Relevant bibliographies by topics / Polynomial Shape Functions

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Polynomial Shape Functions'

Author: Grafiati
Published: 4 June 2021
Last updated: 6 September 2023

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Journal articles on the topic "Polynomial Shape Functions"

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Papanicolopulos, S. A., and A. Zervos. "Polynomial C1 shape functions on the triangle." Computers & Structures 118 (March 2013): 53–58. http://dx.doi.org/10.1016/j.compstruc.2012.07.003.

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Stachó, László. "Locally generated Hermitian C1-splines on triangular meshes." Miskolc Mathematical Notes 23, no. 2 (2022): 897. http://dx.doi.org/10.18514/mmn.2022.3399.

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We classify all possible local linear procedures on triangular meshes resulting in polynomial C1-spline functions with affinely uniform shape conditions for the basic functions at the edges, and fitting the 9 value and gradient data at the vertices of the mesh triangles. There is a unique procedure among them with shape functions and basic polynomials of degree 5 and all other admissible procedures are its perturbations with higher degree.
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Zurru, Marco. "Dynamics of cable truss via polynomial shape functions." Meccanica 54, no. 3 (2018): 353–79. http://dx.doi.org/10.1007/s11012-018-00930-z.

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Sit, Atilla, Julie C. Mitchell, George N. Phillips, and Stephen J. Wright. "An Extension of 3D Zernike Moments for Shape Description and Retrieval of Maps Defined in Rectangular Solids." Computational and Mathematical Biophysics 1 (April 16, 2013): 75–89. http://dx.doi.org/10.2478/mlbmb-2013-0004.

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AbstractZernike polynomials have been widely used in the description and shape retrieval of 3D objects. These orthonormal polynomials allow for efficient description and reconstruction of objects that can be scaled to fit within the unit ball. However, maps defined within box-shaped regions ¶ for example, rectangular prisms or cubes ¶ are not well suited to representation by Zernike polynomials, because these functions are not orthogonal over such regions. In particular, the representations require many expansion terms to describe object features along the edges and corners of the region. We o
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Wang, Xi, Cui Cui Gao, and Chen Jiang. "Cubic Uniform B-Spline Curve and Surface with Multiple Shape Parameters." Applied Mechanics and Materials 543-547 (March 2014): 1860–63. http://dx.doi.org/10.4028/www.scientific.net/amm.543-547.1860.

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In order to construct B-spline curves with local shape control parameters, a class of polynomial basis functions with two local shape parameters is presented. Properties of the proposed basis functions are analyzed and the corresponding piecewise polynomial curve is constructed with two local shape control parameters accordingly. In particular, the G1 continuous and the shapes of other segments of the curve can remain unchangeably during the manipulation on the shape of each segment on the curve. Numerical examples illustrate that the constructed curve fit to the control polygon very well. Fur
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Catania, Giuseppe, and Silvio Sorrentino. "Spectral modeling of vibrating plates with general shape and general boundary conditions." Journal of Vibration and Control 18, no. 11 (2011): 1607–23. http://dx.doi.org/10.1177/1077546311426593.

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The analysis and design of lightweight plate structures require efficient computational tools, because exact analytical solutions for vibrating plates are currently known only for some standard shapes in conjunction with a few basic boundary conditions. This paper deals with vibration analysis of Kirchhoff plates of general shape with non-standard boundary conditions, adopting a Rayleigh-Ritz approach. Three different coordinate mappings are considered, using different kinds of functions: 1) trigonometric and polynomial interpolation functions for mapping the shape of the plate, 2) trigonometr
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Luintel, Mahesh Chandra. "Development of Polynomial Mode Shape Functions for Continuous Shafts with Different End Conditions." Journal of the Institute of Engineering 16, no. 1 (2021): 151–61. http://dx.doi.org/10.3126/jie.v16i1.36653.

