Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods

2015 ◽  
Vol 104 (13) ◽  
pp. 1173-1199 ◽  
Author(s):  
Sundararajan Natarajan ◽  
Stéphane PA Bordas ◽  
Ean Tat Ooi
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Finite Element Methods

scholarly journals A Brief Review on Polygonal/Polyhedral Finite Element Methods

2018 ◽  
Vol 2018 ◽  
pp. 1-22 ◽  
Author(s):  
Logah Perumal
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Finite Element Methods ◽  
Virtual Node ◽  
Advantages And Disadvantages

This paper provides brief review on polygonal/polyhedral finite elements. Various techniques, together with their advantages and disadvantages, are listed. A comparison of various techniques with the recently proposed Virtual Node Polyhedral Element (VPHE) is also provided. This review would help the readers to understand the various techniques used in formation of polygonal/polyhedral finite elements.

scholarly journals Shape-Free Finite Element Method: Another Way between Mesh and Mesh-Free Methods

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Song Cen ◽  
Ming-Jue Zhou ◽  
Yan Shang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Elements ◽  
Finite Element Methods ◽  
High Performance ◽  
Mesh Quality ◽  
Mesh Free ◽  
Plane Element ◽  
Mesh Free Methods ◽  
Hybrid Stress

Performances of the conventional finite elements are closely related to the mesh quality. Once distorted elements are used, the accuracy of the numerical results may be very poor, or even the calculations have to stop due to various numerical problems. Recently, the author and his colleagues developed two kinds of finite element methods, named hybrid stress-function (HSF) and improved unsymmetric methods, respectively. The resulting plane element models possess excellent precision in both regular and severely distorted meshes and even perform very well under the situations in which other elements cannot work. So, they are calledshape-freefinite elements since their performances are independent to element shapes. These methods may open new ways for developing novel high-performance finite elements. Here, the thoughts, theories, and formulae of aboveshape-freefinite element methods were introduced, and the possibilities and difficulties for further developments were also discussed.

Deep 3D Convolution Neural Network Methods for Brain White Matter Hybrid Computational Simulations

2020 ◽  
Author(s):  
Xuehai Wu ◽  
Assimina A. Pelegri
Keyword(s):  
Neural Network ◽  
Finite Element ◽  
White Matter ◽  
Finite Elements ◽  
Frequency Domain ◽  
Finite Element Methods ◽  
Material Properties ◽  
Output Data ◽  
Micro Scale ◽  
Brain White Matter

Abstract Material properties of brain white matter (BWM) show high anisotropy due to the complicated internal three-dimensional microstructure and variant interaction between heterogeneous brain-tissue (axon, myelin, and glia). From our previous study, finite element methods were used to merge micro-scale Representative Volume Elements (RVE) with orthotropic frequency domain viscoelasticity to an integral macro-scale BWM. Quantification of the micro-scale RVE with anisotropic frequency domain viscoelasticity is the core challenge in this study. The RVE behavior is expressed by a viscoelastic constitutive material model, in which the frequency-related viscoelastic properties are imparted as storage modulus and loss modulus for the composite comprised of axonal fibers and extracellular glia. Using finite elements to build RVEs with anisotropic frequency domain viscoelastic material properties is computationally very consuming and resource-draining. Additionally, it is very challenging to build every single RVE using finite elements since the architecture of each RVE is arbitrary in an infinite data set. The architecture information encoded in the voxelized location is employed as input data and is consequently incorporated into a deep 3D convolution neural network (CNN) model that cross-references the RVEs’ material properties (output data). The output data (RVEs’ material properties) is calculated in parallel using an in-house developed finite element method, which models RVE samples of axon-myelin-glia composites. This novel combination of the CNN-RVE method achieved a dramatic reduction in the computation time compared with directly using finite element methods currently present in the literature.

scholarly journals New perspectives on polygonal and polyhedral finite element methods

2014 ◽  
Vol 24 (08) ◽  
pp. 1665-1699 ◽  
Author(s):  
Gianmarco Manzini ◽  
Alessandro Russo ◽  
N. Sukumar
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Finite Element Methods ◽  
Rates Of Convergence ◽  
Explicit Construction ◽  
Barycentric Coordinates ◽  
Finite Difference Schemes ◽  
Stability Matrix ◽  
Generalized Barycentric Coordinates ◽  
Element Method

Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finite difference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary polygonal meshes was devised. The method was coined as the virtual element method (VEM), since it did not require the explicit construction of basis functions. This advance provides a more in-depth understanding of mimetic schemes, and also endows polygonal-based Galerkin methods with greater flexibility than three-node and four-node finite element methods. In the VEM, a projection operator is used to realize the decomposition of the stiffness matrix into two terms: a consistent matrix that is known, and a stability matrix that must be positive semi-definite and which is only required to scale like the consistent matrix. In this paper, we first present an overview of previous developments on conforming polygonal and polyhedral finite elements, and then appeal to the exact decomposition in the VEM to obtain a robust and efficient generalized barycentric coordinate-based Galerkin method on polygonal and polyhedral elements. The consistent matrix of the VEM is adopted, and numerical quadrature with generalized barycentric coordinates is used to compute the stability matrix. This facilitates post-processing of field variables and visualization in the VEM, and on the other hand, provides a means to exactly satisfy the patch test with efficient numerical integration in polygonal and polyhedral finite elements. We present numerical examples that demonstrate the sound accuracy and performance of the proposed method. For Poisson problems in ℝ2and ℝ3, we establish that linearly complete generalized barycentric interpolants deliver optimal rates of convergence in the L2-norm and the H1-seminorm.

AN HEXAHEDRAL FACE ELEMENT METHOD FOR THE DISPLACEMENT FORMULATION OF STRUCTURAL ACOUSTICS PROBLEMS

2001 ◽  
Vol 09 (03) ◽  
pp. 911-918 ◽  
Author(s):  
ALFREDO BERMÚDEZ ◽  
PABLO GAMALLO ◽  
RODOLFO RODRÍGUEZ
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Numerical Computation ◽  
Finite Element Methods ◽  
Degrees Of Freedom ◽  
Computing Time ◽  
The Other ◽  
Structural Acoustics ◽  
Displacement Potential ◽  
Hexahedral Meshes

Several finite element methods for the numerical computation of elastoacoustic vibrations are compared. They are applied to two formulations based on different variables to describe the fluid: presssure and displacement potential in one case, and displacements in the other. While the first one is discretized by standard Lagrangean finite elements for both variables, the second one is solved by "face" Raviart-Thomas elements. In each case we consider both tetrahedral and hexahedral meshes. Elastoacoustic eigenmodes have been computed for a test example by means of MATLAB implementations of all these methods. The numerical results allow us to compare all of them in terms of error versus number of degrees of freedom and computing time.

scholarly journals A simplified variant of the time finite element methods based on the shape functions of an axial finite bar

2021 ◽  
Vol 15 (4) ◽  
pp. 42-53
Author(s):  
Thanh Xuan Nguyen ◽  
Long Tuan Tran
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Structural Dynamics ◽  
Finite Element Methods ◽  
Time Integration ◽  
Shape Functions ◽  
Runge Kutta Method ◽  
Structural Responses ◽  
The Time Domain ◽  
High Order Time

In the field of structural dynamics, the structural responses in the time domain are of major concern. There already exist many methods proposed previously including widely used direct time integration methods such as ones in the β-Newmark family, Houbolt’s method, and Runge-Kutta method. The time finite element methods (TFEM) that followed the well-posed variational statement for structural dynamics are found to bring about a superior accuracy even with large time steps (element sizes), when compared with the results from methods mentioned above. Some high-order time finite elements were derived with the procedure analogous to the conventional finite element methods. In the formulation of these time finite elements, the shape functions are like the ones for a (spatial) 2-order finite beam. In this article, a simplified variant for the TFEM is proposed where the shape functions similar to the ones for a (spatial) axial bar are used. The accuracy in the obtained results of some numerical examples is found to be comparable with the accuracy in the previous results.

