The CIP method embedded in finite element discretizations of incompressible fluid flows

Quite effective low‐order finite element and finite volume methods for incompressible fluid flows have been established and are widely used. However, higher‐order finite element methods that are stable, have high accuracy and are computationally efficient are still sought. Such discretization schemes could be particularly useful to establish error estimates in numerical solutions of fluid flows. The objective of this paper is to report on a study in which the cubic interpolated polynomial (CIP) method is embedded into 4‐node and 9‐node finite element discretizations of 2D flows in order to stabilize the convective terms. To illustrate the capabilities of the formulations, the results obtained in the solution of the driven flow square cavity problem are given. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical analysis of a new Eulerian–Lagrangian finite element method applied to steady‐state hot rolling processes

A finite element code for steady‐state hot rolling processes of rigid–visco‐plastic materials under plane–strain conditions was developed in a mixed Eulerian–Lagrangian framework. This special set up allows for a direct calculation of the local deformations occurring at the free surfaces outside the contact region between the strip and the work roll. It further simplifies the implementation of displacement boundary conditions, such as the impenetrability condition. When applied to different practical hot rolling situations, ranging from thick slab to ultra‐thin strip rolling, the velocity–displacement based model (briefly denoted as vu‐model) in this mixed Eulerian–Lagrangian reference system proves to be a robust and efficient method. The vu‐model is validated against a solely velocity‐based model (vv‐model) and against elementary methods based on the Kármán–Siebel and Orowan differential equations. The latter methods, when calibrated, are known to be in line with experimental results for homogeneous deformation cases. For a massive deformation it is further validated against the commercial finite‐element software package Abaqus/Explicit. It is shown that the results obtained with the vu‐model are in excellent agreement with the predictions of the vv‐model and that the vu‐model is even more robust than its vv‐counterpart. Throughout the study we assumed a rigid cylindrical work roll; only for the homogeneous test case, we also investigated the effect of an elastically deformable work roll within the frame of the Jortner Green's function method. The new modelling approach combines the advantages of conventional Eulerian and Lagrangian modelling concepts and can be extended to three dimensions in a straightforward manner. Copyright © 2004 John Wiley & Sons, Ltd.     

Parallel computing of high‐speed compressible flows using a node‐based finite‐element method

An efficient parallel computing method for high‐speed compressible flows is presented. The numerical analysis of flows with shocks requires very fine computational grids and grid generation requires a great deal of time. In the proposed method, all computational procedures, from the mesh generation to the solution of a system of equations, can be performed seamlessly in parallel in terms of nodes. Local finite‐element mesh is generated robustly around each node, even for severe boundary shapes such as cracks. The algorithm and the data structure of finite‐element calculation are based on nodes, and parallel computing is realized by dividing a system of equations by the row of the global coefficient matrix. The inter‐processor communication is minimized by renumbering the nodal identification number using ParMETIS. The numerical scheme for high‐speed compressible flows is based on the two‐step Taylor–Galerkin method. The proposed method is implemented on distributed memory systems, such as an Alpha PC cluster, and a parallel supercomputer, Hitachi SR8000. The performance of the method is illustrated by the computation of supersonic flows over a forward facing step. The numerical examples show that crisp shocks are effectively computed on multiprocessors at high efficiency. Copyright © 2003 John Wiley & Sons, Ltd.     

Self-similarity in particle-laden flows at constant volume

This paper deals with the evolution of a localized, constant-volume initial condition on an incline into a spreading descending thin-film solution. Clear fluids in this geometry are known to have a front position that moves according to a t ^1/3 scaling law, based on similarity-solution analysis by Huppert (Nature 300:427–429, 1982). The same dynamics are investigated for particle-laden flow using a recently proposed lubrication model and physical experiments. The analysis includes the role of a precursor in the model. In the lubrication model, the height of the precursor significantly influences the position of the fluid front, independent of particles settling in the direction of flow. By comparing theory with experiments it is shown that the t ^1/3 scaling law persists, to leading order, for particle-laden flows with particle settling. However, additional physics is needed in the existing lubrication models to quantitatively explain departures from clear-fluid self-similarity due to particle settling.     

