An algorithm for the shallow water equations with body fitted meshes

Summary An efficient algorithm based on flux difference splitting is presented for the solution of the two-dimensional shallow water equations in a generalised coordinate system. The scheme is based on solving linearised Riemann problems approximately and in more than one dimension incorporates operator splitting. The scheme has good jump capturing properties and the advantage of using body-fitted meshes. Numerical results are shown for flow past a circular obstruction.     

Flux difference splitting for the Euler equations with axial symmetry

An approximate (linearised) Riemann solver is presented for the solution of the Euler equations of gas dynamics for axially symmetric flows. The method is Roe's flux difference splitting with a technique for dealing with source terms and incorporates operator splitting. Results for the problem of a converging spherical shock are presented. This work forms part of the research programme of the Institute for Computational Fluid Dynamics at the Universities of Oxford and Reading and was funded by A.W.R.E., Aldermaston under contract no. NSN/13B/2A88719.     

Simulation of hypersonic flows on unstructured grids

The paper describes a solver for the compressible inviscid flow equations which is based on a flux vector splitting strategy able to deal with chemical reaction effects. The methodology here adopted is based on a modification of the Flux Vector Splitting technique due to van Leer.¹¹ The scheme operates on completely unstructured grids and has been coupled with an adaptive remeshing procedure to compute high speed flows. Solutions for two-dimensional problems for non-reactive and reactive air in thermodynamic equilibrium are presented.     

A direct boundary integral equation formulation for the Oseen flow past a two-dimensional cylinder of arbitrary cross-section

Summary A novel boundary integral formulation is presented for the direct solution of the classical problem of slow flow past a two-dimensional cylinder of arbitrary cross section in an unbounded viscous medium, the equations of motion having first been linearised by the Oseen approximation. It is shown how the governing partial differential equations of motion, together with the no-slip boundary conditions on the cylinder, may be reformulated as a pair of coupled integral equations of the second kind, which may be manipulated further to yield the lift and drag coefficients explicitly, as well as flow characteristics anywhere in the flowfield.The present formulation requires a non-iterative numerical solution procedure which is applicable to low Reynolds number flows. The method is not restricted in its ability to deal with complicated cylinder geometries, as the discretisation of only the cylinder surface is required.Results of the present method are shown to be in good agreement with those of previous analytical and numerical investigations.With 2 Figures     

Flux difference splitting for the Euler equations in generalised coordinates using a local parameterisation of the equation of state

An efficient algorithm based on flux difference splitting is presented for the solution of the three-dimensional Euler equations of gas dynamics in a generalised coordinate system with a general equation of state. The scheme is based on solving linearised Riemann problems approximately and in more than one dimension incorporates operator splitting. The algorithm uses a local parameterisation of the equation of state and as a consequence requires only one function evaluation in each computational cell. The scheme has good shock capturing properties and the advantage of using body-fitted meshes. Numerical results are shown for Mach 8 flow of equilibrium air past a circular cylinder.This work forms part of the research programme for the Institute of Computational Fluid Dynamics at the Universities of Oxford and Reading and was funded by AWRE, Aldermaston under Contract No. NSN/13B/2A88719.     

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the Engquist–Osher approximation for the flux and explicit time-stepping. An adaptive multiresolution scheme with cell averages is then used to speed up CPU time and meet memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for the sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier–thickener model illustrate the efficiency of this method.     

