A stabilized volume‐averaging finite element method for flow in porous media and binary alloy solidification processes

A stabilized equal‐order velocity–pressure finite element algorithm is presented for the analysis of flow in porous media and in the solidification of binary alloys. The adopted governing macroscopic conservation equations of momentum, energy and species transport are derived from their microscopic counterparts using the volume‐averaging method. The analysis is performed in a single domain with a fixed numerical grid. The fluid flow scheme developed includes SUPG (streamline‐upwind/Petrov–Galerkin), PSPG (pressure stabilizing/Petrov–Galerkin) and DSPG (Darcy stabilizing/Petrov–Galerkin) stabilization terms in a variable porosity medium. For the energy and species equations a classical SUPG‐based finite element method is employed. The developed algorithms were tested extensively with bilinear elements and were shown to perform stably and with nearly quadratic convergence in high Rayleigh number flows in varying porosity media. Examples are shown in natural and double diffusive convection in porous media and in the directional solidification of a binary‐alloy. Copyright © 2004 John Wiley & Sons, Ltd.     

Effects of dendritic arm coarsening on macroscopic modelling of solidification of binary alloys

Abstract

A pure macroscopic two-dimensional numerical model has been developed, capable of capturing the effects of dendritic arm coarsening on the transport phenomena occurring during a binary alloy solidification process. The general continuum conservation equations are aptly modified to take into account shrinkage induced fluid flow. Simultaneously, the effective permeability of the mushy zone is numerically modelled according to the microscopic coarsening kinetics. Moreover, a new nodal latent heat updating algorithm is proposed that takes into account dendritic arm coarsening considerations. The numerical results are first tested against experimental results reported in the literature, corresponding to the solidification of an Al-Cu alloy in a bottom cooled cavity. It is concluded that dendritic arm coarsening leads to an increased effective permeability of the mushy region as well as an enhanced eutectic fraction of the solidified ingot. Consequently, an enhanced macrosegregation is predicted, compared with that dictated by shrinkage induced fluid flow alone. Physical insights are also developed regarding the effects of various parameters on the overall macrosegregation.     

Continuum model for predicting macrosegregation in dendritic alloys

“Macrosegregation” represents a class of defects in cast products of serious concern to both alloy producers and users. Many types of macrosegregation result from thermosolutal convection in the solid plus liquid and all-liquid regions of a solidifying alloy, and this has spurred modeling and simulations, which treat the solid plus liquid region (i.e., the mushy zone) as a porous medium of variable porosity and permeability. Simulations include scenarios in which the convection is strong enough to make channels in the mushy zone region, and these channels lead to localized segregates known as “freckles”. Using Pb-10 wt.% Sn as a model alloy, we simulated vertical solidification with various solidification rates. By sufficiently increasing the cooling rate at the bottom surface, convection can be suppressed enough to prevent the formation of freckles. The simulation is an example of relating microstructural metrics to a macroscopic property of the porous medium used in continuum theory. In this case, the property is the permeability, which is governed by two microstructural metrics: the volume fraction of liquid and a characteristic length in the dendritic microstructure. Permeability data, relevant to columnar dendritic solidification, are reviewed, and recommendations for future work on determining the permeability in terms of microstructural metrics are given.     

Eulerian elasto‐visco‐plastic formulations for residual stress prediction

A stabilized, Galerkin finite element formulation for modeling the elasto‐visco‐plastic response of quasi‐steady‐state processes, such as welding, laser surfacing, rolling and extrusion, is presented in an Eulerian frame. The mixed formulation consists of four field variables, such as velocity, stress, deformation gradient and internal variable, which is used to describe the evolution of the material's resistance to plastic flow. The streamline upwind Petrov–Galerkin method is used to eliminate spurious oscillations, which may be caused by the convection‐type of stress, deformation gradient and internal variable evolution equations. A progressive solution strategy is introduced to improve the convergence of the Newton–Raphson solution procedure. Two two‐dimensional numerical examples are implemented to verify the accuracy of the Eulerian formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

