42 research outputs found
Numerical methods for integral equations of Mellin type
We present a survey of numerical methods (based on piecewise polynomial approximation) for integral equations of Mellin type, including examples arising in boundary integral methods for partial differential equations on polygonal domains
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A comparison of the finite element and boundary element methods for solving partial differential equations associated with engineering problems
Two numerical methods, the finite element method and the boundary element method, have been compared by studying the elastic torsion problem for various shaped cross sections including ones where there are boundary singularities. A series of numerical experiments was performed illustrating the effects of grid refinement on convergence for each method and a comparison of the amount of work involved in using each method made
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
On orthogonal collocation solutions of partial differential equations
In contrast to the h-version most frequently used, a p-version of the Orthogonal Collocation Method as applied to differential equations in two-dimensional domains is examined. For superior accuracy and convergence, the collocation points are chosen to be the zeros of a Legendre polynomial plus the two endpoints. Hence the method is called the Legendre Collocation Method. The approximate solution in an element is written as a Lagrange interpolation polynomial. This form of the approximate solution makes it possible to fully automate the method on a personal computer using conventional memory.
The Legendre Collocation Method provides a formula for the derivatives in terms of the values of the function in matrix form. The governing differential equation and boundary conditions are satisfied by matrix equations at the collocation points. The resulting set of simultaneous equations is then solved for the values of the solution function using LU decomposition and back substitution.
The Legendre Collocation Method is applied further to the problems containing singularities. To obtain an accurate approximation in a neighborhood of the singularity, an eigenfunction solution is specifically formulated to the given problem, and its coefficients are determined by least-squares or minimax approximation techniques utilizing the results previously obtained by the Le Legendre Collocation Method. This combined method gives accurate results for the values of the solution function and its derivatives in a neighborhood of the singularity.
All results of a selected number of example problems are compared with the available solutions. Numerical experiments confirm the superior accuracy in the computed values of the solution function at the collocation points
Nyström’s Method and Iterative Solvers for the Solution of the Double-Layer Potential Equation over Polyhedral Boundaries
In this paper we consider a quadrature method for the solution of the double-layer potential equation corresponding to Laplace’s equation in a three-dimensional polyhedron. We prove the stability for our method in the case of special triangulations over the boundary of the polyhedron. For the solution of the corresponding system of linear equations, we consider a two-grid iteration and a further simple iteration procedure. Finally, we establish the rates of convergence and complexity and discuss the effect of mesh refinement near the corners and edges of the polyhedron
Finite element solution for elliptic partial differential equations
The contents of this thesis are a detailed study of the implementation
of Finite Element method for solving linear and non-linear elliptic
partial differential equations. It commences with a description and
classification of partial differential equations, the related matrix and
eigenvalue theory and the related matrix methods to solve the linear and
non-linear systems of equations.
In Chapter Three, we discuss the development of the, finite element
method and its application with a full description of an orderly step-by-step
process. In Chapter Four, we discuss the implementation of developing
an efficient easy-to-use finite element program for the general two-dimensional
problem along with the capability of handling problems for
different domains and boundary conditions and with a fully automated mesh
generation and refinement technique along with a description of generalised
pre- and post-processors for the Finite Element Method. [Continues.
Nyström's method and iterative solvers for the solution of the double layer potential equation over polyhedral boundaries.
In this paper we consider a quadrature method for the solution of the double layer potential equation corresponding to Laplace's equation in a three-dimensional polyhedron. We prove the stability for our method in case of special triangulations over the boundary of the polyhedron. For the solution of the corresponding system of linear equations, we consider a two-grid iteration and a further simple iteration procedure. Finally, we establish the rates of convergence and complexity and discuss the effect of mesh refinement near the corners and edges of the polyhedron
An unsteady, accelerated, high order panel method with vortex particle wakes
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2006.Includes bibliographical references (leaves 125-138).Potential flow solvers for three dimensional aerodynamic analysis are commonly used in industrial applications. The limitation on the number of discretization elements and the user expertise and effort required to specify the wake location are two significant drawbacks preventing the even more widespread use of these codes. These drawbacks are addressed by the hands off, accelerated, unsteady, panel method with vortex particle wakes which is described. In the thesis, an unsteady vortex particle representation of the domain vorticity is coupled to several boundary element method potential flow formulations. Source-doublet, doublet-Neumann membrane (doublet lattice), and source-Neumann boundary integral equation formulations are implemented. A precorrected-FFT accelerated Krylov subspace iterative solution technique is implemented to efficiently solve the boundary element method linear system of equations. Similarly, a Fast Multipole Tree algorithm is used to accelerate the vortex particle interactions. Additional simplification of the panel method setup is achieved through the introduction of a body piercing wake discretization for lifting bodies with thickness.(cont.) Linear basis functions on flat panel surface triangulations are implemented in the accelerated potential flow framework. The advantages of linear order basis functions outweigh the increased complexity of the implementation when compared with traditional constant collocation approaches. Panel integration approaches for the curved panel, double layer self term are presented. A quadratic curved panel, quadratic basis function, Green's theorem direct potential flow solver is presented.by David Joe Willis.Ph.D