A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part I: The variational formulation for a single evolution equation

The objective of the research presented here was to develop a generic adaptive computational method for porous media evolution problems that involve coupled heat flow, fluid flow and species transport processes with sharply defined phase-change interfaces. In this paper we examine the general least squares variational approach and develop the conceptual framework for a rate least squares variational formulation of a continuously deforming mixed variable finite element method for solving highly non-linear time-dependent partial differential equations. In Part II of this paper¹ we extend the formulation given here for a single evolution equation to a system of coupled evolution equations. In Part III² we discuss in detail the numerical procedures that were implemented in a computer program and present several numerical examples that demonstrate the performance of this computational method.     

A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part II: The coupled problem of phase-change in porous media

The conceptual framework of a least squares rate variational approach to the formulation of continuously deforming mixed-variable finite element computational scheme for a single evolution equation was presented in Part I.¹ In this paper (Part II), we extend these concepts and present an adaptively deforming mixed variable finite element method for solving general two-dimensional transport problems governed by a system of coupled non-linear partial differential evolution equations. In particular, we consider porous media problems that involve coupled heat and mass transport processes that yield steep continuous moving fronts, and abrupt, discontinuous, moving phase-change interfaces. In this method, the potentials, such as the temperature, pressure and species concentration, and the corresponding fluxes, are permitted to jump in value across the phase-change interfaces. The equations, and the jump conditions, governing the physical phenomena, which were specialized from a general multiphase, multiconstituent mixture theory, provided the basis for the development and implementation of a two-dimensional numerical simulator. This simulator can effectively resolve steep continuous fronts (i.e. shock capturing) without oscillations or numerical dispersion, and can accurately represent and track discontinuous fronts (i.e. shock fitting) through adaptive grid deformation and redistribution. The numerical implementation of this simulator and numerical examples that demonstrate the performance of the computational method are presented in Part III² of this paper.     

Continuously deforming finite elements

A general class of time-dependent co-ordinate transformations is introduced in a variational formulation for evolution problems. The variational problem is posed with respect to both solution and transformation field variables. An approximate analysis using finite elements is developed from the continuous variational form. Modified forms of the variational functional are considered to ensure the deforming mesh is not top irregular. ODE system integrators are utilized to integrate the resulting semidiscrete systems. In Part I we consider the formulation for problems in one spatial dimension and time, including, in particular, convection-dominated flows described by the convection-diffusion, Burgers' and Buckley–Leverett equations. In Part II the extension of the method to two dimensions and supporting numerical experiments are presented.     

A SPACE-TIME COUPLED p-VERSION LEAST SQUARES FINITE ELEMENT FORMULATION FOR UNSTEADY TWO-DIMENSIONAL NAVIER–STOKES EQUATIONS

This paper presents a p-version least squares finite element formulation for two-dimensional unsteady fluid flow described by Navier–Stokes equations where the effects of space and time are coupled. The dimensionless form of the Navier–Stokes equations are first cast into a set of first-order differential equations by introducing auxiliary variables. This permits the use of C⁰ element approximation. The element properties are derived by utilizing the p-version approximation functions in both space and time and then minimizing the error functional given by the space–time integral of the sum of squares of the errors resulting from the set of first-order differential equations. This results in a true space–time coupled least squares minimization procedure. The application of least squares minimization to the set of coupled first-order partial differential equations results in finding a solution vector {δ} which makes gradient of error functional with respect to {δ} a null vector. This is accomplished by using Newton's method with a line search. A time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step. Equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance. The space–time coupled p-version approximation functions provide the ability to control truncation error which, in turn, permits very large time steps. What literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space–time coupled approach. The generality, success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity. The results are compared with the analytical solutions and those reported in the literature. The formulation presented here is ideally suited for space–time adaptive procedures. The element error functional values provide a mechanism for adaptive h, p or hp refinements.     

On finite element modelling of surface tension Variational formulation and applications – Part I: Quasistatic problems

This paper describes variational formulation and finite element discretization of surface tension. The finite element formulation is cast in the Lagrangian framework, which describes explicitly the interface evolution. In this context surface tension formulation emerges naturally through the weak form of the Laplace–Young equation.The constitutive equations describing the behaviour of Newtonian fluids are approximated over a finite time step, leaving the governing equations for the free surface flow function of geometry change rather than velocities. These nonlinear equations are then solved by using Newton-Raphson procedure.Numerical examples have been executed and verified against the solution of the ordinary differential equation resulting from a parameterization of the Laplace-Young equation for equilibrium shapes of drops and liquid bridges under the influence of gravity and for various contact angle boundary conditions.     