Non‐linear finite element analysis of mechanical electrochemical phenomena in hydrated soft tissues based on triphasic theory

The well‐accepted triphasic theory for modelling the mechano‐electrochemical phenomena of charged hydrated soft tissue has been limited to infinitesimal deformation problems due to the difficulty of defining a common reference configuration for the whole tissue. In this paper, an imaginary reference configuration for soft tissue under large deformation is established based on the reference configuration of a solid matrix and a Piola transformation of the relative velocities of the fluid and ionic phases. A non‐linear finite element analysis formulation is proposed by applying a weighted residual method to the reformulated governing equations of triphasic theory reformulated in the imaginary reference configuration, with the displacement of the solid, fluid flows, ionic molar flows, hydrostatic pressure, and electrical potential as the unknown variables. After verifying the proposed finite‐element formulation by comparing the results of a linear‐confined compression problem with those obtained by the finite difference method, the numerical analysis of a three‐dimensional free‐swelling problem of articular cartilage with large deformation, and a strong non‐linearity in the material properties is carried out to reproduce the curling behaviour of articular cartilage strips in vitro when submerged in solution baths of various concentrations. The results obtained by finite element analysis are in agreement with those measured experimentally. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical modelling of the propagation of fast landslides using the finite element method

This paper proposes a two‐dimensional (2D) model for the analysis of the propagation of fast landslides involving a fluidized material such as debris and mud flows, flowslides and avalanching flows. The model is based on the Navier–Stokes depth‐integrated equations. To incorporate the effect of steep slopes and centrifugal forces due to the high velocities characterizing the flowslides and the bed curvature, a curvilinear system of reference is used. The corresponding equations of motion are complemented by depth‐averaged constitutive equations and bed friction laws. The resulting set of differential equations are solved using the two‐step Taylor–Galerkin algorithm. This algorithm has been used by the authors to solve hydraulic and dam‐break problems using the finite element method. Owing to the importance of the source term compared to the advection component, the proposed algorithm follows a splitting scheme using a fourth‐order Runge–Kutta method for integrating the friction and slope components. The performance of the overall approach has been checked in a number of examples. The analysis of the results provides insights into the key elements of the model and shows the adequacy of the method to solve real problems where merging and splitting of the flow occur. Copyright © 2004 John Wiley & Sons, Ltd.     

Two‐fluid model with interface sharpening

Two‐fluid models are applicable for simulations of all types of two‐phase flows ranging from separated flows with large characteristic interfacial length scales to highly dispersed flows with very small characteristic interfacial length scales. The main drawback of the two‐fluid model, when used for simulations of stratified flows, is the numerical diffusion of the interface. Stratified flows can be easily and more accurately solved with interface tracking methods; however, these methods are limited to the flows, that do not develop into dispersed types of flows. The present paper describes a new approach, where the advantage of the two‐fluid model is combined with the conservative level set method for interface tracking. The advection step of the volume fraction transport equation is followed by the interface sharpening, which preserves the thickness of the interface during the simulation. The proposed two‐fluid model with interface sharpening was found to be more accurate than the existing two‐fluid models. The mixed flow with both: stratified and dispersed flow, is simulated with the coupled model in this paper. In the coupled model, the dispersed two‐fluid model and two‐fluid model with interface sharpening are used locally, depending on the parameter which recognizes the flow regime. Copyright © 2010 John Wiley & Sons, Ltd.     

The effect of grease on the activated sludge process in wastewater treatment

Abstract

The purpose of this paper is to introduce the major potentials and perspectives of applications of finite element analysis in solving the problems of shallow water wave equations. One‐dimensional and two‐dimensional shallow water wave equations will both be incorporated into the modeling procedures. For one‐dimensional flows, the models will cover the typical single channels, confluence channels system, division channels system, and natural river systems. As far as two‐dimensional flows are concerned, the overland flows are investigated. The simulation results are compared with the data obtained by physical modeling and field observation and with the results of other existing literature. The models were found to be very feasible in modeling the complex flow fields of shallow water wave equation problems.     

