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TITLE: SPATIAL AND ANGULAR VARIATION AND DISCRETIZATION OF THE SELF-ADJOINT TRANSPORT OPERATOR

AUTHOR(S): Randy M. Roberts

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i

Spatial and Angular Variation and Discretization of the Self- Adj oint Transport Operator

Randy bl. Roberts

March 11, 1996

1

1 Variational Derivation 1.1 Self-Adjoint Equation The first-order equation for the angular flux, $(h, 2) is

This equation may be rearranged as

$ = (at - s)-1 [q - A . d$] .

si. a [(at - S ) - l ( q - 0 . a$)] +at$ - S$ - q = 0.

(3)

Combining these equations we obtain the second-order, self-adjoint, form of the transport equation,

(4

1.2 Space Variational Functional Now that we have the self-adjoint form of the transport equation we can find a functional, r[$], whose minimization solves the transport equation, i.e.

br[$] = o ( 5 ) implies $ is a solution of Equation 4. Additionally the functional's minimization should also lead to $ having the correct boundary conditions. The correct boundary conditions for the transport equation will be explained later.

A Ritz procedure will be used to obtain the desired functional. We define the functional, W[$, S$3, as

WW, WI = / J ~ G L(+) d3x dfi

= 0 VS$,

where L($) = 0 corresponds to Equation 4. For the self-adjoint equation W[V, b$] becomes,

w[$, ~ $ 1 = JJS$ fi . a [(at - s)-' (q - . a$)] d3x dfi

+ /J6$ (at$ - S$ - q) d3xdfl

2

The boundary conditions for the integral are derived from the term inside the surface integral,

(ut - s y ( q - fi . e$) , which from Equation 3 is a $-like term. If we choose the value of $(A) on the boundary to be $,, for fl . ii < 0 and $ otherwise, the equation for W[$, SI)] becomes,

where 6- is defined as the set of angles at the surfaces where f l . ii < 0, and fi+ are angles at the surfaces where fl . ii > 0.

This equation W[$,d$] = 0 VS$ can be obtained from minimizing the func- tional, r[$],

where we have made use of the self-adjoint property of the operators, S, and (at - S)-'.

2 1-D Example Let's choose a 1-D example with S = 0 and q = 0 to demonstrate the choice of boundary conditions. We will look at the boundary conditions from the left and right sides of the problem, with p fi . 2 > 0. At x = 21, ii = -2, and at x = x,, ii = 2. The next vertex to the right of x = xz is x = x, = x1 + h. The values of the angular flux at x = x1 and x = x, are $1 and &, respectively.

We can ignore the area perpendicular to the x-axis. Our expression for I?[$] becomes,

3

f

Defining r z crt h / p the above expression becomes,

The solution for $1 and gr that minimizes I?[$] is

$b (1 + 3- + g) $1 =

$7 =

l+r+$+$

1 + 7 + $ + $. @b

The value of can be compared with the exponential of r by,

3 SN Discretized Functional

r U [ $ l ,

For this section we will concentrate on the volume contribution to the functional,

We will be expressing the discretized functional in terms of $(fl, Z) evaluated at the corners of the cells,

For the i’th corner of a cell we will define an approximation to the gradient of

$(A, 5) + $i(flt). (19)

(20) 111 as

Q*$ = C($j(Q - $ i ( f l ) ) d j ,

3

with the sum extending over the three neighbors sharing an edge with the i’th corner. The expression &i]e is defined by

where k and 1 are the two neighb_or corners of i that are not j . convenience, the expression for fi . Vi$ may be expressed as

For later

4

where the sum over j has been extended to all corners in a cell. The expression,

fi ' 6 k i ,

implies a 3-vector dot product of a 3-vector with a 3-vector of matrices, i.e.

With these definitions the volume functional (over one cell)' can be approx- imated by

In the above expression the corner weights, U c k , evaluated in the cell, sum to the total volume of the cell, and the qck expression is evaluated within the cells2. The extrema of rV is found by letting '$ contain a variation around the extema value,

and setting the coefficient of first order term in E to 0, $ i ( f i ) -+ + i ( f i ) + c u i ( f i ) , ( 2 5 )

Since the ' $ i l S extremize I?, Equation 26 should hold for any choice of u i ' s . A set of simultaneous equations can be found by setting ui(6) = 1 for a particular choice of i with uj(6) = 0 for j # i.

