A finite element procedure for viscoelastic flows

A finite element method for the simulation of viscoelastic flows has been developed. It uses a weak formulation of the method of characteristics to treat the viscoelastic constitutive law. Numerical results in a 4:1 contraction are presented and are discussed with respect to previous computations. New phenomena are put in evidence and new questions are opened in this already controversial problem.     

An incompressible SPH method for simulation of unsteady viscoelastic free-surface flows

In this paper, an incompressible smoothed particle hydrodynamics (SPH) method is presented to solve unsteady free-surface flows. Both Newtonian and viscoelastic fluids are considered. In the case of viscoelastic fluids, both the Maxwell and Oldroyd-B models are investigated. The proposed SPH method uses a Poisson pressure equation to satisfy the incompressibility constraints. The solution algorithm is an explicit predictor-corrector scheme and employs an adaptive smoothing length based on density variations. To alleviate the numerical difficulties encountered when fluid is highly stretched, an artificial stress term is incorporated into the momentum equation which reduces the risk of unrealistic fractures in the material. Two challenging test cases, the impacting drop and the jet buckling problems, are solved to demonstrate the capability of the proposed scheme in handling viscoelastic flows with complex free surfaces. The jet buckling test case was solved for a wide range of Weissenberg numbers. It was shown that in all cases the method is stable and fairly accurate and agrees well with the available data.     

Numerical simulation method for viscoelastic flows with free surfaces—fringe element generation method

To simulate filling flow in injection moulding for viscoelastic fluids, a numerical method, based on a finite element method and a finite volume method, has been developed for incompressible isothermal viscoelastic flow with moving free surfaces. The advantages of this method are, first, good applicability to arbitrarily shaped mould geometries and, second, accurate treatment for boundary conditions on the free surface. Typical filling flows are simulated, namely filling flow into a 1:4 expansion cavity with and without an obstacle. Numerical results predict the position of weld lines and air-traps. The method also indicates the effects of elongational flow on molecular orientation.     

A mixed finite element scheme for viscoelastic flows with XPP model

A mixed finite element formulation for viscoelastic flows is derived in this paper, in which the FIC （finite incremental calculus） pressure stabilization process and the DEVSS （discrete elastic viscous stress splitting） method using the Crank-Nicolson-based split are introduced within a general framework of the iterative version of the fractional step algorithm. The SU （streamline-upwind） method is particularly chosen to tackle the convective terms in constitutive equations of viscoelastic flows. Thanks to the proposed scheme the finite elements with equal low-order interpolation approximations for stress-velocity-pressure variables can be successfully used even for viscoelastic flows with high Weissenberg numbers. The XPP （extended Pom-Pom） constitutive model for describing viscoelastic behaviors is particularly integrated into the proposed scheme. The numerical results for the 4：1 sudden contraction flow problem demonstrate prominent stability, accuracy and convergence rate of the proposed scheme in both pressure and stress distributions over the flow domain within a wide range of the Weissenberg number, particularly the capability in reproducing the results, which can be used to explain the ＂die swell＂ phenomenon observed in the polymer injection molding process.     

Lattice Boltzmann method for the simulation of viscoelastic fluid flows

The simulation of viscoelastic fluids is a challenging task from the theoretical and numerical points of view. This class of fluids has been extensively studied with the help of classical numerical methods. In this paper we propose a new approach based on the lattice Boltzmann method in order to simulate linear and non-linear viscoelastic fluids and in particular those described by the Oldroyd-B and FENE-P constitutive equations. We study the accuracy and stability of our model on three different benchmarks: the 3D Taylor–Green vortex decay, the simplified 2D four-rolls mill, and the 2D Poiseuille flow. To our knowledge, the methodology described in this work is a first attempt for the simulation of non-trivial flows of viscoelastic fluids using the lattice Boltzmann method to discretize the constitutive and conservation equations.     

An efficient three-dimensional semi-implicit finite element scheme for simulation of free surface flows

An efficient semi-implicit finite element model is proposed for the simulation of three-dimensional flows in stratified seas. The body of water is divided into a number of layers and the two horizontal momentum equations for each layer of water are first integrated vertically. Nine-node Lagrangian quadratic isoparametric elements are employed for spatial discretization in the horizontal domain. The time derivatives are approximated using a second-order-accurate semi-implicit time-stepping scheme. The distinguishing feature of the proposed numerical scheme is that only nodal values on the same vertical line are coupled. Two test cases for which analytic solutions are available are employed to test the proposed scheme. The test results show that the scheme is efficient and stable. A numerical experiment is also included to compare the proposed scheme with a finite difference scheme.     

