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.

Generalized barycentric coordinates and applications

Acta Numerica ◽  
2015 ◽  
Vol 24 ◽  
pp. 161-214 ◽  
Author(s):  
Michael S. Floater
Keyword(s):  
Finite Element ◽  
Computer Graphics ◽  
Finite Element Methods ◽  
Surface Deformation ◽  
Geometric Modelling ◽  
Barycentric Coordinates ◽  
Surface Mesh ◽  
Generalized Barycentric Coordinates

This paper surveys the construction, properties, and applications of generalized barycentric coordinates on polygons and polyhedra. Applications include: surface mesh parametrization in geometric modelling; image, curve, and surface deformation in computer graphics; and polygonal and polyhedral 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 Exponential convergence in $$H^1$$ of hp-FEM for Gevrey regularity with isotropic singularities

2019 ◽  
Vol 144 (2) ◽  
pp. 323-346 ◽  
Author(s):  
M. Feischl ◽  
Ch. Schwab
Keyword(s):  
Finite Element ◽  
Finite Number ◽  
Finite Element Methods ◽  
Exponential Convergence ◽  
Rates Of Convergence ◽  
Gevrey Regularity ◽  
Galerkin Finite Element ◽  
Galerkin Finite Element Methods ◽  
Continuous Galerkin ◽  
Exponential Rates

AbstractFor functions $$u\in H^1(\Omega )$$u∈H1(Ω) in a bounded polytope $$\Omega \subset {\mathbb {R}}^d$$Ω⊂Rd$$d=1,2,3$$d=1,2,3 with plane sides for $$d=2,3$$d=2,3 which are Gevrey regular in $$\overline{\Omega }\backslash {\mathscr {S}}$$Ω¯\S with point singularities concentrated at a set $${\mathscr {S}}\subset \overline{\Omega }$$S⊂Ω¯ consisting of a finite number of points in $$\overline{\Omega }$$Ω¯, we prove exponential rates of convergence of hp-version continuous Galerkin finite element methods on affine families of regular, simplicial meshes in $$\Omega $$Ω. The simplicial meshes are geometrically refined towards $${\mathscr {S}}$$S but are otherwise unstructured.

The Application of a Finite Element Method to the Evaluation of oil Whirl Characteristics

1974 ◽  
Vol 16 (2) ◽  
pp. 101-108 ◽  
Author(s):  
P. Shelly ◽  
C. Ettles
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Elements ◽  
Parabolic Type ◽  
Pressure Profile ◽  
Combined Effects ◽  
Axial Pressure ◽  
Oil Whirl ◽  
Element Method

A method using finite elements for the whirl analysis of plain bearings is outlined. The special properties of an exponentially shaped element are linked to a parabolic type of approximation for the axial pressure profile. The method is developed and applied to the prediction of whirl paths of a heavy rotor operating both in the horizontal and vertical modes. Several locus paths are presented to show the separate and combined effects of rotor unbalance and unidirectional loading over a range of rotational speeds.

Teaching Electromagnetics through Finite Elements. Part I: The Rationale

1988 ◽  
Vol 25 (1) ◽  
pp. 33-49 ◽  
Author(s):  
S. Ratnajeevan H. Hoole
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Elements ◽  
Special Solution ◽  
The Finite Element Method ◽  
Solution Methods ◽  
Element Method

The rationale for teaching undergraduate electromagnetics partly through the finite element method, is put forward. Properly presented, the finite element method, easily within the ken of the engineering undergraduate, promotes clarity and helps to replace large portions of syllabi devoted to special solution methods, with problems of industrial magnitude and character.

scholarly journals Some advances in high-performance finite element methods

2019 ◽  
Vol 36 (8) ◽  
pp. 2811-2834 ◽  
Author(s):  
Song Cen ◽  
Cheng Jin Wu ◽  
Zhi Li ◽  
Yan Shang ◽  
Chenfeng Li
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
High Performance ◽  
Displacement Function ◽  
Content Type ◽  
Reissner Plate ◽  
Long Time ◽  
History Of ◽  
Function Element ◽  
Element Method

Purpose The purpose of this paper is to give a review on the newest developments of high-performance finite element methods (FEMs), and exhibit the recent contributions achieved by the authors’ group, especially showing some breakthroughs against inherent difficulties existing in the traditional FEM for a long time. Design/methodology/approach Three kinds of new FEMs are emphasized and introduced, including the hybrid stress-function element method, the hybrid displacement-function element method for Mindlin–Reissner plate and the improved unsymmetric FEM. The distinguished feature of these three methods is that they all apply the fundamental analytical solutions of elasticity expressed in different coordinates as their trial functions. Findings The new FEMs show advantages from both analytical and numerical approaches. All the models exhibit outstanding capacity for resisting various severe mesh distortions, and even perform well when other models cannot work. Some difficulties in the history of FEM are also broken through, such as the limitations defined by MacNeal’s theorem and the edge-effect problems of Mindlin–Reissner plate. Originality/value These contributions possess high value for solving the difficulties in engineering computations, and promote the progress of FEM.

scholarly journals Construction of Multivariate Interpolation Hermite Polynomials for Finite Element Method

2020 ◽  
Vol 226 ◽  
pp. 02007
Author(s):  
Galmandakh Chuluunbaatar ◽  
Alexander A. Gusev ◽  
Ochbadrakh Chuluunbaatar ◽  
Vladimir P. Gerdt ◽  
Sergue I. Vinitsky ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Elements ◽  
Hermite Polynomials ◽  
Analytical Form ◽  
Multivariate Interpolation ◽  
Partial Derivatives ◽  
Helmholtz Problem ◽  
Finite Element Schemes ◽  
Element Method

A new algorithm for constructing multivariate interpolation Hermite polynomials in analytical form in a multidimensional hypercube is presented. These polynomials are determined from a specially constructed set of values of the polynomials themselves and their partial derivatives with continuous derivatives up to a given order on the boundaries of the finite elements. The effciency of the finite element schemes, algor thms and programs is demonstrated by solving the Helmholtz problem for a cube.

Simulation of Axisymmetric Forming Processes, With Friction, Coupling Finite Elements and Boundary Elements

2006 ◽  
Author(s):  
Dominique Bigot ◽  
Hocine Kebir ◽  
Jean-Marc Roelandt
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Element Method ◽  
Finite Elements ◽  
Boundary Element ◽  
Symmetric Part ◽  
Forming Process ◽  
Optimal Shapes ◽  
Numerical Software ◽  
Element Method

Nowadays, the simulation of forming processes is rather well integrated in the industrial numerical codes. However, to take into account the possible modifications of the tool during cycle of working, we develop dedicated numerical software. This one more particularly will allow the identification of the fatigue criteria of the tool. With the view to conceiving the optimal shapes of tool allowing increasing their lifespan while ensuring a quality required of the part thus manufactured. This latter uses coupling with friction finite element method — for modelling the axi-symmetric part — and boundary element method — for modelling the tool. For the validation, we modeled forming process.

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