An implicit method for the analysis of transient flows in pipe networks

Existing methods for the analysis of transient flows in pipe networks are often geared towards certain types of flows such as gas flows vis‐à‐vis liquid flows or isothermal flows vis‐à‐vis non‐isothermal flows. Also, simplifying assumptions are often made which introduce inaccuracies when the method is applied outside the domain for which it was originally intended. This paper describes an implicit finite difference method based on the simultaneous pressure correction approach which is valid for both liquid and gas flows, for both isothermal and non‐isothermal flows and for both fast and slow transients. The problematic convective acceleration term in the momentum equation, often neglected in other methods, is retained but eliminated by casting the momentum equation in an alternative form. The accuracy and stability of the method, depending on a time‐step weighing factor α, are illustrated by analysing fast transients in a pipeline and simple branching network. The proposed method compares very well with the second‐order‐method of characteristics and the two‐step Lax–Wendroff method. The advantages of the present method is its speed over a range of problems including both fast and slow transients, its accuracy, its stability and its flexibility. Copyright © 2001 John Wiley & Sons, Ltd.     

Finite element analysis of earthquake-induced sloshing in axisymmetric tanks

A finite element anaylysis to predict the sloshing dislacements and hydrodynamic pressures in liquid-filled tanks subjected to earthquake ground motions is presented. Finite element equations were derived using the Galerkin formulation, and the predicted results were checked against the test data, showing a good agreement between the test and finite element results.     

An essentially non‐oscillatory semi‐Lagrangian method for tidal flow simulations

We develop an essentially non‐oscillatory semi‐Lagrangian method for solving two‐dimensional tidal flows. The governing equations are derived from the incompressible Navier–Stokes equations with assumptions of shallow water flows including bed frictions, eddy viscosity, wind shear stresses and Coriolis forces. The method employs the modified method of characteristics to discretize the convective term in a finite element framework. Limiters are incorporated in the method to reconstruct an essentially non‐oscillatory algorithm at minor additional cost. The central idea consists in combining linear and quadratic interpolation procedures using nodes of the finite element where departure points are localized. The resulting semi‐discretized system is then solved by an explicit Runge–Kutta Chebyshev scheme with extended stages. This scheme adds in a natural way a stabilizing stage to the conventional Runge–Kutta method using the Chebyshev polynomials. The proposed method is verified for the recirculation tidal flow in a channel with forward‐facing step. We also apply the method for simulation of tidal flows in the Strait of Gibraltar. In both test problems, the proposed method demonstrates its ability to handle the interaction between water free‐surface and bed frictions. Copyright © 2009 John Wiley & Sons, Ltd.     

Modelling convection in solidification processes using stabilized finite element techniques

Solidification of dendritic alloys is modelled using stabilized finite element techniques to study convection and macrosegregation driven by buoyancy and shrinkage. The adopted governing macroscopic conservation equations of momentum, energy and species transport are derived from their microscopic counterparts using the volume‐averaging method. A single domain model is considered with a fixed numerical grid and without boundary conditions applied explicitly on the freezing front. The mushy zone is modelled here as a porous medium with either an isotropic or an anisotropic permeability. The stabilized finite‐element scheme, previously developed by authors for modelling flows with phase change, is extended here to include effects of shrinkage, density changes and anisotropic permeability during solidification. The fluid flow scheme developed includes streamline‐upwind/Petrov–Galerkin (SUPG), pressure stabilizing/Petrov–Galerkin, Darcy stabilizing/Petrov–Galerkin and other stabilizing terms arising from changes in density in the mushy zone. For the energy and species equations a classical SUPG‐based finite element method is employed with minor modifications. The developed algorithms are first tested for a reference problem involving solidification of lead–tin alloy where the mushy zone is characterized by an isotropic permeability. Convergence studies are performed to validate the simulation results. Solidification of the same alloy in the absence of shrinkage is studied to observe differences in macrosegregation. Vertical solidification of a lead–tin alloy, where the mushy zone is characterized by an anisotropic permeability, is then simulated. The main aim here is to study convection and demonstrate formation of freckles and channels due to macrosegregation. The ability of stabilized finite element methods to model a wide variety of solidification problems with varying underlying phenomena in two and three dimensions is demonstrated through these examples. Copyright © 2005 John Wiley & Sons, Ltd.     

