On numerical stabilization in the solution of Saint-Venant equations using the finite element method

Solving the Saint-Venant equations by using numerical schemes like finite difference and finite element methods leads to some unwanted oscillations in the water surface elevation. The reason for these oscillations lies in the method used for the approximation of the nonlinear terms. One of the ways of smoothing these oscillations is by adding artificial viscosity into the scheme. In this paper, by using a suitable discretization, we first solve the one-dimensional Saint-Venant equations by a finite element method and eliminate the unwanted oscillations without using an artificial viscosity. Second, our main discussion is concentrated on numerical stabilization of the solution in detail. In fact, we first convert the systems resulting from the discretization to systems relating to just water surface elevation. Then, by using M-matrix properties, the stability of the solution is shown. Finally, two numerical examples of critical and subcritical flows are given to support our results.     

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 stable finite element for the stokes equations

We present in this paper a new velocity-pressure finite element for the computation of Stokes flow. We discretize the velocity field with continuous piecewise linear functions enriched by bubble functions, and the pressure by piecewise linear functions. We show that this element satisfies the usual inf-sup condition and converges with first order for both velocities and pressure. Finally we relate this element to families of higer order elements and to the popular Taylor-Hood element.     

On the automatic solution of nonlinear finite element equations

An algorithm for the automatic incremental solution of nonlinear finite element equations in static analysis is presented. The procedure is designed to calculate the pre- and post-buckling/collapse response of general structures. Also, eigensolutions for calculating the linearized buckling response are discussed. The algorithms have been implemented and various experiences with the techniques are given.     

On numerical error in the finite element method

Various sources of errors, physical and numerical, in the finite element method are analysed. A new type of iterative improvement is introduced where the residual is calculated in single precision. The iteration scheme is analysed with respect to round-off errors and found to give significant improvement over existing direct approaches.     

A transient finite element solution method for the Navier-Stokes equations

This paper is concerned with the discrete finite element formulation and numerical solution of transient incompressible viscous flow in terms of the primitive variables. A restricted variational principle is introduced as equivalent to the momentum equations and the Poisson equation for pressure. The latter is introduced to replace the continuity equation, and thus the incompressibility condition is realized only asymptotically; i.e. through the iterative process. An incomplete cubic interpolation function is used for both the velocities and pressure within a triangular finite element. The discrete equations are integrated in time with backward finite differences. We illustrate the similarity between the (ψ,ζ) finite difference method and the ($\text{u}\text{,p}$) finite element method by calculations on the driven square cavity problem.     

A hybrid finite element method for stokes flow: Part I—Formulation and numerical studies

A hybrid finite element scheme, based on assumed deviatoric fluid stresses in the element and continuous velocity fields at the element-boundaries, is presented. The deviatoric stress and hydrostatic pressure field are subject a priori to the constraints of balance of momenta, the advantages of the present scheme are discussed, and its versatility is demonstrated through a few numerical examples. Studies of convergence and stability of the method are included in Part II of the paper.     

An iterative penalty method for the finite element solution of the stationary Navier-Stokes equations

The objective of this paper is to analyse an iterative procedure for the finite element solution of the Stokes and Navier-Stokes stationary problems. For the latter case, the usual condition on the viscosity and the data that ensures uniqueness is assumed. The method is based on the iterative imposition of the incompressibility condition via penalization. Theoretical and numerical results show that this constraint can be approximated iteratively within the same iterative loop used to deal with the nonlinear term of the equations. Two particular iterative schemes are analysed, namely those based on the Picard and Newton-Raphson algorithms.     

On the numerical solution of state equations

Two new methods for the solution of state equations of a linear time-invariant system are suggested. These methods are based on Romberg's algorithm and utilize the special form of the function to be integrated. The suggested methods are compared with existing ones.     

On a finite element method for the equations of one-dimensional nonlinear thermoviscoelasticity

A computationally uncoupled numerical scheme for the equations of one-dimensional nonlinear thermoviscoelasticity is proposed. The scheme makes use of the finite element method for the space variable and the different method for the time variable. The existence and uniqueness of the approximate solutions are proved, and bounds of the error are analyzed.     

A parallel finite element solution method

New parallel computer architectures have revolutionized the design of computer algorithms, and promise to have significant influence on algorithms for structural engineering computations. In this paper, a parallel finite element solution method is presented. The solution method proposed does not require the formation of global system equations, but computes directly the element distortions, as opposed to solving a system of nodal equations. An element or substructure is mapped on to a processor of an MIMD multiprocessing system. Each processor stores only the information relevant to the element or substructure for which the processor represents. The finite element computations can be performed in parallel, in that a processor generates the local stiffness, computes the element distortions and determines the stress-strain characteristics for the element or substructure associated with the processor.     

