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.     

Numerical solution of a phase field model for incompressible two-phase flows based on artificial compressibility

We present an artificial compressibility based numerical method for a phase field model for simulating two-phase incompressible viscous flows. The phase model was proposed by Liu and Shen [Physica D. 179 (2003) 211–228], in which the interface between two fluids is represented by a thin transition region of fluid mixture that stores certain amount of mixing energy. The model consists of the Navier–Stokes equations coupled with the Allen–Cahn equation (phase field equation) through an extra stress term and a transport term. The extra stress in the momentum equations represents the phase-induced capillary effect for the mixture due to the surface tension. The coupled equations are cast into a conservative form suitable for implementation with the artificial compressibility method. The resulting hyperbolic system of equations are then discretized with weighted essentially non-oscillatory (WENO) finite difference scheme. The dual-time stepping technique is applied for obtaining time accuracy at each physical time step, and the approximate factorization algorithm is used to solve the discretized equations. The effectiveness of the numerical method is demonstrated in several two-phase flow problems with topological changes. Numerical results show the present method can be used to simulate incompressible two-phase flows with small interfacial width parameters and topological changes.     

Simulation of unsteady compressible flow in a channel with vibrating walls - Influence of the frequency

This study deals with the numerical solution of a 2D unsteady flow of a compressible viscous fluid in a channel for low inlet airflow velocity. The unsteadiness of the flow is caused by a prescribed periodic motion of a part of the channel wall with large amplitudes, nearly closing the channel during oscillations. The channel is a simplified model of the glottal space in the human vocal tract and the flow can represent a model of airflow coming from the trachea, through the glottal region with periodically vibrating vocal folds to the human vocal tract.The flow is described by the system of Navier–Stokes equations for laminar flows. The numerical solution is implemented using the finite volume method (FVM) and the predictor–corrector MacCormack scheme with Jameson artificial viscosity using a grid of quadrilateral cells. Due to the motion of the grid, the basic system of conservation laws is considered in the Arbitrary Lagrangian–Eulerian (ALE) form.The authors present the numerical simulations of flow fields in the channel, acquired from a program developed exclusively for this purpose. The numerical results for unsteady flows in the channel are presented for inlet Mach number M_∞ = 0.012, Reynolds number Re_∞ = 4.5 × 10³ and the wall motion frequency 20 and 100 Hz.     

结构变形对整流罩分离轨迹的影响

为研究弹性体在稠密大气中的分离问题,基于非结构网格,采用运动网格与局部网格重构相结合的方法求解大位移相对运动的流场,并耦合6自由度刚体运动方程得到整流罩的运动.非定常流动方程使用格心有限体积法进行空间离散,并运用LU-SGS进行求解.应用标准算例验证该方法的准确性,并用于某整流罩飞行轨迹的计算.结果表明结构变形可能会使...     

A contact force model considering constant external forces for impact analysis in multibody dynamics

The adjoint method is an elegant approach for the computation of the gradient of a cost function to identify a set of parameters. An additional set of differential equations has to be solved to compute the adjoint variables, which are further used for the gradient computation. However, the accuracy of the numerical solution of the adjoint differential equation has a great impact on the gradient. Hence, an alternative approach is the discrete adjoint method, where the adjoint differential equations are replaced by algebraic equations. Therefore, a finite difference scheme is constructed for the adjoint system directly from the numerical time integration method. The method provides the exact gradient of the discretized cost function subjected to the discretized equations of motion.     

Finite element analysis of viscous,incompressible fluid flow: Part 1 : Basic methodology

A numerical procedure is developed for the analysis of general two-dimensional flows of viscous, incompressible fluids using the finite element method. The partial differential equations describing the continuum motion of the fluid are discretized by using an integral energy balance approach in conjunction with the finite element approximation. The nonlinear algebraic equations resulting from the discretization process are solved using a Picard iteration technique.A number of computational procedures are developed that allow significant reductions to be made in the computational effort required for the analysis of many flow problems. These techniques include a coarse-to-fine-mesh rezone procedure for the detailed study of regions of particular interest in a flow field and a special finite element to model far-field regions in external flow problems.     

具有端点质量的一维波动方程半离散格式的一致指数稳定性

无穷维系统主要由偏微分方程描述, 可是大部分用偏微分方程描述的控制系统, 无论是单纯的数值实验还是需要应用到实际的问题中去, 都需要对方程进行有限数值离散. 本文考虑了端点带有质量的波动方程在边界反馈控制下半离散格式的一致指数稳定性. 首先, 原闭环系统通过降阶法变成低阶的等价系统, 通过一种间接Lyapunov函数方法证明了降阶等价的连续系统是一致指数稳定的. 其次, 对等价系统空间变量离散得到半离散的差分格式.平行于连续系统, 间接Lyapunov函数方法证明了半离散系统的一致指数稳定性. 数值实验证明了基于降阶法的一致指数稳定性和经典半离散格式的非一致指数稳定性.     

