A group-theoretic formulation for symmetric ﬁnite elements

Finite Elements in Analysis and Design41 (2005) 615–635

www.elsevier.com/locate/finel

A group-theoretic formulation for symmetric finite elementsA. Zingoni∗

Department of Civil Engineering, University of Cape Town, Rondebosch 7701, Cape Town, South Africa

Received 8 March 2004; received in revised form 22 September 2004; accepted 4 October 2004Available online 15 December 2004

Abstract

After a brief review of the literature on applications of group theory to problems exhibiting symmetry in structuralmechanics, an efficient formulation is presented for the computation of matrices for symmetric finite elements. Thetheory of symmetry groups and their representation allows the vector spaces of the element nodal freedoms tobe decomposed into independent subspaces spanned by symmetry-adapted element nodal freedoms. Similarly, thedisplacement field of the elements is decomposed into components with the same symmetry types as the subspaces.In this way, element shape functions and element matrices are computed separately within each subspace, throughthe solution of a much smaller system of equations and the integration of a much simpler set of functions. Theprocedure is illustrated with reference to the computation of consistent-mass matrices for truss, beam, plane-stress,plate-bending and solid elements.A numerical example has been considered in order to demonstrate the effectivenessof the approach. Overall, the formulation leads to significant reductions in computational effort in comparisonwith the conventional computation of element matrices. However, the proposed formulation only becomes reallyadvantageous in the case of finite elements with a high degree of symmetry (typically solid hexahedral elements),and a large number of nodes and nodal degrees of freedom.� 2004 Elsevier B.V. All rights reserved.

Keywords:Symmetry groups; Group theory; Subspaces; Idempotents; Displacement-field decomposition; Finite elements;Element matrices; Consistent-mass matrices

∗ Tel.: +27 21 650 2601; fax: +27 21 689 7471.E-mail address:[email protected](A. Zingoni).

0168-874X/$ - see front matter� 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.finel.2004.10.004

http://www.elsevier.com/locate/finel

mailto:[email protected]

616 A. Zingoni / Finite Elements in Analysis and Design41 (2005) 615–635

1. Introduction

A body or a structure is said to exhibit symmetry if one part of it is similar or identical to an-other part relative to a centre, an axis or a plane. More formally, a body or a structure is said toexhibit symmetry if it can be turned into one or more new configurations physically indistinguish-able from the initial configuration through the application of one or more symmetry operations suchas reflections in planes, rotations about axes, or inversions about some centre. Symmetry abounds innature at both the microscopic and macroscopic levels, from molecular and crystal structures to cos-mic structures in the physical world, and from tiny viruses to larger plant and animal structures in thebiological world.

In the analysis of physical systems, group theory provides the mathematical tool for exploiting symme-try in a systematic and comprehensive manner. In general, a set of elements{a, b, c, . . . , g, . . .} comprisesa groupG if the following axioms are satisfied:

(i) The productc of any two elementsa andb of the group, denoted byc=ab, must be a unique elementwhich also belongs to the group.

(ii) Among the elements ofG, there must exist an identity elemente which, when multiplied with anyelementa of the group, leaves the element unchanged:ea = ae = a.

(iii) For every elementa of G, there must exist another elementd also belonging to the groupG, suchthatad = da = e; d is referred to as the inverse ofa, and denoted bya−1.

(iv) The order of the multiplication of three or more elements ofG does not affect the result (that is,multiplication is associative):(ab)c = a(bc).

When all elements ofG are symmetry operations, then the groupG is called asymmetry group. Sym-metry operations are transformations which bring an object into coincidence with itself, and leaves itindistinguishable from its original configuration. For finite objects (whose symmetry groups are referredto aspoint groups), which will be the main concern of this paper, symmetry operations may, of course,be one of the following types:

(i) reflections in planes of symmetry, which we will denote by�l , wherel is the plane of symmetry;(ii) rotations about an axis of symmetry, which we will denote byCn, if the angle of rotation is 2�/n;

(iii) rotation-reflections, which we will denote bySn; these represent a rotation through an angle 2�/n,combined with a reflection in the plane perpendicular to the axis of rotation;

(iv) an inversion through thecentre of symmetry(that is, the one point of a finite object which remainsunmoved by all symmetry operations), which we will denote byi; an inversion is a special case ofSn with n= 2.

If a physical system possesses symmetry properties which can be described by a group, decompo-sition of the vector spaceV of the system on the basis of representation theory of symmetry groupsresults in a block-diagonal form of the matrix of equations describing the behaviour of the system,allowing separation into independent sets of equations each corresponding to a particular subspace ofV .In eigenvalue problems, all eigenvalues and eigenvector components of the spaceV of the system can begenerated by separately solvingk smaller eigenvalue problems, each corresponding to a subspaceU(i)

(i = 1,2, . . . , k) of the problem, resulting in large savings in computational effort in comparison withconventional methods.

A. Zingoni / Finite Elements in Analysis and Design41 (2005) 615–635 617

2. Literature review

Applications of group theory to problems in physics and chemistry, within the branches of quantummechanics, molecular chemistry and crystallography, among others, are well known[1–13].

Within the engineering sciences of solid and structural mechanics, a number of problems have beentackled on the basis of Lie groups. Such problems have included the Timoshenko beam equations[14], thevon Karman equations for large deflections of thin elastic plates[15], and various theories of rods, platesand shells[16–18]. It must be noted, however, that these group-theoretic applications, being based onLie groups, exploit the symmetry properties of sets of differential equations for physical problems, ratherthan the symmetry of the physical problems themselves, which will be the concern of the present paper.Problems in solid and structural mechanics exhibiting symmetry properties, and hence suited to treatmentby group-theoretic methods, encompass bifurcation and vibration (which will be briefly reviewed here),and finite-element formulations (the subject of the rest of the paper).

