Convergence Improvement and Qualification of a Model for Fission

Gas Behaviour in Nuclear Fuel

G A Ë L D U B U S

Master of Science Thesis Stockholm, Sweden 2007

Convergence Improvement and Qualification of a Model for Fission

Gas Behaviour in Nuclear Fuel

G A Ë L D U B U S

Master’s Thesis in Numerical Analysis (20 credits) at the School of Computer Science and Engineering Royal Institute of Technology year 2007 Supervisor at CSC was Ninni Carlsund Examiner was Axel Ruhe TRITA-CSC-E 2007:060 ISRN-KTH/CSC/E--07/060--SE ISSN-1653-5715 Royal Institute of Technology School of Computer Science and Communication KTH CSC SE-100 44 Stockholm, Sweden URL: www.csc.kth.se

CONVERGENCE IMPROVEMENT AND QUALIFICATION OF A MODELFOR FISSION GAS BEHAVIOUR IN NUCLEAR FUEL

Abstract :During the irradiation in a fuel assembly of a Pressurized Water Reactor (PWR), the thermal dif-fusion processes are not the only active mechanisms: the effects of irradiation are significant, in-volving variations of the diffusion coefficients as well as creation and evolution of point defects andcavities in the material.In this document we present the work we have done on MOGADOR, a numerical fission gas beha-viour model for nuclear fuel under irradiation in a PWR focusing on the modelling of the behaviour ofirradiation defects, fission gases and as-fabricated pores. This work has been accomplished withinthe frame of a six-month internship, from September 2006 to March 2007 at the LSC laboratory,located at the Cadarache research centre.The first task was to optimize the model and to improve its convergence in a simplified case, whichwas a necessary condition to go further with the study of a complete case. Then we started totackle the physical qualification of MOGADOR, that is to say verify the behaviour of some phy-sical quantities and their dependencies to some parameters. We present also a short review ofnumerical methods commonly used for solving ordinary differential equations.

KONVERGENSFÖRBÄTTRING OCH KVALIFICERING AV EN MODELLFÖR FISSIONSGASERNAS BETEENDE I KÄRNBRÄNSLET

Sammanfattning :Under bestrålning i en tryckvattenreaktors bränsleelement är de termiska diffusionsprocessernainte de enda aktiva mekanismerna: strålningseffekterna är viktiga, och involverar ändringar av dif-fusionskoefficienterna samt förekomst och evolution av punktdefekter och hålrum i materialet.I det här dokumentet presenterar vi vårt examensarbete om MOGADOR, en numerisk modell förfissionsgasernas beteende i kärnbränslet vid bestrålning i en tryckvattenreaktor, som är specielltdesignat för modelleringen av bestrålningsdefekter, fissionsgaser och tillverkningsporer. Arbetetutfördes inom ramen för en sex månader lång praktik på LSC-laboratoriet, som ligger på franskaatomenergikommissionens förskningsanläggning i Cadarache.Arbetets första steg handlade om att optimisera modellen och förbättra sin konvergens. Sedan harvi kunnat komma med MOGADOR:s fysiska kvalificering, det vill säga kontrollera det bra beteendetav olika fysiska storheter samt sina beroende på somliga parametrar. Vi presenterar också en kortöversikt över numeriska metoder som oftast används för att lösa ordinära differentialekvationer.

T A B L E O F C O N T E N T S

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 CONTEXT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 THE FRENCH ATOMIC ENERGY COMMISSION (CEA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 THE LABORATORY FOR FUEL BEHAVIOUR STUDIES AND SIMULATION (LSC) . . . . . . . . . . . . . . . . . . . . 2

2 NUMERICAL METHODS FOR SOLVING AN ORDINARY DIFFERENTIAL EQUATION . . . . . . . . . . . . . . . . . . . . . 2

2.1 BASIC CONCEPTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Explicit or implicit ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.2 Stability and stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.3 Order, consistency and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.4 One-step vs. multi-step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 RUNGE-KUTTA METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 ADAMS METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.1 Adams-Bashforth methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.2 Adams-Moulton methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 BACKWARD DIFFERENTIATION FORMULAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 DESCRIPTION OF MOGADOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1 PHYSICAL BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 NUMERICAL MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.1 Space discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.2 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 THE SOLVER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4 MOGADOR AND THE PLEIADES RESEARCH PROGRAM AT THE LSC . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.5 INITIAL OBJECTIVES OF THE INTERNSHIP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 OPTIMIZATION AND CORRECTION OF MOGADOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.1 MAKEFILE OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 AUTOMATIC DIVISION OF THE MACROSCOPIC TIMESTEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3 CONVERGENCE TESTS DEALING WITH TOLERANCES IN THE INTRAGRANULAR CASE . . . . . . . . . . . . . 16

4.4 IDENTIFICATION OF THE CONVERGENCE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.5 MODIFICATION OF THE TRAPPING TERMS OF GAS ATOMS AND POINT DEFECTS BY CAVITIES . . . . . 18

4.5.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.5.2 Implementation of a generic function for modification the trapping terms . . . . . . . . . . . . . . . . . . 19

4.6 VERIFICATION OF THE GAS BALANCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 PHYSICAL QUALIFICATION OF MOGADOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.1 VALIDATION OF THE GAS BEHAVIOUR DEPENDING ON TEMPERATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2 ANIMATED HISTOGRAM FOR POROSITY RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.3 CONVERGENCE TESTS IN A COMPLETE CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.4 STUDY OF DEPENDENCY TO PARAMETERS (INTRAGRANULAR CASE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.4.1 First tests : unitary dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.4.2 Further tests : global behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.4.3 Comparing with experimental values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 CONCLUSIONS AND PROSPECTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

L IST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

L IST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

L IST OF APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

APPENDIX A : P HYSICAL PHENOMENA MODELLED BY MOGADOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

APPENDIX B : E QUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

B.1 INTRAGRANULAR EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

B.1.1 Diffusion equation for vacancies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

B.1.2 Diffusion equation for interstitials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

B.1.3 Diffusion equation for intragranular gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

B.1.4 Conservation equation for dislocation loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

B.1.5 Evolution equation for dislocation density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

B.1.6 Conservation equation for intragranular cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

B.2 INTERGRANULAR EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

B.2.1 Diffusion equation for intergranular gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

B.2.2 Conservation equation for intergranular cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

APPENDIX C : RESULTS OF THE INTRAGRANULAR TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

C.1 WITHOUT INITIAL PORE PROFILE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

C.1.1 Concentration of vacancies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

C.1.2 Concentration of interstitials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

C.1.3 Concentration of intragranular gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

C.1.4 Gas behaviour : number of cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

C.1.5 Gas behaviour : mean radius of cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

C.1.6 Gas behaviour : swelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

C.1.7 Gas balance : ratio of gas in the bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

C.1.8 Gas balance : ratio of dissolved gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

C.1.9 Gas balance : ratio of intergranular gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

C.2 WITH INITIAL PORE PROFILE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

C.2.1 Concentration of vacancies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

C.2.2 Concentration of interstitials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

C.2.3 Concentration of intragranular gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

C.2.4 Gas behaviour : number of cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

C.2.5 Gas behaviour : mean radius of cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

C.2.6 Gas behaviour : swelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

APPENDIX D : RESULTS OF THE INTERGRANULAR TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

D.1 NUMBER OF INTRAGRANULAR CAVITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

D.2 NUMBER OF INTERGRANULAR CAVITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

D.3 MEAN RADIUS OF INTRAGRANULAR CAVITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

D.4 MEAN RADIUS OF INTERGRANULAR CAVITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

D.5 CONCENTRATION OF INTRAGRANULAR GAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

D.6 INTRAGRANULAR SWELLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

D.7 INTERGRANULAR SWELLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

D.8 RATIO OF GAS IN THE BUBBLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

D.9 RATIO OF DISSOLVED GAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

D.10 RATIO OF INTERGRANULAR GAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

APPENDIX E : STUDY OF PARAMETER DEPENDENCY : ENVELOPE CURVES AT 800K . . . . . . . . . . . . . . . . . . . . 53

E.1 NUMBER OF CAVITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

E.2 MEAN RADIUS OF CAVITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

E.3 MEAN PRESSURE INSIDE THE CAVITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

E.4 RATIO OF FISSION GAS DISSOLVED IN THE GRAIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

E.5 RATIO OF FISSION GAS IN THE INTRAGRANULAR CAVITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

E.6 RATIO OF FISSION GAS AT THE GRAIN BOUNDARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

E.7 TOTAL LENGTH OF DISLOCATION LOOPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

E.8 TOTAL LENGTH OF DISLOCATION LINES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

E.9 INTRAGRANULAR SWELLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

INTRODUCTION

This internship was held under the direction of Antoine Bouloré at the Cadarache research centre inFrance, with help from Philippe Garcia and the staff of the LSC. We were two students hired to work onthis model, Mathieu Scotto di Perta and I. A large part of the work has been done in common at first, mainlyconcerning the identification of the convergence problem, then I focused on the physical qualification ofthe model while Mathieu worked on the integration of the model into the software environment PLEIADES.At first we present the working environment of the internship, then we write a short review of numericalmethods commonly used to solve ordinary differential equations. After this theoretical section we introducethe model MOGADOR and we notify the objectives of the internship. The first objective was to improvethe model in order to solve the convergence issue in the intragranular case, which has been fulfilled and isdescribed in part 4. Then we performed tests activating all the intergranular equations, which unfortunatelyled to other convergence difficulties. Therefore we decided that it was more important to focus on thequalification of the intragranular tests before going further with the complete case, since those difficultiescould originate from badly adjusted parameters, so a study of the dependency to some parameters ispresented in the last part.

1 CONTEXT

In this part we describe the working environment of the internship : the French Atomic Energy Commission(CEA) and the laboratory which we worked for.

