Shape Function, Coordinate Transformation, Isoparametric Element, and Infinite Element

2018 ◽  
pp. 125-200 ◽  
Keyword(s):  
Coordinate Transformation ◽  
Shape Function ◽  
Isoparametric Element ◽  
Infinite Element

scholarly journals Direct determination of shape functions for isoparametric elements with arbitrary node configuration

2015 ◽  
Vol 5 (1) ◽  
Author(s):  
Julian Hoth ◽  
Wojciech Kowalczyk
Keyword(s):  
Shape Function ◽  
Direct Determination ◽  
Correction Procedure ◽  
Shape Functions ◽  
Isoparametric Element ◽  
Lagrange Polynomial ◽  
Regular Node ◽  
Element Type ◽  
Single Formula

AbstractShape functions have been derived to describe different forms of elements, notably triangles and rectangles in 2-D, and tetrahedrons, cuboids, and triangular prisms in 3-D. There are generalised solutions for some regular node configurations, and hierarchical correction algorithms help with more difficult node distributions. But to this point there is no single formula or set of formulae that allows the direct determination of shape functions for any node configuration without restrictions. This paper shows how a general set of formulae can be derived which is applicable to any isoparametric element type with arbitrary node configuration. This formulation is in such a form that it is clear and concise. The approach is based on the Lagrange polynomial considering up to three Cartesian and four volume coordinates. Additionally, the correction procedure that is inherent in the formulation to guarantee an appropriate evaluation of the generalised shape functions and to fulfil all four isoparametric shape function criteria is discussed. The proof of validity illustrates the correctness of the method.

Dynamic Inhomogeneous Isoparametric Element with Coordinate Transformation

2015 ◽  
Vol 32 (3) ◽  
pp. 289-296
Author(s):  
Z.-L. Yang ◽  
J.-W. Zhang ◽  
Y. Wang
Keyword(s):  
Finite Element ◽  
Coordinate Transformation ◽  
Material Parameter ◽  
Transformation Method ◽  
Isoparametric Element ◽  
Coordinate Form ◽  
Isoparametric Finite Element ◽  
Mass Matrices ◽  
Parameter Distributions ◽  
Simplified Form

AbstractBased on the coordinate transformation method, the formula of the dynamic inhomogeneous isoparametric finite element method is presented for generating element stiffness, damping and mass matrices. First, the global coordinate form and simplified form of dynamic inhomogeneous finite element are given in this paper. Then, the discrete material parameter distributions under the isoparametric coordinate system are obtained by using the transformation relationship between the global coordinates and the isoparametric coordinates. The simplified form with the discrete material parameter distributions is obtained for generating the element stiffness and mass matrices of the dynamic inhomogeneous isoparametric element. The numerical examples show that the scheme proposed in present paper has high precision.

scholarly journals Isoparametric Element Analysis of Two-Dimensional Unsaturated Transient Seepage

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Yang Luo ◽  
Yuan Liu ◽  
Juanjuan Wang
Keyword(s):  
Shape Function ◽  
Jacobi Matrix ◽  
Two Dimensional ◽  
Isoparametric Element ◽  
Weighted Residual Method ◽  
Quadrilateral Element ◽  
Element Analysis ◽  
Seepage Problem ◽  
Transient Seepage ◽  
Control Equation

A FEM for unsaturated transient seepage is established by using a quadrilateral isoparametric element, considering the fact that the main permeability does not coincide with the axis situation. It creates a function by using the element’s node hydraulic head and shape function instead of the real head in the Richard seepage control equation. With the help of the Galerkin weighted residual method, a FEM equation is given for analyzing 2-dimensional transient seepage problem. Further, based on the Jacobi matrix and Gauss numerical integral, it determines the elements of stiffness and capacitance matrices. This FEM equation considers not only the anisotropic of soil but also the uncoincidence between permeability and the axis. It is a common form of transient seepage. In the end, two examples illustrate the node accuracy of the quadrilateral element and the correctness of this FEM equation.

Condition Number of Acoustic Infinite Element

2011 ◽  
Vol 181-182 ◽  
pp. 926-931 ◽  
Author(s):  
Rui Liang Yang ◽  
Cai Xia Zhu
Keyword(s):  
Finite Element ◽  
Condition Number ◽  
Shape Function ◽  
Unbounded Domains ◽  
Research Literature ◽  
Bounded Domains ◽  
Node Number ◽  
Infinite Element ◽  
Infinite Element Method ◽  
Element Method

Various shape function and weight function of infinite element are researched and summarized into eight methods, and then various infinite element methods can be summarized as general equation, the condition number of which can reflect merits of infinite method. Condition number of various methods versus frequency and the node number are calculated in this paper. Finally, most optimal infinite element method is summed up. The infinite element method [1-12] is among the most successful techniques used to solve boundary-value problems on unbounded domains and whose solutions satisfy some condition at infinity. Two ideas make the infinite element method attractive: the idea of partition and the idea of approximation. The partition idea covers unbounded domains by attaching infinite strips to finite element partitions of bounded domains. More mature versions of infinite element method involved the approximation idea. These ideas make it possible that the finite element/infinite element method yields significantly greater computational efficiency than other methods such as the boundary element method. There have been a large number of infinite element methods, in which some methods have obvious advantages and some methods have fewer advantages. However, there is less research literature about merits of various infinite element methods appear at home and abroad. Thus, condition number of matrix equation is applied to verify merits of various infinite methods in this paper.