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Common methods used to determine the solutions for vibration response of continuous systems are assumed mode method, Rayleigh-Ritz method, Galerkin Method, finite element method, etc. Each of these methods requires the shape functions which satisfy the boundary conditions. Shape functions derived in most of the classical textbooks are simple trigonometric functions for some end conditions but are very complex transcendental functions for many end conditions. It is very difficult to determine the vibration response of a continuous system analytically by using such transcendental shape functions
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Baitsch, Matthias, and Dietrich Hartmann. "Piecewise polynomial shape functions for hp-finite element methods." Computer Methods in Applied Mechanics and Engineering 198, no. 13-14 (2009): 1126–37. http://dx.doi.org/10.1016/j.cma.2008.05.016.

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Nwachukwu, K., J. Ezeh, H. Ozioko, J. Eiroboyi, and D. Nwachukwu. "Formulation Of The Total Potential Energy Functional Relevant To The Stability Analysis Of A Doubly Symmetric Single (DSS) Cell Thin- Walled Box Column In Line With Raleigh- Ritz Method." American Journal of Computing and Engineering 4, no. 1 (2021): 57–82. http://dx.doi.org/10.47672/ajce.815.

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Purpose: This work is concerned with the formulation of peculiar Total Potential Energy Functional (TPEF) for a Doubly Symmetric Single (DSS) cell Thin -walled Box Column (TWBC). The formulated Energy Functional Equations support the stability analysis of a DSS cell thin-walled box (closed) column cross-section using Raleigh - Ritz Method (RRM) with polynomial shape functions.
 Methodology: This present formulation is based on the governing TPEF developed by Nwachukwu and others (2017). The polynomial shape functions (only the first two coordinate polynomial shape functions) for different
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Krajnović, S. "Shape optimization of high-speed trains for improved aerodynamic performance." Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 223, no. 5 (2009): 439–52. http://dx.doi.org/10.1243/09544097jrrt251.

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A new procedure for the optimization of aerodynamic properties of trains is presented, where simple response surface (RS) models are used as a basis for optimization instead of a large number of evaluations of the Navier—Stokes solver. The suggested optimization strategy is demonstrated in two flow optimization cases: optimization of the train's front for the crosswind stability and optimization of vortex generators (VGs) for the purpose of drag reduction. Besides finding the global minimum for each aerodynamic objective, a strategy for finding a set of optimal solutions is demonstrated. This
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Dissertations / Theses on the topic "Polynomial Shape Functions"

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Smaïli, Nasser-Eddine. "Les polynômes e-semi-classiques de classe zéro." Paris 6, 1987. http://www.theses.fr/1987PA066081.

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Guerfi, Malik. "Les polynômes de Laguerre-Hahn affines discrets." Paris 6, 1988. http://www.theses.fr/1988PA066275.

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On montre que les polynômes semiclassiques sont des polynômes de Laguerre-Hahn affines et réciproquement. On étudie les polynômes corécursifs et les polynômes associés des polynômes de Laguerre Hahn affines. On généralise tous ces résultats aux polynômes semiclassiques discrets (avec l'opérateur de Hahn)
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Vazquez, Thais Godoy. "Funções de interpolação e regras de integração tensorizaveis para o metodo de elementos finitos de alta ordem." [s.n.], 2008. http://repositorio.unicamp.br/jspui/handle/REPOSIP/263500.

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Orientador: Marco Lucio Bittencourt<br>Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica<br>Made available in DSpace on 2018-08-10T12:57:32Z (GMT). No. of bitstreams: 1 Vazquez_ThaisGodoy_D.pdf: 11719751 bytes, checksum: c6d385d6a6414705c9f468358b8d3bea (MD5) Previous issue date: 2008<br>Resumo: Este trabalho tem por objetivo principal o desenvolvimento de funções de interpolaçao e regras de integraçao tensorizaveis para o Metodo dos Elementos Finitos (MEF) de alta ordem hp, considerando os sistemas de referencias locais dos elementos. Para isso, primeira
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Narayan, Shashi. "Smooth Finite Element Methods with Polynomial Reproducing Shape Functions." Thesis, 2013. http://etd.iisc.ac.in/handle/2005/3332.

