Parallel-vector computations for geometrically nonlinear finite element analysis

Existing procedures for nonlinear finite element analysis are reviewed. Common computational steps among existing methods are identified. Parallel-vector solution strategies for the generation and assembly of element matrices, solution of the resulting system of linear equations, calculations of the unbalanced loads, displacements and stresses are all incorporated into the Newton-Raphson (NR), modified Newton-Raphson (mNR), and BFGS methods. Furthermore, a mixed parallel-vector Choleski-Preconditioned Conjugate Gradient (C-PCG) equation solver is also developed and incorporated into the piecewise linear procedure for nonlinear finite element analysis. Numerical results have indicated that the Newton-Raphson method is the most effective nonlinear procedure and the mixed C-PCG equation solver offers substantial computational advantages in a parallel-vector computer environment.     

A Unified Analysis of Several Mixed Methods for Elasticity with Weak Stress Symmetry

We give a unified error analysis of several mixed methods for linear elasticity which impose stress symmetry weakly. We consider methods where the rotations are approximated by discontinuous polynomials. The methods we consider are such that the approximate stress spaces contain standard mixed finite element spaces for the Laplace equation and also contain divergence free spaces that use bubble functions.     

Degenerate Two-Phase Incompressible Flow IV: Local Refinement and Domain Decomposition

This is the fourth paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we describe a finite element approximation for this system on locally refined grids. This adaptive approximation is based on a mixed finite element method for the elliptic pressure equation and a Galerkin finite element method for the degenerate parabolic saturation equation. Both discrete stability and sharp a priori error estimates are established for this approximation. Iterative techniques of domain decomposition type for solving it are discussed, and numerical results are presented.     

Mixed finite element method for analysis of viscoelastic fluid flow

In the present paper, numerical analysis of incompressible viscoelastic fluid flow is discussed using mixed finite element Galerkin method. Because Maxwellian viscoelasticity is assumed as the constitutive equation, stress components could not be eliminated from the governing equation system. Because of this, mixed finite element method is utilized to discretize the basic equations. For the solution procedures to solve discretized equation system, Newton-Raphson method for steady flow and perturbation method for unsteady flow is employed. As the numerical examples, comparison was made on the finite element computational results between by direct method and by mixed method. Effects of the viscoelasticity is analyzed for the flows at Reynold's numbers 30, 50 and 70.     

New fast super-dashpot time-dependent techniques for the numerical simulation of steady flows—I: Numerical formulation

This paper describes new fast artificial time-dependent methods leading asymptotically, after a sufficiently long time, to the solution of any steady system of first-order equations. They can, namely, be very useful and efficient for computing steady inviscid transonic mixed flows, as well as for solving the steady hybrid equations of subsonic rotational flows. The time-dependent equations used by the new methods are constructed by adding a purely artificial unsteady operator to the steady physical equations. That operator introduces a strong internal damping of the perturbation waves similar to that due to dashpots on the surface of a vibrating membrane. As a result, a very large rate of convergence of the same order of magnitude as that of the over-relaxation techniques is obtained. The new methods can be applied to any conservative finite differences, finite volumes or finite element discretization of the steady equations. Their level of generality is comparable to that of the classical time-dependent techniques using the unsteady Euler equations, but they are much faster.     

Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem

In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by semilinear parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. By applying some error estimates techniques of mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present a numerical example which confirms our theoretical results.     

Continuous and discontinuous finite element methods for burgers' equation

We apply two discontinuous finite element methods to the inviscid Burgers' equation and to the full equation with viscosity. In both cases we compare with a continuous space-time finite element method previously studied. For v = 0 discontinuous methods give better results, while the reverse prevails for the viscous equation.     

A Mixed Finite Element Method for the Biharmonic Problem Using Biorthogonal or Quasi-Biorthogonal Systems

We consider a finite element method based on biorthogonal or quasi-biorthogonal systems for the biharmonic problem. The method is based on the primal mixed finite element method due to Ciarlet and Raviart for the biharmonic equation. Using different finite element spaces for the stream function and vorticity, this approach leads to a formulation only based on the stream function. We prove optimal a priori estimates for both stream function and vorticity, and present numerical results to demonstrate the efficiency of the approach.     

An Adaptive Moving Mesh Finite Element Solution of the Regularized Long Wave Equation

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a \(C^0\) finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution.     

A stabilized MITC element for accurate wave response in Reissner–Mindlin plates

Residual based finite element methods are developed for accurate time-harmonic wave response of the Reissner–Mindlin plate model. The methods are obtained by appending a generalized least-squares term to the mixed variational form for the finite element approximation. Through judicious selection of the design parameters inherent in the least-squares modification, this formulation provides a consistent and general framework for enhancing the wave accuracy of mixed plate elements. In this paper, the mixed interpolation technique of the well-established MITC4 element is used to develop a new mixed least-squares (MLS4) four-node quadrilateral plate element with improved wave accuracy. Complex wave number dispersion analysis is used to design optimal mesh parameters, which for a given wave angle, match both propagating and evanescent analytical wave numbers for Reissner–Mindlin plates. Numerical results demonstrates the significantly improved accuracy of the new MLS4 plate element compared to the underlying MITC4 element.     

Error Analysis of Mixed Finite Element Methods for Nonlinear Parabolic Equations

In this paper, we prove a discrete embedding inequality for the Raviart–Thomas mixed finite element methods for second order elliptic equations, which is analogous to the Sobolev embedding inequality in the continuous setting. Then, by using the proved discrete embedding inequality, we provide an optimal error estimate for linearized mixed finite element methods for nonlinear parabolic equations. Several numerical examples are provided to confirm the theoretical analysis.     

