An Exact Method for Computing the Area Moments of Wavelet and Spline Curves

We present a method for the exact computation of the moments of a region bounded by a curve represented by a scaling function or wavelet basis. Using Green's Theorem, we show that the computation of the area moments is equivalent to applying a suitable multidimensional filter on the coefficients of the curve and thereafter computing a scalar product. The multidimensional filter coefficients are precomputed exactly as the solution of a two-scale relation. To demonstrate the performance improvement of the new method, we compare it with existing methods such as pixel-based approaches and approximation of the region by a polygon. We also propose an alternate scheme when the scaling function is sinc(x)

Doctor of Philosophy

dissertationWhile boundary representations, such as nonuniform rational B-spline (NURBS) surfaces, have traditionally well served the needs of the modeling community, they have not seen widespread adoption among the wider engineering discipline. There is a common perception that NURBS are slow to evaluate and complex to implement. Whereas computer-aided design commonly deals with surfaces, the engineering community must deal with materials that have thickness. Traditional visualization techniques have avoided NURBS, and there has been little cross-talk between the rich spline approximation community and the larger engineering field. Recently there has been a strong desire to marry the modeling and analysis phases of the iterative design cycle, be it in car design, turbulent flow simulation around an airfoil, or lighting design. Research has demonstrated that employing a single representation throughout the cycle has key advantages. Furthermore, novel manufacturing techniques employing heterogeneous materials require the introduction of volumetric modeling representations. There is little question that fields such as scientific visualization and mechanical engineering could benefit from the powerful approximation properties of splines. In this dissertation, we remove several hurdles to the application of NURBS to problems in engineering and demonstrate how their unique properties can be leveraged to solve problems of interest

Finite element solutions to boundary value problems

The finite element solution of certain two-point boundary value problems is discussed. In order to obtain more accuracy than the linear finite element method can give, an order-h[superscript 4] global superconvergence technique is studied. This technique, which uses a quasi-inverse of the Rayleigh-Ritz-Galerkin (finite element) method, is motivated by the papers of C. de Boor and G. J. Fix [14] and P. 0. Frederickson [25]. The Peano kernel theorem is generalized and used to approximate the rate of convergence of the global superconvergence. Following Sard’s theory on best quadrature formulae [50], with some generalization, several quadrature formulae are derived. These quadrature formulae are shown to be consistent, and have some advantages over those obtained by Herbold, Schultz and Varga [34]. For solution of large linear systems which result from the finite element method, LU decomposition (Gaussian Elimination Method) is fast and accurate. However, when it comes to a singular or a nearly singular system, LU decomposition fails. The algorithm FAPIN developed by P. 0. Frederickson for 2-dimensional systems is able to solve singular systems as we demonstrate. We found FAPIN will work more efficiently in 1-dimensional case if we replace the DB[subscript q] approximate inverse C, developed by Benson [3], with other approximate inverses. For the sake of verifying the theory, appropriate numerical experiments are carried out

PAN AIR: A computer program for predicting subsonic or supersonic linear potential flows about arbitrary configurations using a higher order panel method. Volume 4: Maintenance document (version 1.1)

The Maintenance Document is a guide to the PAN AIR software system, a system which computes the subsonic or supersonic linear potential flow about a body of nearly arbitrary shape, using a higher order panel method. The document describes the over-all system and each program module of the system. Sufficient detail is given for program maintenance, updating and modification. It is assumed that the reader is familiar with programming and CDC (Control Data Corporation) computer systems. The PAN AIR system was written in FORTRAN 4 language except for a few COMPASS language subroutines which exist in the PAN AIR library. Structured programming techniques were used to provide code documentation and maintainability. The operating systems accommodated are NOS 1.2, NOS/BE and SCOPE 2.1.3 on the CDC 6600, 7600 and Cyber 175 computing systems. The system is comprised of a data management system, a program library, an execution control module and nine separate FORTRAN technical modules. Each module calculates part of the posed PAN AIR problem. The data base manager is used to communicate between modules and within modules. The technical modules must be run in a prescribed fashion for each PAN AIR problem. In order to ease the problem of supplying the many JCL cards required to execute the modules, a separate module called MEC (Module Execution Control) was created to automatically supply most of the JCL cards. In addition to the MEC generated JCL, there is an additional set of user supplied JCL cards to initiate the JCL sequence stored on the system