Multiphase image segmentation using a phase-field model

In this paper, we propose a new, fast, and stable hybrid numerical method for multiphase image segmentation using a phase-field model. The proposed model is based on the Allen-Cahn equation with a multiple well potential and a data-fitting term. The model is computationally superior to the previous multiphase image segmentation via Modica-Mortola phase transition and a fitting term. We split its numerical solution algorithm into linear and a nonlinear equations. The linear equation is discretized using an implicit scheme and the resulting discrete system of equations is solved by a fast numerical method such as a multigrid method. The nonlinear equation is solved analytically due to the availability of a closed-form solution. We also propose an initialization algorithm based on the target objects for the fast image segmentation. Finally, various numerical experiments on real and synthetic images with noises are presented to demonstrate the efficiency and robustness of the proposed model and the numerical method.     

Numerical solution of potential flow equations with a predictor-corrector finite difference method

We develop a numerical solution algorithm of the nonlinear potential flow equations with the nonlinear free surface boundary condition.A finite difference method with a predictor-corrector method is applied to solve the nonlinear potential flow equations in a two-dimensional (2D) tank.The irregular tank is mapped onto a fixed square domain with rectangular cells through a proper mapping function.A staggered mesh system is adopted in a 2D tank to capture the wave elevation of the transient fluid.The finite difference method with a predictor-corrector scheme is applied to discretize the nonlinear dynamic boundary condition and nonlinear kinematic boundary condition.We present the numerical results of wave elevations from small to large amplitude waves with free oscillation motion,and the numerical solutions of wave elevation with horizontal excited motion.The beating period and the nonlinear phenomenon are very clear.The numerical solutions agree well with the analytical solutions and previously published results.     

Numerical simulation of interaction between turbulent flow and a vibrating airfoil

The subject of this paper is the numerical simulation of the interaction of two-dimensional incompressible viscous flow and a vibrating airfoil. A solid elastically supported airfoil with two degrees of freedom, which can rotate around the elastic axis and oscillate in the vertical direction, is considered. The numerical simulation consists of the stabilized finite element treatment of the Reynolds averaged Navier–Stokes (RANS) approach, the use of turbulence models and the solution of the system of ordinary differential equations describing the airfoil motion. The time dependent computational domain and a moving grid are taken into account with the aid of the Arbitrary Lagrangian–Eulerian (ALE) formulation of the Navier–Stokes equations. High Reynolds numbers up to 10⁶ require to use a suitable stabilization of the finite element discretization and the application of a turbulence model. We apply the algebraic turbulence model, which was designed by Baldwin and Lomax and modified by Rostand. The developed technique was tested by the simulation of flow past a flat rigid plate and the computation of pressure distribution around a rotating airfoil with prescribed motion. Finally, the method was applied to the simulation of flow induced airfoil vibrations. This research was supported under the Grant No. IAA200760613 of the Grant Agency of Academy of Sciences of the Czech Republic. The research of M. Feistauer was partly supported by the research project MSM 0021620839 financed by the Ministry of Education of the Czech Republic and the research of L. Dubcová was partly supported by the grant No. 48607 of the Grant Agency of the Charles University. The authors acknowledge the support of these institutions.     

Transient analysis of flexible multi-body systems. Part II: Application to aircraft landing

This investigation presents a method for transient analysis of a large-scale multi-body aircraft consisting of interconnected rigid and flexible bodies that undergo large angular rotations. Elastic components of the aircraft are discretized using the finite element method. The system equations of motion and nonlinear algebraic constraint equations describing joints between different components are written in the Lagrangian formulation using a finite set of coupled reference and modal coordinates. The system differential equations of motion and algebraic constraint equations are computer-generated and integrated forward in time using an explicit-implicit direct numerical integration algorithm coupled with a Newton-Raphson type iteration in order to check on constraint violations. Impact and intermittent motion events are accounted for by using a generalized momentum balance that predict jump discontinuities in the generalized velocities as well as jump discontinuities in the system reaction forces. The formulation presented and the computer program developed are used to simulate the impact between the landing gear and the runway. The method is also used to predict the dynamic behavior of the aircraft during the traverse of an abrupt elevation change in the runway.     