Early applications of group theory to problems of stability include the work of Renton[19], whoconsidered the buckling of symmetrical frameworks. The subject of bifurcation has provided one of themost fruitful areas of application of group-theoretic methods in structural mechanics[20–22]. Buzano[23] has studied the secondary bifurcation (post-buckling) behaviour of a prismatic rod with rectangularcross section under axial compression, and employed group-theoretic concepts to simplify the analysisand generate the required bifurcation diagrams. A number of investigations have focussed onglobal (asopposed tolocal) bifurcation analysis[24–28]. It is recognised that symmetric structures have a muchmore complex bifurcation behaviour than non-symmetric structures. Multiple critical points, where morethan one eigenvalue simultaneously vanishes, are inherent in such structures owing to symmetry[26],and the group-theoretic approach is well-suited to simplifying the analysis. Characteristically, it caststhe tangent-stiffness matrix into block-diagonal form, eliminating problems of numerical ill-conditioningusually associated with the bifurcation analysis of thin shells[28], or facilitating tests for singularity ofthe tangent-stiffness matrix[27].

Another noteworthy area of application of group theory in structural mechanics has been vibrationanalysis which, of course, is closely related to bifurcation analysis. For large eigenvalue problems ofskeletal structures with symmetry, Healey and Treacy[29] have proposed a computational scheme com-bining group-theoretic ideas and substructuring techniques, to block-diagonalise the matrices. Zlokovic[30] has developed effective group-theoretic procedures not only for problems of the vibration analysisof symmetric systems (such as lumped-parameter models of beams), but also for the static and stabilityanalysis of symmetric structural configurations. The approach involves the decomposition of the vectorspace of the normal variables of a problem into a number of independent subspaces spanned by symmetry-adapted variables, permitting quantities of interest to be separately obtained for each subspace throughthe solution of a system of equations with only a fraction of the number of unknowns in the originalproblem. The overall result is a substantial reduction in computational effort, in comparison with conven-tional procedures. A similar approach has been adopted in formulating and tackling the specific problemsof the vibration analysis of high-tension cable nets[31], the natural-frequency determination of sym-metric grid-mass systems[32], and the vibration analysis of symmetric plates using the finite-differencemethod[33].

A more detailed discussion of group-theoretic formulations of the general problems of bifurcationand vibration in the context of solid and structural mechanics may be seen in a recent state-of-the-artreview[34].


In the domain of finite-element analysis, group-theoretic formulations permit a more efficient compu-tation of element matrices for symmetric finite elements. One such formulation has been presented[35]for several types of symmetric finite elements, and successfully applied to the computation of elementstiffness matrices.

3. The group-theoretic formulation for finite elements

In this paper, the group-theoretic formulation for the efficient computation of basic matrices for sym-metric finite elements is presented, and then applied to the derivation of consistent-mass matrices for avariety of such elements, namely the 2-node 2-d.o.f. truss element, the 2-node 4-d.o.f. beam element, the4-node 8-d.o.f. rectangular plane-stress element, the 4-node 12-d.o.f. rectangular plate-bending elementand the 8-node 24-d.o.f. rectangular hexahedral element. Nowhere in the literature have consistent-massmatrices for dynamic applications been computed using this approach, and this contribution is there-fore an extension of earlier work[35], and further demonstrates the effectiveness of the group-theoreticformulation.

The key feature of the formulation is the decomposition of the general displacement field of theelement into subfields of symmetry types corresponding to those of the irreducible representations ofthe symmetry group describing the nodal configuration of the element. In this way, the vector spaceof the original problem is decomposed into independent subspaces, permitting element shape functionsof very simple form to be written down for each subspace, and associated element matrices to be computedthrough the integration of much simpler terms in comparison with their conventional counterparts.

Overall, even allowing for the cost of decomposition of the problems and final superposition andtransformation of subspace quantities, the computational effort expended in arriving at the results issubstantially smaller than that associated with the conventional approach. The method is particularlyadvantageous in the computation of matrices for elements with a large number of nodes and degreesof freedom.

4. Choice of origin, node numbering and positive directions

The group-theoretic formulation requires a special choice of origin, sequence of node numbering,and positive directions for freedoms.Figs. 1–3indicate the symmetry adapted conventions for nodenumbering and positive directions for nodal freedoms for truss, beam, rectangular plane-stress, rectangular

Fig. 1. Symmetry-adapted conventions for straight line elements: (a) node numbering; (b) positive directions for nodal freedomsfor a truss element; and (c) positive directions for nodal freedoms for a beam element.


Fig. 2. Symmetry-adapted conventions for rectangular surface elements: (a) node numbering; (b) positive directions for nodalfreedoms for a plane-stress element; and (c) positive directions for nodal freedoms for a plate-bending element.

Fig. 3. Symmetry-adapted conventions for rectangular solid elements: (a) node numbering; and (b) positive directions for nodalfreedoms for a hexahedral element.

plate-bending and rectangular solid elements. The origin of the local coordinate system of the elementshould be chosen at the centre of symmetryO of the configuration, that is, the point through which allaxes of rotation pass, and/or all planes of reflection intersect.


For line elements in thexy plane, oriented along thex-axis and with their perpendicular bisectordefining they-axis, Node 1 must be chosen on the positive side of the perpendicular bisector, that is, onthe positive branch of thex-axis. For rectangular flat elements, Node 1 is chosen in the positive–positivequadrant of thexy Cartesian coordinate system. For rectangular solid elements, Node 1 is chosen in thepositive–positive–positive octant of thexyz coordinate system.

