{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/311429"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/311429","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Numerical stability of Monte Carlo neutron transport and isotopic depletion for nuclear reactor analysis","abstract":"Coupling Monte Carlo neutron transport with isotopic depletion is known to produce non-physical results for large reactor geometries. This thesis begins with a survey of the stable methods used to avoid this and highlights some of their drawbacks. Chapter 2 introduces the known phenomenon of neutron clustering in Monte Carlo as a factor strongly contributing to previous reports of instability. It is shown that attempting to minimise clustering effects produces more stable burn-up calculations. Furthermore, accounting for clustering is shown to allow for the accurate simulation of xenon transients in simple systems which have been noted to pose a challenge for burn-up simulations. Finally, it is demonstrated that neutron clustering can also affect burn-up simulations substantially even when xenon equilibrium is enforced, namely by way of the previously hypothesised `gadolinium instabilities'. Chapter 3 begins by highlighting how implicit burn-up schemes may be viewed as root-finding schemes for a discrete map. It is shown that, for reasonably long time-steps, the corrector step of predictor-corrector schemes does not succeed in locating the root (or the stable solution) of this map. Hence, relaxation schemes are introduced; relaxation schemes have been applied to neutronics/depletion coupling previously in the form of the stochastic approximation, although this is relatively inefficient. The relaxation scheme proposed here uses a fixed relaxation factor (rather than the variable factor previously used) and demonstrates its improved stability and computational efficiency compared to the stochastic approximation. The same investigations are applied to a depletion problem where equilibrium xenon is enforced -- it is seen that this, too, can be unstable, but is resolvable through relaxation. Chapter 4 performs a Von Neumann stability analysis of a simple coupled neutron diffusion-depletion system. Extending a previous analysis, this chapter provides a justification for applying a relaxation to predictor-corrector schemes, shows the possibility of obtaining symmetric burn-up instabilities, and demonstrates that no neutronics-depletion coupling scheme, without relaxation, is assuredly more stable than another, depending on the depletion system in question. Finally, Chapter 5 summarises the findings, proposes an explanation regarding general cases of burn-up instability, and suggests future work.","abstract_html":"Coupling Monte Carlo neutron transport with isotopic depletion is known to produce non-physical results for large reactor geometries. This thesis begins with a survey of the stable methods used to avoid this and highlights some of their drawbacks. Chapter 2 introduces the known phenomenon of neutron clustering in Monte Carlo as a factor strongly contributing to previous reports of instability. It is shown that attempting to minimise clustering effects produces more stable burn-up calculations. Furthermore, accounting for clustering is shown to allow for the accurate simulation of xenon transients in simple systems which have been noted to pose a challenge for burn-up simulations. Finally, it is demonstrated that neutron clustering can also affect burn-up simulations substantially even when xenon equilibrium is enforced, namely by way of the previously hypothesised `gadolinium instabilities'. Chapter 3 begins by highlighting how implicit burn-up schemes may be viewed as root-finding schemes for a discrete map. It is shown that, for reasonably long time-steps, the corrector step of predictor-corrector schemes does not succeed in locating the root (or the stable solution) of this map. Hence, relaxation schemes are introduced; relaxation schemes have been applied to neutronics/depletion coupling previously in the form of the stochastic approximation, although this is relatively inefficient. The relaxation scheme proposed here uses a fixed relaxation factor (rather than the variable factor previously used) and demonstrates its improved stability and computational efficiency compared to the stochastic approximation. The same investigations are applied to a depletion problem where equilibrium xenon is enforced -- it is seen that this, too, can be unstable, but is resolvable through relaxation. Chapter 4 performs a Von Neumann stability analysis of a simple coupled neutron diffusion-depletion system. Extending a previous analysis, this chapter provides a justification for applying a relaxation to predictor-corrector schemes, shows the possibility of obtaining symmetric burn-up instabilities, and demonstrates that no neutronics-depletion coupling scheme, without relaxation, is assuredly more stable than another, depending on the depletion system in question. Finally, Chapter 5 summarises the findings, proposes an explanation regarding general cases of burn-up instability, and suggests future work.","abstract_has_math":false,"creators":["Cosgrove, Paul"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Shwageraus, Eugene"],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-04-27","date_published":"2020-04-27","updated_at":"2026-07-22T22:24:03Z","subjects":["Nuclear","Monte Carlo","Reactor","Neutron transport","Isotopic depletion"],"languages":["eng"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/ea3050b9-cb57-4823-aca8-f717685240b8/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000328296299"],"render_values":[{"text":"0000-0003-2829-6299","href":"https://orcid.org/0000-0003-2829-6299","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.58521","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Shwageraus, Eugene"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["EPSRC