ReviewArticle A Brief Review on Polygonal/Polyhedral...

Review ArticleA Brief Review on Polygonal/Polyhedral Finite Element Methods

Logah Perumal

Faculty of Engineering and Technology, Multimedia University, Jalan Ayer Keroh Lama, Bukit Beruang, 75450 Melaka, Malaysia

Correspondence should be addressed to Logah Perumal; [email protected]

Received 25 May 2018; Revised 26 August 2018; Accepted 13 September 2018; Published 4 October 2018

Academic Editor: Roberto G. Citarella

Copyright © 2018 Logah Perumal. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1. Introduction

Element equations are obtained by incorporating nodalconditions of the element geometry into the shape functions.One of the requirements is that the field variable obtainedfrom the element equations should be linear on the ele-ment boundaries. This requirement is met for triangularand quadrilateral elements, by selecting suitable linear (orbilinear) shape functions from the Pascal triangle [1]. It is alsonoted that the variation can be of higher order when a highernumber of nodes are used on each side of the element.

However, suitable first-order shape functions were notavailable for element geometries with more than four sidesuntil around the 1970s. Wachspress [2, 3] introduced a newtype of shape functions based on principles of perspectivegeometry known as Wachspress shape functions. Linearrelations within shape functions for elements with morethan four nodes are obtained by using rational functions.It can be seen that the shape functions consist of com-plex rational functions, which requires special integrationtechniques to solve. Wachspress method was revisited andgained more attention around the year 2000. Meanwhile,various methods have been proposed over the years to formpolygonal/polyhedral finite elements and to solve problemswithin polygonal/polyhedral meshes. These methods are asfollows:

(1) Voronoi cell finite element method (VCFEM)and polygonal finite element based on parametric

variational principle and the parametric quadraticprogramming method

(2) Hybrid polygonal element (HPE)(3) Conforming polygonal finite element method based

on barycentric coordinates (conforming PFEM, orPFEM)

(4) n-Sided polygonal smoothed finite element method(nSFEM)

(5) Polygonal scaled boundary finite element method(PSBFEM)

(6) Mimetic finite difference (MFD) and virtual elementmethod (VEM)

(7) Virtual node method (VNM)(8) Discontinuous Galerkin finite element method

(DGFEM)(9) Trefftz/Hybrid Trefftz polygonal finite element (T-

FEMorHT-FEM) and Boundary element based FEM(BEM-based FEM)

(10) Hybrid stress-function (HS-F) polygonal element(11) Base forces element method (BFEM)(12) Other recent techniques/schemes

The methods above are briefly highlighted in the followingsections.

HindawiMathematical Problems in EngineeringVolume 2018, Article ID 5792372, 22 pageshttps://doi.org/10.1155/2018/5792372

http://orcid.org/0000-0002-1241-1320https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2018/5792372

2 Mathematical Problems in Engineering

PrescribedTraction, T

Freeboundary

PrescribedDisplacement

Interelementboundaries

ΩmΩℎ

Ωe

Ωe

Ωh

Figure 1: A Voronoi cell element.

2. Review on Various Techniques ofPolygonal/Polyhedral Finite Elements

2.1. Voronoi Cell Finite Element Method (VCFEM). Aroundthe 1990s, Ghosh and Mukhopadhyay [4] and Ghosh andMoorthy [5] proposed a technique to model and simulatepolycrystalline ferroelectrics by using polygonal elements,which are formulated by using Voronoi cell. This method isknown as Voronoi cell finite element method (VCFEM).Thegrains in the microstructure are represented by Voronoi cells,which are generated through Voronoi tessellation. Each ofthese Voronoi cells (in the form of polygons with arbitrarynumber of sides) contains heterogeneity (in the form of voidor inclusions) and is treated as a single finite element. Figure 1shows a Voronoi cell element with heterogeneity (within theelement) and boundary conditions.

The matrix phase is denoted as Ω𝑚 and heterogeneityphase as Ωℎ. The heterogeneity surface and element surfaceare denoted as 𝜕Ωℎ and 𝜕Ω𝑒, respectively. The elementboundaries consist of three types, which are prescribedtraction/displacement boundary, free boundary, and interele-ment boundary.

VCFEM combines assumptions from micromechanicstheories and adaptive enhancements. This technique is com-putationally efficient compared to the conventional FEM(displacement based triangular or quadrilateral elements),since each polygonal grain is represented by a single finiteelement and no further subdivision of the domain is required.Stress functions for the interior of the element are obtained interms of polynomial expansions of the global coordinates andthese polynomial functions are formulated in such a way thatthey satisfy equilibrium within the element.

An example of VCFEM formulation for analysis ofCosserat materials based on the parametric minimum com-plementary energy principle is given as [6]

𝑒∏Ω

= ∫Ω

12𝑑𝜎𝑇𝑆𝑑𝜎 𝑑Ω + ∫Ω 𝜆𝑇𝑄𝑑𝜎𝑑Ω− ∫𝜕Ω𝑑𝑇𝑇𝑑𝑢𝑑𝑠,

(1)

whereΩ is the computational region of the element𝜕Ω represents boundary of the element𝜎 is a 6-by-1 matrix containing equivalent stress compo-nents𝑆 is a 6-by-6 inverse of the material property matrix(elastic compliance matrix)𝜆 is a matrix containing plastic flow parameters𝑄 is a matrix of partial derivative of flow potentialfunction with respect to the stress𝑢 and 𝑇 are the matrices for displacement and tractioncomponents along the element boundary, respectively𝑠 represents boundary surface

Stress functions for the interior of this element is definedby using Airy’s stress function, Φ and the Mindlin stressfunction, Ψ in terms of polynomial expansions of the globalcoordinates [6]:

Φ = 𝛽1𝑥2 + 𝛽2𝑥𝑦 + 𝛽3𝑦2 + 𝛽6𝑥3 + 𝛽7𝑥2𝑦 + 𝛽8𝑥𝑦2+ 𝛽9𝑦3 + 𝛽13𝑥4 + 𝛽14𝑥3𝑦 + 𝛽15𝑥2𝑦2 + 𝛽16𝑥𝑦3+ 𝛽17𝑦4 + ⋅ ⋅ ⋅

(2)

Ψ = 𝛽4𝑥 + 𝛽5𝑦 + 𝛽10𝑥2 + 𝛽11𝑥𝑦 + 𝛽12𝑦2 + 𝛽18𝑥3+ 𝛽19𝑥2𝑦 + 𝛽20𝑥𝑦2 + 𝛽21𝑦3 + ⋅ ⋅ ⋅ , (3)

where 𝛽𝑖 (𝑖 = 1, 2, . . . 𝑛) are the undetermined coefficients.VCFEM has been found to perform poorly when the

heterogeneity is in the form of voids (but works well forinclusions), due to the poorly defined stress functions withinthe interior of the element. This problem is solved bytaking into account the geometry effects through conformalmapping [7].

Later, Sze and Sheng [8] included other characteristicfeatures to the VCFEM by incorporating the electromechan-ical Hellinger–Reissner principle. Since then, the VCFEMhas been revisited, extended, and implemented in differentapplications such as in analysis of heterogeneous materials [5,9, 10], determination of the effective elastic properties/crackanalysis of functionally graded materials [11, 12], analysisof microstructural Representative Volume Element (RVE)[13], simulation of the crack and analysis/failure analysis ofcompositematerials [7, 14–19], andmultiscale simulations formicrostructural modelling [9]. The method is also extendedto 3D [14, 20, 21].

Hybrid of VCFEMwith other methods such as numericalconformal mapping (NCM) method can be seen in thework by Tiwary, Hu, and Ghosh [22]. The hybrid techniqueis known as NCM-VCFEM and it is developed so thatreal micrographs of heterogeneous materials with irregu-lar shapes can be analyzed effectively. The effectiveness isattained by using NCMsuch as Schwarz–Christoffel mappingto convert arbitrary/irregular shapes of micrographs into aunit circle. Another example is coupling of VCFEM withparametric variational principle and the parametric quadraticprogramming method [6, 23].This approach is implementedin order to incorporate the constitutive relations of thephysical phenomenon and to generalize the variational

Mathematical Problems in Engineering 3

C1C2

b

a

y

x

r=1

Figure 2: Mapping of a Voronoi cell element.

principle. The new formulations are found to be able toproduce good solutions, competitive with that of conven-tional FEM package in ANSYS and with fewer nodes (whichreduces the computational cost). Simultaneously, Zhang et al.[23] developed a polygonal finite element by using similarapproach and named it as parametric variational principlebased polygonal finite element method (PFEM). PFEM issimilar to conventional displacement based finite elementsin the sense that PFEM is applicable for macroanalysisand compatible interpolation functions are used for theentire element domain (boundaries as well as interior of theelements).

VCFEM is generally suitable for micromechanical analy-sis and multiscale modelling [24].

