The mesh of 6-cubes is defined by the Mefisto commands:
Option 61: Mefisto file of the 6-cube (length=20)
This a regular mesh of sub 6-cubes of one 6-cube with 5 EDGES in each of the 6 DIRECTIONS (of type finite differences)
The drawing is done with a projection into R3.
With only 6 EDGES the following error is obtained:
NUMBER of VERTICES = 6**6 = 46 656 (6 edges => 7**6 = 117 649 vertices)
NUMBER of 6-CUBES = 5**6 = 15 625
=> ERROR in thevvp: NOT ENOUGH MEMORY or SWAP for MG + KG + Aux
115 222 032 DECLARED MEMORY WORDS
1 451 343 744 NECESSARY WORDS to store MG + KG + Aux
=> REDUCE the MESH
To construct a finer mesh around the origin and growing with the distance from the origin to reduce the number of finite elements, the 6-cubes are generated by an homothetical mapping of the initial center 6-cube through its 12 5-faces with a number of layers. The mesh of 6-cubes is defined by the Mefisto commands:
Option 62: Mefisto file of the 6-cube mesh construction (length=20)
This a regular mesh of sub 6-cubes of one 6-cube with 50 layers.
But, this time, the number of finite elements is not enough around the origin!
Option 63: Fichier Mefisto of the 6-cube (length=20, 4 layers)
This third mesh is the combination of the 2 previous meshes.
A kernel of 3 finite difference 6-cubes (to obtain more finite elements around the origin) with layers of homothetical 6-cubes through the 5-faces of their boundary (to obtain a progress of the mesh with not too more elements).
For n finite differences of the kernel, there are - (n+1)**6 vertices - (n+1)**6 - (n-1)**6 vertices by layer - n**6 + nblayers ( 12 * n**5 ) 6-cubes. Kernel n=1 => 2**6= 64 vertices and 665 vertices by layer Kernel n=2 => 3**6= 729 vertices and 4 032 vertices by layer Kernel n=3 => 4**6= 4 096 vertices and 14 896 vertices by layer Kernel n=4 => 5**6= 15 625 vertices and 42 560 vertices by layer Kernel n=5 => 6**6= 46 656 vertices and 102 024 vertices by layer Kernel n=6 => 7**6= 117 649 vertices and 215 488 vertices by layer Kernel n=7 => 8**6= 262 144 vertices and 413 792 vertices by layer Kernel n=8 => 9**6= 531 441 vertices and 737 856 vertices by layer Kernel n=9 =>10**6=1 000 000 vertices and 1 240 120 vertices by layer !... A good accuracy requires a large value of n => a VERY LARGE MEMORY is necessary to solve the problem!
The mesh with n=3 finite differences and 5 layers:
The mesh with 5 layers does not permit the computation of energies, by default of a sufficient main memory.
The computation of energies with the 3 meshes is defined by the same Mefisto commands:
Mefisto file to compute the energies of He atom
With the third mesh, the necessary main memory is
INITIAL SKYLINE MATRIX = 85 335 188 x 8 BYTES FINAL SKYLINE MATRIX = 65 103 823 x 8 BYTES = 520 827 404 BYTES 3 SKYLINE MATRICES ARE NECESSARY => 1.6 Gigabytes!
The first 14 computed energies with this last mesh:
cube exact error
ENERGY 1 = -2.816518 -2.90358 < 3.0 %
ENERGY 2 = -2.050233 -2.17522 < 5.7 %
ENERGY 3 = -1.830378 -2.14596 < 15 %
ENERGY 4 = -1.830232 -2.13316 < 15 %
ENERGY 5 = -1.689611
ENERGY 6 = -1.582237
ENERGY 7 = -1.582237
ENERGY 8 = -1.197280
ENERGY 9 = -1.194597
ENERGY 10 = -1.097584
ENERGY 11 = -1.090540
ENERGY 12 = -1.087805
ENERGY 13 = -1.055419
ENERGY 14 = -1.037078
CONCLUSION:
It is evident that the TOO SMALL NUMBER of finite elements of the mesh must be augmented to obtain a better and correct accuracy.
Technically, this augmentation needs a powerful computer with a very large memory and compilers able to manage adresses greater than 2**32. We hope to obtain these ones in the next months.
