Finite element approximations of singular parabolic problems

An efficient implementation of finite element methods for free boundary parabolic problems in general two dimensional space domains is presented. The Stefan problem, the Hele-Shaw problem and the porous medium equation are included. Backward differences or linearization techniques are used for the time discretization of the problem. The performances of these schemes are discussed with several numerical tests.     

Finite element approximations for near-incompressible and near-inextensible transversely isotropic bodies

This work comprises a detailed theoretical and computational study of the boundary value problem for transversely isotropic linear elastic bodies. General conditions for well-posedness are derived in terms of the material parameters. The discrete form of the displacement problem is formulated for conforming finite element approximations. The error estimate reveals that anisotropy can play a role in minimising or even eliminating locking behaviour for moderate values of the ratio of Young's moduli in the fibre and transverse directions. In addition to the standard conforming approximation, an alternative formulation, involving under-integration of the volumetric and extensional terms in the weak formulation, is considered. The latter is equivalent to either a mixed or a perturbed Lagrangian formulation, analogously to the well-known situation for the volumetric term. A set of numerical examples confirms the locking-free behaviour in the near-incompressible limit of the standard formulation with moderate anisotropy, with locking behaviour being clearly evident in the case of near-inextensibility. On the other hand, under-integration of the extensional term leads to extensional locking-free behaviour, with convergence at superlinear rates.     

Finite element method for piezoelectric vibration

A finite element formulation which includes the piezoelectric or electroelastic effect is given. A strong analogy is exhibited between electric and elastic variables, and a ‘stiffness’ finite element method is deduced. The dynamical matrix equation of electroelasticity is formulated and found to be reducible in form to the well-known equation of structural dynamics, A tetrahedral finite element is presented, implementing the theorem for application to problems of three-dimensional electroelasticity.     

Finite element formulation for a digital image correlation method

A finite element formulation for a digital image correlation method is presented that will determine directly the complete, two-dimensional displacement field during the image correlation process on digital images. The entire interested image area is discretized into finite elements that are involved in the common image correlation process by use of our algorithms. This image correlation method with finite element formulation has an advantage over subset-based image correlation methods because it satisfies the requirements of displacement continuity and derivative continuity among elements on images. Numerical studies and a real experiment are used to verify the proposed formulation. Results have shown that the image correlation with the finite element formulation is computationally efficient, accurate, and robust.     

Finite element method for orthotropic micropolar elasticity

The total potential energy for a body composed of an anisotropic micropolar linear elastic material is developed and used to formulate a displacement type finite element method of analysis. As an example of this formulation triangular plane stress (and plane couple stress) elements are used to analyze several problems. The program is verified by computing the stress concentration around a hole in an isotropic micropolar material for which an exact analytical solution exists. Several anisotropic material cases are presented which demonstrate the dependence of the stress concentration factor on the micropolar material parameters.     

Finite element analysis of flows in secondary settling tanks

The equations governing unsteady flows in secondary settling tanks, a component of the wastewater treatment process, are analysed using the finite element method. The model corresponding to such liquid–solid flows is highly non‐linear and coupled, and incorporates the effects of turbulence. The results of numerical simulations are compared against experimental results from tests on full‐scale settling tanks, and against results obtained from a finite difference code based on an idealized one‐dimensional flux theory. The results compare well with the test results, over the range of applicability of the model. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite element analysis of internal flows with heat transfer

This paper presents a finite element-based model for the prediction of 2-D and 3-D internal flow problems. The Eulerian velocity correction method is used which can render a fast finite element code comparable with the finite difference methods. Nine different models for turbulent flows are incorporated in the code. A modified wall function approach for solving the energy equation with high Reynolds number models is presented for the first time. This is an extension of the wall function approach of Benim and Zinser and the method is insensitive to initial approximation. The performance of the nine turbulent models is evaluated by solving flow through pipes. The code is used to predict various internal flows such as flow in the diffuser and flow in a ribbed channel. The same Eulerian velocity correction method is extended to predict the 3-D laminar flows in various ducts. The steady state results have been compared with benchmark solutions and the agreement appears to be good.     