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A couple of discretization schemes, based on an FE-like tessellation of the domain and polynomial reproducing, globally smooth shape functions, are considered and numerically explored to a limited extent. The first one among these is an existing scheme, the smooth DMS-FEM, that employs Delaunay triangulation or tetrahedralization (as approximate) towards discretizing the domain geometry employs triangular (tetrahedral) B-splines as kernel functions en route to the construction of polynomial reproducing functional approximations. In order to verify the numerical accuracy of the smooth DMS-FEM v
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Narayan, Shashi. "Smooth Finite Element Methods with Polynomial Reproducing Shape Functions." Thesis, 2013. http://etd.iisc.ernet.in/2005/3332.

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A couple of discretization schemes, based on an FE-like tessellation of the domain and polynomial reproducing, globally smooth shape functions, are considered and numerically explored to a limited extent. The first one among these is an existing scheme, the smooth DMS-FEM, that employs Delaunay triangulation or tetrahedralization (as approximate) towards discretizing the domain geometry employs triangular (tetrahedral) B-splines as kernel functions en route to the construction of polynomial reproducing functional approximations. In order to verify the numerical accuracy of the smooth DMS-FEM v
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Devaraj, G. "Schemes for Smooth Discretization And Inverse Problems - Case Study on Recovery of Tsunami Source Parameters." Thesis, 2016. http://etd.iisc.ac.in/handle/2005/2719.

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This thesis deals with smooth discretization schemes and inverse problems, the former used in efficient yet accurate numerical solutions to forward models required in turn to solve inverse problems. The aims of the thesis include, (i) development of a stabilization techniques for a class of forward problems plagued by unphysical oscillations in the response due to the presence of jumps/shocks/high gradients, (ii) development of a smooth hybrid discretization scheme that combines certain useful features of Finite Element (FE) and Mesh-Free (MF) methods and alleviates certain destabilizing facto
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Devaraj, G. "Schemes for Smooth Discretization And Inverse Problems - Case Study on Recovery of Tsunami Source Parameters." Thesis, 2016. http://hdl.handle.net/2005/2719.

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This thesis deals with smooth discretization schemes and inverse problems, the former used in efficient yet accurate numerical solutions to forward models required in turn to solve inverse problems. The aims of the thesis include, (i) development of a stabilization techniques for a class of forward problems plagued by unphysical oscillations in the response due to the presence of jumps/shocks/high gradients, (ii) development of a smooth hybrid discretization scheme that combines certain useful features of Finite Element (FE) and Mesh-Free (MF) methods and alleviates certain destabilizing facto
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Chak, Pok Man. "Approximation and consistent estimation of shape-restricted functions and their derivatives." 2001. http://wwwlib.umi.com/cr/yorku/fullcit?pNQ67896.

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Thesis (Ph. D.)--York University, 2001. Graduate Programme in Economics.<br>Typescript. Includes bibliographical references (leaves 116-121). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ67896.
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Books on the topic "Polynomial Shape Functions"

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Shape-preserving approximation by real and complex polynomials. Birkhäuser, 2008.

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Solymar, L., D. Walsh, and R. R. A. Syms. Bonds. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198829942.003.0005.

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Mechanical properties of bonds are discussed, with the aid of a simple phenomenological model in which the variation of energy as a function of distance between the elements is described in terms of polynomials. The properties of various kinds of bonds (ionic bond, metallic bond, covalent bond, van der Waals bond) are explained with the aid of simple models. Carbon is discussed with two examples: bonds between 60 atoms that lead to the formation of a three-dimensional molecule known as Buckminsterfullerene, and the alternative sheet-shaped configuration known as graphene, that has recently bec
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Book chapters on the topic "Polynomial Shape Functions"

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Jena, Subrat Kumar, Dineshkumar Harursampath, Vinyas Mahesh, and Sathiskumar A. Ponnusami. "Comparing different polynomials-based shape functions in the Rayleigh–Ritz method for investigating dynamical characteristics of nanobeam." In Polynomial Paradigms. IOP Publishing, 2022. http://dx.doi.org/10.1088/2053-2563/ac9580ch007.