A Comparison of Coupling Molecular Dynamics to General Finite Elements

2009 ◽  
Author(s):  
Michael F. Macri
Keyword(s):  
Molecular Dynamics ◽  
Finite Element ◽  
Finite Elements ◽  
Finite Element Methods ◽  
Continuum Model ◽  
Partition Of Unity ◽  
Multiscale Model ◽  
Moving Least Squares ◽  
Interpolation Functions ◽  
The Continuum

In this paper, we assess the ability of three interpolation functions in a discretized continuum model to capture and accurately represent the solution. In particular we examine the differences between the partition of unity, moving least squares and finite element methods in the continuum part of the multiscale model.

scholarly journals CONVERGENCE OF ADAPTIVE FINITE ELEMENT METHODS FOR EIGENVALUE PROBLEMS

2009 ◽  
Vol 19 (05) ◽  
pp. 721-747 ◽  
Author(s):  
EDUARDO M. GARAU ◽  
PEDRO MORIN ◽  
CARLOS ZUPPA
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Finite Element Methods ◽  
Eigenvalue Problems ◽  
Second Order ◽  
Adaptive Finite Element ◽  
Adaptive Finite Element Methods ◽  
Multiple Eigenvalues ◽  
Elliptic Eigenvalue Problems

In this paper we prove convergence of adaptive finite element methods for second-order elliptic eigenvalue problems. We consider Lagrange finite elements of any degree and prove convergence for simple as well as multiple eigenvalues under a minimal refinement of marked elements, for all reasonable marking strategies, and starting from any initial triangulation.

Simplified Finite-Element Models for Reservoir Flow Problems

1979 ◽  
Vol 19 (05) ◽  
pp. 333-343 ◽  
Author(s):  
Vilgeir Dalen
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Finite Difference ◽  
Variational Methods ◽  
Finite Element Methods ◽  
Hyperbolic Equations ◽  
Finite Difference Methods ◽  
Flow Problems ◽  
Reservoir Flow ◽  
Difference Methods

Abstract This paper summarizes some research that was conducted to construct finite-element models for reservoir flow problems. The models are based on Galerkin's method, but the method is applied in an unorthodox manner to simplify calculation of coefficients and to improve stability. Specifically, techniques of compatibility relaxation, capacity lumping, and upstream mobility weighting are used, and schemes are obtained that seem to combine the simplicity and high stability of conventional finite-difference models with the generality and modeling flexibility of finite-element methods.The development of a model for single-phase gas flow and a two-phase oil/water model is described. Numerical examples are included to illustrate the usefulness of finite elements. In particular, the triangular element with linear interpolation is shown to be an attractive alternative to the standard five-point, finite-difference approximation for two-dimensional analysis. Introduction During the past decades, finite-element methods have been developed to a high level of sophistication and have gained wide popularity within several branches of engineering science. In some fields, such methods have replaced to some extent the older finite-difference methods in engineering practice because they have been regarded as a more convenient tool for numerical analysis. An increasing interest in finite-element methods, or variational methods in general, also may be noticed in the field of numerical reservoir simulation, but so far no definitive breakthrough has occurred in this field.One reason for this probably is the complexity of reservoir flow problems. Reservoir flow equations in most cases are nonlinear, and for multiphase flow, they are usually found on the borderline between parabolic and hyperbolic equations. For such parabolic and hyperbolic equations. For such problems, the dissimilarities between problems, the dissimilarities between finite-difference and finite-element methods are much more pronounced than for linear problems of the elliptic type. This means that the finite-element method may not be looked upon as easily as an extension or generalization of finite-difference methods. Second, one can question whether all the advantages that are gained in other instances by using finite elements may be realized at all.Applications of variational methods to single-phase flow problems, or diffusion-type problems in general, have been studied extensively. The merits of finite elements for such problems are apparently well established, at least as far as linear problems are concerned.The literature on variational methods in multiphase flow is comparatively sparse, and so far the results are inconclusive regarding the relative advantages of variational methods and finite- difference methods in this field. In summary, variational methods offer the potential advantages of (1) easy implementation of higher-order approximations, (2) a more proper treatment of variable coefficients, and (3) greater modeling flexibility.Previously, attention was focused on Aspects 1 and 2. Several authors used cubic Hermitian basis functions. SPEJ P. 333

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