An adaptive finite element method for transient compressible flows

An adaptive finite element method for the solution of time dependent strongly compressible flows in two dimensions is described. The computational domain is represented by an unstructured assembly of linear triangular elements and the mesh adaptation is achieved by local regeneration of the grid, using an error estimation procedure coupled to an automatic triangular mesh generator. Problems involving shock propagation are solved to illustrate the numerical performance of the proposed approach.     

A hybrid element method for steady linearized free-surface flows

It is first shown that the two-dimensional linearized ship wave problem can be recast as the sum of a radiation and a diffraction problem for simple harmonic waves. Each problem can be solved by a hybrid element method (HEM) where conventional finite elements are used near the body and analytical solutions are used in the remaining infinite regions (super-elements). Variational principles which incorporate the matching conditions between regular and super-elements as natural conditions are derived. Numerical examples are presented. The theoretical aspects for extending the above ideas to a three-dimensional ship wave problems are also described.     

A three-step finite element method for unsteady incompressible flows

This paper describes a three-step finite element method and its applications to unsteady incompressible fluid flows. The stability analysis of the one-dimensional purely convection equation shows that this method has third-order accuracy and an extended numerical stability domain in comparison with the Lax-Wendroff finite element method. The method is cost effective for incompressible flows, because it permits less frequent updates of the pressure field with good accuracy. In contrast with the Taylor-Galerkin method, the present three-step finite element method does not contain any new higher-order derivatives, and is suitable for solving non-linear multi-dimensional problems and flows with complicated outlet boundary conditions. The three-step finite element method has been used to simulate unsteady incompressible flows, such as the vortex pairing in mixing layer. The properties of the flow fields are displayed by the marker and cell technique. The obtained numerical results are in good agreement with the literature.     

A penalty finite element method for non-Newtonian creeping flows

In this paper we present an iterative penalty finite element method for viscous non-Newtonian creeping flows. The basic idea is solving the equations for the difference between the exact solution and the solution obtained in the last iteration by the penalty method. For the case of Newtonian flows, one can show that for sufficiently small penalty parameters the iterates converge to the incompressible solution. The objective of the present work is to show that the iterative penalization can be coupled with the iterative scheme used to deal with the non-linearity arising from the constitutive law of non-Newtonian fluids. Some numerical experiments are conducted in order to assess the performance of the approach for fluids whose viscosity obeys the power law.     

Finite element method for solving mhd flow in a rectangular duct

A finite element method is given to obtain the solution in terms of velocity and induced magnetic field for the steady MHD (magnetohydrodynamic) flow through a rectangular pipe having arbitrarily conducting walls. Linear and then quadratic approximations have been taken for both velocity and magnetic field for comparison and it is found that with the quadratic approximation it is possible to increase the conductivity and Hartmann number M (M ≤ 100). A special solution procedure has been used for the resulting block tridiagonal system of equations. Computations have been carried out for several values of Hartmann number (5 ≤ M ≤ 100) and wall conductivity. It is also found that, if the wall conductivity increases, the flux decreases. The same is the effect of increasing the Hartmann number. Selected graphs are given showing the behaviour of the velocity field and induced magnetic field.     

Finite element method for spectral modelling of tides

A spectral method for modelling of tides is proposed and applied to the calculation of the M₂ component of the tide in the English Channel. The classical non-linear hyperbolic problem of long wave propagation in shallow waters is transformed into a sequence of elliptic problems by looking at a multiperiodic solution the frequencies of which are previously known. The method is based upon a perturbation technique, but the principal difficulties arise from the non-analytical form of the quadratic friction term: the main conclusions of the corresponding study only are given here, because the purpose of this paper is to present a practical application of the method to the calculation of a tidal component by the finite element method. The variational formulation of the problem is presented, and the finite element package used is described. Some results are given for the M₂ tide in the English Channel: cotidal maps and current fields.     

Finite element multigrid solution of Euler flows past installed aero-engines

A finite element based procedure for the solution of the compressible Euler equations on unstructured tetrahedral grids is described. The spatial discretisation is accomplished by means of an approximate variational formulatin, with the explicit addition of a matrix form of artificial viscosity. The solution is advanced in time by means of an explicit multi-stage time stepping procedure. The method is implemented in terms of an edge based representation for the tetrahedral grid. The solution procedure is accelerated by use of a fully unstructured multigrid algorithm. The approach is applied to the simulation of the flow past an installed aero-engine nacelle, at three different free stream conditions. Comparison is made between the numerical predictions and experimental pressure observations.     