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Jakóbczak, Dariusz Jacek. "Probabilistic Nodes Combination (PNC)." In Natural Language Processing. IGI Global, 2020. http://dx.doi.org/10.4018/978-1-7998-0951-7.ch050.

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The method of Probabilistic Nodes Combination (PNC) enables interpolation and modeling of two-dimensional curves using nodes combinations and different coefficients γ: polynomial, sinusoidal, cosinusoidal, tangent, cotangent, logarithmic, exponential, arc sin, arc cos, arc tan, arc cot or power function, also inverse functions. This probabilistic view is novel approach a problem of modeling and interpolation. Computer vision and pattern recognition are interested in appropriate methods of shape representation and curve modeling. PNC method represents the possibilities of shape reconstruction and curve interpolation via the choice of nodes combination and probability distribution function for interpolated points. It seems to be quite new look at the problem of contour representation and curve modeling in artificial intelligence and computer vision. Function for γ calculations is chosen individually at each curve modeling and it is treated as probability distribution function: γ depends on initial requirements and curve specifications.
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Schubert, Till, and Wolf-Dieter Schuh. "A Flexible Family of Compactly Supported Covariance Functions Based on Cutoff Polynomials." In International Association of Geodesy Symposia. Springer Berlin Heidelberg, 2023. http://dx.doi.org/10.1007/1345_2023_200.

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AbstractIn time series analysis, signal covariance modeling is an essential part of stochastic methods like prediction or filtering. In geodetic applications, covariance functions are rarely treated as true compactly supported functions although large amounts of data would approve such. Covariance models for complex correlation shapes are also rare. Ideally, general families of covariance functions with a large flexibility are desirable to model complex correlations structures like negative correlations. In this paper, we derive isotropic finite covariance functions that are parametrized in a way that positive definiteness is guaranteed. These are based on cutoff polynomials which are derived from operations such as autoconvolution and autocorrelation. Next to the compact support, the resulting autocovariance models share the advantages of (a) positive definiteness by design, (b) extensibility to arbitrary orders and (c) extensive flexibility by employing multiple tunable shape parameters. All these realize various correlation shapes such as negative correlations (the so-called hole effect) and several oscillations. The methodological concepts are derived for homogeneous and isotropic random fields in $$\mathbb {R}^d$$ ℝ d . The family of covariance functions is then derived for one-dimensional applications. A data example demonstrates the covariance modeling approach using stationary time-series data.
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Sah, Tanishq Kumar. "Extension of Theories." In Building on the Past to Prepare for the Future, Proceedings of the 16th International Conference of The Mathematics Education for the Future Project, King's College,Cambridge, Aug 8-13, 2022. WTM-Verlag, 2022. http://dx.doi.org/10.37626/ga9783959872188.0.087.

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From an atom to this universe, from a bowl of water to the cosmic ocean this constant is present everywhere. This constant is π ( periodicity of the tangent function). For tangent function we know that tan(tan-1(x))=x, but the expression tan(ntan-1(x)) looks very complicated but is actually an expression of the type polynomial divided by another polynomial. The sine function is very important not only for graphs but for geometry too. There are some inputs whose behavior is very strange from the usual ones. Geometrical shapes and their relations are very important for many thing such as for vectors and many more but the triangle is very special because it is the least sided polygon. Riemann zeta function is very crucial for prime numbers. Infinite series related to them may be a game changer for it. Wallis’s integral formula is a boon but its domain is very constrained and needs another solution to it.
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Aschenbrenner, Matthias, Lou van den Dries, and Joris van der Hoeven. "Asymptotic Fields and Asymptotic Couples." In Asymptotic Differential Algebra and Model Theory of Transseries. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691175423.003.0010.

