## Abstract

Many structures in nature and engineering are symmetric. Depending on the degree of symmetry, it is possible to simplify the computations considerably by block diagonalizing the stiffness matrices. Closed-form solutions of transformation matrices for such block diagonalizations can be derived using group theory for arbitrary symmetry groups. This paper presents closed-form solutions of transformation matrices based on an alternative derivation. It is shown that transformation matrices for *C*_{nv} and *D*_{nh} groups can be obtained from a finite Fourier series decomposition of load and displacement vectors. Furthermore, it is shown that structures with tetrahedral, octahedral and icosahedral symmetries can be block diagonalized in an elegant way using vector spherical harmonics.

## 1. Introduction

It is well known in engineering that for structures with a bilateral symmetry, it is possible to decompose an arbitrary loading into symmetric and anti-symmetric parts (figure 1*a*). The gain in calculation time then results from the possibility of structural simplification as shown in figure 1*b*. However, in case of computer simulations, there is, in general, no advantage if the loads are non-symmetric since both substructures have to be computed separately for different boundary conditions.

It is less well known that stiffness matrices, for example from a finite-element simulation, of a symmetric structure can be block diagonalized if the load and displacement vectors are expressed in terms of symmetric and antisymmetric orthogonal coordinate systems. The generation of these coordinate systems is relatively simple so long as the structures have a small degree of symmetry. This is shown in figure 2, where all possible symmetric and antisymmetric vector combinations of the structure from figure 1 are shown. If we write a matrix where each column corresponds to one normed coordinate system and separate the symmetric from the antisymmetric part, then we get the transformation matrix ** V** for the block diagonalization. This matrix is shown in table 1, where the horizontal lines separate the single nodes and assign them to the symmetry planes

**,**

*A***,**

*B***and**

*C***as defined in figure 1. It should be noted that this matrix has seven symmetric and seven antisymmetric coordinate systems, where each coordinate system is described by the vectors of one picture from figure 2. The block diagonalization can now be written as(1.1)with**

*D***as the original and as the block-diagonalized stiffness matrix. The load vector**

*K***and the displacement vector**

*f***transform according to(1.2)**

*u*Figure 3 shows the result of the block diagonalization of the structure from figure 1. The left picture illustrates the non-zero entries of the original stiffness matrix ** K**, whereas the right picture displays the block-diagonalized stiffness matrix . Note that the boundary conditions have not yet been introduced into the stiffness matrices. It can be seen that the structure splits into two blocks, where the first block corresponds to the symmetric coordinate systems and the second block to the antisymmetric coordinate systems. The advantage of a block-diagonalized stiffness matrix is that the system of linear equations can be solved separately for each block. This is especially interesting if more than one computer is involved in the solution process. Furthermore, depending on the degree of symmetry, the sparsity increases considerably due to less coupling between the degrees of freedom. Nevertheless, it should be noted that this method works, in general, only for simulations that are based on a linear deformation theory, since the symmetry of the structure is normally lost for an updated configuration.

Unfortunately, it is much harder to find all symmetry coordinate systems if the degree of symmetry is higher. Therefore, a mathematical description is needed that gives a general relationship between the structural symmetry and the transformation matrix that block diagonalizes the corresponding stiffness matrix. The described problem can be expressed using group theory. The first application of group theory for structural analysis was by Zloković (1973) who formulated the so-called G-vector analysis. Since then, a lot of work has been done by numerous authors (e.g. Bossavit 1986). A different approach for introducing symmetry transformation matrices and an extensive review can be found in Kangwai *et al.* (1999). It should be remarked that symmetry is widely used in the natural sciences to simplify calculations. Hence, its use is not restricted to structural analysis. Another field that greatly benefits from symmetry considerations is, for example, the investigation of molecular vibration (James & Liebeck 1993). However, the disadvantage of existing methods is that they are rather complicated and hence difficult to use. The purpose of this paper is to present an alternative derivation of closed-form solutions for symmetry transformation matrices that are based on Fourier series and vector spherical harmonics.

