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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.

Element equations are obtained by incorporating nodal conditions of the element geometry into the shape functions. One of the requirements is that the field variable obtained from the element equations should be linear on the element boundaries. This requirement is met for triangular and quadrilateral elements, by selecting suitable linear (or bilinear) shape functions from the Pascal triangle [

However, suitable first-order shape functions were not available for element geometries with more than four sides until around the 1970s. Wachspress [

Voronoi cell finite element method (VCFEM) and polygonal finite element based on parametric variational principle and the parametric quadratic programming method

Hybrid polygonal element (HPE)

Conforming polygonal finite element method based on barycentric coordinates (conforming PFEM, or PFEM)

Polygonal scaled boundary finite element method (PSBFEM)

Mimetic finite difference (MFD) and virtual element method (VEM)

Virtual node method (VNM)

Discontinuous Galerkin finite element method (DGFEM)

Trefftz/Hybrid Trefftz polygonal finite element (T-FEM or HT-FEM) and Boundary element based FEM (BEM-based FEM)

Hybrid stress-function (HS-F) polygonal element

Base forces element method (BFEM)

Other recent techniques/schemes

The methods above are briefly highlighted in the following sections.

Around the 1990s, Ghosh and Mukhopadhyay [

A Voronoi cell element.

The matrix phase is denoted as

VCFEM combines assumptions from micromechanics theories and adaptive enhancements. This technique is computationally efficient compared to the conventional FEM (displacement based triangular or quadrilateral elements), since each polygonal grain is represented by a single finite element and no further subdivision of the domain is required. Stress functions for the interior of the element are obtained in terms of polynomial expansions of the global coordinates and these polynomial functions are formulated in such a way that they satisfy equilibrium within the element.

An example of VCFEM formulation for analysis of Cosserat materials based on the parametric minimum complementary energy principle is given as [

Stress functions for the interior of this element is defined by using Airy’s stress function,

VCFEM has been found to perform poorly when the heterogeneity is in the form of voids (but works well for inclusions), due to the poorly defined stress functions within the interior of the element. This problem is solved by taking into account the geometry effects through conformal mapping [

Later, Sze and Sheng [

Hybrid of VCFEM with other methods such as numerical conformal mapping (NCM) method can be seen in the work by Tiwary, Hu, and Ghosh [

VCFEM is generally suitable for micromechanical analysis and multiscale modelling [

On the other hand, Zhang and Katsube [

Figure

Mapping of a Voronoi cell element.

An example of a hybrid functional

The interior stresses are obtained by using interpolation functions (in the form of trigonometric functions) through the following relation [

Similar functions are used to determine the displacements within the element.

Application of HPEs for the analysis of heterogeneous media in 2D as well as 3D can be seen in works by Zhang and Katsube [

Around the year of 2000, the Wachspress method gained more attention and was revisited alongside with other techniques to formulate interpolation or shape functions for polygonal elements. These techniques are known as barycentric coordinates method [

Inverse bilinear coordinates were developed for quadrilaterals, based on bilinear mapping of a unit square to convex quadrilaterals. Rational functions are used for the mapping and their inverses were studied to develop the inverse bilinear coordinates for quadrilaterals. Wachspress developed rational polynomial functions which can be used to produce conforming shape functions for arbitrary polygons. Meyer et al. [

This shortage is avoided in mean value coordinates, which are written in terms of trigonometric functions. Mean value coordinates can be adapted to complex arbitrary polygons, especially star-shaped geometries (concave polygons). It is noted that shape functions for concave polygonal elements cannot be represented by rational polynomial functions, since convex shapes (can be represented by rational polynomial functions) cannot be mapped to concave shapes [

Motivated by mesh-free method, authors of references [

Polygonal elements based on barycentric coordinates have been implemented in various areas such as computer graphics, animation and geometric modelling [

However, evaluation of barycentric coordinates is neither simple nor efficient compared to the conventional displacement based FEM, due to the complex functions which arise in the former techniques. Furthermore, barycentric coordinates are not efficient for assembling the stiffness matrices associated with weak solutions of Poisson equations [

Construction of shape functions

Construction of shape functions based on Wachspress.