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This chapter deals with asymptotic differential fields and their asymptotic couples. Asymptotic fields include Rosenlicht's differential-valued fields and share many of their basic properties. A key feature of an asymptotic field is its asymptotic couple. The chapter first defines asymptotic fields and their asymptotic couples before discussing H-asymptotic couples. It then considers asymptotic couples independent of their connection to asymptotic fields, along with the behavior of differential polynomials as functions on asymptotic fields. It also describes asymptotic fields with small derivation and the operations of coarsening and specialization, algebraic and immediate extensions of asymptotic fields, and differential polynomials of order one. Finally, it proves some useful extension results about asymptotic couples and establishes a property of closed H-asymptotic couples.
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Jakóbczak, Dariusz Jacek. "Applications of PNC in Artificial Intelligence." In Natural Language Processing. IGI Global, 2020. http://dx.doi.org/10.4018/978-1-7998-0951-7.ch026.

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Interpolation methods and curve fitting represent so huge problem that each individual interpolation is exceptional and requires specific solutions. Presented method is such a new possibility for curve fitting and interpolation when specific data (for example handwritten symbol or character) starts up with no rules for polynomial interpolation. The method of Probabilistic Nodes Combination (PNC) enables interpolation and modeling of two-dimensional curves using nodes combinations and different coefficients γ. This probabilistic view is novel approach a problem of modeling and interpolation. Computer vision and pattern recognition are interested in appropriate methods of shape representation and curve modeling. PNC method represents the possibilities of shape reconstruction and curve interpolation via the choice of nodes combination and probability distribution function for interpolated points. It seems to be quite new look at the problem of contour representation and curve modeling in artificial intelligence and computer vision.
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Wang, Hui, Lei Zhang, Shilei Liang, Yanyan Wang, and Weidong Zhu. "One-Dimensional High-Order Dynamic Model of U-Shaped Thin-Wall Arm Segment of Telescopic Boom of Crane." In Advances in Transdisciplinary Engineering. IOS Press, 2022. http://dx.doi.org/10.3233/atde220473.

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In order to accurately analyse the dynamic performance of the arm segment, a dynamic model of u-shaped thin-walled beam based on one-dimensional high-order beam theory is proposed to predict the three-dimensional displacement of the beam at any point. First, a one-dimensional high-order model is established using Hamilton’s principle. The high-order model considers the displacement field by linear superposition of a set of basis functions that vary axially along the beam. A basis function represents a deformation mode, and interpolation polynomials are used to approximate the three-dimensional displacements of nodes on the center line of the section. At the same time, different section discretization methods are analysed, which have different influences on the precision of the model by discretization of the curved surface part of u-shaped section by straight transposition. Finally, the generalized characteristic of the model is worth to obtain the natural frequency, which is compared with the ANASYS plate and shell theory. The error range of the first 16 orders is within 1.5%. The results show that the discrete mode of the model has a certain influence on the frequency error, and the more discrete nodes of the circular arc part, the higher the accuracy.
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Ciulla, Carlo. "On the Implications of the Sub-Pixel Efficacy Region and the Bridging Concept of the Unifying Theory." In Improved Signal and Image Interpolation in Biomedical Applications. IGI Global, 2009. http://dx.doi.org/10.4018/978-1-60566-202-2.ch020.