2.2. Hybrid Polygonal Element (HPE). On the other hand,Zhang and Katsube [25] proposed a different method toanalyze micromechanical properties of heterogeneous mate-rials. The authors used the hybrid stress element method [26]withMuskhelishvili’s complex analysis approach to formulatepolygonal elements. This method is known as the hybridpolygonal element (HPE) method [27] and this technique ispursued, since the stress variation around the heterogeneity(inclusions) is not well defined in the VCFEM [28]. Knowingthis, HPE has later been incorporated into VCFEM as well[20]. In the HPE method, the compatible interpolationfunctions are formulated along the boundaries only, whilethe interior of the element is represented by self-equilibratingstress field.

Figure 2 shows a HPE in global 𝑥-𝑦 coordinate systemand its equivalent mapping to a standard ellipse in reference𝜂-𝜉 coordinate system.

An example of a hybrid functional ∏(𝐻)𝑒 for a HPE withhole (void) is given as [25]

(𝐻)∏𝑒

= 12 (∮𝐶1 𝑇𝑇𝑢𝑑𝑠 − ∮𝐶2 𝑇𝑇𝑢𝑑𝑠) − ∮𝐶1 𝑇𝑇�̃�(1)𝑑𝑠, (4)where𝑢 and 𝑇 are the matrices for displacement and tractioncomponents along the element𝐶1 and 𝐶2 represent the element’s outer boundary withadjacent elements and the inner interface between the matrixand heterogeneity, respectively

�̃�(1) represents the components of the specified displace-ments along 𝐶1𝑠 represents boundary surface

The interior stresses are obtained by using interpolationfunctions (in the form of trigonometric functions) throughthe following relation [25]:

{{{{{𝜎11𝜎22𝜎12}}}}}= 12𝜇

⋅ 𝑚𝑢∑𝑘=𝑚𝑏

[[[2𝐶1 − 𝐶2 + 𝐶3 −2𝐷1 + 𝐷2 − 𝐷3 −𝐶1 𝐷12𝐶1 + 𝐶2 − 𝐶3 −2𝐷1 − 𝐷2 + 𝐷3 𝐶1 −𝐷1𝐷2 − 𝐷3 𝐶2 − 𝐶3 𝐷1 𝐶1

]]]

⋅{{{{{{{{{{{{{

𝑎𝑘𝑎𝑘�̃�𝑘�̂�𝑘

}}}}}}}}}}}}},

(5)

where 𝐶1 to 𝐶3 and 𝐷1 and 𝐷3 are the functions of 𝑘, 𝑟, and𝜃. 𝑚𝑢 and 𝑚𝑏 are the upper and low limits of the series. 𝑟and 𝜃 are the parameters that resulted due to the mappingof the computational/matrix region Ω to the elliptical regionin reference system. Expression for 𝐶1 is shown here as anexample:

𝐶1 (𝑘, 𝑟, 𝜃)= 𝑘𝑟𝑘−1 (𝑓1 cos (𝑘 − 1) 𝜃 − 𝑓2 sin (𝑘 − 1) 𝜃) , (6)

where 𝑓1 and 𝑓2 are functions in terms of 𝑟, 𝜃, 𝑎, and 𝑏. 𝑎and 𝑏 represent the major andminor semiaxes of the mappedellipse.

Similar functions are used to determine the displace-ments within the element.

Application of HPEs for the analysis of heterogeneousmedia in 2D as well as 3D can be seen in works by Zhang andKatsube [25] and Kaliappan and Andreas [29]. Wachspressshape functions were not utilized in these elements (in HPEand VCFEM), due to the difficulties in integrating the com-plex rational functions [28]. A limitation of the VCFEM andHPE methods is that the resulting polygonal elements cancontain only one irregular phase (void/inclusion) within theelement. Due to this, the extended multiscale finite elementmethod [30] has been developed to analyze mechanicalbehaviors of heterogeneous materials with randomly dis-tributed polygonal microstructure. Another version of HPEfor plane linear elasticity problems with better performancecompared to conventional FEM is presented in [31]. Thepolygonal meshes are generated based on the MATLAB codePolyMesher which operates based on Voronoi diagrams.

2.3. Conforming Polygonal Finite Element Method Based onBarycentric Coordinates (Conforming PFEM). Around theyear of 2000, the Wachspress method gained more atten-tion and was revisited alongside with other techniques toformulate interpolation or shape functions for polygonal


elements. These techniques are known as barycentric coor-dinates method [32, 33] and they yield complex shape func-tions consisting of rational, logarithmic, and trigonometricfunctions. Recently, polynomial spline functions (BernsteinBezier functions) have been proposed to be included in thebarycentric coordinates method [34]. Polygonal elementswhich are formed by these methods are implemented inFEM and known as conforming polygonal finite elementmethod (conforming PFEM, or simply PFEM). Some of thebarycentric coordinates methods used in the formulationof conforming PFEM are inverse bilinear coordinates [35],Wachspress [32, 36–40], mean value coordinates [41, 42],harmonic coordinates [43], maximum entropy coordinates[44–46],metric coordinates [47], and natural neighbor-basedcoordinates (Laplace shape functions) [33]. Some of themethods such as Wachspress, mean value, and harmoniccoordinates have been extended to 3D [48–52]. The abovementioned barycentric coordinate methods are describedbelow.

Inverse bilinear coordinates were developed for quadri-laterals, based on bilinear mapping of a unit square to convexquadrilaterals. Rational functions are used for the mappingand their inverses were studied to develop the inversebilinear coordinates for quadrilaterals.Wachspress developedrational polynomial functions which can be used to produceconforming shape functions for arbitrary polygons. Meyeret al. [32] modified the Wachspress coordinates by replacingthe adjoint with triangle areas and rewrote the barycentriccoordinates in a simpler form. Similarly, other simplificationswere carried out onto the Wachspress coordinates suchas representing the Wachspress coordinates in terms ofperpendicular distance between two points [32], redefiningthe adjoint polynomials by other means [48], and so on.Advantages of Wachspress coordinates over inverse bilinearcoordinates are that the Wachspress method is applicablefor arbitrary polygons and do not contain square root terms[35]. However, Wachspress’ rational shape functions do notperform well for concave polygonal elements.

This shortage is avoided inmean value coordinates, whichare written in terms of trigonometric functions. Mean valuecoordinates can be adapted to complex arbitrary polygons,especially star-shaped geometries (concave polygons). It isnoted that shape functions for concave polygonal elementscannot be represented by rational polynomial functions, sinceconvex shapes (can be represented by rational polynomialfunctions) cannot be mapped to concave shapes [47]. Meanvalue coordinates are useful and vastly applied in param-eterizing triangular meshes and surface fitting [35]. Thesebarycentric coordinates are found to be robust and applicablefor concave polygons as well, even though the method doesnot guarantee positive functions for all the cases, since itis bounded only for star-shaped geometries [47]. On theother hand, apart from being linearly precise, harmonicand maximum-entropy coordinates are guaranteed positivefor both convex and concave polygons [46]. Another suit-able barycentric coordinate which satisfies the boundednessrequirement for both convex and concave polygonal elementsis the metric coordinate method [53]. A general framework toconstruct barycentric coordinates was proposed by Floater,

Hormann, and Kos [54]. This framework reproduces thebarycentric coordinates under various values of coefficient 𝑐in the formulation.

Motivated by mesh-free method, authors of references[33, 47] developed natural neighbor-based coordinates (alsoknown as Laplace shape functions) for arbitrary polygonalelements. The development is based on natural neighbor-based schemes within a Voronoi cell. The method was latertested for utilization in FEM, together with Wachspress, met-ric, and mean value coordinates. Simulation results showedthat the Laplace interpolant is simpler and computationallyattractive and yields more accurate results compared tothe rest for convex and weakly convex polygonal elements.Nonetheless, the Laplace interpolant is not suitable forconcave elements and best results for convex elements areattained through mapping of parent element [33, 47].

Polygonal elements based on barycentric coordinateshave been implemented in various areas such as computergraphics, animation and geometric modelling [55, 56], topol-ogy optimization [57], surface parameterization [58], geo-metric modelling [41], analysis of a plate with a circular hole[59], crack growth modelling [60], contact-impact problems[61], mesh generation, material fracture [62], finite elasticityproblems [63], modelling of rock materials [64], and so on.Recently barycentric coordinates have been implementedin static and free vibration analyses of laminated compos-ite plates [65], multimaterial topology optimization [66],Reissner-Mindlin plate problems [67], and transient heatconduction problems [68]. Other barycentric coordinatesmethods have been investigated such as Poisson coordinates[69], Green coordinates, reconstructions of Green coordi-nates by using Cauchy’s theorem, moving least squares coor-dinates, and attempt to designnewmethods through complexrepresentation of real-valued barycentric coordinates [35].

However, evaluation of barycentric coordinates is neithersimple nor efficient compared to the conventional displace-ment based FEM, due to the complex functions which arise inthe former techniques. Furthermore, barycentric coordinatesare not efficient for assembling the stiffness matrices associ-ated with weak solutions of Poisson equations [34].