For a finite element having a nodal configuration belonging to the symmetry groupG, let the collectionof all the nodal positions of the element that are permuted, or interchanged among themselves, by sym-metry operations ofG, be referred to as anodal set. An element may therefore have one or more nodalsets. Finite elements having a large number of nodes with configurations belonging to symmetry groupsof low order (that is, having a small number of symmetry elements) generally exhibit a multiplicity ofnodal sets.

The rest of the nodes belonging to the same nodal set as Node 1 are numbered consecutively in theorder generated by permuting Node 1 through each symmetry operation ofG, such symmetry operationsbeing performed in the same order as appears across the top of the character table of the symmetry group[4]. For symmetry groupsC1v ,C2v andD2h describing the configurations of straight line elements, planerectangular elements and solid rectangular elements, respectively, the symmetry operations ordered in thismanner are{E,C2}, {E,C2, �x, �y}, and{E,Cz2, Cy2 , Cx2 , i, �xy, �xz, �yz}, respectively. HereE is theidentity operation,C2 is a rotation through 180◦, �x is a reflection in the perpendicular plane containingthex-axis,�y is a reflection in the perpendicular plane containing they-axis,Cz2 is a rotation through180◦ about thez-axis,Cy2 is a rotation through 180◦ about they-axis,Cx2 is a rotation through 180◦ aboutthex-axis,i is an inversion about the centre of symmetryO, �xy is a reflection in thexy plane,�xz is areflection in thexz plane, and�yz is a reflection in theyz plane.

After all the nodal positions of the first nodal set have been numbered, say from 1 up ton1, another node(if any) on the positive branch of thex-axis in the case of line elements, or in the positive–positive quadrantof thexy coordinate system in the case of rectangular plane elements, or in the positive–positive–positiveoctant of thexyz coordinate system in the case of rectangular solid elements, is selected and numberedn1 + 1, and the rest of the nodal positions in this second nodal set are numbered consecutively upwardsfrom n1 + 2 following the same sequence described for the first nodal set. The procedure is repeated forall nodal sets of the element’s nodal positions until all then individual nodes of the element have beennumbered consecutively from 1 up ton.

Having numbered all the nodes of a finite element with a symmetric configuration of nodes in accordancewith the above procedure, the positive directions of the freedoms and loads at Node 1 may be chosenarbitrarily, but having fixed these, the positive directions for the rest of the nodal freedoms and nodal loadsof the element should be chosen such that the ensuing overall pattern of directions of nodal quantities hasthe symmetry type of the firstirreducible representation[4] of the symmetry groupG. In other words,the pattern of nodal directions must preserve the full symmetry of the pattern of nodes itself. Accordingto this scheme, corresponding nodes of the various nodal sets will have the same positive directions ofnodal quantities.

5. Symmetry-adapted nodal freedoms

In representation theory for symmetry groups, an idempotent� of a symmetry groupG is a lin-ear combination of symmetry elements of the group which, when applied upon all the vectors of thevector space of the problem, has the property of nullifying all vectors of the vector space other than


those which belong to a particular subspace associated with a specific symmetry type. In this sense, anidempotent acts as aprojection operator. Thus, given any physical system with a configuration con-forming to a symmetry groupG, we can use the idempotents ofG to transform the normal variablesof the problem spanning the vector spaceV , into orthogonal sets of symmetry-adapted variables, eachset spanning a subspaceS of the vector spaceV . Any given symmetry groupG has a certain fixednumber of irreducible representations, each associated with its own idempotent[4,5]. The idempotents�(j), corresponding to the irreducible representationsL(j) of symmetry groupsC1v,C2v andD2h (notingthatj = {1,2} for groupC1v, j = {1,2,3,4} for groupC2v, andj = {1,2, . . . ,8} for groupD2h), areas follows:

[�(1)

�(2)

]= 1

2

[1 1

1 −1

] [E

�y

],

�(1)

�(2)

�(3)

�(4)

= 1

4

1 1 1 1

1 1 −1 −1

1 −1 1 −1

1 −1 −1 1

E

C2

�x

�y

, (1, 2)

�(1)

�(2)

�(3)

�(4)

�(5)

�(6)

�(7)

�(8)

= 1

8

1 1 1 1 1 1 1 1

1 1 −1 −1 1 1 −1 −1

1 −1 1 −1 1 −1 1 −1

1 −1 −1 1 1 −1 −1 1

1 1 1 1 −1 −1 −1 −1

1 1 −1 −1 −1 −1 1 1

1 −1 1 −1 −1 1 −1 1

1 −1 −1 1 −1 1 1 −1

E

Cz2

Cy2

Cx2

i

�xy

�xz

�yz

. (3)

The general form of the above relations may be written as� = T�, where� is the column vector ofidempotents ofG, � is the column vector of symmetry operations ofG, andT is the square matrix linkingthe two vectors.

Applying the idempotents�(j) to the nodal-freedom setsD(i) of a finite-element configuration yieldssymmetry-adaptednodal-freedom sets�(j) corresponding to the subspacesS(j) associated with the sym-metry group in question. For instance, considering a rectangular hexahedral element with an arbitrarynumbernof nodes, applying the first idempotent�(1) of symmetry groupD2h to each of the nodal-freedomsetsD(i) (i=1, . . . , n) of the configuration, we obtainn linear combinations of theD(i) (with up to eightdifferentD(i) per linear combination), from which we can identify a set ofk1 (k1<n) independentlinear

combinations, namely�(1)1 ,�(1)2 , . . . ,�(1)k1, as the basis vectors or symmetry-adapted nodal-freedom sets

for subspaceS(1). Similarly, the second idempotent�(2) yieldsk2 symmetry-adapted nodal-freedom setsfor subspaceS(2), namely�(2)1 ,�(2)2 , . . . ,�(2)k2

, and so on. Thekj (j = 1,2, . . . ,8) symmetry-adapted

nodal-freedom sets for subspaceS(j), namely�(j)1 ,�(j)2 , . . . ,�(j)kj , may be collected together as the col-

umn vector�(j). Thus the dimensions of subspacesS(1), S(2), . . . , S(8) arek1, k2, . . . , k8 respectively,andk1 + k2 + · · · + k8 = n.