A stable and adaptive semi-Lagrangian potential model for unsteady and nonlinear ship-wave interactions

We present an innovative numerical discretization of the equations of inviscid potential flow for the simulation of three-dimensional, unsteady, and nonlinear water waves generated by a ship hull advancing in water.The equations of motion are written in a semi-Lagrangian framework, and the resulting integro-differential equations are discretized in space via an adaptive iso-parametric collocation boundary element method, and in time via implicit backward differentiation formulas (BDF) with adaptive step size and variable order.When the velocity of the advancing ship hull is non-negligible, the semi-Lagrangian formulation (also known as Arbitrary Lagrangian Eulerian formulation or ALE) of the free-surface equations contains dominant transport terms which are stabilized with a streamwise upwind Petrov–Galerkin (SUPG) method.The SUPG stabilization allows automatic and robust adaptation of the spatial discretization with unstructured quadrilateral grids. Preliminary results are presented where we compare our numerical model with experimental results on a Wigley hull advancing in calm water with fixed sink and trim.     

Analysis on fluid permeability of dendritic mushy zone during peritectic solidification in a temperature gradient

Compared with the growing applications of peritectic alloys,none research on the fluid permeability K of dendritic network during peritectic solidification has been reported before.The fluid permeability K of dendritic network in the mushy zone during directional solidification of Sn-Ni peritectic alloy was investigated in this study.Examination on the experimental results demonstrates that both the temperature gradient zone melting (TGZM) and Gibbs-Thomson (G-T) effects have obvious influences on the morphology of dendritic network during directional solidification.This is realized through different stages of liquid diffusion within dendritic mushy zone by these effects during directional solidification.The TGZM effect is demonstrated to play a more important role as compared with the G-T effect during directional solidification.Besides,it is shown that the evolution of dendrite network is more complex during peritectic solidification due to the involvement of the peritectic phase.Through the specific surface Sv,analytical expression based on the Carman-Kozeny model was proposed to analyze the fluid permeability of dendritic mushy zone in directionally solidified peritectic alloys.In addition,it is interesting to find a rise in permeability K after peritectic reaction in both theoretical predication and experimental results,which is different from that in other alloys.The theoretical predictions show that this rise in fluid permeability K after pedtectic reaction is caused by the remelting/resolidification process on dendritic structure by the TGZM and G-T effects during peritectic solidification.     

Thermo‐elasto‐plastic finite element analysis of quasi‐state processes in Eulerian reference frames

An Eulerian finite element formulation for quasi‐state one way coupled thermo‐elasto‐plastic systems is presented. The formulation is suitable for modeling material processes such as welding and laser surfacing. In an Eulerian frame, the solution field of a quasi‐state process becomes steady state for the heat transfer problem and static for the stress problem. A mixed small deformation displacement elasto‐plastic formulation is proposed. The formulation accounts for temperature dependent material properties and exhibits a robust convergence. Streamline upwind Petrov–Galerkin (SUPG) is used to remove spurious oscillations. Smoothing functions are introduced to relax the non‐differentiable evolution equations and allow for the use of gradient (stiffness) solution scheme via the Newton–Raphson method. A 3‐dimensional simulation of a laser surfacing process is presented to exemplify the formulation. Results from the Eulerian formulation are in good agreement with results from the conventional Lagrangian formulation. However, the Eulerian formulation is approximately 15 times faster than the Lagrangian. Copyright © 2001 John Wiley & Sons, Ltd.     