Hierarchical parallelisation for the solution of stochastic finite element equations

As an example application the elliptic partial differential equation for steady groundwater flow is considered. Uncertainties in the conductivity may be quantified with a stochastic model. A discretisation by a Galerkin ansatz with tensor products of finite element functions in space and stochastic ansatz functions leads to a certain type of stochastic finite element system (SFEM). This yields a large system of equations with a particular structure. They can be efficiently solved by Krylov subspace methods, as here the main ingredient is the multiplication with the system matrix and the application of the preconditioner. We have implemented a “hierarchical parallel solver” on a distributed memory architecture for this. The multiplication and the preconditioning uses a—possibly parallel—deterministic solver for the spatial discretisation as a building block in a black-box fashion. This paper is concerned with a coarser grained level of parallelism resulting from the stochastic formulation. These coarser levels are implemented by running different instances of the deterministic solver in parallel. Different possibilities for the distribution of data are investigated, and the efficiencies determined. On up to 128 processors, systems with more than 5 × 10⁷ unknowns are solved.     

Penalty finite element method for the Navier-Stokes equations

We present an analysis of a penalty formulation of the stationary Navier-Stokes equations for an incompressible fluid. Subject to restrictions on the viscosity and prescribed body force, it is shown that there exists a unique solution to this penalty problem. The solution to the penalty problem is shown to converge to the solution of the Navier-Stokes problem as O(ε) where ε → 0 is the penalty parameter.Existence, uniqueness and stability properties for the approximate problem are then developed and we derive estimates for finite element approximation of the penalized Navier-Stokes problem presented here. Numerical studies are conducted to examine rates of convergence and sample numerical results presented for test cases.     

Some practical procedures for the solution of nonlinear finite element equations

Procedures for the solution of incremental finite element equations in practical nonlinear analysis are described and evaluated. The methods discussed are employed in static analysis and in dynamic analysis using implicit time integration. The solution procedures are implemented, and practical guidelines for their use are given.     

An explicit finite element method for convection-diffusion equations using rational basis functions

An explicit Galerkin method is formulated by using rational basis functions. The characteristics of the rational difference scheme are investigated with regard to consistency, stability and numerical convergence of the method. Numerical results are also presented.     

Direct numerical simulation of turbulent channel flows using a stabilized finite element method

Direct numerical simulations (DNS) of incompressible turbulent channel flows at Re_τ = 180 and 395 (i.e., Reynolds number, based on the friction velocity and channel half-width) were performed using a stabilized finite element method (FEM). These simulations have been motivated by the fact that the use of stabilized finite element methods for DNS and LES is fairly recent and thus the question of how accurately these methods capture the wide range of scales in a turbulent flow remains open. To help address this question, we present converged results of turbulent channel flows under statistical equilibrium in terms of mean velocity, mean shear stresses, root mean square velocity fluctuations, autocorrelation coefficients, one-dimensional energy spectra and balances of the transport equation for turbulent kinetic energy. These results are consistent with previously published DNS results based on a pseudo-spectral method, thereby demonstrating the accuracy of the stabilized FEM for turbulence simulations.     

Characteristic-based operator-splitting finite element method for Navier-Stokes equations

A new finite element method, which is the characteristic-based operator-splitting (CBOS) algorithm, is developed to solve Navier-Stokes (N-S) equations. In each time step, the equations are split into the diffusive part and the convective part by adopting the operator-splitting algorithm. For the diffusive part, the temporal discretization is performed by the backward difference method which yields an implicit scheme and the spatial discretization is performed by the standard Galerkin method. The convective...     

On the numerical solution of some algebraic integral equations

We discuss the numerical solution of some algebraic integral equations and integral equations of Lichtenstein type by means of a variational method and by using the subspace ofL-splines. For these approximate solutions the proofs of existence, uniqueness, and convergence as well as error estimates are given.     

Some aspects on three-dimensional numerical modelling of reinforced concrete structures using the finite element method

This paper deals with some aspects related to three-dimensional numerical modelling of reinforced concrete structures using the Finite Element Method (FEM). Some subjects such as the solution technique of the non-linear equilibrium equations and the constitutive model for concrete and reinforcement steel are emphasised and commented. A robust method for the evaluation of the intersecting points of the embedded reinforcement bars into the three-dimensional finite element mesh is also presented. The main advantages of the Generalised Displacement Control Method with the Generalised Displacement Parameter to improve the response of the concrete and reinforced concrete analyses are highlighted. Finally, a series of numerical examples related to the above-mentioned aspects are presented.