Analysis of flux flow and the formation of oscillation marks in the continuous caster

In the industrial process of continuous steel casting, flux added at the top of the casting mould melts and forms a lubricating layer in the gap between the steel and the oscillating mould walls. The flow of flux in the gap plays an essential role in smoothing the casting operation. The aim of the present work is to better understand the mechanics of flux flow, with an emphasis on such problems as how the flux actually moves down the mould, the physical parameters governing the consumption rate of the flux and the geometry of the lubricating layer. The problem considered is a coupled problem of liquid flow and multi-phase heat transfer. In the first part of the paper, the formation of the lubricating layer is analysed and a set of equations to describe the flux flow is derived. Then, based on an analysis of the heat transfer from the molten steel through the lubricating layer to the mould wall, a system of equations correlating the temperature field in the steel and flux with the geometry of the lubricating layer is derived. Subsequently, the equations for the flux flow are coupled with those arising from heat-transfer analysis and then a numerical scheme for the calculation of the consumption rate of flux, the geometry of the lubricating layer and the solidification surface of the steel is presented.     

Geometrically non-linear method of incompatible modes in application to finite elasticity with independent rotations

We discuss a geometrically non-linear method of incompatible modes. The model problem chosen for the discussion is the finite elasticity with independent rotations. The conditions which ensure the convergence of the method and the methodology to construct incompatible modes are presented. A detailed derivation of variational equations and their linearized form is given for a two-dimensional plane problem. A couple of geometrically non-linear two-dimensional elements with independent rotational freedoms are proposed based on the presented methodology. The elements exhibit a very satisfying performance over a set of problems in finite elasticity.     

One-dimensional modelling of a vascular network in space-time variables

In this paper a one-dimensional model of a vascular network based on space-time variables is investigated. Although the one-dimensional system has been more widely studied using a space-frequency decomposition, the space-time formulation offers a more direct physical interpretation of the dynamics of the system. The objective of the paper is to highlight how the space-time representation of the linear and nonlinear one-dimensional system can be theoretically and numerically modelled. In deriving the governing equations from first principles, the assumptions involved in constructing the system in terms of area-mass flux (A,Q), area-velocity (A,u), pressure-velocity (p,u) and pressure-mass flux(p,Q) variables are discussed. For the nonlinear hyperbolic system expressed in terms of the (A,u) variables the extension of the single-vessel model to a network of vessels is achieved using a characteristic decomposition combined with conservation of mass and total pressure. The more widely studied linearised system is also discussed where conservation of static pressure, instead of total pressure, is enforced in the extension to a network. Consideration of the linearised system also allows for the derivation of a reflection coefficient analogous to the approach adopted in acoustics and surface waves. The derivation of the fundamental equations in conservative and characteristic variables provides the basic information for many numerical approaches. In the current work the linear and nonlinear systems have been solved using a spectral/hp element spatial discretisation with a discontinuous Galerkin formulation and a second-order Adams-Bashforth time-integration scheme. The numerical scheme is then applied to a model arterial network of the human vascular system previously studied by Wang and Parker (To appear in J. Biomech. (2004)). Using this numerical model the role of nonlinearity is also considered by comparison of the linearised and nonlinearised results. Similar to previous work only secondary contributions are observed from the nonlinear effects under physiological conditions in the systemic system. Finally, the effect of the reflection coefficient on reversal of the flow waveform in the parent vessel of a bifurcation is considered for a system with a low terminal resistance as observed in vessels such as the umbilical arteries.     

Boltzmann schemes for continuum gas dynamics

Many problems arising in the aerodynamic design of aerospace vehicles require the numerical solution of the Euler equations of gas dynamics. These are nonlinear partial differential equations admitting weak solutions such as shock waves and constructing robust numerical schemes for these equations is a challenging task. A new line of research called Boltzmann or kinetic schemes discussed in the present paper exploits the connection between the Boltzmann equation of the kinetic theory of gases and the Euler equations for inviscid compressible flows. Because of this connection, a suitable moment of a numerical scheme for the Boltzmann equation yields a numerical scheme for the Euler equations. This idea called the “moment method strategy” turns out to be an extremely rich methodology for developing robust numerical schemes for the Euler equations. The richness is demonstrated by developing a variety of kinetic schemes such as kinetic numerical method, kinetic flux vector splitting method, thermal velocity based splitting, multidirectional upwind method and least squares weak upwind scheme. A 3-D time-marching Euler code calledbheema based on the kinetic flux vector splitting method and its variants involving equilibrium chemistry have been developed for computing hypersonic reentry flows. The results obtained from the codebheema demonstrate the robustness and the utility of the kinetic flux vector splitting method as a design tool in aerodynamics. The work presented in this paper is based on the research work done by several graduate students at our laboratory and collaborators from research and development organizations within the country.     

