CBS versus GLS stabilization of the incompressible Navier–Stokes equations and the role of the time step as stabilization parameter

In this work we compare two apparently different stabilization procedures for the finite element approximation of the incompressible Navier–Stokes equations. The first is the characteristic‐based split (CBS). It combines the characteristic Galerkin method to deal with convection dominated flows with a classical splitting technique, which in some cases allows us to use equal velocity–pressure interpolations. The second approach is the Galerkin‐least‐squares (GLS) method, in which a least‐squares form of the element residual is added to the basic Galerkin equations. It is shown that both formulations display similar stabilization mechanisms, provided the stabilization parameter of the GLS method is identified with the time step of the CBS approach. This identification can be understood from a formal Fourier analysis of the linearized problem

Finite volume VS finite element. Is there really a choice

The finite volume method appears to be a particular case of finite elements with a non Galerkin weighting. It is course less accurate for self adjoint problems but has some computationally useful features for first order equations involving only surface integrals. For certain problems this is a substational economy and leads to computationally useful approximations. &nbsp

A general algorithm for compressible and incompressible flows. Part III: The semi‐implicit form

In this paper we consider some particular aspects related to the semi‐implicit version of a fractional step finite element method for compressible flows that we have developed recently. The first is the imposition of boundary conditions. We show that no boundary conditions at all need to be imposed in the first step where an intermediate momentum is computed. This allows us to impose the real boundary conditions for the pressure, a point that turns out to be very important for compressible flows. The main difficulty of the semi‐implicit form of the scheme arises in the solution of the continuity equation, since it involves both the density and the pressure. These two variables can be related through the equation of state, which in turn introduces the temperature as a variable in many cases. We discuss here the choice of variables (pressure or density) and some strategies to solve the continuity equation. The final point that we study is the behaviour of the scheme in the incompressible limit. It is shown that the method has an inherent pressure dissipation that allows us to reach this limit without having to satisfy the classical compatibility conditions for the interpolation of the velocity and the pressur

A hierarchical finite element method based on the partition of unity

In this paper we consider the application of hierarchical functions to base approximations which are a partition of unity. The particular hierarchical functions used are added to base finite element interpolations which, for Co approximations, are a particular case of the partition of unity. We also show how the functions may be constructed to preserve the interpolation property of the base finite element functions. An application to linear elasticity is used to illustrate the properties and stability of the approximation.&nbsp

The Characteristic‐Based Split (CBS) scheme—a unified approach to fluid dynamics

This paper presents a comprehensive overview of the characteristic‐based methods and Characteristic‐Based Split (CBS) scheme. The practical difficulties of employing the original characteristic schemes are discussed. The important features of the CBS scheme are brought out by studying several problems of compressible and incompressible flows. All special consideration necessary for solving these problems are thoroughly discussed. The CBS scheme is presented in such a way that any interested researcher should be able to develop a code using the information provided. Several invicid and viscous flow examples are also provided to demonstrate the unified CBS approach. For sample two‐dimensional codes, input files and instructions, the readers are referred to ‘www.nithiarasu.co.uk’&nbsp

Numerical modelling of compressible laminar and turbulent flow. The Cbs algorithm

This work is about the development of a general algorithm for the numerical solution of flow equations: the Navier-Stokes set. This set of differential equations models the time dependent behavior of fluids. It is formed by continuity, linear momentum and an energy transport equations. The algorithm here described is a general one since it can handle equally a great variety of problems, ranging from incompressible to compressible flows, viscous to inviscid, stationary and transient, all of them phisically modeled by the same set of differential equations. In the present work, a quest for a general algorithm is described, following one of many possible ways to tackle the problem. In general, this is done extending methods either from compressible to incompressible flows or from incompressible to compressible ones