On the reinitialization procedure in a narrow‐band locally refined level set method for interfacial flows

A local level set algorithm for simulating interfacial flows described by the two‐dimensional incompressible Navier–Stokes equations is presented. The governing equations are solved using a finite‐difference discretization on a Cartesian grid and a second‐order approximate projection method. The level set transport and reinitialization equations are solved in a narrow band around the interface using an adaptive refined grid, which is reconstructed every time step and refined using a simple uniform cell‐splitting operation within the band. Instabilities at the border of the narrow band are avoided by smoothing the level set function in the outer part of the band. The influence of different PDE‐based reinitialization strategies on the accuracy of the results is investigated. The ability of the proposed method to accurately compute interfacial flows is discussed using different tests, namely the advection of a circle of fluid in two different time‐reversed vortex flows, the advection of Zalesak's rotating disk, the propagation of small‐amplitude gravity and capillary waves at the interface between two superposed viscous fluids in deep water, and a classical test of Rayleigh–Taylor instability with and without surface tension effects. The interface location error and area loss for some of the results obtained are compared with those of a recent particle level set method. Copyright © 2005 John Wiley & Sons, Ltd.     

A conservative embedded boundary method for an inviscid compressible flow coupled with a fragmenting structure

We present an embedded boundary method for the interaction between an inviscid compressible flow and a fragmenting structure. The fluid is discretized using a finite volume method combining Lax–Friedrichs fluxes near the opening fractures, where the density and pressure can be very low, with high‐order monotonicity‐preserving fluxes elsewhere. The fragmenting structure is discretized using a discrete element method based on particles, and fragmentation results from breaking the links between particles. The fluid‐solid coupling is achieved by an embedded boundary method using a cut‐cell finite volume method that ensures exact conservation of mass, momentum, and energy in the fluid. A time explicit approach is used for the computation of the energy and momentum transfer between the solid and the fluid. The embedded boundary method ensures that the exchange of fluid and solid momentum and energy is balanced. Numerical results are presented for two‐dimensional and three‐dimensional fragmenting structures interacting with shocked flows. Copyright © 2015 John Wiley & Sons, Ltd.     

Simulation of two‐ and three‐dimensional viscoplastic flows using adaptive mesh refinement

This paper presents a finite element solver for the simulation of steady non‐Newtonian flow problems, using a regularized Bingham model, with adaptive mesh refinement capabilities. The solver is based on a stabilized formulation derived from the variational multiscale framework. This choice allows the introduction of an a posteriori error indicator based on the small scale part of the solution, which is used to drive a mesh refinement procedure based on element subdivision. This approach applied to the solution of a series of benchmark examples, which allow us to validate the formulation and assess its capabilities to model 2D and 3D non‐Newtonian flows.     

A Lagrangian finite element approach for the analysis of fluid–structure interaction problems

A Lagrangian finite element method for the analysis of incompressible Newtonian fluid flows, based on a continuous re‐triangulation of the domain in the spirit of the so‐called Particle Finite Element Method, is here revisited and applied to the analysis of the fluid phase in fluid–structure interaction problems. A new approach for the tracking of the interfaces between fluids and structures is proposed. Special attention is devoted to the mass conservation problem. It is shown that, despite its Lagrangian nature, the proposed combined finite element‐particle method is well suited for large deformation fluid–structure interaction problems with evolving free surfaces and breaking waves. The method is validated against the available analytical and numerical benchmarks. Copyright © 2010 John Wiley & Sons, Ltd.     