The outline of this paper is as follows. Section 2 introduces the motivation that led to this paper. Section 3 gives an outline of the used notation and §4 introduces a closed-form solution for the transformation matrices of structures with *n*-fold rotation symmetry. An example of a framework with *D*_{nh} symmetry is given in Section 5. Finally, §6 derives transformation matrices for structures with tetrahedral, octahedral and icosahedral symmetries on the basis of vector spherical harmonics.

## 2. Motivation

This paper arose during the simulation of the stability of large scientific Ultra Long Duration Balloons (ULDBs) as they are currently developed by NASA (figure 4; Pagitz & Pellegrino in press). Since such balloons have around 290 lobes, a complete model is discretized with more than 10^{6} degrees of freedom. Therefore, a simulation based on a standard personal computer becomes impossible without further considerations. Owing to the fact that these balloons belong to the symmetry group *D*_{nh}, the derivations in this paper concentrate mainly on this group. However, it is shown in §6 how block diagonal stiffness matrices of structures with tetrahedral, octahedral and icosahedral symmetries can be obtained on the basis of vector spherical harmonics.

## 3. Notation

The following derivations are based on a sphere that belongs to the *D*_{nh} symmetry group. The sphere possesses an *n*-fold rotation axis, *n*-twofold axes perpendicular to the principal axis, *n* vertical and one horizontal mirror planes. It has a radius *R* and is centred on the origin such that the points (*R*, 0, 0) and (−*R*, 0, 0) are the *poles* and the great circle that results from an intersection of the *Y, Z* plane with the sphere is the *equator* (figure 5). The sphere is divided into *n* equal segments by *meridians* which are *n* half great circles running between the poles and intersecting the equator at the points (0, *R* sin (2*πk*/*n*), *R* cos(2*πk*/*n*)) for some *k*∈{0, …, *n*−1}.

The structures we are considering are represented by *nodes* connected together by structural elements. These nodes are defined by the intersection of the meridians with pairs of parallel planes defined by *X*=*x* and *X*=−*x*, for some values of *x* between 0 and *R* (including 0 and *R* themselves, so there are nodes at the poles and on the equator). The symmetry group of the nodes of two corresponding planes is again *D*_{nh} and the orbit of a node lying in the plane *X*=*x* is all the nodes contained in that plane and all the nodes in the plane *X*=−*x*.

## 4. Geometrical derivations

### (a) Tropical orbits of *D*_{∞h}

Since the orbits of a structure that belongs to *D*_{∞h} are circles, it is possible to obtain a set of orthogonal vectors for the transformation matrices from a Fourier series. Figure 6 shows half of the orthogonal vectors of an orbit for *f*=3. A rotation of these vectors by *ϕ*=π/(2*f*) around the *X*-axis leads to the second half. Furthermore, it is possible to divide them into symmetric and antisymmetric vectors according to a reflection in the *YZ*-plane. In the following, all linear independent (orthogonal) vectors of the upper planes of the tropical orbits of *D*_{∞h} are given for the first four frequencies. These vectors are divided into two categories, black/black–white and grey/grey–white, as shown in figure 7, where each category consists of radial, tangential and axial vectors. It will be shown later that each category is the basis for two blocks in the transformed stiffness matrix. The lower parts of the black and grey orbits can be obtained by a reflection of the upper parts at the *YZ*-plane, those of the black–white and grey–white orbits by an inversion and reflection. This leads to a complete set of orthogonal tropical orbits.

