A segregated formulation of Navier-Stokes equations with finite elements

A finite element procedure is presented for the solution of the Navier-Stokes equations, adopting a segregated velocity-pressure formulation. The procedure is based on the derivation of an approximate pressure equation, which allows the uncoupling of pressure and velocity solutions. This equation is derived from the discretized continuity equation by considering the relationships established between velocity and pressure in the discretized momentum equations. The numerical performance of the proposed method is demonstrated by investigating several examples. The results are compared with finite difference predictions.     

Numerical prediction of discontinuous central bursting in axisymmetric forward extrusion by continuum damage mechanics

The prediction of the central burst defects in axisymmetric cold extrusion is analyzed numerically by using 2D finite element analysis (FEA) accounting for the ductile damage effect. The coupling between the ductile damage and the thermoelastoplastic constitutive equations is formulated in the framework of the thermodynamics of irreversible processes together with the continuum damage mechanics (CDM) theory. A simple isotropic ductile damage model is fully coupled with thermoelastoplastic constitutive equations of Prandtl-Reuss type including non-linear isotropic hardening and thermal effects. A modified ductile damage criterion based on linear combination of the stress tensor invariants is proposed in order to predict the occurrence of micro-crack initiation as a discontinuous central bursts along the bar axis. The implicit integration scheme of the fully coupled constitutive equations and the iterative resolution scheme to solve the associated thermomechanical equilibrium problem are presented. A three fields (velocity, hydrostatic pressure and temperature) variational formulation is used to solve the resulting algebraic system. The effects of various process parameters, namely, the diameter reduction ratio, the die semi-angle, the friction coefficient and the material ductility, …, on the central bursts occurrence are discussed. The quantitative effects of ductile damage on the extrusion parameters are studied and qualitative comparison with some available experimental data are given.     

A numerical solution of the Navier-Stokes equations using the finite element technique

The finite element discretisation technique is used to effect a solution of the Navier- Stokes equations. Two methods of formulation are presented, and a comparison of the effeciency of the methods, associated with the solution of particular problems, is made. The first uses velocity and pressure as field variables and the second stream function and vorticity. It appears that, for contained flow problems the first formulation has some advantages over previous approaches using the finite elemental method[1,2].     

A multi-director formulation for nonlinear elastic-viscoelastic layered shells

A C⁰ finite element formulation for nonlinear analysis of multi-layered shells comprised of elastic and viscoelastic layers is presented for applications involving small strains but finite rotations. The elastic and viscoelastic layers may occupy arbitrary layer locations and the formulation is applicable to thick and thin shells. The formulation utilizes a three-dimensional variational approach in which the layered shell is represented as a multi-director field. The incorporated kinematic theory describes, within individual layers, the effects of transverse shear and transverse normal strain to arbitrary orders in the layer thickness coordinate. Stresses are computed through the three-dimensional constitutive equations and the usual “zero normal stress” shell hypothesis is not employed. Sufficiently general constitutive equations for the viscoelastic layers are proposed in objective rate form and a product algorithm, based on an operator split in the complete set of constitutive equations, is used for the temporal integration of the rate equations. The definition of the tangent operator, used in Newton's method for the solution of the nonlinear equations, is derived consistently from the product algorithm. Observations on the use of reduced/selective integration in the presence of high order kinematics are made and a number of numerical examples are presented to illustrate the capability of the formulation.     

Application of lower order integration schemes in linear shell problems

A method is presented by which steady flow solutions may be obtained to problems which involve non-Newtonian memory fluids. The finite element method is used in conjunction with a Galerkin form of the equations of motion and continuity. Integral constitutive laws are directly employed without extra-stress differential equations. The stress is computed by construction of the portion of the streamline lying upstream of element quadrature points. This construction is shown to be quite simple, owing to the special form of finite element trial velocity fields. Two test problems are analyzed which use the integral form of the Maxwell constitutive law. The interaction between the fluid elasticity and solution procedures for the discrete equations is discussed.     