The method of moments and its optimization

The method of moments is a semidiscrete numerical method for solving partial differential equations. The method approximates the solution of a partial differential equation by a finite sum of products of two functions. One function in the product is an unknown function of a single variable and the other function (moment function) is a prescribed function in the remaining variables. Using variational technique we obtain a finite system of boundary value problems of ordinary differential equations for the unknown functions. The main goal of this paper is the study of the theoretical background and numerical effectiveness of the method of moments for solving linear partial differential equations on rectangular-like domains. The mathematical formulation of the method together with error estimates and the theory of optimal moment functions are given. If for the one-dimensional moment functions piecewise polynomials of degree K are used then finite element type error bounds are obtained for the approximate solution in two dimensions. We also consider the numerical implementation of the method through the factorization method and efficient initial value methods. Several numerical examples showing the efficiency of the method are presented.     

Least squares finite element analysis of laminar boundary layer flows

Based on the least squares error criterion, a class of finite element is formulated for the numerical analysis of steady state viscous boundary layer flow problems. The method is essentially a discrete element-wise minimization of square and weighted residuals which arise from the attempts in approximately satisfying boundary layer equations. An iterative linearization scheme is developed to circumvent the mathematical difficulties posed by the non-linear boundary layer equations. It results in a process of successive least squares minimizations of residual errors arising from satisfying a set of linear differential equations. A mathematical justification for the method is presented. A major feature of the method lies in the linearization approach which renders non-linear differential equations amenable to linear least squares finite element analysis. Another important feature rests on the proposed finite element formulation which preserves the symmetric nature of finite element matrix equations through the use of the least squares error criterion. Numerical examples of viscous flow along a flat plate are presented to demonstrate the applicability of the method as well as to illuminate discussions on the theoretical aspects of the method.     

Meshless local boundary integral equation method for 2D elastodynamic problems

A new meshless method for solving transient elastodynamic boundary value problems, based on the local boundary integral equation (LBIE) method and the moving least squares approximation (MLS), is proposed in this paper. The LBIE with the MLS is applied to both transient and steady‐state (Laplace transformed) elastodynamics. Applying the MLS approximation for spatially dependent terms in the first approach, the LBIEs are transformed into a system of ordinary differential equations for nodal unknowns. This system of ordinary differential equations is solved by the Houbolt finite difference scheme. In the second formulation, the time variable is eliminated by using the Laplace transformation. Unknown Laplace transforms of displacements and traction vectors are computed from the LBIEs with the MLS approximation. The time‐dependent values are obtained by the Durbin inversion technique. Copyright © 2003 John Wiley & Sons, Ltd.     

A least-squares front-tracking finite element method analysis of phase change with natural convection

This paper focuses on the numerical modelling of phase-change processes with natural convection. In particular, two-dimensional solidification and melting problems are studied for pure metals using an energy preserving deforming finite element model. The transient Navier–Stokes equations for incompressible fluid flow are solved simultaneously with the transient heat flow equations and the Stefan condition. A least-squares variational finite element method formulation is implemented for both the heat flow and fluid flow equations. The Boussinesq approximation is used to generate the bulk fluid motion in the melt. The mesh motion and mesh generation schemes are performed dynamically using a transfinite mapping. The consistent penalty method is used for modelling incompressibility. The effect of natural convection on the solid/liquid interface motion, the solidification rate and the temperature gradients is found to be important. The proposed method does not possess some of the false diffusion problems associated with the standard Galerkin formulations and it is shown to produce accurate numerical solutions for convection dominated phase-change problems.     

Numerical analysis of an elasto‐piezoelectric problem with damage

In this paper, we analyze an algorithm for the quasistatic evolution of the mechanical state of an elasto‐piezoelectric body with damage. Both damage, caused by the development and the growth of internal microcracks, and piezoelectric effects, are included in the model. The mechanical problem is expressed as an elliptic system for the displacement field coupled with a non‐linear parabolic partial differential equation for the damage field and a linear partial differential equation for the electric potential. The variational formulation leads to a coupled system composed of two linear variational equations for the displacement field and the electric potential, and a non‐linear parabolic variational equation for the damage field. The existence of a unique weak solution is stated. Then, a fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, some numerical simulations are performed, in one, two and three dimensions, to demonstrate the accuracy of the scheme and the behaviour of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

A ψ–ω finite element formulations for the Navier-Stokes equations

A new method of solving the two-dimensional Navier–Stokes equations by using the finite element method is proposed. The flow is represented by the stream function–vorticity formulation and the no-slip boundary conditions are explicitly introduced in the nonlinear equations. This formulation coupled with the Newton-Raphson method enables the study of stationary flows for high Reynolds number, without any convergence problem. A number of flow problems are analysed in order to demonstrate the efficiency of the present formulation.     