If we define an angle 0≤*α*<2*π*, measured from the *Y*-axis, we are able to write for the direction of the radial vectors(4.1)and their un-normed vector lengths are(4.2)

It should be realized that it is not important whether *α* is measured clockwise or anticlockwise. For the direction of the tangential vectors, we get(4.3)with their un-normed vector lengths(4.4)

Therefore, the sum of the in-plane orbits of one block, , results in constant vector lengths,(4.5)

Hence, for *f*=1, the sum of the in-plane vectors of the black and grey orbits describe rigid body translations in the *Y* and *Z* direction. Finally, the direction of the axial vectors is(4.6)with their un-normed vector lengths equal to those of the radial vectors given by equation (4.2). As can be seen in figure 7, each frequency generates four blocks (black, grey, black–white and grey–white) of the symmetry transformation matrix. Furthermore, it can be seen that the grey/grey–white orbits of the last three frequencies could be obtained by rotating the black/black–white orbits by *ϕ*=π/(2*f*) around the *X*-axis. Looking at *f*=0, it becomes clear that this is not true for blocks with constant vector lengths since they would not generate linearly independent orbits for any rotations around the *X*-axis. Hence, there exist only half as many coordinate systems for *f*=0. Nevertheless, the black and black–white blocks split into two parts such that we have again four blocks for the zero frequency.

### (b) Derivation of *D*_{nh} from *D*_{∞h}

Taking the interrelations of the different orbits into account, given by equations (4.1)–(4.4), we are able to construct all symmetry coordinate systems from the black normal orbits. Therefore, we will only consider them from now on, to keep the illustrations as simple as possible. As can be seen in figure 7, the cross-sections of orbits of *D*_{∞h} are always circles. This is not true for finite groups. In the following, we will display the orbits of finite *D*_{nh} groups as regular polygons with *n*-edges. Figure 8 shows the first four frequencies of *D*_{∞h} together with a hexagon that symbolizes *D*_{6h}. A set of orthogonal vectors of *D*_{6h} can now be obtained by taking the vectors of *D*_{∞h} at the intersections between the circle and hexagon. This is true not only for the in-plane radial vectors but also for all other orbits. Therefore, we can rewrite equations (4.1)–(4.4) for finite groups. The direction of the radial vectors can be derived as(4.7)with 0≤*k*<*n* selecting the according node (intersection). The normalized lengths of the radial vectors are(4.8)where *L* is used to norm the length of the resulting vectors.(4.9)

We can derive for the direction of the tangential vectors(4.10)with their normed vector lengths resulting in(4.11)

The direction of axial vectors for finite *D*_{nh} groups is given by equation (4.6) and their vector lengths are prescribed by equation (4.8). It is interesting to note that if *n* is even and *f*=*f*_{max}=*n*/2, then sin((2*π*/*n*)*kf*)=0, so half the coordinate systems vanish. Hence, as for *f*=0, they generate only half as many coordinate systems.

Therefore, we can make the following conclusions. The transformation matrix of a structure with *D*_{nh} symmetry and *p* degrees of freedom has *p* orthogonal coordinate systems. Furthermore, it can be said that there are only half as many tropical orbits for *f*=0 and *f*=*f*_{max}, if *n* is even. Finally, the size of the single blocks depends on the number of equatorial, tropical and polar orbits. However, the number of blocks is constant no matter how many different orbits are present in the structure since the degree of symmetry does not depend on *p*.

To show that this answer agrees with that derived using group theory (and hence substantially simplifies the stiffness matrix), it is enough to show that the orthonormal basis can be partitioned into bases of the homogeneous components (Definition 32.12, James & Liebeck 1993). In fact, we can show that there is a finer partition such that each part is a basis of an irreducible *D*_{nh}-module. Each vector corresponding to the zero frequency (and the frequency *n*/2 if *n* is even) is either preserved or sent to its negative by every element in *D*_{nh}. Therefore, each of these vectors spans a one-dimensional *D*_{nh}-module (which is by definition irreducible). The vectors corresponding to the other frequencies (1≤*f*<*n*/2) occur in pairs (either black and grey or black–white and grey–white) and every element of *D*_{nh} maps these vectors to a linear combination of the pair. This follows from the addition formula for sine and cosine; furthermore, this allows us to calculate the character of the module. By observing that the inner product of this character with itself is 1, we conclude that the module is irreducible.