1.1 THE FRENCH ATOMIC ENERGY COMMISSION (CEA)

The French Atomic Energy Commission (Commissariat à l’Energie Atomique) is an industry-oriented re-search organization which aims at developing applications of atomic energy, both in civil and militarydomain. It was created in 1945 and employs approximately 15,000 people in 9 research centres. It isorganized in 5 divisions :– The Division of Nuclear Energy (DEN)– The Division of Technological Research (DRT)– The Division of Life Sciences (DSV)– The Division of Materials Sciences (DSM)– The Division of Military Applications (DAM)The Cadarache research centre has been carrying studies on nuclear energy since 1959, mainly aboutdesign of nuclear power reactors, reprocessing and long-term storage of radioactive waste. The site waschosen in 2005 to host the international project ITER for an experimental full-scale fusion power reactor.

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1.2 THE LABORATORY FOR FUEL BEHAVIOUR STUDIES AND SIMULATION (LSC)

The Laboratory for Fuel behaviour studies and Simulation (LSC) is a branch of the Service for Fuel be-havior studies and Simulation (SESC), which includes 3 other laboratories. The SESC is part of the Fuelbehaviour Studies Department (DEC), which is included in the Division of Nuclear Energy. The missionsof the LSC are :– To identify the physical phenomena and to model the behaviour of the fuels and other elements of the

core under and after irradiation– To ensure the development, the validation and the maintenance of the reference software for fuel com-

putation, linked to other units of the DEN. In particular, it is in charge of developing the platform fornumerical simulation PLEIADES

– To develop, maintain and administrate the data bases related to the behaviour of the fuel elements (rods,compacts, pebbles, plates, ...) under irradiation in effective or experimental reactors and to the physicalproperties of the fuel materials.

– To notify the experimental programs required to create new models and to validate the codesThere are about 25 permanent employees working at the LSC, which is located at the Cadarache researchcentre.

2 NUMERICAL METHODS FOR SOLVING AN ORDINARY DIFFERENTIAL

EQUATION

In this part we describe a few methods commonly used to solve an initial value problem for an ordinarydifferential equation. Almost all the methods in the literature are designed for solving equations written inan explicit form :

{

X = F (t,X)

X(t0) = X0

(∗)

However, in some cases the equation appears to be defined implicitly :{

G(t,X, X) = 0

X(t0) = X0

(∗∗)

If it is obvious that (*) can be written in this last form - as for MOGADOR - proceeding the other way roundis not always possible. In 2.4 and 3.3 we present a method for implicitely defined ordinary differentialequations used in that case.

The basic idea for most of the methods is to use the relation X(t+h) = X(t)+

∫ t+h

t

F (s,X(s))ds, then to

compute the integral term as∫ t+h

t

g(s)ds = h

m∑

i=1

big(t + cih) where bi are given weights and 0 ≤ ci ≤ 1.

The term g(t + cih) is evaluated with help from a quadrature formula.

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2.1 BASIC CONCEPTS

2.1.1 Explicit or implicit ?

The explicit character of a numerical method should not be confused with the explicit definition of theequation : one can use implicit methods to solve (*). A method is said to be explicit when one does notneed to solve a system at each timestep whereas an implicit method requires such a resolution. Implicitmethods are thus more expensive in terms of computational time but are often used because they may bemore stable than explicit methods.

2.1.2 Stability and stiffness

A numerical algorithm solving differential equations is stable if at a fixed time, the solution remains boundedwhen the timestep tends to zero. Stability is an essential condition for a numerical algorithm to be efficientlyconvergent : two different algorithms equivalent in theory could give acceptable or unacceptable resultsregarding their stability. The problem of an unstable algorithm is basically that it propagates and amplifiesany error - e.g. truncature error inherently linked to the execution of any numerical algorithm. An algorithmcan be unconditionally stable or unstable, but there is almost always a stability condition (generally on thetimestep) needed.A problem is said to be stiff if there are variables in the equation causing strong variations in the solution.This can occur if the orders of magnitude of coupled variables are very different. A stiff problem leadsfrequently to the unstability of numerical methods unless the timestep is very small.

2.1.3 Order, consistency and convergence

A numerical method, defined as Xn+1 = Ψ(tn+1,Xn−k, ...,Xn+l), is consistent if limh→0

δn

h= 0

where δn = Ψ(tn,X(tn−k−1), ...,X(tn+l−1)) − X(tn) is the local error of the method. It has order p ifδn = O(hp+1) as h → 0.The obvious aim of any numerical scheme is to be convergent, i.e. the numerical solution should approachthe exact solution as the timestep goes to zero, which can be written as :

limh→0

maxn

‖Xn − X(tn)‖ = 0

The Lax equivalence theorem states that being consistent and numerically stable is a sufficient conditionfor a numerical scheme to be convergent.

2.1.4 One-step vs. multi-step

A one-step method is an algorithm using a formula with an incremental function Φ of the type :

Xn+1 = Xn + hnΦ(hn, tn,Xn)

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where hn = tn+1 − tn.A one-step method only uses Xn, the last computed value of X, whereas a multi-step method uses severalformerly computed values of X.

2.2 RUNGE-KUTTA METHODS

A s-stage Runge-Kutta method for solving (*) is defined as :

Xn+1 = Xn + hΦ(h, tn,Xn)

with the incremental function Φ(h, t,X) =s

∑

i=1

biki where ki = F (t + hci,X + hs

∑

j=1

aijkj).

The method is explicit when aij = 0 if j ≥ i and implicit otherwise. The values given for bi, ci and aij

should ensure the consistence of the method. For explicit methods, the method is consistent ifi−1∑

j=1

aij = ci

for 2 ≤ i ≤ s. There are additional conditions if we require the method to have order p.The only consistent explicit Runge-Kutta method with 1 stage is the forward Euler method, defined withc1 = 0 and b1 = 1 :

Xn+1 = Xn + hF (tn,Xn)

Another commonly used Runge-Kutta method is the 4-stage method of order 4, often referred to as ‘RK4‘,defined by :

c1 0

c2 a21 1/2 1/2

c3 a31 a32 = 1/2 0 1/2

c4 a41 a42 a43 1 0 0 1

b1 b2 b3 b4 1/6 1/3 1/3 1/6

2.3 ADAMS METHODS

Adams methods are multi-steps methods that use the relation Xn+1 = Xn +∫ t+h

tF (s,X(s))ds, replacing

F with an interpolation polynomial.

2.3.1 Adams-Bashforth methods

The Adams-Bashforth methods are explicit methods, so the interpolation polynomial uses only previouslycomputed values for X :

Xn+1 = Xn + hs

∑

j=0

bjF (tn−j , yn−j)

where bj =(−1)j

j!(s − j)!

∫ 1

0

s∏

i=0i6=j

(u + i)du.

The polynomial interpolates the s + 1 data points (ti, f(ti, xi))n−s≤i≤n, so its degree is s.

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2.3.2 Adams-Moulton methods

Adams-Moulton methods are similar to Adams-Bashforth methods, but with an implicit scheme :

Xn+1 = Xn + h

s−1∑

j=−1

bjF (tn−j , yn−j)

where bj =(−1)j+1

(j + 1)!(s − j − 1)!

∫ 1

0

s−1∏

i=−1i6=j

(u + i)du.

In this case the polynomial interpolates the s + 1 data points (ti, f(ti, xi))n−s+1≤i≤n+1.These two families of methods are often used together as a predictor-corrector scheme : a first approxi-mation of the next value Xn+1 is guessed at first by an explicit method, then one uses an implicit method(more stable) to improve the initial guess.

2.4 BACKWARD DIFFERENTIATION FORMULAS

In this class of methods we use an interpolation polynomial Q interpolating the points (tj ,Xj)n−k+1≤j≤n+1

(Xn+1 being unknown) and we consider the equation :

Q′(tn+1) = F (tn+1, Q(tn+1))

Solving this equations gives the polynomial Q and then Xn+1 = Q(tn+1).The Backward Differentiation Formulas (BDF) are often used for stiff problems. Moreover, they allow animplicit definition of the ordinary differential equation (as in (**)). It is a BDF method which is implementedin MOGADOR and described more precisely in 3.3.

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3 DESCRIPTION OF MOGADOR

MOGADOR is a model for the behaviour of nuclear fuel under irradiation in a Pressurized Water Reactor(PWR), which aims at modelling :– The evolution of the cationic point defects (vacancies and interstitials) created by irradiation– The evolution of the crystalline line defects (dislocation lines and loops) responsible of the “rim” or High

Burn-up Structure (HBS) of the oxide fuels at a high burnup, according to the literature– The evolution of the cavities (gas bubbles and as-fabricated pores) under irradiation that induces swel-

ling (nucleation of cavities) and densification phenomena (reduction of the volume of the as-fabricatedpores)

Basically, the model consists in 8 coupled partial differential equations (diffusion or conservation) withsource and sink terms, which are described in appendix A. For the computation, these equations havebeen discretized by the finite volumes method, and the discrete system has been written as a first orderimplicit differential system (Cauchy problem) :

{

F (Y, Y , t) = 0

Y (t = 0) = Y0

where Y represents the vector of the discrete unknowns.

3.1 PHYSICAL BACKGROUND

In this section we describe the physical phenomena happening in a UO2 fuel pellet under irradiation ina nuclear reactor. The physical experiments lead us to look carefully at some species to modelize. Theaim of MOGADOR (MOdélisation du GAz et des Défauts pour les Oxydes en Réacteur) is to describe theevolution of the irradiation defects in the oxide, the behaviour of the fission gases and of the as-fabricatedpores.During the irradiation in a PWR, the thermal diffusion processes are not the only active mechanisms : theeffects of irradiation are significant. There are three major mechanisms due to irradiation :– The increase of diffusion coefficients, causing a quickening of densification– The fission fragments crossing a cavity can eliminate it with a mechanism similar to the re-solution for

the fission gas atoms– The fission fragment create point defects in the material : vacancies and interstitials, which can recom-

binate, gather in defect clusters (like dislocation loops) or disappear due to sinks (dislocations, grainboundaries, cavities...)

