Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

462 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsi.e., the (analog) expected score varies approximately exponentially with x. In the majorityof practical deep-penetration problems, the spatial drop of the particle density can be approximatedquite well by an exponential, thus, approximation (7.53) is reasonable. Thevalue of X in Equation (7.53) can be determined by two ways. One possibility is to startparticles from different positions (characterized by different x values) and fit Equation (7.53)to the scores. Alternatively, one may score the flux integral in small regions about selectedx values and fit an exponential of the form Be ^xto the estimated values. Having obtainedan estimate of X in Equation (7.53), the score in the transformed game will be approximatelyindependent of the position ifTH 1(P) = e~ bx M,(P) = Ae ( " ~ b)x= eousyi.e., ifb = X (7.54)Accordingly, a quasi-optimum stretched cross-section is(r(P) --- cr(P) • A/A (7.55)where A is the coefficient of x in the exponential spatial drop of the particle density in theanalog game and p is the cosine of the angle between the actual flight and the preferreddirection.Note that in certain cases the stretched cross section in Equation (7.55) may becomenegative, which poses specific problems in the simulation. This matter will be discussed inChapter 7.III.The optimal procedure outlined above was successfully applied in practical problems. 36It will be seen in Chapter 7.111 that by defining a transformed game in which not only thetransition kernel but also the collision kernel is biased, the direct statistical approach appliedabove will determine an approximation to a zero-variance game.II. OPTIMIZATION OF GEOMETRICAL SPLITTINGGeometrical splitting is one of the simplest variance-reducing and also efficiency-increasingtechniques, and is used in almost all general and special-purpose Monte Carlocodes. It is especially favored in deep-penetration calculations, but is also efficient inenhancing the particle population in regions where the analog particle density is low. Inspite of its conceptual simplicity, its use is still based mainly on intuition, practice, or "rulesof thumb". Optimization of geometrical splitting has recently gained considerable attention,mainly because of the powerful mathematical tools provided by the Monte Carlo momentequations and the concept of the direct statistical approach.Early results in optimum splitting schemes 14 ' 15are based on very simple models, similarto those presented in previous Chapters. The variance of the score in the parallel use ofsplitting and exponential transformation was investigated by Sarkar and Prasad 46on the basisof the moment equations. Juzaitis 21extended the investigations to the efficiency of a gamewith splitting and proposed to solve the moment equations by a standard S ncode. The resultsso obtained for single-surface splitting in monoenergetic homogeneous simulation may serveas reference values for more sophisticated schemes. Dubi. Elperin, and Dudziak 91 " and laterDubi" presented a very detailed description of a general fixed-surface splitting game throughthe direct statistical approach. Practical applicability of their analysis is still hindered by thegreat number of bulk parameters to be determined in the model. Nevertheless, simplificationof the model may yield feasible automatic optimization schemes.