scholarly journals A quadratic finite element wavelet Riesz basis

2018 ◽  
Vol 16 (04) ◽  
pp. 1850033 ◽  
Author(s):  
Nikolaos Rekatsinas ◽  
Rob Stevenson
Keyword(s):  
Finite Element ◽  
Linear Combination ◽  
Riesz Basis ◽  
Basis Functions ◽  
Condition Numbers ◽  
Vanishing Moments ◽  
Unit Square ◽  
Nodal Basis

In this paper, continuous piecewise quadratic finite element wavelets are constructed on general polygons in [Formula: see text]. The wavelets are stable in [Formula: see text] for [Formula: see text] and have two vanishing moments. Each wavelet is a linear combination of 11 or 13 nodal basis functions. Numerically computed condition numbers for [Formula: see text] are provided for the unit square.

Two-dimensional empirical mode decomposition by finite elements

2006 ◽  
Vol 462 (2074) ◽  
pp. 3081-3096 ◽  
Author(s):  
Y Xu ◽  
B Liu ◽  
J Liu ◽  
S Riemenschneider
Keyword(s):  
Finite Element ◽  
Linear Combination ◽  
Empirical Mode Decomposition ◽  
Spline Interpolation ◽  
Basis Functions ◽  
Two Dimensional ◽  
Element Analysis ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Mode Decomposition

Empirical mode decomposition (EMD) is a powerful tool for analysis of non-stationary and nonlinear signals, and has drawn significant attention in various engineering application areas. This paper presents a finite element-based EMD method for two-dimensional data analysis. Specifically, we represent the local mean surface of the data, a key step in EMD, as a linear combination of a set of two-dimensional linear basis functions smoothed with bi-cubic spline interpolation. The coefficients of the basis functions in the linear combination are obtained from the local extrema of the data using a generalized low-pass filter. By taking advantage of the principle of finite-element analysis, we develop a fast algorithm for implementation of the EMD. The proposed method provides an effective approach to overcome several challenging difficulties in extending the original one-dimensional EMD to the two-dimensional EMD. Numerical experiments using both simulated and practical texture images show that the proposed method works well.

Ray tracing in a finite-element domain using nodal basis functions

2014 ◽  
Vol 53 (24) ◽  
pp. F10 ◽  
Author(s):  
Karl N. Schrader ◽  
Samuel R. Subia ◽  
John W. Myre ◽  
Kenneth L. Summers
Keyword(s):  
Finite Element ◽  
Ray Tracing ◽  
Basis Functions ◽  
Nodal Basis

scholarly journals An Online Generalized Multiscale Finite Element Method for Unsaturated Filtration Problem in Fractured Media

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1382
Author(s):  
Denis Spiridonov ◽  
Maria Vasilyeva ◽  
Aleksei Tyrylgin ◽  
Eric T. Chung
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Model Reduction ◽  
Basis Functions ◽  
Computational Domain ◽  
Filtration Problem ◽  
Spectral Problems ◽  
Multiscale Finite Element Method ◽  
Multiscale Finite Element ◽  
Element Method

In this paper, we present a multiscale model reduction technique for unsaturated filtration problem in fractured porous media using an Online Generalized Multiscale finite element method. The flow problem in unsaturated soils is described by the Richards equation. To approximate fractures we use the Discrete Fracture Model (DFM). Complex geometric features of the computational domain requires the construction of a fine grid that explicitly resolves the heterogeneities such as fractures. This approach leads to systems with a large number of unknowns, which require large computational costs. In order to develop a more efficient numerical scheme, we propose a model reduction procedure based on the Generalized Multiscale Finite element method (GMsFEM). The GMsFEM allows solving such problems on a very coarse grid using basis functions that can capture heterogeneities. In the GMsFEM, there are offline and online stages. In the offline stage, we construct snapshot spaces and solve local spectral problems to obtain multiscale basis functions. These spectral problems are defined in the snapshot space in each local domain. To improve the accuracy of the method, we add online basis functions in the online stage. The construction of the online basis functions is based on the local residuals. The use of online bases will allow us to get a significant improvement in the accuracy of the method. We present results with different number of offline and online multisacle basis functions. We compare all results with reference solution. Our results show that the proposed method is able to achieve high accuracy with a small computational cost.

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.

