Computers and Structures 112-113 (2012) 266–282

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Computers and Structures

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A group-theoretic finite-difference formulation for plate eigenvalue problems

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

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 January 2012Accepted 28 August 2012Available online 12 October 2012

Keywords:Group theorySymmetryFinite-difference methodPlate vibrationEigenvalue analysis

0045-7949/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compstruc.2012.08.009

⇑ Tel.: +27 21 650 2601; fax: +27 21 650 5864.E-mail address: alphose.zingoni@uct.ac.za

We consider the problem of the free vibration of plates, and develop an efficient group-theoretic formu-lation for the solution of the problem by the method of finite differences. The procedure requires the writ-ing down of the finite-difference equations for a relatively small number of nodes. These equations arethen transformed into uncoupled sets of symmetry-adapted equations for the various subspaces, whichare solved independently to generate the eigenvalues in the original space of the problem. Not only doesthe procedure reduce computational effort, but it also allows valuable insights to be gained on the vibra-tion properties of the system.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Group theory provides a mathematical tool for the study ofphysical systems possessing symmetry. Applications to problemsin physics and chemistry are well-known [1–4]. Within the domainof solid and structural mechanics, group theory has been applied tosimplify problems of the vibration of a variety of structures [5–12],the bifurcation of rods, thin shells and skeletal space structures[13–16], and the statics and kinematics of trusses and frames[6,17–22]. These techniques have also been extended to thefinite-element method [23–26]. As may be seen from a recent sur-vey [27], applications of group theory to problems in solid andstructural mechanics are growing all the time. Apart from thegroup-theoretic methods, other approaches have also been devel-oped for exploiting symmetry in structures, such as those basedon concepts of graphs [28–30], or combinations of graphs andgroup theory [31,32].

The characteristic feature of all group-theoretic methods is thedecomposition of the vector space of the problem into a number ofsmaller subspaces spanned by symmetry-adapted variables as ba-sis vectors. The matrix equation describing the behaviour of thesystem is rendered into block-diagonal form, each block beingassociated with a particular subspace of the problem. The subspaceproblems are of much smaller dimension than the original prob-lem; they are solved-for independently of each other, thus achiev-ing considerable reductions in computational effort. The essentialattribute in all these problems is symmetry. It is the prerequisitefor the application of group theory.

ll rights reserved.

In this paper, and as a further extension of the application ofgroup theory to numerical methods, we consider the eigenvalueproblem of the free vibration of rectangular plates by the finite-difference method, and develop a group-theoretic formulation forthe analysis of the problem. This work is an improvement of a for-mulation that was originally presented by the author at a past con-ference [33]. In the previous work, the full set of finite-differenceequations for all the nodes of the finite-difference mesh had tobe written down first, before being reduced to smaller sets of equa-tions using symmetry-adapted operators. In the present improvedformulation, it is only necessary to write down the finite-differenceequations for the nodes corresponding to the first components ofthe basis vectors of each subspace. These nodes are usually (butnot always) the first nodes of the nodal sets of the mesh, a nodalset being the collection of all the nodes that are permuted by thesymmetry elements of the group. Here, a more efficient numberingscheme for the nodes of the mesh is also presented.

Although illustrated on the basis of rectangular and squareplates, it is evident that the whole approach is applicable to anyshape of plate (triangular, hexagonal, pentagonal, octagonal, etc.),as long as both the geometry of the plate and prescription ofboundary conditions conform to a recognisable symmetry group.The adopted pattern of the finite difference mesh would also needto conform to the same symmetry type as the global shape of theplate (that is, we choose a rectangular mesh in the case of rectan-gular plates, a square mesh in the case of square plates, a triangularmesh in the case of triangular plates, a hexagonal mesh in the caseof hexagonal plates, etc.). Furthermore, although we focus consid-erations on the vibration of plates, it is also evident that thepresented formulation (particularly the process by which thesymmetry-adapted finite-difference equations are generated) is

A. Zingoni / Computers and Structures 112-113 (2012) 266–282 267

applicable to a wide range of finite-difference problems in solidand structural mechanics, including the solution of the differentialequations of elasticity, plates, shells, heat flow, fluid flow, etc.

In these days where finite-element programs are readily avail-able, it may be argued that the finite-difference approach is nowobsolete. However, unlike a finite-element programme that doesnot easily allow the user to have control of the governing equationsof the problem, the finite-difference approach permits the user towork directly with the governing differential equations of theproblem, allowing one to have better mathematical control of thesolution procedure. In the finite-difference method, if the differen-tial equations describing the behaviour of a physical system areaccurate, then the finite-difference solution will be accurate, pro-vided a sufficiently fine mesh is adopted. On the other hand, theaccuracy of finite-element modelling will depend on a wider rangeof factors, such as the ability of the chosen elements to simulatethe real behaviour of the material at element level, or the abilityof the chosen formulation to satisfy the conditions of compatibilitybetween elements; furthermore, convergence of the solution maysometimes be elusive, even if one adopts a very fine mesh.

Despite this greater range of uncertainties, the advantage of thefinite-element approach is its greater flexibility (one does not needto know the governing equations of the system at the structural le-vel). Group-theoretic finite-element formulations [25] have alreadybeen developed to take advantage of this flexibility. In this paper,we focus attention on the development of group-theoretic finite-difference formulations, for the reasons given above.

In general, regardless of the type of problem in question (static,dynamic, stability, etc.), the group-theoretic approach, based onthe theory of symmetry groups and associated representation the-ory, has the capability of taking into account all the symmetryproperties of a configuration in a systematic manner. Even in thosecases where engineering structures exhibit small deviations fromperfect symmetry, group theory may still be employed to simplifythe analysis [34]. Indeed, various other techniques are available fordealing with small departures from symmetry and other forms ofregularity [35].

2. Finite-difference representation of the equation for platevibration

The equation of motion for the undamped free vibration of aplate may be written as [36]

@4w@x4 þ 2

@4w@x2@y2 þ

@4w@y4 þ

qD@2w@t2 ¼ 0 ð1Þ

where w is the transverse displacement at a point defined by thecoordinates {x,y} at any given time t, D is the flexural rigidity ofthe plate and q is the mass of the plate per unit area of its surface.For a plate of constant thickness h and material properties E(Young’s modulus of elasticity) and m (Poisson’s ratio), the flexuralrigidity D is given by

D ¼ Eh3

12ð1� m2Þ ð2Þ

Assuming harmonic vibration, we may write

wðx; y; tÞ ¼Wðx; yÞ sinxt ð3Þ

where W(x,y) is a shape function satisfying the boundary conditionsand describing the shape of the deflected middle surface of thevibrating plate, and x is a natural circular frequency of the plate.Substituting for w in Eq. (1), we obtain

@4W@x4 þ 2

@4W@x2@y2 þ

@4W@y4 � gW ¼ 0 ð4Þ

where

g ¼ qx2

Dð5Þ

The finite-difference representation of Eq. (4) at a pivotal point(m,n) of the mesh, based on central differences and taking equalmesh intervals d = Dx = Dy (in the x and y directions), is as follows[36]:

20Wm;n � 8 Wm�1;n þWmþ1;n þWm;n�1 þWm;nþ1ð Þþ 2 Wm�1;n�1 þWm�1;nþ1 þWmþ1;n�1 þWmþ1;nþ1ð Þ þWm�2;n

þWmþ2;n þWm;n�2 þWm;nþ2 � kWm;n ¼ 0 ð6Þ

where

k ¼ gd4 ð7Þ

To illustrate the implementation of the above equation, Fig. 1shows the 12 mesh points around a pivotal point denoted by O.Application of Eq. (6) to this arrangement yields the central-differ-ence equation for Point O as follows:

20W0 � 8ðW1 þW2 þW3 þW4Þ þ 2ðW6 þW8 þW10 þW12ÞþW5 þW7 þW9 þW11 � kW0 ¼ 0 ð8Þ

For mesh points adjacent to boundaries, fictitious points areintroduced in the usual way. Consideration of the boundaryconditions along the edges of the plate gives the deflectionvalues to be used in the central-difference equation for therespective fictitious points. For instance, the condition of zeroslope across a clamped edge (Fig. 2a) requires that the imagi-nary deflection at a given fictitious point be the same in mag-nitude and sign as that at the corresponding real mesh point,while the condition of zero moment and zero deflection acrossa simply supported edge (Fig. 2b) requires that the imaginarydeflection at a given fictitious point be the same in magnitudebut of opposite sign to that at the corresponding real meshpoint.

