Appendix C. Efficient computing methods and the analysis of multiplication operations
When evaluating the fourth order orientation tensor components, aijkl , it is important to reduce the
computing cost by using an efficient computing method for the closure approximation.
The
equations in this appendix describe an efficient computational method, and provide a detailed analysis
of the corresponding multiplication operations for both the IBOF-5 and EBOF-5 closures.
Using the
symmetries and the normalization relations of aijkl , only nine independent components need to be
calculated out of the 81 components of aijkl , regardless of which closure approximation is used.
(The other 72 components can be evaluated based on the symmetries and normalization conditions,
with negligible computing effort).
Table C.1 summarizes the computation analysis for IBOF-5 and
EBOF-5.
Invariant-based system (IBOF-5)
Introducing the symmetric part, as indicated in Eq. 8, explicitly into Eq. 7 results in the following
equation:

aijkl  1  ij kl   ik  jl   il jk



  3 aij akl  aik a jl  ail a jk 
  4  ij bkl   klbij   ik b jl   il b jk   jl bik   jk bil 
  5 aij bkl  a kl bij  aik b jl  ail b jk  a jl bik  a jk bil 
  6 bij bkl  bik b jl  bil b jk .
  2  ij akl   kl aij   ik a jl   il a jk   jl aik   jk ail
(C.1)
where the second order tensor bij represents the square of aij , i.e.,
bij  aim am j
(C.2)
1  1 / 3,  2   2 / 6,  3   3 / 3,  4   4 / 6,  5   5 / 6,  6   6 / 3
(C.3)
and  i is defined as follows:
Eq. C.1 can be further reduced and categorized according to the number of identical indices in aijkl , as
follows:
Group 1: a1111, a2222,
aiiii  31  6 2aii  33aii 2  6 4bii  6 5aiibii  3 6bii 2 ,
1
(C.4)
where i=1,2 (no sum on i).
Group 2: a1122,


a1122  1   2 a22  a11    3 a11a22  2a12   4 b22  b11 
2


(C.5)
  5 a11b22  a22b11  4a12b12    6 b11b22  2b12 .
2
Group 3: a1123 , a2231,
aijkl   2 akl   3 (aij akl  2aik a jl )   4bkl
(C.6)
 5 (aijbkl  akl bij  2aik b jl  2a jl bik )   6 (bijbkl  2bik b jl ).
Group 4: a1131 , a1112, a2223 , a2212,


aijkl  32akl  33aij akl  3 4bkl  5 3aijbkl  3aklbij  36bijbkl .
(C.7)
The computational burden in terms of the number of multiplication operations required to obtain
the nine independent components of aijkl can be determined from Eqs. C.4 - C.7.
In IBOF-5, the
second and third invariants must first be evaluated:
 
1
1  aij a ji ,
2
  a11 (a22a33  a23a32 )  a12 (a23a31  a21a33 )  a13 (a21a32  a22a31 ).
(C.8)
Evaluation of II and III requires 10 and 9 multiplication operations, respectively. (Henceforth, the
multiplication operation is indicated by appending M behind the number of operations: for instance,
10M and 9M for II and III, respectively.) Therefore, the calculation of the invariants requires 19M in
total. To calculate the  i ’s requires the following number of multiplication operations:
Each  i (i  3,4,6) requires 30M from Eq. 10;
1 ,  2 ,  5 require 15M, 11M, 5M, respectively, from Eq. 9.
Thus 121M are required in total.
The five components of the symmetric bij must be evaluated ( b11 , b12 , b13 , b22 , b23 ). Thus,
aim am j requires 3M for each component, and the evaluation of bij requires 15M in total.
The nine independent components of aijkl can now be calculated using Eqs. C.4 - C.7.
Each
component of group 1, group 2, group 3, and group 4 requires 14M, 15M, 17M and 15M, respectively,
totaling 137M.
Therefore, the number of multiplication operations required to obtain the nine components of
aijkl totals 292M.
If the order of the polynomial expansion for  i changes, the number of operations will only
change accordingly for  i (i  3,4,6) .
remains the same.
The number of operations for the rest of the calculations
This demonstrates the insignificant computational burden that results from using a
2
fifth order polynomial expansion for  i in Eq. 10.
Eigenspace-based system (EBOF-5)
The transformation rule to change the fourth order tensor components from the global to the
eigenspace coordinate systems is:
aijkl  VimV jnVkpVlq a m npq  Lim jn Lkplq a m npq ,
p
p
(C.9)
where Vij is the rotation tensor for coordinate transformation, and aijkl and aijkl
p
represent the
fourth order orientation tensor components in the global coordinate system and in the eigenspace
coordinate system, respectively.
Lijkl is defined as follows:
Lijkl  VijVkl .
(C.10)
For efficient computation, Eq. C.9 can be explicitly expressed for the nine independent
components of aijkl , as for the invariant-based system:
Group 1: a1111, a2222,
aijkl  Li1 j1 Lk1l1a1111p  Li 2 j 2 Lk 2l 2 a 2222 p  Li 3 j 3 Lk 3l 3 a3333p
 6 Li 2 j 3 Lk 2l 3 a 2323p  6 Li 3 j1 Lk 3l1a3131p  6 Li1 j 2 Lk1l 2 a1212 p
(C.11)
Group 2: a1122,
aijkl  Li1 j1 Lk1l1a1111  Li 2 j 2 Lk 2l 2 a 2222  Li 3 j 3 Lk 3l 3 a3333
p
p
p
 Li 2 k 3 4 Li 3k 2  Li 2 k 3   Li 3i 3 Lk 2 k 2 a 2323p
 Li 3k1 4 Li1k 3  Li 3k1  

