow water equations is applied to study the evolution of the potential vorticity field. The height and velocity field of the shallow water equations are discretized on a fixed Eulerian grid and time-stepped with a fractional step method recently reported in [M. Shoucri, Comput. Phys. Comm. 164 (2004) 396; M. Shoucri, A. Qaddouri, M. Tanguay, J. Côté, A Fractional Steps Method for the Numerical Solution of the Shallow Water Equations, International Workshop on Solution of Partial Differential Equations, The Fields Institute, Toronto, August 2002], where the Riemann invariants of the equations are interpolated at each time step along the characteristics using a cubic spline interpolation. The potential vorticity, which develops steep gradients and evolve into thin filaments during the evolution, is nicely calculated at every time-step from the solution of the code. The method is efficient and has lower numerical diffusion than other methods, since it evolves the equations without the iterative steps involved in the multi-dimensional interpolation problem, and without the iteration associated with the intermediate step of solving a Helmholtz equation, usually associated with other methods like the semi-Lagrangian method. The absence of iterative steps in the present technique makes it very suitable for problems in which small time steps and grid sizes are required, as for instance in the present problem where steepness of the gradients and small scale structures are the main features of the potential vorticity, and more generally for problems of regional climate modeling. The simplicity of the method makes it very suitable for parallel computer.     

An immersed boundary technique for simulating complex flows with rigid boundary

A new immersed boundary (IB) technique for the simulation of flow interacting with solid boundary is presented. The present formulation employs a mixture of Eulerian and Lagrangian variables, where the solid boundary is represented by discrete Lagrangian markers embedding in and exerting forces to the Eulerian fluid domain. The interactions between the Lagrangian markers and the fluid variables are linked by a simple discretized delta function. The numerical integration is based on a second-order fractional step method under the staggered grid spatial framework. Based on the direct momentum forcing on the Eulerian grids, a new force formulation on the Lagrangian marker is proposed, which ensures the satisfaction of the no-slip boundary condition on the immersed boundary in the intermediate time step. This forcing procedure involves solving a banded linear system of equations whose unknowns consist of the boundary forces on the Lagrangian markers; thus, the order of the unknowns is one-dimensional lower than the fluid variables. Numerical experiments show that the stability limit is not altered by the proposed force formulation, though the second-order accuracy of the adopted numerical scheme is degraded to 1.5 order. Four different test problems are simulated using the present technique (rotating ring flow, lid-driven cavity and flows over a stationary cylinder and an in-line oscillating cylinder), and the results are compared with previous experimental and numerical results. The numerical evidences show the accuracy and the capability of the proposed method for solving complex geometry flow problems both with stationary and moving boundaries.     

一致高精度KFVS方法用于多分量流的计算

５１．引言 很多传统的守恒型差分格式用于多组分流体的数值计算时,如果比热比１在不同流体间的界面附近不为常数,则数值解容易产生数值误差,并可能导致非物理解．文［１,４,７］就一些具体的格式提出了相应的减少物质界面附近数值误差的处理方法．它们的主要思想是对原来的算法作相应的非守恒校正, Ｋａｒｎｉ在文［５］中使用了原始变量算法求解多组分流,在文［６１又进一步研究了原始变量方法和 Ｌｅｖｅｌ Ｓｅｔ（位标）方法混合的算法．董素琴等［’］研究了多组分流体的二维非守恒型差分格式,结果表明,计算解在界面附近的误…     

On energy conservation for finite element approximation of flow-induced airfoil vibrations

In this paper the numerical approximation of a two-dimensional fluid–structure interaction problem is addressed. The fully coupled formulation of incompressible viscous fluid flow interacting with a flexibly supported airfoil is considered. The flow is described by the incompressible system of Navier–Stokes equations, where large values of the Reynolds number are considered. The Navier–Stokes equations are spatially discretized by the finite element method and stabilized with a modification of the Galerkin Least Squares (GLS) method; cf. [T. Gelhard, G. Lube, M.A. Olshanskii, J.-H. Starcke, Stabilized finite element schemes with LBB-stable elements for incompressible flows, Journal of Computational and Applied Mathematics 177 (2005) 243–267]. The motion of the computational domain is treated with the aid of Arbitrary Lagrangian Eulerian (ALE) method and the stabilizing terms are modified in a consistent way with the ALE formulation.     