Construction of shape functions 0𝑗 based on Wachspress[47] for a polygonal element in the 𝑥-𝑦 coordinate systemaccording to Figure 3 is given as

0𝑗 (𝑥, 𝑦) = 𝑤𝑗 (𝑥, 𝑦)∑𝑛𝑘=1𝑤𝑘 (𝑥, 𝑦) ,𝑤𝑗 (𝑥, 𝑦) = 𝐴 (𝑝𝑖, 𝑝𝑗, 𝑝𝑘)𝐴 (𝑝, 𝑝𝑖, 𝑝𝑗)𝐴 (𝑝, 𝑝𝑗, 𝑝𝑘) ,

(7)

where𝐴 represents area enclosed by the three nodes within thebracket𝑃𝑖, 𝑃𝑗, and 𝑃𝑘 represent a particular external/surface nodeof the polygonal element𝑃 represents inner node of the polygonal element


ΨＤ Pj

P

Pk

Pi

Ｄ

Figure 3: Construction of shape functions based on Wachspress.

i

j

i

j

Pj

P

Pk

Pi

j

i

Figure 4: Construction of shape functions based on mean valuecoordinates.

The expression for 𝑤𝑗(𝑥, 𝑦) in (7) can be rewritten byusing the angles formed (𝜑 and Ψ) between the nodes [47]as

𝑤𝑗 (𝑥, 𝑦) = 2( cot𝜑𝑗 + cotΨ𝑗(𝑥 − 𝑥𝑗)2 + (𝑦 − 𝑦𝑗)2) (8)The expression for 𝑤𝑗(𝑥, 𝑦) in (7) based on mean valuecoordinates [47] for a polygonal element is given as

𝑤𝑗 (𝑥, 𝑦) = tan (∝𝑖/2) + tan (∝𝑗/2)√(𝑥 − 𝑥𝑗)2 + (𝑦 − 𝑦𝑗)2, (9)

where ∝ is the angle that is formed within the triangularpartitions at the inner node, as shown in Figure 4.

The expression for 𝑤𝑗(𝑥, 𝑦) in (7) based on the conceptof natural neighbors (Laplace shape functions) [47] for apolygonal element according to Figure 5 is given as

𝑤𝑗 (𝑥, 𝑦) = 𝑠𝑗 (𝑥, 𝑦)ℎ𝑗 (𝑥, 𝑦) (10)where𝑠𝑗 represents length of the Voronoi edge and ℎ𝑗 =√(𝑥 − 𝑥𝑗)2 + (𝑦 − 𝑦𝑗)2.

1

h6

h5

h4 h3h2

h1

s6

s5

s4 s3

s2

s1

6

5

43

2

Figure 5: Construction of shape functions based on the concept ofnatural neighbors.

1

6

5

43

2Ωsk

0

Figure 6: Partitioning of a nSFEM element into subtriangles.

2.4. 𝑛-Sided Polygonal Smoothed Finite Element Method(nSFEM). Another attempt to form polygonal finite elementmethod can be seen within the smoothed finite elementmethod (SFEM). SFEM is formed by merging conventionalFEM with meshless methods. SFEM was initially formedfor quadrilateral elements. Later, Dai, Liu, and Nguyen [70]extended the four-node quadrilateral smoothed elements toarbitrary sides termed as n-sided polygonal smoothed finiteelements (nSFEM) and implemented the method in solidmechanics (macrolevel).

In nSFEM, the polygonal element is divided into severalsmoothing cells in the form of triangles, which share acommon node at the center of the polygon. These trianglesknown as smoothing cells are then subjected to smoothingtechniques onto the strain components.

An example of nSFEM element is shown in Figure 6.Point 0 in Figure 6 represents the center of the polygonal

element. The displacement at this point, 𝑑0 is taken as theaverage of displacement of all the external nodes, given bythe following equation [71]:

𝑑0 = 1𝑛𝑛∑𝑝=1

𝑑𝑝, (11)where 𝑑𝑝 represents displacement at a particular node 𝑝and 𝑛 represents total number of nodes of the polygon. The


displacement within a particular subtriangle 0-1-6 (subtrian-gle 1) can then be represented by [71]:

𝑢𝑖,1 = 𝑁1𝑑1 + 𝑁2𝑑2 + 𝑁3𝑑0, (12)where 𝑁𝑗 (𝑗 = 1, 2, 3) represents the conventional shapefunctions for a 3 nodes’ triangular element. Substituting (12)into (11) and simplifying gives

𝑢𝑖,1 = (𝑁1 + 1𝑛𝑁3) 𝑑1 + (𝑁2 + 1𝑛𝑁3) 𝑑2 + 1𝑛𝑁3𝑑3+ ⋅ ⋅ ⋅ + 1𝑛𝑁3𝑑𝑛−1 + 1𝑛𝑁3𝑑𝑛

(13)

The shape functionmatrix for subtriangle 0-1-6 then becomes

𝑁= {(𝑁1 + 1𝑛𝑁3) (𝑁2 + 1𝑛𝑁3) (1𝑛𝑁3) ⋅ ⋅ ⋅ ( 1𝑛𝑁3) (1𝑛𝑁3)} (14)

The strain components are smoothed according to [71]

𝜀 (𝑥) = 𝑛∑𝐼=1

𝐵𝐼 (𝑥, 𝑦) 𝑑𝐼 (15)𝐵𝐼 (𝑥, 𝑦) = 1𝐴𝑠

𝑘

∫Ω𝑠𝑘

𝐵𝐼 (𝑥, 𝑦) 𝑑Ω, (16)where𝐴𝑠𝑘 represents area of smoothing domain,𝐵𝐼 representsthe conventional compatible strain displacement matrix, andΩ𝑠𝑘 represents the smoothing domain.

There are three types of smoothing techniques applicablefor these smoothing cells, which are cell, node, and edgebased.These techniques are known as n-sided polygonal cell-based smoothed FEM (nCS-FEM) [70], n-sided polygonaledge-based smoothed FEM (nES-FEM) [72], and n-sidedpolygonal node-based smoothed FEM (nNS-FEM) [73, 74],respectively.

In case of nCS-FEM, the smoothing is carried out byintegrating the gradient of displacement over the particularsmoothing cell’s triangular area (for 2D as shown in Figure 6),individually [71]. When combined with the conventionalFEM to obtain the displacements, the area integration isreduced to surface integration for the case of a constantsmoothing function. The smoothed stiffness matrix for anelement is then obtained by summing up individual stiffnessmatrices of all the smoothing cells within the polygonalelement. Shape functions are derived for each side/edge of thesmoothing cells by using the two nodes that make up the par-ticular side/edge. These shape functions for the sides/edgesshould be compatible, since these sides coincide with eachother to form the polygonal element. Shape functions for theinterior of the smoothing cells are obtained by using othermethods such as PFEM or mesh-free techniques [70]. Incase of nES-FEM, the smoothing is done based on particularedge/side (instead of cell as in nCS-FEM) of the smoothingcells within the polygonal element as shown in Figure 7. Thesmoothing is done by performing the integration over severalcell domains which are linked to the particular edge or side.

Ωsk

0

CK

D

Figure 7: Smoothing domain Ω𝑠𝑘 (OCDKO) for nES-FEM.

Ωsk

C

B

A

F

DE

Figure 8: Smoothing domain Ω𝑠𝑘 (ABCDEFA) for nNS-FEM.

Therefore, the integration domain may extend to smoothingcells of adjacent elements as well (since a particular side/edgecan be shared by adjacent elements) [71]. In case of nNS-FEM, the smoothing is done based on a particular node ofthe smoothing cells within the polygonal element as shownin Figure 8. Similarly, the smoothing is done by performingthe integration over several cell domains which are linkedto the particular node and therefore, the integration domainmay also extend to smoothing cells of adjacent elements(since a particular node can be shared by adjacent elements)[71].

nCS-FEM has many advantages over the conventionalFEM such as the fact that stability provides accurate resultsas compared to the conventional FEM. This is because thestiffness matrices of nCS-FEM tend to be less stiff and can beapplied for nearly incompressible materials by using selectiveintegration schemes to avoid volumetric locking phenomena[70]. However, for solid mechanics, nCS-FEM is proposedto be used only for regions near the boundary or veryirregular parts. This is because use of these elements forinterior regions would increase the number of nodes and


eventually increases the computational cost [70]. Advantageof nNS-FEM is that it is immune from volumetric lockingphenomena. Disadvantage of nNS-FEM is that the compu-tational time is longer compared to conventional FEM forthe same number of global nodes, due to larger bandwidthof stiffness matrices. Disadvantage of nES-FEM is that thereis a tendency to overestimate or underestimate the strainenergy of themodel for some cases. Apart from that, similarlyto nNS-FEM, nES-FEM requires more computational timecompared to conventional 3-node triangular elements dueto the larger bandwidth [72]. Comparison between the threetypes of nSFEM is provided by Nguyen et al. [72], for solidmechanics problem. It is shown that nES-FEM providesmost accurate solution compared to the others and thestiffness/softness of the model of nES-FEM is in between theother two techniques. Combination of nES-FEM and nNS-FEM (termed as nES/NS-FEM) to avoid volumetric lockingand to achieve faster convergence can be seen in the literature[72, 75]. Applications of nSFEM can be seen in determinationof upper bound solutions to solid mechanics problems [73],fluid-solid interaction problems [75], and new application inanalysis of elastic solids subjected to torsion [76]. Recently,nSFEM has been implemented for the analysis of fluid-solid interaction (FSI) problems in viscous incompressibleflows together with sliding mesh [77]. Simulation resultsshowed that the method performs better compared to theconventional finite elements. Major advantage of the nSFEMin FSI is that it is capable of performing independent domaindiscretization.