Fig. 1. Mesh points for the central-difference equation for an interior point O.

(a) (b)Fig. 2. Mesh points adjacent to boundaries: (a) clamped edge: w (at P) = w 0 (at P 0); (b) simply-supported edge: w (at P) = �w 0 (at P 0).

268 A. Zingoni / Computers and Structures 112-113 (2012) 266–282

3. Symmetry groups and idempotents

A set of elements {a,b,c, . . .,g, . . .} comprises a group G with re-spect to a binary operation (for example, multiplication) if the fol-lowing axioms are satisfied [2]:

(i) The product c of any two elements a and b of the group,denoted by c = ab, must be a unique element which alsobelongs to the group.

(ii) Among the elements of G, there must exist an identity ele-ment e which, when multiplied with any element a of thegroup, leaves the element unchanged: ea = ae = a.

(iii) For every element a of G, there must exist another element dalso belonging to the group G, such that ad = da = e; d isreferred to as the inverse of a, and denoted by a�1.

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

When all elements of G are symmetry operations, then thegroup G is called a symmetry group. Symmetry operations are trans-formations which bring an object into coincidence with itself, andleaves it indistinguishable from its original configuration. For finiteobjects, symmetry operations are typically of the following types:

(i) reflections in planes of symmetry, which we will denote byrl, where l is the plane of symmetry;

(ii) rotations about an axis of symmetry, which we will denoteby Cn, if the angle of rotation is 2p/n;

(iii) rotation-reflections, which we will denote by Sn; these rep-resent a rotation through an angle 2p/n, combined with areflection in the plane perpendicular to the axis of rotation.The special case of Sn when n = 2 is sometimes referred to asan ‘‘inversion’’ (symbol i). It represents a reflection throughthe centre of symmetry (that is, the one point of a finite objectwhich remains unmoved by all symmetry operations).

According to representation theory [8], idempotents P(i) of asymmetry group G are linear combinations of its elements satisfy-ing the relation P(i)P(i) = P(i), and P(i)P(j) = 0 if i – j. By operating onvectors of the space Q, an idempotent P(i) nullifies every vectorwhich does not belong to the subspace S(i), and selects all vectorsbelonging to the subspace S(i) (which will all have a definite sym-metry type characteristic of the subspace). It therefore acts as aprojection operator [8] of the subspace S(i). When applied uponthe functions /1, /2, . . .,/n of an n-dimensional physical problem,idempotents generate the symmetry-adapted functions for theirrespective subspaces, enabling basis vectors for the various

subspaces to be written down. As examples, idempotents of thegroups C2v, C3v, C4v and C6v, describing the symmetry propertiesof a rectangle, an equilateral triangle, a square and a regular hexa-gon, respectively, are as follows [10]:

Group C2v (applicable to rectangular plates)

Pð1Þ ¼ 14ðeþ C2 þ rx þ ryÞ ð9aÞ

Pð2Þ ¼ 14ðeþ C2 � rx � ryÞ ð9bÞ

Pð3Þ ¼ 14ðe� C2 þ rx � ryÞ ð9cÞ

Pð4Þ ¼ 14ðe� C2 � rx þ ryÞ ð9dÞ

Group C3v (applicable to triangular plates)

Pð1Þ ¼ 16ðeþ C3 þ C�1

3 þ r1 þ r2 þ r3Þ ð10aÞ

Pð2Þ ¼ 16ðeþ C3 þ C�1

3 � r1 � r2 � r3Þ ð10bÞ

Pð3Þ ¼ 13ð2e� C3 � C�1

3 Þ ð10cÞ

Group C4v (applicable to square plates)

Pð1Þ ¼ 18ðeþ C4 þ C�1

4 þ C2 þ rx þ ry þ r1 þ r2Þ ð11aÞ

Pð2Þ ¼ 18ðeþ C4 þ C�1

4 þ C2 � rx � ry � r1 � r2Þ ð11bÞ

Pð3Þ ¼ 18ðe� C4 � C�1

4 þ C2 þ rx þ ry � r1 � r2Þ ð11cÞ

Pð4Þ ¼ 18ðe� C4 � C�1

4 þ C2 � rx � ry þ r1 þ r2Þ ð11dÞ

Pð5Þ ¼ 12ðe� C2Þ ¼ Pð5;1Þ þ Pð5;2Þ ð11eÞ

where for Eq. (11e), either of Eqs. (12) or (13) below hold:

Pð5;1Þ ¼ 14ðe� C2 þ r1 � r2Þ;

Pð5;2Þ ¼ 14ðe� C2 � r1 þ r2Þ ð12a;bÞ

Pð5;1Þ ¼ 14ðe� C2 þ rx � ryÞ;

Pð5;2Þ ¼ 14ðe� C2 � rx þ ryÞ ð13a;bÞ

A. Zingoni / Computers and Structures 112-113 (2012) 266–282 269

Group C6v (applicable to hexagonal plates)

Pð1Þ ¼ 112ðeþ C6 þ C�1

6 þ C3 þ C�13 þ C2 þ ra þ rb

þ rc þ r1 þ r2 þ r3Þ ð14aÞ

Pð2Þ ¼ 112ðeþ C6 þ C�1

6 þ C3 þ C�13 þ C2 � ra � rb

� rc � r1 � r2 � r3Þ ð14bÞ

Pð3Þ ¼ 112ðe� C6 � C�1

6 þ C3 þ C�13 � C2 þ ra þ rb

þ rc � r1 � r2 � r3Þ ð14cÞ

Pð4Þ ¼ 112ðe� C6 � C�1

6 þ C3 þ C�13 � C2 � ra � rb

� rc þ r1 þ r2 þ r3Þ ð14dÞ

Pð5Þ ¼ 16ð2eþ C6 þ C�1

6 � C3 � C�13 � 2C2Þ ð14eÞ

Pð6Þ ¼ 16ð2e� C6 � C�1

6 � C3 � C�13 þ 2C2Þ ð14fÞ

A symmetry group G with k idempotents {P(1),P(2), . . .,P(k)} gen-erally decomposes an n-dimensional problem (i.e. one with n de-

(a)

(c)Fig. 3. Symmetry-conforming finite-difference mesh patterns: (a) square plate with a striangular mesh; (d) hexagonal plate with a hexagonal mesh. All plates are simply-supp

grees of freedom in the case of vibration problems) into kcorresponding independent subspaces {S(1),S(2), . . .,S(k)}, each S(i)(-i = 1,2, . . .,k) spanned by a number ri of symmetry-adapted func-tions that is typically small in comparison with n, sincer1 + r2 + � � � + rk = n. Problems belonging to the symmetry groupsC2v,C3v,C4v and C6v therefore potentially decompose into 4, 3, 5and 6 subspaces respectively.

For the symmetry group C4v associated with square meshes,the conceptual splitting of idempotent P(5) of subspace S(5) intotwo independent operators P(5,1) and P(5,2) is not a standard resultfrom representation theory, but a means for decomposing thesubspace S(5) into two identical subspaces S(5,1) and S(5,2) whichare spanned by physically indistinguishable symmetry-adaptedfunctions. For the two independent operators P(5,1) and P(5,2), wemay adopt the pair given as Eqs. (12) or Eqs. (13) but not both.Adopting either pair, it may readily be seen thatP(5,1)P(5,1) = P(5,1) and P(5,2)P(5,2) = P(5,2). Moreover, and more impor-tantly, it is established that P(5,1)P(5,2) = 0 (the operators are mutu-ally orthogonal) and P(5,1)P(j) = P(5,2)P(j) = 0 for j = {1,2,3,4} (eachoperator is also orthogonal to the idempotents of the first foursubspaces of the group).

(b)

(d)quare mesh; (b) rectangular plate with a square mesh; (c) triangular plate with aorted at the corners.

