Stabilized finite element formulation for incompressible flow on distorted meshes

Flow computations frequently require unfavourably meshes, as for example highly stretched elements in regions of boundary layers or distorted elements in deforming arbitrary Lagrangian Eulerian meshes. Thus, the performance of a flow solver on such meshes is of great interest. The behaviour of finite elements with residual‐based stabilization for incompressible Newtonian flow on distorted meshes is considered here. We investigate the influence of the stabilization terms on the results obtained on distorted meshes by a number of numerical studies. The effect of different element length definitions within the elemental stabilization parameter is considered. Further, different variants of residual‐based stabilization are compared indicating that dropping the second derivatives from the stabilization operator, i.e. using a streamline upwind Petrov–Galerkin type of formulation yields better results in a variety of cases. A comparison of the performance of linear and quadratic elements reveals further that the inconsistency of linear elements equipped with residual‐based stabilization introduces significant errors on distorted meshes, while quadratic elements are almost unaffected by moderate mesh distortion. Copyright © 2008 John Wiley & Sons, Ltd.     

A moving discontinuous Galerkin finite element method for flows with interfaces

A moving discontinuous Galerkin finite element method with interface condition enforcement is formulated for flows with discontinuous interfaces. The underlying weak formulation enforces the interface condition separately from the conservation law, so that the residual only vanishes upon satisfaction of both. In this formulation, the discrete grid geometry is treated as a variable, so that, in contrast to the standard discontinuous Galerkin method, this method has both the means to detect interfaces, via interface condition enforcement, and to satisfy, via grid movement, the conservation law and its associated interface condition. The method therefore directly fits interfaces, including shocks, preserving a high-order representation up to the interface without requiring shock capturing or an upwind numerical flux to achieve stability. It can be generalized to flows with a priori unknown interfaces with nontrivial topology and curved interface geometry as well as to an arbitrary number of spatial dimensions. Unsteady flows are represented in a manner similar to steady flows using a space-time formulation. In addition to computing flows with interfaces, the method can represent point singularities in a flow field by degenerating cuboid elements. In general, the method works in conjunction with standard local grid operations, including edge collapse, to ensure that degenerate cells are removed. Test cases are presented for up to three-dimensional flows that provide an initial assessment of the stability and accuracy of the method.     

Stabilized Bubble Function Method for Shallow Water Long Wave Equation

A relationship between the stabilized bubble function method and the stabilized finite element method is shown in this paper. The Petrov–Galerkin formulation with bubble function, i.e. a stabilized bubble function method, is proposed for the shallow water long wave equation. The Petrov–Galerkin formulation with the bubble function formulation possesses better stability than the Bubnov–Galerkin formulation with the bubble function.     

Parallel finite element computation of unsteady incompressible flows

A parallel semi-explicit iterative finite element computational procedure for modelling unsteady incompressible fluid flows is presented. During the procedure, element flux vectors are calculated in parallel and then assembled into global flux vectors. Equilibrium iterations which introduce some ‘local implicitness’ are performed at each time step. The number of equilibrium iterations is governed by an implicitness parameter. The present technique retains the advantages of purely explicit schemes, namely (i) the parallel speed-up is equal to the number of parallel processors if the small communication overhead associated with purely explicit schemes is ignored and (ii) the computation time as well as the core memory required is linearly proportional to the number of elements. The incompressibility condition is imposed by using the artificial compressibility technique. A pressure-averaging technique which allows the use of equal-order interpolations for both velocity and pressure, this simplifying the formulation, is employed. Using a standard Galerkin approximation, three benchmark steady and unsteady problems are solved to demonstrate the accuracy of the procedure. In all calculations the Reynolds number is less than 500. At these Reynolds numbers it was found that the physical dissipation is sufficient to stabilize the convective term with no need for additional upwind-type dissipation. © 1998 John Wiley & Sons, Ltd.     

NEW ALGORITHM OF COUPLING ELEMENT-FREE GALERKIN WITH FINITE ELEMENT METHOD

IntroductionMeshless methods, as a special numerical method, originated from1970s. Since thediffuse element method was proposed by Nayroleset al.[1]in1992, the meshless methodshave received wide attentions in the mechanics area, and have shown obvious adv…     

A control volume finite element method for three-dimensional three-phase flows

A novel control volume finite element method with adaptive anisotropic unstructured meshes is presented for three-dimensional three-phase flows with interfacial tension. The numerical framework consists of a mixed control volume and finite element formulation with a new P₁DG-P₂ elements (linear discontinuous velocity between elements and quadratic continuous pressure between elements). A “volume of fluid” type method is used for the interface capturing, which is based on compressive control volume advection and second-order finite element methods. A force-balanced continuum surface force model is employed for the interfacial tension on unstructured meshes. The interfacial tension coefficient decomposition method is also used to deal with interfacial tension pairings between different phases. Numerical examples of benchmark tests and the dynamics of three-dimensional three-phase rising bubble, and droplet impact are presented. The results are compared with the analytical solutions and previously published experimental data, demonstrating the capability of the present method.     

