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,

Spain 2 7 Tank containing the PCM and a section of the material ,

3 9 Empty layer generated by the loss of volume in liquid state ,

An example of a triangular mesh ,

Non-conforming interface passing through unstructured (triangular) mesh ,

18 1.5 llustration of the procedure for generating a Voronoi diagram based computational mesh ,

, Quadree image representations: example of union, p.26

27 2.3 Construction of a quadtree for a square domain with an internal subdomain ,

, Hilbert filling curve ordering on quadree data structure, p.30

, Gray code filling curve ordering on quadtree data structure, p.30

, Morton filling curve ordering on quadtree data structure, p.30

, Internal Morton code of an octant through faces and vertices, p.31

, Morton filling curve ordering on octree data structure, p.31

, Creation of Morton index by binary interleaving, p.32

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,

,

35 2.14 A parallel partition of a structured grid and its global indexing along the cores intersections, Surface indices vs. number of processes: four tests ,

, 3D PETSc operator matrix draw. 36128 points, octree level 6, p.36

, A parallel partition of a structured grid on 16 cores, p.37

, Possible neighbourhood cases through a face, p.39

Identifying a configuration ,

Uniform mesh ,

, A two dimensional nodes centred discretization, p.52

53 3.4 Uniform mesh configuration. The weights are enumerated following the internal Z-Order of a 0 through faces, then through vertices ,

62 4.3 Examples of error distribution on a grid corresponding to two different levels. Simple repeat test configuration for each level of refinement ,

,

, Error distribution example. Unbalanced case mesh, p.67

, Error distribution example, level 8, sinus analytical function, p.68

69 4.9 Distribution of error on a AMR following a central sphere with radius 0.15 ,

, Section of the domain studying the residual, p.71

Distribution of error around and inside the sphere. Tree's levels 9 inside and 6 outside ,

, Distribution of error around and inside the sphere. Tree's levels 9 inside and 6 outside, p.75

, Zoom of error near the penalized spherical zone, p.76

, Approximated boundary conditions convergence study, p.77

Error distribution when Dirichlet condition is imposed on a circle of radius 0 ,

Zoom of the error next to the circle ,

82 4.22 Numerical result obtained on uniform mesh ,

Levels 8 and 9 of the AMR along the mollification with uniform mesh outside at level 7 ,

, Comparison between uniform and AMR resolutions, p.86

, Time comparison for different parallel cases (strong scalability, p.91

, Time partition for a complete execution. 48 cores distributed on 2 nodes. Total execution time 29 sec(s), p.92

, Time comparison for different parallel cases (weak scalability, p.93

, Empty layers generated by the liquid phase, p.98

Hybrid media simplified domain ,

, One dimensional sub-interval containing the fictitious layer, p.99

103 5.5 Boundary layer zone behaviour in order with the level of refinement, Double mollified function ?( x) from 1 to 10 ,

, 105 5.7 Numerical domain, host media containing three PCM capsules, p.110

Propagation front of temperature u f in time and liquid fraction, vol.113 ,

, Liquid fraction propagation front: three capsules, p.113

, Liquid fraction propagation front: six capsules, p.114

, u f propagation front for three capsules case, p.114

u f propagation front for six capsules case, p.115 ,

117 List of Tables 4.1 Error norms and order of the scheme. Proof of consistency.. 63 4.2 Error norms and order of the scheme. A second proof of consistency, vol.115 ,

, Error norms and order. Unbalanced mesh, p.67

, Error norms and order. Unbalanced circle mesh, p.68

,

Errors of Laplacian resolution AMR in a sphere. Two-dimensional sinus function ,

Errors of Laplacian resolution AMR in a sphere. Three-dimensional sinus function ,

Error norms and order of the scheme ,

80 4.14 Convergence results. Difference between AMR along the mollification and external mesh: five levels, Times ,

, Convergence results with ? = ?x

, , vol.104