Generally, the nSFEM is advantageous over the con-ventional FEM since it produces more accurate solutions,able to tolerate volumetric locking and does not requireisoparametric mapping (which enables the elements to takearbitrary shape: concave and convex forms). Apart from that,the shape functions consist of polynomials, which are easierto evaluate compared to the conforming PFEM based onbarycentric coordinates. Extension of the method to 3D canbe seen in literatures [78–80].

2.5. Polygonal Scaled Boundary FEM (PSBFEM). Scaledboundary FEM is a semianalytical method which combinesthe boundary element method (BEM) and FEM. It wasfirst introduced by Song and Wolf [81], and it was soonextended to polygonal elements and named as PolygonalScaled Boundary FEM (PSBFEM). Example of a PSBFEMelement is shown in Figure 9. PSBFEM works based onscaling center, which is located at the center of the polygonalelement. The scaling center is located within the polygonalelement (usually at the center) in such a way that all theboundaries/sides of the polygonal element are visible fromthis scaling center. Radial lines are formed from the scalingcenter to the outer nodes of the polygonal element, and theselines are assigned value of zero at the center (scaling node)and reach value of 1 at each node. This is accomplishedthrough implementation of radial coordinate system. Simi-larly, each boundary/side of the polygonal element is assignedvalue of 1 to -1, through implementation of local coordinatesystem. The boundaries are represented by conventionalnumerical line elements of FEM.

Radial line

Line element

Similar curve for = 0.5

Scaling center

=

0.5

=

1

1

2

3

4

5

6

0

(x2 , y2)

(x1 , y1)

s = 1

s = -1

s

Figure 9: A PSBFEM element.

Solution along the radial direction is obtained by ana-lytical expressions by using m number of shape functions,where m represents number of nodes of the polygonal ele-ment. Transformation between theCartesian coordinates andscaled boundary coordinates (radial coordinate system andlocal coordinate system) is accomplished through isopara-metric mapping similar to the conventional FEM. The map-ping which describes scaling of the boundary has led to thenameof themethod. Equation (17) [82] shows transformationof coordinate system (transformation between the scaledboundary coordinate system 𝜉, 𝑠 and Cartesian coordinatesystem x, y) for a point within the triangular subdomain0-1-2 (as shown in Figure 9). Similar transformation isdone for any point within any particular triangular subdo-main.

𝑥 = 𝜉 [𝑁 (𝑠)] {𝑥𝑏}𝑦 = 𝜉 [𝑁 (𝑠)] {𝑦𝑏} , (17)

where𝑁(𝑠) = [ 𝑁1(𝑠) 𝑁2(𝑠) ] = [ (1/2)(1 − 𝑠) (1/2)(1+ 𝑠) ]represents shape functions of the line element and {𝑥𝑏} ={ 𝑥1𝑥2 } and {𝑦𝑏} = { 𝑦1𝑦2 } represent the coordinates of the bound-ary nodes enclosed by the specific triangular subdomain(subdomain 0-1-2).

The displacement 𝑢(𝜉, 𝑠) within a particular triangularsubdomain of the element is interpolated through utilizationof similar shape functions as shown in (18) below [82]:

𝑢 (𝜉, 𝑠) = [𝑁 (𝑠)] {𝑢 (𝜉)} , (18)where 𝑢(𝜉) = { 𝑢1(𝜉)𝑢2(𝜉) } represent the displacements on the lines(radial lines) passing through the scaling center, 0, and nodes1 and 2, respectively.

Various applications of PSBFEM can be seen in literaturesuch as in linear elasticity [83], crack propagation [84–88],applications within geotechnical structures [89], dynamicfracture simulation [90], analysis of cracked functionallygraded materials [91], polygonal mesh creation (Song et al.,2017), elastoplastic analysis of structures [92], predictionof structural responses with randomly distributed materialproperties [93], simulation of crack surface contact problems


[94], and analysis of mesoscale concrete samples [95]. Exten-sion of themethod to 3D can be seen in the literature [96–98].Advantages of PSBFEM compared to BEM and conventionalFEM are that, in PSBFEM, analytical solutions are achievedinside the domain, discretization of free and fixed boundariesand interfaces between different materials are not required,and the calculation of stress concentrations and intensityfactors based on their definition is straightforward [82].PSBFEM exhibits some disadvantages [82]. PSBFEM is notdirectly applicable for unbounded domains with stronglyinclined interfaces, due to the difficulty in selecting a scalingcenter within the body which is visible to all sides of thedomain. Some modifications are needed for these casessuch as the introduction of redundant nodes for subdomaincreation or by moving the boundaries upward in orderto create a single scaling center (alteration of the actualphysical problem). In case of time dependent problems,PSBFEM cannot be directly used to process transient excita-tion as opposed to BEM. Unit impulse response matrices arerequired and there is a need for convolution integrals whichincreases the computational effort. It is also found that thismethod is not as efficient as conventional FEM or BEMwhensolving problems involving smooth stress variations withinbounded/enclosed domain. However, PSBFEM yields highlyaccurate solutions for problems involving stress singularities.The stress singularity refers to a particular point within thedomain in which the stress does not converge to a specificvalue.

PSBFEM has been found to be superior to other tech-niques such as nSFEM and conforming PFEM within thecontext of linear elasticity and the linear elastic fracturemechanics [83].

2.6. Mimetic Finite Difference (MFD) Method and VirtualElement Method (VEM). One of the difficulties faced in theconstruction of polygonal finite elements is the developmentof interpolation functions which extends to the interior ofthe element. Beir ao da Veiga, Gyrya, Lipnikov, and Manzini[99] implemented a mimetic finite difference (MFD)methodto polygonal mesh and showed that the method is efficientin solving problems involving polygonal meshes, since themethod uses only the surface representation of discreteunknowns and therefore the formulations are simpler.

For example, consider a heat conduction phenomenonwhich is governed by the following governing equation (19)[100]:

−div (𝐾∇𝑢) = 𝑞, (19)where K represents the full conductivity tensor, u representsthe temperature, and q is the forcing term indicating thesource of heat. The crucial step in MFD method is to mimicthe essential properties of the physical and mathematicalmodel above, which can be achieved through Green formula:

∫Ω𝐾−1 (𝐾∇𝑢) ⋅ →𝐹𝑑𝑥 = −∫

Ω𝑢 div→𝐹𝑑𝑥

+ ∫𝜕Ω𝑢0→𝐹 ⋅ →𝑛𝑑𝑥,

(20)

Figure 10: Low order degrees of freedom for MFD method. Thearrows represent fluxes and the center represents the temperature.

where →𝐹 represents the heat flux. Subsequent step isimplementation of the finite difference discretization, whichinvolves discretization of scalar functions, vector functions,and the differential operators div and 𝐾∇. Discretizationof scalar and vector functions is accomplished throughintroduction of degree of freedom. Example of degrees offreedom for low order MFD method is shown in Fig-ure 10.

An additional step is necessary, which is the discretizationof the integrals. This step is required in order to approximatediv and 𝐾∇ (differential operators) according to mimeticapproach.

The method is also shown to be useful for meshes withdegenerate and nonconvex polygonal elements. Since then,MFD method has been implemented in various problems(diffusion/convection-diffusion [101–104], electromagneticfield problems [105, 106], elasticity problems, Lagrangianhydrodynamics problems [107], and solving wave equations[108]) and for modelling fluid flows [109, 110]. MFD hasbeen extended to higher order [100, 102] and also to 3D[102, 105, 111, 112].

Beir ao Da Veiga, Brezzi, Cangiani, Manzini, Marini, andRusso [113] mentioned that it was quite difficult to presentMFD due to nonexistence of trial functions for the interiorof the element. Therefore, the MFD has been generalizedand reintroduced as virtual element method (VEM). InVEM, the unknown degrees of freedom are attached totrial functions within the interior of the polygonal domain(which do not exist earlier in MFD). Three variants ofVEM are presented by Russo [114], with different numberof these internal degrees of freedom. This approach is nowsimilar to conventional PFEM. This similarity opened thepossibility of coupling VEMwith the conforming PFEM.Theresulting hybrid method (VEM-PFEM) has been found topossess high accuracy and, most importantly, the difficultiesfaced in the integration of complex functions resulting frombarycentric coordinates in PFEM are entirely avoided [62].Applications of VEM [115] can be seen in plate bending prob-lems, elasticity problems, Stokes problems, Steklov eigenvalueproblems, finite strain plasticity problems [116], hyperbolicproblems [117], and topology optimization problems [118].Extension of VEM to 3D can be seen in literatures [119,120].