An upwind finite element scheme for high-Reynolds-number flows

A new upwind finite element scheme for the incompressible Navier-Stokes equations at high Reynolds number is presented. The idea of the upwind technique is based on the choice of upwind and downwind points. This scheme can approximate the convection term to third-order accuracy when these points are located at suitable positions. From the practical viewpoint of computation, the algorithm of the pressure Poisson equation procedure is adopted in the framework of the finite element method. Numerical results of flow problems in a cavity and past a circular cylinder show excellent dependence of the solutions on the Reynolds number. The influence of rounding errors causing Karman vortex shedding is also discussed in the latter problem.     

Numerical simulation of loading edge cracks by edge impact using the extended finite element method

The extended finite element method is used to analyze a plate with two parallel edge cracks impacted by a cylindrical projectile. The influence of the impact speed, crack length,plate thickness and notch tip radius on the crack initiation and propagation is studied. Dynamics equations are solved by an implicit time integration scheme which is unconditionally stable. Very good agreement is achieved between numerical predictions and experimental results. The critical velocity of the crack initiation under different conditions is examined. The influence of the crack length is greater than that of the impact speed, plate thickness and notch tip radius.     

An extended finite element method for modeling near-interfacial crack propagation in a layered structure

The extended finite element method (XFEM) is applied for the simulation of near-interfacial crack propagation in a metal–ceramic layered structure. An experimental evidence indicates that, in a ceramic–metal–ceramic sandwich structure, a near-interfacial crack in the ceramic layer can be drawn to or deflect away from the metal layer depending on the difference in elastic properties across the interface. To model near-interfacial fracture, only the Heaviside functions are used for the XFEM, and the vector level set method is employed for efficient evaluation of the enrichment functions. The crack propagation paths predicted by the XFEM simulation are found to be consistent with the experimental observation.     

A least squares finite element method for viscoelastic fluid flow problems

Viscoelastic flows remain a demanding class of problems for approximate analysis, particularly at increasing Weissenberg numbers. Part of the difficulty stems from the convective behavior and in the treatment of the stress field as a primary unknown. This latter aspect has led to the use of higher-order piecewise approximations for the stress approximation spaces in recent finite element research. The computational complexity of the discretized problem is increased significantly by this approach but at present it appears the most viable technique for solving these problems. Motivated by recent success in treating mixed systems and convective problems, we formulate here a least squares finite element method for the viscoelastic flow problem. Numerical experiments are conducted to test the method and examine its strengths and limitations. Some difficulties and open issues are identified through the numerical experiments. We consider the use of high degree elements (p refinement) to improve performance and accuracy.     

A 3D finite element method for the simulation of thermoconvective flows and its performances on a vector-parallel computer

A three-dimensional finite element method for the simulation of thermoconvective flows is presented. Vector-parallel performances of some preconditioned conjugate gradient methods are compared for solving both large linear systems and the Stokes problem. As significant examples, numerical experiments on the steady two- and three-dimensional Rayleigh-Bénard convection at high Prandtl number are reported.     

A finite element method for the solution of two-dimensional transonic flows in cascades

Steady two-dimensional transonic flow is calculated in cascades of compressor and turbine blades using a mesh of triangular finite elements. A velocity potential is used, the equations being solved by the Newton-Raphson technique. The resulting computer program is fast, and is shown to give good accuracy. Shock waves are well represented, provided they are not too strong.     

Thermoelastic analysis of multiple defects with the extended finite element method

In this paper, the extended finite element method (XFEM) is adopted to analyze the interaction between a sin-gle macroscopic inclusion and a single macroscopic crack as well as that between multiple macroscopic or micro-scopic defects under thermal/mechanical load. The effects of different shapes of multiple inclusions on the material thermomechanical response are investigated, and the level set method is coupled with XFEM to analyze the interaction of multiple defects. Further, the discretized extended finite element approximations in relation to thermoelastic prob-lems of multiple defects under displacement or temperature field are given. Also, the interfaces of cracks or materials are represented by level set functions, which allow the mesh assignment not to conform to crack or material interfaces. Moreover, stress intensity factors of cracks are obtained by the interaction integral method or the M-integral method, and the stress/strain/stiffness fields are simulated in the case of multiple cracks or multiple inclusions. Finally, some numer-ical examples are provided to demonstrate the accuracy of our proposed method.     