The least-squares finite element method for low-mach-number compressible viscous flows

The present paper reports the development of the Least-Squares Finite Element Method (LSFEM) for simulating compressible viscous flows at low Mach numbers in which the incompressible flows pose as an extreme. The conventional approach requires special treatments for low-speed flows calculations: finite difference and finite volume methods are based on the use of the staggered grid or the preconditioning technique, and finite element methods rely on the mixed method and the operator-splitting method. In this paper, however, we show that such a difficulty does not exist for the LSFEM and no special treatment is needed. The LSFEM always leads to a symmetric, positive-definite matrix through which the compressible flow equations can be effectively solved. Two numerical examples are included to demonstrate the method: driven cavity flows at various Reynolds numbers and buoyancy-driven flows with significant density variation. Both examples are calculated by using full compressible flow equations.     

Finite element method for thermomechanical response of near-incompressible elastomers

Summary The present study addresses finite element analysis of the coupled thermomechanical response of near-incompressible elastomers such as natural rubber. Of interest are applications such as seals, which often involve large deformations, nonlinear material behavior, confinement, and thermal gradients. Most published finite element analyses of elastomeric components have been limited to isothermal conditions. A basic quantity appearing in the finite element equation for elastomers is thetangent stiffness matrix. A compact expression for theisothermal tangent stiffness matrix has recently been reported by the first author, including compressible, incompressible, and near-incompressible elastomers. In the present study a compact expression is reported for the tangent stiffness matrix under coupled thermal and mechanical behavior, including pressure interpolation to accommodate near-incompressibility. The matrix is seen to have a computationally convenient structure and to serve as a Jacobian matrix in a Newton iteration scheme. The formulation makes use of a thermoelastic constitutive model recently introduced by the authors for near-incompressible elastomers. The resulting relations are illustrated using a near-incompressible thermohyperelastic counterpart of the conventional Mooney-Rivlin model. As an application, an element is formulated to model the response of a rubber rod subjected to force and heat.Notation A _i n _i /c - C Cauchy strain tensor - c VEC (C) - C ₁,C ₂ constants in Mooney-Rivlin model for elastomer - c ₂ VEC (C ²) - c _i eigenvalues ofC - c _e ,c _e , _e specific heat at constant strain - D _nl stiffness matrix due to the geometric nonlinearity - D _T ,D _T isothermal tangent modulus matrices - e VEC () - e _d VEC ( _d ) - f, f(T) thermal expansion function, =1/[1+(T-T ₀)/3] - f combined vector of nodal forces and heat fluxes - f _M consistent nodal force vector - f _T ,f _1T ,f _2T ,f _3T ,f _4T consistent heat flux vector - F deformation gradient tensor - g related tof _T - h time step - I _i invariants ofC - I 9×9 identity tensor - I identity tensor - i vectorial counterpart ofI:VEC(I) - J the Jacobian matrix in Newton iteration scheme - J determinant ofF - J _i invariants ofI ₁ ^–1/3 C - k, k(T) thermal conductivity - K tangent stiffness matrix - K _MM ,K _MT ,K _MP tangent stiffness submatrices - L,L _M ,L _P ,L _S lower triangular matrices related toLU decomposition ofK - M ₁,M ₂ strain-displacement matrices - N interpolation matrix - n surface normal vector - n _i (I _i /c) ^T - P matrix arising in theLU decomposition ofK - P the tension applied to the rubber rod - p (true) pressure - Q heat flux - q heat flux vector - r,r _M ,r _T ,r residual vectors - R,R ₁,R ₂,R ₃ matrices from thermal boundary conditions - R _s 1/2(R+R ^T ) - R _a 1/2(R–R ^T ) - R –R ₁+R ₂+R - R _s 1/2(R+R ^T ) - R _a 1/2(R-R ^T ) - s VEC() - S surface in undeformed configuration - t time - t traction referred to undeformed configuration - T, T ₀ temperature, reference temperature - T upper-shelf temperature in the surface convective relation - U upper triangular matrix inLU decompositionK - u displacement vector - v combined vector of nodal parameters - v _n value ofv at thenth time step - V volume in undeformed configuration - w strain energy density per unit undeformed volume - x position vector in deformed configuration - X position vector in undeformed configuration - volumetric thermal expansion coefficient - _c coefficient in the surface convective relation - ₁ strain-displacement matrix - _T interpolation matrix for thermal gradient: T - vector of nodal displacements - Lagrangian strain tensor - _d deviatoric Lagrangian strain tensor - interpolation function for - entropy per unit mass in the undeformed configuration - vector of nodal temperatures - þ isothermal bulk modulus - interpolation function forT - temperature-adjusted pressure,p/f ³(T) - mass density in the undeformed configuration - matrix arising inLU decomposition ofK - 2nd Piola-Kirchhoff stress - Cauchy stress tensor - , _M , _T , _c , ₀ Helmholtz free energy density function per unit mass - _i - _ij - vector of nodal values of - matrix arising in theLU decomposition ofK - near-incompressibility constraint function - internal energy density per unit mass - (·) variational operator - VEC(·) vectorization operator - symbol for Kronecker product of two tensors - tr(·) trace of a tensor - det(·) determinant of a tensor     