A variational multiscale method for particle-cloud tracking in turbomachinery flows

We present a computational method for simulation of particle-laden flows in turbomachinery. The method is based on a stabilized finite element fluid mechanics formulation and a finite element particle-cloud tracking method. We focus on induced-draft fans used in process industries to extract exhaust gases in the form of a two-phase fluid with a dispersed solid phase. The particle-laden flow causes material wear on the fan blades, degrading their aerodynamic performance, and therefore accurate simulation of the flow would be essential in reliable computational turbomachinery analysis and design. The turbulent-flow nature of the problem is dealt with a Reynolds-Averaged Navier–Stokes model and Streamline-Upwind/Petrov–Galerkin/Pressure-Stabilizing/Petrov–Galerkin stabilization, the particle-cloud trajectories are calculated based on the flow field and closure models for the turbulence–particle interaction, and one-way dependence is assumed between the flow field and particle dynamics. We propose a closure model utilizing the scale separation feature of the variational multiscale method, and compare that to the closure utilizing the eddy viscosity model. We present computations for axial- and centrifugal-fan configurations, and compare the computed data to those obtained from experiments, analytical approaches, and other computational methods.     

Eulerian formulation using stabilized finite element method for large deformation solid dynamics

This paper describes an Eulerian formulation for large deformation solid dynamics. In the present Eulerian approach, an advective equation is solved using the Stream‐Upwind/Petrov–Galerkin finite element method. The Eulerian finite element method is applied to path‐dependent solid analyses such as impact bar and ductile necking problems. These computational results using the Eulerian finite element method are compared with the results obtained from using the Lagrangian finite element method and an Eulerian formulation based on a finite difference method. Copyright © 2007 John Wiley & Sons, Ltd.     

Petrov–Galerkin finite element methods with a hinged test space for singularly perturbed problems

In this paper we introduce finite element methods of Petrov–Galerkin type for the approximate solution of two-point boundary-value problems for singularly perturbed, second-order, ordinary, linear differential equations. We write down Petrov–Galerkin methods on a uniform mesh which have asymptotic error estimates, as the mesh size tends to zero, whose magnitude is independent of the singular perturbation parameter. This is in marked contrast to standard finite element methods which do not possess such a property on a uniform mesh. For these, typically, the error on a fixed uniform mesh blows up as the singular perturbation parameter tends to zero. This robust behaviour of these Petrov–Galerkin methods for singularly perturbed problems is achieved by choosing trial spaces of standard piecewise polynomial type, while the test spaces consist of hinged piecewise polynomials. We consider self-adjoint and non-self-adjoint two-point boundary-value problems with Dirichlet boundary conditions. We define hinged test spaces for both types of problem. We then introduce a number of sample problems and we present numerical solutions of these sample problems using a Petrov–Galerkin method with the appropriate hinged test space.     

Finite element simulation of thin liquid film flow and heat transfer including a hydraulic jump

A numerical simulation has been performed to investigate planar and radial flows of thin liquid film subject to constant wall temperature or constant wall heat flux, considering the surface tension effect. To simulate the variation of the film height including a hydraulic jump, an Arbitrary Lagrangian–Eulerian (ALE) method is adopted in describing the governing equations. An iterative split algorithm is used to improve the continuity constraint in time marching of the governing equations which are discretized by Streamline Upwind Petrov–Galerkin (SUPG) finite element method. It has been shown clearly that the surface tension has to be considered in order to describe realistically a hydraulic jump preceded by a capillary ripple. The variation of the film height is in good agreement with the existing experimental data. Physical aspects of how the flowrate as well as temperature‐dependent fluid properties affect the formation of the hydraulic jump and the variation of the Nusselt number are discussed rationally. Copyright © 1999 John Wiley & Sons, Ltd.     

Homogenized free surface flow in porous media for wet‐out processing

This paper presents a novel porous media model for homogenized free surface flow, representing wet‐out composites processing. The model is derived from concepts of homogenization applied to a compressible two‐phase flow, accounting for capillary effects and the concept of relative permeability. Based on mass balance considerations, we obtain a nonlinear set of equations of convection‐diffusion type involving the mixture (fluid) pressure and the degree of saturation as primary fields. A staggered Galerkin finite element approach is employed to decouple the solution. Moreover, the streamline upwind/Petrov‐Galerkin technique is applied to attenuate the oscillations in the saturation solutions. The model accuracy and convergence of the finite element solutions are demonstrated through 1‐dimensional and 2‐dimensional examples, representing resin transfer molding flow processes.     