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The last chapter of the book reports on concluding remarks recalling to the reader the message given through these works and also recalling the proposed novelty. The novelty is discussed within the context of the current literature with specific attention to other works devoted to the improvement of the interpolation error. The reader is acknowledged that the methodological approach outlined through the theory can be seen as a viable pathway to follow in order to conceptualize interpolation in an innovative and alternative manner. This descends from the adoption of the mathematical formulation that is dependent on the joint information content of node intensity and curvature of the interpolation function and has brought to the determination of a viable option to adopt when re-sampling the signal (image). Re-sampling inherent to interpolation can be performed at sub-pixel locations that are not necessarily the same as the given misplacement neither are they necessarily the same pixel-by-pixel. This is because of the variability of pixel intensity and curvature of the interpolation function at the neighborhood and between neighborhoods and also because such variability corresponds to various signal shape characteristics. The reader is informed that within the context of a paradigm to be used for the improvement of the interpolation error it is of relevance to include the curvature in the methodology that is chosen to improve the approximation capability of a given interpolation function. The focus is also towards the evidence that local re-sampling is capable of changing the band-pass filtering property of the interpolation functions. Also, a study is undertaken to determine how beneficial is the application of the Sub-pixel Efficacy Region in the estimation of signals at unknown time-space locations and this is done for the one-dimensional interpolation functions presented in the book. It is also shown that the SRE-based interpolation functions are capable to determine error improvement and to be more accurate with respect to the classic interpolation functions in the estimation of signals at locations that are not captured by the sampling frequency because of the Nyquist’s theorem constraint. Also, based on the same data, the effect of the sampling resolution is studied on the interpolation error and the interpolation error improvement. Consequentially, it is outlined the licit conclusion that under the umbrella of the unifying framework proposed within the theory, the sampling resolution influences both interpolation error and interpolation error improvement obtained from the SRE-based functions. Finally the chapter reports on the investigation of the influence of the SCALE parameter and on the performance comparison across classic and SRE-based interpolation functions. The SCALE parameter is employed to scale the convolution of the pixel intensities determined through the polynomials forms: quadratic and cubic B-Splines, Lagrange, and also to scale the numerical values of the sums of cosine and sine functions of the Sinc interpolation function.
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Conference papers on the topic "Polynomial Shape Functions"

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Surampudi, R., and A. Gupta. "Application of Harmonic Shape Function in Finite Element Modal Analysis." In ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/detc1995-0484.

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Abstract Existing element types available in finite element codes typically utilize polynomial shape functions to define the displacement field in the problem of interest. The polynomial shape functions serve the purpose adequately in static analysis where the displacements and the stresses in a structure are of primary interest. These shape functions give rise to increasing inaccuracy for higher natural frequencies. It is shown that harmonic shape functions yield better results for the higher natural frequencies with same element count. Axial vibrations of bars and transverse vibrations of be
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Klein, Stanley A., and Brent Beutter. "Hermite functions maximize the spacespatial frequency uncertainty of Gaborlike functions." In OSA Annual Meeting. Optica Publishing Group, 1991. http://dx.doi.org/10.1364/oam.1991.ww3.

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Wavelets that are localized both in space and in spatial frequency are useful for vision modeling and image processing. For example, in image compression one would like localization in spatial frequency so that the small spatial filters are insensitive to low spatial frequencies to which the visual system is highly sensitive. One would also like spatially localized filters to reduce interference between adjacent features. Several investigators, including Gabor, asked what particular wavelet shape minimizes the joint space-spatial frequency uncertainty (the Heisenberg uncertainty). Consider the
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Batura, Anatolii, Igor Orynyak, and Andrii Oryniak. "Semianalytical Method for the SIF Calculation for a Crack of Arbitrary Shape in Infinite Body." In ASME 2014 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/pvp2014-28383.

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The exact analytical approach for stress intensity factor calculation for an arbitrary shape mode I crack loaded by the polynomial stresses is proposed. The approach is based on the calculation of the crack faces displacement at given loading. The displacement field is presented as a shape function multiplied by an adjustment polynomial. At that the key problem is the solution of well-known inverse task: obtaining the stresses field at the crack faces on the base of a given displacements field. Multiply solution of such task for a whole set of certain displacements base functions (e.g., for th
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Mohamed, Abdel-Nasser A. "ANCF Tetrahedral Solid Element Using Shape Functions Based on Cartesian Coordinates." In ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/detc2016-59183.

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This paper introduces a new solid, four-node, fully parameterized, tetrahedron element. The tetrahedron element is based on the absolute nodal coordinate formulations in which the absolute position vector and three slope vectors are considered nodal coordinates. The linear transformation of the shape functions from the tetrahedron coordinates to Cartesian coordinates leads to very stiff elements. On the other hand, the nonlinear transformation from tetrahedron coordinates to Cartesian coordinates is very complicated. Therefore, the use of the barycentric coordinates was bypassed and the Cartes
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Peng, Chengyang, Octavian Donca, Guillermo Castillo, and Ayonga Hereid. "Safe Bipedal Path Planning via Control Barrier Functions for Polynomial Shape Obstacles Estimated Using Logistic Regression." In 2023 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2023. http://dx.doi.org/10.1109/icra48891.2023.10160671.