Therefore, we can summarize the following consequences. The character of the *D*_{nh}-module of the displacement vectors based on an orbit of tropical nodes is easy to calculate (it is a tensor product of the ‘natural’ three-dimensional representation with the permutation module derived from the action on the nodes). Hence, the size of each homogeneous component can easily be found by calculating the inner product of the character with every irreducible character. Applying this method shows that each orbit of tropical nodes contributes 6 to the upper bound of the largest block size in the factorization. Similarly, we can show that the equatorial nodes contribute 4 and that the polar nodes contribute 2. In particular, the size of the blocks is independent of *n*.

## 5. Example

### (a) Structure

In the following, a framework with *D*_{6h} symmetry is used for demonstrating the block diagonalization of its stiffness matrix (figure 9). It can be seen that this structure has one equatorial (** A**), tropical (

**) and polar (**

*B***) orbit.**

*C*### (b) Tropical orbits

The tropical orbits of *D*_{6h} with constant vector lengths are shown in figure 10. The left part, which corresponds to a zero frequency, exists for all *D*_{nh} groups. The right part is only present for groups where *n* is even and is associated with the maximum frequency, *f*_{max}=*n*/2. Figure 11 displays the coordinate systems for the rest of the frequencies. Each row of the different frequencies of figures 10 and 11 is the basis for one block of the transformed stiffness matrix. It can be seen that there are three blocks, with one or more coordinate systems, where the sum of all vectors results in a non-zero value. Furthermore, there are three other blocks that contain coordinate systems where the moment of all vectors around the inversion centre (centre of the structure) is non-zero. These six blocks model rigid body translations and rotations, and the resulting force and moment vectors are indicated in figures 10 and 11 by a chain-dotted line.

### (c) Equatorial and polar orbits

The corresponding equatorial coordinate systems of each block can be obtained by adding the vectors of both tropical planes and norming them. The apex coordinate systems are derived in a similar manner, but this time all vectors of each plane are added, merged into one node (figure 12).

### (d) Symmetry transformation matrix

If we write each symmetry coordinate system as a (normed) column vector, then we get the symmetry transformation matrix. It should be noted that it is important that all symmetry coordinate systems of one block are written together in this matrix in order to get blocks that are decoupled (table 2). Figure 13 shows the non-zero entries of the original and block-diagonalized stiffness matrices of the structure shown in figure 9. It should be noted that the matrix splits into 4×*f*_{tot}=16 independent blocks, where *f*_{tot} is the total number of frequencies of *D*_{6h}. Furthermore, it is interesting to note that the sparsity of the matrix increases considerably after the transformation.

## 6. Tetrahedral, octahedral and icosahedral symmetries

It was shown in §4 that it is possible to derive the transformation matrix of a structure with *D*_{nh} symmetry on the basis of Fourier series that go around a circle. The question arises now whether there exist a set of functions around a sphere that can be exploited to derive the symmetry transformation matrices of groups which do not have a major rotation axis. In the following, we show how the transformation matrices for structures with tetrahedral, octahedral and icosahedral1 symmetry (figure 14) can be derived on the basis of vector spherical harmonics.

### (a) Vector spherical harmonics

It was shown by Hill (1954) that vector spherical harmonics can be derived on the basis of their scalar counterparts. Spherical harmonics themselves are obtained by calculating the angular portion of the solution to Laplace's equation in spherical coordinates (Arfken 1985). It should be remarked that the notation used in the following is identical with Hill's notation (figure 15). The first few spherical harmonics are

According to Hill (1954), the vector spherical harmonics result inwhere *f*∈[0, ∞] and *m*∈[0,±*f*]. The radial (*r*_{1}) and tangential [Θ_{1}, *φ*_{1}] vectors are

It can be seen that the vector spherical harmonics contain real and imaginary parts. A set of real vectors can be obtained by adding and subtracting . Hence, we can writewhere *m*∈[0, *f*]. Furthermore, it is possible to add and subtract and , such that we purely get radial (** R**) and tangential (

