Non-linear dynamics of a drill-string with uncertain model of the bit–rock interaction

The drill-string dynamics is difficult to predict due to the non-linearities and uncertainties involved in the problem. In this paper a stochastic computational model is proposed to model uncertainties in the bit–rock interaction model. To do so, a new strategy that uses the non-parametric probabilistic approach is developed to take into account model uncertainties in the bit–rock non-linear interaction model. The mean model considers the main forces applied to the column such as the bit–rock interaction, the fluid–structure interaction and the impact forces. The non-linear Timoshenko beam theory is used and the non-linear dynamical equations are discretized by means of the finite element method.     

A variational multiscale Newton–Schur approach for the incompressible Navier–Stokes equations

In the following paper, we present a consistent Newton–Schur (NS) solution approach for variational multiscale formulations of the time‐dependent Navier–Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three‐dimensional problems and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non‐symmetric matrices. In addition to the quadratic convergence characteristics of a Newton–Raphson‐based scheme, the NS approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two‐level approach to stabilizing the incompressible Navier–Stokes equations based on a coarse and fine‐scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three‐dimensional problems for Reynolds number up to 1000 including steady and time‐dependent flows. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the time‐dependent Navier–Stokes equations

This article considers numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the two‐dimensional non‐stationary Navier–Stokes equations. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the Crank–Nicolson scheme for the linear term and the explicit Adams–Bashforth scheme for the nonlinear term. Comparison with other methods, through a series of numerical experiments, shows that this method is almost unconditionally stable and convergent, i.e. stable and convergent when the time step is smaller than a given constant. Copyright © 2009 John Wiley & Sons, Ltd.     

Consideration of the yarn–yarn interactions in meso/macro discrete model of fabric. Part I: Single yarn behaviour

A mesoscopic discrete model of dry fabric has been developed, based on the yarn–yarn interactions occurring at the yarns crossing points. The fabric yarns, described initially by a Fourier series development, are discretized into elastic straight bars represented by stretching springs and connected at frictionless hinges by rotational springs. The motion of each node is described by a lateral displacement and a rotation. The expression of the reaction force exerted by the transverse yarns at the contact points is assessed, from which the work of the reaction forces is established. The equilibrium shape of the yarn is obtained as the minimum of its total potential energy, accounting for the work of the reaction forces due to the transverse yarns. Simulations of a traction curve of a single yarn are performed, that evidence the effect of the yarn interactions. The two principal deformation mechanisms, the variation of undulation and the yarn stretching, are separately analysed.     

Raviart–Thomas and Brezzi–Douglas–Marini finite‐element approximations of the shallow‐water equations

An analysis of the discrete shallow‐water equations using the Raviart–Thomas and Brezzi–Douglas–Marini finite elements is presented. For inertia–gravity waves, the discrete formulations are obtained and the dispersion relations are computed in order to quantify the dispersive nature of the schemes on two meshes made up of equilateral and biased triangles. A linear algebra approach is also used to ascertain the possible presence of spurious modes arising from the discretization. The geostrophic balance is examined and the smallest representable vortices are characterized on both structured and unstructured meshes. Numerical solutions of two test problems to simulate gravity and Rossby modes are in good agreement with the analytical results. Copyright © 2007 John Wiley & Sons, Ltd.     

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection‐dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two‐dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.     

A reappraisal of Taylor–Galerkin algorithm for drying–wetting areas in shallow water computations

Non‐linear shallow water equations are a useful approximation to phenomena such as estuary dynamics, tidal propagation, breaking of a dam, flood waves, etc. Quite frequently they involve propagation over dry beds and drying of wet zones, for which boundary changes. To solve this problem either special techniques such as remeshing and Arbitrary Lagrangian Eulerian formulations or algorithms initially developed for flows exhibiting shocks have been proposed in past years. The purpose of this paper is to show how classical finite elements formulations, such as Taylor–Galerkin can be applied to solve the problem to wetting–drying areas in a simple yet efficient manner. Copyright © 2002 John Wiley & Sons, Ltd.     

Finite element solution of the Navier–Stokes equations by a velocity–vorticity method

A velocity–vorticity formulation of the Navier–Stokes equations is presented as an alternative to the primitive variables approach. The velocity components and the vorticity are solved for in a fully coupled manner using a Newton method. No artificial viscosity is required in this formulation. The pressure is updated by a method allowing natural imposition of boundary conditions. Incompressible and subsonic results are presented for two-dimensional laminar internal flows up to high Reynolds numbers.     

Space–time integrated least squares: a time‐marching approach

A time‐marching formulation is derived from the space–time integrated least squares (STILS) method for solving a pure hyperbolic convection equation and is numerically compared to various known methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical results for the conservation of energy of Maxwell media for the Rayleigh–Stokes problem

This publication continues our studies of analytical solutions of the Rayleigh–Stokes problem for Maxwell fluids [J. Zierep, C. Fetecau, Energetic balance for the Rayleigh–Stokes problem of a Maxwell fluid, Int. J. Eng. Sci. 45 (2007) 617–627]. We start from the Fourier sine transform. The numerical result is given and discussed for the velocity u, the power of the wall shear stresses L, the dissipation Φand the boundary layer thickness δ. These new results are important for nature and technology.     

Asymptotic–Newton method for solving incompressible flows

In this paper we present a comparative study of three non-linear schemes for solving finite element systems of Navier–Stokes incompressible flows. The first scheme is the classical Newton–Raphson linearization, the second one is the modified Newton–Raphson linearization and the last one is a new scheme called the asymptotic–Newton method. The relative efficiency of these approaches is evaluated over a large number of examples. © 1997 John Wiley & Sons, Ltd.     

