A partial upwind finite element approximation for the stationary Navier-Stokes equations

A new partial upwind finite element method for steady viscous incompressible flow over a wide range of Reynolds numbers (Re) is proposed and its applications to a cavity flow problem are demonstrated. Numerical experiments show that the partial upwind scheme is effective for Re = 10³ when the conventional Galerkin type approximation produces poor results. The partial upwind scheme for the finite element method was originally developed by Ikeda (1983) for approximation of the advection diffusion equation. The advantage of the scheme is its effectiveness over a wide range of Peclet numbers due to the automatic switch in the approximation techniques of the advection term.     

Physically correct penalty-like formulations for accurate pressure calculation in finite element algorithms of the Navier-Stokes equations

Finite element algorithms for incompressible fluids, in particular the penalty function method, are re-examined and re-formulated based on physically consistent approximations to the mass conservation equation. These approximations lead to methods that satisfy all fluid conditions under the assumption of slightly compressible flow and directly yield approximations to the pressure field that are free from spurious oscillations. Penalty function-type algorithms are developed for direct steady-state calculations and for time-dependent simulations of viscous incompressible flows that are based on the bilinear isoparametric quadrilateral element and circumvent the Ladyszhenskaya, Babuska and Brezzi (LBB) condition. Further-more, methods that eliminate pressure oscillations in the low Mach number region of high-speed compressible flows are obtained. These methods are tested in examples that show their superior performance in approximating the pressure field.     

A stream function-vorticity finite element formulation for Navier-Stokes equations in multi-connected domain

In this paper we present a stream function-vorticity finite element formulation for Navier-Stokes equations which does not require an iterative procedure for satisfying the boundary conditions, and show that this formulation is of great advantage in solving the flow problems in a multi-connected domain.     

On the stream function-vorticity finite element solutions of Navier-Stokes equations

The stream function-vorticity method for solving Navier-Stokes equations with finite elements is often disregarded in view of the necessity of an iterative proceudre for satisfying the boundary conditions. The present work shows briefly that such iterations are unnecessary. Sample solutions for flows in a square cavity and in a channel with a step show good agreement with existing solutions.     

Comparative study of finite element formulations for the semiconductor drift-diffusion equations

A number of transient and steady-state finite element formulations of the semiconductor drift-diffusion equations are studied and compared with respect to their accuracy and efficiency on a simple test structure (the Mock diode). A new formulation, with a consistent interpolation function used to represent the electron and hole carrier densities throughout the set of semiconductor drift-diffusion and Poisson's equations, is introduced. Results highlight the advantages in using consistent interpolation functions showing an increased accuracy in the calculated values and a saving in data storage and execution time. The results also illustrate how the use of different time integration methods affect the number of time steps required during transient simulations. The combination of the fully consistent DFUS with appropriate time integration methods is found to yield a saving of up to 80 per cent of the execution time required for standard spatial/temporal discretization techniques. © 1997 by John Wiley & Sons, Ltd.     

Turbulent transition simulation using the k–ω model

This paper describes a novel approach in simulating laminar to turbulent transition by using two-equation models. The Total Stresses Limitation (TSL) concept is used to make the two-equation model capable of predicting high-Reynolds-number transitional flow. In order to handle the transition triggered by laminar separation at a low Reynolds number location, which commonly occurs in high speed flow, a sensor is introduced to detect separation and trigger transition in the separated zone. Test cases include the classical flat-plate turbulent boundary flow, and low-pressure turbine cascade flows at design and off-design conditions. © 1998 John Wiley & Sons, Ltd.     