270 A. Zingoni / Computers and Structures 112-113 (2012) 266–282

4. Group-theoretic computational procedure

The steps involved in the group-theoretic implementation ofthe finite-difference method have been developed in detail in thecontext of this study, and will be illustrated by reference to a rect-angular mesh and a square mesh. Before we consider these specificmesh configurations, we summarise the developed group-theoreticprocedure as follows:

1. Identify the symmetry elements (and hence the symmetrygroup G) of the plate configuration. In general, rectangularplates will belong to the symmetry group C2v with four symme-try elements, square plates will belong to the symmetry groupC4v with eight symmetry elements, plates with the shape ofan equilateral triangle will belong to the symmetry group C3vwith six symmetry elements, and plates with the shape of a reg-ular hexagon will belong to the symmetry group C6v with 12symmetry elements, and so forth.

2. Choose a suitably fine finite-difference mesh whose patternconforms to the overall symmetry of the plate (e.g. a square gridfor a square plate; a square or a rectangular grid for a rectangu-lar plate; a triangular grid for a triangular plate; a triangular or ahexagonal grid for a hexagonal plate). Fig. 3 illustrates somesymmetry-conforming finite-difference mesh patterns.

3. Choose the centre of symmetry of the plate as the origin of theco-ordinate system, and plot all symmetry axes of the plate andlabel these x,y,z,1,2,3,a,b,c as appropriate. For rectangular andsquare plates, x and y are the reflection planes of symmetrycoinciding with the co-ordinate axes, while z is the axis of rota-tional symmetry passing through the origin of the co-ordinatesystem and perpendicular to the plate midsurface. The squareplate has the additional reflection planes of symmetry 1 and 2coinciding with the diagonals of the plate.

4. Number the nodes of the mesh as follows: If a node coincideswith the centre of symmetry of the plate, label this as Node 1.Limiting our considerations to rectangular and square plates,choose a node in the positive-positive quadrant of the coordi-nate system (or on the positive branch of the x or y axis), andnearest to the centre of symmetry, and label this Node 2 (thiswould be Node 1 if no node coincides with the centre of sym-metry). Then number all nodes belonging to the same permuta-tion set as this node, in the order generated by permuting thenode through successive symmetry operations of the group Gas given by the character table of the group. Pick the next meshnode in the positive-positive quadrant (or on the positivebranch of the x or y axis), and continue the numbering in theorder of permutation of the node. Repeat (generally movingoutward from the centre of symmetry) until all nodes of themesh are numbered from 1 up to n.

5. For each subspace of the problem, apply the associated idempo-tent to each nodal function /1,/2, . . .,/n, and select the indepen-dent basis vectors for the subspace.

6. Write down the conventional finite-difference equations foronly the nodes corresponding to the first components of the basisvectors of each subspace (these are usually the first nodes ofthe nodal sets of the mesh). The first components of the basisvectors of the various subspaces are usually the same, allowingone set of conventional finite-difference equations to serve forall the subspaces of the problem.

7. For each subspace of the problem, use the associated basis vec-tors to transform the conventional finite-difference equations(for the nodes corresponding to the first components of thebasis vectors of the subspace) into symmetry-adapted finite-difference equations for the subspace.

8. From the vanishing condition for the determinant of the ensu-ing eigenvalue matrix for a given subspace, solve for the eigen-

values for that subspace. These are also eigenvalues of theconventional finite-difference system. Generate all the eigen-values (hence the natural frequencies of vibration) of the sys-tem by solving for the eigenvalues for the individualsubspaces independently of each other (a much easier task thantackling the full problem).

9. Substitute a particular eigenvalue of a given subspace into thereduced eigenvalue equation for the subspace, to obtain aneigenvector in the r-dimensional subspace (r� n). Convert thisto an eigenvector in the original n-dimensional vector space ofthe problem by allocating the calculated values of the subspaceeigenvector components to all nodes associated with the basisvector. In this way, we obtain the conventional mode shapesof vibration, and this completes the analysis.

The above procedure is amenable to computer programmingand, indeed, efforts are already underway to develop algorithmsfor implementation in general group-theoretic computationalschemes. An algorithm has already been proposed for the auto-matic recognition of symmetry [37], as the first step in the comput-erised implementation of the group-theoretic procedure.

5. Application to rectangular and square plates

Fig. 4(a) shows a rectangular plate simply supported on all 4edges, with a regular grid of mesh lines in the x and y directionsgiving a total of 24 mesh points on the plate. Fig. 4(b) shows asquare plate clamped on all four edges, with a regular grid of meshlines in the x and y directions giving a total of 25 mesh points onthe plate. In both cases, the origin of the x, y coordinate systemis located at the centre of symmetry of the plate, and in the caseof the square plate, the additional diagonal planes of symmetryare labelled 1 � 1 and 2 � 2. Clearly, the mesh configuration forthe rectangular plate belongs to symmetry group C2v, while themesh configuration for the square plate belongs to symmetrygroup C4v. Mesh points on the plates have been numbered in accor-dance with the group-theoretic procedure outlined in the previoussection. Also shown outside the boundary of the plate (as mirrorimages of the adjacent nodes on the plate) are the relevant ficti-tious nodes, for use in the finite-difference equations for the realnodes.

5.1. Basis vectors for the rectangular plate

As per Step 5 of the previous section, applying the idempotentsP(1), P(2), P(3) and P(4) of the symmetry group C2v (Eqs. 9), for sub-spaces S(1), S(2), S(3) and S(4) respectively, to each of the 24 nodalfunctions /1,/2, . . .,/24 associated with the mesh points of the rect-angular plate, we obtain 24 linear combinations of these functionsfor each subspace, not all of which are independent (some areidentical to each other or linearly dependent). For a given sub-space, we then select a set of linearly independent combinationsof functions as the basis vectors for the subspace. It is found thatall four subspaces are 6-dimensional (i.e. have 6 basis vectorseach). The basis vectors may be taken as follows:

Subspace S(1)

Uð1Þ1 ¼ /1 þ /2 þ /3 þ /4 ð15aÞUð1Þ2 ¼ /5 þ /6 þ /7 þ /8 ð15bÞUð1Þ3 ¼ /9 þ /10 þ /11 þ /12 ð15cÞUð1Þ4 ¼ /13 þ /14 þ /15 þ /16 ð15dÞUð1Þ5 ¼ /17 þ /18 þ /19 þ /20 ð15eÞUð1Þ6 ¼ /21 þ /22 þ /23 þ /24 ð15fÞ

(a)

(b)

Fig. 4. Illustrative examples of rectangular and square plates: (a) rectangular plate simply-supported along all four edges and having a square mesh with 24 mesh points;(b) square plate clamped along all four edges and having a square mesh with 25 mesh points.

A. Zingoni / Computers and Structures 112-113 (2012) 266–282 271

Subspace S(2)

Uð2Þ1 ¼ /1 þ /2 � /3 � /4 ð16aÞUð2Þ2 ¼ /5 þ /6 � /7 � /8 ð16bÞUð2Þ3 ¼ /9 þ /10 � /11 � /12 ð16cÞUð2Þ4 ¼ /13 þ /14 � /15 � /16 ð16dÞUð2Þ5 ¼ /17 þ /18 � /19 � /20 ð16eÞUð2Þ6 ¼ /21 þ /22 � /23 � /24 ð16fÞ

Subspace S(3)

Uð3Þ1 ¼ /1 � /2 þ /3 � /4 ð17aÞUð3Þ2 ¼ /5 � /6 þ /7 � /8 ð17bÞUð3Þ3 ¼ /9 � /10 þ /11 � /12 ð17cÞUð3Þ4 ¼ /13 � /14 þ /15 � /16 ð17dÞUð3Þ5 ¼ /17 � /18 þ /19 � /20 ð17eÞUð3Þ6 ¼ /21 � /22 þ /23 � /24 ð17fÞ

(a) (b)

(c) (d)Fig. 5. Symmetry types of the four subspaces of the rectangular-plate problem illustrated by the 6th basis vector of each subspace: (a) Uð1Þ6 of subspace S(1); (b) Uð2Þ6 ofsubspace S(2); (c) Uð3Þ6 of subspace S(3); (d) Uð4Þ6 of subspace S(4). Filled (black) circles denote positive co-ordinates of the basis vectors, while unfilled circles (rings) denotenegative co-ordinates.