The virtual element space on a polygonal domain (thatdiscretizes the problem domain) is defined as [113]

𝑉𝐾,𝑘= {V ∈ 𝐻1 (𝐾) : V|𝜕𝐾 ∈ B𝑘 (𝜕𝐾) , V|𝐾 ∈ P𝑘−2 (𝐾)} , (21)

where 𝑉𝐾,𝑘 represents the finite dimensional space, K rep-resents a generic polygonal element within the domain, krepresents polynomial degree of the virtual element scheme,V represents displacement field, 𝐻1(𝐾) represents the spacecontaining K, 𝜕𝐾 represents a generic edge on the polygon,B𝑘(𝜕𝐾) represents the set of polynomials of degree less thanor equal to k on 𝜕𝐾, and P𝑘−2(𝐾) denotes the space ofpolynomials of degree less than or equal to k-2 on K. For k= 1, the trial functions are linear on each edge and the insideof the element is represented by harmonic functions. For k =2, the trial functions on each edge are made of polynomialsof degree less than or equal to 2. The inside of the element iscomposed of polynomials with constant Laplacian.

A similarity between VCFEM, HPE, MFD, and VEM isthat thesemethods donot require the extension of compatibleinterpolation functions to the interior of the element. Thecompatible interpolation functions are required for the ele-ment boundaries only, which simplifies the formulation andenables the formulation of elements with arbitrary numberof sides/nodes. Disadvantage of MFD and VEM is thatthey involve complex procedures and therefore require highcomputational effort [121].

2.7. Virtual Node Method (VNM). Another attempt to over-come the difficulty in forming compatible shape functionsfor polygonal FEM can be seen in the literature [122]. Theauthors presented a novel polygonal FEMwhich uses a virtualnode at the center of the element to formulate shape functionsconsisting of simple polynomials. The method was proposedas an alternative to the conforming PFEM which sufferedfrom inaccuracy resulting from the integration of complexfunctions in the stiffness matrix. First, a polygonal elementwith arbitrary number of nodes is formed. The polygonaldomain is then divided into several triangular regions, whichshare a common node at the center of the element (denotedas the virtual node), as shown in Figure 11.

Compatibility among each triangular region is fulfilled byusing the conventional FEM shape functions of a 3 nodes’triangular element.These triangular regions are then coupledtogether (to form the polygonal element) by representing thevirtual node at the center in terms of mean least square shapefunctions, as shown by (22) [122, 123]:

𝜑[𝑇V] = 𝜙[𝑇V]𝑖 𝜑𝑖 + 𝜙[𝑇V]𝑗 𝜑𝑗 + 𝜙[𝑇V]𝑘 𝜑𝑘, (22)where 𝜑 represents unknown field variable at a particularpoint/node, 𝜑 represents the field variable at the virtual node,𝜙𝑖, 𝜙𝑗, and 𝜙𝑘 are the conventional FEM shape functions ofa 3 nodes’ triangular element, and 𝑇V represents a particulartriangular element within the polygonal domain. For triangle

1

6

5

43

2

T

T

1

4

7T5

T3

T6

T2

Figure 11: Partitioning of a polygon according to VNM.

𝑇1 (an example), 𝑇V = 𝑇1, 𝑖 = 1, 𝑗 = 2, and 𝑘 = 7. Equation(22) above can be written as

𝜑 = 𝜙𝑖𝜑1 + 𝜙𝑗𝜑2 + 𝜙𝑘𝜑7 (23)Field variable at the internal node (virtual node) 𝜑7 can berepresented by the least mean square [123]:

𝜑7 = 𝑛∑𝑚=1

𝑁𝑚𝜑𝑚 = 𝑁𝑇𝜑, (24)where 𝑛 represents total number of nodes of the polygonalelement,𝑁𝑇 = 𝑝𝑇(𝑥, 𝑦)𝐴−1𝐵,𝑝 is a set of basis functions, and𝐴 and 𝐵 are basis matrices. Equation (24) can be rewritten interms of shape function for a particular node:

𝜙[𝑇V]𝑚 = (𝜙[𝑇V]𝑖 + 𝜙[𝑇V]𝑗 )⋅ [(𝜙[𝑇V]𝑖 𝜑𝑖 + 𝜙[𝑇V]𝑗 𝜑𝑗) + 𝜙[𝑇V]𝑘 𝑁𝑚 (𝑥𝑘, 𝑦𝑘)]+ 𝜙[𝑇V]𝑘𝑁𝑚

𝜑𝑖 = {{{1, 𝑖𝑓 𝑚 = 𝑖,0, 𝑖𝑓 𝑚 ̸= 𝑖;

𝜑𝑗 = {{{1, 𝑖𝑓 𝑚 = 𝑗,0, 𝑖𝑓 𝑚 ̸= 𝑗.

(25)

Stiffnessmatrix for a particular polygonal element is obtainedby summing up stiffness matrices of the individual triangularregions. Next, global stiffness matrix is obtained by summingup all the stiffness matrices of the individual polygonalelements. These are shown by (26) and (27) below:

𝑘𝑒 = 𝑛∑V=1𝑘𝑇V (26)

𝐾 = 𝑙∑𝑒=1

𝑘𝑒, (27)where l represents total number of elements in the mesh. Itcan be seen that this method is quite similar to nCS-FEM


in terms of the partitioning of the domain into triangularregions/cells, integration is performed on each triangularregion/cell individually, and summing of stiffness matricesof each triangular region/cell to obtain the element stiffnessmatrix. The difference is that strain smoothing is not carriedout in VNM.Themethod is later extended to higher order byOh and Lee [124].

The method is advantageous compared to compatiblePFEM due to the simple polynomial shape functions (whichis easier to work with). VNM is found to be efficient inadaptive computation, by using quadtree or octree mesh.Themethod has been extended to 3D polyhedral (VPHE) andhexahedral forms and implemented in adaptive computationsas can be seen in the literature [123, 125, 126]. Recently,the method has been coupled with extended FEM (XFEM)[127, 128].

2.8. Discontinuous Galerkin FEM (DGFEM). This methodwas proposed due to the difficulties faced in executing otherpolygonal FEMs such as compatible PFEM, MFD, and VEM.The difficulties arise due to the complicated shape functionsin compatible PFEM and complex procedures involved inMFD and VEM. These complicated entities demand highcomputational effort as well [121]. DGFEM, on the otherhand, does not require any conforming shape function andthe method is simple. Application of DGFEM in polygonalmeshes can be seen in the literature [129–135]. Extension ofthe method to 3D can be seen in the literature [136, 137].

In DGFEM, the problem domain is discretized into sev-eral polygonal cells which represent the polygonal elements.The interpolation for the elements is carried out based ona set of monomial functions which are totally independentof the element. These interpolation functions do not complyto shape function requirements of conventional FEM andtherefore they are not continuous across different elements(not compatible). Due to this, integrations are carried out onthe boundaries of each cell. This step is an addition com-pared to the conventional FEM. The degrees of freedom areobtained based on the coefficients of the linear combinationof the monomial functions [138]. Additional advantages ofthe method are that the adjacent elements can be of differentorder and the elements do not need to be conforming.

Interpolation of field variables Θ within the element isachieved through [130]

Θ = {𝑁} {Θ} , (28)where {Θ} = [[

Θ1

Θ2

...Θ𝑁𝑒

]] represents matrix of the local degreesof freedom, Ne represents the number of polygonal elements,{𝑁} = [𝑠1𝑏1 𝑠2𝑏2 . . . 𝑠𝑁𝑒𝑏𝑁𝑒] represents matrix containingthe incompatible interpolation functions b, and 𝑠 representsthe element support [130]:

𝑠 = {{{1 𝑓𝑜𝑟 𝑝𝑜𝑖𝑛𝑡𝑤𝑖𝑡ℎ𝑖𝑛 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛0 𝑓𝑜𝑟 𝑝𝑜𝑖𝑛𝑡 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 (29)

The interpolation functions can be made of different kindsof functions. For instance, the functions can be polynomialswhich are obtained from Taylor expansion as shown belowfor various degrees [130]:

First order: 𝑏 = [1 𝑥 𝑦]Second order: 𝑏 = [1 𝑥 𝑦 𝑥𝑦 𝑥2 𝑦2]3rd order: 𝑏 = [1 𝑥 𝑦 𝑥𝑦 𝑥2 𝑦2 𝑥3 𝑥2𝑦 𝑥𝑦2 𝑦3]

where 𝑥 and 𝑦 are the local coordinates. The interpolationfunctions can also be made of radial basis functions [130]:

𝑏 = 𝑒−𝑟21 𝑒−𝑟22 ⋅ ⋅ ⋅ 𝑒−𝑟2𝑛 , (30)where 𝑟2𝑖 = (𝑥 − 𝑥𝑖)2 + (𝑦 − 𝑦𝑖)2 and (𝑥𝑖, 𝑦𝑖) representscoordinates of the polygon vertices as well as the mass center.