The results for symmetry-adapted nodal-freedom sets for the 2-node 2-d.o.f. truss element, 2-node 4-d.o.f. beam element, 4-node 8-d.o.f. rectangular plane-stress element, 4-node 12-d.o.f. rectangular plate-bending element and 8-node 24-d.o.f. rectangular hexahedral element, are given below as

[�(1)

�(2)

]=

[�(1)1

�(2)1

]= 1

2

[(u1 + u2)

(u1 − u2)

]= 1

2

[1 1

1 −1

] [u1

u2

]= 1

2

[1 1

1 −1

] [D(1)

D(2)

], (4)

[�(1)

�(2)

]=

[ [�(1)1 �(1)2 ]T[�(2)1 �(2)2 ]T

]= 1

2

[ [ (w1 + w2) (�1 + �2) ]T[ (w1 − w2) (�1 − �2) ]T

]= 1

2

[1 1

1 −1

] [D(1)

D(2)

], (5)

�(1)

�(2)

�(3)

�(4)

=

[�(1)1 �(1)2 ]T[�(2)1 �(2)2 ]T[�(3)1 �(3)2 ]T[�(4)1 �(4)2 ]T

= 1

4

[ (u1 + u2 + u3 + u4) (v1 + v2 + v3 + v4) ]T[ (u1 + u2 − u3 − u4) (v1 + v2 − v3 − v4) ]T[ (u1 − u2 + u3 − u4) (v1 − v2 + v3 − v4) ]T[ (u1 − u2 − u3 + u4) (v1 − v2 − v3 + v4) ]T

= 1

4

1 1 1 1

1 1 −1 −1

1 −1 1 −1

1 −1 −1 1

D(1)

D(2)

D(3)

D(4)

, (6)

�(1)

�(2)

�(3)

�(4)

=

[�(1)1 �(1)2 �(1)3 ]T[�(2)1 �(2)2 �(2)3 ]T[�(3)1 �(3)2 �(3)3 ]T[�(4)1 �(4)2 �(4)3 ]T

= 1

4

[ (w1 + w2 + w3 + w4) (�x1 + �x2 + �x3 + �x4) (�y1 + �y2 + �y3 + �y4) ]T[ (w1 + w2 − w3 − w4) (�x1 + �x2 − �x3 − �x4) (�y1 + �y2 − �y3 − �y4) ]T[ (w1 − w2 + w3 − w4) (�x1 − �x2 + �x3 − �x4) (�y1 − �y2 + �y3 − �y4) ]T[ (w1 − w2 − w3 + w4) (�x1 − �x2 − �x3 + �x4) (�y1 − �y2 − �y3 + �y4) ]T

= 1

4

1 1 1 1

1 1 −1 −1

1 −1 1 −1

1 −1 −1 1

D(1)

D(2)

D(3)

D(4)

, (7)


�(1)

�(2)

�(3)

�(4)

�(5)

�(6)

�(7)

�(8)

=

[�(1)1 �(1)2 �(1)3 ]T[�(2)1 �(2)2 �(2)3 ]T[�(3)1 �(3)2 �(3)3 ]T[�(4)1 �(4)2 �(4)3 ]T[�(5)1 �(5)2 �(5)3 ]T[�(6)1 �(6)2 �(6)3 ]T[�(7)1 �(7)2 �(7)3 ]T[�(8)1 �(8)2 �(8)3 ]T

= 1

8

1 1 1 1 1 1 1 11 1 −1 −1 1 1 −1 −11 −1 1 −1 1 −1 1 −11 −1 −1 1 1 −1 −1 11 1 1 1 −1 −1 −1 −11 1 −1 −1 −1 −1 1 11 −1 1 −1 −1 1 −1 11 −1 −1 1 −1 1 1 −1

D(1)

D(2)

D(3)

D(4)

D(5)

D(6)

D(7)

D(8)

, (8a)

where

�(1)1

�(2)1

�(3)1

�(4)1

�(5)1

�(6)1

�(7)1

�(8)1

= 1

8

(u1 + u2 + u3 + u4 + u5 + u6 + u7 + u8)

(u1 + u2 − u3 − u4 + u5 + u6 − u7 − u8)

(u1 − u2 + u3 − u4 + u5 − u6 + u7 − u8)

(u1 − u2 − u3 + u4 + u5 − u6 − u7 + u8)

(u1 + u2 + u3 + u4 − u5 − u6 − u7 − u8)

(u1 + u2 − u3 − u4 − u5 − u6 + u7 + u8)

(u1 − u2 + u3 − u4 − u5 + u6 − u7 + u8)

(u1 − u2 − u3 + u4 − u5 + u6 + u7 − u8)

(8b)

and exactly similar relations apply for the�(j)2 (combinations ofv1, v2, . . . , v8) and�(j)3 (combinations ofw1, w2, . . . , w8), wherej = 1, . . . ,8. All the above five relations (Eqs. (4)–(8)) are of the form� = TD,where� is the column vector of symmetry-adapted nodal freedoms,D is the column vector of normal(that is, actual) nodal freedoms, andT is as previously defined.