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.     

The BFGS algorithm for a nonlinear least squares problem arising from blood flow in arteries

Using the Arbitrary Lagrangian Eulerian coordinates and the least squares method, a two-dimensional steady fluid structure interaction problem is transformed in an optimal control problem. Sensitivity analysis is presented. The BFGS algorithm gives satisfactory numerical results even when we use a reduced number of discrete controls.     

A physically based satellite rainfall estimation method using fluid dynamics modelling

A cloud motion winds (CMW) method is presented for improving quantitative rainfall estimation advection schemes that use both infrared (IR) and passive microwave (PMW) satellite data. Advection schemes are used to provide quantitative rainfall estimates by combining more direct PMW rainfall estimates with more frequent IR cloud top temperature measures using a two‐step technique: (1) PMW estimates are transported along CMW trajectories calculated with an advection scheme at subpixel resolution; and (2) PMW estimates are calibrated using the IR gradient along those trajectories. These schemes outperform traditional methods of satellite rainfall estimation but no clear physical basis for the procedure has yet been described. Here, the physical basis for the image processing techniques used in advection techniques is described. It is shown that geostationary satellite‐derived CMW from IR sensors can be modelled in terms of fluid dynamics using Navier–Stokes equations. This approach allows for modelling the problem as equivalent to the flow of a brightness temperature field, also providing subpixel resolution and unlimited rotation/deformation possibilities. The method is illustrated with rainfall estimates from a numerical weather prediction (NWP) model and with 3‐hourly PMW products as simulation data, obtaining consistent results.     

On the immersed interface method for solving time-domain Maxwell's equations in materials with curved dielectric interfaces

This paper deals with accurate numerical simulation of two-dimensional time-domain Maxwell's equations in materials with curved dielectric interfaces. The proposed fully second-order scheme is a hybridization between the immersed interface method (IIM), introduced to take into account curved geometries in structured schemes, and the Lax-Wendroff scheme, usually used to improve order of approximations in time for partial differential equations. In particular, the IIM proposed for two-dimensional acoustic wave equations with piecewise constant coefficients [C. Zhang, R.J. LeVeque, The immersed interface method for acoustic wave equations with discontinuous coefficients, Wave Motion 25 (1997) 237-263] is extended through a simple least squares procedure to such Maxwell's equations. Numerical results from the simulation of electromagnetic scattering of a plane incident wave by a dielectric circular cylinder appear to indicate that, compared to the original IIM for the acoustic wave equations, the augmented IIM with the proposed least squares fitting greatly improves the long-time stability of the time-domain solution. Semi-discrete finite difference schemes using the IIM for spatial discretization are also discussed and numerically tested in the paper.     

数值求解中网格自适应加密和合并技术的研究

在计算流体力学(CFD)领域,几乎所有的方法都离不开网格,网格是各种数值方法的基础,网格质量的好坏直接影响数值结果的精度,甚至影响到数值计算的成败,随着计算流体力学解决的问题越来越复杂,对网格质量的要求也越来越高,传统的统一网格技术已不能适应这一不断发展的需求,为此CFD工作者发展了许多方法,如迭合网格、贴体网格和非     

Textured Liquids based on the Marker Level Set

In this work we propose a new Eulerian method for handling the dynamics of a liquid and its surface attributes (for example its color). Our approach is based on a new method for interface advection that we term the Marker Level Set (MLS). The MLS method uses surface markers and a level set for tracking the surface of the liquid, yielding more efficient and accurate results than popular methods like the Particle Level Set method (PLS). Another novelty is that the surface markers allow the MLS to handle non-diffusively surface texture advection, a rare capability in the realm of Eulerian simulation of liquids. We present several simulations of the dynamical evolution of liquids and their surface textures.     

The method of fundamental solutions for Helmholtz-type equations in composite materials

In this paper, we introduce and develop the method of fundamental solutions (MFS) for solving Helmholtz-type elliptic partial differential equations in composite materials. This study builds upon the previous developments and applications of the MFS to linear and nonlinear heat conduction, elasticity, and functionally graded composite layered materials. Numerical results are presented and discussed for four examples involving both the modified Helmholtz and the Helmholtz equations in two-dimensional or three-dimensional, bounded or unbounded, smooth or non-smooth composite domains. It was found that the method produces numerical results which are in good agreement with the analytical solutions, where available.     