We have 8 unknowns :– The concentration of vacancies in the standard grain Cl (mol.m-3)– The concentration of interstitials in the standard grain Ci (mol.m-3)– The concentration of gas dissolved in the standard grain Cxedag (mol.m-3)– The density distribution for dislocation loops in the standard grain Sb (m-4)– The density for dislocation lines in the standard grain ρd (m/m3)

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– The density distribution for cavities in the standard grain Sag (mol-1.m-6)– The concentration of gas in the equivalent sphere Cxedeg (mol.m-3)– The density distribution for intergranular cavities Seg (mol-1.m-6)

The corresponding 8 equations are described in appendix B. They are coupled with the variables accordingto this table :

XX

XX

XX

XX

XX

XX

EquationVariable

Cl Ci Cxedag Sb ρd Sag Cxedeg Seg

(1) Cl G L G G G G G

(2) Ci L G G G G G

(3) Cxedag G G G G

(4) Sb L

(5) ρd G G L G

(6) Sag G G G

(7) Cxedeg L G G

(8) Seg G G G

TABLE 1 : COUPLING TABLE BETWEEN VARIABLES AND EQUATIONS

The rows correspond to the equations and the columns correspond to the variables. G corresponds to aglobal coupling of the variables (over the whole mesh) and L to a local coupling (over one cell).

MOGADOR models all the physical phenomena overviewed in Fig.1, which we exhaustively write down inappendix A.

The irradiation leads to the creation of fission products, among which noble gases (Xenon, Krypton, He-lium) inducing a swelling of the fuel pellet. We calculate the swelling by the volume integral of all thecavities.

The swelling is a macroscopic quantity which can be observed and measured during experiments, thus itis very interesting to compute it and compare the results to experimental values. Along with the swellinga few integral quantities are significant : the number of cavities per unit of volume, the mean radius of thecavities, the mean pressure inside the cavities and the ratios of gas dissolved in the grain, in the cavi-ties and at the grain boundaries. It is also meaningful to compute the total length of dislocation lines anddislocation loops, though it is more difficult to get experimental values. Indeed, new methods are beingdeveloped for this kind of measures and some experimental results will be available in near future.

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FIGURE 1 : General overview of the physical phenomena modelled in MOGADOR

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3.2 NUMERICAL MODELS

3.2.1 Space discretization

The modelling of the oxide uses the widely used equivalent sphere model. There are two scales used tocompute the unknown variables :– A scale corresponding to a standard grain (supposed to be spherical) for most of the unknowns– A scale corresponding to a sphere simulating the distance between the considered point and a free sur-

face, this sphere being made of identical grains, used for intergranular quantities such as the concen-tration of intergranular gas

In order to ensure the conservation of mass for all the modelled species, the Cauchy problem is solved bya finite volume method. This method guarantees the conservation of mass of the system, unlikely to thefinite elements method.In 1D we consider a mesh of cells C1 , C2 , ... , Cn, using the finite volume method we assume that thevalue of the variable is constant over each cell and we compute the flux (mass transfer) coming from orgoing to its neighbouring cells.Since we assume a spherical symmetry in the grain all the meshes are 1-Dimensional except for thedensity distributions of the cavities which depend on two parameters : the volume of the cavity u and thenumber of moles of gas m contained in the cavity. This mesh is described in Fig.2.All the discretized terms for all the equations are fully described in [11].

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FIGURE 2 : Mesh for the density distributions for cavities

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3.2.2 Time discretization

The timestep is automatically handled by the solver described in 3.3, but since we want to compute untila high burnup (e.g. 30 GWd/tM, corresponding to an irradiation over three years) we divide the time intomacroscopic steps. This is also useful in order to deal with the interpolation of the solution by the solveras well as the choice of the initial timestep for each call to the solver.The initial division into macroscopic timesteps was equidistant but it turned out that the characteristic timeresponse of the model was quite small and in order to make out the transient state, one had to choosea very fine macroscopic subdivision. Therefore we decided to implement an option determining whetherthe subdivision is equidistant or refined at the beginning (in this case the final times of each macroscopictimestep follow a geometric progression with a predefined ratio) . This work was achieved and describedby Mathieu Scotto di Perta (see [10]).

The algorithm used to solve the Cauchy problem, described in 3.3, is an implicit global resolution me-thod. The problem to solve is written as :

{

F (Y n+1, Y n+1, tn+1) = 0(

Y i)

n−k≤i≤nknown, n ≥ k ≥ 0

3.3 THE SOLVER

Whereas the main program is written in C++, it calls the DASSL solver that is written in Fortran 77. DASSL(Differential Algebraic System Solver), described in [9], was developed in 1983 by Linda Petzold and is stillone of the most used solvers nowadays. It uses automatic selection of methods for solving stiff and non-stiff systems of differential equations. We tried a more recent solver : DASPK, in which a preconditioningmodule is implemented (see [10]). Yet all the results were similar.

Here is the algorithm used to solve the Cauchy problem :We want to compute Y n+1, the solution vector approximating Y (tn+1) given Y n approximating Y (tn). Asthe function F is non-linear, this system is solved by Gear Backward Differentiation Formula method (Cf.[9, 8]), which is caracterized by :– its predictive/corrective algorithm– variable timestep and orderThis method is based on polynomial interpolations. The prediction is done by a Lagrange interpolation :let Pp be the polynomial of order k interpolating the points

(

Y i, ti)

n−k≤i≤n. A prediction of the solution is

defined by :Y n+1

p = Pp(tn+1)

The next step of the method is the correction. The correction polynomial Pc is implicitely defined as theinterpolation of

(

Y i)

n−k+1≤i≤n+1and is found by solving the problem

G(Y, t) = F (Y, Y , t) = 0

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To write Yn+1 = P ′c(t

n+1) as a function of Yn+1, we write the Lagrange interpolation formula for Pc :

Pc(t) =n+1∑

i=n−k+1

Yi

n+1∏

j=n−k+1j 6=i

(t − tj)

n+1∏

j=n−k+1j 6=i

(ti − tj)

then :

P ′c(t) =

n+1∑

i=n−k+1

Yi

n+1∑

l=n−k+1l 6=i

n+1∏

j=n−k+1j 6=ij 6=l

(t − tj)

n+1∏

j=n−k+1j 6=i

(ti − tj)

P ′c(t

n+1) =

n+1∑

i=n−k+1

Yi

n+1∑

l=n−k+1l 6=i

n+1∏

j=n−k+1j 6=ij 6=l

(tn+1 − tj)

n+1∏

j=n−k+1j 6=i

(ti − tj)

= ωYn+1 + B

where ω =

n∑

l=n−k+1

n∏

j=n−k+1j 6=l

(tn+1 − tj)

n∏

j=n−k+1

(tn+1 − tj)

=

n∑

l=n−k+1

1

tn+1 − tl

and B is function of the previous computed solutions (Yi)n−k+1≤i≤n

The problem to solve is then :{

G(Y, t) = F (Y, ωY + B, t) = 0

Y 0 = Y n+1p

It is solved by a quasi-Newton algorithm, which requires the computation of :– The residual G

– The Jacobian matrixdG

dY(Y, t) =

∂F

∂Y+ ω

∂F

∂YThe evaluation of the Jacobian matrix can be done numerically with the successive computations of G, butthis is very costly in terms of computational time. Therefore one would rather use the analytical computa-tion, which is possible since all the partial derivatives have been calculated formally.If the residual G is larger than the tolerance, the solution is not accepted and the solver performs another

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iteration of the algorithm with a closer tn+1. This is the reason why the timestep is variable.The tolerance is computed as ε = RTOL|Y | + ATOL, where RTOL is the relative tolerance and ATOL

the absolute tolerance. As we performed test computation, we were surprised that the absolute tolerancecould not be set to zero (it led to convergence errors). We contacted Linda Petzold and she explained thereason for that :

The problem with setting absolute tolerances to zero is that if a solution

component becomes very small, say because it happens to be changing sign,

then it is impossible to satisfy the relative error tolerance. Hence you

should always set the absolute tolerance to some positive number, even if it

happens to be very small. The code will work best if you set the absolute

tolerances so that for a nominal value of component yi, RTOL|yi| is

approximately the same size as ATOLi. Of course, this is not always possible

because you don’t know the solution values without solving the problem first.

But if you have any information about the relative sizes of the solution

components, you should try to incorporate that into the choice of ATOL for

that component.

3.4 MOGADOR AND THE PLEIADES RESEARCH PROGRAM AT THE LSC

The MOGADOR project started in 1998 and was set off by Philippe Garcia. The main idea was to developa new model for densification and fission gas release in the UO2 material handling all the cavities in thefuel (as-fabricated pores and fission gas bubbles) the same way. A simplified version of the model wasdeveloped within the framework of Antoine Bouloré’s Ph. D. thesis [2] until 2001. From 2002 the completecase has been developed, its qualification being now in progress - and our work is part of this process.An objective for MOGADOR is to be integrated into PLEIADES, which is the software environment for multi-dimensional fuel performance modelling developed at the LSC. Having a unified environment sharing databases and models represents a huge benefit in terms of coherence and computational resources.

There are several applications dedicated to fuel simulation in PLEIADES :– ALCYONE for multidimensional PWR studies– CYRANO3, the EDF code for industrial PWR studies– ATLAS for High Temperature Reactor studies– MAIA for Materials Test Reactor studies– GERMINAL for Sodium-cooled Fast Reactor studies– CELAENO for the Gas-cooled Fast Reactor studiesSince it is a model dealing with UO2 fuel, MOGADOR can be used in all the applications dealing with thiskind of fuel, whatever the reactor type. The first part of the integration of MOGADOR into PLEIADES hasalso been done within the framework of Mathieu Scotto’s internship [10].