Dynamic simulation of free surfaces in capillaries with the finite element method

The mathematical formulation of the dynamics of free liquid surfaces including the effects of surface tension is governed by a non-linear system of elliptic differential equations. The major difficulty of getting unique closed solutions only in trivial cases is overcome by numerical methods. This paper considers transient simulations of liquid–gas menisci in vertical capillary tubes and gaps in the presence of gravity. Therefore the CFD code FIDAP 7.52 based on the Galerkin finite element method (FEM) is used. Calculations using the free surface model are presented for a variety of contact angles and cross-sections with experimental and theoretical verification. The liquid column oscillations are compared for numerical accuracy with a mechanical mathematical model, and the sensitivity with respect to the node density is investigated. The efficiency of the numerical treatment of geometric non-trivial problems is demonstrated by a prismatic capillary. Present restrictions limiting efficient transient simulations with irregularly shaped calculational domains are stated. © 1998 John Wiley & Sons, Ltd.     

固体非傅立叶温度场的时域间断Galerkin有限元法

运用时域间断Galerkin有限元法[1],对高频非傅立叶热波动问题[2-3]进行分析。其主要特点是:取温度及温度的时间导数为基本未知量,对其分别进行3次Hermite插值和线性插值。在保证节点温度自动保持连续的基础上,温度的时间导数在离散时域存在间断。数值结果表明所提出的方法能够滤掉虚假的数值震荡,能够良好地模拟固体中的非傅立叶热波动行为。     

Numerical investigations of stabilized finite element computations for acoustics

Least-squares stabilization stands out among the numerous approaches that have been proposed for relaxing resolution requirements of Galerkin computations for acoustics, by combining substantial improvement in performance with extremely simple implementation. The Galerkin/least-squares and Galerkin-gradient/least-squares methods are quite similar for structured meshes of linear finite elements. A series of numerical tests compares the two methods for several configurations with different kinds of boundary conditions employing structured and unstructured meshes. Various definitions of the resolution-dependent stability parameters are considered, along with different definitions of the mesh size upon which they depend.     

Gauge finite element method for incompressible flows

A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E, Liu J‐G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and differs from the standard gauge variable that arises from decomposing a compressible velocity field. It has the advantage that an additional boundary condition can be assigned to the gauge variable, thus eliminating the issue of a pressure boundary condition associated with the original primitive variable formulation. The computational task is then reduced to solving standard heat and Poisson equations, which are approximated by straightforward, piecewise linear (or higher‐order) finite elements. This method can achieve high‐order accuracy at a cost comparable with that of solving standard heat and Poisson equations. It is naturally adapted to complex geometry and it is much simpler than traditional finite element methods for incompressible flows. Several numerical examples on both structured and unstructured grids are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Some improvements on free surface simulation by the particle finite element method

The Lagrangian method has become increasingly popular in numerical simulation of free surface problems. In this paper, after a brief review of a recent Lagrangian method, namely the particle finite element method, some issues are discussed and some improvements are made. The least‐square finite element method is adopted to simplify the solving of the Navier–Stokes equations. An adaptive time method is derived to obtain suitable time steps. A mass correction procedure is imported to improve the mass conservation in long time calculations and time discretization scheme is adopted to decrease the pressure oscillations during the calculations. Finally, the method is used to simulate a series of examples and the results are compared with the commercial FLOW3D code. Copyright © 2008 John Wiley & Sons, Ltd.     

随机结构有限元分析的递推求解方法

提出了一种新的谱随机有限元分析方法——递推求解方法。该方法将随机结构的随机响应表示成非正交多项式展式,建立了和摄动法类似的一系列确定的递推方程,并通过确定性有限元方法对这些递推方程进行静力问题求解。算例表明,当随机量出现较大涨落时,计算结果相对于传统摄动法有不小的改进。     

Consistent Petrov–Galerkin finite element simulation of channel flows

In this paper, Navier–Stokes fluid flows in curved channels are considered. Upstream of the backward‐facing step, there exists a channel with a 90° bend and a fixed curvature of 2.5. The purpose of conducting this study was to apply a finite element code to study the effect of the distorted upstream velocity profile developing over the bend on laminar expansion flows behind the step. The size of the eddies formed downstream of the step is addressed. The present work employs primitive velocities, which stagger the pressure working variable, to assure satisfaction of the inf–sup stability condition. In quadratic elements, spatial derivatives are approximated within the consistent Petrov–Galerkin finite element framework. Use of this method aids stability in the sense that artificial damping is solely added to the direction parallel to the flow direction. Through analytical testing, in conjunction with two other benchmark tests, the integrity of applying the computer code in quadratic elements is verified. Copyright © 1999 John Wiley & Sons, Ltd.     

