Construction of shape functions for theh- andp-versions of the FEM using tensorial product

2007 ◽  
Vol 71 (5) ◽  
pp. 529-563 ◽  
Author(s):  
M. L. Bittencourt ◽  
M. G. Vazquez ◽  
T. G. Vazquez
Keyword(s):  
Shape Functions ◽  
Tensorial Product

scholarly journals Higher Order Multiscale Finite Element Method for Heat Transfer Modeling

Materials ◽  
2021 ◽  
Vol 14 (14) ◽  
pp. 3827
Author(s):  
Marek Klimczak ◽  
Witold Cecot
Keyword(s):  
Heat Transfer ◽  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
Higher Order ◽  
Shape Functions ◽  
Multiscale Finite Element Method ◽  
Multiscale Finite Element ◽  
Steady State Heat ◽  
Steady State Heat Transfer

In this paper, we present a new approach to model the steady-state heat transfer in heterogeneous materials. The multiscale finite element method (MsFEM) is improved and used to solve this problem. MsFEM is a fast and flexible method for upscaling. Its numerical efficiency is based on the natural parallelization of the main computations and their further simplifications due to the numerical nature of the problem. The approach does not require the distinct separation of scales, which makes its applicability to the numerical modeling of the composites very broad. Our novelty relies on modifications to the standard higher-order shape functions, which are then applied to the steady-state heat transfer problem. To the best of our knowledge, MsFEM (based on the special shape function assessment) has not been previously used for an approximation order higher than p = 2, with the hierarchical shape functions applied and non-periodic domains, in this problem. Some numerical results are presented and compared with the standard direct finite-element solutions. The first test shows the performance of higher-order MsFEM for the asphalt concrete sample which is subject to heating. The second test is the challenging problem of metal foam analysis. The thermal conductivity of air and aluminum differ by several orders of magnitude, which is typically very difficult for the upscaling methods. A very good agreement between our upscaled and reference results was observed, together with a significant reduction in the number of degrees of freedom. The error analysis and the p-convergence of the method are also presented. The latter is studied in terms of both the number of degrees of freedom and the computational time.

MESHFREE CELL-BASED SMOOTHED POINT INTERPOLATION METHOD USING ISOPARAMETRIC PIM SHAPE FUNCTIONS AND CONDENSED RPIM SHAPE FUNCTIONS

2011 ◽  
Vol 08 (04) ◽  
pp. 705-730 ◽  
Author(s):  
G. Y. ZHANG ◽  
G. R. LIU
Keyword(s):  
Computational Efficiency ◽  
Interpolation Method ◽  
Shape Functions ◽  
Generalized Gradient ◽  
Singularity Problem ◽  
Numerical Examples ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
System Equations ◽  
Gradient Smoothing

This paper presents two novel and effective cell-based smoothed point interpolation methods (CS-PIM) using isoparametric PIM (PIM-Iso) shape functions and condensed radial PIM (RPIM-Cd) shape functions respectively. These two types of PIM shape functions can successfully overcome the singularity problem occurred in the process of creating PIM shape functions and make the constructed CS-PIM models work well with the three-node triangular meshes. Smoothed strains are obtained by performing the generalized gradient smoothing operation over each triangular background cells, because the nodal PIM shape functions can be discontinuous. The generalized smoothed Galerkin (GS-Galerkin) weakform is used to create the discretized system equations. Some numerical examples are studied to examine various properties of the present methods in terms of accuracy, convergence, and computational efficiency.

Rayleigh-Ritz Analysis of Vibrating Plates Based on a Class of Eigenfunctions

2009 ◽  
Author(s):  
Giuseppe Catania ◽  
Silvio Sorrentino
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Boundary Conditions ◽  
Linear Combination ◽  
Equation Of Motion ◽  
Extensive Literature ◽  
Shape Functions ◽  
Element Analysis ◽  
Vibrating Plates ◽  
General Basis

In the Rayleigh-Ritz condensation method the solution of the equation of motion is approximated by a linear combination of shape-functions selected among appropriate sets. Extensive literature dealing with the choice of appropriate basis of shape functions exists, the selection depending on the particular boundary conditions of the structure considered. This paper is aimed at investigating the possibility of adopting a set of eigenfunctions evaluated from a simple stucture as a general basis for the analysis of arbitrary-shaped plates. The results are compared to those available in the literature and using standard finite element analysis.

Nonlinear Winkler-based beam element with improved displacement shape functions

2013 ◽  
Vol 17 (1) ◽  
pp. 192-201 ◽  
Author(s):  
Suchart Limkatanyu ◽  
Kittisak Kuntiyawichai ◽  
Enrico Spacone ◽  
Minho Kwon
Keyword(s):  
Beam Element ◽  
Shape Functions
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