Mathematical Basis of G Spaces

2016 ◽  
Vol 13 (04) ◽  
pp. 1641007 ◽  
Author(s):  
Meng Chen ◽  
Ming Li ◽  
G. R. Liu
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Weak Formulation ◽  
Mathematical Basis ◽  
Numerical Examples ◽  
Interpolation Methods ◽  
Point Interpolation ◽  
Theoretical Frame ◽  
Lower Boundedness ◽  
Numerical Approaches

This paper represents some basic mathematic theories for G[Formula: see text] spaces of functions that can be used for weakened weak (W2) formulations, upon which the smoothed finite element methods (S-FEMs) and the smoothed point interpolation methods (S-PIMs) are based for solving mechanics problems. We first introduce and prove properties of G[Formula: see text] spaces, such as the lower boundedness and convergence of the norms, which are in contrast with H1spaces. We then prove the equivalence of the Gsnorms and its corresponding semi-norms. These mathematic theories are important and essential for the establishment of theoretical frame and the development of relevant numerical approaches. Finally, numerical examples are presented by using typical S-FEM models known as the NS-FEM and [Formula: see text]S-FEM to examine the properties of a smoothed method based on Gsspaces, in comparison with the standard FEM with weak formulation.

Mesh Refinement Formulations in Adaptive Finite Element Methods

1996 ◽  
Vol 210 (4) ◽  
pp. 353-361 ◽  
Author(s):  
L-Y Li ◽  
P Bettess ◽  
J W Bull ◽  
T Bond
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Mesh Refinement ◽  
Adaptive Finite Element ◽  
Global Level ◽  
Adaptive Finite Element Methods ◽  
Element Level ◽  
Adaptive Remeshing ◽  
Numerical Examples ◽  
New Ideas

This paper presents some new ideas for developing adaptive remeshing strategies. It is shown that correct mesh refinement formulations should be defined at an element level rather than a global level. To accomplish this, permissible element errors are required to be defined. This paper describes the methods to determine the permissible element errors. Two mesh refinement formulations are derived according to different accuracy definitions and are compared with the conventional mesh refinement formulation derived at the global level. Numerical examples are shown to explain the features of these mesh refinement formulations. Recommendations are made for use of these mesh refinement formulations.

The Mathematical Basis of Finite Element Methods.

1986 ◽  
Vol 47 (175) ◽  
pp. 383
Author(s):  
W. G. ◽  
David F. Griffiths
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Mathematical Basis

Force Transfer for High Order Finite Element Methods Using Intersected Meshes

2007 ◽  
Author(s):  
Stefan Kollmannsberger ◽  
Alexander Du¨ster ◽  
Ernst Rank
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Data Transfer ◽  
High Order ◽  
Integration Scheme ◽  
Numerical Examples ◽  
Structure Interaction ◽  
Force Transfer ◽  
New Type ◽  
Structural Code

High order Finite Element Methods have been shown to be an efficient approach for computing the behavior of fluids and structures alike. However the coupling of such methods in a framework for a partitioned fluid-structure interaction is still in its early stages. A difficulty hereby is a conservative transfer of the loads from the fluid to the solid and an appropriate transfer of the structural displacements back to the boundary of the fluid. This contribution describes the coupling of a high order finite element structural code to the commercial finite volume fluid solver CFX and focuses on the transfer of the loads. For this purpose, the fluid mesh and the structural mesh are intersected. The force acting on the solid is then computed by a composed integration scheme performed on the intersected mesh. The approach can be interpreted as a projection method taking into account the discretization on both sides, i.e. fluid and solid. Numerical examples will demonstrate the basic properties of this new type of data transfer.

scholarly journals Numerical Solution of Some Differential Equations with Henstock–Kurzweil Functions

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
D. A. León-Velasco ◽  
M. M. Morín-Castillo ◽  
J. J. Oliveros-Oliveros ◽  
T. Pérez-Becerra ◽  
J. A. Escamilla-Reyna
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Numerical Solution ◽  
Elliptic Problem ◽  
The Finite Element Method ◽  
Weak Formulation ◽  
Numerical Examples ◽  
Integrable Functions ◽  
Bounded Variation Functions

In this work, the Finite Element Method is used for finding the numerical solution of an elliptic problem with Henstock–Kurzweil integrable functions. In particular, Henstock–Kurzweil high oscillatory functions were considered. The weak formulation of the problem leads to integrals that are calculated using some special quadratures. Definitions and theorems were used to guarantee the existence of the integrals that appear in the weak formulation. This allowed us to apply the above formulation for the type of slope bounded variation functions. Numerical examples were developed to illustrate the ideas presented in this article.

scholarly journals Finite element methods for surface PDEs

Acta Numerica ◽  
2013 ◽  
Vol 22 ◽  
pp. 289-396 ◽  
Author(s):  
Gerhard Dziuk ◽  
Charles M. Elliott
Keyword(s):  
Finite Element ◽  
Numerical Analysis ◽  
Finite Element Methods ◽  
Geometric Analysis ◽  
Diffuse Interface ◽  
Implicit Surface ◽  
Numerical Examples ◽  
Surface Finite Elements ◽  
Triangulated Surfaces ◽  
Evolving Surfaces

In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples.

Equilibrium Profile of Modern Belted Radial Ply Tires: Its Determination and Performance Benefits

2013 ◽  
Vol 41 (2) ◽  
pp. 127-151
Author(s):  
Rudolf F. Bauer
Keyword(s):  
Finite Element ◽  
Aspect Ratio ◽  
Finite Element Methods ◽  
Mathematical Analysis ◽  
Internal Stresses ◽  
Stress Concentrations ◽  
Equilibrium Profiles ◽  
Equilibrium Profile ◽  
And Performance ◽  
Beneficial Property

ABSTRACT The benefits of a tire's equilibrium profile have been suggested by several authors in the published literature, and mathematical procedures were developed that represented well the behavior of bias ply tires. However, for modern belted radial ply tires, and particularly those with a lower aspect ratio, the tire constructions are much more complicated and pose new problems for a mathematical analysis. Solutions to these problems are presented in this paper, and for a modern radial touring tire the equilibrium profile was calculated together with the mold profile to produce such tires. Some construction modifications were then applied to these tires to render their profiles “nonequilibrium.” Finite element methods were used to analyze for stress concentrations and deformations within all tires that did or did not conform to equilibrium profiles. Finally, tires were built and tested to verify the predictions of these analyses. From the analysis of internal stresses and deformations on inflation and loading and from the actual tire tests, the superior durability of tires with an equilibrium profile was established, and hence it is concluded that an equilibrium profile is a beneficial property of modern belted radial ply tires.

Finite element methods for internal flow calculations

1983 ◽  
Author(s):  
W. HABASHI ◽  
M. HAFEZ ◽  
P. KOTIUGA
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Internal Flow
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