p
 Li1k 2 4 Li 2 k1  Li1k 2   Li 2i 2 Lk1k1 a1212
p
Li1i1 Lk 3k 3 a3131
(C.12)
Group 3: a1123 , a2231,
aijkl  Li1 j1 Lk1l1a1111p  Li 2 j 2 Lk 2l 2 a 2222 p  Li 3 j 3 Lk 3l 3 a3333p


 2 Li 3 j1 Lk 3l1  Lk1l 3   Li 3i 3 Lk1l1  Li1i1 Lk 3l 3 a3131p
 2 Li1 j 2 Lk1l 2  Lk 2l1   Li1i1 Lk 2l 2  Li 2i 2 Lk1l1 a1212 p .
 2 Li 2 j 3 Lk 2l 3  Lk 3l 2   Li 2i 2 Lk 3l 3  Li 3i 3 Lk 2l 2 a 2323p
Group 4: a1131 , a1112, a2223 , a2212,
3
(C.13)
aijkl  Li1 j1 Lk1l1a1111p  Li 2 j 2 Lk 2l 2 a 2222 p  Li 3 j 3 Lk 3l 3 a3333p
 3Li 2 j 3 Lk 2l 3  Lk 3l 2  a 2323p
 3Li1 j 3 Lk1l 3 
(C.14)

 3Li1 j 2 Lk1l 2  Lk 2l1  a1212 p .
Lk 3l1 a3131p
First, the eigenvalues and eigenvectors of the second order tensor aij are required. When they
are calculated using an iteration method, as is generally adopted by many researchers, the number of
multiplication operations is typically in the range of 18n3 ~ 30n3 for an n  n symmetric matrix
(Press et al., 1992).
Approximately 486M ~ 810M operations are required.
p
The three independent principal values of aijkl must be determined.
For EBOF-5, a fifth order
polynomial expansion in terms of a1 and a2 is used for A11, A22 and A33 instead of Eq. 6 (which
closure
is written for EBOF-2). Thus, 30M are required for each Am m
( m  1,2,3, no sum on m ).
The other three Aii (i  4,5,6) require only 1M via the normalization condition.
This portion of the
calculation totals 91M.
Lijkl requires 81M operations.
Finally, each of the nine independent components of aijkl can be determined from Eqs. C.11 C.14. Each component of group 1, group 2, group 3, and group 4 requires 15M, 18M, 21M and 15M,
respectively, totaling 150M for the nine components.
Therefore, the number of whole multiplication operations required to obtain the nine components
of aijkl in EBOF-5 is at least 808M.
closure
Once again, if the order of polynomial expansion for Am m
changes, the number of
operations will only change accordingly for m  1,2,3 . The number of operations required for the
rest of the calculations remains the same.
Summary
Therefore, IBOF requires 30 ~ 40% of the EBOF computational time.
The major difference in the computation time is in the invariant calculation (IBOF) vs. eigenvalues /
eigenvectors calculation (EBOF).
4
Table. C.1. Summary of Multiplication Operations for IBOF and EBOF
IBOF
EBOF
Terms
Multiplication
Equation
Terms
Multiplication
Equation
calculated
operations
used
calculated
operations
used

10
(C.8)
a1, a2 , a3
486~810
Press et al.

9
Vij
(1992)
3 ,  4 , 6
30x3=90*
(10)
A11, A22 , A33
1 ,  2 ,  5
15, 11, 5
(9)
A44 , A55, A66
1
b11 , b12 , b13
3x5=15
(C.2)
Lijkl
81
(C.10)
a1111, a2222
14x2=28
(C.4)
a1111, a2222
15x2=30
(C.11)
a1122
15
(C.5)
a1122
18
(C.12)
a1123 , a2231
17x2=34
(C.6)
a1123 , a2231
21x2=42
(C.13)
a1131, a1112
15x4=60
(C.7)
a1131, a1112
15x4=60
(C.14)
30x3=90*
(6)
b22 , b 23
a2223 , a2212
Total
a2223 , a2212
292
Total
* valid for fifth order polynomial expansion (IBOF-5, EBOF-5)
5
808~1132