Adaptive mixed least squares Galerkin/Petrov finite element method for stationary conduction convection problems

An adaptive mixed least squares Galerkin/Petrov finite element method (FEM) is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. Using the general theory of Verfürth, the posteriori error estimates of the residual type are derived. Finally, numerical tests are presented to illustrate the effectiveness of the method.     

Adaptive mixed least squares Galerkin/Petrov finite element method for stationary conduction convection problems

An adaptive mixed least squares Galerkin/Petrov finite element method (FEM) is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. Using the general theory of Verfürth, the posteriori error estimates of the residual type are derived. Finally, numerical tests are presented to illustrate the effectiveness of the method.     

Semi-analytical finite element method for fictitious crack model in fracture mechanics of concrete

Based on the Hamiltonian governing equations of plane elasticity for sectorial domain, the variable separation and eigenfunction expansion techniques were employed to develop a novel analytical finite element for the fictitious crack model in fracture mechanics of concrete. The new analytical element can be implemented into FEM program systems to solve fictitious crack propagation problems for concrete cracked plates with arbitrary shapes and loads. Numerical results indicate that the method is more efficient and accurate than ordinary finite element method.     

A finite element model for the simulation of lost foam casting

In this paper, we present a numerical model to simulate the lost foam casting process. We introduce this particular casting first in order to capture the different physical processes in play during a casting. We briefly comment on the possible physical and numerical models used to envisage the numerical simulation. Next we present a model which aims to solve ‘part of’ the complexities of the casting, together with a simple energy budget that enables us to obtain an equation for the velocity of the metal front advance. Once the physical model is established we develop a finite element method to solve the governing equations. The numerical and physical methodologies are then validated through the solution of a two‐ and a three‐dimensional example. Finally, we discuss briefly some possible improvements of the numerical model in order to capture more physical phenomena. Copyright © 2004 John Wiley & Sons, Ltd.     

Difference scheme and numerical simulation based on mixed finite element method for natural convection problem

ntroductionLetΩ R2 beaboundeddomain .Weconsiderthefollowingnon_stationarynaturalconvectionproblem :Problem (Ⅰ )　Findu =(u1,u2 ) ,p ,andTsuchthat,foranyt1>0 ,ut- μΔu +(u· )u + p=λjT　　 ((x ,y ,t) ∈Ω× (0 ,t1) ) ,divu =0　　　　　　　　　 ((x ,y,t) ∈Ω× (0 ,t1) ) ,Tt-ΔT +λu· T =0　　 ((x,y,t) ∈Ω× (0 ,t1) ) ,u =0 ,T =0　　　　　　 ((x,y,t)∈ Ω× (0 ,t1) ) ,u(x ,y ,0 ) =0 ,　T(x,y,0 ) =f(x,y)　　 ((x,y) ∈Ω) ,whereuisthefluidvelocityvectorfield ,pthepressurefield ,Tthet…     

A least-squares finite element method for time-dependent incompressible flows with thermal convection

The time-dependent Navier–Stokes equations and the energy balance equation for an incompressible, constant property fluid in the Boussinesq approximation are solved by a least-squares finite element method based on a velocity–pressure–vorticity–temperature–heat-flux ( u –P–ω–T– q ) formulation discretized by backward finite differencing in time. The discretization scheme leads to the minimization of the residual in the l²-norm for each time step. Isoparametric bilinear quadrilateral elements and reduced integration are employed. Three examples, thermally driven cavity flow at Rayleigh numbers up to 10⁶, lid-driven cavity flow at Reynolds numbers up to 10⁴ and flow over a square obstacle at Reynolds number 200, are presented to validate the method.     

Inverse asymptotic solution method for finite deformation elasto-plasticity

I.IntroductionTheaccuracyofcomputermethodforsolvingengineeringproblemsfirstdependsonreliabilityofthetheorywhichisbeingapplied.Althoughfiniteelementmethodhasgainedgreatsuccessesinengineeringapplication.However,duetothetheoreticallimitationsofclassicalmecha…