A streamfunction–velocity approach for 2D transient incompressible viscous flows

Electrodeposition is a widely used technique for the fabrication of high aspect ratio microstructures. In recent years, much research has been focused within this area aiming to understand the physics behind the filling of high aspect ratio vias and trenches on substrates and in particular how they can be made without the formation of voids in the deposited material. This paper reports on the fundamental work towards the advancement of numerical algorithms that can predict the electrodeposition process in micron scaled features. Two different numerical approaches have been developed, which capture the motion of the deposition interface and 2‐D simulations are presented for both methods under two deposition regimes: those where surface kinetics is governed by Ohm's law and the Butler–Volmer equation, respectively. In the last part of this paper the modelling of acoustic forces and their subsequent impact on the deposition profile through convection is examined. Copyright © 2009 John Wiley & Sons, Ltd.     

Artificial neural network prediction to the hot compressive deformation behavior of Al–Cu–Mg–Ag heat-resistant aluminum alloy

The behavior of the flow stress of Al-Cu-Mg-Ag heat-resistant aluminum alloys during hot compression deformation was studied by thermal simulation test. The temperature and the strain rate during hot compression were 340-500 °C, 0.001 s⁻¹ to 10 s⁻¹, respectively. Constitutive equations and an artificial neural network (ANN) model were developed for the analysis and simulation of the flow behavior of the Al-Cu-Mg-Ag alloys. The inputs of the model are temperature, strain rate and strain. The output of the model is the flow stress. Comparison between constitutive equations and ANN results shows that ANN model has a better prediction power than the constitutive equations.     

Pseudostress–velocity formulation for incompressible Navier–Stokes equations

This paper presents a numerical algorithm using the pseudostress–velocity formulation to solve incompressible Newtonian flows. The pseudostress–velocity formulation is a variation of the stress–velocity formulation, which does not require symmetric tensor spaces in the finite element discretization. Hence its discretization is greatly simplified. The discrete system is further decoupled into an H ( div ) problem for the pseudostress and a post‐process resolving the velocity. This can be done conveniently by using the penalty method for steady‐state flows or by using the time discretization for nonsteady‐state flows. We apply this formulation to the 2D lid‐driven cavity problem and study its grid convergence rate. Also, computational results of the time‐dependent‐driven cavity problem and the flow past rectangular problem are reported. Copyright © 2009 John Wiley & Sons, Ltd.     

A new stable space–time formulation for two‐dimensional and three‐dimensional incompressible viscous flow

A space–time finite element method for the incompressible Navier–Stokes equations in a bounded domain in ?^d (with d=2 or 3) is presented. The method is based on the time‐discontinuous Galerkin method with the use of simplex‐type meshes together with the requirement that the space–time finite element discretization for the velocity and the pressure satisfy the inf–sup stability condition of Brezzi and Babu?ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two‐dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

A fractional-step method for the incompressible Navier–Stokes equations related to a predictor–multicorrector algorithm

An implicit fractional-step method for the numerical solution of the time-dependent incompressible Navier–Stokes equations in primitive variables is studied in this paper. The method, which is first-order-accurate in the time step, is shown to converge to an exact solution of the equations. By adequately splitting the viscous term, it allows the enforcement of full Dirichlet boundary conditions on the velocity in all substeps of the scheme, unlike standard projection methods. The consideration of this method was actually motivated by the study of a well-known predictor–multicorrector algorithm, when this is applied to the incompressible Navier–Stokes equations. A new derivation of the algorithm in a general setting is provided, showing in what sense it can also be understood as a fractional-step method; this justifies, in particular, why the original boundary conditions of the problem can be enforced in this algorithm. Two different finite element interpolations are considered for the space discretization, and numerical results obtained with them for standard benchmark cases are presented. © 1998 John Wiley & Sons, Ltd.     

Third‐order‐accurate semi‐implicit Runge–Kutta scheme for incompressible Navier–Stokes equations

A semi‐implicit three‐step Runge–Kutta scheme for the unsteady incompressible Navier–Stokes equations with third‐order accuracy in time is presented. The higher order of accuracy as compared to the existing semi‐implicit Runge–Kutta schemes is achieved due to one additional inversion of the implicit operator I‐τγL, which requires inversion of tridiagonal matrices when using approximate factorization method. No additional solution of the pressure‐Poisson equation or evaluation of Navier–Stokes operator is needed. The scheme is supplied with a local error estimation and time‐step control algorithm. The temporal third‐order accuracy of the scheme is proved analytically and ascertained by analysing both local and global errors in a numerical example. Copyright © 2005 John Wiley & Sons, Ltd.     

Strain localization analysis using a large deformation anisotropic elastic–plastic model coupled with damage

Sheet metal forming processes generally involve large deformations together with complex loading sequences. In order to improve numerical simulation predictions of sheet part forming, physically-based constitutive models are often required. The main objective of this paper is to analyze the strain localization phenomenon during the plastic deformation of sheet metals in the context of such advanced constitutive models. Most often, an accurate prediction of localization requires damage to be considered in the finite element simulation. For this purpose, an advanced, anisotropic elastic–plastic model, formulated within the large strain framework and taking strain-path changes into account, has been coupled with an isotropic damage model. This coupling is carried out within the framework of continuum damage mechanics. In order to detect the strain localization during sheet metal forming, Rice’s localization criterion has been considered, thus predicting the limit strains at the occurrence of shear bands as well as their orientation. The coupled elastic–plastic-damage model has been implemented in Abaqus/implicit. The application of the model to the prediction of Forming Limit Diagrams (FLDs) provided results that are consistent with the literature and emphasized the impact of the hardening model on the strain-path dependency of the FLD. The fully three-dimensional formulation adopted in the numerical development allowed for some new results – e.g. the out-of-plane orientation of the normal to the localization band, as well as more realistic values for its in-plane orientation.