Putting Equation 24 into Equation 26 we have,

where we have again made use of the self-adjoint properties of S and (et - S)-'. The (et - S)-' [ (fl . $k'$) - q c k ] term in the above equation may be evaluated inside the cell using Equation 3. The above equation becomes,

'The sum over IC can be viewed as a double sum, first over the cells, and then over the

2For the steady-state problem qck is evaluated on the vertices. For time-dependent prob- corners of the cell. The other sums over corners can then be restricted to the current cell.

lems &k will include QL,k/cAt, evaluated at the previous iteration.

5

where $ c k ( f i ) is evaluated within the cell. \Ye will use Equation 2 to evaluate $ck in Equation 28,

The superscripts represent iterations, since S is not easily invertable in the S, representation. If we introduce a new source term to represent the previous iteration the above equation may be expressed as

it is implied that $k?b is evaluated within the cell. Additionally, in order to decouple the angles, the S $ k term in Equation 28 will be evaluated at the previous iteration, resulting in a Qvk = S$p' + q p ) term. These equations are inserted into Equation 28 to yield

d f i uck ( O t u k $k - Qvk u k ) = 0. (31) +' k

we introduce the matrix expression for f i . d k from Equation 22,

where D k i ( f i ) fi . 5 k i (c.f. Equation 23). To obtain a set of simultaneous equations we set the uil 's,

and Equation 32 becomes,

6

r _1

# We continue by introducing the Hessian matrix, H ,

c)

where Hij is the dyadic,

In matrix form SN equation becomes, r 1

where the functions Aij(f2) and b i ( f i ) do not explicitly depend on the $'s from this iteration.

If we assume some angular discretization, the matrix equation becomes

where

and w1 are the weightings for the integral over h. Since there is no explicit coupling between different values of E , Equation 40 must hold separately for each I, i.e.

Gjl = Gj (hd, (41)

Afj$jl - bil = 0. (42) j

4 PRJ Discretized Functional The PN discretization begins as does the SN up to Equation 27,

7

' i

I

where qck: is evaluated within the cell. There is no need to substitute 4 c k into this equation3. In the PN formalism (ot - S)-' is easily invertible.

The next step is instead to define $!Jk by its spherical harmonic series,

1

1

In this equation the 1 index represents both the polar and azimuthal moments, i.e. I represents the (1, m) tuple. The spherical harmonics are orthonormal,

Before we begin to include the spherical harmonic series in the discretization we need to find the expansion for 6.5 We may express f in terms of its spherical harmonics, A,

1 I'

where A111 is a matrix of vectors representing the E'th coefficient of (6) fi. With these definitions Equation 43 becomes,

km' I'

The l ' , m, and mi indices are also single index representations of both the polar and azimuthal spherical moments. In the above equation we used the property that S is diagonal in the space of X ' s , i.e.

S X = a l x

3For time-dependent problems you will still need to evaluate and store the spherical mo- ments of $ck (c.f. Equation 29). The moments of qck terms will include the corresponding moments of Qck / cAt , evaluated at the previous iteration.

8

The only remaining functions off? are the K (6)’s. Due to the orthonormality conditions the variation equation becomes,

(54)

1

f X V c k u k l ’ [(fft - f f l j )$kl’ - q k l ’ ] k 1’

It is important to realize that the expression,

’ 6 k i ,

( 5 5 )

implies a 3-vector dot product of two 3-vectors of matrices, i.e.

and the (i, 1)’th variation equation becomes,

+ v c i [(nt - al)4i~ - qil]

Consolidating terms in 4 we arrive at

9

The above equation is written in terms of AT, even though A is symmetric, in order to expose the symmetry of the final matrix. We continue by introducing the Hessian matrix, H ,

*

tf where Hij is the dyadic,

k

This definition is identical to that used in the SN formalism. Like Equation 37, Equation 61 may be written as a matrix equation,

j m

where A$' and bil do not explicitly depend on the 4's from this iteration.

5 Surface Discretization From Equation 11 the surface contribution the I? functional, rs, is

We shall discretize rS over boundary faces as follows,

where the sum over j extends first to all boundary faces, and then to the corners of faces, ZCj is the area vector associated with that face and corner.

5.1 SN Surface Angular Discretiztion For the SN treatment the integrals over angle are replaced by weighted sums,

10

Minimization of this equation with respect to & l , on the boundary, for a par- ticular value of 61 leads to

where the sum is over all boundary surfaces containing vertex i. The first of these surface terms will be added to the b,l term of Equation 42, the second term is added to the E, Af,$,i of that equation.