*T*_{1},

*T*_{2}) vectors. The first few vectors are

*f*=0, *m*=0:

*f*=1, *m*=0:

*f*=1, *m*=+1:

*f*=1, *m*=–1:where the scaling factors have been omitted for the sake of brevity. Figure 16 shows the vector spherical harmonics for *f*∈[0, 1]. It should be noted that the rigid body translations and rotations are captured by the harmonics with *f*=1. The translations can be expressed as a sum of ** R** and

**1, whereas the rotations are given by**

*T***2 for each**

*T**m*. As in the case of the Fourier series in §4, we obtain a set of orthogonal vector fields that are exploited to derive the symmetry transformation matrices.

### (b) Geometrical derivations

It was shown in §4 that the transformation matrices for finite *D*_{nh} groups can be obtained by computing the vectors of *D*_{∞h} orbits at the intersections with a regular polygon that possesses *D*_{nh} symmetry. The procedure for the construction of transformation matrices for structures with tetrahedral, octahedral and icosahedral symmetries is similar. Since the nodes of all structures that belong to one of these symmetry groups are located on a sphere, it is possible to construct the transformation matrices by calculating the vectors of the spherical vector harmonics at the intersections between the surrounding sphere and the nodes of the structure.

### (c) Examples

In order to demonstrate the validity of the described method, we generate stiffness matrices of pin-jointed structures with tetrahedral, octahedral and icosahedral symmetries. This is done by attaching bar elements along the edges of the Platonic solids shown in figure 14 and connecting them at the common nodes. If we calculate the vector harmonics that are summarized in table 4 at the nodes of the three solids from figure 14 and write them as normed column vectors in the order shown in table 3, then we get the transformation matrices for the pin-jointed structures with tetrahedral, octahedral and icosahedral symmetries. It should be noted that contains both tangential vectors, so that there are in total 12 orthogonal column vectors for the tetrahedron, 18 for the octahedron and 36 for the icosahedron. The non-zero entries of the stiffness matrices before and after the transformation are shown in figure 17. In order to keep this paper concise, only the transformation matrix of the octahedron is given in table 5, and figure 18 shows the corresponding symmetry coordinate systems. It should be noted that the eigenvalues of the stiffness matrix are of course identical before and after the symmetry transformation and result inif we assume an octahedron that fits into a unit sphere and the axial stiffness of the bars to be one.

## 7. Conclusion

This paper introduced closed-form solutions of symmetry transformation matrices on the basis of orthogonal vector fields. The derivations are straightforward and easy to understand since they are not directly based on representation or group theory. It was shown that the symmetry transformation matrix for the block diagonalization of the stiffness matrix of a structure with *D*_{nh} symmetry can be obtained from a Fourier series. A closed-from solution of the corresponding transformation matrices was derived in §4 and the usefulness of the block diagonalization was demonstrated in §5 using an example. The proposed method is easy to implement into finite element codes and it is shown in Pagitz & Pellegrino (in press) that, together with static condensation, the block diagonalization can reduce the computational effort enormously. Finally, we have shown in §6 that it is possible to extend this methodology to structures with tetrahedral, octahedral and icosahedral symmetries using vector spherical harmonics.

## Acknowledgements

M.P. would like to express his gratitude to his supervisor Professor S. Pellegrino. Furthermore, he would also like to thank Dr S.D. Guest, Professor C.R. Calladine and Dr N.G. Kingsbury from the Cambridge University Engineering Department for their helpful discussions. J.J. would like to thank his supervisor Professor J. Saxl for suggesting useful references. MP gratefully acknowledges the financial support from EPSRC, Corpus Christi College and Cambridge European Trust. Finally, both authors would like to thank Corpus Christi College for Leckhampton House without which this collaboration would never have happened.

## Footnotes

↵These are the underlying symmetries of the five Platonic solids.

- Received June 9, 2006.
- Accepted February 27, 2007.

- © 2007 The Royal Society