Predicting residual stresses in steady-state forming processes

A two-dimensional, Eulerian finite element formulation for modeling isotropic, elasto-viscoplastic, steady-state deformations which is capable of predicting residual stresses is presented in this paper. This problem is solved in two parts, namely, solution of the boundary value problem by a mixed finite element formulation for the velocity and pressure fields, and integration of the constitutive equations along pathlines across the domain. In this formulation, a discontinuous pressure field is used in the finite element formulation to reduce the system of equations to a system for only the velocity field. A new method for integrating the constitutive equations is also presented which improves the efficiency of the algorithm.     

Strong and weak Lagrange-Galerkin spectral element methods for the shallow water equations

The Lagrange-Galerkin spectral element method for the two-dimensional shallow water equations is presented. The equations are written in conservation form and the domains are discretized using quadrilateral elements.Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the nonlinearities introduced by the advection operator of the fluid dynamics equations.Two types of Lagrange-Galerkin methods are presented: the strong and weak formulations. The strong form relies mainly on interpolation to achieve high accuracy while the weak form relies primarily on integration. Lagrange-Galerkin schemes offer an increased efficiency by virtue of their less stringent CFL condition. The use of quadrilateral elements permits the construction of spectral-type finite-element methods that exhibit exponential convergence as in the conventional spectral method, yet they are constructed locally as in the finite-element method; this is the spectral method.In this paper, we show how to fuse the Lagrange-Galerkin methods with the spectral element method and present results for two standard test cases in order to compare and contrast these two hybrid schemes.     

Numerical modeling of elastoplastic flows by the Godunov method on moving Eulerian grids

The paper proposes a numerical method for calculating elastoplastic flows on adaptive Eulerian computational grids. Elastoplastic processes are described using the Prandtl-Reuss model. The spatial discretization of the Euler equations is carried out by the Godunov method on a moving grid. In order to improve the accuracy of the scheme, piecewise linear reconstruction of the grid functions is employed using a MUSCL-type interpolation scheme generalized to unstructured grids. The basic idea of the method is to split the system of governing equations into a hydrodynamic and an elastoplastic component. The hydrodynamic equations are solved by an absolutely stable explicit-implicit scheme, and the constitutive equations (elastoplastic component) are solved by a two-stage Runge-Kutta scheme. Theoretical analysis is performed and analytical solutions are obtained for a one-dimensional model describing the structures of a shock wave and a rarefaction wave in an elastoplastic material in the approximation of uniaxial strains. The proposed method is verified by the obtained analytical solutions and the solutions calculated using alternative approaches.     

A hybrid method of semi-Lagrangian and additive semi-implicit Runge–Kutta schemes for gyrokinetic Vlasov simulations

A hybrid method of semi-Lagrangian and additive semi-implicit Runge–Kutta schemes is developed for gyrokinetic Vlasov simulations in a flux tube geometry. The time-integration scheme is free from the Courant–Friedrichs–Lewy condition for the linear advection terms in the gyrokinetic equation. The new method is applied to simulations of the ion-temperature-gradient instability in fusion plasmas confined by helical magnetic fields, where the parallel advection term severely restricts the time step size for explicit Eulerian schemes. Linear and nonlinear results show good agreements with those obtained by using the explicit Runge–Kutta–Gill scheme, while the new method substantially reduces the computational cost.     

A combination of proper orthogonal decomposition–discrete empirical interpolation method (POD–DEIM) and meshless local RBF-DQ approach for prevention of groundwater contamination

The main presented idea is to reduce the used CPU time for employing the local radial basis functions-differential quadrature (LRBF-DQ) method. To this end, the proper orthogonal decomposition–discrete empirical interpolation method (POD–DEIM) has been combined with the LRBF-DQ technique. For checking the ability of the new procedure, the groundwater equation is solved. This equation has been classified in category of system of advection–diffusion equations. The solutions of advection equations have some shock, thus, special numerical methods should be applied for example discontinuous Galerkin and finite volume methods. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes.