Showing papers on "Piecewise published in 1981"

Approximation theory and methods

Abstract: Most functions that occur in mathematics cannot be used directly in computer calculations. Instead they are approximated by manageable functions such as polynomials and piecewise polynomials. The general theory of the subject and its application to polynomial approximation are classical, but piecewise polynomials have become far more useful during the last twenty years. Thus many important theoretical properties have been found recently and many new techniques for the automatic calculation of approximations to prescribed accuracy have been developed. This book gives a thorough and coherent introduction to the theory that is the basis of current approximation methods. Professor Powell describes and analyses the main techniques of calculation supplying sufficient motivation throughout the book to make it accessible to scientists and engineers who require approximation methods for practical needs. Because the book is based on a course of lectures to third-year undergraduates in mathematics at Cambridge University, sufficient attention is given to theory to make it highly suitable as a mathematical textbook at undergraduate or postgraduate level.

The p-Version of the Finite Element Method

Abstract: In the p-version of the finite element method, the triangulation is fixed and the degree p, of the piecewise polynomial approximation, is progressively increased until some desired level of precision is reached.In this paper, we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation. These properties lead to an a priori estimate of the asymptotic rate of convergence of the p-version. The estimate shows that the p-version gives results which are not worse than those obtained by the conventional finite element method (called the h-version, in which h represents the maximum diameter of the elements), when quasi-uniform triangulations are employed and the basis for comparison is the number of degrees of freedom. Furthermore, in the case of a singularity problem, we show (under conditions which are usually satisfied in practice) that the rate of convergence of the p-version is twice that of the h-version with quasi-uniform mesh. Inve...

Numerical treatment of delay differential equations by Hermite Interpolation

Abstract: A class of numerical methods for the treatment of delay differential equations is developed. These methods are based on the wellknown Runge-Kutta-Fehlberg methods. The retarded argument is approximated by an appropriate multipoint Hermite Interpolation. The inherent jump discontinuities in the various derivatives of the solution are considered automatically. Problems with piecewise continuous right-hand side and initial function are treated too. Real-life problems are used for the numerical test and a comparison with other methods published in literature.

Analysis and optimal control of time-varying linear systems via block-pulse functions

Abstract: Succinct and clear-sighted operational properties of block-pulse functions are fully applied to the analysis and optimal control of time-varying linear systems with quadratic performance index. Piecewise constant solutions equally distributed, which are simple in form and convenient for use or implementation, are consequently obtained. Another advantage of this method is that any positive integer can be chosen as the number of sub-intervals, whereas in the case of Walsh function approximation the choice can only be made from 2, 4, 8, 16, 32, and so on. The algorithms developed in the paper are illustrated by appropriate examples.

An analysis of a uniformly accurate difference method for a singular perturbation problem

Abstract: It will be proven that an exponential tridiagonal difference scheme, when applied with a uniform mesh of size h to: euXx + b(x)ux = f(x) for O O, b and f smooth, e in (0, 1], and u(O) and u(I) given, is uniformly second-order accurate (i.e., the maximum of the errors at the grid points is bounded by Ch2 with the constant C independent of h and e). This scheme was derived by El-Mistikawy and Werle by a C' patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point. The behavior of the approximate solution in between the grid points will be analyzed, and some numerical results will also be given.

Zero sets of multiparameter functions and stability of multidimensional systems

Abstract: New theorems, which can be used to find the zero set of a multiparameter rational function of a complex variable where the distinguished boundary of the parameters' closed domain of definition is composed of piecewise smooth Jordan arcs or curves, are stated and proved. The results are used to derive a new simplified procedure for a multi-dimensional stability test, presented for the continuous-analog case, as well as for the discrete-digital case. Now proofs to Huang's and Strintzis' theorems on multidimensional stability are also provided, as an application of the theorems on zero sets.

High bandwidth evaluation of elementary functions

Abstract: Among the requirements currently being imposed on high-performance digital computers to an increasing extent are the high-bandwidth computations of elementary functions, which are relatively time-consuming procedures when conducted in software. In this paper, we elaborate on a technique for computing piecewise quadratric approximations to many elementary functions. This method permits the effective use of large RAMs or ROMs and parallel multipliers for rapidly generating single-precision floating-point function values (e.g., 30–45 bits of fraction, with current RAM and ROM technology). The technique, based on the use of Taylor series, may be readily pipelined. Its use for calculating values for floating-point reciprocal, square root, sine, cosine, arctangent, logarithm, exponential and error functions is discussed.

The structure of piecewise monotonic transformations

Abstract: Transformations on [0, 1] which are piecewise monotonic and piecewise continuous are considered. Using symbolic dynamics, the structure of their nonwandering set is determined. This is then used to prove results about maximal and absolutely continuous invariant measures.

Some remarks on the discrete maximum-principle for finite elements of higher order

Abstract: The discrete maximum principle for finite element approximations of standard elliptic problems in the plane is discussed. Even in the case Δu=0 a slightly stronger version of the principle does not hold with piecewise quadratic elements for all but some very special triangularisation geometries.