6. Displacement field decomposition

In the present formulation, terms of the assumed displacement polynomial for the symmetric finiteelement are allocated to the subspaces of the symmetry group of the configuration of the finite elementin accordance with the symmetry types of the displacement fields associated with the terms. Since the


assumed displacement polynomials foru andv in the case of rectangular plane-stress elements, and foru,v andw in the case of rectangular solid elements, are usually of the same form, we need only consider onedisplacement component, sayu, and allocate the terms of the displacement polynomial to the subspacesof the respective symmetry groups. For the 2-node 2-d.o.f. truss element, 2-node 4-d.o.f. beam element,4-node 8-d.o.f. rectangular plane-stress element, 4-node 12-d.o.f. rectangular plate-bending element and8-node 24-d.o.f. rectangular hexahedral element, let us assume the displacement polynomials are as givenbelow by

u(x)= �1 + �2x, (9)

w(x)= �1 + �2x + �3x2 + �4x

3, (10a)

�(x)= dw(x)

dx= �2 + 2�3x + 3�4x

2, (10b)

u(x, y)= �1 + �2x + �3y + �4xy, (11)

w = �1 + �2x + �3y + �4x2 + �5xy + �6y

2 + �7x3 + �8x

2y

+ �9xy2 + �10y

3 + �11x3y + �12xy

3, (12a)

�x = �w

�x= �2 + 2x�4 + �5y + 3x2�7 + 2xy�8 + �9y

2 + 3x2y�11 + �12y3, (12b)

�y = �w

�y= �3 + �5x + 2y�6 + �8x

2 + 2yx�9 + 3y2�10 + �11x3 + 3y2x�12, (12c)

u(x, y, z)= �1 + �2x + �3y + �4z+ �5xy + �6yz+ �7xz+ �8xyz. (13)

Terms of the displacement polynomials in Eq. (9) for the truss element and Eq. (10) for the beam elementare allocated to the two subspacesS(1) andS(2) of the symmetry groupC1v, as shown below in Eqs. (14)and (15), respectively. Terms of the displacement polynomials in Eq. (11) for the rectangular plane-stresselement and Eqs. (12) for the rectangular plate-bending element are allocated to the four subspacesS(1),S(2), S(3) andS(4) of the symmetry groupC2v, as indicated below in Eqs. (16) and (17). Finally, terms ofthe displacement polynomial in Eq. (13) for the rectangular hexahedral element are allocated to the eightsubspaces of the symmetry groupD2h, as shown below in Eq. (18)

[u(1)(x)

u(2)(x)

]=

[x 0

0 1

] [�2

�1

], (14)

(15)


u(1)(x, y)

u(2)(x, y)

u(3)(x, y)

u(4)(x, y)

=

x

y

1xy

�2�3�1�4

, (16)

(17)

u(1)(x, y, z)

u(2)(x, y, z)

u(3)(x, y, z)

u(4)(x, y, z)

u(5)(x, y, z)

u(6)(x, y, z)

u(7)(x, y, z)

u(8)(x, y, z)

=

x

y

z

xyz

yz

xz

xy

1

�2�3�4�8�6�7�5�1

. (18)

7. Subspace shape functions

Substituting the nodal coordinates of the first nodes of the nodal sets of the element into the decomposedpolynomial automatically yields the sets�(j) of the symmetry-adapted nodal displacements for subspaceS(j), in terms of the polynomial coefficients. Rearrangement then yields the polynomial coefficientsin terms of the symmetry-adapted nodal displacements�(j), enabling the symmetry-adapted internaldisplacementsU(j) (that is, the above subspace displacement fields) to be written down in terms ofthe symmetry-adapted nodal displacements�(j) in the formU(j) = N(j)�(j), whereN(j) is the matrixof symmetry-adapted shape functions for subspaceS(j). For the 2-node 2-d.o.f. truss element, 2-node4-d.o.f. beam element, 4-node 8-d.o.f. rectangular plane-stress element, 4-node 12-d.o.f. rectangularplate-bending element and 8-node 24-d.o.f. rectangular hexahedral element, the symmetry-adapted shapefunctions for the various subspaces of the respective problems are given by

N(1) = x

l, N(2) = 1, (19a, b)

N(1)1 = 1, N

(1)2 = − l

2+ x2

2l, (20a)


N(2)1 = 3x

2l− x3

2l3, N

(2)2 = −x

2+ x3

2l2, (20b)

N(1) = x

a, N(2) = y

b, N(3) = 1, N(4) = xy

ab, (21a–d)

N(1)1 = 1, N

(1)2 = −a

2+ x2

2a, N

(1)3 = −b

2+ y2

2b, (22a)

N(2)1 = 2xy

ab− x3y

2a3b− xy3

2ab3 , N(2)2 = −xy

2b+ x3y

2a2b, N

(2)3 = −xy

2a+ xy3

2ab2 , (22b)

N(3)1 = 3x

2a− x3

2a3 , N(3)2 = −x

2+ x3

2a2 , N(3)3 = −bx

2a+ xy2

2ab, (22c)

N(4)1 = 3y

2b− y3

2b3 , N(4)2 = −ya

2b+ x2y

2ab, N

(4)3 = −y

2+ y3

2b2 , (22d)

N(1) = x

a, N(2) = y

b, N(3) = z

c, N(4) = xyz

abc,

N(5) = yz

bc, N(6) = xz

ac, N(7) = xy

ab, N(8) = 1. (23a–h)

Notice how much simpler the subspace shape functions are in comparison with conventional shapefunctions for these elements.