272 A. Zingoni / Computers and Structures 112-113 (2012) 266–282

Subspace S(4)

Uð4Þ1 ¼ /1 � /2 � /3 þ /4 ð18aÞUð4Þ2 ¼ /5 � /6 � /7 þ /8 ð18bÞUð4Þ3 ¼ /9 � /10 � /11 þ /12 ð18cÞUð4Þ4 ¼ /13 � /14 � /15 þ /16 ð18dÞUð4Þ5 ¼ /17 � /18 � /19 þ /20 ð18eÞUð4Þ6 ¼ /21 � /22 � /23 þ /24 ð18fÞ

The symmetry types associated with the above four subspacesof the problem may be visualised by reference to Fig. 5, in whichthe sixth basis vector of each subspace has been plotted. Since allbasis vectors of a given subspace have the same symmetry type, thereis no point in plotting all of them in order to illustrate the symme-try. We have therefore chosen one representative basis vector foreach subspace.

5.2. Basis vectors for the square plate

We implement Step 5 of the previous section by applying theidempotents P(1), P(2), P(3), P(4), P(5,1) and P(5,2) of the symmetrygroup C4v (Eqs. (10)–(12)), for subspaces S(1), S(2), S(3), S(4), S(5,1)

and S(5,2) respectively, to each of the 25 nodal functions

/1,/2, . . .,/25 associated with the mesh points of the square plate.This results in 25 linear combinations of these functions for eachsubspace, not all of which are independent. Selecting for each sub-space a set of linearly independent combinations of functions asthe basis vectors for the subspace, we obtain the results:

Subspace S(1)

Uð1Þ1 ¼ /1 ð19aÞUð1Þ2 ¼ /2 þ /3 þ /4 þ /5 ð19bÞUð1Þ3 ¼ /6 þ /7 þ /8 þ /9 ð19cÞUð1Þ4 ¼ /10 þ /11 þ /12 þ /13 ð19dÞUð1Þ5 ¼ /14 þ /15 þ /16 þ /17 þ /18 þ /19 þ /20 þ /21 ð19eÞUð1Þ6 ¼ /22 þ /23 þ /24 þ /25 ð19fÞ

Subspace S(2)

Uð2Þ1 ¼ /14 þ /15 þ /16 þ /17 � /18 � /19 � /20 � /21 ð20Þ

Subspace S(3)

Uð3Þ1 ¼ /2 � /3 � /4 þ /5 ð21aÞUð3Þ2 ¼ /10 � /11 � /12 þ /13 ð21bÞUð3Þ3 ¼ /14 � /15 � /16 þ /17 þ /18 þ /19 � /20 � /21 ð21cÞ

(a) (b)

(c) (d)Fig. 6. Symmetry types of the first four subspaces of the square-plate problem illustrated by one basis vector of each subspace: (a) Uð1Þ5 of subspace S(1); (b) Uð2Þ1 of subspaceS(2); (c) Uð3Þ3 of subspace S(3); (d) Uð4Þ2 of subspace S(4). Filled (black) circles denote positive co-ordinates of the basis vectors, while unfilled circles (rings) denote negative co-ordinates.

A. Zingoni / Computers and Structures 112-113 (2012) 266–282 273

Subspace S(4)

Uð4Þ1 ¼ /6 � /7 � /8 þ /9 ð22aÞUð4Þ2 ¼ /14 � /15 � /16 þ /17 � /18 � /19 þ /20 þ /21 ð22bÞUð4Þ3 ¼ /22 � /23 � /24 þ /25 ð22cÞ

Subspace S(5,1)

Uð5;1Þ1 ¼ /2 þ /3 � /4 � /5 ð23aÞUð5;1Þ2 ¼ /7 � /8 ð23bÞUð5;1Þ3 ¼ /10 þ /11 � /12 � /13 ð23cÞUð5;1Þ4 ¼ /14 � /17 þ /20 � /21 ð23dÞUð5;1Þ5 ¼ /15 � /16 þ /18 � /19 ð23eÞUð5;1Þ6 ¼ /23 � /24 ð23fÞ

Subspace S(5,2)

Uð5;2Þ1 ¼ /2 � /3 þ /4 � /5 ð24aÞUð5;2Þ2 ¼ /6 � /9 ð24bÞUð5;2Þ3 ¼ /10 � /11 þ /12 � /13 ð24cÞUð5;2Þ4 ¼ /14 � /17 � /20 þ /21 ð24dÞUð5;2Þ5 ¼ /15 � /16 � /18 þ /19 ð24eÞUð5;2Þ6 ¼ /22 � /25 ð24fÞ

The symmetry types associated with the first four subspaces(S(1)–S(4)) are depicted in Fig. 6, in which only one representativebasis vector of each subspace has been selected for plotting. Forsubspaces S(5,1) and S(5,2), we have plotted all the basis vectors of

Fig. 7. Basis vectors of subspaces S(5,1) and S(5,2) of the square-plate problemshowing the symmetry type of each subspace, and orientation of the symmetry andantisymmetry planes of each vector plot: (a) subspace S(5,1):

Uð5;1Þ1 ;Uð5;1Þ2 ;Uð5;1Þ3 ;Uð5;1Þ4 ;Uð5;1Þ5 ;Uð5;1Þ6

n o; (b) subspace S(5,2): Uð5;2Þ1 ;Uð5;2Þ2 ;Uð5;2Þ3 ;Uð5;2Þ4 ;

nUð5;2Þ5 ;Uð5;2Þ6 g.

274 A. Zingoni / Computers and Structures 112-113 (2012) 266–282

each subspace (see Fig. 7), not only to show the type of symmetryassociated with these subspaces, but more importantly, to illus-trate the fact that subspaces S(5,1) and S(5,2) always yield sets of ba-sis vectors that are identical except for orientation. The plots forsubspace S(5,1) are symmetrical about the diagonal axis 1 � 1 andantisymmetrical about the diagonal axis 2 � 2, while those for sub-space S(5,2) are symmetrical about the axis 2 � 2 and antisymmet-rical about the axis 1 � 1. Clearly, orientation of the plots of thebasis vectors does not affect the physical properties of the system(frequencies and mode shapes). It follows therefore that subspacesS(5,1) and S(5,2) will yield identical sets of solutions for the eigen-values, so it will be sufficient to consider only one of these sub-spaces. The matching solutions in subspaces S(5,1) and S(5,2)

correspond to doubly-repeating roots in the full space of the con-ventional problem.

5.3. Nodal sets

We will define a nodal set of a finite-difference mesh as the setof all nodes that are permuted by the symmetry elements of thegroup G of the system. For instance, if we apply the symmetry ele-ments {e,C2,rx,ry} of the group C2v to Node 1 of the mesh of therectangular plate (Fig. 4a), we generate the nodal positions{1,2,3,4}, which therefore constitute a nodal set of the system. No-dal sets of the finite-difference mesh of the square plate (Fig. 4b)are the sets of all positions that are permuted by the symmetry ele-ments fe; C4;C

�14 ;C2;rx;ry;r1;r2g of the group C4v. For our two

examples, the nodal sets are obtained as follows:

Rectangular plate:

f1;2;3;4gf5;6;7;8gf9;10;11;12gf13;14;15;16gf17;18;19;20gf21;22;23;24g

Square plate:

f1gf2;3;4;5gf6;7;8;9gf10;11;12;13gf14;15;16;17;18;19;20;21gf22;23;24;25g

The consecutive numbering within the nodal sets is a directconsequence of the rules for node numbering that were definedin Step 4 of Section 4.

