Hermite Type Spline Spaces over Rectangular Meshes with Complex Topological Structures

AbstractMotivated by the magneto hydrodynamic (MHD) simulation for Tokamaks with Isogeometric analysis, we present splines defined over a rectangular mesh with a complex topological structure, i.e., with extraordinary vertices. These splines are piecewise polynomial functions of bi-degree (d,d) and parameter continuity. And we compute their dimension and exhibit basis functions called Hermite bases for bicubic spline spaces. We investigate their potential applications for solving partial differential equations (PDEs) over a physical domain in the framework of Isogeometric analysis. For instance, we analyze the property of approximation of these spline spaces for the L2-norm; we show that the optimal approximation order and numerical convergence rates are reached by setting a proper parameterization, although the fact that the basis functions are singular at extraordinary vertices.

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On the Convexity Preservation of a Quasi C3 Nonlinear Interpolatory Reconstruction Operator on σ Quasi-Uniform Grids

Mathematics ◽

10.3390/math9040310 ◽

2021 ◽

Vol 9 (4) ◽

pp. 310 ◽

Cited By ~ 1

Author(s):

Pedro Ortiz ◽

Juan Carlos Trillo

Keyword(s):

Initial Data ◽

Numerical Experiments ◽

Approximation Order ◽

Piecewise Polynomial ◽

Nonuniform Grids ◽

Uniform Grids ◽

Reconstruction Operator ◽

Convexity Properties ◽

Second Degree

This paper is devoted to introducing a nonlinear reconstruction operator, the piecewise polynomial harmonic (PPH), on nonuniform grids. We define this operator and we study its main properties, such as its reproduction of second-degree polynomials, approximation order, and conditions for convexity preservation. In particular, for σ quasi-uniform grids with σ≤4, we get a quasi C3 reconstruction that maintains the convexity properties of the initial data. We give some numerical experiments regarding the approximation order and the convexity preservation.

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Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

Mathematical Problems in Engineering ◽

10.1155/2019/8152136 ◽

2019 ◽

Vol 2019 ◽

pp. 1-10

Author(s):

Darae Jeong ◽

Yibao Li ◽

Chaeyoung Lee ◽

Junxiang Yang ◽

Yongho Choi ◽

...

Keyword(s):

Parabolic Equations ◽

Convergence Rates ◽

Numerical Solutions ◽

Second Order ◽

Order Convergence ◽

Numerical Convergence ◽

Verification Method ◽

First Order ◽

Convergence Test ◽

Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

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Spline Based Design of Cam Contours for Improved Dynamic Performance

13th Computers in Engineering Conference ◽

10.1115/cie1993-0031 ◽

1993 ◽

Author(s):

Holly K. Ault ◽

James C. Wilkinson

Keyword(s):

Dynamic Performance ◽

Integrated Design ◽

Polynomial Functions ◽

Piecewise Polynomial ◽

Cam Design ◽

Piecewise Polynomial Functions ◽

Cam Follower ◽

Cam Profile ◽

Pitch Curve ◽

Design And Manufacture

Abstract A method for the integrated design and manufacture of radial plate cams is discussed. Currently, a cam-follower system is designed by specifying constraints on the motion of the follower. The physical cam contour or cam pitch curve are not mathematically defined. The cam is manufactured from the discretized follower motion program. A new method for cam design is proposed which will produce a smooth, mathematically defined cam pitch curve while maintaining the proper constraints on the follower motion. Piecewise polynomial functions in the form of rational and/or non-rational splines may be used. Cams will be manufactured using smoothed profiles and tested for improved dynamic performance. The results of initial investigations of cam profile design for this research are presented.

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Piecewise polynomial functions, convex polytopes and enumerative geometry

Banach Center Publications ◽

10.4064/-36-1-25-44 ◽

1996 ◽

Vol 36 (1) ◽

pp. 25-44 ◽

Cited By ~ 6

Author(s):

Michel Brion

Keyword(s):

Convex Polytopes ◽

Enumerative Geometry ◽

Polynomial Functions ◽

Piecewise Polynomial ◽

Piecewise Polynomial Functions

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Adaptive Numerical Modeling of Engineering Problems Using Hierarchical Fup Basis Functions and Control Volume IsoGeometric Analysis

10.31534/doct.051.kamg ◽

2021 ◽

Author(s):

◽

Grgo Kamber ◽

Keyword(s):