Finite elements in space and time for generalized viscoelastic maxwell model

 The generalization of a new numerical approach with simultaneous space–time finite element discretization for viscoelastic problems developed in the papers by Buch et al. (1999) and Idesman et al. (2000) is presented for the case of the generalized viscoelastic Maxwell model. New non-symmetric variational and discretized formulations are derived using the continuous Galerkin method (CGM) and discontinuous Galerkin method (DGM). Viscoelastic behaviour described by the generalized Maxwell model is represented by means of internal variables. It allows to use only differential equations for the constitutive equations instead of integrodifferential ones. The variational formulation reduces to two types of equations with total displacements and internal displacements (internal variables) as unknowns, namely to the equilibrium equation and the evolution equations for the internal displacements which are fulfilled in the weak form. Using continuous test functions in space and time, a continuous space–time finite element formulation is obtained with simultaneous discretization in space and time. Subdividing the total observation time interval into appropriate time slabs and introducing discontinuous trial functions, being continuous within time slabs and allowing jumps across time interfaces, a more general discontinuous finite element formulation is obtained. The difference between these two formulations for one time slab consists in the satisfaction of initial conditions which are fulfilled exactly for the continuous formulation and in a weak form for the discontinuous case. The proposed approach has some very attractive advantages with respect to semidiscretization methods, regarding the possibility of adaptive space–time refinements and efficient parallel processing on MIMD-parallel computers. The considered numerical examples show the effectiveness of simultaneous space–time finite element calculations and a high convergence rate for adaptive refinement. Numerical efficiency is an advantage of DGM in comparison with CGM for discontinuously changing (e.g. piecewise constant) boundary conditions in time and for solutions with high gradients. Received 7 February 2000     

Free and transient vibration analysis of an un-symmetric cross-ply laminated plate by spectral finite elements

In this study, a spectral finite element formulation for free and transient vibration analysis of a rectangular un-symmetric cross-ply laminated composite plate of Levy-type is developed. Plate kinematics obeys classical lamination plate theory. The formulation is based on dynamic shape functions in frequency-domain derived from the exact solution of the governing wave equations. Based on Levy-type solution and finite strip method, 2/D space-time partial differential equations (PDEs) of motion are transformed to 1/D space-time PDEs. Spectral elements are used to transform 1/D space-time PDEs to temporal ordinary differential equations. For investigating validity and accuracy of the model presented, numerical solutions are established for both free and transient vibrations. Comparison of spectral finite element method (SFEM) time responses with an exact analytical method demonstrates the superiority of SFEM in both reducing computational cost and increasing numerical accuracy. It is correct especially for excitations of high-frequency contents. A comparison of SFEM and FEM shows the same result.     

A space–time coupled p-version least-squares finite element formulation for unsteady fluid dynamics problems

This paper presents a p-version least-squares finite element formulation for unsteady fluid dynamics problems where the effects of space and time are coupled. The dimensionless form of the differential equations describing the problem are first cast into a set of first-order differential equations by introducing auxiliary variables. This permits the use of C° element approximation. The element properties are derived by utilizing p-version approximation functions in both space and time and then minimizing the error functional given by the space–time integral of the sum of squares of the errors resulting from the set of first-order differential equations. This results in a true space–time coupled least-squares minimization procedure. A time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step. The space–time coupled p-version approximation functions provide the ability to control truncation error which, in turn, permits very large time steps. What literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space–time coupled approach. For non-linear problems the non-linear algebraic equations resulting from the least-squares process are solved using Newton's method with a line search. This procedure results in a symmetric Hessian matrix. Equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance. The generality, success and superiority of the present formulation procedure is demonstrated by presenting specific formulations and examples for the advection–diffusion and Burgers equations. The results are compared with the analytical solutions and those reported in the literature. The formulation presented here is ideally suited for space–time adaptive procedures. The element error functional values provide a mechanism for adaptive h, p or hp refinements. The work presented in this paper provides the basis for the extension of the space–time coupled least-squares minimization concept to two- and three-dimensional unsteady fluid flow.     