6. Finite-difference equations for generator nodes of the basisvectors

6.1. Generator nodes

In the present group-theoretic procedure, we do not need towrite down the finite-difference equations for all the nodes ofthe mesh, and then operate on these to reduce the number of equa-tions. Instead, we need only write down the finite-difference equationsfor the nodes corresponding to the first components of the basisvectors of each subspace, and operate on this greatly reduced set ofequations in order to generate all the required symmetry-adapted

A. Zingoni / Computers and Structures 112-113 (2012) 266–282 275

finite-difference equations for the various subspaces of the problem.This represents a major improvement over previous group-theo-retic finite-difference procedures [33]. We will refer to these spe-cial nodes as generator nodes of the basis vectors.

For the rectangular plate (C2v symmetry), all subspaces S(i)(i =1,2,3,4) each have 6 basis vectors UðiÞ1 ;U

ðiÞ2 ;U

ðiÞ3 ;U

ðiÞ4 ;U

ðiÞ5 ;U

ðiÞ6

n owhose first components are {/1,/5,/9,/13,/17,/21} respectively,irrespective of the subspace. The nodes corresponding to the firstcomponents of the basis vectors (that is, the generator nodes) aretherefore {1,5,9,13,17,21} for all four subspaces. We note thatthese are also the first nodes of the nodal sets (see Section 5.3).We will therefore need to write down finite-difference relation-ships for these 6 nodes only (not for the entire 24 nodes of theproblem), and be able to derive reduced sets of equations for allsubspaces on the basis of these.

For the square plate (C4v symmetry), the first components of thebasis vectors UðiÞ1 ;U

ðiÞ2 ; . . . ;UðiÞr , for each subspace S(i) of dimension r,

are as follows (these are given in the respective order of the basisvectors):

Subspace S(1): {/1,/2,/6,/10,/14,/22}Subspace S(2): {/14}Subspace S(3): {/2,/10,/14}Subspace S(4): {/6,/14,/22}Subspace S(5,1): {/2,/7,/10,/14,/15,/23}Subspace S(5,2): {/2,/6,/10,/14,/15,/22}

For subspace S(1), the generator nodes of the basis vectors aretherefore {1,2,6,10,14,22}, which are also the first nodes of the no-dal sets (see Section 5.3). The sets of generator nodes for subspacesS(2),S(3) and S(4) are all subsets of the set of generator nodes for sub-space S(1), so the set of finite-difference equations written down forsubspace S(1) will also cover subspace S(2) (where we will need theequation for Node 14 only), subspace S(3) (where we will need theequations for Nodes 2, 10 and 14 only) and subspace S(4) (where wewill need the equations for Nodes 6, 14 and 22 only).

Subspaces S(5,1) and S(5,2) need to be treated a little differently.Subspace S(5,1) will require the finite-difference equations forNodes 2, 10 and 14, which are already included in the set of equa-tions for subspace S(1), and the additional finite-difference equa-tions for Nodes 7, 15 and 23. On the other hand, subspace S(5,2)

will require the finite-difference equations for Nodes 2, 6, 10, 14and 22 (all of which are already included in the set of equationsfor subspace S(1)), and only one additional finite-difference equa-tion for Node 15. Since subspaces S(5,1) and S(5,2) yield identical setsof eigenvalues for reasons already explained, we need consideronly one of them. We choose subspace S(5,2) since this requires onlyone additional finite-difference equation (for Node 15).

In summary, for the rectangular plate, the nodes for which fi-nite-difference equations need to be written down are:

f1;5;9;13;17;21g

while for the square plate, the nodes for which finite-differenceequations need to be written down are:

f1;2;6;10;14;15;22g:

6.2. Basis finite-difference equations for the rectangular plate

All edges are simply supported, implying that W = 0 for allnodes lying on the edges of the plate. For the fictitious nodes ofthe finite-difference mesh (see Fig. 4(a)), the deflections W areequal in magnitude but opposite in sign to those of the correspond-ing real nodes. The finite-difference equations for the relevantnodes of the problem (i.e. Nodes {1,5,9,13,17,21}) are as follows:

Node 1

ð20� kÞW1 þ 2W2 � 8W3 � 8W4 � 8W5 þ 2W7 þW8

þW9 � 8W13 þW15 þ 2W16 þ 2W17 ¼ 0 ð25aÞ

Node 5

� 8W1 þ 2W3 þW4 þ ð20� kÞW5 � 8W7 � 8W9 þ 2W11

þ 2W13 � 8W17 þW19 þ 2W21 ¼ 0 ð25bÞ

Node 9

W1 � 8W5 þ 2W7 þ ð19� kÞW9 � 8W11 þ 2W17

� 8W21 þW23 ¼ 0 ð25cÞ

Node 13

� 8W1 þW3 þ 2W4 þ 2W5 þ ð19� kÞW13 � 8W16

� 8W17 þW20 þW21 ¼ 0 ð25dÞ

Node 17

2W1 � 8W5 þW7 þ 2W9 � 8W13 þW16

þ ð19� kÞW17 � 8W21 ¼ 0 ð25eÞ

Node 21

2W5 � 8W9 þW11 þW13 � 8W17 þ ð18� kÞW21 ¼ 0 ð25fÞ

6.3. Basis finite-difference equations for the square plate

All edges are fixed, implying that W = 0 for all nodes lying on theedges of the plate. For the fictitious nodes of the finite-differencemesh (see Fig. 4(b)), the deflections W are equal in magnitudeand of the same sign as those of the corresponding real nodes.The finite-difference equations for the relevant nodes of the prob-lem (i.e. Nodes {1,2,6,10,14,15,22}) are:

Node 1

ð20� kÞW1 � 8W2 � 8W3 � 8W4 � 8W5 þ 2W6 þ 2W7

þ 2W8 þ 2W9 þW10 þW11 þW12 þW13 ¼ 0 ð26aÞ

Node 2

� 8W1 þ ð20� kÞW2 þ 2W3 þ 2W4 þW5 � 8W6

� 8W7 � 8W10 þ 2W14 þW15 þ 2W18 þW21 ¼ 0 ð26bÞ

Node 6

2W1 � 8W2 � 8W4 þ ð20� kÞW6 þW7 þW8 þ 2W10

þ 2W12 � 8W14 � 8W21 þ 2W22 ¼ 0 ð26cÞ

Node 10

W1 � 8W2 þ 2W6 þ 2W7 þ ð21� kÞW10 � 8W14

� 8W18 þW22 þW23 ¼ 0 ð26dÞ

Node 14

2W2 þW4 � 8W6 � 8W10 þ ð21� kÞW14 þW18

þ 2W21 � 8W22 ¼ 0 ð26eÞ

Node 15

W2 þ 2W3 � 8W7 � 8W11 þ ð21� kÞW15 þ 2W18

þW20 � 8W23 ¼ 0 ð26fÞ

Node 22

2W6 þW10 þW12 � 8W14 � 8W21 þ ð22� kÞW22 ¼ 0

ð26gÞ

276 A. Zingoni / Computers and Structures 112-113 (2012) 266–282

7. Symmetry-adapted finite-difference equations and systemeigenvales

7.1. General

For any given basis vector UðiÞj of subspace S(i), the coefficients ofthe components / are either + 1 or � 1 for all subspaces of the C2vproblem (rectangular plate) and the C4v problem (square plate).These coefficients give the relative values of the transversedisplacements associated with the nodes of Uj. For a givenr-dimensional subspace, we will denote the amplitude of thedisplacements associated with the nodes of Uj (j = 1,2, . . .,r) bythe parameter fj. This amplitude will be the same for all nodes ofUj. This formulation results in an r-dimensional eigenvalue prob-lem within the subspace S(i), which upon solving yields the r eigen-values (or natural circular frequencies of vibration of the plate) forthat subspace. Very significantly, these subspace eigenvalues arealso eigenvalues of the full vector space of the vibration problem(no further computations are required), and by collecting togetherall the sets of eigenvalues yielded by the various subspaces of theproblem, we obtain the full set of natural frequencies for the plate.

The detailed computations and obtained results for the twoexamples of present considerations are given in the sections below.