Finite element solution for advection and natural convection flows

A new equal order velocity—pressure finite element procedure is presented for the calculation of 2-D viscous, incompressible flows of a recirculating nature. As in the finite difference procedures, velocity and pressure e uncoupled and the equations are solved one after the other. In this splitting-up method, an auxilary velocity field is computed first, which accounts for all contributions to the acceleration, except pressure, and satisfies the velocity boundary conditions. Then, the final velocities are evaluated by adding to the auxilary velocities pressure contributions which are computed to satisfy the continuity equation. The effectiveness is illustrated via example problems of 2-D advection and natural convection flows.     

Rosenbrock-type methods applied to finite element computations within finite strain viscoelasticity

In this article time-adaptive high-order Rosenbrock-type methods are applied to the system of differential–algebraic equations which results from the space-discretization using finite elements based on a constitutive model of finite strain viscoelasticity. It is shown that in this smooth problem more efficient finite element computations result in comparison to classical finite element approaches since the time integration on the basis of Rosenbrock-type methods does not lead to a system of non-linear equations. In other words, all aspects of implicit finite elements as local iterations on Gauss-point level and global equilibrium iterations do not occur. The first introduction to this approach proposed by Hartmann and Wensch [22] is extended here to the case of finite strain applications, where the geometrical non-linear deformation has an essential contribution to the non-linearities. Additionally, a clear decomposition into local (element or Gauss-point) work and global computational work using the Schur-complement is introduced to exploit the classical finite element character. Moreover, the extension to the reaction force computation, which is different to the classical approach, and the influence to mixed element formulations, here, the three-field formulation for displacements, pressure and dilatation, are discussed. The performance of various Rosenbrock-type methods is investigated and shows that for low accuracy requirements as in order one methods, the proposal yields a drastic reduction of the computational time.     

Hybrid numerical methods to solve shallow water equations for hurricane induced storm surge modeling

In this paper an efficient numerical method based on hybrid finite element and finite volume techniques to solve hurricane induced storm surge flow problem is presented. A segregated implicit projection method is used to solve the 2D shallow water equations on staggered unstructured meshes. The governing equations are written in non-conservation form. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centered finite volume method. The nonlinear wave equation is solved by the node-based Galerkin finite element method. This staggered-mesh scheme is distinct from other conventional approaches in that the velocity components and auxiliary variables are stored at cell centers and vertices, respectively. The present model uses an implicit method, which is very efficient and can use a large time step without losing accuracy and stability.The hurricane induced wind stress and pressure, bottom friction, Coriolis effect, and tidal forcing conditions are implemented in this model. The levee overtopping option is implemented in the model as well. Hurricane Katrina (2005) storm surge has been simulated to demonstrate the robustness and applicability of the model.     

Nonlinear finite element analysis of shells: Part I. three-dimensional shells

A nonlinear finite element formulation is presented for the three-dimensional quasistatic analysis of shells which accounts for large strain and rotation effects, and accommodates a fairly general class of nonlinear, finite-deformation constitutive equations. Several features of the developments are noteworthy, namely: the extension of the selective integration procedure to the general nonlinear case which, in particular, facilitates the development of a ‘heterosis-type’ nonlinear shell element; the presentation of a nonlinear constitutive algorithm which is ‘incrementally objective’ for large rotation increments, and maintains the zero normal-stress condition in the rotating stress coordinate system; and a simple treatment of finite-rotational nodal degrees-of-freedom which precludes the appearance of zero-energy in-plane rotational modes. Numerical results indicate the good behavior of the elements studied.     

Finite element solution of incompressible flows using an explicit segregated approach

Summary An explicit finite element method for solving the incompressible Navier-Stokes equations for laminar and turbulent, newtonian, nonisothermal flow is presented. This method is based on the segregated velocity pressure formulation which has seen considerable development in the last decade. An endeavour has been made to include beneficial features from much of the relevant published work in the developed code. Some of the main features include, the use of the velocity correction method (segregation at the differential equation level), equal order interpolation of velocity and pressure, splitting of advection and diffusion terms, Taylor-Galerkin method for discretizing the advection terms, lumped-explicit solution of diffusion, and iterative-explicit solution of advection. In addition to these a consistent treatment of the natural boundary conditions for the pressure Poisson equation has been presented. Full details of the formulation are given with examples demonstrating the method.     