Computations of high‐speed compressible flows with adaptive cell‐centered finite element method

Abstract

An upwind cell‐centered finite element formulation is combined with an adaptive meshing technique to solve Navier‐Stokes equations for high‐speed inviscid and viscous compressible flows. The finite element formulation and the computational procedure are described. An adaptive meshing technique is applied to increase the analysis solution accuracy, as well as to minimize the computational time and the computer memory requirement. The efficiency of the combined method is evaluated by the examples of Mach 2.6 inviscid flow in a channel with compression and expansion ramps, Mach 6.47 inviscid and viscous flows past a cylinder, and Mach 4 viscous flow over a flat plate.     

Large‐eddy simulation of separated leading‐edge flow in general co‐ordinates

An incompressible separated transitional boundary‐layer flow on a flat plate with a semi‐circular leading edge has been simulated and a very good agreement with the experimental data has been obtained, demonstrating how this technique may be applied even when finite difference formulae are used in the periodic direction. The entire transition process has been elucidated and vortical structures have been identified at different stages during the transition process. Efficient numerical methods for the large‐eddy simulation (LES) of turbulent flows in complex geometry are developed. The methods used are described in detail: body‐fitted co‐ordinates with the contravariant velocity components of the general Navier–Stokes equations discretized on a staggered mesh with a dynamic subgrid‐scale model in general co‐ordinates. The main source of computational expense in simulations for incompressible flows is due to the solution of a Poisson equation for pressure. This is especially true for flows in complex geometry. Fourier techniques can be employed to speed up the pressure solution significantly for a flow which is periodic in one dimension. With simple conditions fulfilled, it is possible to Fourier transform a discrete elliptic equation such as the Poisson equation for the pressure field, decomposing the problem into a set of two‐dimensional problems of similar type (Poisson‐like). Even when a complex geometry and body‐fitted curvilinear co‐ordinates are used in the other two dimensions, as in the present case, the resulting Fourier‐transformed 2D problems are much more efficiently solved than the 3D problem by iterative means. Copyright © 2000 John Wiley & Sons, Ltd.     

An over‐current protection module for telephone network line cards—an analysis of electro‐thermal properties

In this paper the results of a finite‐element analysis of the electro‐thermal behaviour of an over‐current protection thick‐film hybrid module are presented. The module was designed for protecting the line card of a telephone network against an abnormal surge of current, resulting from accidental shorts between adjacent power feed lines. The switching time of the device is crucial to its effectiveness as a protective element. A transient finite‐element thermal analysis was performed in order to predict the dynamic temperature states at the critical points of the circuit design and to evaluate the influence on the switching characteristics. A comparison between simulated and practical results is given. Copyright © 2002 John Wiley & Sons, Ltd.     

A Jacobian‐free‐based IIM for incompressible flows involving moving interfaces with Dirichlet boundary conditions

In this paper, a finite difference marker‐and‐cell (MAC) scheme is presented for the steady Stokes equations with moving interfaces and Dirichlet boundary condition. The moving interfaces are represented by Lagrangian control points and their position is updated implicitly using a Jacobian‐free approach within each time step. The forces at the moving interfaces are calculated from the position of the interfaces and interpolated using cubic splines and then applied to the fluid through the related jump conditions. The proposed Jacobian‐free Newton–generalized minimum residual (GMRES) method avoids the need to form and store the matrix explicitly in the computation of the inverse of the Jacobian and betters numerical stability. The Stokes equations are discretized on a MAC grid via a second‐order finite difference scheme with the incorporation of jump contributions and the resulting saddle point system is solved by the conjugate gradient Uzawa‐type method. Numerical results demonstrate very well the accuracy and effectiveness of the proposed method. The present algorithm has been applied to solve incompressible Navier–Stokes flows with moving interfaces. Copyright © 2010 John Wiley & Sons, Ltd.