Piecewise Polynomials, Interpolation and Applications

This article was supposed to be on ‘multivariate splines». An informal survey, taken recently by asking various people in Approximation Theory what they consider to be a ‘multivariate spline’, resulted in the answer that a multivariate spline is a possibly smooth piecewise polynomial function of several arguments. In particular the potentially very useful thin-plate spline was thought to belong more to the subject of radial basis funtions than in the present article. This is all the more surprising to me since I am convinced that the variational approach to splines will play a much greater role in multivariate spline theory than it did or should have in the univariate theory. Still, as there is more than enough material for a survey of multivariate piecewise polynomials, this article is restricted to this topic, as is indicated by the (changed) title.

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Estimation of Distributed Unbalance of Rotors

Volume 4: Manufacturing Materials and Metallurgy; Ceramics; Structures and Dynamics; Controls, Diagnostics and Instrumentation; Education; IGTI Scholar Award ◽

10.1115/2001-gt-0245 ◽

2001 ◽

Author(s):

Tachung Yang ◽

Chunyi Lin

Keyword(s):

Finite Element ◽

Singular Value Decomposition ◽

Element Model ◽

Singular Value ◽

Lumped Mass ◽

Piecewise Polynomials ◽

Mass Unbalance ◽

Thick Disks ◽

Value Decomposition ◽

Thin Disks

Mass unbalance commonly causes vibration of rotor-bearing systems. Lumped mass modeling of unbalance was adapted in most previous research. The lumped unbalance assumption is adequate for thin disks or impellers, but not for thick disks or shafts. Lee et al. (1993) proposed that the unbalance of shafts should be continuously distributed. Balancing methods based on discrete unbalance models may not be very appropriate for rotors with distributed unbalance. A better alternative is to identify the distributed unbalance of shafts before balancing. In this study, the eccentricity distribution of the shaft is assumed in piecewise polynomials. A finite element model for the distributed unbalance is provided. Singular value decomposition is used to identify the eccentricity curves of the rotor. Numerical validation of this method is presented and examples are given to show the effectiveness of the identification method.

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Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

Journal of Scientific Research ◽

10.3329/jsr.v2i2.4483 ◽

2010 ◽

Vol 2 (2) ◽

pp. 264-272 ◽

Cited By ~ 7

Author(s):

A. Shirin ◽

M. S. Islam

Keyword(s):

Integral Equation ◽

Galerkin Method ◽

Integral Equations ◽

Fredholm Integral Equation ◽

Numerical Solutions ◽

Bernstein Polynomials ◽

Fredholm Integral Equations ◽

Matrix Formulation ◽

Piecewise Polynomials ◽

The Galerkin Method

In this paper, Bernstein piecewise polynomials are used to solve the integral equations numerically. A matrix formulation is given for a non-singular linear Fredholm Integral Equation by the technique of Galerkin method. In the Galerkin method, the Bernstein polynomials are used as the approximation of basis functions. Examples are considered to verify the effectiveness of the proposed derivations, and the numerical solutions guarantee the desired accuracy. Keywords: Fredholm integral equation; Galerkin method; Bernstein polynomials. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i2.4483 J. Sci. Res. 2 (2), 264-272 (2010)

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