Mixed finite element methods — Reduced and selective integration techniques: A unification of concepts

The equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established. This enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method. Applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles. These include thin beams and plates, and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics.     

Fully discrete mixed finite element approximations for non-Darcy flows in porous media

A numerical approximation procedure is proposed to solve equations describing non-Darcy flow of a single-phase fluid in a porous medium in two or three spacial dimensions, including the generalized Forchheimer equation. Fully discrete mixed finite element methods are considered and analyzed for the approximation. Existence and uniqueness of the approximation are discussed and optimal order error estimates in L² are derived for the three relevant functions.     

A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows

In this paper we introduce and analyze new mixed finite element schemes for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. The methods are based on a non-standard mixed approach in which the velocity, the pressure, and the pseudostress are the original unknowns. However, we use the incompressibility condition to eliminate the pressure, and set the velocity gradient as an auxiliary unknown, which yields a twofold saddle point operator equation as the resulting dual-mixed variational formulation. In addition, a suitable augmented version of the latter showing a single saddle point structure is also considered. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well-posed. In particular, we show that Raviart–Thomas elements of order k ? 0 for the pseudostress, and piecewise polynomials of degree k for the velocity and its gradient, ensure the well-posedness of the associated Galerkin schemes. Moreover, we prove that any finite element subspace of the square integrable tensors can be employed to approximate the velocity gradient in the case of the augmented formulation. Then, we derive a reliable and efficient residual-based a posteriori error estimator for each scheme. Finally, we provide several numerical results illustrating the good performance of the resulting mixed finite element methods, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithms.     

A state space finite element for laminated composite plates

This paper presents a semi-analytical finite element solution for the stress analysis of cross-ply laminated composite plates. The method is based on a mixed variational principle that includes the variations of both displacements and stresses. Finite element approximation is introduced only for the in-plane variations of displacements and stresses, while the through-thickness distributions of them are obtained by using the method of state equation. Numerical tests show that the results obtained approach the analytical three-dimensional solutions. Moreover, the use of the recursive formulation of the state equation leads to the solution of an algebra equation system whose order does not depend on the number of material layers of the laminate. Compared with the traditional finite element method, the new solution always provides continuous distributions of both displacements and transverse stresses across material interfaces.     

Formulation and analysis of fully-mixed methods for stress-assisted diffusion problems

This paper is devoted to the mathematical and numerical analysis of a mixed-mixed PDE system describing the stress-assisted diffusion of a solute into an elastic material. The equations of elastostatics are written in mixed form using stress, rotation and displacements, whereas the diffusion equation is also set in a mixed three-field form, solving for the solute concentration, for its gradient, and for the diffusive flux. This setting simplifies the treatment of the nonlinearity in the stress-assisted diffusion term. The analysis of existence and uniqueness of weak solutions to the coupled problem follows as combination of Schauder and Banach fixed-point theorems together with the Babu?ka–Brezzi and Lax–Milgram theories. Concerning numerical discretization, we propose two families of finite element methods, based on either PEERS or Arnold–Falk–Winther elements for elasticity, and a Raviart–Thomas and piecewise polynomial triplet approximating the mixed diffusion equation. We prove the well-posedness of the discrete problems, and derive optimal error bounds using a Strang inequality. We further confirm the accuracy and performance of our methods through computational tests.     

L ∞-error estimates of higher order mixed finite element approximations for elliptic optimal control problems

In this paper, we investigate the L ^∞-error estimates of the numerical solutions of linear-quadratic elliptic control problems by using higher order mixed finite element methods. The state and co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order k (k≥1). Optimal L ^∞-error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for optimal control problems.     

Numerical solutions of the one-phase classical Stefan problem using an enthalpy green element formulation

The enthalpy method is exploited in tackling a heat transfer problem involving a change of state. The resulting governing equation is then solved with a hybrid finite element - boundary element technique known as the Green element method (GEM). Two methods of approximation are employed to handle the time derivative contained in the discrete element equation. The first involves a finite difference method, while the second utilizes a Galerkin finite element approach. The performance of both methods are assessed with a known closed form solution. The finite element based time discretization, despite its greater challenge, yields less reliable numerical results. In addition a numerical stability test of both methods based on a Fourier series analysis explain the dispersive characters of both techniques, and confirms that replication of correct results is largely attributed to their ability to handle the harmonics of small wavelengths which are usually dominant in the vicinity of a front.     

Finite difference and finite element methods

The relationships between and relative advantages of finite difference and finite element methods are discussed. The less familiar finite element methods are described first for equilibrium problems: it is shown how quadratic elements on right triangles lead to natural generalisations of the powerful, fourth order accurate nine-point difference scheme for the Laplacian. For evolutionary problems, the recent development of more accurate difference methods is considered together with that of Galerkin methods. It is shown how conservation properties are best preserved by the latter methods and, in particular, how the supression of non-linear instabilities in the advection equation is achieved by the Arakawa schemes. Finally, an error analysis is described which is applicable to both finite difference and finite element methods.     

Numerical methods for calculating pressure distribution in gas bearings

In gas bearings, the pressure distribution is governed by a non-linear Reynolds equation. In order to solve this equation two numerical methods, the conservative difference scheme and the finite element method, are provided in this paper. They are superior to the finite difference method of Colemman [2]. Use of the finite element method is advocated because of its flexibility in solving the Reynolds equation.