A three-field formulation for incompressible viscoelastic fluids

This paper presents a new stabilized finite element method for incompressible viscoelastic fluids. A three-field formulation is developed wherein Oldroyd-B model is coupled with the mass and momentum conservation equations for an incompressible viscous fluid. The variational multiscale (VMS) framework is employed to develop a stabilized formulation for the coupled momentum, continuity and stress equations. Based on the new stabilized method a family of linear and higher-order triangle and quadrilateral elements with equal-order velocity-pressure-stress fields is developed. Stability and convergence property of the various elements is studied and optimal rates are attained in the norms considered. The method is applied to some benchmark problems and accuracy and computational economy of the formulation is investigated for various flow conditions.     

On finite element modelling of surface tension: Variational formulation and applications – Part II: Dynamic problems

In Part I, a finite element model of surface tension has been discussed and used to solve some quasi-static problems. The quasi-static analysis is often required to find not only the initial shape of the liquid but also the static equilibrium state of a liquid body before a dynamic analysis can be carried out. In general, natural and industrial processes in which surface tension force is dominant are of dynamic nature. In this second part of this work, the dynamic effects will be included in the finite element model described in Part I.A fully Lagrangian finite element method is used to solve the free surface flow problem and Newtonian constitutive equations describing the fluid behaviour are approximated over a finite time interval. As a result the momentum equations are function of nodal position instead of velocities. The resulting ordinary differential equation is integrated using Newmark algorithm. To avoid overly distorted elements an adaptive remeshing strategy is adopted. The adaptive strategy employs a remeshing indicator based on viscous dissipation functional and incorporates an appropriate transfer operator.The validation of the model is performed by comparing the finite element solutions to available analytical solutions of a droplet oscillations and experimental results pertaining to stretching of a liquid bridge.     

Use of the least squares criterion in the finite element formulation

This paper presents a new method of formulating the finite element relationships based on the least squares criterion. To overcome the high degree of inter-element continuity, a method of reducing the original governing differential equation to a set of equivalent system of first-order differential equations is proposed. The validity of the method is demonstrated by means of several numerical examples. In particular, application of the method to problems with unknown variational functionals is considered.     

Meshless local Petrov–Galerkin (MLPG) method in combination with finite element and boundary element approaches

The meshless local Petrov–Galerkin (MLPG) method is an effective truly meshless method for solving partial differential equations using moving least squares (MLS) interpolants. It is, however, computationally expensive for some problems. A coupled MLPG/finite element (FE) method and a coupled MLPG/boundary element (BE) method are proposed in this paper to improve the solution efficiency. A procedure is developed for the coupled MLPG/FE method and the coupled MLPG/BE method so that the continuity and compatibility are preserved on the interface of the two domains where the MLPG and FE or BE methods are applied. The validity and efficiency of the MLPG/FE and MLPG/BE methods are demonstrated through a number of examples. Received 6 June 2000     

Evolutionary topology and shape design for general physical field problems

 As a practical tool for design engineers, evolutionary techniques for structural topology, shape and size optimisation have successfully resolved the whole range of structural problems from frames to 2D and 3D continuums with design criteria of stress, stiffness, frequency and buckling. In view of the generality of the finite element formulation using either a variational calculus or weighted residual approach, it is logical to extend its applications to other steady state field problems in mathematical physics governed by partial differential equations. The range of physical problems falling to this category includes heat conduction, incompressible fluid flow, elastic torsion, electrostatics and magnetostatics, etc. This paper discusses the general principles involved in setting up the adaptive evolutionary algorithms that have finite element techniques as the analysis engines. To avoid the complexity of classical solutions, the proposed method develops a simple outer loop procedure consisting of finite element analysis and design modifications. Illustrative examples are presented to demonstrate the capability in solving the above-mentioned physical field situations.     

Finite element analysis of transient free convection flow over vertical plate

A finite element method is used to study the heat-transfer response of an incompressible, laminar, transient free convection flow over a semi-infinite vertical plate. The non-dimensional governing equations are solved by the finite element method (FEM). The resulting non-linear integral equations are linearized and solved using the Newton–Raphson iteration. The resulting first-order ordinary differential equations with respect to time are solved using the implicit Euler scheme. Numerical results for the details of the velocity and temperature contours and profiles as well as heat transfer rate in terms of Nusselt number which are shown on graphs have been presented.