Analysis of unsteady compressible boundary layer flow via finite elements

This paper is concerned with the discrete formulation and numerical solution of unsteady compressible boundary layer flows using the Galerkin-finite element method. Linear interpolation functions for the velocity, density, temperature and pressure are used in the momentum equation and equations of continuity, energy and state. The coupled nonlinear finite element equations are approximated by a third order Taylor series expansion as temporal operator to integrate in time with Newton-Raphson type iterations performed until convergence within each time step. As an example, a boundary layer problem of a perfect gas behind a normal shock wave is solved. A comparison of the results with those by other method indicates a favorable agreement.     

Simulation of a viscoelastic flexible multibody system using absolute nodal coordinate and fractional derivative methods

This paper employs a new finite element formulation for dynamics analysis of a viscoelastic flexible multibody system. The viscoelastic constitutive equation used to describe the behavior of the system is a three-parameter fractional derivative model. Based on continuum mechanics, the three-parameter fractional derivative model is modified and the proposed new fractional derivative model can reduce to the widely used elastic constitutive model, which meets the continuum mechanics law strictly for pure elastic materials. The system equations of motion are derived based on the absolute nodal coordinate formulation (ANCF) and the principle of virtual work, which can relax the small deformation assumption in the traditional finite element implementation. In order to implement the viscoelastic model into the absolute nodal coordinate, the Grünwald definition of the fractional derivative is employed. Based on a comparison of the HHT-I3 method and the Newmark method, the HHT-I3 method is used to solve the equations of motion. Another particularity of the proposed method based on the ANCF method lies in the storage of displacement history only during the integration process, reducing the numerical computation considerably. Numerical examples are presented in order to analyze the effects of the truncation number of the Grünwald series (fading memory phenomena) and the value of several fractional model parameters and solution convergence aspects. An erratum to this article can be found at     

On the numerical solution of the stokes equations using the finite element method

A numerical solution of the stationary Stokes equations is considered based on the work of Crouzeix and Raviart [1]. The finite element method is used to discretize the partial differential equations, and a direct discretization of the velocity field and pressure is given which is applicable in both two and three dimensions. It is shown that not every arbitrary element can be used, and a condition is given to check whether or not an element is admissible. The system of linear equations is solved using the method of Powell and Hestenes for constrained optimization (see [2]).     

Preconditioned Uzawa Algorithm for the Velocity-Pressure-Stress Formulation of Viscoelastic Flow Problems

This paper is devoted to the development of efficient iterative methods for solving the systems of algebraic equations arising from a spectral element discretization of the equations governing the flow of an Oldroyd B fluid. The governing equations are written in terms of velocity, pressure and extra-stress, giving rise to the so-called three-field formulation. The convection terms are treated using an operator-integration-factor splitting method. The remaining terms in the system of equations constitute a generalized Stokes problem. This problem is formulated in terms of an Uzawa operator applied to the pressure. Efficient preconditioners are developed for inverting this operator that are independent of the numerical and physical parameters of the problem.     

A finite element formulation for elastoplasticity based on a three-field variational equation

A three-field variational equation, which expresses the momentum balance equation, the plastic consistency condition, and the dilatational constitutive equation in a weak form, is proposed as a basis for finite element computations in hardening elastoplasticity. The finite element formulation includes algorithms for the integration of the elastoplastic rate constitutive equations which are similar to members of the “return mapping” family of algorithms employed in displacement formulations, except that the proposed algorithms are not required to explicitly satisfy the plastic consistency condition at the end of each time step. This condition is imposed globally by the inclusion of a variational equation that suitably constrains the solution. The plastic incompressibility constraint is also treated in an appropriate variational sense. Solution of the nonlinear finite element equations is obtained by use of Newton's method and details of the linearization of the variational equation are given. The formulation is developed for an associative von Mises plasticity model with general nonlinear isotropic and kinematic strain hardening. A number of numerical test examples are provided.     