Isogeometric Analysis ◽

Numerical Solutions ◽

Control Volume ◽

Spatial Scales ◽

Basis Functions ◽

Approximation Properties ◽

Unified Framework ◽

B Splines ◽

Engineering Problems ◽

Resolution Level

The main objective of this thesis is to utilize the powerful approximation properties of Fup basis functions for numerical solutions of engineering problems with highly localized steep gradients while controlling spurious numerical oscillations and describing different spatial scales. The concept of isogeometric analysis (IGA) is presented as a unified framework for multiscale representation of the geometry and solution. This fundamentally high-order approach enables the description of all fields as continuous and smooth functions by using a linear combination of spline basis functions. Classical IGA usually employs Galerkin or collocation approach using B-splines or NURBS as basis functions. However, in this thesis, a third concept in the form of control volume isogeometric analysis (CV-IGA) is used with Fup basis functions which represent infinitely smooth splines. Novel hierarchical Fup (HF) basis functions is constructed, enabling a local hp-refinement such that they can replace certain basis functions at one resolution level with new basis functions at the next resolution level that have a smaller length of the compact support (h-refinement), but also higher order (p-refinement). This hp-refinement property enables spectral convergence which is significant improvement in comparison to the hierarchical truncated B-splines which enable h-refinement and polynomial convergence. Thus, in domain zones with larger gradients, the algorithm uses smaller local spatial scales, while in other region, larger spatial scales are used, controlling the numerical error by the prescribed accuracy. The efficiency and accuracy of the adaptive algorithm is verified with some classic 1D and 2D benchmark test cases with application to the engineering problems with highly localized steep gradients and advection-dominated problems.

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The Pierce-Birkhoff Conjecture for Curves

Canadian Journal of Mathematics ◽

10.4153/cjm-1992-075-0 ◽

1992 ◽

Vol 44 (6) ◽

pp. 1262-1271 ◽

Cited By ~ 5

Author(s):

Murray Marshall

Keyword(s):

Real Closed Field ◽

Noetherian Rings ◽

Closed Field ◽

Polynomial Functions ◽

Piecewise Polynomial ◽

Algebraic Set ◽

Dedekind Domains ◽

Piecewise Polynomial Functions ◽

Relative Case

AbstractThe results obtained extend Madden’s result for Dedekind domains to more general types of 1-dimensional Noetherian rings. In particular, these results apply to piecewise polynomial functions t:C → R where R is a real closed field and C ⊆ Rn is a closed 1-dimensional semi-algebraic set, and also to the associated “relative” case where t, C are defined over some subfield K ⊆ R.

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Advanced Zig-Zag Beam Theories for Sandwich Structures Analyses

Volume 1: Advances in Aerospace Technology ◽

10.1115/imece2018-86783 ◽

2018 ◽

Author(s):

Matteo Filippi ◽

Erasmo Carrera

Keyword(s):

Sandwich Structures ◽

Higher Order ◽

Polynomial Functions ◽

Piecewise Polynomial ◽

Displacement Fields ◽

New Class ◽

Dynamic Analyses ◽

Piecewise Polynomial Functions ◽

Carrera Unified Formulation ◽

Beam Theories

This work aims at evaluating the capabilities of several higher-order beam formulations for stress and dynamic analyses of layered sandwich structures. The structural models are conceived within the framework of the Carrera Unified Formulation (CUF) that allows one to generate (and compare) an infinite number of displacement fields. The number and the type of functions that are selected to generate the kinematic expansions are input parameters of the problem. Besides the well-known Taylor- and Lagrange-type expansions, great attention is paid to a new class of advanced higher-order zig-zag theories, which are written as combinations of continuous piecewise polynomial functions. Numerical simulations are performed on laminated and sandwich beams with very low length-to-depth ratio values. Also, structures with soft layers made of viscoelastic materials are considered to investigate the different dissipation mechanisms.

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On the integral representation of the endomorphisms of Mikuśinski's operator field

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500009640 ◽

1971 ◽

Vol 17 (4) ◽

pp. 351-367 ◽

Cited By ~ 2

Author(s):

András Bleyer

Keyword(s):

Integral Representation ◽

Polynomial Functions ◽

Piecewise Polynomial ◽

Operator Field ◽

Piecewise Polynomial Functions

We proved in (1) that every continuous endomorphism can be generated on a subring of the field M. More precisely, the ring H of piecewise polynomial functions has the property that every isomorphism from H into M, continuous in the sequential topology of H, can be extended to a continuous endomorphism of M where the notion of continuity in M is the usual sequential one.

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Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

Computational Mechanics ◽

10.1007/s00466-019-01796-y ◽

2019 ◽

Vol 65 (3) ◽

pp. 807-838 ◽

Cited By ~ 3

Author(s):

F. de Prenter ◽

C. V. Verhoosel ◽

E. H. van Brummelen ◽

J. A. Evans ◽

C. Messe ◽

...

Keyword(s):

Finite Element ◽

Finite Element Methods ◽

Isogeometric Analysis ◽

Convergence Rates ◽

Computational Cost ◽

Iterative Solvers ◽

System Matrix ◽

B Splines ◽

Immersed Finite Element ◽

Immersed Finite Element Methods

AbstractIll-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral properties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines.

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