Lecture 11: Variance reduction/ adjoint overview

1. Lecture 11: Variance reduction/adjoint overview • Review line/disk sources with buildup • Project defined • Cell-weight variance reduction technique in MCNP • Use of adjoint fluxes (descriptive)

2. Buildup Factor Illustration

3. Line and disk sources with buildup “Examining this says that you can solve a buildup problem (a total dose problem) just as easily as an uncollided dose problem. You just have to interpret A1 as a source multiplication factor and (1+a1) as a mean free path multiplication factor. (In practice, a1 is negative—but no lower than -1—so that it can be interpreted as the fraction by which mu is reduced.)”

4. Project • Posted in Public area

5. Surface 1 2 3 4 5 Source Population Control Biasing • Using the POPULATION information (which is a summary of MC particles, not physical particles) we can increase the codes FOCUS on regions of interest by making them more important.

6. Description of Problem • A hollow (thick) iron ball:

7. Tutorial 3 Base Code Tutorial 3, base case c ********************************************************************* c * c * c * c ********************************************************************* c * c Cells * c * c ********************************************************************* 1 0 -1 imp:p=1 2 1 -7.87 1 -2 imp:p=1 99 0 2 imp:p=0 c ********************************************************************* c * c Surfaces * c * c ********************************************************************* 1 sph 0 0 0 10 2 sph 0 0 0 110 c ********************************************************************* c * c Data cards * c * c ********************************************************************* mode p sdefpos = 0. 0 0 erg=3 par=2 m1 26000 1 f1:p 1 2 ctme .25 PRINT

8. Tutorial 3 Base Code Tutorial 3, base case c ********************************************************************* c * c * c * c ********************************************************************* c * c Cells * c * c ********************************************************************* 1 0 -1 imp:p=1 2 1 -7.87 1 -2 imp:p=1 3 1 -7.87 2 -3 imp:p=1 4 1 -7.87 3 -4 imp:p=1 5 1 -7.87 4 -5 imp:p=1 6 1 -7.87 5 -6 imp:p=1 7 1 -7.87 6 -7 imp:p=1 8 1 -7.87 7 -8 imp:p=1 9 1 -7.87 8 -9 imp:p=1 10 1 -7.87 9 -10 imp:p=1 11 1 -7.87 10 -11 imp:p=1 99 0 11 imp:p=0 c ********************************************************************* c * c Surfaces * c * c ********************************************************************* 1 sph 0 0 0 10 2 sph 0 0 0 20 3 sph 0 0 0 30 4 sph 0 0 0 40 5 sph 0 0 0 50

9. Tutorial 3 Base Code 6 sph 0 0 0 60 7 sph 0 0 0 70 8 sph 0 0 0 80 9 sph 0 0 0 90 10 sph 0 0 0 100 11 sph 0 0 0 110 c ********************************************************************* c * c Data cards * c * c ********************************************************************* mode p sdefpos = 0. 0 0 erg=3 par=2 m1 26000 1 f1:p 1 2 3 4 5 6 7 8 9 10 11 ctme .25 PRINT

10. Use of Adjoint in Source/Detector • There are several uses of adjoint B.E. • We will examine it in a Source/Detector problem: Volume V Source Detector Surface S

11. Adjoint or forward? Q: If either way gives the same answer, how do you choose between the forward and adjoint modes? A: Both methods involve two steps: • Solve for forward or adjoint flux (expensive) • Integrate flux over response function or source (cheap) Therefore, you should choose the mode that minimizes expense: • FORWARD if you have 1 source, many detectors • ADJOINT if you have 1 detector, many sources

12. A little more abstract • A FORWARD calculation involves DEDUCING the consequences of the presence of a SOURCE. It results in a particle flux distribution that can be used to make a DOSE MAP. • On the other hand, an ADJOINT calculation involves deducing the consequences of the placement of a DETECTOR. It results in a adjoint flux distribution that can be translated into a DETECTOR SENSITIVITY MAP. (i.e., a map of what source locations a detector can “see”).

13. Adjoints and boundary sources • An important class of shielding problems do not fit the source/detector geometry we have been using, but instead involve boundary fluxes serving as sources: Volume V Boundary flux as a surface source Detector Surface S

14. Adjoints/boundary sources (2) • Solve the “normal” adjoint problem: and use the adjoint flux and boundary fluxes to get the detector response using: • The adjoint path is especially useful in “do-it-yourself” response function creation: Run an adjoint on a “detector” object you define (e.g., human phantom, battle tank) and then subsequently use the surface adjoint current as a response function for the object.

15. Surface scoring • This “surface source” methodology can even be applied to “normal” source/detector problems by partitioning the problem into a “source side” (V1) and a “detector side” (V2) : Volume V1 Volume V2 Source Detector Surface S Partitioning surface PS

16. Surface scoring (2) • The detector response is given by the surface integral: where the integral need only be taken over the Partitioning Surface PS because (like before) the external boundary conditions guarantee that the product y*y is 0 on all external surfaces. • Note that the partitioning surface can be ANY surface that completely separates the source and detector

17. List of other uses of adjoint • Other uses of the adjoint in nuclear engineering include: • As an importance function for emerging particles in a Monte Carlo calculation (NE582) • As the optimum weight function for perturbation theory in absence of flux information for final state (NE571) • As the optimum weight function for the generation of point-kinetics parameters (NE571?) • As a graphical tool for illustrating the important particle paths in a shielding analysis (i.e., the product y*y is the so-called “contributon” flux, the distribution of particles that are “destined” to be detected).