5.2 PN Surface Angular Discretiztion For the PN treatment we mininize Equation 65 using

$j(fi) -+ Ilrj(fi2) + obtaining ,

0 = volume terms + h . &j $bj uj + fi . &j $ j U j . (69) j

The equation is now expanded into its spherical harmonics,

0 = volume terms

= volume terms

= volume terms

Setting ujp = SijSlp we arrive at

0 = volume terms

c m

11

6 7

1 4

Figure 1: The corner numbering scheme for the hexahedron.

where the sum over c indicates a sum over all boundary surfaces containing vertex i. The first of these surface terms will be added t o the bil term of Equation 63, the second term (multiplied by 6 i j and summed over j ) is added to the Cj AfY4jm of that equation.

6 Specific Element Geometries *

In this section we will display the explicit forms of the 6 and H matrices that were defined above.

One important property of the 6 matrix is that its only non-zero elements are between indices that share an edge in the element in question. This also implies that the 6 matrix is “structurally” symmetric, meaning that if 3ij # 0 then 6 j i # 0. The H matrix is truly symmetric.

*

6.1 Hexahedron For the hexahedron we will look at the matrix elements associated with corner 1 of the hex (c.f. Figure 1). All other matrix elements may be derived by a suitable permutation of corner indices, corresponding to a rotation of the hex.

For corner 1 the only non-zero elements of 6 correspond to corners 1, 2, 4, and 5,

-(& + 314 + 615) IC = 1

otherwise 6 l k = { zlk k E { 2 , 4 , 5 } . (74)

0 c)

For the H matrix there are 4 distinct types of elements associated with

12

corner 1. The diagonal element is

(75) cf

H11 = u c k g k l f i k l k € { 1.2,4,5}

u c l ( 6 1 2 f 6 1 4 f 6 1 5 ) ( 6 1 2 + 6 1 4 f 6 1 5 ) =

f uc2 6 2 1 6 2 1 + uc4 6 4 1 6 4 1 f vc5 6 5 1 6 5 1 - (76)

The matrix elements from corner 1 to its nearest neighbors, corners 2, 4, and 5, are represented by H 1 2 ,

ti

cf

H I 2 = u c k f i k l z k 2 (77) k€{1,2}

-Ucl (612 f 6 1 4 f 6 1 ~ ) 6 1 2 - U C ~ 6 2 1 ( 6 2 1 f 6 2 3 f &G). = (78)

The matrix elements from corner 1 to its next-nearest neighbors, corners 3, 6 , and 8, are represented by H 1 3 ,

t)

The matrix element from corner 1 to its third-nearest neighbor, corner 7, is

This is because there are no nearest neighbor corners in common for corners 1 and 7.

6.2 Wedge For the wedge we will look at the matrix elements associated with corner 1 (c.f. Figure 2). All other matrix elements may be derived by a suitable permutation of corner indices, corresponding to a rotation of the wedge.

For corner 1 the only non-zero elements of 6 correspond to corners 1, 2, 3, and 4,

-(&2 + 613 + 6 1 4 ) k = 1

otherwise k € (2,3,4} . (82)

tf For the H matrix there are 3 distinct types of elements associated with

corner 1. The diagonal element is c,

Hi1 =

13

5

6

1 3

Figure 2: The corner numbering scheme for the wedge.

The matrix elements from corner 1 to its nearest neighbors, corners 2, 3, and 4, are represented by H l 2 ,

tt

c1

H i 2 = u ~ k d k i 5 k 2 (85) ~ ( 1 ~ 3 )

f uc3 6 3 1 s 3 2 . (86)

- - -uc1 (dl2 + 6 1 3 + &14)612 - uc2 z 2 1 ( 6 2 1 + z 2 3 f z 2 5 )

The matrix elements from corner 1 to its next-nearest neighbors, corners 5, and 6, are represented by HIS,

+t

tt

H 1 5 = uck d k l d k S (87)

= 21~2 z216!25 + 21~4 Z 4 1 6 4 5 - (88) k ~ { 2 , 4 )

6.3 Tetrahedron For the tetrahedron we will look at the matrix elements associated with cor- ner 1 (c.f. Figure 3). All other matrix elements may be derived by a suitable permutation of corner indices, corresponding to a rotation of the tet.

For corner 1 all matrix elements of d are non-zero,

f+ For the H matrix there are 2 distinct types of elements associated with

corner 1. The diagonal element is

14

4

3

1 2

Figure 3: The corner numbering scheme for the tetrahedron.

The matrix elements from corner 1 to its neighbors, corners 2, represented by H12,

ct

(91)

3, and 4, are

6.4 Pyramid

15

t

5

1 2

Figure 4: The corner numbering scheme for the pyramid.

16