Numerical modelling of subcritical open channel flow using the K-ε turbulence model and the penalty function finite element technique

A numerical model has been developed that employs the penalty function finite element technique to solve the vertically averaged hydrodynamic and turbulence model equations for a water body using isoparametric elements. The full elliptic forms of the equations are solved, thereby allowing recirculating flows to be calculated. Alternative momentum dispersion and turbulence closure models are proposed and evaluated by comparing model predictions with experimental data for strongly curved subcritical open channel flow. The results of these simulations indicate that the depth-averaged two-equation k-ε turbulence model yields excellent agreement with experimental observations. In addition, it appears that neither the streamline curvature modification of the depth-averaged k-ε model, nor the momentum dispersion models based on the assumption of helicoidal flow in a curved channel, yield significant improvement in the present model predictions. Overall model predictions are found to be as good as those of a more complex and restricted three-dimensional model.     

Finite element modelling of solidification in sand castings employing an implicit—explicit algorithm

The finite element method is used to model the solidification process in a sand casting. With a relatively coarse mesh, there is good agreement between numerical and experimental results. An implicit-explicit time-stepping algorithm demonstrated significant computer time savings when the appropriate solver was implemented. Consideration is also given to the problem of capacitance matrix lumping when quadratic elements are employed.     

Symmetrization of matrix technique in a finite element context

A finite element Galerkin-based formulation of the mass conservation and momentum equations can require, if convective type terms are retained in the coefficient matrix, a non-symmetric solver. The resulting increase in core storage for efficient utilization of CPU time can be considerable. The current paper advocates a simple symmetrization of matrix technique, at element level which results in a considerable reduction in core requirement. The increase in CPU time required when solving linear systems of equations is considerable. However, for nonlinear systems the penalty can be negligible.     

Modelling of laminate structures in elasto-plasticity with a mixed finite element

The main advantage of the mixed finite element (displacements-stresses) is that, because of its continuity, it gives a good approximate stress field, without needing a high degree of interpolation in displacements. This degree of approximation is essential for elastic-plastic computations. However, the total continuity of stresses is too strong in a laminate structure along the interfaces. We show a method of achieving the correct level of continuity without losing the advantage of a good approximation. Some examples of laminate plane structures with plastic areas along the interfaces are given.     

Finite element modelling of anisotropic elastic–viscoplastic behaviour of metals

An implementation of the unified theory of visco-plasticity of Bodner in a three-dimensional finite element program for the analysis of anisotropic inelastic behaviour of selected metals is presented in this paper. A derivation of an effective hardening parameter for the anisotropic (directional) deformation state is also given in this paper using some basic assumptions introduced by Bodner. The effect of the imposed strain rate on the level of the stress–strain curve is also investigated. A comparison of the results of the present finite element model with some published theoretical and experimental results for pure titanium and 2024-T4 aluminium alloy is also made.     

Finite element approximation to a contact problem in linear thermoelasticity

A finite element approximation to the solution of a one-dimensional linear thermoelastic problem with unilateral contact of the Signorini type and heat flux is proposed. An error bound is derived and some numerical experiments are performed.

     

Efficient non-hydrostatic modelling of surface waves interacting with structures

An efficient three-dimensional non-hydrostatic model is applied to simulate free-surface waves interacting with structures. The model employs an implicit Crank–Nicholson scheme to discretize the Navier–Stokes equations under a Cartesian staggered grid framework. An integration method is introduced to account for the full effects of non-hydrostatic pressure at the free-surface layer. A domain decomposition method is proposed to effectively solve the resulting matrix system. The model is first validated by simulating three-dimensional sloshing waves in a container. The model is then applied to simulate waves propagating over two-dimensional and three-dimensional submerged structures, in which the effects of non-linearity and dispersion are important. The model results show that the model using only two vertical layers are in all favorable agreements with experimental data, demonstrating the efficiency and accuracy of the model on simulating surface waves interacting with structures.     

On surface waves in diffraction gratings

We discuss the existence criterion of surface waves based on the augmented scattering matrices. Such matrices arise if one takes into account not only oscillating waves but also those which grow (attenuate) in amplitude far from the grating. A family of planar dielectric gratings with periodic modulation of the refraction index is considered. Asymptotic and numerical analysis of the model are given. We represent various examples of gratings which support (or do not support) surface waves. Copyright © 2000 John Wiley & Sons, Ltd.     

An a priori error estimate for the finite element modelling of electromagnetic waves interacting with a periodic diffraction grating

An a priori error estimate using a so called α,β‐ periodic transformation to study electromagnetic waves in a periodic diffraction grating is derived. It has been reported for single scattering that there is an instability in numerical methods for high wavenumbers. To address this problem, the analytical solution of the scattering problem when the domain is scatterer free and an unknown function called the α,β‐quasi periodic solution are used to transform the associated Helmholtz problem. The well‐posedness of the resulting continuous problem is analysed before approximating its solution using a finite element discretization. To guarantee the uniqueness of this approximate solution, an a priori error estimate is derived. Finally, numerical results are presented that suggest that the α,β‐quasi periodic method converges at a far lower number of degrees of freedom than the α,0‐quasi periodic method reported previously; especially for high wavenumbers. This is particularly true when the incident wave only undergoes a small perturbation because of the presence of the scatterer. Copyright © 2012 John Wiley & Sons, Ltd.     