8. Subspace element matrices

Symmetry-adapted element matrices for subspaceS(j) then follow by integration of the respective shapefunctions over only the positive half of the element for line elements, the positive–positive quadrantfor rectangular flat elements, and the positive–positive–positive octant for rectangular solid elements.For instance, the symmetry-adapted element stiffness and consistent-mass matrices,K(j) andM(j) forsubspaceS(j), are obtained as

K(j) =∫V

[B(j)]T[D][B(j)] dV, M(j) =∫V

[N(j)]T[N(j)] dV, (24)

whereB(j) andN(j) are the subspace strain-displacement and shape-function matrices,D is a matrixof material stiffnesses and is the material mass density. For the 2-node 2-d.o.f. truss element, 2-node4-d.o.f. beam element, 4-node 8-d.o.f. rectangular plane-stress element, 4-node 12-d.o.f. rectangularplate-bending element and 8-node 24-d.o.f. rectangular hexahedral element, the subspace consistent-mass matrices are obtained as given below in Eqs. (25)–(29), respectively, where the material density ofall elements, cross-sectional areaA of the two line elements, and thicknesst of the two surface elementshave been assumed to be constant

M(1) = A

∫ l

0N(1)N(1) dx = Al

3, M(2) = A

∫ l

0N(2)N(2) dx = Al, (25a, b)


M(1) = A

∫ l

0[N(1)]T[N(1)] dx = Al

105

[105 −35l−35l 14l2

], (26a)

M(2) = A

∫ l

0[N(2)]T[N(2)] dx = Al

105

[51 −9l−9l 2l2

], (26b)

M(1) = t

∫ b

0

∫ a

0N(1)N(1) dx dy = tab

3, (27a)

M(2) = t

∫ b

0

∫ a


3, (27b)

M(3) = t

∫ b

0

∫ a

0N(3)N(3) dx dy = tab, (27c)

M(4) = t

∫ b

0

∫ a


9, (27d)

M(1) = t

∫ b

0

∫ a

0[N(1)]T[N(1)] = tab

135

[ 135 −45a −45b−45a 18a2 15ab−45b 15ab 18b2

], (28a)

M(2) = t

∫ b

0

∫ a

0[N(2)]T[N(2)] = tab

1575

[ 349 −52a −52b−52a 10a2 7ab−52b 7ab 10b2

], (28b)

M(3) = t

∫ b

0

∫ a

0[N(3)]T[N(3)] = tab

315

[ 153 −27a −42b−27a 6a2 7ab−42b 7ab 14b2

], (28c)

M(4) = t

∫ b

0

∫ a

0[N(4)]T[N(4)] = tab

315

[ 153 −42a −27b−42a 14a2 7ab−27b 7ab 6b2

], (28d)

M(1) =

∫ c

0

∫ b

0

∫ a

0N(1)N(1) dx dy dz= abc

3, (29a)

M(2) =

∫ c

0

∫ b

0

∫ a


3, (29b)

M(3) =

∫ c

0

∫ b

0

∫ a


3, (29c)

M(4) =

∫ c

0

∫ b

0

∫ a


27, (29d)

M(5) =

∫ c

0

∫ b

0

∫ a


9, (29e)


M(6) =

∫ c

0

∫ b

0

∫ a


9, (29f)

M(7) =

∫ c

0

∫ b

0

∫ a


9, (29g)

M(8) =

∫ c

0

∫ b

0

∫ a

0N(8)N(8) dx dy dz= abc. (29h)

9. Final element matrices

The results of the preceding section may readily be converted into conventional form through superpo-sition of the subspace matrices, followed by some simple coordinate transformations. For each element,we begin by writing down the permutation table for all the nodes of the element, which is a table showinghow the individual nodes are permuted under the operations of the symmetry group of the configurationin question. The results for the 2-node configurations of the truss and the beam elements (Fig. 1a), the4-node configurations of the rectangular plane-stress and plate-bending elements (Fig. 2a), and the 8-nodeconfiguration of the rectangular solid element (Fig. 3a), respectively, are as follows:

The element consistent-mass matrices in conventional form may then be assumed to be of the forms

M =[m1 m2m2 m1

], M =

m1 m2 m3 m4m2 m1 m4 m3m3 m4 m1 m2m4 m3 m2 m1

,

M =

m1 m2 m3 m4 m5 m6 m7 m8m2 m1 m4 m3 m6 m5 m8 m7m3 m4 m1 m2 m7 m8 m5 m6m4 m3 m2 m1 m8 m7 m6 m5m5 m6 m7 m8 m1 m2 m3 m4m6 m5 m8 m7 m2 m1 m4 m3m7 m8 m5 m6 m3 m4 m1 m2m8 m7 m6 m5 m4 m3 m2 m1

, (30–32)

respectively, for the 2-node configurations of the truss and the beam elements (Eq. (30)), the 4-nodeconfigurations of the rectangular plane-stress and plate-bending elements (Eq. (31)), and the 8-node


configuration of the rectangular solid element (Eq. (32)), the patterns of subscripting being given bythe respective permutation tables. The elementsmi of these matrices are then linear combinations of therespective subspace matricesM(j) (refer to Eqs. (25)–(29)), assembled with coefficients given by rowi ofthe corresponding matricesT (refer to Eqs. (1)–(3)). The results for the 2-node 2-d.o.f. truss element, the2-node 4-d.o.f. beam element, the 4-node 8-d.o.f. rectangular plane-stress element, the 4-node 12-d.o.f.rectangular plate-bending element and the 8-node 24-d.o.f. rectangular hexahedral element are given by

m1 = 1

2(M(1) +M(2))= 2Al

3, (33a)

m2 = 1

2(M(1) −M(2))= −Al

3, (33b)

m1 = 1

2(M(1) +M(2))= Al

210

[156 −22l−22l 4l2

], (34a)

m2 = 1

2(M(1) −M(2))= Al

210

[54 −13l

−13l 3l2

], (34b)

m1 = 1

4(M(1) +M(2) +M(3) +M(4))= 4tab

9, (35a)

m2 = 1

4(M(1) +M(2) −M(3) −M(4))= −tab

9, (35b)

