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

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34 Description of the simulation method The complementary problem has a correction term such as σcorrec = (LP −LM) : ɛ1 in VP . With this correction term, the current stress field in the particle volume can be constructed by the superposi- tion of two fields (σ = σ1 +σ2 in VP ). By replacing ɛ1 with L −1 M : σ1, this correction term equals to (LP : L −1 M − I) : σ1, where I represents the fourth-order unit tensor. (LP : L −1 M − I) is expressed by the Young’s modulii and the Poisson’s ratios of the matrix (E, ν) and the particle (E ∗ , ν ∗ ) as follows. (LP : L −1 M ⎛ ⎞ a1 − 1 ⎜ − I) = ⎜ ⎝ a2 a1 − 1 a2 a2 a1 − 1 0 0 0 a3 − 1 0 0 0 0 a3 − 1 0 0 0 0 0 a3 − 1 ⎟ ⎠ with a1 being E∗ (1−ν∗−2νν ∗ ) E(1+ν∗ )(1−2ν∗ ) , a2 being E∗ (ν∗−ν) E(1+ν∗ )(1−2ν∗ ) and a3 being E∗ (1+ν) E(1+ν∗ ) . The complementary problem is solved using CAST∃M( 12 ), a finite element code developed in France by the Commissariat à l’Energie Atomique. The FEM formulation in the particle volume of the second problem can be written as follows with a strain-displacement matrix (B) : VP B T · L · BdV · u + VP B T · LP : L −1 M − I · σ1dV = 0 (2.22) In order to compute the body force-like term (right-hand side of Eq. 2.22) within the precipitate volume, first the stresses σ1 due to the dislocations at the points in VP , e.g., at the stress integration Gauss points, are computed. Then the correction term is computed within each finite element inside the particle. The computed stresses σcorrec are then changed into a nodal body force field, f b . This can be easily done with the ’BSIG’ operator in CAST∃M. These forces are applied to VP and the FEM gives the solution of a two phases boundary problem where forces Fap −FD and displacements Uap − UD are imposed at the boundary and body forces f b are applied inside the particle. The continuity of displacement and normal stresses at the particle/matrix interface are enforced by the FEM. 12 Finite element code developed by Commissariat à l’Energie Atomique, CEA-DRN/DMT/SEMT
2.5 Acceleration of the DDD code 35 Figure 2.16: Decomposition of the problem into the problem of dislocations in infinite media and the complementary problem of inhomogeneous finite volume without dislocations. Forces(Fap) and displacements(Uap) are applied on the boundary. In the complementary problem, boundary condi- tions are modified with forces(FD), displacements(UD) and a nodal body force field(f b ) generated by dislocations. 2.5 Acceleration of the DDD code 2.5.1 Problem description and review of literatures Internal stress computation is the most computationally intensive part in the DDD method. This is due to the fact that the stress field at a distance r from a dislocation line is proportional to 1/r. The stress field of a dislocation line is thus long-ranged. Another time consuming part in the DDD method is handling the dislocation segments interactions. Segment motion involves examination of possible interactions, between dislocations or between dislocation and internal interfaces. In a programming perspective, the two parts can be represented as follows in pseudo-code. Internal stress computation DO I=1,Nsegm DO J=1, Nsegm if(J=I and I’s neighbor) ENDDO ENDDO Compute σ int I←J Nsegm: Number of segments Nfacets: Number of facets Segment motion DO I=1,Nsegm DO J=1, Nsegm Examine interaction with segment J ENDDO DO K=1, Nfacets Examine interaction with facet K ENDDO Move segment I ENDDO
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
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Page 77 and 78: 3.3 Parallelization of the serial D
Page 91 and 92: 3.4 Performance improvment 81 Speed
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3.4 Performance improvment 85 Speed
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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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