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3.5 INITIAL OBJECTIVES OF THE INTERNSHIP

The aim for this model is to calculate physical quantities in good agreement with the experimental results.This requires first a physical qualification, i.e. to verify the activation of the modelled physical phenomenain the different operational temperature slots.

At the point where we got the source code of the program, it was mainly implemented in its autonomousversion and there were a lot of cases (sets of parameters, sets of tolerances, while activating or not differentterms and different equations) that did not lead to convergence. Therefore the obligatory preliminary workwas to find the origin of the non-convergence, which was conditioned by the good comprehension ofall the numerical models exposed in 3.2 and of the behaviour of the solver (Cf. 3.3). Thus the first partof the internship planned tasks was to read the documentation, then to familiarize ourselves with thesource code and perform a few tests with the given program. After that, we had to study and resolvethe convergence problem. Once the first goal had been achieved, we could tackle the qualification of theautonomous model, at first activating only the intragranular equations, then activating all the equations(intragranular and intergranular). Regarding the quality of the results, we could either solve the remainingproblems or improve the model and integrate it into the software environment PLEIADES at first, then intoa multiprocessor system using parallel calls to modelize a full slice of a fuel pellet under irradiation.Since the complete tests were not satisfying in terms of computational time, we focused on the qualificationof the intragranular case. Moreover, it turned out that the parallel hardware was not ready to be used atthe LSC, so we could not tackle the last part of the software integration.

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4 OPTIMIZATION AND CORRECTION OF MOGADOR

4.1 MAKEFILE OPTIMIZATION

This was not the longest part of the work but it was very useful : adding the optimization option at thecompilation reduced considerably the computational time. We corrected also a few errors to get an effectivemake clean.

4.2 AUTOMATIC DIVISION OF THE MACROSCOPIC TIMESTEP

As explained formerly, at some point of the computation the solver could not converge and returned anerror. An idea was to improve the program implementing a module that recalled the solver at the initial timeof the current macroscopic timestep (where the solver failed to converge) with larger tolerances. Thus weimplemented a recursive function managing the calls to the solver with a backtracking in case of diver-gence, using larger tolerances.Unfortunately, the results were not as good as expected : the modification did not help the solver toconverge in the critical cases. Furthermore, the idea of an automatic (and therefore without control) changeof the tolerance was not fully acceptable.However, we kept the idea of a backtracking in case of divergence, dividing recursively the macroscopictimestep into two equal parts. This did not change any variables or tolerances automatically, only the timemacroscopic subdivision, and therefore was more convenient for the user. It also gave better results : afew former critical cases did work, and the computation terminated in almost all the cases. Yet some casesstill not converged (even whithin a large recursion depth) and a lot of computations gave a discontinuousresult for many variables (see Fig.3).

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0.1108

0.111

0.1112

0.1114

0.1116

0.1118

0.112

0.1122

0.1124

0 500000 1e+06 1.5e+06 2e+06 2.5e+06 3e+06 3.5e+06 4e+06 4.5e+06

Cla

c

t

Concentration of vacancies at the first node

FIGURE 3 : A discontinuous case

Even if the jump was quite small in absolute value, these solutions were numerically (and obviously physi-cally) not acceptable. In fact, we did not tackle the origin of the instability of the model but we tried to makeit more resistant to ill-conditioned cases - which was nevertheless a good thing to do.

4.3 CONVERGENCE TESTS DEALING WITH TOLERANCES IN THE INTRAGRANULAR CASE

We noticed that the model was quite sensitive to a change of some absolute tolerances (see 3.3 for the roleof absolute and relative tolerances) in the unstable cases. As told before, the most important convergenceproblem occured when activating the destruction term for cavities in equation (4). As a consequence, thetolerance term to be considered was the absolute tolerance related to the conservation equation for thecavities (4). We performed a lot of convergence tests changing this tolerance as well as the absolute tole-rances for the first instable variables, i.e. those which were struck by discontinuity at first - vacancies andinterstitials. All the relative tolerances were set at 10−6.

The computations were long (a few hours) or very long (a few days), depending on the value of theseabsolute tolerances. Yet there was no logical behaviour of the program regarding the variations of tole-rances : it could take a longer time to converge, it could even fail to converge, both with larger and smallervalue for a given tolerance. Furthermore, there were a minimum (one cannot set the absolute tolerancesto zero) and a maximum (a too large tolerance gives obviously rise to a bad computation) for acceptableabsolute tolerances so we had to keep in mind that the values for acceptable tolerances might change inother computations with different parameters.

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That behaviour of the solver was later explained by Linda Petzold (Cf. 3.3). Here is a set of absolute tole-rances that worked in most of the cases :

Equation Value for the Absolute Tolerance

(1) Diffusion equation for vacancies 10−6

(2) Diffusion equation for interstitials 10−12

(3) Diffusion equation for gas 10−3

(4) Conservation equation for dislocation loops 1030

(5) Evolution equation for dislocation density 101

(6) Conservation equation for intragranular cavities 1040

(7) Diffusion equation for intergranular gas 10−8

(8) Conservation equation for intergranular cavities 1060

TABLE 2 : SET OF ABSOLUTE TOLERANCES

It is important to note that all the computations were done only in the intragranular case : the intergranu-lar equations had not been activated yet, so the tolerances relative to these equations are not relevant(actually, they were set to their initial default value). Other convergence tests were performed later in acomplete case (see 5.3).

Still some critical cases either did not work or took a very long time to converge, so the convergenceproblem was not yet identified.

4.4 IDENTIFICATION OF THE CONVERGENCE PROBLEM

We tried the numerial evaluation of the Jacobian matrix (Cf. 3.3) to make sure that there was no error inthe formulas for partial derivatives. The computational time was much larger than for the analytical com-putation of the Jacobian matrix, but the results were about the same, so we decided to get back to theanalytical computation. We also tried a limitation of the order of the interpolation polynomials but it did notchange anything.

Then we added a row in the mesh for intragranular cavities, which surprisingly induced convergence forformer critical cases. This led us to look at the destruction term for the cavities :In the conservation equation for intragranular cavities (6) there is a destruction function pd, which is definedas a step function

pd(u,m) =1

2

(

tanh(−1

Vcdes(u − Vd)) + 1

)

where Vd is the critical volume under which a cavity is destructed by a fission fragment with an appropriate

energy level and Vcdes a parameter for the destruction function, initially set toVd

100. We changed the value

of the parameter toVd

10to get a smoother destruction function (Cf. Fig.4), which did not affect the physical

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0

0.2

0.4

0.6

0.8

1

1e-25 2e-25 3e-25 4e-25 5e-25 6e-25 7e-25 8e-25 9e-25 1e-24

Pd(

u)

u

Vcdes = Vd/100

0

0.2

0.4

0.6

0.8

1

1e-25 2e-25 3e-25 4e-25 5e-25 6e-25 7e-25 8e-25 9e-25 1e-24

Pd(

u)

u

Vcdes = Vd/10

FIGURE 4 : The destruction function for intragranular cavities

phenomenon modelled.

The results were very good : all the computations took much less time than before (from 1 to 10 mi-nutes) and we had no longer discontinuity problems, neither convergence difficulties - provided that weused an acceptable set of absolute tolerances. Yet the results in the cases that formerly converged wereall similar for both of the destruction functions, therefore we obviously chose the second one.In the meantime Philippe Garcia, who defined the physical equations of the model, changed the model-ling of the destruction of cavities, introducing new trapping terms instead of a destruction function pd withre-solution. Mathieu Scotto di Perta made a first implementation of this new model (detailed in [10]) andwe kept both of the versions of the model in order to compare the results, which were similar (see thegraphs in appendix of [10] for the computation with trapping terms and those of the present report for thecomputation with destruction function and re-solution). The alternative modelling of trapping is describedin 4.5, as well as the later implementation.

4.5 MODIFICATION OF THE TRAPPING TERMS OF GAS ATOMS AND POINT DEFECTS BY

CAVITIES

In this part we describe the modification of the modelling of the trapping mechanism : at first the flux ofdefects (vacancies, interstitials, gas atoms) was defined as a simple diffusion flux Φ = 4πRbD0 where Rb isthe radius of the cavity and D0 the diffusion coefficient of the defect in the UO2 material. But this modellingdid not take into account the influence of the stress induced by the cavity in the material, which dependson its internal pressure. Moreover in order to get results in agreement with the experiments a destructionmechanism was modelled by the previously introduced destruction function pd, which was not physicallymotivated and gave rise to numerical instability. In this section we sum up the new modelling introducedby Philippe Garcia in [7].

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4.5.1 Motivations

The basic idea is that the local diffusion coefficient depends on the local stress induced by the cavity inthe material. Therefore, the hyperpressurized bubbles do not act as sinks for the point defects. In fact,one should use a local diffusion coefficient D, taking into account the local stress σrr, depending on

the pressure inside the cavity : D = D0exp

(

σrr∆Vα

kT

)

, where ∆Vα is the volume variation induced by

the introduction of the defect into the material. An analytical resolution of the local diffusion equation∇.(D

−→∇C) = 0 leads to the expression of the defect flux towards the cavity Φ = 4πRbD0 × correction

where correction =−3exp

[

− fR3

b

+(

AR3

b

+ B)

∆Vα

kT

] (

fR3

b

)1/3

∫ f/R3

b

0

t−2/3e−tdt

, A = R3b

(

−σh − Pb +2γ

Rb

)

, B = σh and

f =A∆Vα

kT.

Pb is the pressure inside the cavity, σh is the macroscopic hydrostatic stress infinitely far from the cavity, T

is the local temperature and k and γ are physical constants.

