Lesson 7 Objectives

1. Lesson 7 Objectives • Reduction to one-group analysis: Stand-in for space-direction calculations • Expected convergence rate • Review • Reduction of B.E. to 1D space • 1D Quadratures

2. Analysis of 1-group eqn. • Basic idea: We can attack the B.E. as a series of ONE GROUP problems in space and direction • The effects of other groups will be BURIED in the scattering-in source term • Therefore, each group’s solution for spatial/direction flux will use methods (e.g., FORTRAN subroutines) that IGNORE energy • This is of enormous value in MODULARIZING the computer codes

3. 1-group analysis (2) • WithOUT the infinite medium approximation, we re-introduce space and direction into the group g equation to get: • Notice that I am once again playing games with the source term. It now contains (1) external source, (2) scattering from OTHER groups, (3) contribution from fission neutrons (where F is the total neutron source as a function of position):

4. 1-group analysis (2) • With flux on the right side AND the left side, so we procede by introducing ANOTHER LEVEL of iteration, the INNER iteration: where k is the INNER iteration counter and I have dropped the ‘g’ subscripts since each term has one. • This inner iteration has an interesting physical interpretation. If you set the initial flux (which is traditionally referred to “Iteration Zero”) to zero: then

5. Expected inner convergence rate • This physical analogy leads us to expect that the convergence rate of the inner iterations (i.e., how quickly we expect the flux to settle down) will depend on how important within-group scattering is: • Therefore: • High absorption problems=fast convergence • High scattering, low absorption=slow convergence • REMEMBER: We are inside a group, so the scattering term is the “within-group” scattering.

6. Basic fixed source solution procedure • Set initial flux guess to zero in each group • For each energy group (high to low): • Set RHS to fixed source • Add the scattering source from other groups to RHS • Using latest group flux, add within-group scattering source to RHS • Solve for new group flux [REST OF COURSE] • Repeat steps A-D to convergence of flux • Repeat step 2 to convergence of flux. Be sure NOT to reset the fluxes to zero (i.e., do NOT return to step 1!). Start with the best fluxes you have for each group.

7. Recap of where we are • The forward Boltzmann equation in space, energy, angle equation was derived to be: • The left side held the flux-dependent terms, the right side held the non-flux-dependent term. • We also derived the adjoint form of the equation (which we will ignore the rest of the course)

8. Recap of where we are (2) • In the absence of an external source, we found that we had to add an eigenvalue to the equation. We learned four variations on this, the most common being the k-effective (or lambda) eigenvalue: • The fission neutron source term became our new “source”, so we moved it to the right hand side.

9. Recap of where we are (3) • We then began attacking the equation, beginning with energy. We approximated the energy dependence of the flux and cross sections using the MULTIGROUP equation with (after proper approximation of the group parameters), gave us (for each group): • This “theme” of throwing more and more over to the “source” term will continue as we go!

10. Recap of where we are (4) • Since the only non-g term of the left hand side was the scattering from other groups, we then (what else) relegated all of the non-g scattering to the right-hand-side to get: • We label the new source “out” because it becomes the source for each group in an OUTER iteration, where each of the energy groups are treated one-at-a-time. • We learned two ways of computing the source, depending on whether the inscattering sources from groups that have already been handled in the current outer iteration use the OLD iteration fluxes (“Jacobi”) or the NEW iteration fluxes (“Gauss-Seidel”)

11. Recap of where we are (5) • Adding in the outer iteration “counters”, , makes this: • The “?” is there because the iteration counter depends on whether you are using Jacobi or Gauss-Seidel (and, if G-S, whether g’ is a group that has already been calculated in the current outer iteration) • I gave you some practice using this in HW#6

12. Recap of where we are (6) • Just for a little practice in the numerical treatment of outer iterations, we reduced this to the infinite-medium-form, where the flux does not depend on either space or direction, giving us: • Following the traditional notation, this equation was written in terms of the SCALAR flux (the direction-integrated ANGULAR flux) because, for isotropic infinite-medium fluxes, they are the same value (as long as you use unit solid angle):

13. Recap of where we are (7) • Whenever you use an iterative method, if the numerical method is convergent, the fluxes will change less and less with each iteration. The user has to decide when close enough is good enough. • Usually this is done with a convergence criterion of the MAXIMUM FRACTIONAL CHANGE of any iterating variable. • After each iteration, you loop over each flux value and determine the fractional amount it has changed during this iteration. You find the maximum ABSOLUTE VALUE of any change and that becomes your “error” value for the iteration. • You continue iterating until either: • The absolute value of your iteration error is less than some preset CONVERGENCE CRITERION; or • You have exceeded the maximum number of iterations you will allow (necessary so that a divergent problem doesn’t run forever) • In setting the convergence criterion, you must be careful not to demand more digits than the computer uses for a variable! (This is why FRACTIONAL change is used.) I like 1.0e-06

14. Recap of where we are (8) • Concentrating our attention on the solution within each group (dropping the outer iteration counter since they are all ): we again simplified by throwing the more complicated within-group scattering term to the right-hand-side to get: • Using an INNER iteration counter of k, this becomes:

15. Recap of where we are (9) • Notice that I did NOT use the removal cross section since the total cross section and within-group scattering are no longer on the left-hand-side of the equation • YOU MUST BE SURE to start each inner iteration with the initial fluxes set to the BEST AVAILABLE fluxes for that group (i.e., the results of the last time this group was calculated). DO NOT reset them to zero; they are only set to 0 the first time.