Approximation of boundary values in the boundary element method

Various techniques may be applied to the approximation of the unknown boundary functions involved in the boundary element method (BEM). Several techniques have been examined numerically to find the most efficient. Techniques considered were: Lagrangian polynomials of the zeroth, first and second orders; spline functions; and the novel weighted minimization technique used successfully in the finite difference method (FDM) for arbitrarily irregular meshes. All these approaches have been used in the BEM for the numerical analysis of plates with various boundary conditions.Both coarse and fine grids on the boundary have been assumed. Maximal errors of the deflections of each plate and the bending moments have been found and the effective computer CPU times determined.Analysis of the results showed that, for the same computer time, the greatest accuracy was obtained by the weighted FDM approach. In the case of the Lagrange approximation, higher order polynomials have proved more efficient. The spline technique yielded more accurate results, but with a higher CPU time.Two discretization approaches have been investigated: the least-squares technique and the collocation method. Despite the fact that the simultaneous algebraic equations obtained were not symmetric, the collocation approach has been confirmed as clearly superior to the least-squares technique, because of the amount of computation time used.     

Finite element Galerkin methods for multi-phase stefan problems

The modelling of heat conduction with phase transitions in terms of integral indentities in Galerkin form is actually based directly on physical concepts like conservation of heat, the conduction law etc. In this way we obtain a formulation which apparently covers general physical situations, for instance with melting and freezing of layered materials. Choosing finite elements in a fixed grid we can still track the moving boundaries by an accurate treatment of the nonlinear enthalpy terms. This makes it possible to use standard finite element packages to build an efficient programme for Stefan-like problems.     

Solution of two-phase flow problems in porous media via an alternating-direction finite element method

The application of an alternating-direction finite element solution procedure to two-phase immiscible displacement problems in porous media is illustrated. This solution scheme provides for rapid solution of the discrete problem, due to the narrow banded matrices involved, with an accuracy which is comparable to that of standard finite element approximations. The governing partial differential equations for immiscible two-phase porous media flow are given and their discretization, via a Laplace-modified time stepping scheme, is presented. Iterative improvement of the time stepping scheme is also considered and numerical examples are provided which demonstrate the saving in computational time which can be achieved.     

Exact evaluation of ∫dsr2 over a circular element

A simple analytical scheme to evaluate the essential integral in the Boundary Element Method (BEM) is derived. It is shown that better accuracy is achieved whilethe computer time is considerably reduced. Application to noncircular elements is also discussed and an example problem is included.     

Finite element approximation of coupled seismic and electromagnetic waves in fluid‐saturated poroviscoelastic media

This work presents a collection of global and iterative finite element procedures for the numerical approximation of coupled seismic and electromagnetic waves in 2D bounded fluid‐saturated porous media, with absorbing boundary conditions at the artificial boundaries. The equations being analyzed are the coupled Biot's equations of motion and Maxwell equations in the diffusive range. Both seismoelectric and electroseismic coupling are simultaneously included and analyzed in the model. The case of compressional and vertically polarized shear waves coupled with the transverse magnetic polarization (PSVTM‐mode) is analyzed in detail, including the derivation of a priori error estimates on the global finite element procedure and results on the convergence of a domain decomposition iterative algorithm. Later, the corresponding results for the case of horizontally polarized shear waves coupled with the transverse electric polarization (SHTE‐mode) are stated. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Subsonic and supersonic disturbances in the generation of surface waves in a liquid layer adjacent to a high-speed gas-stream

In an effort to shed further light upon the nature of “supersonic” disturbances as distinct from that of ‘subsonic’ disturbances in parallel compressible flows, this paper makes an investigation of the stability characteristics of the surface waves generated in a liquid layer adjacent to a high-speed gas-stream. It turns out that the nature of the surface waves generated in the liquid layer depends markedly upon the type of disturbances present in the high-speed gas-stream. For the case of ‘subsonic’ disturbances it is shown that the energy transfer from the gas stream to the surface waves is contributed predominantly by the Fourier component of the normal gas-pressure force-field in phase with the slope of the wavy surface. For the case of ‘supersonic’ disturbances, this energy transfer is shown to be predominantly due to the component of the pressure-field in phase with the surface-wave displacement and is related to the presence of travelling periodic waves in the gas-stream—this energy transfer is shown to promote always the growth of the surface waves.     

Sub-optimal constant-volume control for open channel networks

A state variable mathematical model for use in the synthesis of automatic control systems for open-channel networks is presented. The system considered here consists of n-cascaded reaches joined by control gates.The linear time invariant model consists of a controllable and observable representation where the state variables are the stored water volume variations in each reach and the control signals are the variations of the control gates opening sections. The model derives, through appropriate simplifications, from a more complex one in terms of transfer functions which was derived by linearizing the Saint-Venant equations.The problem of a linear quadratic optimal regulator is formulated in classical terms for the canal system and the constant-volume control laws obtained for the simplified model have been imposed on the complex one: such a control is therefore to be considered sub-optimal.The results of a digital simulation of the controlled system behaviour indicate that the system operates with practically constant stored water volumes in each reach and that such behaviour is fairly close to that of a pressure-water pipe system.