4.5.2 Implementation of a generic function for modification the trapping terms

Since this modification of the trapping mechanism could be effective not only for the diffusing gas atomsbut also for the other defects (vacancies and interstitials) we decided to implement a generic functioncomputing the correction factor of the defect flux in trapping terms, with the defect type as a parameter.The correction of the trapping mechanim can be activated independently for each defect type by a keyword.The tests when activating the correction for the fission gas atoms were satisfying since they gave similarresults to those with the artificial destruction mechanism.

4.6 VERIFICATION OF THE GAS BALANCES

An important result to look at is the behaviour of the gas produced by the irradiation, inducing the swelling.The gas is distributed into three species :– The intragranular cavities (Sag)– The gas dissolved in the crystalline structure of the grain (Cxedag)– The gas at the boundaries of the grain (Cxedeg)The sum of the quantities of these three species should be equal to the total amount of produced gas.While verifying the gas balances, it turned out that it was not the case. Indeed, there was an error in thecomputation of an integral corresponding to a source term (Kg4 in equation (3)). Once this bug had beenfixed we could begin the validation of the gas behaviour. We merged also two files computing Kg2 andKg3, corresponding to integral terms on the same domain of integration

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5 PHYSICAL QUALIFICATION OF MOGADOR

Once there were only very few computations that did not converge in the intragranular case - yet it wasgenerally possible to make the critical cases converge changing the absolute tolerances - we could tacklethe first part of the qualification of the model : we checked that the main variables obtained with the modelbehaved in concordance with the experiments or the expectations.

5.1 VALIDATION OF THE GAS BEHAVIOUR DEPENDING ON TEMPERATURE

All these tests were only performed in the intragranular case at first (tests for the complete case are des-cribed in 5.3), that is to say that we did not activate the last two equations (7) and (8), both with and withoutan initial profile of as-fabricated pores (cavities). The complete set of graphs for the main variables is givenin appendix C.

We can see that the number of cavities is larger if the temperature is lower, on the other hand the bubblesare larger at high temperature, which is in agreement with the physically observed results. The swelling islarger at high temperature, which is also physically acceptable. In the case of an initial as-fabricated poresprofile, we can make out that the swelling decreases at first, which corresponds to the densification of thepores, and then increases due to the production of gas under irradiation and its precipitation in bubbles.

5.2 ANIMATED HISTOGRAM FOR POROSITY RESULTS

We used the results in the output files to create short animations representing the temporal evolution ofthe histograms of repartition of cavities function of their size. We created the snapshots with gnuplot andripped the movie with ffmpeg. A snapshot of the animation is shown in Figure 5.

These histograms can be directly compared with experimental measurements and are convenient to makeout the temporal evolution of the spectrum of cavities. At a low temperature (800K) the bubbles remainsmall whereas if we increase the temperature the spectrum shifts slowly to larger values.It is important to note that since the output files contain the values of the step variables at the end ofeach macroscopic timestep one has to choose the equidistant time subdivision in order to get a coherentanimation (Cf. 2.2.2).

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FIGURE 5 : Snapshot of the animated histogram for porosity results

5.3 CONVERGENCE TESTS IN A COMPLETE CASE

As the model was working satisfyingly in the intragranular case we started to perform convergence testsin a complete case, activating the intergranular equations (7) and (8). The set of tests was a case withoutinitial pore profile, for a local temperature of 800K, 1000K, 1200K, 1500K and 2000K. The first observationwe could make was that the computations took much more time than in the intragranular case, as expec-ted. For a test at 800K the computation was about ten times longer than in the intragranular case and tookless than one hour to finish, which remained an acceptable computation time for the model.Unfortunately at a higher temperature, the timestep - automatically handled by the solver - decreaseddrastically at some critical point of the computation, depending on the temperature. Here is a table sho-wing the critical time for all the temperatures when we tried a final time corresponding to three irradiationcycles (6.107 s) :

Temperature Critical time (s)

800K Finished1000K Finished1200K 5.107

1500K 4.105

2000K 6.103

TABLE 3 : CRITICAL TIMES IN THE COMPLETE CASE

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As shown in this table, the higher the temperature the sooner the computation slowed down. Since thecritical time for the 1200K test turned out to be just before the final time of computation, we let the modelrun until it finished, which took about one week. Therefore we got the complete results for the three lowertemperatures, and the results only until the critical times for 1500K and 2000K.

A positive side is that the behaviour of the variables regarding the temperature is coherent, which wecan make out on graphs given in appendix D. While focusing on the time interval where the variablesare defined in the 2000K test (resp. 1500K) we observe that the activation of all the mechanisms due totemperature is working (Cf. Fig. 6 for the example of the intragranular swelling)

0

5e-11

1e-10

1.5e-10

2e-10

2.5e-10

3e-10

3.5e-10

4e-10

4.5e-10

5e-10

0 1000 2000 3000 4000 5000 6000 7000 8000

t

Intragranular swelling - Complete case - Without initial pore profile

800K1000K1200K1500K2000K

0

5e-08

1e-07

1.5e-07

2e-07

2.5e-07

3e-07

3.5e-07

4e-07

4.5e-07

5e-07

0 50000 100000 150000 200000 250000 300000 350000 400000 450000

t

Intragranular swelling - Complete case - Without initial pore profile

800K1000K1200K1500K

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

t

Intragranular swelling - Complete case - Without initial pore profile

800K1000K1200K

FIGURE 6 : Intragranular swelling in the complete case, different critical times

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The intragranular variables behave generally the same way than in the intragranular case studied in 5.1 :there are more cavities at a lower temperature, which are smaller. The graphs giving the observable resultsat 2000K are not always in accordance with the expected results (Cf. graphs D.1 and D.3 for example) butone has to keep in mind that the time scale is not the same : the transient state is not finished in thiscase. However, it gives a good argument of why the solver is much slower : we can make out that all themechanisms are active more quickly. Moreover, the size of the unknown vector Y is approximatively twicethe size of Y in the simplified intragranular case (≈ 1000 unknowns).

We had many attempts to improve the behaviour of the model, unfortunately without success. We tried inparticular to change every absolute and relative tolerance - both increasing and decreasing, we refinedthe meshes for the equivalent sphere, the cavities spectra and the dislocations to enlight hypotheticaldependencies, we looked carefully at the way the dislocations loops were created and destroyed, twoantagonist mechanisms similarly modelled than for the cavities. Our conclusion was that changing themodelling of the evolution of the dislocations may improve the convergence, since it may not be definitelystated, but the main problem remains the stiffness of the problem. Indeed, the model is dealing withmany quantities, which have different orders of magnitude : for example the interstitial concentration Ci isabout 10−8 mol.m-3 whereas the cavity spectra can reach 10+60 m-6.mol-1. This is doubtlessly a source ofstiffness and, as explained in 2.1.2, this can force the solver to choose unacceptably small timesteps inorder to guarantee the stability. We explain in the conclusion how this might be arranged.

5.4 STUDY OF DEPENDENCY TO PARAMETERS (INTRAGRANULAR CASE)

Since the convergence of the complete case was not satisfying, we decided to focus on the qualificationof the intragranular case. The idea was to study the dependency to 5 parameters, the value of whichbeing not necessarily definitely stated. This study could be used later to perform a calibration of theseparameters with respect to the experimental results. Having well-calibrated parameters might help themodel to converge, and is anyway a good improvement. The considered parameters are :– Dlv, the diffusion coefficient for vacancies– Div, the diffusion coefficient for interstitials– Dxedag, the diffusion coefficient for intraganular gases– Zib, a characteristic factor of the interaction interstitial/dislocation loop– Zlb, a characteristic factor of the interaction vacancy/dislocation loopTheir default values are specified in [6]. The range of test values for each parameter is given in table 4.

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Value by default Minimum value Maximum value

Dlv Dlv/10 10 × Dlv

Div Div/10 10 × Div

Dxedag Dxedag/10 10 × Dxedag

Zib 0.995 × Zib 1.005 × Zib

Zlb 0.995 × Zlb 1.005 × Zlb

TABLE 4 : RANGE OF TEST VALUES FOR THE STUDIED PARAMETERS

We wanted to determine the influence of these parameters on the following macroscopic variables :– The number of cavities per unit of volume Ncav (m-3)– The mean radius of the cavities Rcav (m)– The mean pressure inside the cavities Pcav (Pa)– The ratio of gas dissolved in the grain %xedag

– The ratio of gas in the intragranular cavities %cav

– The ratio of gas at the grain boundaries %inter

– The total length of dislocation loops per unit of volume Lloops (m/m3)– The total length of dislocation lines per unit of volume Llines (m/m3)– The intragranular swelling Sintra

We performed a first series of tests changing only one parameter value, both increasing and decreasingthe parameter, while the other parameters remained unchanged. This could give us the unitary depen-dency of each variable to each considered parameter, which we present in 5.4.1. Then we tried all thecombinations of extreme values, all the parameters changing (25 = 32 computations, from all to the mini-mum values to all to the maximum values), which gave us three useful informations. Firstly we could checkthat the program converged for all the sets of parameter values : the code was stable enough to bear aconsequential parameter calibration. Moreover, we checked that all the previous computations gave resultcurves situated within the envelope curves, which was due to the fact that they corresponded to minorchanges of the set of parameters regarding the second collection of tests. The last advantage was thatwe could determine the general dependency of the macroscopic variables to the parameters, taking intoaccount the couplings of parameter variations and the way they affect the results. All the computationswere done at 800K and 1200K.

5.4.1 First tests : unitary dependencies

We sum up the observations of unitary dependencies in Table 5. We can make out that Div does not seemto have any influence on the macroscopic variables, whereas a change for Dxedag, Zlb or Zib affects allthese quantities. The diffusion coefficient Dlv affects only the dislocations.