Error Comparison of Different Acoustic Infinite Element

2011 ◽  
Vol 128-129 ◽  
pp. 1448-1451
Author(s):  
Rui Liang Yang ◽  
Long Gao
Keyword(s):  
Weight Function ◽  
Boundary Value Problems ◽  
Condition Number ◽  
General Equation ◽  
Shape Function ◽  
Boundary Value ◽  
Unbounded Domains ◽  
Node Number ◽  
Surface Error ◽  
Infinite Element

On the basis of previous studies, various shape function and weight function of infinite element are researched and summarized into eight methods, and then various infinite element methods can be summarized as general equation, the condition number of various infinite element methods is researched to judge the merits of infinite method. Surface error of selected methods versus frequency and the node number are calculated in this paper. Finally, relatively optimal infinite element methods are summed up according error comparison, which helps to apply appropriate infinite method to solve boundary-value problems on unbounded domains.

SOME CONSIDERATIONS ON NUMERICAL METHOD OF CONVECTIVE EQUATION BY MEANS OF SPATIAL MOMENT METHOD WITH LINEAR SHAPE FUNCTION

2018 ◽  
Vol 74 (4) ◽  
pp. I_769-I_774
Author(s):  
Takashi HOSODA ◽  
Fumiya YUZAWA ◽  
Feng YE ◽  
Shinichiro ONDA ◽  
Hidekazu SHIRAI
Keyword(s):  
Numerical Method ◽  
Moment Method ◽  
Shape Function ◽  
Spatial Moment ◽  
Linear Shape Function ◽  
Linear Shape

scholarly journals Modification to the Hawking temperature of a dynamical black hole by a flow-induced supertranslation

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Hsu-Wen Chiang ◽  
Yu-Hsien Kung ◽  
Pisin Chen
Keyword(s):  
Black Hole ◽  
Coordinate Transformation ◽  
Information Loss ◽  
Hawking Temperature ◽  
Asymptotic Region ◽  
Unruh Effect ◽  
Anisotropic Flow ◽  
Information Loss Paradox ◽  
History Of ◽  
Entire History

Abstract One interesting proposal to solve the black hole information loss paradox without modifying either general relativity or quantum field theory, is the soft hair, a diffeomorphism charge that records the anisotropic radiation in the asymptotic region. This proposal, however, has been challenged, given that away from the source the soft hair behaves as a coordinate transformation that forms an Abelian group, thus unable to store any information. To maintain the spirit of the soft hair but circumvent these obstacles, we consider Hawking radiation as a probe sensitive to the entire history of the black hole evaporation, where the soft hairs on the horizon are induced by the absorption of a null anisotropic flow, generalizing the shock wave considered in [1, 2]. To do so we introduce two different time-dependent extensions of the diffeomorphism associated with the soft hair, where one is the backreaction of the anisotropic null flow, and the other is a coordinate transformation that produces the Unruh effect and a Doppler shift to the Hawking spectrum. Together, they form an exact BMS charge generator on the entire manifold that allows the nonperturbative analysis of the black hole horizon, whose surface gravity, i.e. the Hawking temperature, is found to be modified. The modification depends on an exponential average of the anisotropy of the null flow with a decay rate of 4M, suggesting the emergence of a new 2-D degree of freedom on the horizon, which could be a way out of the information loss paradox.

scholarly journals Application of the topological sensitivity method to the reconstruction of plasma equilibrium domain in a tokamak

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohamed Abdelwahed ◽  
Nejmeddine Chorfi ◽  
Maatoug Hassine ◽  
Imen Kallel
Keyword(s):  
Inverse Problem ◽  
Asymptotic Expansion ◽  
Numerical Algorithm ◽  
Shape Function ◽  
Optimization Technique ◽  
Plasma Equilibrium ◽  
Topological Sensitivity ◽  
Sensitivity Method

AbstractThe topological sensitivity method is an optimization technique used in different inverse problem solutions. In this work, we adapt this method to the identification of plasma domain in a Tokamak. An asymptotic expansion of a considered shape function is established and used to solve this inverse problem. Finally, a numerical algorithm is developed and tested in different configurations.

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