A finite element method for compressible viscous fluid flows

This work presents a mixed three‐dimensional finite element formulation for analyzing compressible viscous flows. The formulation is based on the primitive variables velocity, density, temperature and pressure. The goal of this work is to present a ‘stable’ numerical formulation, and, thus, the interpolation functions for the field variables are chosen so as to satisfy the inf–sup conditions. An exact tangent stiffness matrix is derived for the formulation, which ensures a quadratic rate of convergence. The good performance of the proposed strategy is shown in a number of steady‐state and transient problems where compressibility effects are important such as high Mach number flows, natural convection, Riemann problems, etc., and also on problems where the fluid can be treated as almost incompressible. Copyright © 2010 John Wiley & Sons, Ltd.     

A DOMAIN DECOMPOSITION ALGORITHM WITH FINITE ELEMENT-BOUNDARY ELEMENT COUPLING

A domain decomposition algorithm coupling the finite element and the boundary element was presented. It essentially involves subdivision of the analyzed domain into sub-regions being independently modeled by two methods, i.e., the finite element method (FEM) and the boundary element method (BEM). The original problem was restored with continuity and equilibrium conditions being satisfied on the interface of the two sub-regions using an iterative algorithm. To speed up the convergence rate of the iterative algorithm, a dynamically changing relaxation parameter during iteration was introduced. An advantage of the proposed algorithm is that the locations of the nodes on the interface of the two sub-domains can be inconsistent. The validity of the algorithm is demonstrated by the consistence of the results of a numerical example obtained by the proposed method and those by the FEM, the BEM and a present finite element-boundary element (FE-BE) coupling method.     

A Hermite finite element method for incompressible fluid flow

We describe some Hermite stream function and velocity finite elements and a divergence‐free finite element method for the computation of incompressible flow. Divergence‐free velocity bases defined on (but not limited to) rectangles are presented, which produce pointwise divergence‐free flow fields (∇· u _h≡0). The discrete velocity satisfies a flow equation that does not involve pressure. The pressure can be recovered as a function of the velocity if needed. The method is formulated in primitive variables and applied to the stationary lid‐driven cavity and backward‐facing step test problems. Copyright © 2009 John Wiley & Sons, Ltd.     

A simple finite element method for Stokes flows with surface tension using unfitted meshes

This paper presents a simple finite element method for Stokes flows with surface tension. The method uses an unfitted mesh that is independent of the interface. Due to the surface force, the pressure has a jump across the interface. Based on the properties of the level set function that implicitly represents the interface, the jump of the pressure is removed, and a new problem without discontinuities is formulated. Then, classical stable finite element methods are applied to solve the new problem. Some techniques are used to show that the method is equivalent to an easy‐to‐implement method that can be regarded as a traditional method with a modified pressure space. However, the matrix of the resulting linear system of equations is the same as that of the traditional method. Optimal error estimates are derived for the proposed method. Finally, some numerical tests are presented to confirm the theoretical results. Copyright © 2015 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.     

无穷域势流问题的有限元／差分线法混合求解

提出了一种将有限元和差分线法相结合求解无穷域势流问题的算法。用两同心圆将求解域划分为存在重叠的有限和无限两个区域,在有限和无限域上分别用有限元和差分线法求解Laplace方程边值问题。用差分线法推导出的关系式修正有限元方程,求解该方程组从而得到原问题的解。本算法将求解无穷域问题转化为代数特征值问题和有限域内线性方程组的...     

A conservative flux-corrected continuous finite element method for fluid interface dynamics

This paper presents a continuous finite element solution for fluid flows with interfaces. The method is founded on the sign-preserving flux correction transport methodology and extends nonoscillatory finite element algorithm capabilities to predict interface motion efficiently. The procedure is composed of three main stages, along the lines of the conservative level set method: transport of phase function, reconstruction of phase function, and solution of equations of motion of two incompressible fluids. The flux correction technique takes action on the three steps. Limiting process incorporates a straightforward refinement to remove global mass residuals present in the earliest version of the algorithm. This is of particular importance in the transport step. Moreover, new method retains the efficacy of the original. To reconstruct the phase function after transport, a novel nonlinear (and conservative) streamlined diffusion equation is proposed, with an anisotropic diffusivity comprising artificial compression and diffusive fluxes along interface displacements direction. A substantial reduction of unphysical overshoots along the interface is reached by an improved bound estimation that includes interface information. Complete operation of the correction algorithm for two incompressible fluids flows requires two pressure solutions. We explore a reduced form to circumvent this extra burden. Numerical experiments verify the formulation by reproducing stringent benchmarks both for transport/reinitialization and for two-fluid interface propagation.     

Multiphase flow in heterogeneous porous media: A classical finite element method versus an implicit pressure–explicit saturation‐based mixed finite element–finite volume approach

Various discretization methods exist for the numerical simulation of multiphase flow in porous media. In this paper, two methods are introduced and analyzed—a full‐upwind Galerkin method which belongs to the classical finite element methods, and a mixed‐hybrid finite element method based on an implicit pressure–explicit saturation (IMPES) approach. Both methods are derived from the governing equations of two‐phase flow. Their discretization concepts are compared in detail. Their efficiency is discussed using several examples. Copyright © 1999 John Wiley & Sons, Ltd.     