强磁场下Cu-50%Ag合金定向凝固过程中的组织及固液界面形貌演变

目的 研究强磁场下Cu-50%(质量分数)Ag合金定向凝固过程中的组织演变、固液界面形貌变化及溶质迁移行为,分析强磁场对金属凝固过程的作用机制,为强磁场下的金属材料制备提供理论借鉴和指导。方法 在不同的凝固速率与磁场条件下进行定向凝固和淬火实验,对合金的定向凝固组织、糊状区与固液界面形貌以及溶质分布行为进行考察。结果 强磁场破坏了凝固组织的定向生长,使凝固组织转变为枝晶与等轴晶共存的形貌;强磁场诱发了熔体对流,减少了糊状区中溶质的含量;强磁场改变了固液界面处的溶质分布和固液界面形貌,破坏了固液界面的稳定性。结论 强磁场通过洛伦兹力和热电磁力的共同作用,诱发了糊状区内液相的纵向环流,改变了固液界面及糊状区中的组织形貌与元素分布。     

Two‐dimensional finite element method solution of a class of integro‐differential equations: Application to non‐Fickian transport in disordered media

A finite element method is developed to solve a class of integro‐differential equations and demonstrated for the important specific problem of non‐Fickian contaminant transport in disordered porous media. This transient transport equation, derived from a continuous time random walk approach, includes a memory function. An integral element is the incorporation of the well‐known sum‐of‐exponential approximation of the kernel function, which allows a simple recurrence relation rather than storage of the entire history. A two‐dimensional linear element is implemented, including a streamline upwind Petrov–Galerkin weighting scheme. The developed solver is compared with an analytical solution in the Laplace domain, transformed numerically to the time domain, followed by a concise convergence assessment. The analysis shows the power and potential of the method developed here. Copyright © 2017 John Wiley & Sons, Ltd.     

A DRD finite element formulation for computing turbulent reacting flows in gas turbine combustors

An effective multiscale treatment of turbulent reacting flows is presented with the use of a stabilized finite element formulation. The method proposed is developed based on the streamline-upwind/Petrov–Galerkin (SUPG) formulation, and includes discontinuity capturing in the form of a new generation “DRD” method, namely the “DRDJ” technique. The stabilized formulation is applied to finite-rate chemistry modelling based on mixture-fraction approaches with the so-called presumed-PDF technique. The turbulent combustion process is simulated for an aero-engine combustor configuration of RQL concept in non-premixed flame regime. The comparative analysis of the temperature and velocity fields demonstrate that the proposed SUPG+DRDJ formulation outperforms the stand-alone SUPG method. The improved accuracy is demonstrated in terms of the combustor overall performance, and the mechanisms involved in the distribution of the numerical diffusivity are also discussed.     

A control volume capacitance method for solidification modelling with mass transport

Capacitance methods are popular methods used for solidification modelling. Unfortunately, they suffer from a major drawback in that energy is not correctly transported through elements and so provides a source of inaccuracy. This paper is concerned with the development and application of a control volume capacitance method (CVCM) to problems where mass transport and solidification are combined. The approach adopted is founded on theory that describes energy transfer through a control volume (CV) moving relative to the transporting mass. An equivalent governing partial differential equation is established, which is designed to be transformable into a finite element system that is commonly used to model transient heat‐conduction problems. This approach circumvents the need to use the methods of Bubnov–Galerkin and Petrov–Galerkin and thus eliminates many of the stability problems associated with these approaches. An integration scheme is described that accurately caters for enthalpy fluxes generated by mass transport. Shrinkage effects are neglected in this paper as all the problems considered involve magnitudes of velocity that make this assumption reasonable. The CV approach is tested against known analytical solutions and is shown to be accurate, stable and computationally competitive. Copyright © 2002 John Wiley & Sons, Ltd.     