The stiffness matrix 𝐾 for a heat transfer phenomenon isobtained through the formula [130]

𝐾= ∫Ω𝑘𝐵𝑇𝐵𝑑Ω

− ∫Γ𝑠

𝑘 ⟦𝑁⟧𝑇 𝑛𝑠𝑇 ⟨𝐵⟩ 𝑑Γ + 𝜅∫Γ𝑠

𝑘 ⟨𝐵⟩𝑇 𝑛𝑠 ⟦𝑁⟧𝑑Γ+ ∫Γ𝑠

𝜎 ⟦𝑁⟧𝑇 ⟦𝑁⟧ 𝑑Γ + ∫ΓΘ

𝜎𝑁𝑇𝑁𝑑Γ,(31)

where ⟦𝑁⟧ represents discontinuity jump, ⟨𝐵⟩ representsmean value of 𝐵, Ω represents the domain space, B repre-sents matrix containing the differentiation of interpolationfunctions, k represents thermal conductivity, 𝑛𝑠 representsunit vector which is tangent to the boundary, ΓΘ representsboundary of the polygon, ΓΘ represents boundary in whichthe unknown variable (temperature) is prescribed, 𝜎 repre-sents discontinuity penalization parameter, and parameter 𝜅has the value +1, -1, or 0, depending on the scheme.

2.9. Trefftz/Hybrid Trefftz Polygonal Finite Element (T-FEMor HT-FEM) and Boundary Element Based FEM (BEM-BasedFEM). T-FEM utilizes two different sets of functions toapproximate the solutions, one for the boundary and theother is for the interior domain. For the interior of theelement, a series of homogeneous solutions of the governingequation (problem equation to be solved) is used as basisfunctions. These basis functions are known as T-complete setand they are not conforming across the element boundaries[139]. The boundary is represented by other independentsets of conforming basis functions. An example of T-FEMelement is shown in Figure 12.

The displacement 𝑢 within a domain (interior) can beinterpolated by using

𝑢 = �̌� + 𝑚∑𝑖=1

𝑁𝑖𝑐𝑖, (32)where �̌� represents the known function on the boundary,𝑁𝑖 is the trial function, 𝑐𝑖 is the unknown coefficient, and


1

6

5

43

2

u = N(x)d

u = ǔ +m

∑i=1

Nici

Figure 12: An example of T-FEM with its shape functions utilizedfor the case of solid mechanics.

m represents number of trial functions. The trial functions𝑁𝑖 can be generated from Muskhelishvili’s complex variableformulation [139].The displacement �̃� on the boundary of thedomain can be interpolated by using (33) below:

�̃� = �̃� (𝑥) 𝑑, (33)where 𝑑 represents vector of nodal displacements and�̃�(𝑥) = { (1−𝜉)/2(1+𝜉)/2 representsmatrix of the corresponding one-dimensional shape functions for the boundary in terms of thelocal coordinate system, 𝜉.

Continuity or boundary conditions are incorporated intothe interior domain by various ways, in which one of the waysis by hybrid method known as hybrid Trefftz finite elementmethod (HT-FEM). This method uses the conforming func-tions of the element boundary (also known as frame) to linkthe interior of the elements together [140].

HT-FEM has been successfully applied in linear elasticityproblems [140] and found to be able to producemore accurateresults and higher convergence rates compared to the con-forming PFEM with Laplace/Wachspress shape functions.One of the advantages of the HT-FEM is that elements withembedded cracks or voids can be constructed. This leadsto the development of new VCFEMs for microstructuralanalysis. The new method is known as T-Trefftz VoronoiCell Finite Elements (VCFEM-TTs) [141] and was soonextended to 3D [142]. Recently another novel hybrid FEMhas been formulated, by coupling HT-FEM with the idea oftheMethod of Fundamental Solution (MFS) known as hybridfundamental solution based FEM (HFS-FEM) [143].

The advantage of HT-FEM compared to conventionalFEM [144] is that this method is able to handle geom-etry induced singularities and stress/force concentrationsefficiently without mesh adjustment. This is achieved byemploying special purpose Trefftz functions that satisfy boththe governing equations and boundary conditions associatedwith the singularities. Apart from that, general polygonalelements with curved sides can be generated and the elementsare tolerant to mesh distortions. Advantages of HT-FEMcompared to BEM [144] are that this method is applicable forproblems involving different and heterogeneous materials,and boundary integration can be avoided when field variablesare to be computed inside an element and the calculation of

1

6

5

43

2

12

7

11

109

8

y

x

i

kj

= +1 = 0

= -1

Figure 13: An example of a HS-F element.

coefficient matrices are simpler. Disadvantage of the methodis that the T-complete set for some problems are eithercomplex or difficult to formulate. Application of the methodto 3D is described by Copeland, Langer, and Pusch [139].Soon, another version of polygonal FEM emerged known asBoundary element-based FEM (BEM-based FEM) [139, 145–148].This method is developed based on the Trefftz function.

2.10. Hybrid Stress-Function (HS-F) Polygonal Element.Another method has been proposed by combining the prin-ciple of minimum complementary energy (similar principalused in VCFEM) with the Airy stress function which isknown as hybrid stress-function (HS-F) element method.It was developed for quadrilaterals and triangles [149–154].These elements are found to possess excellent performancecompared to the conventional elements and especially inde-pendent of the element geometry (immune to mesh distor-tion). Later Zhou and Cen [155] expanded the method topolygonal elements. An example of HS-F polygonal elementis shown in Figure 13.Nodes 1-6 are corner nodes and nodes 7-12 are the midside nodes. Local coordinate system 𝜉 is shownin Figure 13 for the edge 6-7-1.

The displacement along a particular element edge d isgiven by [155]

𝑑 = 𝑁𝑞, (34)where 𝑁 = {𝑁𝑖 𝑁𝑗 𝑁𝑘} represents the vector of shapefunctions for the three nodes (i, j, k) along a particular edgeof the element and 𝑞 = {𝑢𝑖 V𝑖 𝑢𝑗 V𝑗 𝑢𝑘 V𝑘} represents thedisplacement vector for the nodes. The shape functions aregiven as [155]

𝑁1 = −0.5𝜉 (1 − 𝜉)𝑁2 = 1 − 𝜉2𝑁3 = 0.5𝜉 (1 + 𝜉) ,

(35)

where 𝜉 represents local coordinate system along each ele-ment boundary. The element stress fields are given as [155]

𝜎 = 𝑆𝛽 + 𝜎∗, (36)where S is the stress solution matrix (with dimensions 3 by k)which is derived from k number of analytical solutions of the


y

x

1

6

5

43

2

TK

TI

TJ

TL

TNTM

J

K

I

L

NM

Figure 14: An example of a BFEM element.

Airy stress functions, 𝛽 represents matrix (with dimensionsk by 1) of unknown stress parameters and 𝜎∗ representsparticular solution corresponding to body forces.

2.11. Base Forces ElementMethod (BFEM). Stress based FEMssuch as HPE and HS-F are not well desired for mostof the engineering applications, due to the difficulties inobtaining suitable/compatible stress functions. Furthermore,it is difficult to acquire nodal displacements in stress basedFEMs [156]. BFEM on the other hand was formulated basedon the concept of “base forces” which was introduced byGao [157]. This method replaces the stress functions in thestress based FEMs with base forces which are easier to obtain(obtained directly from strain energy).

Conventional FEM exhibits major drawback for nonlin-ear analysis.The conventional FEM is not able to approximatethe strain and force fields accurately since these terms aredependent on the interpolation of the displacement field(displacement shape functions). This problem is avoided inBFEM which is directly based on interpolation of the inter-nal force fields (force shape functions) [158]. Performanceof BFEM is also found to be superior than conventionalFEM when analyzing large strain contact problems and thenonlinear problems. This is because the deformation fieldin conventional FEM is complex and discontinuous nearnonlinear regions and, in the case of large strain problems,the steep gradients of deformation cannot be representedaccurately by the conventional shape functions [159]. BFEMhas been later extended to polygonal elements based on thecomplementary energy principle [160] and potential energyprinciple [161]. Figure 14 shows an example of a BFEMelement. 𝐼, 𝐽,𝐾, 𝐿,𝑀,𝑁 represent the edges of the elementand 𝑇𝐼, 𝑇𝐽, 𝑇𝐾, 𝑇𝐿, 𝑇𝑀, 𝑇𝑁 represent the force vectors whichact on the element edges.

The stresses corresponding to a point I on the element canbe represented as [160]

𝜎 = 1𝐴[[[[[[

4∑𝐼=1

𝑇𝐼1𝑃𝐼1 4∑𝐼=1

𝑇𝐼1𝑃𝐼24∑𝐼=1

𝑇𝐼2𝑃𝐼1 4∑𝐼=1

𝑇𝐼2𝑃𝐼2]]]]]], (37)

where 𝑇𝐼1, 𝑇𝐼2 are the components of force vectors whichact on the center of edge I, 𝑃𝐼1, 𝑃𝐼2 are the componentsof position vector of point I, and A represents area of theelement.