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Scano, D. "Analysis of composite beams, plates, and shells using Jacobi polynomials and NDK models." In Aerospace Science and Engineering. Materials Research Forum LLC, 2023. http://dx.doi.org/10.21741/9781644902677-22.

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Abstract. In this work, hierarchical Jacobi-based expansions are explored for the static analysis of multilayered beams, plates, and shells as structural theories as well as shape functions. Jacobi polynomials, denoted as P_p^((γ,θ) ), belong to the family of classical orthogonal polynomials and depend on two scalars parameters 𝛾 and 𝜃, with p being the polynomial order. Regarding the structural theories, layer wise and equivalent single-layer approaches can be used. It is demonstrated that the parameters 𝛾 and 𝜃 of the Jacobi polynomials are not influential for the calculations. These polynom
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Wong, Chun Nam, Hong-Zhong Huang, Jingqi Xiong, and Tianyou Hu. "Weibull Distributed Stress-Dependent Strength Analysis of Aeroengine Alloy Using Lagrange Factor Polynomial." In ASME 2010 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2010. http://dx.doi.org/10.1115/detc2010-28090.

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In this paper, the unilateral dependency of strength on stress is taken into account. And the stress-dependent strength is represented by a discrete random variable that has different conditional probability mass functions under different stress amplitudes. Then the Lagrange factor polynomial technique is developed to generate the stress-strength interference model with stress-dependent strength. This model assumes that the strength probability mass function is Weibull distributed, while the stress probability mass function is Normal distributed. Accuracy of this method is investigated by an a
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Lai, Ye-Chen, Timothy C. S. Liang, and Zhenxue Jia. "Implementation of p and h-p Versions of the Finite Element Method." In ASME 1992 International Computers in Engineering Conference and Exposition. American Society of Mechanical Engineers, 1992. http://dx.doi.org/10.1115/cie1992-0114.

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Abstract Based on hierarchic shape functions and an effective convergence procedure, the p-version and h-p adaptive analysis capabilities were incorporated into a finite element software system, called COSMOS/M. The range of the polynomial orders can be varied from 1 to 10 for two dimensional linear elastic analysis. In the h-p adaptive analysis process, a refined mesh are first achieved via adaptive h-refinement. The p-refinement is then added on to the h-version designed mesh by uniformly increasing the degree of the polynomials. Some numerical results computed by COSMOS/M are presented to i
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Orynyak, Igor, Anatolii Batura, Andrii Oryniak, and Igor Lokhman. "Oore-Burns Function of Form Application in Numerical Treatment of Mode I Flat Crack Problem in Infinite Body." In ASME 2016 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/pvp2016-63304.

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Abstract:
The general approach of numerical treatment of integro-differential equation of the flat crack problem is considered. It consists in presenting the crack surface loading as the set of the polynomial functions of two Cartesian coordinates while the corresponding crack surface displacements are chosen as the similar polynomials multiplied by the function of form (FoF) which reflects the required singularity of their behavior. To find the relations matrixes between these two sets a new effective numerical procedure for the integration over the area of arbitrary shape crack is developed. In based
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10

Alanbay, Berkan, Karanpreet Singh, and Rakesh K. Kapania. "Vibration of Curvilinearly Stiffened Plates Using Ritz Method With Orthogonal Jacobi Polynomials." In ASME 2018 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/imece2018-86871.

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Abstract:
This paper presents a general approach for the free vibration analysis of curvilinearly stiffened rectangular and quadrilateral plates using Ritz method employing classical orthogonal Jacobi polynomials. Both the plate and stiffeners are modeled using first-order shear deformation theory (FSDT). The displacement and rotations of the plate and a stiffener are approximated by separate sets of Jacobi polynomials. The ease of modification of the Jacobi polynomials enables the Jacobi weight function to satisfy geometric boundary conditions without loss of orthogonality. The distinctive advantage of
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