7.2. Rectangular plate

Subspace S(1)

From the coefficients of the Uð1Þj ðj ¼ 1;2; . . . ;6Þ, let

W1 ¼W2 ¼W3 ¼W4 ¼ f1

W5 ¼W6 ¼W7 ¼W8 ¼ f2

W9 ¼W10 ¼W11 ¼W12 ¼ f3

W13 ¼W14 ¼W15 ¼W16 ¼ f4

W17 ¼W18 ¼W19 ¼W20 ¼ f5

W21 ¼W22 ¼W23 ¼W24 ¼ f6

Making the above substitutions into each of Eqs. (25a)–(25f), we ob-tain six equations in {f1, f2, f3, f4, f5, f6}:

ð6� kÞ �5 1 �5 2 0�5 ð12� kÞ �6 2 �7 21 �6 ð11� kÞ 0 2 �7�5 2 0 ð11� kÞ �7 12 �7 2 �7 ð19� kÞ �80 2 �7 1 �8 ð18� kÞ

2666666664

3777777775

f1

f2

f3

f4

f5

f6

2666666664

3777777775¼

000000

2666666664

3777777775

ð27Þ

The vanishing condition for the determinant of the above 6x6 ma-trix yields a 6th-degree polynomial equation in k, whose roots(the required eigenvalues) are obtained as:

k1 ¼ 0:336; k2 ¼ 3:752; k3 ¼ 7:930;

k4 ¼ 13:169; k5 ¼ 17:414; k6 ¼ 34:398

Subspace S(2)

From the coefficients of the Uð2Þj ðj ¼ 1;2; . . . ;6Þ, let

W1 ¼W2 ¼ �W3 ¼ �W4 ¼ f1

W5 ¼W6 ¼ �W7 ¼ �W8 ¼ f2

W9 ¼W10 ¼ �W11 ¼ �W12 ¼ f3

W13 ¼W14 ¼ �W15 ¼ �W16 ¼ f4

W17 ¼W18 ¼ �W19 ¼ �W20 ¼ f5

W21 ¼W22 ¼ �W23 ¼ �W24 ¼ f6

Making the above substitutions into each of Eqs. (25a)–(25f), weobtain:

ð38� kÞ �11 1 �11 2 0�11 ð28� kÞ �10 2 �9 2

1 �10 ð27� kÞ 0 2 �9�11 2 0 ð27� kÞ �9 1

2 �9 2 �9 ð19� kÞ �80 2 �9 1 �8 ð18� kÞ

2666666664

3777777775

f1

f2

f3

f4

f5

f6

2666666664

3777777775¼

000000

2666666664

3777777775

ð28Þ

The vanishing condition for the determinant of the above 6 � 6 ma-trix yields a 6th-degree polynomial equation in k, whose roots (therequired eigenvalues) are obtained as:

k1 ¼ 4:558; k2 ¼ 14:646; k3 ¼ 19:106;

k4 ¼ 26:873; k5 ¼ 36:761; k6 ¼ 55:056

Subspace S(3)

From the coefficients of the Uð3Þj ðj ¼ 1;2; . . . ;6Þ, let

W1 ¼ �W2 ¼W3 ¼ �W4 ¼ f1

W5 ¼ �W6 ¼W7 ¼ �W8 ¼ f2

W9 ¼ �W10 ¼W11 ¼ �W12 ¼ f3

W13 ¼ �W14 ¼W15 ¼ �W16 ¼ f4

W17 ¼ �W18 ¼W19 ¼ �W20 ¼ f5

W21 ¼ �W22 ¼W23 ¼ �W24 ¼ f6

Making the above substitutions into each of Eqs. (25a)–(25f), weobtain:

ð18� kÞ �7 1 �9 2 0�7 ð12� kÞ �6 2 �7 21 �6 ð11� kÞ 0 2 �7�9 2 0 ð27� kÞ �9 12 �7 2 �9 ð19� kÞ �80 2 �7 1 �8 ð18� kÞ

2666666664

3777777775

f1

f2

f3

f4

f5

f6

2666666664

3777777775¼

000000

2666666664

3777777775

ð29Þ

The vanishing condition for the determinant of the above 6 � 6 ma-trix yields a 6th-degree polynomial equation in k, whose roots (therequired eigenvalues) are obtained as:

k1 ¼ 1:288; k2 ¼ 7:992; k3 ¼ 11:364;

k4 ¼ 17:505; k5 ¼ 25:635; k6 ¼ 41:216

Subspace S(4)

From the coefficients of the Uð4Þj ðj ¼ 1;2; . . . ;6Þ, let

W1 ¼ �W2 ¼ �W3 ¼W4 ¼ f1

W5 ¼ �W6 ¼ �W7 ¼W8 ¼ f2

W9 ¼ �W10 ¼ �W11 ¼W12 ¼ f3

W13 ¼ �W14 ¼ �W15 ¼W16 ¼ f4

W17 ¼ �W18 ¼ �W19 ¼W20 ¼ f5

W21 ¼ �W22 ¼ �W23 ¼W24 ¼ f6

Making the above substitutions into each of Eqs. (25a)–(25f), weobtain:

ð18� kÞ �9 1 �7 2 0�9 ð28� kÞ �10 2 �9 21 �10 ð27� kÞ 0 2 �9�7 2 0 ð11� kÞ �7 12 �9 2 �7 ð19� kÞ �80 2 �9 1 �8 ð18� kÞ

2666666664

3777777775

f1

f2

f3

f4

f5

f6

2666666664

3777777775¼

000000

2666666664

3777777775

ð30Þ

The vanishing condition for the determinant of the above 6 � 6 ma-trix yields a 6th-degree polynomial equation in k, whose roots (therequired eigenvalues) are obtained as:

A. Zingoni / Computers and Structures 112-113 (2012) 266–282 277

k1 ¼ 2:496; k2 ¼ 8:626; k3 ¼ 14:563;

k4 ¼ 21:427; k5 ¼ 26:760; k6 ¼ 47:128

7.3. Square plate

Subspace S(1)

From the coefficients of the Uð1Þj ðj ¼ 1;2; . . . ;6Þ, let

W1 ¼ f1

W2 ¼W3 ¼W4 ¼W5 ¼ f2

W6 ¼W7 ¼W8 ¼W9 ¼ f3

W10 ¼W11 ¼W12 ¼W13 ¼ f4

W14 ¼W15 ¼W16 ¼W17 ¼W18 ¼W19 ¼W20 ¼W21 ¼ f5

W22 ¼W23 ¼W24 ¼W25 ¼ f6

Making the above substitutions into each of Eqs. (26a)–(26e) and(26g), we obtain:

ð20� kÞ �32 8 4 0 0�8 ð25� kÞ �16 �8 6 02 �16 ð22� kÞ 4 �16 21 �8 4 ð21� kÞ �16 20 3 �8 �8 ð24� kÞ �80 0 2 2 �16 ð22� kÞ

2666666664

3777777775

f1

f2

f3

f4

f5

f6

2666666664

3777777775¼

000000

2666666664

3777777775

ð31Þ

The vanishing condition for the determinant of the above 6 � 6 ma-trix yields a 6th-degree polynomial equation in k, whose roots (therequired eigenvalues) are obtained as:

k1 ¼ 0:796; k2 ¼ 6:829; k3 ¼ 16:698;

k4 ¼ 18:683; k5 ¼ 34:568; k6 ¼ 56:426

Subspace S(2)

From the coefficients of the Uð2Þj ðj ¼ 1Þ, let

W14 ¼W15 ¼W16 ¼W17 ¼ �W18 ¼ �W19 ¼ �W20

¼ �W21 ¼ f1

Making the above substitutions into Eq. (26e) with all otherWi(i = 1,2, . . .,13; 22,23,24,25) set equal to zero, we obtain

½ð18� kÞ�½f1� ¼ ½0� ð32Þ

yielding the solution k = 18 when the determinant of the 1 � 1 ma-trix is set equal to zero. This is the eigenvalue (natural frequency)corresponding to the one mode of vibration with the symmetry typeof subspace S(2). It is noted that for this mode, the four axes of sym-metry of the plate {x,y,1,2} are all nodal lines experiencing notransverse displacements.