Advances in FE explicit formulation for simulation of metalforming processes

This paper presents some advances of finite element explicit formulation for simulation of metal forming processes. Because of their computational efficiency, finite element programs based on the explicit dynamic formulation proved to be a very attractive tool for the simulation of metal forming processes. The use of explicit programs in the sheet forming simulation is quite common, the possibilities of these codes in bulk forming simulation in our opinion are still not exploited sufficiently. In our paper, we will consider aspects of bulk forming simulation. We will present new formulations and algorithms developed for bulk metal forming within the explicit formulation. An extension of a finite element code for the thermomechanical coupled analysis is discussed. A new thermomechanical constitutive model developed by the authors and implemented in the program is presented. A new formulation based on the so-called split algorithm allows us to use linear triangular and tetrahedral elements in the analysis of large plastic deformations characteristic to forming processes. Linear triangles and tetrahedra have many advantages over quadrilateral and hexahedral elements. Linear triangles and tetrahedra based on the standard formulations exhibit volumetric locking and are not suitable for large plastic strain simulation. The new formulation allows to overcome this difficulty. New formulations and algorithms have been implemented in the finite element code Stampack developed at the International Centre for Numerical Methods in Engineering in Barcelona. Numerical examples illustrate some of the possibilities of the finite element code developed and validate new algorithms

Split, characteristic based semi‐implicit algorithm for laminar/turbulent incompressible flows

In an earlier paper, Zienkiewicz and Codina (Int. j. numer. methods fluids, 20, 869–885 (1995)) presented a general algorithm for the solution of both compressible and incompressible Navier–Stokes equations. The algorithm, based on operator splitting, permits arbitrary interpolation functions to be used while avoiding the Babŭska–Brezzi restriction. In addition, its characteristic based approach introduces a form of rational dissipation. Zienkiewicz et al. (Int. j. numer. methods fluids, 20, 887–913 (1995)) presented the application of this algorithm in its fully explicit form to various inviscid compressible flow problems. They also presented two incompressible flow problems solved by the fully explicit form, employing a pseudo compressibility. The present work deals with the application of the above algorithm it its semi‐implicit form to some incompressible flow benchmark problems. Further, it extends the methodology to turbulent flows by employing both one, and two equation turbulence models. A comparison of results with earlier investigations is presented. Other issues addressed in this study include the effect of additional diffusion terms present in the scheme for both laminar and turbulent flow problems and some practical difficulties associated with local time stepping

A finite point method in computational mechancis. Applications to convective transport and fluid flow.

The paper presents a fully meshless procedure fo solving partial differential equations. The approach termed generically the ‘finite point method’ is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals. Some examples showing the accuracy of the method for solution of adjoint and non‐self adjoint equations typical of convective‐diffusive transport and also to the analysis of compressible fluid mechanics problem are presented

The characteristic‐based‐split procedure: an efficient and accurate algorithm for fluid problems

In 1995 the two senior authors of the present paper introduced a new algorithm designed to replace the Taylor–Galerkin (or Lax–Wendroff) methods, used by them so far in the solution of compressible flow problems. The new algorithm was applicable to a wide variety of situations, including fully incompressible flows and shallow water equations, as well as supersonic and hypersonic situations, and has proved to be always at least as accurate as other algorithms currently used. The algorithm is based on the solution of conservation equations of fluid mechanics to avoid any possibility of spurious solutions that may otherwise result. The main aspect of the procedure is to split the equations into two parts, (1) a part that is a set of simple scalar equations of convective–diffusion type for which it is well known that the characteristic Galerkin procedure yields an optimal solution; and (2) the part where the equations are self‐adjoint and therefore discretized optimally by the Galerkin procedure. It is possible to solve both the first and second parts of the system explicitly, retaining there the time step limitations of the Taylor–Galerkin procedure. But it is also possible to use semi‐implicit processes where in the first part we use a much bigger time step generally governed by the Peclet number of the system while the second part is solved implicitly and is unconditionally stable. It turns out that the characteristic‐based‐split (CBS) process allows equal interpolation to be used for all system variables without difficulties when the incompressible or nearly incompressible stage is reached. It is hoped that the paper will help to make the algorithm more widely available and understood by the profession and that its advantages can be widely realised