2.12. Other Recent Techniques/Schemes. Recently, more newschemes have been proposed for the development of polygo-nal/polyhedral finite elements. They are the Compatible Dis-crete Operator (CDO) scheme [162, 163], Hybrid High-Order(HHO) scheme [164–166], Weak Galerkin (WG) scheme[167–171], gradient correction scheme [172], and vertex-basedschemes [173].

Other recent techniques/methods include analysis ofpolygonal carbon nanotubes reinforced composite plates byusing the first-order shear deformation theory (FSDT) andthe element-free IMLS-Ritz method [174], an adaptive polyg-onal finite element method using the techniques of cut-celland quadtree refinement [175], new adaptivemesh generationfor polygonal element [176], and ultraweak formulations forhigh-order polygonal finite element methods [177]. Newtechnique for 3-dimensional polyhedral elements can be seenwithin the framework of the finite volume method [178].Another new approach to form polyhedral elements is bycutting a regular hexahedral element with CAD surfaces[179].

3. Comparison of the Various Methods

Various techniques described in Section 2 above are com-pared with the recently proposed polyhedral element,known as Virtual Node Polyhedral Element, VPHE (three-dimensional version of Virtual Node Method, as mentionedin Section 2.7 above). The comparison is given in Table 1.

4. Software Packages

Each method described above has been tested and analyzedby using computer programs such as MATLAB/Abaqus.These computer programs have been developed specificallyfor the purpose of testing and analyzing the proposedtechniques. However, some of the methods have been welldeveloped and made available as commercial software. Thissection described some of the software packages (eithercommercially available or for intended use only) which havebeen developed for polygonal/polyhedral FEM.

VCFEM has been incorporated into a software packageknown as Palmyra [180]. This software can be used to designcomposite materials and also to determine physical proper-ties of heterogeneous materials. Three-dimensional Voronoicell software library (an open source software) is availablein the form of MATLAB code [181]. PFEM techniques havebeen incorporated into computer codes by using Fortran andMATLAB as well as Java [182]. Abaqus package is availablefor nSFEM [183]. VEM has been developed and tested inMATLAB and Abaqus packages [184, 185]. MATLAB code onPSBFEM is used in [186].

It is seen that currently there are few commercial softwarepackages which are available for polygonal/polyhedral finite


Table1:Com

paris

onof

thee

xisting

metho

dswith

thep

ropo

sed/presentelement.

Metho

dElem

entF

ormulation

Advantages

Disa

dvantages

Specialty

ofVPH

Eelem

entcom

paredto

othertechn

iques

Applicationfieldsa

ndtypo

logy

VCFE

MPrincipleo

fminim

umcomplem

entary

energy.

1.Com

putatio

nally

efficientcom

pared

tothec

onventionalF

EM.

1.Perfo

rmpo

orlywhentheh

eterogeneity

isin

theform

ofvoids.

2.Po

orlydefin

edstr

essfun

ctions

with

intheinterioro

fthe

elem

ent.

Stressfunctio

nswith

inthee

lement

arew

elld

efinedby

mon

omials.

Simulationof

microstr

uctures

(grains)andmultiscale

mod

ellin

g.Ap

plicableforb

oth

2Dand3D

prob

lems.

Applicableforb

othlin

eara

ndno

nlineara

nalyses.

NCM

-VC

FEM

Hybrid

ofVC

FEM

with

otherm

etho

dsuch

asnu

mericalconformal

mapping

.

1.Th

evariatio

nalprin

cipleis

generalized.

2.Solutio

naccuracy

iscompetitive

with

thatof

conventio

nalF

EMpackageinANSY

S.3.Re

ducesthe

compu

tatio

nalcost

whencomparedto

FEM

packagein

ANSY

S.

1.Th

eNCM

-based

stressfun

ction

constructio

nisexpensiveincomparis

onwith

conformalmapping

ofregu

lar

shapes

such

asellip

sesa

ndcircles.

2.NCM

-based

stressfun

ctions

intro

duce

singu

larity.Specialtechn

ique

(suchas

divergence

theorem)isn

eededto

redu

cetheo

rder

oftheses

ingu

larities.

Specialtechn

iquesare

notn

eededto

hand

lesin

gularitiesinthe

shapefun

ctions.

Realmicrographs

ofheterogeneou

smaterialswith

irregular

shapes

canbe

analyzed

effectiv

ely.

Applicablefor2

Dprob

lems

fortim

ebeing

.App

licablefor

both

lineara

ndno

nlinear

analyses.

HPE

Hybrid

stresse

lement

metho

dtogether

with

Muskh

elish

vili’s

complex

analysis.

1.Stressfunctio

nswith

intheinterior

ofthee

lementare

defin

edby

self-equilib

ratin

gstr

essfi

eld.

2.Be

tterp

erform

ance

comparedto

conventio

nalF

EMforp

lane

linear

elastic

ityprob

lems.

1.Can

containon

lyon

eirregular

phase

(void/inclu

sion)

with

inthee

lement.

-

Simulationof

microstr

uctures

(grains)with

heterogeneity.

Applicableforb

oth2D

and3D

prob

lems.Ap

plicableforb

oth

lineara

ndno

nlineara

nalyses.

PFEM

Barycentric

Coo

rdinates.

1.Ab

leto

take

arbitraryform

,with

arbitrarynu

mbero

fsides

andno

des.

1.Ev

aluatio

nof

barycentric

coordinates

(com

plex

ratio

nalfun

ctions)isn

either

simplen

oreffi

cientcom

paredto

the

conventio

nald

isplacementb

ased

FEM.

2.Not

efficientfor

assemblingthe

stiffn

essm

atric

esassociated

with

weak

solutio

nsof

Poiss

onequatio

ns.

Thes

hape

functio

nsconsist

ofsim

ple

mon

omials

irrespectiveo

nnu

mbero

fplanes/sides.

Solid

mechanics

andheat

transfe

rpheno

mena.

Com

puterg

raph

ics,

anim

ationandgeom

etric

mod

ellin

g.Quadtree/Octree

meshgeneratio

n.Ap

plicable

forb

oth2D

and3D

prob

lems.

Applicableforb

othlin

eara

ndno

nlineara

nalyses.

nSFE

MCou

plingof

conventio

nal

FEM

with

meshless

metho

d.

1.nSFE

Misadvantageous

over

the

conventio

nalF

EMsin

ceitprod

uces

morea

ccuratesolutions,ableto

toleratevolumetric

lockinganddo

not

requ

ireiso

parametric

mapping

.

1.nC

S-FE

Mincreasesthe

compu

tatio

nal

costforsolid

mechanics.

2.Com

putatio

naltim

eofn

NS-FE

Mand

nES-FE

Mislonger

comparedto

conventio

nalF

EMforthe

samen

umber

ofglob

alno

des,du

etolarger

band

width

ofstiffn

essm

atric

es.

3.Disa

dvantage

ofnE

S-FE

Misthatthere

istend

ency

tooverestim

ateo

run

derestimatethe

strainenergy

ofthe

mod

elforsom

ecases.

Form

ulationof

the

prop

osed/present

elem

entissim

ilarto

nSFE

M.Th

ecurrent

techniqu

ecan

beim

proved

bycarrying

outsmoo

thing

techniqu

e,which

will

bethen

similarto

nCS-FE

M.

Solid

mechanics

andheat

transfe

rpheno

mena.

Fluid-solid

interaction(FSI)

prob

lems.Ap

plicableforb

oth

2Dand3D

prob

lems.

Applicableforb

othlin

eara

ndno

nlineara

nalyses.

14 Mathematical Problems in EngineeringTa

ble1:Con

tinued.

Metho

dElem

entF

ormulation

Advantages

Disa

dvantages

Specialty

ofVPH

Eelem

entcom

paredto

othertechn

iques

Applicationfieldsa

ndtypo

logy

PSBF

EM

Semianalytic

almetho

dwhich

combinesb

ound

ary

elem

entm

etho

d(BEM

)andFE

M.

1.Analytic

alsolutio

nsarea

chieved

insid

ethe

domain,

discretizationof

freea

ndfixed

boun

darie

sand

interfa

cesb

etwe

endifferent

materials

aren

otrequ

ired,andthec

alculatio

nof

stressc

oncentratio

nsandintensity

factorsb

ased

ontheird

efinitio

nis

straigh

tforw

ard.

2.Yields

high

lyaccuratesolutio

nsfor

prob

lemsinvolving

stresssingu

larities.

3.Superio

rtoothertechn

iquessuchas

nSFE

MandconformingPF

EMwith

inthec

ontextof

lineare

lasticityandthe

lineare

lasticfracturem

echanics.

1.Not

directlyapplicableforu

nbou

nded

domains

with

stron

glyinclin

edinterfa

ces.

2.PS

BFEM

cann

otbe

directlyused

toprocesstransient

excitatio

nas

oppo

sedto

BEM.