16. Recap of where we are (10) • Whenever you are dealing with BOTH inner and outer iterations, you have to decide how finely to converge the inner iterations before moving on to the next group. • Basically, this comes down to setting the maximum number of inner iterations to run for each outer sweep through a group. • The two limiting cases of this parameter are 1 and infinity, i.e., • Running just one inner iteration per outer iteration; and • Converging the inner iterations FULLY before moving on, no matter how long it takes. • Although this decision is problem dependent, usually a good value can be found. (At SRP, we used 4.) • FYI, it is frequently done that the INNER iteration convergence criterion is smaller than the OUTER iteration convergence criterion (I like to use half). This helps keep the number of outer iterations lower.

17. Treatment of Space and Direction • Now we move on to solve the group equations in both direction and space • Direction this lecture • Space next lecture • You might THINK that we would treat one of them first and then move on to the next one. BUT, since multidimensional problems are more difficult than one-dimensional problems, we shall instead attack: • Both space and energy in one-dimensional problems and THEN • Both space and energy in multi-dimensional problems • So, next is the treatment of one dimensional problems. We shall also break THIS up by difficulty, looking at: • One dimensional SLAB geometries; and then • One dimensional CURVED geometries (cylindrical and spherical).

18. Reduction to 1D Slab • To concentrate our attention on our angular treatments, we will simplify space to 1D slab geometry: • where the second relation reminds that the angular dependence of the source is given as Legendre source moments.

19. Quadrature integration • In a minute, we are going to represent the angular dependence of the angular flux as a quadrature. • But, since this is a relatively obscure mathematical practice, let’s review what a quadratureis and the specific properties of a Gauss-Legendre quadrature. • Whereas an expansion is used to approximate a function at all points of the domain: a quadrature is used to approximate the integral of a function between a and b by sampling specific points of the function and using a weighted sum (with weights summing to 1):

20. Quadrature integration (2) • The different quadratures differ in the number of points sampled, the prescription of the values to be sampled, and how much weight each is given: • The most common quadratures use equally spaced x values, but use specially chosen weights: • Reimann sum: N cell-centered x’s, weights=1/N • Trapezoidal rule: N+1 cell-edged x’s, weights=(0.5,1,1,…,1,0.5)/N • Simpson’s rule: N+1 cell-edged x’s, weights=(1,4,2,4,2,…,2,4,1)/(3N) • Example: • Reimann: 2(0.5^3+1.5^3)=3.5 • Trapezoidal: 2(0.5*0+1*1+0.5*8)=5 • Simpson’s: 2(1*0+4*1+1*8)/3)=4 (Simpson’s rule is exact up to 3rd order)

21. Gauss-Legendre Quadrature • An Nth-order Gauss-Legendre quadrature is RESTRICTED to integrals between -1 and 1 (although they can be adapted for other domains). • This quadrature defines the wn’s and the mn’s by the following rules: • The mn’s are chosen to be the roots of the Nth order Legendre polynomial: • The wn’s are chosen so that the Legendre moments up to order N are satisfied exactly, i.e,

22. Gauss-Legendre Quadrature (2) • The advantage of a Gauss-Legendre quadrature is that an Nth-order Gauss-Legendre quadrature integrates a (2N-1)th order polynomial exactly. • The resulting quadrature sets are given in text on Table 3-1. • NOTE: Only the positive values of the xn are given. You must include the negative values as well (with the same weights.) • NOTE: The ORNL code convention is to have all weights add up to 1. The text follows the convention that the weights add up to the width of the domain of integration (i.e., 2 for our problems). In effect, they incorporate the (b-a) term in the weights.

23. Gauss-Legendre Quadrature (3)

24. Gauss-Legendre Quadrature (3) • Quick sample problems for you to check your understanding: • N=2 • N=4

25. Application of SN quadrature to direction • We will now apply the Gaussian quadrature to our direction variable • As engineers, we are seldom interested in the angular flux itself. We are much more interested in REACTION RATES, which are determined from the scalar flux (using unit :

26. Applic. of SN quadrature to direct’n (2) • This approach also fits well with our definition of the angular source, for which we use flux moments:

27. Resulting equation • The final result is that we calculate the angular flux ONLY IN PARTICULAR DIRECTIONS (“discrete ordinates”): • We will begin the spatial attack next lesson

28. Homework 7-1 • Solve the problem from HW 6-1 again, but this time utilizing BOTH outer and inner iterations (in 6-1 you implicitly used 1 inner/outer) Group 1 2 3 Total 0.6 0.8 1.0 Scat to 1 0.4 0.1 0.1 Scat to 2 0.1 0.3 0.1 Scat to 3 0.05 0.3 0.4 Source 1 0 0 • Use a convergence criterion of 1.e-6. • Experiment with varying the maximum number of inners/outer to find the most efficient (in terms of TOTAL NUMBER OF INNER ITERATIONS)

29. Homework 7-2 • Solve for : a. b. c. using the Legendre-based quadratures from Table 3-1 of the text. For each of them, use the 2, 4, 6, 8, 10, and 12 order quadrature parameters.