P 1-nonconforming triangular finite element method for elliptic and parabolic interface problems

The lowest order P₁-nonconforming triangular finite element method (FEM) for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optimal order error estimates are obtained in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.     

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.     

Co-located equal-order control-volume finite element method for two-dimensional axisymmetric incompressible fluid flow

The formulation of a control-volume-based finite element method (CVFEM) for axisymmetric, two-dimensional, incompressible fluid flow and heat transfer in irregular-shaped domains is presented. The calculation domain is discretized into torus-shaped elements and control volumes. In a longitudinal cross-sectional plane, these elements are three-node triangles, and the control volumes are polygons obtained by joining the centroids of the three-node triangles to the mid-points of the sides. Two different interpolation schemes are proposed for the scalar-dependent variables in the advection terms: a flow-oriented upwind function, and a mass-weighted upwind function that guarantees that the discretized advection terms contribute positively to the coefficients in the discretized equations. In the discretization of diffusion transport terms, the dependent variables are interpolated linearly. An iterative sequential variable adjustment algorithm is used to solve the discretized equations for the velocity components, pressure and other scalar-dependent variables of interest. The capabilities of the proposed CVFEM are demonstrated by its application to four different example problems. The numerical solutions are compared with the results of independent numerical and experimental investigations. These comparisons are quite encouraging.     

A novel twice-interpolation finite element method for solid mechanics problems

Formulation and numerical evaluation of a novel twice-interpolation finite element method （TFEM） is presented for solid mechanics problems. In this method, the trial function for Galerkin weak form is constructed through two stages of consecutive interpolation. The primary interpolation follows exactly the same procedure of standard FEM and is further reproduced according to both nodal values and averaged nodal gradients obtained from primary interpolation. The trial functions thus constructed have continuous nodal gradients and contain higher order polynomial without increasing total freedoms. Several benchmark examples and a real dam problem are used to examine the TFEM in terms of accuracy and convergence. Compared with standard FEM, TFEM can achieve significantly better accuracy and higher convergence rate, and the continuous nodal stress can be obtained without any smoothing operation. It is also found that TFEM is insensitive to the quality of the elemental mesh. In addition, the present TFEM can treat the incompressible material without any modification.     

A two‐phase flow interface capturing finite element method

A new interface capturing algorithm is proposed for the finite element simulation of two‐phase flows. It relies on the solution of an advection equation for the interface between the two phases by a streamline upwind Petrov–Galerkin (SUPG) scheme combined with an adaptive mesh refinement procedure and a filtering technique. This method is illustrated in the case of a Rayleigh–Taylor two‐phase flow problem governed by the Stokes equations. Copyright © 2006 John Wiley & Sons, Ltd.     

Nonlinear quasi-conforming finite element method

The nonlinear quasi-conforming FEM is presented based on the basic concept of the quasiconforming finite element. First, the incremental principle of stationary potential energy is discussed. Then, the formulation process of the nonlinear quasi-conforming FEM is given. Lastly, two computational examples of shells are given.     

A parallel two-level finite element method for the Navier-Stokes equations

Based on domain decomposition, a parallel two-level finite element method for the stationary Navier-Stokes equations is proposed and analyzed. The basic idea of the method is first to solve the Navier-Stokes equations on a coarse grid, then to solve the resulted residual equations in parallel on a fine grid. This method has low communication complexity. It can be implemented easily. By local a priori error estimate for finite element discretizations, error bounds of the approximate solution are derived. Numerical results are also given to illustrate the high efficiency of the method.     

考虑粘结面厚度的局部粘结单元法

传统无厚度粘结单元法CFEM (Cohesive finite element method)在模拟脆性材料断裂方面具有很强的优势,但也存在很大问题.一是单元尺寸增大,收敛性变差;二是单元尺寸变小,模型刚度发生折减.为了克服这两个问题,发展了考虑厚度的局部粘结单元法,即在裂纹可能扩展区插入具有一定厚度的粘结面单元.粘结面单元采用拓展虚内键本构(Augmented virtual internal bond)描述.由于考虑了厚度,粘结面交叉处会形成多边形空缺.为了弥补这一空缺,将其看作多边形键元胞,采用离散虚内键模型(Discretized virtual internal bond)对其建模,保证了模型的几何完整性.模拟结果表明,本文方法有效,克服了传统CFEM方法的刚度折减问题,提高了计算稳定性和收敛性.     

A new finite element formulation for both compressible and nearly incompressible fluid dynamics

A new finite element formulation designed for both compressible and nearly incompressible viscous flows is presented. The formulation combines conservative and non‐conservative dependent variables, namely, the mass–velocity (density * velocity), internal energy and pressure. The central feature of the method is the derivation of a discretized equation for pressure, where pressure contributions arising from the mass, momentum and energy balances are taken implicitly in the time discretization. The method is applied to the analysis of laminar flows governed by the Navier–Stokes equations in both compressible and nearly incompressible regimes. Numerical examples, covering a wide range of Mach number, demonstrate the robustness and versatility of the new method. Copyright © 2000 John Wiley & Sons, Ltd.