ANM for stationary Navier–Stokes equations and with Petrov–Galerkin formulation

This paper deals with the use of the asymptotic numerical method (ANM) for solving non‐linear problems, with particular emphasis on the stationary Navier–Stokes equation and the Petrov–Galerkin formulation. ANM is a combination of a perturbation technique and a finite element method allowing to transform a non‐linear problem into a succession of linear ones that admit the same tangent matrix. This method has been applied with success in non‐linear elasticity and fluid mechanics. In this paper, we apply the same kind of technique for solving Navier–Stokes equation with the so‐called Petrov–Galerkin weighting. The main difficulty comes from the fact that the non‐linearity is no more quadratic and it is not evident, in this case, to be able to compute a large number of terms of the perturbation series. Several examples of fluid mechanic are presented to demonstrate the performance of such a method. Copyright © 2001 John Wiley & Sons, Ltd.     

Consistent higher degree Petrov–Galerkin methods for the solution of the transient convection–diffusion equation

The solution of the convection–diffusion equation for convection dominated problems is examined using both N + 1 and N + 2 degree Petrov–Galerkin finite element methods in space and a Crank–Nicolson finite difference scheme in time. While traditional N + 1 degree Petrov–Galerkin methods, which use test functions one polynomial degree higher than the trial functions, work well for steady-state problems, they fail to adequately improve the solution for the transient problem. However, using novel N + 2 degree Petrov–Galerkin methods, which use test functions two polynomial degrees higher than the trial functions, yields dramatically improved solutions which in fact get better as the Courani number increases to 1·0. Specifically, cubic test functions with linear trial functions and quartic test functions in conjunction with quadratic trial functions are examined. Analysis and examples indicate that N + 2 degree Petrov–Galerkin methods very effectively eliminate space and especially time truncation errors. This results in substantially improved phase behaviour while not adversely affecting the ratio of numerical to analytical damping.     

Dispersion analysis of stabilized finite element methods for acoustic fluid interaction with Reissner–Mindlin plates

The application of stabilized finite element methods to model the vibration of elastic plates coupled with an acoustic fluid medium is considered. A complex‐wavenumber dispersion analysis of acoustic fluid interaction with Reissner–Mindlin plates is performed to quantify the accuracy of stabilized finite element methods for fluid‐loaded plates. Results demonstrate the improved accuracy of a recently developed hybrid least‐squares (HLS) plate element based on a modified Hellinger–Reissner functional, consistently combined with residual‐based methods for the acoustic fluid, compared to standard Galerkin and Galerkin gradient least‐squares plate elements. The technique of complex wavenumber dispersion analysis is used to examine the accuracy of the discretized system in the representation of free waves for fluid‐loaded plates. The influence of fluid and coupling matrices resulting from consistent implementation of pressure loading in the residual for the plate equation is examined and clarified for the different finite element approximations. Copyright © 2001 John Wiley & Sons, Ltd.     

An object oriented implementation of a front tracking finite element method for directional solidification processes

This paper focuses on the numerical simulation of phase‐change processes using a moving finite element technique. In particular, directional solidification and melting processes for pure materials and binary alloys are studied. The melt is modelled as a Boussinesq fluid and the transient Navier–Stokes equations are solved simultaneously with the transient heat and mass transport equations as well as the Stefan condition. The various streamline‐upwind/Petrov–Galerkin‐based FEM simulators developed for the heat, flow and mass transport subproblems are reviewed. The use of classes, virtual functions and smart pointers to represent and link the particular simulators in order to model a phase change process is discussed. The freezing front is modelled using a spline interpolation, while the mesh motion is defined from the freezing front motion using a transfinite mapping technique. Various two‐ and three‐dimensional numerical tests are analysed and discussed to demonstrate the efficiency of the proposed techniques. Copyright © 1999 John Wiley & Sons, Ltd.