A well posed boundary value problem in transonic gas dynamics

A boundary value problem for the Tricomi equation was studied in connection with transonic gas dynamics. The transformed equation delta u plus 1/3Y u sub Y equals 0 in canonical coordinates was considered in the complex domain of two independent complex variables. A boundary value problem was then set by prescribing the real part of the solution on the boundary of the real unit circle. The Dirichlet problem in the upper unit semicircle with vanishing values of the solution at Y = 0 was solved explicitly in terms of the hypergeometric function for the more general Euler-Poisson-Darboux equation. An explicit representation of the solution was also given for a mixed Dirichlet and Neumann problem for the same equation and domain

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed

Higher-order finite-difference methods for partial differential equations

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis develops two families of numerical methods, based upon rational approximations having distinct real poles, for solving first- and second-order parabolic/ hyperbolic partial differential equations. These methods are thirdand fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods first- and second-order spatial derivatives are approximated by finite-difference approximations which produce systems of ordinary differential equations expressible in vector-matrix forms. Solutions of these systems satisfy recurrence relations which lead to the development of parallel algorithms suitable for computer architectures consisting of three or four processors. Finally, the methods are tested on advection, advection-diffusion and wave equations with constant coefficients

A multiscale method for the double layer potential equation on a polyhedron

This paper is concerned with the numerical solution of the double layer potential equation on polyhedra. Specifically, we consider collocation schemes based on multiscale decompositions of piecewise linear finite element spaces defined on polyhedra. An essential difficulty is that the resulting linear systems are not sparse. However, for uniform grids and periodic problems one can show that the use of multiscale bases gives rise to matrices that can be well approximated by sparse matrices in such a way that the solutions to the perturbed equations exhibits still sufficient accuracy. Our objective is to explore to what extent the presence of corners and edges in the domain as well as the lack of uniform discretizations affects the performance of such schemes. Here we propose a concrete algorithm, describe its ingredients, discuss some consequences, future perspectives, and open questions, and present the results of numerical experiments for several test domains including non-convex domains

ANALYSIS OF ITERATIVE METHODS FOR THE SOLUTION OF BOUNDARY INTEGRAL EQUATIONS WITH APPLICATIONS TO THE HELMHOLTZ PROBLEM

This thesis is concerned with the numerical solution of boundary integral equations and the numerical analysis of iterative methods. In the first part, we assume the boundary to be smooth in order to work with compact operators; while in the second part we investigate the problem arising from allowing piecewise smooth boundaries. Although in principle most results of the thesis apply to general problems of reformulating boundary value problems as boundary integral equations and their subsequent numerical solutions, we consider the Helmholtz equation arising from acoustic problems as the main model problem. In Chapter 1, we present the background material of reformulation of Helmhoitz boundary value problems into boundary integral equations by either the indirect potential method or the direct method using integral formulae. The problem of ensuring unique solutions of integral equations for exterior problems is specifically discussed. In Chapter 2, we discuss the useful numerical techniques for solving second kind integral equations. In particular, we highlight the superconvergence properties of iterated projection methods and the important procedure of Nystrom interpolation. In Chapter 3, the multigrid type methods as applied to smooth boundary integral equations are studied. Using the residual correction principle, we are able to propose some robust iterative variants modifying the existing methods to seek efficient solutions. In Chapter 4, we concentrate on the conjugate gradient method and establish its fast convergence as applied to the linear systems arising from general boundary element equations. For boundary integral equalisations on smooth boundaries we have observed, as the underlying mesh sizes decrease, faster convergence of multigrid type methods and fixed step convergence of the conjugate gradient method. In the case of non-smooth integral boundaries, we first derive the singular forms of the solution of boundary integral solutions for Dirichlet problems and then discuss the numerical solution in Chapter 5. Iterative methods such as two grid methods and the conjugate gradient method are successfully implemented in Chapter 6 to solve the non-smooth integral equations. The study of two grid methods in a general setting and also much of the results on the conjugate gradient method are new. Chapters 3, 4 and 5 are partially based on publications [4], [5] and [35] respectively.Department of Mathematics and Statistics, Polytechnic South Wes

An Integrated Nonlinear Wind-Waves Model for Offshore Wind Turbines

This thesis presents a numerical model capable of simulating offshore wind turbines exposed to extreme loading conditions. External condition-based extreme responses are reproduced by coupling a fully nonlinear wave kinematic solver with a hydro-aero-elastic simulator. First, a two-dimensional fully nonlinear wave simulator is developed. The transient nonlinear free surface problem is formulated assuming the potential theory and a high-order boundary element method is implemented to discretize Laplace's equation. For temporal evolution a second-order Taylor series expansion is used. The code, after validation with experimental data, is successfully adopted to simulate overturning plunging breakers which give rise to dangerous impact loads when they break against wind turbine substructures. Emphasis is then placed on the random nature of the waves. Indeed, through a domain decomposition technique a global simulation framework embedding the numerical wave simulator into a more general stochastic environment is developed. The proposed model is meant as a contribution to meet the more and more pressing demand for research in the offshore wind energy sector as it permits taking into account dangerous effects on the structural response so as to increase the global structural safety level

Meshless Hemodynamics Modeling And Evolutionary Shape Optimization Of Bypass Grafts Anastomoses

Objectives: The main objective of the current dissertation is to establish a formal shape optimization procedure for a given bypass grafts end-to-side distal anastomosis (ETSDA). The motivation behind this dissertation is that most of the previous ETSDA shape optimization research activities cited in the literature relied on direct optimization approaches that do not guaranty accurate optimization results. Three different ETSDA models are considered herein: The conventional, the Miller cuff, and the hood models. Materials and Methods: The ETSDA shape optimization is driven by three computational objects: a localized collocation meshless method (LCMM) solver, an automated geometry pre-processor, and a genetic-algorithm-based optimizer. The usage of the LCMM solver is very convenient to set an autonomous optimization mechanism for the ETSDA models. The task of the automated pre-processor is to randomly distribute solution points in the ETSDA geometries. The task of the optimized is the adjust the ETSDA geometries based on mitigation of the abnormal hemodynamics parameters. Results: The results reported in this dissertation entail the stabilization and validation of the LCMM solver in addition to the shape optimization of the considered ETSDA models. The LCMM stabilization results consists validating a custom-designed upwinding scheme on different one-dimensional and two-dimensional test cases. The LCMM validation is done for incompressible steady and unsteady flow applications in the ETSDA models. The ETSDA shape optimization include single-objective optimization results in steady flow situations and bi-objective optimization results in pulsatile flow situations. Conclusions: The LCMM solver provides verifiably accurate resolution of hemodynamics and is demonstrated to be third order accurate in a comparison to a benchmark analytical solution of the Navier-Stokes. The genetic-algorithm-based shape optimization approach proved to be very effective for the conventional and Miller cuff ETSDA models. The shape optimization results for those two models definitely suggest that the graft caliber should be maximized whereas the anastomotic angle and the cuff height (in the Miller cuff model) should be chosen following a compromise between the wall shear stress spatial and temporal gradients. The shape optimization of the hood ETSDA model did not prove to be advantageous, however it could be meaningful with the inclusion of the suture line cut length as an optimization parameter