3D DISCRETE DISLOCATION DYNAMICS APPLIED TO ... - NUMODIS

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38 Description of the simulation method as Eq.(2.24). ib1 ib2 1 ib3 3 ib4 5 15 14 7 11 16 6 212 9 10 8 13 4 indexb ib1 7 1 ib2 10 4 ib3 3 5 ib4 12 2 Figure 2.18: Linked list of segments min a ≥ 1 √ ld, a ≥ 2 2 √ vmaxδt 6 ib1 ib2 ib3 ib4 2 4 5 5 6 16 14 11 15 13 9 8 (2.24) The first term in Eq.(2.24) states that neighbors of a segment are located inside the first neighbor- hood, hence it is given by the discretization length ld. The second term states that a segment is not allowed to move across the first neighboring boxes in one step, thus it is a function of the maximum velocity vmax (see Sec 2.3.2) and time step ∆t. Fig. 2.19 shows the criterion of the minimum edge length of a box. The use of the box sizes larger than the minimum size of the box reduces computing cost in ’Segment motion’ part. This is because only interactions within a maximum free-flight distance of a segment need to be considered instead of taking all the segments and facets into consideration. This will reduce the number of segments and facets to be inspected without any approximation. The internal stress acting on a segment is divided into a long-distance stress (σ LR ), which varies rather slowly over time steps and a short-distance stress (σ SR ), which shows large fluctuation over single time steps. σ SR of a segment in the box ib is computed by taking all the segments into account in the L th neighboring boxes at every simulation step. σ LR is adopted by the stress at the center point of ib from all the segments outside of the L th neighboring boxes. All segments in the same box, therefore, have the same σ LR . This approximation is valid if σ LR has a wave length larger than the box size. The computation of σ LR for all the boxes is updated every f step. The parameters involved in the box method are listed in Table 2.3. The maximum number of boxes M is given by the minimum box size (Eq.(2.24)) and the simulation volume size. The other parameters should be chosen based on the numerical accuracy and the speedup, and will be the issue of the following section.
2.5 Acceleration of the DDD code 39 [100] [001] Slip direction, [112] (1) (2) Slip plane, (11-1) Screw segment Figure 2.19: Minimum box size Parameter Description [010] M number of boxes along each side of a simulation volume f frequency of σ LR update L number of layers for σ SR Table 2.3: Parameters of the box method The pseudo-code of internal stress computation in Sec. 2.5.1 is then substituted by the following pseudo-code. Internal stress computation DO I=1,Nsegm Identify the box ’ib’ of the segment ’I’ Compute the stresses σ SR by segments within short-distance boxes Add σ LR (ib) ENDDO Long-distance stress computation (every f step) DO iz=1,M DO iy=1,M DO ix=1,M compute the box index ’ib’ compute the long-distance stresses σ LR (ib) ENDDO ENDDO ENDDO at the center of ’ib’
Page 1 and 2: INSTITUT NATIONAL POLYTECHNIQUE DE
Page 3: Acknowledgements First of all, I ex
Page 6 and 7: Résumé La dynamique des dislocati
Page 8 and 9: viii CONTENTS 2.3.5 Plastic strain
Page 11 and 12: Chapter 1 Introduction 1.1 Computat
Page 13 and 14: 1.2 Dislocation dynamics 3 (a) Weak
Page 15 and 16: 1.3 Scope of Thesis 5 Recently the
Page 17: 1.3 Scope of Thesis 7 coupling meth
Page 20 and 21: 10 Description of the simulation me
Page 61 and 62: Chapter 3 Parallelization of the Di
Page 63 and 64: 3.1 An introduction to Supercomputi
Page 71 and 72: 3.2 Towards a parallel DDD code 61
Page 77 and 78: 3.3 Parallelization of the serial D
Page 91 and 92: 3.4 Performance improvment 81 Speed
Page 97 and 98: 3.4 Performance improvment 87 Y Z I
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3.5 Application to Stage I-II trans
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Chapter 4 Dislocation-precipitate i
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4.1 Image stresses due to a 3D part
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4.2 A simple case of dislocation-pa
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Chapter 5 Conclusions and perspecti
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namics codes into a parallel versio
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Bibliography [Abraham 97] Abraham F
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BIBLIOGRAPHY 155 [Devincre & Robert
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BIBLIOGRAPHY 157 [Khraishi et al. 0
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BIBLIOGRAPHY 159 [Risbet et al. 03]
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