A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel

Abstract: We construct collocation methods with an arbitrary degree of accuracy for integral equations with logarithmically or algebraically singular kernels. Superconvergence at collocation points is obtained. A grid is used, the degree of non-uniformity of which is in good conformity with the smoothness of the solution and the desired accuracy of the method.

Prediction of subsonic aerodynamic characteristics - A case for low order panel methods

Abstract: A low-order panel method is presented for the calculation of subsonic aerodynamic characteristics of general configurations. The method is based on piecewise constant doublet and source singularities. Two forms of the internal Dirichlet boundary condition are discussed and the source distribution is determined by the external Neumann boundary condition. Calculations are compared with higher-order solutions for a number of cases. It is demonstrated that for comparable density of control points where the boundary conditions are satisfied, the low-order method gives comparable accuracy to the higher-order solutions. It is also shown that problems associated with some earlier low-order panel methods, e.g., leakage in internal flows and junctions and also poor trailing-edge solutions, do not appear for the present method. Further, the application of the Kutta condition is extremely simple; no extra equation or trailing-edge velocity point is required. The method has very low computing costs and this has made it practical for application to nonlinear problems requiring iterative solutions and to three-dimensional unsteady problems using a time-stepping approach. In addition, the method has been extended to model separated flows in three dimensions, using free vortex sheets to enclose the separated zone.

The Somigliana identity on piecewise smooth surfaces

Abstract: TheSomigliana identity is an integral representation of the displacement field of a body, i.e. the solution of the Cauchy-Navier equation in the classical theory of elasticity is represented by influence functions. This well established identity for bodies with smooth surfaces is extended in the present paper to cover two- and three-dimensional bodies with piecewise smooth surfaces.

Extensions to nonlinear and minimax approaches

Abstract: Publisher Summary This chapter presents a nonparametric approach requiring the determination of the weighting function for linear and linearized models. The linear plant is identified using correlation methods of random functions. To overcome the difficulties of solving the integral equation, a method using decomposition of the correlation function is developed. A statistical approach to identification based on the use of the first two moment functions of random functions immediately leads to correlation and dispersion methods for identification. Piecewise approximation does not entail difficulties of designing polynomial models because of poor conditionality of the matrix of the set of normal equations in using the method of least squares. Finally, speed of response of iterative procedures in piece wise approximation does not depend much on the dimension of the input space and the form of the desired response.

Superconvergence of collocation methods for Volterra and Abel integral equations of the second kind

Abstract: This paper deals with the question of the attainable order of convergence in the numerical solution of Volterra and Abel integral equations by collocation methods in certain piecewise polynomial spaces and which are based on suitable interpolatory quadrature for the resulting moment integrals. The use of a (nonlinear) variation of constants formula for the representation of the error function in terms of the defect allows for a unified treatment of equations with continuous and weakly singular kernels.

An approximately divergence‐free 9‐node velocity element (with variations) for incompressible flows

Abstract: Explicit basis functions are constructed for 9-node biquadratic velocity fields which guarantee that a weak form of the continuity equation is satisfied. The corresponding pressure approximations are either piecewise constant, piecewise linear or piecewise bilinear. These results are extended to give bases for bilinear velocity/piecewise constant pressure elements and also to some three-dimensional brick elements.

A Suggested Extension of Special Ordered Sets to Non-Separable Non-Convex Programming Problems*

Abstract: This paper suggests a branch and bound method for solving non-separable non-convex programming problems where the nonlinearities are piecewise linearly approximated using the standard simplicial subdivision of the hypercube. The method is based on the algorithm for Special Ordered Sets, used with separable problems, but involves using two different types of branches to achieve valid approximations.

Analysis and optimal control of linear systems via single term Walsh series approach

Abstract: This paper considers a simple method of analysis to find an optimal control of linear time-invariant and time-varying systems by the single term Walsh series approach (STWS). The value of the STWS approach in linear system analysis is established. The time varying gains in optimal control are approximated by piecewise constant gains which are easily determined by the STWS method for linear time-invariant and time-varying systems with quadratic performance criteria. The cumbersome Kronecker product is eliminated, and inversion of large matrices is avoided. Several examples are presented to demonstrate the power and simplicity of the STWS method.

The Polarizabilities of Electrically Small Apertures of Arbitrary Shape

Abstract: Numerical procedures, based on a method of moments approach, are given for computing the magnetic and electric polarizabilities of electrically small apertures of arbitrary shape. The magnetic polarizability density is determined through the use of pulse expansion functions defined over quadrilateral subdomains, while the electric polarizability density is obtained by using basis functions, each of which consists of a piecewise arrangement of simple linear functions defined over triangles and having an area coordinate representation. All the subdomains are generated automatically by applying either the Gordon-Hall or the Zienkiewicz-Phillips subdivision techniques. Computed results are obtained for several aperture shapes, including the circle and the ellipse. The calculations for the last two cases are in excellent agreement with the exact values.

HOLOMORPHIC INEQUIVALENCE OF SOME CLASSES OF DOMAINS IN $\mathbf{C}^n$

Abstract: This article proves the biholomorphic inequivalence of some classes of domains in : in particular, bounded pseudoconvex domains with smooth and piecewise smooth (but not smooth) boundaries. It is shown that a proper holomorphic mapping of one strictly pseudoconvex domain onto another is locally biholomorphic. Bibliography: 20 titles.