m3 = 1

4(M(1) −M(2) +M(3) −M(4))= 2tab

9, (35c)

m4 = 1

4(M(1) −M(2) −M(3) +M(4))= −2tab

9, (35d)

m1 = 1

4(M(1) +M(2) +M(3) +M(4))= tab

3150

[ 1727 −461a −461b−461a 160a2 126ab−461b 126ab 160b2

], (36a)

m2 = 1

4(M(1) +M(2) −M(3) −M(4))= tab

3150

[ 197 −116a −116b−116a 60a2 56ab−116b 56ab 60b2

], (36b)

m3 = 1

4(M(1) −M(2) +M(3) −M(4))= tab

3150

[ 613 −199a −274b−199a 80a2 84ab−274b 84ab 120b2

], (36c)

m4 = 1

4(M(1) −M(2) −M(3) +M(4))= tab

3150

[ 613 −274a −119b−274a 120a2 84ab−199b 84ab 80b2

], (36d)


m1 = 1

8(M(1) +M(2) +M(3) +M(4) +M(5) +M(6) +M(7) +M(8))= 8abc

27, (37a)

m2 = 1

8(M(1) +M(2) −M(3) −M(4) +M(5) +M(6) −M(7) −M(8))= −2abc

27, (37b)

m3 = 1

8(M(1) −M(2) +M(3) −M(4) +M(5) −M(6) +M(7) −M(8))= −2abc

27, (37c)

m4 = 1

8(M(1) −M(2) −M(3) +M(4) +M(5) −M(6) −M(7) +M(8))= 2abc

27, (37d)

m5 = 1

8(M(1) +M(2) +M(3) +M(4) −M(5) −M(6) −M(7) −M(8))= −abc

27, (37e)

m6 = 1

8(M(1) +M(2) −M(3) −M(4) −M(5) −M(6) +M(7) +M(8))= 4abc

27, (37f)

m7 = 1

8(M(1) −M(2) +M(3) −M(4) −M(5) +M(6) −M(7) +M(8))= 4abc

27, (37g)

m8 = 1

8(M(1) −M(2) −M(3) +M(4) −M(5) +M(6) +M(7) −M(8))= −4abc

27. (37h)

The matrices of Eqs. (30)–(32), with elements as defined in Eqs. (33)–(37), are, of course, based onthe group-theoretic system of node numbering and positive directions for nodal quantities as depicted inFigs. 1–3, which in general differ from conventional systems. They look similar, but are not quite identicalto the conventional results one normally encounters in standard finite-element texts. To render these finalresults exactly coincident with the element consistent-mass matrices for any system of conventional nodenumbering and freedom directions, all that requires to be done is to compare the group-theoretic systemfor the element in question (as depicted inFigs. 1–3) with the conventional system and, where necessary,to interchange nodes and reverse directions of freedoms until the two systems coincide. Correspond-ing simple matrix transformations on the results of Eqs. (30)–(32) then, of course, convert the elementmatrices into the required conventional form. These last steps, being of a re-numbering/re-orientationnature, are relatively trivial, and we may thus regard the results represented by Eqs. (30)–(32) (with allparameters fully known as given by Eqs. (33)–(37)) as the element consistent-mass matrices in conven-tional form.

10. Numerical example

The presented formulation is intended to simplify and hence accelerate the numerical computationof element matrices. Once these are obtained, the remainder of the FEM steps are the same as for theconventional method. Hence to demonstrate the effectiveness of the formulation, we need only considerthe computation of element matrices based on a prescribed displacement field, and simply compare thecomputational time associated with the group-theoretic method against that associated with the conven-tional method.

We will consider the numerically challenging example of the 64-node 192 d.o.f solid hexahedralelement (Fig. 4). This example typifies the case of a symmetric element with a large number of nodes and


Fig. 4. Numerical example: the 64-node solid hexahedral element.

a large number of degrees of freedom, for which the group-theoretic method is expected to be particularlyadvantageous. Clearly, the size of this problem is not amenable to closed-form derivation of results, sonumerical solution procedures have to be adopted in computing element matrices for both the conventionaland the group-theoretic methods.

For this element, the following 64-term polynomial may be assumed for the displacement field:

u(x, y, z)= �1 + �2x + �3y + �4z+ �5x2 + �6xy + �7xz+ �8y

2 + �9yz+ �10z2 + �11x

3

+�12x2y+�13x

2z+�14xy2+�15xyz+�16y

3+�17y2z+�18z

2x+�19z2y

+�20z3+�21x

3y+�22x3z+�23x

2y2+�24x2yz+�25x

2z2+�26xy3

+�27xy2z+�28xyz

2+�29xz3+�30y

3z+�31y2z2+�32yz

3+�33x3y2

+�34x3yz+�35x

3z2+�36x2y3+�37x

2y2z+�38x2yz2+�39x

2z3

+�40xy3z+�41xy

2z2+�42xyz3+�43y

3z2+�44y2z3+�45x

3y3+�46x3y2z

+�47x3yz2+�48x

3z3+�49x2y3z+�50x

2y2z2+�51x2yz3+�52xy

3z2

+�53xy2z3+�54y

3z3+�55x3y3z+�56x

3y2z2+�57x3yz3+�58x

2y3z2

+�59x2y2z3+�60xy

3z3+�61x3y3z2+�62x

3y2z3+�63x2y3z3+�64x

3y3z3. (38)