An Angular Finite Element Method for the Solution of the Even-Parity Equation

2009 ◽  
Author(s):  
R. Becker ◽  
R. Koch ◽  
M. F. Modest ◽  
H.-J. Bauer
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Basis Functions ◽  
New Method ◽  
Discrete Ordinates ◽  
Test Case ◽  
Element Approximation ◽  
Promising Alternative ◽  
Element Discretization ◽  
Spatial Coordinates

The present article introduces a new method to solve the radiative transfer equation (RTE). First, a finite element discretization of the solid angle dependence is derived, wherein the coefficients of the finite element approximation are functions of the spatial coordinates. The angular basis functions are defined according to finite element principles on subdivisions of the octahedron. In a second step, these spatially dependent coefficients are discretized by spatial finite elements. This approach is very attractive, since it provides a concise derivation for approximations of the angular dependence with an arbitrary number of angular nodes. In addition, the usage of high-order angular basis functions is straightforward. In the current paper the governing equations are first derived independently of the actual angular approximation. Then, the design principles for the angular mesh are discussed and the parameterization of the piecewise angular basis functions is derived. In the following, the method is applied to two-dimensional test cases which are commonly used for the validation of approximation methods of the RTE. The results reveal that the proposed method is a promising alternative to the well-established practices like the Discrete Ordinates Method (DOM) and provides highly accurate approximations. A test case known to exhibit the ray effect in the DOM verifies the ability of the new method to avoid ray effects.

scholarly journals On linear independence for integer translates of a finite number of functions

1993 ◽  
Vol 36 (1) ◽  
pp. 69-85 ◽  
Author(s):  
Rong-Qing Jia ◽  
Charles A. Micchelli
Keyword(s):  
Finite Number ◽  
Linear Combination ◽  
Linear Independence ◽  
Sufficient Conditions ◽  
Basis Functions ◽  
Partial Difference ◽  
Integer Translates ◽  
First Case ◽  
Compactly Supported Functions ◽  
Necessary And Sufficient

We investigate linear independence of integer translates of a finite number of compactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some lp space (1 ≦ p ≦ ∞) and we are interested in bounding their lp-norms in terms of the Lp-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

Two-Stage Robust Tracking Controller for Linear Systems With Known Uncertainty Using Filtered Basis Functions

2020 ◽  
Author(s):  
Keval S. Ramani ◽  
Chinedum E. Okwudire
Keyword(s):  
Linear Combination ◽  
Linear Systems ◽  
Tracking Error ◽  
Basis Functions ◽  
Frobenius Norm ◽  
Control Input ◽  
Two Stage ◽  
Nominal Model ◽  
Control Of Linear Systems ◽  
Optimal Set

Abstract There is growing interest in the use of the filtered basis functions (FBF) approach to track linear systems, especially nonminimum phase (NMP) plants, because of the distinct advantages it presents as compared to other popular methods in the literature. The FBF approach expresses the control input to the plant as a linear combination of basis functions. The basis functions are forward filtered through the plant dynamics and the coefficients of the linear combination are selected such that the tracking error is minimized. This paper proposes a two-stage robust filtered basis functions approach for tracking control of linear systems in the presence of known uncertainty. In the first stage, the nominal model for filtering the basis functions is selected such that a Frobenius norm metric which considers the known uncertainty is minimized. In the second stage, an optimal set of basis functions is selected such that the effect of uncertainty is minimized for the nominal model selected in the first stage. Experiments on a 3D printer, demonstrate up to 7 times improvement in tracking performance using the proposed method as compared to the standard FBF approach.

Effects of sections added to multi-cell square tubes on crash performance

2020 ◽  
Vol 62 (5) ◽  
pp. 471-480 ◽  
Author(s):  
Emre İsa Albak
Keyword(s):  
Finite Element ◽  
Genetic Algorithms ◽  
Radial Basis Functions ◽  
Axial Loading ◽  
Basis Functions ◽  
Finite Element Analyses ◽  
Energy Absorber ◽  
Radial Basis ◽  
Crash Performance

Abstract In this study, the effects of sections added to multi-cell square tubes on crash performance are examined. Square, hexagonal and circular sections are added to multi-cell square tubes and their results are examined. Finite element analyses under axial loading are performed to examine the crash performance of the multi-cell tubes. Analyses show that by adding a section to the multi-cell square tubes. the crash behavior of the tubes is improved. According to the results, S5H multi-cell square tube reveals the best crash performance. The optimization of S5H is carried out by using genetic algorithms and radial basis functions. The S5H tube presents a good crashworthiness performance and could be used as an energy absorber.

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