The expression for

Construction of shape functions based on mean value coordinates.

The expression for

Construction of shape functions based on the concept of natural neighbors.

Another attempt to form polygonal finite element method can be seen within the smoothed finite element method (SFEM). SFEM is formed by merging conventional FEM with meshless methods. SFEM was initially formed for quadrilateral elements. Later, Dai, Liu, and Nguyen [

In

An example of

Partitioning of a

Point

There are three types of smoothing techniques applicable for these smoothing cells, which are cell, node, and edge based. These techniques are known as

In case of

Smoothing domain

Smoothing domain

Generally, the

Scaled boundary FEM is a semianalytical method which combines the boundary element method (BEM) and FEM. It was first introduced by Song and Wolf [

A PSBFEM element.

Solution along the radial direction is obtained by analytical expressions by using

The displacement

Various applications of PSBFEM can be seen in literature such as in linear elasticity [

PSBFEM has been found to be superior to other techniques such as

One of the difficulties faced in the construction of polygonal finite elements is the development of interpolation functions which extends to the interior of the element. Beir ao da Veiga, Gyrya, Lipnikov, and Manzini [

For example, consider a heat conduction phenomenon which is governed by the following governing equation (

Low order degrees of freedom for MFD method. The arrows represent fluxes and the center represents the temperature.

An additional step is necessary, which is the discretization of the integrals. This step is required in order to approximate div and

The method is also shown to be useful for meshes with degenerate and nonconvex polygonal elements. Since then, MFD method has been implemented in various problems (diffusion/convection-diffusion [

Beir ao Da Veiga, Brezzi, Cangiani, Manzini, Marini, and Russo [

The virtual element space on a polygonal domain (that discretizes the problem domain) is defined as [

A similarity between VCFEM, HPE, MFD, and VEM is that these methods do not require the extension of compatible interpolation functions to the interior of the element. The compatible interpolation functions are required for the element boundaries only, which simplifies the formulation and enables the formulation of elements with arbitrary number of sides/nodes. Disadvantage of MFD and VEM is that they involve complex procedures and therefore require high computational effort [

Another attempt to overcome the difficulty in forming compatible shape functions for polygonal FEM can be seen in the literature [

Partitioning of a polygon according to VNM.

Compatibility among each triangular region is fulfilled by using the conventional FEM shape functions of a 3 nodes’ triangular element. These triangular regions are then coupled together (to form the polygonal element) by representing the virtual node at the center in terms of mean least square shape functions, as shown by (

The method is advantageous compared to compatible PFEM due to the simple polynomial shape functions (which is easier to work with). VNM is found to be efficient in adaptive computation, by using quadtree or octree mesh. The method has been extended to 3D polyhedral (VPHE) and hexahedral forms and implemented in adaptive computations as can be seen in the literature [

This method was proposed due to the difficulties faced in executing other polygonal FEMs such as compatible PFEM, MFD, and VEM. The difficulties arise due to the complicated shape functions in compatible PFEM and complex procedures involved in MFD and VEM. These complicated entities demand high computational effort as well [

In DGFEM, the problem domain is discretized into several polygonal cells which represent the polygonal elements. The interpolation for the elements is carried out based on a set of monomial functions which are totally independent of the element. These interpolation functions do not comply to shape function requirements of conventional FEM and therefore they are not continuous across different elements (not compatible). Due to this, integrations are carried out on the boundaries of each cell. This step is an addition compared to the conventional FEM. The degrees of freedom are obtained based on the coefficients of the linear combination of the monomial functions [

Interpolation of field variables

First order:

Second order:

where

The stiffness matrix

T-FEM utilizes two different sets of functions to approximate the solutions, one for the boundary and the other is for the interior domain. For the interior of the element, a series of homogeneous solutions of the governing equation (problem equation to be solved) is used as basis functions. These basis functions are known as T-complete set and they are not conforming across the element boundaries [

An example of T-FEM with its shape functions utilized for the case of solid mechanics.

