The purely geometric symmetries of a suitable equation lead to a decomposition (2.16) of an arbitrary function in a function space, and thus the decomposition of the function space itself. The decomposed elements are linearly independent and can be arranged to form an orthonormal system. This can be exploited in the calculations.

In a homogeneous material one can readily construct trial functions that fulfill the diffusion equation at each point of V. For example consider

V2H(r) + Af(r)= 0 (4.12)

£f — Ls + £f D 1. The general solution to (4.12) takes the form of

E tk eiXkr e Wk(e)de k=l lel=1
Hr)	(4.13)

where the weight functions Wk (e) eigenvectors of matrix A:

are arbitrary suitable functions, i2

Afk = Л-k fk.

(4.16)

where Og maps the interval 0 < в < 2n/G into the interval Ig. In this manner we get the irreducible components of the solution as a linear combination of |G| exponential function, it is only the coefficients in the linear combination that determine the irreducible components. The weight function Wk(6) makes it possible to match the entering currents at given points of the boundary. Let в = 0 correspond to the middle of a side. Then choosing

Wk ((e))(6) = Wk 5(6), (4.17)

we get by (4.15) the solution at face midpoints. The last step is the formation of the irreducible components. Observe that in projection (2.20) the solutions at different images of r are used in a linear combination, the coefficients of the linear combinations are the rows of the character table. But in the images (4.16), only the weight function changes. In each Ig interval the image of Wk (e) is involved, which is a Dirac-delta function, only the place of the singularity changes as the group elements map the place of the singularity. A symmetry of the square maps a face center into another face center thus there will be four distinct positions and the space dependent part of the irreducible component of fa will contain four exponentials:

± eiXkX, ±e—iXkX, ±eiXky, ±e—iXky. (4.18)

From these expressions the following irreducible combinations can be formed:

A : cos XkX + cos XkV’; A2 : cos XkX — cos Xky; £1 : sin AkX; £2 : sin Xyy. (4.19)

It is not surprising that when we represent a side by its midpoint the odd functions along the side are missing.

The above method may serve as a starting point for developing efficient numerical methods. The only approximation is in the continuity of the partial currents at the boundary of adjacent homogeneous nodes.

If elements of the function space are defined for all r Є V, and if f, /2 Є Lу, then the following inner product is applicable:

(/1, /2) = Jvft( r)/2(r)d3r. (4.20)

Let /a (r), £ = 1,…, na be a regular representation of group G. Then

furthermore, for the reactions rates formed with the help of the cross-sections in (4.1), similar orthogonality relation holds. For the volume integrated reaction rates we have

Ш const) = 6a,1 $l,1, (4.22)

in other words: solely the most symmetric, one dimensional representation contributes to the volume integrated reaction rates. Note that as a result of the decomposition of the solution or its approximation into irreducible components not only that irreducible components of a given physical quantity (like flux, reaction rate, net current) but also the given irreducible component of every physical quantity fall into the same linearly independent irreducible subspace. As a consequence, the operators (matrices) mapping the flux into net currents (or vice versa) fall into the same irreducible subspace, therefore the mapping matrix automatically becomes diagonal.

Example 4.3 (Symmetry components of boundary fluxes). Consider the flux given along the boundary of a square. The flux is given by four functions corresponding to the flux along the four sides of the square. The flux along a given face is the sum of an even and an odd function with respect to the reflection through the midpoint of the face. The decomposition (2.21) gives the eight irreducible components shown in Figure 2. Note that the irreducible

-1.0

Fig. 2. Irreducible components on the boundary of a square

subspaces а,-, г < 5 are one-dimensional whereas the subspace a = 5 is two-dimensional, and in a two-dimensional representation there are two pairs of basis functions that are identical as to symmetry properties. Thus, we have altogether eight linearly independent basis functions.

The physical meaning of the irreducible components is that the flux distribution of a node is a combination of the flux distributions established by eight boundary condition types. The
component A represents a complete symmetry that is the same even distribution along each side. Component A2 is also symmetric, but the boundary condition is an odd function on each side. Components Bj and B2 represent entering neutrons along one axes and exiting neutrons along the perpendicular axes, a realization of a second derivative with even functions over a face. B2 is the same but with odd functions along a face. Ej and £4 represent streaming in the x and y directions with even distributions along a face, whereas E2 and E3 with odd distributions along a face.

The symmetry transformations of the square, map the functions given along the half faces into each other but they do not say anything about the function shape along a half face. Therefore, the functions in Fig. 2 serve only as patterns, the function shape is arbitrary along a half face. The corresponding mathematical term is the direct product; each function may be multiplied byafunction f (£), — h/2 < £ < +h/2. It is well known that a function along an interval can be approximated by a suitable polynomial (Weierstrass’s theorem). We know from practice that in reactor calculations a second order polynomial suffices on a face for the precision needed in a power plant.

The invariant subspace means that the boundary flux, the net current, the partial currents must follow one of the patterns shown in Figure 2, the only difference may be in the shape function f (£), — h/2 < £ < +h/2. This means the a constant flux may create a quadratic position dependent current, but the global structure of the flux, and current should belong to the same pattern of Figure 2.

Moreover, if we are interested in the solution inside the square, its pattern must also be the same although there the freedom allows a continuous function along 1/8-th of the square. These features are exploited in the calculation. □