Allocating the terms of the displacement field to the subspacesS(j) (j = 1,2, . . . ,8) associated with thesymmetry groupD2h, we obtain

u(1)(x, y, z)= �1 + �5x2 + �8y

2 + �10z2 + �23x

2y2 + �25x2z2 + �31y

2z2 + �50x2y2z2, (39a)

u(2)(x, y, z)= �6xy + �21x3y + �26xy

3 + �28xyz2 + �45x

3y3 + �47x3yz2

+ �52xy3z2 + �61x

3y3z2, (39b)

u(3)(x, y, z)= �7xz+ �22x3z+ �27xy

2z+ �29xz3 + �46x

3y2z+ �48x3z3

+ �53xy2z3 + �62x

3y2z3, (39c)

u(4)(x, y, z)= �9yz+ �24x2yz+ �30y

3z+ �32yz3 + �49x

2y3z+ �51x2yz3

+ �54y3z3 + �63x

2y3z3, (39d)

u(5)(x, y, z)= �15xyz+ �34x3yz+ �40xy

3z+ �42xyz3 + �55x

3y3z+ �57x3yz3

+ �60xy3z3 + �64x

3y3z3, (39e)

u(6)(x, y, z)= �4z+ �13x2z+ �17y

2z+ �20z3 + �37x

2y2z+ �39x2z3

+ �44y2z3 + �59x

2y2z3, (39f)

u(7)(x, y, z)= �3y + �12x2y + �16y

3 + �19yz2 + �36x

2y3 + �38x2yz2

+ �43y3z2 + �58x

2y3z2, (39g)

u(8)(x, y, z)= �2x + �11x3 + �14xy

2 + �18xz2 + �33x

3y2 + �35x3z2

+ �41xy2z2 + �56x

3y2z2. (39h)

For the numerical example, was assigned a value of 7800 kg/m3, and the element dimensions weretaken asa= 0.60 m,b= 0.45 m andc= 0.30 m. The origin of thex, y, z coordinate system was taken atthe centre of the element, as shown in the figure. Along a given direction (x, y or z), nodes were assumedto be equally spaced (thus coordinates for all the 64 nodes are fully known). Basically, the conventionalmethod required the computation of 64 constants{�1, �2, . . . , �64} first, followed by the integration of2080 distinct termsNiNj (i = 1,2, . . . ,64; j = 1,2, . . . ,64;mij =mji for i �= j ). On the other hand,the group-theoretic decomposition required the computation of 8 constants for each subspace, followedby the integration of 36 distinct termsNiNj (i = 1,2, . . . ,8; j = 1,2, . . . ,8;mij =mji for i �= j ) foreach subspace, which amount to 288 terms for the eight subspaces. The program employed standard pro-cedures for the solution of sets of simultaneous equations and for the numerical integration of functions.For the latter, 27-point three-dimensional Gauss quadrature was employed. Sample coefficientsmij of thecomputed element mass matrixM are given inTable 1. The computational times, normalised with respectto the time taken to perform the calculations of the conventional method, are summarised inTable 2.

The numerical values are practically the same between the methods, confirming the validity of thegroup-theoretic method. The relative computational times show that the group-theoretic method is 11times faster than the conventional approach in generating the elements of the mass matrix of the 64-nodesolid hexahedral element. For a 2 GHz Pentium-4 processor with 1 Gb of memory and running on Linex,the parameterst1 andt2 were timed at 0.0017 and 1.1919 s, respectively.


Table 1Sample coefficientsmij of the element mass matrix for the numerical example

i, j mij : conventional method mij : group-theoretic method

1,1 1.657 1.6541,32 0.140 0.1381,64 0.032 0.0337,7 35.830 35.8187,21 −5.635 −5.66423,7 41.892 41.91023,23 166.595 166.37832,32 7.706 7.69532,64 −1.037 −1.03964,64 1.657 1.654

Table 2Relative computational times of the conventional method and the group-theoretic method

Conventional method Group-theoretic method

Solution for constants 1.0t1 0.14t1Integration ofNiNj terms 1.0t2 0.09t2Total computational time 1.0(t1 + t2) ≈ 0.09(t1 + t2)

11. Concluding remarks

This paper has begun by reviewing various applications of group theory in the fields of solid andstructural mechanics, and then proceeded to focus on applications to finite-element analysis. Based onthe theory of symmetry groups and their representation, a general formulation for the efficient computationof matrices for symmetric finite elements has been presented. The key feature of the procedure is thedecomposition of the displacement field of the element into subfields of the same symmetry types as thesubspaces of the symmetry group associated with the element. This leads to the derivation of relativelysimple symmetry-adapted shape functions for these subspaces. Symmetry-adapted element matrices (suchas stiffness and consistent-mass matrices) are obtained separately for each subspace through the integrationof much simpler quantities (in comparison with the conventional approach).

Considerations have encompassed the 2-node 2-d.o.f. truss element, 2-node 4-d.o.f. beam element,4-node 8-d.o.f. rectangular plane-stress element, 4-node 12-d.o.f. rectangular plate-bending element and8-node 24-d.o.f. rectangular hexahedral element, for which subspace consistent-mass matrices have beenderived, and then linearly combined to give element consistent-mass matrices, which are the real matricesin the full vector space of the original problem, but based on the group-theoretic pattern of node numberingand positive directions of freedoms. Conversion of these results to suit conventional numbering of nodesand positive directions of freedoms may be effected through relatively trivial matrix transformationoperations. Consideration of a numerical example has demonstrated the effectiveness of the formulationin reducing computational time.


Overall, the subspace formulation for symmetric finite elements results in considerable reductions incomputational effort in comparison with the conventional approach. The decomposition feature of themethod may possibly allow the use of parallel processors, making the computations even faster, but theactual merit of this needs to be weighed against the costs and complexities of distributed computing.However, the formulation only becomes really advantageous in the case of finite elements with a highdegree of symmetry (typically solid hexahedral elements), and a large number of nodes and nodal degreesof freedom. This would then justify the cost of the decomposition inherent in the procedure.

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A group-theoretic formulation for symmetric ﬁnite elements

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