3.Not

aseffi

cientasc

onventionalF

EMor

BEM

whensolvingprob

lemsinvolving

smoo

thstr

essv

ariatio

nswith

inbo

unded/enclosed

domain.

-

Solid

mechanics

and

polygonalm

eshcreatio

n.Ap

plicableforb

oth2D

and3D

prob

lems.Ap

plicableforb

oth

lineara

ndno

nlineara

nalyses.

MFD

and

VEM

Surfa

cerepresentatio

nof

discreteun

know

ns(M

FD)

andun

know

ndegreeso

ffre

edom

area

ttached

totrialfun

ctions

with

ininterio

rofthe

polygonal

domain(V

EM).

1.Effi

cientinsolvingprob

lems

involvingpo

lygonalm

eshes.

2.Diffi

culties

facedin

integrationof

complex

functio

nsresulting

from

barycentric

coordinatesinPF

EMare

entirely

avoided.

3.Doesn

otrequ

ireextensionof

compatib

leinterpolationfunctio

nsto

theinterioro

fthe

elem

ent.

1.Quitedifficultto

presentM

FDdu

eto

nonexiste

nceo

ftria

lfun

ctions

forthe

interio

rofthe

elem

ent.

2.Involvec

omplex

procedures

and

thereforer

equire

high

compu

tatio

nal

effort.

Easie

rtoexecuted

ueto

simpler

elem

ent

form

ulation.

Electro

magnetic

field

prob

lems,

convectio

n-diffu

sion

prob

lems,flu

idflo

ws

prob

lems,hydrod

ynam

ics

prob

lems,eigenvalue

prob

lems,solid

mechanics,

heattransfe

r,andtopo

logy

optim

ization.

Applicablefor

both

2Dand3D

prob

lems.

Applicableforb

othlin

eara

ndno

nlineara

nalyses.

VNM

Thep

olygon

aldo

mainis

dividedinto

several

triang

ular

region

swhich

usethe

conventio

nalF

EMshapefun

ctions.Th

ese

triang

ular

region

sare

then

coup

ledtogether

byusing

meanleastsqu

ares

hape

functio

ns.

1.Dono

trequire

form

ulationof

complex

stressfun

ctions

forthe

elem

ent(which

couldbe

difficultfor

somec

ases,asreportedforstre

ssbased

FEMssuchas

HPE

andHS-F

2.Num

ericalintegrationforthe

elem

entsissim

plea

ndexact,as

oppo

sedto

compatib

lePF

EM.

3.Simpler

andeasie

rfor

compu

ter

applications

comparedto

MFD

and

VEM

.

1.Integrationwith

ineach

tetrahedron

canbe

simplified

bymapping

,but

the

mapping

procedureimpo

sesrestrictio

nto

elem

entgeometry

duetohigh

aspect

ratio

(limitedtolerancetow

ards

mesh

disto

rtion).

2.Pron

etoelem

entlocking

.

-

Adaptiv

ecom

putatio

n,solid

mechanics,and

heattransfe

rph

enom

ena.Ap

plicablefor

both

2Dand3D

prob

lems.

Applicableforb

othlin

eara

ndno

nlineara

nalyses.

DGFE

M

Prob

lem

domainis

discretized

into

several

polygonalcellswhich

representthe

polygonal

elem

ents.

Theinterpo

latio

nforthe

elem

entsiscarried

outb

ased

onas

etof

mon

omialfun

ctions

which

aretotallyindepend

ento

fthee

lement.

1.Doesn

otrequ

ireanyconforming

shapefun

ctionandthem

etho

dis

simple.

2.Ad

jacent

elem

entscanbe

ofdifferent

ordera

ndthee

lementsdo

not

need

tobe

conforming.

1.Interpolationfunctio

nsdo

notcom

ply

with

shapefun

ctionrequ

irementsof

conventio

nalF

EM(N

otcompatib

le).

2.Integrations

arec

arrie

dou

tonthe

boun

darie

sofeachcell.Th

isste

pisan

additio

ncomparedto

thec

onventional

FEM.

Thee

lementfulfillsall

ther

equirementsof

tradition

alFE

M.

Solid

mechanics,heattransfer,

andeigenvalue

prob

lemso

npo

lygonalm

eshes.Ap

plicable

forb

oth2D

and3D

prob

lems.

Applicableforb

othlin

eara

ndno

nlineara

nalyses.


Table1:Con

tinued.

Metho

dElem

entF

ormulation

Advantages

Disa

dvantages

Specialty

ofVPH

Eelem

entcom

paredto

othertechn

iques

Applicationfieldsa

ndtypo

logy

T-FE

Mor

HT-FE

M

Utilizes

twodifferent

seto

ffunctio

nsto

approxim

ate

thes

olutions,one

forthe

boun

dary

andtheo

ther

isforthe

interio

rdom

ain.

1.Ab

leto

prod

ucem

orea

ccurate

results

andhigh

erconvergencer

ates

compare

tothec

onform

ingPF

EMwith

Laplace/Wachspressshape

functio

ns.

2.Elem

entswith

embedd

edcracks

orvoidsc

anbe

constructed.

3.Ab

leto

hand

legeom

etry

indu

ced

singu

laritiesa

ndstr

ess/force

concentrations

efficiently

with

out

meshadjustm

ent.

4.Generalpo

lygonalelementswith

curved

sides

canbe

generatedandthe

elem

entsaretoleranttomesh

disto

rtions.

1.T-completesetsfor

somep

roblem

sare

either

complex

ordifficultto

form

ulate.

-

Solid

mechanics

andheat

transfe

rpheno

mena.

Applicablefor2

Dprob

lems

forthe

timeb

eing

.App

licable

forb

othlin

eara

ndno

nlinear

analyses.

BEM-based

FEM

Trefft

z-lik

ebasis

functio

nsared

efined

implicitlyandtre

ated

locally

bymeans

ofBo

undary

Elem

entM

etho

ds(BEM

s).

1.Ap

plicableto

generalp

olygon

almeshes(im

mun

etosevere

mesh

disto

rtion).

2.Com

putatio

naleffo

rtisredu

ced,

since

onlyon

esub

spaceisn

eededto

approxim

atethe

pricew

iseharm

onic

functio

ns.

Largelinearsystemso

fequ

ations

are

generated

-

Adaptiv

emeshgeneratio

n,tim

edependent

prob

lems,and

boun

dary

valuep

roblem

s.Ap

plicableforb

oth2D

and3D

prob

lems.Ap

plicableforb

oth

lineara

ndno

nlineara

nalyses.

HS-F

Com

binatio

nof

principleo

fminim

umcomplem

entary

energy

(sim

ilarp

rincipal

used

inVC

FEM)w

ithAiry

stressfun

ction.

1.Po

ssesse

xcellent

perfo

rmance

comparedto

thec

onventional

elem

entsandespeciallyindepend

ent

ofthee

lementgeometry.

1.Not

welldesired

form

osto

fthe

engineeringapplications,due

tothe

difficulties

inob

taining

suitable/compatib

lestr

essfun

ctions.

2.Diffi

cultto

acqu

ireno

dal

displacementsin

stressb

ased

FEMs.

Thee

lementfulfillsall

ther

equirementsof

tradition

alFE

M.

Solid

mechanics

phenom

ena.

Applicablefor2

Dprob

lems

forthe

timeb

eing

.App

licable

forb

othlin

eara

ndno

nlinear

analyses.

BFEM

Replaces

thes

tress

functio

nsin

thes

tress

basedFE

Msw

ithbase

forces

which

aree

asierto

obtain.

1.Ab

leto

approxim

atethe

strainand

forcefi

elds

accurately.

2.Fo

undto

besuperio

rthan

conventio

nalF

EMwhenanalyzing

larges

traincontactp

roblem

sand

the

nonlinearp

roblem

s.

1.Th

eLagrangem

ultip

lierm

etho

disused

todealwith

thee

quilibrium

equatio

n.So,

thes

tiffnessmatrix

isafullm

atrix

.-

Solid

mechanics,bon

ding

damaged

etectio

nanddamage

mechanics.A

pplicablefor2

Dprob

lemsfor

thetim

ebeing

.Ap

plicableforb

othlin

eara

ndno

nlineara

nalyses.


elements. However, software packages for other methodscan be easily developed by incorporating the source codesdeveloped by the researchers mentioned above with theavailable commercial polygonalmesh generators. Some of thesoftware packages for polygonal/polyhedral mesh generationare Platypus (MATLAB based code) [187], ReALE [188],PolyMesher [189], PolyTop [190], OpenMesh [191], and more,which can be found in [192].

5. Summary

It can be seen that various finite elements have been proposedfor engineering analysis. These elements have been proposedto facilitate meshing of the problem domain, to facilitate theanalysis of physical phenomena, and to overcome drawbacksor limitations in the existing methods. This review enablesthe readers to identify advantages, disadvantages, and a com-parison between the various techniques used in formation ofpolygonal/polyhedral finite elements.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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