ICO-CDT"]},{"key":"dc:creator","label":"Author","values":["Cosgrove, Paul"]},{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000328296299"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2020-04-27"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/311429"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Nuclear","Monte Carlo","Reactor","Neutron transport","Isotopic depletion"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/ea3050b9-cb57-4823-aca8-f717685240b8/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.58521"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/6d906cdf-1768-4282-8ba2-526f198e8d6b/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Coupling Monte Carlo neutron transport with isotopic depletion is known to produce non-physical results for large reactor geometries. This thesis begins with a survey of the stable methods used to avoid this and highlights some of their drawbacks. Chapter 2 introduces the known phenomenon of neutron clustering in Monte Carlo as a factor strongly contributing to previous reports of instability. It is shown that attempting to minimise clustering effects produces more stable burn-up calculations. Furthermore, accounting for clustering is shown to allow for the accurate simulation of xenon transients in simple systems which have been noted to pose a challenge for burn-up simulations. Finally, it is demonstrated that neutron clustering can also affect burn-up simulations substantially even when xenon equilibrium is enforced, namely by way of the previously hypothesised `gadolinium instabilities'. Chapter 3 begins by highlighting how implicit burn-up schemes may be viewed as root-finding schemes for a discrete map. It is shown that, for reasonably long time-steps, the corrector step of predictor-corrector schemes does not succeed in locating the root (or the stable solution) of this map. Hence, relaxation schemes are introduced; relaxation schemes have been applied to neutronics/depletion coupling previously in the form of the stochastic approximation, although this is relatively inefficient. The relaxation scheme proposed here uses a fixed relaxation factor (rather than the variable factor previously used) and demonstrates its improved stability and computational efficiency compared to the stochastic approximation. The same investigations are applied to a depletion problem where equilibrium xenon is enforced -- it is seen that this, too, can be unstable, but is resolvable through relaxation. Chapter 4 performs a Von Neumann stability analysis of a simple coupled neutron diffusion-depletion system. Extending a previous analysis, this chapter provides a justification for applying a relaxation to predictor-corrector schemes, shows the possibility of obtaining symmetric burn-up instabilities, and demonstrates that no neutronics-depletion coupling scheme, without relaxation, is assuredly more stable than another, depending on the depletion system in question. 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Chapter 2 introduces the known phenomenon of neutron clustering in Monte Carlo as a factor strongly contributing to previous reports of instability. It is shown that attempting to minimise clustering effects produces more stable burn-up calculations. Furthermore, accounting for clustering is shown to allow for the accurate simulation of xenon transients in simple systems which have been noted to pose a challenge for burn-up simulations. Finally, it is demonstrated that neutron clustering can also affect burn-up simulations substantially even when xenon equilibrium is enforced, namely by way of the previously hypothesised `gadolinium instabilities'. Chapter 3 begins by highlighting how implicit burn-up schemes may be viewed as root-finding schemes for a discrete map. It is shown that, for reasonably long time-steps, the corrector step of predictor-corrector schemes does not succeed in locating the root (or the stable solution) of this map. Hence, relaxation schemes are introduced; relaxation schemes have been applied to neutronics/depletion coupling previously in the form of the stochastic approximation, although this is relatively inefficient. The relaxation scheme proposed here uses a fixed relaxation factor (rather than the variable factor previously used) and demonstrates its improved stability and computational efficiency compared to the stochastic approximation. The same investigations are applied to a depletion problem where equilibrium xenon is enforced -- it is seen that this, too, can be unstable, but is resolvable through relaxation. Chapter 4 performs a Von Neumann stability analysis of a simple coupled neutron diffusion-depletion system. Extending a previous analysis, this chapter provides a justification for applying a relaxation to predictor-corrector schemes, shows the possibility of obtaining symmetric burn-up instabilities, and demonstrates that no neutronics-depletion coupling scheme, without relaxation, is assuredly more stable than another, depending on the depletion system in question. Finally, Chapter 5 summarises the findings, proposes an explanation regarding general cases of burn-up instability, and suggests future work."],"dc:format.checksum.md5":["964035d9b5fecc4825a7d2326b053867","353adac0d1ebdfd65ab16480263c3c87"],"dc:identifier.doi":["10.17863/CAM.58521"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/6d906cdf-1768-4282-8ba2-526f198e8d6b/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/311429"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/ea3050b9-cb57-4823-aca8-f717685240b8/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["Nuclear","Monte Carlo","Reactor","Neutron transport","Isotopic depletion"],"dc:title":["Numerical stability of Monte Carlo neutron transport and isotopic depletion for nuclear reactor analysis"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:03Z"}