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XX

XX

XX

XX

XX

XX

VariableParameter

Dlv Div Dxedag Zib Zlb

Ncav – i− (small) d− (small)

Rcav + i+ d+

Pcav + i+ d+

%xedag – i− d−

%cav ? i+ d+

%inter + i− d+

Lloops d− – + –

Llines – + – –

Sintra – i+ d−

+ : an increase of the parameter induces an increase of the variable– : an increase of the parameter induces a decrease of the variablei : the dependency is only visible when increasing the parameter from the value by defaultd : the dependency is only visible when increasing the parameter from the value by default? : there exists a dependency which could not be clearly established with only these tests

TABLE 5 : UNITARY DEPENDENCIES BETWEEN PARAMETERS AND MACROSCOPIC VARIABLES

Note that the ‘+‘ symbol corresponds to a monotonically increasing behaviour of the evolution of the va-riable function of the variation of the parameter : an increase of the parameter implies an increase of thevariable, a decrease implies a decrease. As a consequence, the ‘d+‘ symbol corresponds to the casewhere when the variation is only observable when decreasing the parameter, and since it showed a mo-notonically increasing evolution, this means that a decrease of the variable had been observed.It was not possible to determine properly a general influence of Dxedag on the ratio of fission gas in theintragranular cavities because the results were not the same at 800K and at 1200K, but it is the only casewhere the results were different for different temperatures. This is due to the fact that the ratio of fission gasin the intragranular cavities depends inherently of the other ratios, thus it might be not directly dependenton the variations of the parameters, but only through the variation of the other ratios.

5.4.2 Further tests : global behaviour

We performed more complete tests, taking into account the phenomena happening when changing severalparameters simultaneously. The graphs corresponding to these tests are presented in appendix E. Anexhaustive study of all the cases led us to enlight the following dependencies :

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Number of intragranular cavitiesThe main parameter inducing an increase of the number of cavities is the diffusion coefficient for intra-granular gas Dxedag. Increasing Dxedag leads to a strong decrease of the variable, which is in accordancewith the unitary tests and with the results obtained for different temperatures : at a higher temperature, thediffusion coefficient is indeed larger, as explained in 3.1., and the number of cavities was lower.Another small decrease of the variable is caused by three simultaneous changes : an increase of Dlv, adecrease of Zlb and an increase of Zib.

Ratio of dissolved gasHere again, the main dependency concerns Dxedag : the larger Dxedag, the less there is dissolved gas fromthe beginning. There is also a configuration - a decrease of Zlb and an increase of Zib - that encouragesthe creation of cavities, and therefore implies a later decrease of the variable.

Ratio of gas at the grain boundariesIt is clearly established that the variations of this variable are only governed by Dxedag. Indeed, the mainpart of the gas at the grain boundaries comes from the intragranular cavities coming to these boundaries.The cavities are nucleated at the centre of the grain and they are moving through the grain before theyreach a boundary. They are obviously moving faster with a larger diffusion coefficient, so there are morecavities coming to the boundaries if Dxedag is increased.The unitary dependencies to Zib and Zlb shown in Table 5 are too small to be relevant regarding thedependency to Dxedag.

Total length of dislocation loopsOn the graph E.7 we can make out two different variations for this variable. The first one concerns thevalue to which it stabilizes after the transient state, which is larger as Dlv increases. The second one isthe slope of the curve after the stabilization, which is larger and increasing for a larger Dxedag.

Total length of dislocation linesThis variable is slightly decreasing for all the cases, and there are configurations for which it decreasesmuch faster : an increase of Dlv and either an increase of Zib or an increase of Zlb, but not both at thesame time.

5.4.3 Comparing with experimental values

We got the following experimental results :– Intragranular swelling : from 1% to 3% at 800K, from 3% to 5% at 1200K– Number of cavities : about 1.1023 at 800K, slightly less at 1200K– Radius of cavities : from 1 nm to 2 nm at 800K, larger than 2 nm at 1200K– Pressure in the cavities : over 1.109 Pa– Ratio of gas at the boundaries : about 10% at 800K, about 20% at 1200K

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The simulation with the initial values for the parameters showed that we should decrease %inter, whichonly depends on Dxedag, according to the tests in 5.4.2 : we decreased Dxedag in order to get results inaccordance with the experiments : multiplying Dxedag by 0.3 was quite satisfying. After this change, wetried to decrease Ncav without changing Dxedag, by the simultaneous increase of Dlv, decrease of Zlb andincrease of Zib. This led to an unexpected dependency of %inter, not seen in 5.4.2 : this variable decrea-sed with the last operation. It could not be seen with the maximum/minimum tests because the influence ofextremal values for Dxedag was much stronger than the one of the other parameters. Nevertheless, thesedependencies exist - as Table 5 led us to think they do - and it shows that the future parameter calibrationshould take into account all the possible values for parameter variations in the considered range.

6 CONCLUSIONS AND PROSPECTS

In this internship we studied a complex model for the behaviour of UO2 nuclear fuel. The first part of ourwork was a to correct, to improve and to give the model more robustness against numerical instability. Asthere were still convergence difficulties when activating all the terms in all the equations, we decided tofocus on a simplified case in order to qualify a base work for the model. A reason for this choice is thatafter many attempts to quicken the convergence at a high temperature (from 1500K), we arrived at theconclusion that the next step to improve the model was to reduce the stiffness of the problem by modifyingthe equations in order to get the same orders of magnitude for all the variables. This could be done usingscaling methods for ordinary differential equations.Once it has been fully qualified, the model is intended to be integrated into a parallel environment inorder to compute over a complete slice of a fuel element : the calls to the model will be managed by anapplication in PLEIADES, each parallel call corresponding to different initial conditions (temperature, poreprofile...).I found this work very interesting, though it was not easy at first to familiarize with this complex model,especially because we did not have a strong background in materials science. With this project, I havedoubtlessly gained a good experience, from which will certainly benefit in my professional life. I would liketo thank particularly Antoine Bouloré and Philippe Garcia for having always taken the time to answer allour questions and all the staff from the LSC for their help and the good atmosphere at the laboratory.

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R E F E R E N C E S

[1] H. Bailly, D. Ménessier, and C. Prunier. Le combustible nucléaire des réacteurs à eau sous pressionet des réacteurs à neutrons rapides : conception et comportement. CEA, 1996.

[2] A. Bouloré. Etude et modélisation de la densification en pile des oxydes nucléaires UO2 et MOX.PhD. thesis presented by Antoine Bouloré for the École Nationale Supérieure des Mines de Saint-Étienne (ENSMSE) and the Grenoble Institute of Technology (INPG), March 2001.

[3] A. Bouloré. MOGADOR : première étape de qualification du problème intragranulaire et tests deconvergence. NT/SESC/LSC A-MATAV-01-07, July 2006.

[4] J. Della-Dora and V. Perrier. Méthodes numériques 1. ENSIMAG course notes.

[5] P. Garcia. Cahier des charges pour la résolution numérique du problème général de modélisationdes défauts d’irradiation et des gaz de fission. NT/SESC/LSC 00-040, December 2000.

[6] P. Garcia and A. Bouloré. Modélisation des défauts d’irradiation dans les combustibles UO2 et MOX :Application au comportement des gaz de fission, à la densification en pile et aux effets à fort taux decombustion. NT/SESC/LSC 00-037, December 2000.

[7] P. Garcia, P. Martin, G. Carlot, E. Castelier, M. Ripert, C. Sabathier, C. Valot, F. D’Acapito, J-L. Haze-mann, O. Proux, and V. Nassif. A study of xenon aggregates in uranium dioxide using x-ray absorptionspectroscopy. Journal of Nuclear Materials, 352 :136–143, 2006.

[8] A.C. Hindmarsh. Odepack, a systematized collection of ODE solvers. Scientific Computing, c©IMACS/North-Holland Publishing Company :55–64, 1983.

[9] L. Petzold. Automatic selection of methods for solving stiff and nonstiff systems of ordinary differentialequations. Composite Structures, Vol.4, No.1 :136–148, March 1983.

[10] M. Scotto di Perta. Report on the MOGADOR internship. NT/SESC/LSC A-MATAV-01-07, March2007.

[11] N. Vissyrias. Résolution numérique du problème général de modélisation des défauts d’irradiation etdes gaz de fission : Modèles numériques. Réf. ASSYSTEM : 18444A/032, February 2005.

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L I S T O F F I G U R E S

FIGURE 1 GENERAL OVERVIEW OF THE PHYSICAL PHENOMENA MODELLED IN MOGADOR 8

FIGURE 2 MESH FOR THE DENSITY DISTRIBUTIONS FOR CAVITIES 10

FIGURE 3 A DISCONTINUOUS CASE 16

FIGURE 4 THE DESTRUCTION FUNCTION FOR INTRAGRANULAR CAVITIES 18

FIGURE 5 SNAPSHOT OF THE ANIMATED HISTOGRAM FOR POROSITY RESULTS 21

FIGURE 6 INTRAGRANULAR SWELLING IN THE COMPLETE CASE, DIFFERENT CRITICAL TIMES 22

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L I S T O F T A B L E S

TABLE 1 COUPLING TABLE BETWEEN VARIABLES AND EQUATIONS 7

TABLE 2 SET OF ABSOLUTE TOLERANCES 17

TABLE 3 CRITICAL TIMES IN THE COMPLETE CASE 21

TABLE 4 RANGE OF TEST VALUES FOR THE STUDIED PARAMETERS 24

TABLE 5 UNITARY DEPENDENCIES BETWEEN PARAMETERS AND MACROSCOPIC VARIABLES 25

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L I S T O F A P P E N D I C E S

APPENDIX A : PHYSICAL PHENOMENA MODELLED BY MOGADOR . . . . . . . . . . . . . . . . . . . . 32

APPENDIX B : EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

APPENDIX C : RESULTS OF THE INTRAGRANULAR TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