Vibration analysis of cracked plates using the extended finite element method

In the present paper, the extended finite element method (X-FEM) is adopted to analyze vibrations of cracked plates. Mindlin’s plate theory taking into account the effects of shear deformation and rotatory inertia is included in the development of the model. First, conventional FEM without any discontinuity is carried out, then the enrichment proposed by Moës et al. (Int J Numer Methods Eng 46, 131–150, 1999) of nodal elements containing cracks is added to the FEM formulation. Numerical implementation of enriched elements by discontinuous functions is performed, and thus dynamic equations (stiffness and mass matrices) are established. A FORTRAN computer code based on the X-FEM formulation is hence developed. Rectangular and square plates containing through-edge and central cracks with different boundary conditions are considered. The subspace iteration method is used to solve the eigenvalue problem. Natural frequencies as well as the corresponding eigenfunctions are consequently calculated as a function of the crack length. The obtained results show that the X-FEM is an efficient method in the dynamic analysis of plates containing discontinuities.     

Calculation of quasi-three-dimensional incompressible viscous flows by the finite element method

This paper discusses the calculation of quasi-three-dimensional incompressible viscous flow by FEM. The Reynolds-averaged Navier-Stokes equations are solved in curvilinear co-ordinates by the reduced integration and penalty method (RIP). Streamline upwind artificial viscosity (SUAV) and the Baldwin-Lomax algebraic model of turbulence are used. Time discretization is by the general implicit θ-method.     

A finite element method for the one‐dimensional extended Boussinesq equations

A new finite element method for Nwogu's (O. Nwogu, ASCE J. Waterw., Port, Coast., Ocean Eng., 119 , 618–638 (1993)) one‐dimensional extended Boussinesq equations is presented using a linear element spatial discretisation method coupled with a sophisticated adaptive time integration package. The accuracy of the scheme is compared to that of an existing finite difference method (G. Wei and J.T. Kirby, ASCE J. Waterw., Port, Coast., Ocean Eng., 121 , 251–261 (1995)) by considering the truncation error at a node. Numerical tests with solitary and regular waves propagating in variable depth environments are compared with theoretical and experimental data. The accuracy of the results confirms the analytical prediction and shows that the new approach competes well with existing finite difference methods. The finite element formulation is shown to enable the method to be extended to irregular meshes in one dimension and has the potential to allow for extension to the important practical case of unstructured triangular meshes in two dimensions. This latter case is discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

A finite volume approach for unsteady viscoelastic fluid flows

A finite volume, time‐marching for solving time‐dependent viscoelastic flow in two space dimensions for Oldroyd‐B and Phan Thien–Tanner fluids, is presented. A non‐uniform staggered grid system is used. The conservation and constitutive equations are solved using the finite volume method with an upwind scheme for the viscoelastic stresses and an hybrid scheme for the velocities. To calculate the pressure field, the semi‐implicit method for the pressure linked equation revised method is used. The discretized equations are solved sequentially, using the tridiagonal matrix algorithm solver with under‐relaxation. In both, the full approximation storage multigrid algorithm is used to speed up the convergence rate. Simulations of viscoelastic flows in four‐to‐one abrupt plane contraction are carried out. We will study the behaviour at the entrance corner of the four‐to‐one planar abrupt contraction. Using this solver, we show convergence up to a Weissenberg number We of 20 for the Oldroyd‐B model. No limiting Weissenberg number is observed even though a Phan Thien–Tanner model is used. Several numerical results are presented. Smooth and stable solutions are obtained for high Weissenberg number. Copyright © 2002 John Wiley & Sons, Ltd.     

Direct numerical simulation of particulate flows with a fictitious domain method

Our works on the fictitious domain method for the direct numerical simulation of particulate flows are reviewed, and particularly our recent progresses in the simulations of the motion of particles in Poiseuille flow at moderately high Reynolds numbers are reported. The method is briefly described, and its capability to simulate the motion of spherical and non-spherical particles in Newtonian, non-Newtonian and non-isothermal fluids is demonstrated. In addition, the applications of the fictitious domain method reported in the literature are also reviewed, and some comments on the features of the fictitious domain method and the immersed boundary method are given.     

An Euler-Bernoulli-like finite element method for Timoshenko beams

In this paper a new finite element approach for the solution of the Timoshenko beam is shown. Similarly to the Euler-Bernoulli beam theory, it has been considered a single fourth order differential equation governs the equilibrium of the Timoshenko beam. The results obtained by this approach are very good, both in terms of accuracy and computational effort.