Subspace S(3)

From the coefficients of the Uð3Þj ðj ¼ 1;2;3Þ, let

W2 ¼ �W3 ¼ �W4 ¼W5 ¼ f1

W10 ¼ �W11 ¼ �W12 ¼W13 ¼ f2

W14 ¼ �W15 ¼ �W16 ¼W17 ¼W18 ¼W19 ¼ �W20

¼ �W21 ¼ f3

Making these substitutions into Eqs. (26b), (26d) and (26e) with allthe other Wi(i = 1,6,7,8,9,22,23,24,25) set equal to zero, we obtain

ð17� kÞ �8 2�8 ð21� kÞ �161 �8 ð20� kÞ

264

375

f1

f2

f3

264

375 ¼

000

264

375 ð33Þ

The vanishing condition for the determinant of the above 3 � 3 ma-trix yields a 3rd-degree polynomial equation in k, the roots of whichare the three eigenvalues (hence natural frequencies) corresponding

to the modes of vibration with the symmetry type of subspace S(3).The results are:

k1 ¼ 6:760; k2 ¼ 16:688; k3 ¼ 34:551

Subspace S(4)

From the coefficients of the Uð4Þj ðj ¼ 1;2;3Þ, let

W6 ¼ �W7 ¼ �W8 ¼W9 ¼ f1

W14 ¼ �W15 ¼ �W16 ¼W17 ¼ �W18 ¼ �W19 ¼W20 ¼W21 ¼ f2

W22 ¼ �W23 ¼ �W24 ¼W25 ¼ f3

with

W1 ¼W2 ¼W3 ¼W4 ¼W5 ¼W10 ¼W11 ¼W12 ¼W13 ¼ 0:

Making these substitutions into Eqs. (26c), (26e) and (26g), weobtain

ð18� kÞ �16 2

�8 ð22� kÞ �8

2 �16 ð22� kÞ

2664

3775

f1

f2

f3

2664

3775 ¼

0

0

0

2664

3775 ð34Þ

The vanishing condition for the determinant of the above 3 � 3 ma-trix yields a 3rd-degree polynomial equation in k, the roots of whichare the three eigenvalues (hence natural frequencies) correspond-ing to the modes of vibration with the symmetry type of subspaceS(4). The results are:

k1 ¼ 5:835; k2 ¼ 18:065; k3 ¼ 38:100

Subspace S(5,2)

From the coefficients of the Uð5;2Þj ðj ¼ 1;2; . . . ;6Þ, let

W2 ¼ �W3 ¼W4 ¼ �W5 ¼ f1

W6 ¼ �W9 ¼ f2

W10 ¼ �W11 ¼W12 ¼ �W13 ¼ f3

W14 ¼ �W17 ¼ �W20 ¼W21 ¼ f4

W15 ¼ �W16 ¼ �W18 ¼W19 ¼ f5

W22 ¼ �W25 ¼ f6

with

W1 ¼W7 ¼W8 ¼W23 ¼W24 ¼ 0:

Making these substitutions into Eqs. (26b)–(26g), we obtain

ð19� kÞ �8 �8 3 �1 0

�16 ð20� kÞ 4 �16 0 2

�8 2 ð21� kÞ �8 8 1

3 �8 �8 ð23� kÞ �1 �8

�1 0 8 �1 ð19� kÞ 0

0 2 2 �16 0 ð22� kÞ

266666666666664

377777777777775

f1

f2

f3

f4

f5

f6

266666666666664

377777777777775¼

0

0

0

0

0

0

266666666666664

377777777777775

ð35Þ

The vanishing condition for the determinant of this 6 � 6 matrixyields a 6th-degree polynomial equation in k, the roots of whichare the six eigenvalues (hence natural frequencies) correspondingto the modes of vibration with the symmetry type of subspaceS(5,2). The results are:

k1 ¼ 2:776; k2 ¼ 11:284; k3 ¼ 12:091;

k4 ¼ 23:724; k5 ¼ 27:401; k6 ¼ 46:724

As already explained, subspace S(5,1) has exactly the same set ofeigenvalues as subspace S(5,2).

Table 1Natural circular frequencies x (in rad/s) for the rectangular plate.

GTFDM FEM

Subspace S(1) (1) 22.20a 22.96a

(4) 74.15a 85.14a

(6) 107.80 130.71(10) 138.92 185.25(13) 159.74 252.83(20) 224.52 342.04

Subspace S(2) (5) 81.73a 91.91a

(12) 146.50 210.23(15) 167.33 261.41(19) 198.45 331.60(21) 232.10 368.68(24) 284.05 415.25

Subspace S(3) (2) 43.45a 46.24a

(7) 108.22 139.75(9) 129.05 168.33(14) 160.16 255.62(17) 193.82 296.77(22) 245.77 392.26

Subspace S(4) (3) 60.48a 68.69a

(8) 112.43 145.17(11) 146.09 207.04(16) 177.20 275.92(18) 198.02 314.49(23) 262.80 399.61

a First five modes.

Table 2Natural circular frequencies x (in rad/s) for the square plate.

GTFDM FEM

Subspace S(1) (1) 34.15a 44.04a

(6) 100.04 141.96(12) 156.43 260.28(15) 165.46 318.20(21) 225.07 422.72(25) 287.55 495.85

Subspace S(2) (13) 162.41 261.24

Subspace S(3) (5) 99.53a 141.29a

(11) 156.38 235.98(20) 225.02 398.77

Subspace S(4) (4) 92.47a 115.95a

(14) 162.71 318.20(22) 236.29 423.98

Subspace S(5,1) (2,3) 63.78a 78.67a

(7,8) 128.59 176.98(9,10) 133.11 226.63(16,17) 186.46 333.59(18,19) 200.39 367.19(23,24) 261.67 463.27

Subspace S(5,2) (2,3) 63.78a 78.67a

(7,8) 128.59 176.98(9,10) 133.11 226.63(16,17) 186.46 333.59(18,19) 200.39 367.19(23,24) 261.67 463.27

a First five modes.

278 A. Zingoni / Computers and Structures 112-113 (2012) 266–282

7.4. Remarks on the structure of symmetry-adapted eigenvaluematrices

Unlike the n � n eigenvalue matrix of the conventional finite-difference formulation, the r � r symmetry-adapted eigenvaluematrix for a given subspace S(i) of dimension r is, in general, notsymmetric. This is because the number of /k components makingup each basis vector UðiÞj ðj ¼ 1;2; . . . ; rÞ of the subspace are gener-ally different. The number of components making up a basis vectormay be thought of as the weight of the basis vector.

In the case of the rectangular plate, each subspace S(i)(i = 1,2,3,4)is spanned by six basis vectors UðiÞj ðj ¼ 1;2; . . . ;6Þ, all of which havethe same number of basis-vector components /k (i.e. four compo-nents each). Thus all basis vectors of a given subspace have thesame weight. That is why all the four 6 � 6 symmetry-adaptedeigenvalue matrices for this problem are symmetric.

In the case of the square plate, the various subspaces arespanned by basis vectors with different numbers of components.Take subspace S(3) for example. This is 3-dimensional (i.e. it isspanned by 3 basis vectors). The first basis vector Uð3Þ1 has fourcomponents {/2,/3,/4,/5}, the second basis vector Uð3Þ2 also hasfour components {/10,/11,/12,/13}, but the third basis vector Uð3Þ3

has eight components {/14,/15,/16,/17,/18,/19,/20,/21}. Let uswrite the associated eigenvalue matrix in the symbolic form

A ¼a11 a12 a13

a21 a22 a23

a31 a32 a33

264

375 ð36Þ

where columns 1, 2, 3 correspond to basis vectors Uð3Þ1 ;Uð3Þ2 ;Uð3Þ3 . Ifwe examine the obtained results for this matrix, we observe thata12 = a21(= �8) since Uð3Þ1 and Uð3Þ2 have the same number of basis-vector components (i.e. these basis vectors have the same weight).On the other hand, a13 = 2a31 and a23 = 2a32 since basis vector Uð3Þ3

has twice the number of components of either Uð3Þ1 or Uð3Þ2 (i.e. basisvector Uð3Þ3 has double the weight of Uð3Þ1 or Uð3Þ2 Þ. This explains whyin general symmetry-adapted eigenvalue matrices cannot be ex-pected to be symmetric.