Comparison of different finite element formulations for 3Dmagnetostatic problems

The results obtained with several three-dimensional software packages for magnetostatic field calculation using the finite-element method (FEM) are compared with regard to their accuracy and their computational time requirements. The packages are based on the vector potential (VPOT), the reduced scalar potential (RSP), and the total and reduced scalar potential (TSP+RSP), respectively. Results for an iron cylinder immersed in the field of a cylindrical coil are given. It is found that the finite-element formulation using a total and reduced scalar potential and the direct iteration method are useful for dealing with nonlinear magnetostatic field problems     

A block iterative finite element algorithm for numerical solution of the steady–state,compressible Navier–Stokes equations

An iterative method for numerically solving the time independent Navier–Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss–Seidel principle in block form to the systems of the non-linear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C⁰-cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and symptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.     

A general framework for interpreting time finite element formulations

In this work, an attempt is made at filling the apparent gap existing between the two major approaches evolved in the literature towards formulating space-time finite element methods. The first assumes Hamilton's Law as underlying concept, while the second performs a weighted residual approach on the ordinary differential equations emanating from the semidiscretization in the space dimension.A general framework is proposed in the following pages, where the configuration space and the phase space forms of Hamilton's Law provide the general statements of the problem of motion. Within this framework, different families of integration algorithms are derived, according to different interpretations of the boundary terms. The bi-discontinuous form is obtained as the consequence of a consistent impulsive formulation of dynamics, while the discontinuous Galerkin form is obtained when the boundary terms at the end of the time interval are appropriately approximated.     

Characteristics of piezoceramic and 3–3 piezocomposite hydrophones evaluated by finite element modelling

The transducer characteristics of hydrophones manufactured from porous 3–3 piezocomposites are compared with dense piezoceramic disc hydrophones using finite element modelling. Due to the complex porous structure of the 3–3 piezocomposites, a real-size 3-dimensional model was developed while a 2-dimensional axisymmetric model was constructed for the simple dense disc hydrophone. The electrical impedance and receiving sensitivity of the hydrophones in water were evaluated in the frequency range 10–100 kHz. The model results were compared with the experimental results. The sharp resonance peaks observed for the dense piezoceramic hydrophone were broadened to a large extent for porous piezocomposite hydrophones due to weaker coupling of the structure. The receiving sensitivity of piezocomposite hydrophones is found to be constant over the frequency range studied. The flat frequency response suggests that the 3–3 piezocomposites are useful for wide-band hydrophone applications.     

Different finite element formulations of 3D magnetostatic fields

Several finite-element formulations of three-dimensional magnetostatic fields are reviewed. Both nodal and edge elements are considered. The aim is to suggest remedies to some shortcomings of widely used methods. Various formulations are compared based on results for Problem No. 13 of the TEAM Workshops, a nonlinear magnetostatic problem involving thin iron plates     

A finite volume nodal point scheme for solving two dimensional Navier-Stokes equations

Summary A finite volume nodal point spatial discretization scheme for the computation of viscous fluxes in two dimensional Navier-Stokes equations has been presented here. The present scheme gives second order accurate first derivatives and at least first order accurate second derivatives even for stretched and skewed grid. It takes almost the same numerical efforts to solve full Navier-Stokes equations as that for using thin layer approximation. The scheme has been implemented to solve laminar viscous flows past NACA0012 aerofoil. To advance the solution in time a five stage Runge-Kutta scheme has been used. To accelerate the rate of convergence to steady state, local time stepping, residual averaging and enthalpy damping have been employed. Only a fourth order artificial dissipation has been used here for global stability of the solution. A comparative study of the results obtained by the present scheme for full Navier-Stokes equations and for thin layer approximation have been made with other numerical methods developed earlier.     

Penalty-function iterative procedures for mixed finite element formulations

Iterative methods for solving mixed finite element equations that correct displacement and stress unknowns in ‘staggered’ fashion are attracting increased attention. This paper looks at the problem from the standpoint of allowing fairly arbitrary approximations to be made on both the stiffness and compliance matrices used in solving for the corrections. The resulting iterative processes usually diverge unless stabilized with Courant penalty terms. An iterative procedure previously constructed for equality-constrained displacement models is recast to fit the mixed finite element formulation in which displacements play the role of Lagrange multipliers. The penalty function iteration is shown to reduce to an ordinary staggered stress-displacement iteration if the weight is set to zero. Convergence conditions for these procedures are stated and the potentially troublesome effect of prestress modes noted.     