A model of coupled resonators for calculating the electromagnetic wave diffraction on planar one-dimensional Bragg gratings

The problem of excitation of a one-dimensional planar Bragg grating is studied. A model of coupled resonators is proposed for investigating the diffraction of electromagnetic waves on a waveguide surface with arbitrary corrugation profile. The model is based on the strict relationships derived from a solution to a two-dimensional boundary problem for the Helmholtz equation. A particular form of equations is presented for the case of a rectangular corrugation of plates in the grating.     

A point interpolation mesh free method for static and frequency analysis of two-dimensional piezoelectric structures

 A mesh free method called point interpolation method (PIM) is presented for static and mode-frequency analysis of two-dimensional piezoelectric structures. In the present method, the problem domain and its boundaries are represented by a set of properly scattered nodes. The displacements and the electric potential of a point are interpolated by the values of nodes in its local support domain using shape functions derived based on a point interpolation scheme. Techniques are discussed to surmount the singularity of the moment matrix. Variational principle together with linear constitutive piezoelectric equations is used to establish a set of system equations for arbitrary-shaped piezoelectric structures. These equations are assembled for all quadrature points and solved for displacements and electric potentials. A polynomial PIM program has been developed in MATLAB with matrix triangularization algorithm (MTA), which automatically performs a proper node enclosure and a proper basis selection. Examples are also presented to demonstrate the accuracy and stability of the present method and their results are compared with the conventional FEM results from ABAQUS as well as the analytical or experimental ones. Received: 6 February 2002 / Accepted: 5 August 2002     

An experimental-numerical hybrid technique for two-dimensional stress analysis

A numerical procedure for obtaining the stress components in the interior of a two-dimensional body from the values of stresses measured along the boundary, is presented. In this method the two-dimensional elasticity equations are re-written in a convenient form and solved using the finite difference technique along with the boundary stress values. The application of the method is illustrated with examples.     

Numerical Analysis for a Nonlocal Parabolic Problem

This article is devoted to the study of the finite element approximation for a nonlocal nonlinear parabolic problem. Using a linearised Crank-Nicolson Galerkin finite element method for a nonlinear reaction-diffusion equation, we establish the convergence and error bound for the fully discrete scheme. Moreover, important results on exponential decay and vanishing of the solutions in finite time are presented. Finally, some numerical simulations are presented to illustrate our theoretical analysis.     

A finite difference-Galerkin finite element solution of compressible laminar boundary-layer flows

A finite difference-Galerkinfinite element method is presented for the solution of the two-dimensional compressible laminar boundary-layer flow problem. The streamwise derivatives in the momentum and energy equations are approximated by finite differences. An iterative scheme, due to the non-linearity of the problem, in conjunction with the Galerkin finite element method is then proposed for the solution of the problem through the boundary-layer thickness. Numerical results are presented and these are compared with other numerical and analytical solutions in order to show the applicability and the effectiveness of the proposed formulation. In all the cases here examined, the results obtained attained the same accuracy of other numerical methods for a much smaller number of points in the boundary-layer.     

A flux‐limited numerical method for solving the MHD equations to simulate propulsive plasma flows

For numerical simulations to be effective tools in plasma propulsion research, a high‐order accurate solver that captures MHD shocks monotonically and works reliably for strong magnetic fields is needed. For this purpose, a characteristics‐based scheme for the MHD equations, with flux limiters to improve spatial accuracy, has been developed. In this method, the symmetric form of the MHD equations, accounting for waves propagating in all directions, are solved. The required eigensystem of axisymmetric MHD equations, with appropriate normalization, is presented. This scheme was validated with unsteady (Riemann problem) and force‐free equilibrium (Taylor state) test cases, as well as with measured current density patterns in a magnetoplasmadynamic thruster. Copyright © 2001 John Wiley & Sons, Ltd.     

