Relaxation of alternating iterative algorithms for the Cauchy problem associated with the modified Helmholtz equation

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary, in the case of the alternating iterative algorithm of ` 12 ` 12 `$12 `&12 `#12 `^12 `_12 `%12 `~12 *Kozlov91 applied to Cauchy problems for the modified Helmholtz equation. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed methods.

A procedure for determining a spacewise dependent heat source and the initial temperature

The inverse problem of determining a spacewise dependent heat source, together with the initial temperature for the parabolic heat equation, using the usual conditions of the direct problem and information from two supplementary temperature measurements at different instants of time is studied. These spacewise dependent temperature measurements ensure that this inverse problem has a unique solution, despite the solution being unstable, hence the problem is ill-posed. We propose an iterative algorithm for the stable reconstruction of both the initial data and the source based on a sequence of well-posed direct problems for the parabolic heat equation, which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for a typical benchmark test example, which has the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure gives accurate numerical approximations in relatively few iterations.

Determination of a spacewise dependent heat source

This paper investigates the inverse problem of determining a spacewise dependent heat source in the parabolic heat equation using the usual conditions of the direct problem and information from a supplementary temperature measurement at a given single instant of time. The spacewise dependent temperature measurement ensures that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. For this inverse problem, we propose an iterative algorithm based on a sequence of well-posed direct problems which are solved at each iteration step using the boundary element method (BEM). The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for various typical benchmark test examples which have the input measured data perturbed by increasing amounts of random noise.

An iterative method based on boundary integrals for elliptic Cauchy problems in semi-infinite domains

In this study, we investigate the problem of reconstruction of a stationary temperature field from given temperature and heat flux on a part of the boundary of a semi-infinite region containing an inclusion. This situation can be modelled as a Cauchy problem for the Laplace operator and it is an ill-posed problem in the sense of Hadamard. We propose and investigate a Landweber-Fridman type iterative method, which preserve the (stationary) heat operator, for the stable reconstruction of the temperature field on the boundary of the inclusion. In each iteration step, mixed boundary value problems for the Laplace operator are solved in the semi-infinite region. Well-posedness of these problems is investigated and convergence of the procedures is discussed. For the numerical implementation of these mixed problems an efficient boundary integral method is proposed which is based on the indirect variant of the boundary integral approach. Using this approach the mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing that stable and accurate reconstructions of the temperature field on the boundary of the inclusion can be obtained also in the case of noisy data. These results are compared with those obtained with the alternating iterative method.

Computer analysis of large civil engineering structures

The finite element method is now well established among engineers as being an extremely useful tool in the analysis of problems with complicated boundary conditions. One aim of this thesis has been to produce a set of computer algorithms capable of efficiently analysing complex three dimensional structures. This set of algorithms has been designed to permit much versatility. Provisions such as the use of only those parts of the system which are relevant to a given analysis and the facility to extend the system by the addition of new elements are incorporate. Five element types have been programmed, these are, prismatic members, rectangular plates, triangular plates and curved plates. The 'in and out of plane' stiffness matrices for a curved plate element are derived using the finite element technique. The performance of this type of element is compared with two other theoretical solutions as well as with a set of independent experimental observations. Additional experimental work was then carried out by the author to further evaluate the acceptability of this element. Finally the analysis of two large civil engineering structures, the shell of an electrical precipitator and a concrete bridge, are presented to investigate the performance of the algorithms. Comparisons are made between the computer time, core store requirements and the accuracy of the analysis, for the proposed system and those of another program.

Discrete element modelling of a rotating drum and drum granulation

This thesis reports the results of DEM (Discrete Element Method) simulations of rotating drums operated in a number of different flow regimes. DEM simulations of drum granulation have also been conducted. The aim was to demonstrate that a realistic simulation is possible, and further understanding of the particle motion and granulation processes in a rotating drum. The simulation model has shown good qualitative and quantitative agreement with other published experimental results. A two-dimensional bed of 5000 disc particles, with properties similar to glass has been simulated in the rolling mode (Froude number 0.0076) with a fractional drum fill of approximately 30%. Particle velocity fields in the cascading layer, bed cross-section, and at the drum wall have shown good agreement with experimental PEPT data. Particle avalanches in the cascading layer have been shown to be consistent with single layers of particles cascading down the free surface towards the drum wall. Particle slip at the drum wall has been shown to depend on angular position, and ranged from 20% at the toe and shoulder, to less than 1% at the mid-point. Three-dimensional DEM simulations of a moderately cascading bed of 50,000 spherical elastic particles (Froude number 0.83) with a fractional fill of approximately 30% have also been performed. The drum axis was inclined by 50 to the horizontal with periodic boundaries at the ends of the drum. The mean period of bed circulation was found to be 0.28s. A liquid binder was added to the system using a spray model based on the concept of a wet surface energy. Granule formation and breakage processes have been demonstrated in the system.

A procedure for the reconstruction of a stochastic stationary temperature field

An iterative procedure is proposed for the reconstruction of a temperature field from a linear stationary heat equation with stochastic coefficients, and stochastic Cauchy data given on a part of the boundary of a bounded domain. In each step, a series of mixed well-posed boundary-value problems are solved for the stochastic heat operator and its adjoint. Well-posedness of these problems is shown to hold and convergence in the mean of the procedure is proved. A discretized version of this procedure, based on a Monte Carlo Galerkin finite-element method, suitable for numerical implementation is discussed. It is demonstrated that the solution to the discretized problem converges to the continuous as the mesh size tends to zero.

Leak identification in saturated unsteady flow via a Cauchy problem

This work is an initial study of a numerical method for identifying multiple leak zones in saturated unsteady flow. Using the conventional saturated groundwater flow equation, the leak identification problem is modelled as a Cauchy problem for the heat equation and the aim is to find the regions on the boundary of the solution domain where the solution vanishes, since leak zones correspond to null pressure values. This problem is ill-posed and to reconstruct the solution in a stable way, we therefore modify and employ an iterative regularizing method proposed in [1] and [2]. In this method, mixed well-posed problems obtained by changing the boundary conditions are solved for the heat operator as well as for its adjoint, to get a sequence of approximations to the original Cauchy problem. The mixed problems are solved using a Finite element method (FEM), and the numerical results indicate that the leak zones can be identified with the proposed method.