On solving hybrid optimal control problems with higher index DAEs

The paper deals with hybrid optimal control problems described by higher index differential–algebraic equations (DAEs). We introduce a numerical procedure for solving these problems. The procedure has the following features: it is based on the appropriately defined adjoint equations formulated for the discretized equations being the result of the numerical integration of systems equations by an implicit Runge–Kutta method; the consistent initialization procedure is applied whenever control functions jumps, or state variables transition occurs. The procedure can cope with hybrid optimal control problems which are defined by DAEs with the index not exceeding three. Our approach does not require differentiation of some system equations in order to transform higher index DAEs to the underlying ordinary differential equations (ODEs). The presented numerical examples show that the proposed approach can be used to solve efficiently hybrid optimal control problems with higher index DAEs.     

Mixed finite element method for analysis of viscoelastic fluid flow

In the present paper, numerical analysis of incompressible viscoelastic fluid flow is discussed using mixed finite element Galerkin method. Because Maxwellian viscoelasticity is assumed as the constitutive equation, stress components could not be eliminated from the governing equation system. Because of this, mixed finite element method is utilized to discretize the basic equations. For the solution procedures to solve discretized equation system, Newton-Raphson method for steady flow and perturbation method for unsteady flow is employed. As the numerical examples, comparison was made on the finite element computational results between by direct method and by mixed method. Effects of the viscoelasticity is analyzed for the flows at Reynold's numbers 30, 50 and 70.     

An efficient multigrid preconditioner for Maxwell’s equations in micromagnetism

We consider a system of Maxwell’s and Landau-Lifshitz-Gilbert equations describing magnetization dynamics in micromagnetism. The problem is discretized by a convergent, unconditionally stable finite element method. A multigrid preconditioned Uzawa type method for the solution of the algebraic system resulting from the discretized Maxwell’s equations is constructed. The efficiency of the method is demonstrated on numerical experiments and the results are compared to those obtained by simplified models.     

基于边光滑有限元法的剪切变形板几何非线性分析

为改善在计算板的几何非线性问题时有限元法系统过硬的数值缺陷,提高计算精度,在考虑剪切变形的yon Karman假设下,基于全拉格朗日描述方法,将边光滑有限元法应用于板的几何非线性分析.计算公式基于1阶剪切变形理论,并采用离散剪切间隙有效地消除剪切自锁.在三角形单元的基础上进一步形成边界光滑域,在每个光滑域内对应变进行光...     

A numerical method to price discrete double Barrier options under a constant elasticity of variance model with jump diffusion

We develop a numerical method to price discrete barrier options on an underlying described by the constant elasticity of variance model with jump-diffusion (CEVJD). In particular, the partial integro differential equation associated to this model is discretized in time using an operator splitting scheme whose accuracy is enhanced by repeated Richardson extrapolation. Such an approach allows us to approximate the differential terms and the jump integral by means of two different numerical techniques. Precisely, the spatial derivatives, which exist only in the weak sense, are discretized using a finite element method based on piecewise quadratic polynomials, whereas the jump integral is directly collocated at the mesh points, so that it can be easily evaluated by Simpson numerical quadrature. As shown by extensive numerical simulation, the proposed approach is very efficient from the computational standpoint, and performs significantly better than the finite difference scheme developed in Wade et al. [On smoothing of the Crank–Nicolson scheme and higher order schemes for pricing barrier options, J. Comput. Appl. Math. 204 (2007), pp. 144–158].     

A Numerical Scheme for a Viscous Shallow Water Model with Friction

We consider a particular viscous shallow water model with topography and friction laws, formally derived by asymptotic expansion from the three-dimensional free surface Navier-Stokes equations. Emphasize is put on the numerical study: the viscous system is regarded as an hyperbolic system with source terms and discretized using a second order finite volume method. New steady states solutions for open channel flows are introduced for the whole model with viscous and friction terms. The proposed numerical scheme is validated against these new benchmarks.     

A numerical approach for solving the high-order linear singular differential-difference equations

In this paper, a numerical method which produces an approximate polynomial solution is presented for solving the high-order linear singular differential-difference equations. With the aid of Bessel polynomials and collocation points, this method converts the singular differential-difference equations into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. This method gives the analytic solutions when the exact solutions are polynomials. Finally, some experiments and their numerical solutions are given; by comparing the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method. All of the numerical computations have been performed on a PC using some programs written in MATLAB v7.6.0 (R2008a).     

注射成型弧形流道流场分析

针对在注射成型过程中,注射熔体的均匀程度以及其进入型腔的速度方向对制件的品质会产生很大影响的问题,主要对熔体进入型腔前的流动进行分析.采用弧形流道,用有限差分法离散连续性方程和动量方程,用超松驰迭代法求解离散后的代数方程组,进而求出弧形流道熔体流动的速度和压强分布.结果表明,在弧形流道中,外层熔体可能进入中心层甚至内层,局部有旋涡,内外层熔体有物质和能量交换,所以熔体经过弧形流道后密度和温度场分布更均匀.