APPENDIX D : RESULTS OF THE INTERGRANULAR TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

APPENDIX E : STUDY OF PARAMETER DEPENDENCY : ENVELOPE CURVES AT 800K . 53

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APPENDIX A : P HYSICAL PHENOMENA MODELLED BY MOGADOR

. Creation of vacancies in the grain

. Creation of interstitials in the grain

. Appearance of fission gas atoms in the grain

. Diffusion of vacancies in the grain

. Diffusion of interstitials in the grain

. Diffusion of gas atoms in the grain

. Diffusion of gas atoms at the grain boundaries

. Annihilation vacancy/interstitial by recombination

. Nucleation of intragranular cavities

. Nucleation of intergranular cavities

. Nucleation of intragranular cavities due to aggregation of vacancies

. Nucleation of intragranular cavities due to aggregation of gas atoms

. Nucleation of intergranular cavities due to aggregation of gas atoms at the grain boundaries

. Diffusion of intragranular cavities

. Diffusion of intergranular cavities at the grain boundaries

. Flux of vacancies towards intergranular cavities at the grain boundaries

. Flux of interstitials towards intergranular cavities at the grain boundaries

. Trapping of vacancies by intragranular cavities

. Trapping of vacancies by intergranular cavities

. Trapping of vacancies by dislocation loops

. Trapping of vacancies by dislocation lines

. Trapping of interstitials by intragranular cavities

. Trapping of interstitials by intergranular cavities

. Trapping of interstitials by dislocation loops

. Trapping of interstitials by dislocation lines

. Trapping of gas atoms by intragranular cavities

. Trapping of gas atoms at the grain boundaries by intergranular cavities

. Re-solution of vacancies from intragranular cavities

. Re-solution of vacancies from intergranular cavities

. Re-solution of gas atoms from intragranular cavities

. Re-solution of gas atoms from intergranular cavities

. Re-solution of gas atoms from the grain boundaries

. Destruction of intragranular cavities creating vacancies and gas atoms

. Destruction of intergranular cavities creating vacancies and gas atoms

. Coalescence of intragranular cavities

. Coalescence of intergranular cavities

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APPENDIX B : E QUATIONS

These equations are described more precisely in the schedule of conditions of the model [5].

B.1 INTRAGRANULAR EQUATIONS

B.1.1 Diffusion equation for vacancies

∂Cl

∂t= Pl + Dlv∆Cl + K1 + K2 + K3 + K4 + K5 + K6 + K7 + K8 + K9 + K10. (1)

where :– Cl is the concentration of vacancies– Pl is the production term– Dlv is the diffusion coefficient for vacancies– K1 corresponds to the vacancy-interstitial recombination term– K2 corresponds to the vacancy-dislocation loop interaction term– K3 corresponds to the vacancy-dislocation line interaction term– K4 corresponds to the term of re-solution of an intragranular cavity– K5 corresponds to the sink term due to the nucleation of an intragranular cavity– K6 corresponds to the production term due to the destruction of an intragranular cavity– K7 corresponds to the flux term towards an intragranular cavity– K8 corresponds to the re-solution term of an intragranular cavity– K9 corresponds to the production term due to the destruction of an intergranular cavity– K10 corresponds to the flux term towards an intergranular cavity

B.1.2 Diffusion equation for interstitials

∂Ci

∂t= Pi + Pdis + Div∆Ci + K ′

1 + K ′2 + K ′

3 + K ′4 + K ′

5. (2)

where :– Ci is the concentration of interstitials– Pi is the production term due to the decelaration of the fission fragments– Pdis is the production term due to the destruction of dislocation loops– Div is the diffusion coefficient for interstitials– K ′

1 corresponds to the vacancy-interstitial recombination term– K ′

2 corresponds to the interstitial-dislocation loop interaction term– K ′

3 corresponds to the interstitial-dislocation line interaction term– K ′

4 corresponds to the flux term towards an intragranular cavity– K ′

5 corresponds to the flux term towards an intergranular cavity

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B.1.3 Diffusion equation for intragranular gas

∂Cxedag

∂t= Dxedag∆Cxedag + Kg1 + Kg2 + Kg3 + Kg4 + Kg5 + Kg6 + Kg7 + Kg8. (3)

where :– Cxedag is the concentration of intragranular gas– Dxedag is the diffusion coefficient for intragranular gas– Kg1 corresponds to the creation by fission– Kg2 corresponds to the diffusion term towards an intragranular cavity– Kg3 corresponds to the scanning by a moving cavity– Kg4 corresponds to the nucleation of intragranular bubbles– Kg5 corresponds to the re-solution from an intergranular cavity– Kg6 corresponds to the re-solution due to the destruction of an intragranular cavity– Kg7 corresponds to the re-solution from an intergranular cavity– Kg8 corresponds to the re-solution of the dissolved intergranular gas

B.1.4 Conservation equation for dislocation loops

∂Sb(rbd, t)

∂t+

∂

∂r

(

Sb(rbd, t)drbd

dt

)

= pbc(rbd) − g(rbd)Sb(rbd, t). (4)

where :– Sb(rbd, t) is the density distribution for dislocation loops of radius rbd

– pbc is a production term– g(rbd) is a destruction probability

B.1.5 Evolution equation for dislocation density

dρd

dt= Kd1 + Kd2ρd. (5)

where :– ρd is the dislocation density– Kd1 corresponds to the inflow from the dislocation loops (zero if drbd

dt

)

r=rmax

< 0)– Kd2 corresponds to the interaction with point defects

B.1.6 Conservation equation for intragranular cavities

∂Sag(u,m, t)

∂t= −

∂

∂u(Sag(u,m, t)fv(u,m))−

∂

∂m(Sag(u,m, t)fm(u,m))+Kca1 +Kca2 +Kca3 +Kca4. (6)

where :

34/57

– Sag is the density distribution of intragranular cavities– fv = du

dt is the kinetic evolution for the volume of an intragranular cavity– fm = dm

dt is the kinetic evolution for the number of moles contained in an intragranular cavity– Kc1 corresponds to the nucleation term– Kc2 corresponds to the destruction term– Kc3 corresponds to the sink term for cavities coming to the grain boundaries– Kc4 corresponds to the coalescence term for intergranular cavities

B.2 INTERGRANULAR EQUATIONS

B.2.1 Diffusion equation for intergranular gas

∂Cxedeg

∂t= Dxedeg∆Cxedeg + Kge1 + Kge2 + Kge3 + Kge4 + Kge5. (7)

where :– Cxedeg is the concentration of intergranular gas– Dxedeg is the diffusion coefficient for intergranular gas– Kge1 corresponds to the source term coming from inside the grain– Kge2 corresponds to the scanning by moving intergranular cavities– Kge3 corresponds to the flux towards intergranular cavities– Kge4 corresponds to re-solution of the gas in the grain– Kge5 corresponds to the nucleation on the grain boundary

B.2.2 Conservation equation for intergranular cavities

∂Seg(u,m, t)

∂t= −

∂

∂u(Seg(u,m, t)fvj(u,m))−

∂

∂m(Seg(u,m, t)fmj(u,m))+Kce1+Kce2+Kce3+Kce4+Kce5.

(8)where :– Seg is the density distribution of intergranular cavities– fvj = du

dt is the kinetic evolution for the volume of an intergranular cavity– fmj = dm

dt is the kinetic evolution for the number of moles contained in an intergranular cavity– Kce1 corresponds to the nucleation term– Kce2 corresponds to the destruction term– Kce3 corresponds to the inflow of intragranular cavities– Kce4 corresponds to the coalescence of intragranular cavities– Kce5 corresponds to the coalescence of intergranular cavities