Microscale heat conduction in homogeneous anisotropic media: a dual-reciprocity boundary element method and polynomial time interpolation approach

In this paper, a dual-reciprocity boundary element method based on some polynomial interpolations to the time-dependent variables is presented for the numerical solution of a two-dimensional heat conduction problem governed by a third order partial differential equation (PDE) over a homogeneous anisotropic medium. The PDE is derived using a non-Fourier heat flux model which may account for thermal waves and/or microscopic effects. In the analysis, discontinuous linear elements are used to model the boundary and the variables along the boundary. The systems of algebraic equations are set up to solve all the unknowns. For the purpose of evaluating the proposed method, some numerical examples with known exact solutions are solved. The numerical results obtained agree well with the exact solutions.     

Hydrodynamic pressure on a dam with a periodically corrugated reservoir bed

Summary The problem of determining the hydrodynamic pressure, caused by earthquake forces, on a dam with a vertical upstream face and a periodically corrugated reservoir bed is solved approximately by employing a Fourier cosine transform technique to the linearised equations of inviscid and incompressible flow. A particular case of the present problem giving rise to results valid for dams with flat reservoir beds is shown to produce known results as a check of the method used.With 1 Figure     

A Legendre spectral element method for the Stefan problem

In this paper we present a Legendre spectral element method for solution of multi-dimensional unsteady change-of-phase Stefan problems. The spectral element method is a high-order (p-type) finite element technique, in which the computational domain is broken up into general (curved) quadrilateral macroelements, and the solution, data and geometry are expanded within each element in terms of tensor-product Lagrangian interpolants. The discrete equations are generated by a Galerkin formulation followed by Gauss–Lobatto Legendre quadrature, for which it is shown that exponential convergence to smooth solutions is obtained as the polynomial order of fixed elements is increased. The spectral element equations are inverted by conjugate gradient iteration, in which the matrix-vector products are calculated efficiently using tensor-product sum-factorization. To solve the Stefan problem numerically, the heat equations in the liquid and solid phases are transformed to fixed domains applying an interface-local time-dependent immobilization transformation technique. The modified heat equations are discretized using finite differences in time, resulting at each time step in a Helmholtz equation in space that is solved using Legendre spectral element elliptic discretizations. The new interface position is then computed using a variationally consistent flux treatment along the phase boundary, and the solution is projected upon the corresponding updated mesh. The rapid convergence rate and stability of the method are discussed, and numerical results are presented for a one-dimensional Stefan problem using both a semi-implicit and a fully implicit time-stepping scheme. Finally, a two-dimensional Stefan problem with a complex phase boundary is solved using the semi-implicit scheme.     

Dynamic elasto-plastic response of shells in an acoustic medium—EPSA code

A nonlinear, large deflection, elasto-plastic finite element code (EPSA) has been developed for the analysis of shells in an acoustic medium subjected to dynamic loadings. The nonlinear equations of shells are discretized with the aid of a finite difference/finite element method based upon the principle of virtual work. The resulting system of equations contains the nodal displacements as the generalized co-ordinates of the problem. The integration in time of the equations of motion is done explicitly via a central difference scheme. Shell strain-displacement relations are established by a two-dimensional finite difference scheme. The shell constitutive equations are formulated in terms of the shell stress resultants and the shell strains and curvatures. The fluid-structure interaction is accounted for by means of the doubly asymptotic approximation (DAA) expressed in terms of orthogonal fluid expansion functions. The analytically produced results satisfactorily reproduce available experimental data for dynamically loaded shells.