1,145 research outputs found

    Glosarium Matematika

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    273 p.; 24 cm

    Glosarium Matematika

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    Generalized Finite Element Systems for smooth differential forms and Stokes problem

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    We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously introduced notion of Finite Element Systems, and the examples include conforming mixed finite elements for Stokes' equation. In dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes. In dimension 3 we define one low order complex, which may be branched into Whitney forms at a chosen index. Stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension. The finite element spaces all consist of composite polynomials. The framework guarantees some nice properties of the spaces, in particular the existence of commuting interpolators. It also shows that some of the examples are minimal spaces.Comment: v1: 27 pages. v2: 34 pages. Numerous details added. v3: 44 pages. 8 figures and several comments adde

    Constructing G2 Continuous Curve on Freeform Surface with Normal Projection

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    AbstractThis article presents a new method for G2 continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with prescribed tangent direction and curvature vector, respectively, at every point. First, a G2 continuous curve is constructed in three-dimensional space. Then the curve is projected normally onto the given surface. The desired interpolation curve is just the projection curve, which can be obtained by numerically solving the initial- value problems for a system of first-order ordinary differential equations in the parametric domain for parametric case or in three-dimensional space for implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification so that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface. Numerical experiments demonstrate that it is effective and potentially useful in numerical control (NC) machining, path planning for robotic fibre placement, patterns design on surface and other industrial and research fields

    Free vibration and buckling analysis of higher order laminated composite plates using the isogeometric approach

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    This research paper presents a higher order isogeometric laminated composite plate finite element formulation. The isogeometric formulation is based on NURBS (non-uniform rational B-splines) basis functions of varying degree. Plate kinematics is based on the third order shear deformation theory (TSDT) of Reddy in order to avoid shear locking. Free vibration and the buckling response of laminated composite plates are obtained and efficiency of the method is considered. Numerical results with different element order are presented and the obtained results are compared to analytical and conventional numerical results as well as existing isogeometric plate finite elements

    Constructing G2 Continuous Curve on Freeform Surface with Normal Projection

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    AbstractThis article presents a new method for G2 continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with prescribed tangent direction and curvature vector, respectively, at every point. First, a G2 continuous curve is constructed in three-dimensional space. Then the curve is projected normally onto the given surface. The desired interpolation curve is just the projection curve, which can be obtained by numerically solving the initial- value problems for a system of first-order ordinary differential equations in the parametric domain for parametric case or in three-dimensional space for implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification so that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface. Numerical experiments demonstrate that it is effective and potentially useful in numerical control (NC) machining, path planning for robotic fibre placement, patterns design on surface and other industrial and research fields

    High-Order, Stable, And Efficient Pseudospectral Method Using Barycentric Gegenbauer Quadratures

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    The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the well-conditioning of the numerical integration operators to produce well-conditioned systems of algebraic equations, which can be solved easily using standard algebraic system solvers. The core of the work lies in the derivation of novel and stable Gegenbauer quadratures based on the stable barycentric representation of Lagrange interpolating polynomials and the explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous error and convergence analysis of the proposed quadratures is presented along with a detailed set of pseudocodes for the established computational algorithms. The proposed numerical scheme leads to a reduction in the computational cost and time complexity required for computing the numerical quadrature while sharing the same exponential order of accuracy achieved by Elgindy and Smith-Miles (2013). The bulk of the work includes three numerical test examples to assess the efficiency and accuracy of the numerical scheme. The present method provides a strong addition to the arsenal of numerical pseudospectral methods, and can be extended to solve a wide range of problems arising in numerous applications.Comment: 30 pages, 10 figures, 1 tabl

    High-order adaptive methods for computing invariant manifolds of maps

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    The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

    G1 Range Restricted Data Interpolation Using Bézier Triangular Patch

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    Pembinaan permukaan berparameter G 1 julat terhad kepada data yang semua terletak di sebelah satu satah kekangan dipertimbangkan. Permukaan interpolasi dibina secara cebis demi cebis daripada gabungan cembung tiga tampalan segi tiga Bézier kuartik. Syarat cukup untuk keselanjaran satah tangen sepanjang sempadan dua tampalan Bézier kuartik dibentangkan. The construction of range restricted G 1 parametric surface to data that all lie on one side of a constraint plane is considered. The interpolating surface is developed piecewise as the convex combination of three quartic Bézier triangular patches. Sufficient tangent plane continuity conditions along the common boundary of two adjacent quartic Bézier triangular patches are presented
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