Dual finite element formulations for lumped reluctances coupling

A method for coupling magnetostatic and magnetodynamic finite element formulations with lumped reluctances is developed. Two dual h and b-conform formulations are extended to define the necessary magnetic relations between the magnetic fluxes and the magnetomotive forces. Adequate surface scalar potentials are defined and adequate boundary terms in the weak formulations are used. The magnetic relations can then be included in a reluctance network, in which the coupling of finite element regions and lumped regions aims at higher computational efficiency. The methods are developed in three dimensions, using a coupling of nodal and edge finite element approximations for the unknowns, and can easily be particularized in two dimensions.     

Adaptive Eulerian–Lagrangian finite element method for advection–dispersion

A new adaptive finite element method is proposed for the advection–dispersion equation using an Eulerian–Lagrangian formulation. The method is based on a decomposition of the concentration field into two parts, one advective and one dispersive, in a rigorous manner that does not leave room for ambiguity. The advective component of steep concentration fronts is tracked forward with the aid of moving particles clustered around each front. Away from such fronts the advection problem is handled by an efficient modified method of characteristics called single-step reverse particle tracking. When a front dissipates with time, its forward tracking stops automatically and the corresponding cloud of particles is eliminated. The dispersion problem is solved by an unconventional Lagrangian finite element formulation on a fixed grid which involves only symmetric and diagonal matrices. Preliminary tests against analytical solutions of one- and two-dimensional dispersion in a uniform steady-state velocity field suggest that the proposed adaptive method can handle the entire range of Péclet numbers from 0 to ∞, with Courant numbers well in excess of 1.     

Stability analysis of some finite difference schemes for the Navier-Stokes equations

We study the stability of two standard finite differnce schemes for the unsteady Navier-Stokes equations. We write a scheme for small values of Reynolds number, where teh diffusion is predominant, and another one for large values of Reynolds number, where convection is predominant. The purpose of this results may be extended to unsteady problems leading to mixed hyperbolic-parabolic partia differenctial equations.     

A rational approach for choosing stress terms for hybrid finite element formulations

A new approach for choosing the stress terms for a hybrid stress element is based on the condition of vanishing of the virtual work along the element boundary due to the stress terms higher than constant and the additional incompatible displacement. Examples using 4-node plane stress elements have shown that when the incompatible displacements also satisfy the constant strain patch test the resulting elements will provide the most accurate solutions. Advantages of this approach for the formulation of an axisymmetric solid are also indicated.     

A combined space–time extended finite element method

The Newmark method for the numerical integration of second order equations has been extensively used and studied along the past fifty years for structural dynamics and various fields of mechanical engineering. Easy implementation and nice properties of this method and its derivatives for linear problems are appreciated but the main drawback is the treatment of discontinuities. Zienkiewicz proposed an approach using finite element concept in time, which allows a new look at the Newmark method. The idea of this paper is to propose, thanks to this approach, the use of a time partition of the unity method denoted Time Extended Finite Element Method (TX‐FEM) for improved numerical simulations of time discontinuities. An enriched basis of shape functions in time is used to capture with a good accuracy the non‐polynomial part of the solution. This formulation allows a suitable form of the time‐stepping formulae to study stability and energy conservation. The case of an enrichment with the Heaviside function is developed and can be seen as an alternative approach to time discontinuous Galerkin method (T‐DGM), stability and accuracy properties of which can be derived from those of the TX‐FEM. Then Space and Time X‐FEM (STX‐FEM) are combined to obtain a unified space–time discretization. This combined STX‐FEM appears to be a suitable technique for space–time discontinuous problems like dynamic crack propagation or other applications involving moving discontinuities. Copyright © 2005 John Wiley & Sons, Ltd.