Finite dynamic element formulation for a plane triangular element

The higher order dynamic correction terms for the stiffness and inertia matrices associated with a triangular plane stress-strain finite dynamic element are developed in detail. Numerical results presented indicate that the adoption of these matrices along with a suitable quadratic matrix eigenproblem solver effects a significant economy in the free vibration solution of structures when compared with the analysis based on the usual finite element procedure. A FORTRAN IV computer program listing of the various relevant element matrices is also presented in the Appendix.     

Finite element method for analysis of frozen earth structures

A finite element model employing the “viscoplastic flow rule” and the von Mises yield function was developed for frozen soil in the multiaxial stress state. A weighting procedure, which evaluates “effective creep parameters”, was proposed to account for the substantial differences in creep parameters in tension and compression. For reinforced frozen earth structures, bond behavior between the reinforcement and frozen soil was modeled by bond interface elements with nonlinear properties inferred from experiments. Numerical examples include the creep behavior of an open excavation supported by plain and by reinforced frozen walls.     

Finite element simulation of internal flows with heat transfer using a velocity correction approach

This paper enumerates finite-element based prediction of internal flow problems, with heat transfer. The present numerical simulations employ a velocity correction algorithm, with a Galerkin weighted residual formulation. Two problems each in laminar and turbulent flow regimes are investigated, by solving full Navier-Stokes equations. Flow over a backward-facing step is studied with extensive validations. The robustness of the algorithm is demonstrated by solving a very complex problem viz. a disk and doughnut baffled heat exchanger, which has several obstructions in its flow path. The effect of wall conductivity in turbulent heat transfer is also studied by performing a conjugate analysis. Temporal evolution of flow in a channel due to circular, square and elliptic obstructions is investigated, to simulate the vortex dynamics. Flow past an in-line tube bank of a heat exchanger shell is numerically studied. Resulting heat and fluid flow patterns are analysed. Important design parameters of interest such as the Nusselt number, Strouhal number, skin friction coefficient, pressure drop etc. are obtained. It is successfully demonstrated that the velocity correction approach with a Galerkin weighted residual formulation is able to effectively simulate a wide range of fluid flow features.     

An application of global approximations in the finite element method

A technique is described in which global approximations are used to improve the accuracy of the finite element method. The technique is theoretically based on two corollaries to the Lax–Milgram lemma which are presented in this paper. Basically, the technique consists of factoring the unknown function for which an approximation is desired into the product of a global approximation and a second unknown function. Finite element methods are appropriately applied to obtain an approximation to the second unknown function. The approximation to the original function then consists of the product of the specified global approximation and the approximation to the second unknown function. The advantage that finite element methods possess with respect to obtaining banded matrices is preserved in this technique. In addition, numerical examples indicate that the technique's accuracy is as much as a factor of fifty better than the accuracy obtained by directly applying a finite element method to approximate the original function.