The displacement

Continuity or boundary conditions are incorporated into the interior domain by various ways, in which one of the ways is by hybrid method known as hybrid Trefftz finite element method (HT-FEM). This method uses the conforming functions of the element boundary (also known as frame) to link the interior of the elements together [

HT-FEM has been successfully applied in linear elasticity problems [

The advantage of HT-FEM compared to conventional FEM [

Another method has been proposed by combining the principle of minimum complementary energy (similar principal used in VCFEM) with the Airy stress function which is known as hybrid stress-function (HS-F) element method. It was developed for quadrilaterals and triangles [

An example of a HS-F element.

The displacement along a particular element edge

Stress based FEMs such as HPE and HS-F are not well desired for most of the engineering applications, due to the difficulties in obtaining suitable/compatible stress functions. Furthermore, it is difficult to acquire nodal displacements in stress based FEMs [

Conventional FEM exhibits major drawback for nonlinear analysis. The conventional FEM is not able to approximate the strain and force fields accurately since these terms are dependent on the interpolation of the displacement field (displacement shape functions). This problem is avoided in BFEM which is directly based on interpolation of the internal force fields (force shape functions) [

An example of a BFEM element.

The stresses corresponding to a point

Recently, more new schemes have been proposed for the development of polygonal/polyhedral finite elements. They are the Compatible Discrete Operator (CDO) scheme [

Other recent techniques/methods include analysis of polygonal carbon nanotubes reinforced composite plates by using the first-order shear deformation theory (FSDT) and the element-free IMLS-Ritz method [

Various techniques described in Section

Comparison of the existing methods with the proposed/present element.

Method | Element Formulation | Advantages | Disadvantages | Specialty of VPHE element compared to other techniques | Application fields and typology |
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VCFEM | Principle of minimum complementary energy. | 1. Computationally efficient compared to the conventional FEM. | 1. Perform poorly when the heterogeneity is in the form of voids. | Stress functions within the element are well defined by monomials. | Simulation of microstructures (grains) and multiscale modelling. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |

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NCM-VCFEM | Hybrid of VCFEM with other method such as numerical conformal mapping. | 1. The variational principle is generalized. | 1. The NCM-based stress function construction is expensive in comparison with conformal mapping of regular shapes such as ellipses and circles. | Special techniques are not needed to handle singularities in the shape functions. | Real micrographs of heterogeneous materials with irregular shapes can be analyzed effectively. Applicable for 2D problems for time being. Applicable for both linear and nonlinear analyses. |

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HPE | Hybrid stress element method together with Muskhelishvili’s complex analysis. | 1. Stress functions within the interior of the element are defined by self-equilibrating stress field. | 1. Can contain only one irregular phase (void/inclusion) within the element. | - | Simulation of microstructures (grains) with heterogeneity. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |

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PFEM | Barycentric Coordinates. | 1. Able to take arbitrary form, with arbitrary number of sides and nodes. | 1. Evaluation of barycentric coordinates (complex rational functions) is neither simple nor efficient compared to the conventional displacement based FEM. | The shape functions consist of simple monomials irrespective on number of planes/sides. | Solid mechanics and heat transfer phenomena. Computer graphics, animation and geometric modelling. Quadtree/Octree mesh generation. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |

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| Coupling of conventional FEM with meshless method. | 1. | 1. | Formulation of the proposed/present element is similar to | Solid mechanics and heat transfer phenomena. Fluid-solid interaction (FSI) problems. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |

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PSBFEM | Semi analytical method which combines boundary element method (BEM) and FEM. | 1. Analytical solutions are achieved inside the domain, discretization of free and fixed boundaries and interfaces between different materials are not required, and the calculation of stress concentrations and intensity factors based on their definition is straightforward. | 1. Not directly applicable for unbounded domains with strongly inclined interfaces. | - | Solid mechanics and polygonal mesh creation. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |

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MFD and | Surface representation of discrete unknowns (MFD) and unknown degrees of freedom are attached to trial functions within interior of the polygonal domain (VEM). | 1. Efficient in solving problems involving polygonal meshes. | 1. Quite difficult to present MFD due to nonexistence of trial functions for the interior of the element. | Easier to execute due to simpler element formulation. | Electromagnetic field problems, convection-diffusion problems, fluid flows problems, hydrodynamics problems, eigenvalue problems, solid mechanics, heat transfer, and topology optimization. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |

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VNM | The polygonal domain is divided into several triangular regions which use the conventional FEM shape functions. These triangular regions are then coupled together by using mean least square shape functions. | 1. Do not require formulation of complex stress functions for the element (which could be difficult for some cases, as reported for stress based FEMs such as HPE and HS-F | 1. Integration within each tetrahedron can be simplified by mapping, but the mapping procedure imposes restriction to element geometry due to high aspect ratio (limited tolerance towards mesh distortion). | - | Adaptive computation, solid mechanics, and heat transfer phenomena. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |

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DGFEM | Problem domain is discretized into several polygonal cells which represent the polygonal elements. The interpolation for the elements is carried out based on a set of monomial functions which are totally independent of the element. | 1. Does not require any conforming shape function and the method is simple. | 1. Interpolation functions do not comply with shape function requirements of conventional FEM (Not compatible). | The element fulfills all the requirements of traditional FEM. | Solid mechanics, heat transfer, and eigenvalue problems on polygonal meshes. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |

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T-FEM or HT-FEM | Utilizes two different set of functions to approximate the solutions, one for the boundary and the other is for the interior domain. | 1. Able to produce more accurate results and higher convergence rates compare to the conforming PFEM with Laplace/Wachspress shape functions. | 1. T-complete sets for some problems are either complex or difficult to formulate. | - | Solid mechanics and heat transfer phenomena. Applicable for 2D problems for the time being. Applicable for both linear and nonlinear analyses. |

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BEM-based FEM | Trefftz-like | 1. Applicable to general polygonal meshes (immune to severe mesh distortion). | Large linear systems of equations are generated | - | Adaptive mesh generation, time dependent problems, and boundary value problems. Applicable for both 2D and 3D problems. Applicable for both linear and nonlinear analyses. |

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HS-F | Combination of principle of minimum complementary energy (similar principal used in VCFEM) with Airy stress function. | 1. Possess excellent performance compared to the conventional elements and especially independent of the element geometry. | 1. Not well desired for most of the engineering applications, due to the difficulties in obtaining suitable/compatible stress functions. | The element fulfills all the requirements of traditional FEM. | Solid mechanics phenomena. Applicable for 2D problems for the time being. Applicable for both linear and nonlinear analyses. |

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BFEM | Replaces the stress functions in the stress based FEMs with base forces which are easier to obtain. | 1. Able to approximate the strain and force fields accurately. | 1. The Lagrange multiplier method is used to deal with the equilibrium equation. So, the stiffness matrix is a full matrix. | - | Solid mechanics, bonding damage detection and damage mechanics. Applicable for 2D problems for the time being. Applicable for both linear and nonlinear analyses. |

Each method described above has been tested and analyzed by using computer programs such as MATLAB/Abaqus. These computer programs have been developed specifically for the purpose of testing and analyzing the proposed techniques. However, some of the methods have been well developed and made available as commercial software. This section described some of the software packages (either commercially available or for intended use only) which have been developed for polygonal/polyhedral FEM.

VCFEM has been incorporated into a software package known as Palmyra [

It is seen that currently there are few commercial software packages which are available for polygonal/polyhedral finite elements. However, software packages for other methods can be easily developed by incorporating the source codes developed by the researchers mentioned above with the available commercial polygonal mesh generators. Some of the software packages for polygonal/polyhedral mesh generation are Platypus (MATLAB based code) [

It can be seen that various finite elements have been proposed for engineering analysis. These elements have been proposed to facilitate meshing of the problem domain, to facilitate the analysis of physical phenomena, and to overcome drawbacks or limitations in the existing methods. This review enables the readers to identify advantages, disadvantages, and a comparison between the various techniques used in formation of polygonal/polyhedral finite elements.

The authors declare that they have no conflicts of interest.