# Self-adjusting Importances for the Acceleration of MCBEND

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4 The adjoint estimate generated by this method can be very approximate and generally deteriorates with remoteness from the detector. The method is an alternative when diffusion methods are inapplicable and can form a useful starting point for subsequent improvement. Detector Step 1: Sources ( ) generated in the detector mesh Adjoint known Step 2: Sources ( ) generated in the detector neighbours Adjoint known Importance mesh boundaries Fig.2. Steps in the recursive estimate of importance. Step 3: Sources ( ) generated in the next outer layer of neighbours 3.4 Density adjustments. The principal reason why few particles can travel from source to detector without the aid of an importance function is the attenuating effect of the intervening materials. If the densities of all materials in the problem are sufficiently reduced then particles can reach the detector unaided. For example: if attenuation is approximately exponential then it can be reduced from, say, 10 8 to 10 2 by quartering all the densities. Using the technique described earlier, an adjoint for this non-physical problem can be scored. This adjoint can then be used to accelerate a variation in which the density reduction is less severe. Repetition of this process eventually produces an importance map that can be used in the real problem. In the diluted variation, the probability of scoring is governed by geometrical effects and the method is unsuitable for small detector volumes remote from the source. 4. Example Application. The following simple test configuration has been used to illustrate the techniques described above. The case is small enough to be solved in analogue mode but large enough to benefit from use of an importance map. It contains realistic materials and energy ranges. The objective is to calculate the neutron flux above 1.0 MeV in the detector from a fission source. The importance map is represented in eight energy groups between 10.0 and 0.6 MeV and contains 588 spatial meshes - a total of 4704 space/energy cells. The hatched area of iron is replaced by a void in a later model. The following variations have been used. 1 Execution with no importance map. 2 Using an importance map from solution of the adjoint diffusion equation. 3 Using an importance map generated by the recursive method of 3.3 which took 6.0 minutes. 4 Using an importance map by density adjustments: initial execution (5minutes) with no importance and densities scaled by a factor 0.25: secondary execution (10 minutes) using the scored adjoint from the initial execution and densities scaled by a factor 0.5: final execution using the scored adjoint from the secondary execution and real densities.

5 Y or Z 16.0 Carbon Water 14.0 Iron Carbon 12.0 Iron 10.0 Importance mesh Source Detector X Importance mesh Fig.3. Example test geometry. The efficiencies of the four variations are compared in the following table. The times quoted to reach a variance of 1% on the scored response include the times taken to generate the importance maps. Table 1. Results using various methods of generating importance maps. Variation Description Result Time (minutes) Relative efficiency 1 No importance 5.84E-3 ± 1.0% Adjoint diffusion 5.76E-3 ± 1.0% Recursive 5.83E-3 ± 1.0% Density variation 5.83E-3 ± 1.0% For such a small problem (the attenuation of the response between source and detector is only a factor 180) the gains in efficiency given by use of an importance function are not dramatic. The greatest gain is given by the diffusion solution - a typical conclusion when the geometry is suitable for the use of this method. A variation on this test case was made by replacing the hatched area of Fig. 3 by a void region and adding extra importance mesh boundaries along the new surfaces created. In a diffusion solution, the presence of the void tends to flatten the gradients of the adjoint leading to a less efficient mesh. When the methods described above are applied to this case the results are as follows.

8 using the looping and parameterisation facilities of MCBEND to control the sampling times and density factors in each pass. The variances of the results were compared with those obtained in a 14 hour execution of the original case with a Monte Carlo Adjoint. The comparison showed that the importance map produced by the new method was, as to be expected, less efficient - by about a factor 2.5. The case was also executed using the recursive method with a similar conclusion. The approximate importance maps are not intended to compete in efficiency of execution with a proper adjoint produced by Monte Carlo or transport methods. Their merits lie in their ease of application in routine calculations when user effort has a higher premium than computing time. 8. Summary. A range of methods has been described for the automated production of importance maps for acceleration of Monte Carlo calculations. An algorithm has been defined whereby the adjoint solution of a problem can be estimated during the normal, forward execution. Mechanisms have been described for the revision of an initial importance by the estimated adjoint at intervals during a calculation. The methods have all been implemented in the Monte Carlo code MCBEND to provide a variety of options with a convenient user image. Further development is needed to refine the strategies to be used when applying the new methods. Acknowledgements. The funding for these developments has been provided by the HSE/IMC Nuclear Safety Research Program References. 1 S.J.Chucas et al. Advances in the Monte Carlo Code MCBEND, Proceedings of 1996 Topical Meeting: Radiation Protection and Shielding. No. Falmouth, Massachusetts, USA, P.C Miller et al. The Use of an Inbuilt Importance Generator for Acceleration of the Monte Carlo Code MCBEND, Proceedings of the International Conference on the Physics of Fast Reactors. Marseilles, France, 1990

9 Manuscript information file Self-adjusting Importances for the Acceleration of MCBEND 3 Author 4 Shuttleworth, Edmund 5 AEA Technology, Winfrith, Dorset, DT2 8DH, UK End Author 10 Acceleration 11 Monte Carlo 12 Self-learning

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