Nonlinear dynamic analysis of curved beams

A computational procedure is presented for predicting the dynamic response of curved beams with geometric nonlinearities. A mixed formulation is used with the fundamental unknowns consisting of stress resultants, generalized displacements and velocity components. The governing semidiscrete finite element equations consist of a mixed system of algebraic and differential equations. The temporal integration of the differential equations is performed by using an explicit half-station central difference method. A procedure is outlined for lumping both the flexibilities and masses of the mixed model, thereby uncoupling all the equations of the system. The advantages of the proposed computational procedure over explicit methods used with the displacement formulation are discussed. The effectiveness and versatility of the proposed approach are demonstrated by means of numerical examples.     

Parameter and shape sensitivity of thermo-viscoelastic response

Gradient-based optimization methods are still most efficient methods for solving structural optimization problems. The sensitivity formulation has been one of the central issues in the gradient-based optimization algorithm. Thermo-viscoelastic constitutive and parameter sensitivity formulation are presented in this paper. The model considered is composed of two coupled subproblems: the transient heat transfer problem and a rheological, viscoelastic material model known in literature as the standard model. Design variables considered are with material and shape-defining parameters. The investigation includes a finite element formulation and implementation in an object-oriented finite element environment. Results of numerical analysis are presented.     

Higher order stabilized finite element method for hyperelastic finite deformation

This paper presents a higher order stabilized finite element formulation for hyperelastic large deformation problems involving incompressible or nearly incompressible materials. A Lagrangian finite element formulation is presented where mesh dependent terms are added element-wise to enhance the stability of the mixed finite element formulation. A reconstruction method based on local projections is used to compute the higher order derivatives that arise in the stabilization terms, specifically derivatives of the stress tensor. Linearization of the weak form is derived to enable a Newton–Raphson solution procedure of the resulting non-linear equations. Numerical experiments using the stabilization method with equal order shape functions for the displacement and pressure fields in hyperelastic problems show that the stabilized method is effective for some non-linear finite deformation problems. Finally, conclusions are inferred and extensions of this work are discussed.     

Vibration of thin cylindrical shells based on a mixed finite element formulation

A Hellinger-Reissner functional for thin circular cylindrical shells is presented. A mixed finite element formulation is developed from this functional, which is free from line integrals and relaxed continuity terms. This element is applied to the problem of vibration of rectangular cylindrical shells. Bilinear trial functions are used for all field variables. The element satisfies the compatibility and completeness requirements. The numerical results obtained in this work show that convergence is quite rapid and monotonic for a much smaller number of degrees of freedom than other finite element formulations.     

Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams

Most existing formulations for structural elements such as beams, plates and shells do not allow for the use of general nonlinear constitutive models in a straightforward manner. Furthermore, such structural element models, due to the nature of the generalized coordinates used, do not capture some Poisson modes such as the ones that couple the deformation of the cross section of the structural element and stretch and bending. In this paper, beam models that employ general nonlinear constitutive equations are presented using finite elements based on the nonlinear absolute nodal coordinate formulation. This formulation relaxes the assumptions of the Euler–Bernoulli and Timoshenko beam theories, and allows for the use of general nonlinear constitutive models. The finite elements based on the absolute nodal coordinate formulation also allow for the rotation as well as the deformation of the cross section, thereby capturing Poisson modes which can not be captured using other beam models. In this investigation, three different nonlinear constitutive models based on the hyper-elasticity theory are considered. These three models are based on the Neo–Hookean constitutive law for compressible materials, the Neo–Hookean constitutive law for incompressible materials, and the Mooney–Rivlin constitutive law in which the material is assumed to be incompressible. These models, which allow capturing Poisson modes, are suitable for many materials and applications, including rubber-like materials and biological tissues which are governed by nonlinear elastic behavior. Numerical examples that demonstrate the implementation of these nonlinear constitutive models in the absolute nodal coordinate formulation are presented. The results obtained using the nonlinear and linear constitutive models are compared in this study. These results show that the use of nonlinear constitutive models can significantly enhance the performance and improve the computational efficiency of the finite element models based on the absolute nodal coordinate formulation. The results also show that when linear constitutive models are used in the large deformation analysis, singular configurations are encountered and basic formulas such as Nanson’s formula are no longer valid. These singular deformation configurations are not encountered when the nonlinear constitutive models are used.