A Finite Element Method for Problems in Perfect Plasticity Using Discontinuous Trial Functions

Abstract: It is known (see e.g. [3],[5]) that the displacements in an elastic perfectly-plastic body may be discontinuous. Conventional finite element methods (displacement methods) for plasticity problems are based on using continuous trial functions and are thus not particularly well adapted to the nature of the true solution. In this note we propose a finite element method of displacement type for problems in perfect plasticity where we use a finite element space Vh of piecewise polynomial functions with no requirement on inter-element continuity. In order to be able to approximate a discontinuous solution u of a plasticity problem accurately with functions in Vh, the finite element mesh will have to fit the discontinuities of u. Thus, since the location of these discontinuities is in general not known in advance, one would like to use some kind of adaptive technique where according to the results of computations the finite element mesh is succesively modified. In this note we do not consider this more general problem but concentrate on analyzing the proposed method in the case of a given mesh.

Extensions of distance reducing mappings to piecewise congruent mappings on ℓm

Abstract: In this note I will show any distance reducing mapping f: M → ln, where M is a finite subset of lm (m ≤ n), can be extended to a piecewise conqruent mapping f: lm → ln.

Analysis of piecewise constant delay systems via block-pulse functions

Abstract: The solution of linear piecewise constant delay systems has been derived by using the block pulse series expansion method. A matrix called the delay operational matrix is introduced first to manipulate the time delay. Then the solution is obtained through using the matrix and the properties of block pulse functions.

Three-Dimensional Oscillatory Piecewise Continuous-Kernel Function Method—Part I: Basic Problems

Abstract: The two-dimensional subsonic, piecewise continuous-kernel function method used for studying either oscillatory or steady flows is extended in the present work to three-dimensi onal problems involving finite-span wings. The work treats questions associated with the choice of spanwise pressure polynomials, spanwise collocation points, and numerical integration techniques that need be faced by this method. A subsequent paper contains results which confirm the accuracy of the method, its rapid convergence, and its very high efficiency in terms of computational time. The method is tested for a limited class of geometrical discontinuities (i.e., at the wing root only). A third paper contains additional results which relate to a wider class of problems associated with geometrical discontinuities. EOMETRICAL discontinuities have become common in modern airplane wings, including not only control surface deflections but also wing chord discontinuities such as those existing at the root of a delta wing (discontinuity in the first derivative of the chord along the span), at leading-edge extensions, or at wing surface break points. Since these geometrical discontinuities lead to pressure singularities at the same geometrical locations, it is necessary to know the exact form of these pressure singularities if the use of the kernel function method (KFM) is contemplated. The lattice methods (i.e., the vortex or doublet) can successfully cope with unknown pressure singularities if their location is known, but they require a relatively large number of unknowns ("boxes") for convergence which at times leads to a relatively large residual error at the converged values.1'2 In Refs. 1 and 2, a different method is proposed which represents the pressure distribution by a set of piecewise continuous polynomials spanning the regions between adjoining singularities (also referred to as "boxes") and employs the KFM for solution of the pressure coefficients. It is shown in Refs. 1 and 2 where this two-dimensional problem was treated that such an ap- proach, referred to as the piecewise continuous-kernel func- tion method (PCKFM), has the ability to treat pressure discontinuities in a manner similar to the doublet-lattice method, with the added accuracy and rapid convergence characteristics of the kernel function method. In addition, it is not essential to determine the nature or form of the pressure singularities that might exist along some of the boundaries forming each box. However, to accelerate convergence, pressure singularities are assumed to be known only along the boundaries of the wing; or more specifically, the form of the leading-edge (LE), trailing-edge (TE), and wing-tip pressure singularities are assumed to be known and are treated in the analysis. All other pressure singularities are ignored during the analysis and their consideration is limited to the deter- mination of the boundaries for the different boxes. The basic problems associated with the two-dimensional PCKFM were treated in Refs. 1 and 2. These problems in- cluded the determination of orthogonal pressure polynomials for boxes with different known pressure singularities along their boundaries, the determination of the collocation points associated with the assumed pressure polynomials, the determination of the desired number of boxes, and the number of orthogonal polynomials required in each box. These problems will be addressed again while attempting to extend the PCKFM to wings with finite spans. Additional problems arising in three-dimensi onal flow configurations involve the formulation of numerical techniques which are required for the successful application of the method. These techniques, which are useful beyond the methods described in this work, will be developed herein. Wing-Tip Pressure Singularity and Associated

Uniqueness and Eventual Uniqueness of Optimal Designs in Some Time Series Models

Abstract: : Earlier results on the uniqueness and eventual uniqueness of optimal designs for certain time series models are extended to a wider class of processes which includes those with covariance structures such as that of multiple integrals of Brownian motion and Brownian bridge processes. The relationship between the problems of regression design for time series and piecewise polynomial approximation with free breakpoints is discussed and, consequently, asymptotic results obtained by Sacks and Ylvisaker (1970) are seen to hold under weaker assumptions for these processes.