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 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.

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.

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.

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.

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.

scholarly journals Two-Dimensional, Time-dependent Modelling of an Arbitrarily Shaped Ice Mass with the Finite-Element Technique

1985 ◽  
Vol 31 (109) ◽  
pp. 350-359 ◽  
Author(s):  
Steven M. Hodge
Keyword(s):  
Finite Element ◽  
Numerical Models ◽  
Basis Functions ◽  
Time Dependent ◽  
Small Computer ◽  
Two Dimensional ◽  
Maximum Increase ◽  
Steady State Condition ◽  
Finite Element Technique ◽  
Element Technique

AbstractThe two-dimensional, time-dependent flow of an arbitrarily shaped ice mass can be successfully modeled with the finite-element technique on a small computer. Methods developed for automatically generating the mesh data greatly simplify the data preparation and optimize the numerical simulations. Using quadratic basis functions permits the flow to be approximated quite adequately by only two element rows (five nodes vertically). Mixed-order basis functions, however, must be used so that numerical oscillations do not set in, and the ends of the ice mass, where the thickness tends to zero, must be treated carefully. Time simulations to a steady-state condition are necessary to test such numerical models adequately.South Cascade Glacier, Washington, is currently close to equilibrium. A bedrock sill dominates the bed topography in the lower half of the glacier, rising to a height of about 20% of the ice thickness. This sill produces a maximum increase in the overall thickness of about 6–7% compared to what the thickness would have been if the sill were not present. Finally, this glacier does not appear to be sliding much, if at all, despite its maritime alpine environment. This could help explain the difficulties encountered when trying to measure sliding and basal water pressures on the same glacier (Hodge, 1979), or it could imply that drag exerted by the valley walls has a significantly greater effect than conventional shape-factor concepts imply.

Stiff-String Basis Functions for Vibration Analysis of High Speed Rotating Beams

2008 ◽  
Vol 75 (2) ◽  
Author(s):  
Jagadish Babu Gunda ◽  
Ranjan Ganguli
Keyword(s):  
Finite Element ◽  
High Speed ◽  
Natural Frequencies ◽  
Basis Functions ◽  
Beam Equation ◽  
Rotating Beam ◽  
Rotating Beams ◽  
Centrifugal Stiffening ◽  
Convergence Study ◽  
Stiffening Effect

A new rotating beam finite element is developed in which the basis functions are obtained by the exact solution of the governing static homogenous differential equation of a stiff string, which results from an approximation in the rotating beam equation. These shape functions depend on rotation speed and element position along the beam and account for the centrifugal stiffening effect. Using this new element and the Hermite cubic finite element, a convergence study of natural frequencies is performed, and it is found that the new element converges much more rapidly than the conventional Hermite cubic element for the first two modes at higher rotation speeds. The new element is also applied for uniform and tapered rotating beams to determine the natural frequencies, and the results compare very well with the published results given in the literature.

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