8. Numerical results and discussion

On the basis of the foregoing, natural circular frequencies havebeen evaluated for a rectangular steel plate of dimensions 7 � 5 m(assuming all edges are simply supported) and a square steel plateof dimensions 6 � 6 m (assuming all edges are fixed). The thicknessh of the plate is 25 mm. Material properties are as follows:

c = 7800 kg/m3 (density of steel)E = 200 � 109N/m2 (Young’s modulus)m = 0.3 (Poisson’s ratio)

The flexural rigidity D of the plate follows from Eq. (2) as

D ¼ Eh3

12ð1� m2Þ ¼200� 109 � ð25� 10�3Þ3

12ð1� 0:32Þ¼ 286;172 Nm

while q (the mass per unit area of the plate) follows from the givendensity of the material:

q ¼ c h ¼ 7800� 0:025 ¼ 195 kg=m2:

From Eqs. (5) and (7), we have

x ¼ffiffiffiffiffiffiffigDq

s¼

ffiffiffiffiffiffiffiffikD

d4q

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi286;172k

ð1:04Þ � 195

s¼ 38:309

ffiffiffikp

Thus all the values of k that we calculated for the various sub-spaces may be converted to actual natural circular frequencies xof the plate via this last relationship.

Table 1 (for the rectangular plate) and Table 2 (for the squareplate) show values of x calculated on the basis of the group-theo-retic finite-difference method (GTFDM), adopting a finite-differ-ence mesh of spacing 1 � 1 m in the x and y directions, versusvalues obtained from a finite-element method (FEM) using the fi-nite-element programme ABAQUS. The numbers in brackets infront of the GTFDM values denote the ascending order of the nat-ural frequencies (or mode numbers), from 1 to 24 in the case ofthe rectangular plate, and from 1 to 25 in the case of the square

A. Zingoni / Computers and Structures 112-113 (2012) 266–282 279

plate, bearing in mind that for the square plate, subspaces S(5,1) andS(5,2) possess identical sets of frequencies (for instance, mode 2 maybe regarded as being in subspace S(5,1) and mode 3 in subspace S(5,2),or vice versa, the frequency associated with either mode being63.78rad/s). Mode shapes for the first 24 natural frequencies ofthe retangular plate, and the first 25 natural frequencies of thesquare plate, are shown in Figs. 8 and 9.

For the simply-supported rectangular plate, the agreement be-tween the GTFDM and the FEM results is reasonable (within 10–15%) for the lowest five natural circular frequencies. This level ofaccuracy is consistent with that associated with the conventional

Fig. 8. First 24 vibration modes for the rectang

finite-difference method when a relatively coarse mesh is em-ployed [36]. As expected, the agreement becomes poorer for higherfrequencies, because the chosen finite-difference meshes for theplates, while convenient for the purposes of illustrating thegroup-theoretic computations in a simple and manageable way,are too coarse to be able to accurately model the more rapidlychanging deformation patterns of the higher modes. The valuesfor the square plate show even higher discrepancies owing to themore constrained edge conditions, but the first three frequenciesare still in fair agreement (15–30%). In both cases, the noted rea-sonable agreement in the values of the lowest natural circular

ular plate as plotted by FEM programme.

Fig. 9. First 25 vibration modes for the square plate as plotted by FEM programme.

280 A. Zingoni / Computers and Structures 112-113 (2012) 266–282

frequencies, which is entirely consistent with expected levels ofaccuracy for relatively coarse meshes, is sufficient validation forthe developed group-theoretic finite-difference formulation.

Strictly speaking, in seeking to validate the proposed group-the-oretic procedure, it should be noted that the question is not howthe finite-difference method compares with the finite-element

A. Zingoni / Computers and Structures 112-113 (2012) 266–282 281

method (an issue that has been studied many times in the past),but instead, how the results of the group-theoretic procedure com-pare with those obtained using conventional finite-difference cal-culations. So instead of refining the adopted finite-differencemeshes in order to see if better agreement with the results of thefinite-element analysis would be achieved (refinement whichwould detract from the simplicity of the illustrations), the same fi-nite-difference meshes for the rectangular plate and the squareplate were tackled using the conventional finite-difference ap-proach, and the results for these agreed exactly with those of thepresent group-theoretic formulation (no point in having a tableof comparisons). This confirmed the correctness of the group-the-oretic formulation. The exactness of the match is not unexpected,since the group-theoretic procedure is, after all, solving exactlythe same set of mathematical equations, but only more efficiently.

From Tables 1 and 2, we also observe an important attribute ofthe group-theoretic formulation. Apart from the property of break-ing up the original problem into smaller problems which are mucheasier to solve, we observe that the group-theoretic decompositionalso separates modes whose frequencies are too close to eachother, eliminating the numerical problems usually associated withthe computation of frequencies that are too clustered together. Forinstance, by reference to Table 1 for the rectangular plate, we ob-serve that modes 6, 7 and 8 have closely spaced circular frequenciesof {107.80;108.22;112.43} rad/s respectively, but the fact thatthese frequencies are extracted separately within subspacesS(1),S(3) and S(4) makes the numerical computations more stable.Similarly for the square plate (Table 2), the group-theoretic decom-position separates mode 5 (99.53 rad/s) and mode 6 (100.04 rad/s)into subspaces S(3) and S(1) respectively; mode pairs {11,12} and{20,21} are also conveniently separated in a similar manner.

9. Concluding remarks

In this paper, we have presented a comprehensive group-theoretic procedure for the implementation of the finite-differencemethod in the solution of plate eigenvalue problems. Although thedetailed illustrative procedure has been based on rectangular andsquare plates (which belong to the symmetry groups C2v and C4vrespectively), it is clear that the procedure is quite general, beingalso applicable to triangular and hexagonal finite-difference meshpatterns (which belong to the symmetry groups C3v and C6vrespectively).

Unlike in the conventional approach where we need to writedown the finite-difference equations for all the nodes of the meshand then solve the ensuing (usually large) eigenvalue problem, thegroup-theoretic procedure requires the writing down of the finite-difference equations for a relatively small set of nodes (the gener-ator nodes), which are then transformed into symmetry-adaptedequations for the various subspaces (by means of the basis vectorsfor each subspace). In this way, the original eigenvalue problem isdecomposed into a series of smaller eigenvalue problems corre-sponding to the various subspaces, which can be solved indepen-dently for the eigenvalues. Importantly, the eigenvalues yieldedwithin the various subspaces are in fact the actual eigenvalues associ-ated with the original problem, and no further conversion is required.

Apart from simplifying the problem from a computationalstandpoint, some qualitative benefits of the group-theoretic ap-proach have also been noted. These include: (i) gaining valuable in-sights into the symmetries of vibration modes before anycomputations are actually done; (ii) identification of modes/sym-metries associated with doubly repeating frequencies (these occurin subspaces associated with the 2-dimensional irreducible repre-sentations of symmetry groups C4v, C3v and C6v); (iii) separation(via subspaces) of the calculation of frequencies which are very

close to each other (but with modes of different symmetries),bypassing the numerical problems associated with searching forroots that are clustered in the same neighbourhood.

Finally, the question might well be asked why one should resortto a seemingly complicated formulation like the group-theoretic fi-nite-difference method, when finite-element programmes willsolve the free-vibration problem of plates with ease and good accu-racy. First, it should be pointed out that the procedure that hasbeen developed in this paper is amenable to computer program-ming (a point which was made at the end of Section 4), and currentefforts are directed at the automation of the various group-theo-retic computational steps, which will make the application of theprocedure very easy indeed. This is still work in progress. Secondly,if one happens to know the exact differential equations governing aparticular behaviour of a physical system, the finite-difference ap-proach provides a more direct numerical means for solving thoseequations than the finite-element method (which requires discret-isation and assembly). For this reason, there is still merit in devel-oping more efficient finite-difference formulations, such as thegroup-theoretic formulation for symmetric plates that has beendescribed in the present paper.

Acknowledgements

The author would like to acknowledge the assistance of Mr. AngusRule of the University of Cape Town, who prepared the illustrationsof this paper. The financial assistance of the National ResearchFoundation of South Africa is gratefully acknowledged.

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