35/57

APPENDIX C : RESULTS OF THE INTRAGRANULAR TESTS

C.1 WITHOUT INITIAL PORE PROFILE

C.1.1 Concentration of vacancies

0

10

20

30

40

50

60

70

80

90

0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07

Cl at the first node without initial pore profile

800K1000K1200K1500K

C.1.2 Concentration of interstitials

0

5e-09

1e-08

1.5e-08

2e-08

2.5e-08

3e-08

0 500000 1e+06 1.5e+06 2e+06 2.5e+06 3e+06

Ci at the first node without initial pore profile

800K1000K1200K1500K

36/57

C.1.3 Concentration of intragranular gas

0

10

20

30

40

50

60

0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07

Cxedag at the first node without initial pore profile

800K1000K1200K1500K

C.1.4 Gas behaviour : number of cavities

0

2e+21

4e+21

6e+21

8e+21

1e+22

1.2e+22

0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07

Number of cavities without initial pore profile

800K1000K1200K1500K

37/57

C.1.5 Gas behaviour : mean radius of cavities

1e-09

2e-09

3e-09

4e-09

5e-09

6e-09

7e-09

8e-09

9e-09

1e-08

1.1e-08

1.2e-08

0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07

Mean radius of cavities without initial pore profile

800K1000K1200K1500K

C.1.6 Gas behaviour : swelling

0

0.001

0.002

0.003

0.004

0.005

0.006

0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07

Swelling without initial pore profile

800K1000K1200K1500K

38/57

C.1.7 Gas balance : ratio of gas in the bubbles

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07

Ratio of gas in the bubbles without initial pore profile

800K1000K1200K1500K

C.1.8 Gas balance : ratio of dissolved gas

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07

Ratio of gas dissolved in the grain without initial pore profile

800K1000K1200K1500K

39/57

C.1.9 Gas balance : ratio of intergranular gas

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07

Ratio of intergranular gas without initial pore profile

800K1000K1200K1500K

C.2 WITH INITIAL PORE PROFILE

C.2.1 Concentration of vacancies

0

10

20

30

40

50

60

70

80

90

0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07

Cl at the first node with initial pore profile

800K1100K1400K

40/57

C.2.2 Concentration of interstitials

0

5e-09

1e-08

1.5e-08

2e-08

2.5e-08

3e-08

0 500000 1e+06 1.5e+06 2e+06 2.5e+06 3e+06

Ci at the first node with initial pore profile

800K1100K1400K

C.2.3 Concentration of intragranular gas

0

5

10

15

20

25

30

35

40

45

0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07

Cxedag at the first node with initial pore profile

800K1100K1400K

41/57

C.2.4 Gas behaviour : number of cavities

0

2e+21

4e+21

6e+21

8e+21

1e+22

1.2e+22

1.4e+22

1.6e+22

1.8e+22

2e+22

0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07

Number of cavities with initial pore profile

800K1100K1400K

C.2.5 Gas behaviour : mean radius of cavities

0

2e-07

4e-07

6e-07

8e-07

1e-06

1.2e-06

0 2e+06 4e+06 6e+06 8e+06 1e+07 1.2e+07 1.4e+07

Mean radius of cavities with initial pore profile

800K1100K1400K

42/57

C.2.6 Gas behaviour : swelling

0.005

0.0055

0.006

0.0065

0.007

0.0075

0.008

0.0085

0.009

0.0095

0.01

0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07

Swelling with initial pore profile

800K1100K1400K

43/57

APPENDIX D : RESULTS OF THE INTERGRANULAR TESTS

D.1 NUMBER OF INTRAGRANULAR CAVITIES

0

1e+21

2e+21

3e+21

4e+21

5e+21

6e+21

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

t

Number of intragranular cavities - Complete case - Without initial pore profile

800K1000K1200K

0

5e+17

1e+18

1.5e+18

2e+18

2.5e+18

3e+18

3.5e+18

4e+18

4.5e+18

0 50000 100000 150000 200000 250000 300000 350000 400000 450000

t

Number of intragranular cavities - Complete case - Without initial pore profile

800K1000K1200K1500K

0

2e+15

4e+15

6e+15

8e+15

1e+16

1.2e+16

0 1000 2000 3000 4000 5000 6000 7000 8000

t

Number of intragranular cavities - Complete case - Without initial pore profile

800K1000K1200K1500K2000K

44/57

D.2 NUMBER OF INTERGRANULAR CAVITIES

0

1e+20

2e+20

3e+20

4e+20

5e+20

6e+20

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

t

Number of intergranular cavities - Complete case - Without initial pore profile

800K1000K1200K

D.3 MEAN RADIUS OF INTRAGRANULAR CAVITIES

2e-09

3e-09

4e-09

5e-09

6e-09

7e-09

8e-09

9e-09

1e-08

1.1e-08

1.2e-08

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

t

Mean radius of intragranular cavities - Complete case - Without initial pore profile

800K1000K1200K

45/57

2e-09

2.2e-09

2.4e-09

2.6e-09

2.8e-09

3e-09

3.2e-09

0 50000 100000 150000 200000 250000 300000 350000 400000 450000

t

Mean radius of intragranular cavities - Complete case - Without initial pore profile

800K1000K1200K1500K

2.02e-09

2.04e-09

2.06e-09

2.08e-09

2.1e-09

2.12e-09

2.14e-09

2.16e-09

2.18e-09

2.2e-09

2.22e-09

0 1000 2000 3000 4000 5000 6000 7000 8000

t

Mean radius of intragranular cavities - Complete case - Without initial pore profile

800K1000K1200K1500K2000K

D.4 MEAN RADIUS OF INTERGRANULAR CAVITIES

0

2e-09

4e-09

6e-09

8e-09

1e-08

1.2e-08

1.4e-08

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

t

Mean radius of intergranular cavities - Complete case - Without initial pore profile

800K1000K1200K

46/57

1e-09

2e-09

3e-09

4e-09

5e-09

6e-09

7e-09

8e-09

0 50000 100000 150000 200000 250000 300000 350000 400000 450000

t

Mean radius of intergranular cavities - Complete case - Without initial pore profile

800K1000K1200K1500K

0

2e-09

4e-09

6e-09

8e-09

1e-08

1.2e-08

1.4e-08

0 1000 2000 3000 4000 5000 6000 7000 8000

t

Mean radius of intergranular cavities - Complete case - Without initial pore profile

800K1000K1200K1500K2000K

D.5 CONCENTRATION OF INTRAGRANULAR GAS

0

10

20

30

40

50

60

70

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

t

Cxedag at the first node - Complete case - Without initial pore profile

800K1000K1200K

47/57

D.6 INTRAGRANULAR SWELLING

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

t

Intragranular swelling - Complete case - Without initial pore profile

800K1000K1200K

0

5e-08

1e-07

1.5e-07

2e-07

2.5e-07

3e-07

3.5e-07

4e-07

4.5e-07

5e-07

0 50000 100000 150000 200000 250000 300000 350000 400000 450000

t

Intragranular swelling - Complete case - Without initial pore profile

800K1000K1200K1500K

0

5e-11

1e-10

1.5e-10

2e-10

2.5e-10

3e-10

3.5e-10

4e-10

4.5e-10

5e-10

0 1000 2000 3000 4000 5000 6000 7000 8000

t

Intragranular swelling - Complete case - Without initial pore profile

800K1000K1200K1500K2000K

48/57

D.7 INTERGRANULAR SWELLING

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

t

Intergranular swelling - Complete case - Without initial pore profile

800K1000K1200K

0

2e-06

4e-06

6e-06

8e-06

1e-05

1.2e-05

1.4e-05

1.6e-05

1.8e-05

0 50000 100000 150000 200000 250000 300000 350000 400000 450000

t

Intergranular swelling - Complete case - Without initial pore profile

800K1000K1200K1500K

0

5e-07

1e-06

1.5e-06

2e-06

2.5e-06

3e-06

3.5e-06

0 1000 2000 3000 4000 5000 6000 7000 8000

t

Intergranular swelling - Complete case - Without initial pore profile

800K1000K1200K1500K2000K

49/57

D.8 RATIO OF GAS IN THE BUBBLES

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

t

Ratio of gas in the bubbles - Complete case - Without initial pore profile

800K1000K1200K

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 50000 100000 150000 200000 250000 300000 350000 400000 450000

t

Ratio of gas in the bubbles - Complete case - Without initial pore profile

800K1000K1200K1500K

0

1e-05

2e-05

3e-05

4e-05

5e-05

6e-05

0 1000 2000 3000 4000 5000 6000 7000 8000

t

Ratio of gas in the bubbles - Complete case - Without initial pore profile

800K1000K1200K1500K2000K

50/57

D.9 RATIO OF DISSOLVED GAS

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

t

Ratio of gas dissolved in the grain - Complete case - Without initial pore profile

800K1000K1200K

0.88

0.9

0.92

0.94

0.96

0.98

1

0 50000 100000 150000 200000 250000 300000 350000 400000 450000

t

Ratio of gas dissolved in the grain - Complete case - Without initial pore profile

800K1000K1200K1500K

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0 1000 2000 3000 4000 5000 6000 7000 8000

t

Ratio of gas dissolved in the grain - Complete case - Without initial pore profile

800K1000K1200K1500K2000K

51/57

D.10 RATIO OF INTERGRANULAR GAS

0

0.02

0.04

0.06

0.08

0.1

0.12

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07

t

Ratio of intergranular gas - Complete case - Without initial pore profile

800K1000K1200K

0

0.02

0.04

0.06

0.08

0.1

0.12

0 50000 100000 150000 200000 250000 300000 350000 400000 450000

t

Ratio of intergranular gas - Complete case - Without initial pore profile

800K1000K1200K1500K

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 1000 2000 3000 4000 5000 6000 7000 8000

t

Ratio of intergranular gas - Complete case - Without initial pore profile

800K1000K1200K1500K2000K

52/57

APPENDIX E : STUDY OF PARAMETER DEPENDENCY : ENVELOPE CURVES

AT 800K

E.1 NUMBER OF CAVITIES

0

5e+22

1e+23

1.5e+23

2e+23

2.5e+23

3e+23

3.5e+23

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07 7e+07

Nca

v

t

Total number of cavities

53/57

E.2 MEAN RADIUS OF CAVITIES

0

5e-10

1e-09

1.5e-09

2e-09

2.5e-09

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07 7e+07

Rca

v

t

Mean radius of cavities

E.3 MEAN PRESSURE INSIDE THE CAVITIES

0

1e+09

2e+09

3e+09

4e+09

5e+09

6e+09

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07 7e+07

Pca

v

t

Mean pressure inside the cavities

54/57

E.4 RATIO OF FISSION GAS DISSOLVED IN THE GRAIN

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07 7e+07

%xe

dag

t

Ratio of fission gas dissolved in the grain

E.5 RATIO OF FISSION GAS IN THE INTRAGRANULAR CAVITIES

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07 7e+07

%ca

v

t

Ratio of fission gas in the intragranular cavities

55/57

E.6 RATIO OF FISSION GAS AT THE GRAIN BOUNDARIES

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07 7e+07

%in

ter

t

Ratio of fission gas at the grain boundaries

E.7 TOTAL LENGTH OF DISLOCATION LOOPS

0

1e+15

2e+15

3e+15

4e+15

5e+15

6e+15

7e+15

8e+15

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07 7e+07

L loo

ps

t

Total length of dislocation loops

56/57

E.8 TOTAL LENGTH OF DISLOCATION LINES

9940

9950

9960

9970

9980

9990

10000

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07 7e+07

L lin

es

t

Total length of dislocation lines

E.9 INTRAGRANULAR SWELLING

0

0.002

0.004

0.006

0.008

0.01

0.012

0 1e+07 2e+07 3e+07 4e+07 5e+07 6e+07 7e+07

Sin

tra

t

Intragranular swelling

57/57

TRITA-CSC-E 2007:060 ISRN-KTH/CSC/E--07/060--SE

ISSN-1653-5715

www.kth.se