709 research outputs found

    On the solution of integral equations with strong ly singular kernels

    Get PDF
    In this paper some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m or = 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t,x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results

    Numerical methods for integral equations of Mellin type

    Get PDF
    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

    Enriched and Isogeometric Boundary Element Methods for Acoustic Wave Scattering

    Get PDF
    This thesis concerns numerical acoustic wave scattering analysis. Such problems have been solved with computational procedures for decades, with the boundary element method being established as a popular choice of approach. However, such problems become more computationally expensive as the wavelength of an incident wave decreases; this is because capturing the oscillatory nature of the incident wave and its scattered field requires increasing numbers of nodal variables. Authors from mathematical and engineering backgrounds have attempted to overcome this problem using a wide variety of procedures. One such approach, and the approach which is further developed in this thesis, is to include the fundamental character of wave propagation in the element formulation. This concept, known as the Partition of Unity Boundary Element Method (PU-BEM), has been shown to significantly reduce the computational burden of wave scattering problems. This thesis furthers this work by considering the different interpolation functions that are used in boundary elements. Initially, shape functions based on trigonomet- ric functions are developed to increase continuity between elements. Following that, non-uniform rational B-splines, ubiquitous in Computer Aided Design (CAD) soft- ware, are used in developing an isogeometric approach to wave scattering analysis of medium-wave problems. The enriched isogeometric approach is named the eXtended Isogeometric Boundary Element Method (XIBEM). In addition to the work above, a novel algorithm for finding a uniform placement of points on a unit sphere is presented. The algorithm allows an arbitrary number of points to be chosen; it also allows a fixed point or a bias towards a fixed point to be used. This algorithm is used for the three-dimensional acoustic analyses in this thesis. The new techniques developed within this thesis significantly reduce the number of degrees of freedom required to solve a problem to a certain accuracy—this reduc- tion is more than 70% in some cases. This reduces the number of equations that have to be solved and reduces the amount of integration required to evaluate these equations

    On orthogonal collocation solutions of partial differential equations

    Get PDF
    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

    Spectral methods for exterior elliptic problems

    Get PDF
    Spectral approximations for exterior elliptic problems in two dimensions are discussed. As in the conventional finite difference or finite element methods, the accuracy of the numerical solutions is limited by the order of the numerical farfield conditions. A spectral boundary treatment is introduced at infinity which is compatible with the infinite order interior spectral scheme. Computational results are presented to demonstrate the spectral accuracy attainable. Although a simple Laplace problem is examined, the analysis covers more complex and general cases

    Spectral methods in fluid dynamics

    Get PDF
    Fundamental aspects of spectral methods are introduced. Recent developments in spectral methods are reviewed with an emphasis on collocation techniques. Their applications to both compressible and incompressible flows, to viscous as well as inviscid flows, and also to chemically reacting flows are surveyed. The key role that these methods play in the simulation of stability, transition, and turbulence is brought out. A perspective is provided on some of the obstacles that prohibit a wider use of these methods, and how these obstacles are being overcome

    Spectral methods for partial differential equations

    Get PDF
    Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed. Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type. Fluid dynamical applications are emphasized
    • …
    corecore