Some informations on the drawings:
EIGENVALUE 1 = -2.817
COLOR SECTION=EIGENVECTOR1(X,Y,0,0,0)
EIGENVALUE 1 = -2.817
COLOR SECTION=EIGENVECTOR1(X,Y,0,0,0,0)
EIGENVALUE 1 = -2.817
PROFILE Z=EIGENVECTOR1(X,Y,0,0,0,0)
EIGENVALUE 1 = -2.817
PROFILE Z=EIGENVECTOR1(X,Y,0,u=0,0,0)
EIGENVALUE 1 = -2.817
PROFILE Y=EIGENVECTOR1(X,0,Z,0,0,0)
EIGENVALUE 1 = -2.817
PROFILE X=EIGENVECTOR1(0,Y,Z,0,0,0)
EIGENVALUE 2 = -2.050
COLOR SECTION=EIGENVECTOR1(X,Y,0,0,0)
EIGENVALUE 2 = -2.050
COLOR SECTION=EIGENVECTOR1(X,0,Z,0,0,0)
EIGENVALUE 2 = -2.050
PROFILE Z=EIGENVECTOR1(0,Y,Z,0,0,0)
EIGENVALUE 2 = -2.050
PROFILE Z=EIGENVECTOR1(0,Y,Z,0,0,0)
EIGENVALUE 2 = -2.050
PROFILE Y=EIGENVECTOR1(X,0,Z,0,0,0)
EIGENVALUE 2 = -2.050
PROFILE Z=EIGENVECTOR1(0,Y,0,U,0,0)
EIGENVALUE 2 = -2.050
PROFILE Y=EIGENVECTOR1(X,0,0,U,0,0)
EIGENVALUE 2 = -2.050
PROFILE X=EIGENVECTOR1(X,Y,0,0,0,0)
EIGENVALUE 3 = -1.830
COLOR SECTION=EIGENVECTOR1(X,Y,0,0,0)
BR>
EIGENVALUE 3 = -1.830
PROFILE Z=EIGENVECTOR1(X,Y,0,0,0,0)
EIGENVALUE 3 = -1.830
PROFILE Z=EIGENVECTOR1(0,Y,0,U,0,0)
EIGENVALUE 3 = -1.830
PROFILE Y=EIGENVECTOR1(X,0,0,U,0,0)
EIGENVALUE 4 = -1.830
COLOR SECTION=EIGENVECTOR4(X,Y,0,0,0,0)
EIGENVALUE 4 = -1.830
COLOR SECTION=EIGENVECTOR4(X,0,Z,0,0,0) et (X,Y,0,0,0,0)
EIGENVALUE 4 = -1.830
PROFILE Z=EIGENVECTOR4(X,Y,0,0,0,0)
EIGENVALUE 4 = -1.830
PROFILE Y=EIGENVECTOR4(X,0,Z,0,0,0)
EIGENVALUE 4 = -1.830
PROFILE X=EIGENVECTOR4(0,Y,Z,0,0,0)
EIGENVALUE 4 = -1.830
ISOSURFACES of EIGENVECTOR4(X,Y,Z,0,0,0)
EIGENVALUE 4 = -1.830
ISOSURFACES of EIGENVECTOR4(X,Y,0,U,0,0)
EIGENVALUE 10 = -1.098
PROFILE Z=EIGENVECTOR10(X,Y,0,0,0,0)
EIGENVALUE 13 = -1.055
PROFILE Z=EIGENVECTOR13(X,Y,0,U=0,0,0)
EIGENVALUE 13 = -1.055
COLOR SECTION=EIGENVECTOR13(X,Y,0,U=0,0,0)
EIGENVALUE 13 = -1.055
ISOSURFACES of EIGENVECTOR13(X,Y,0,U,0,0)
EIGENVALUE 14 = -1.037
PROFILE Z=EIGENVECTOR14(X,Y,0,0,0,0)
EIGENVALUE 14 = -1.037
PROFILE Z=EIGENVECTOR14(X,Y,0)
EIGENVALUE 14 = -1.037
COLOR SECTION=EIGENVECTOR14(X,Y,0)
EIGENVALUE 14 = -1.037
ISOSURFACES of EIGENVECTOR14(X,Y,Z)
Page written by Alain Perronnet
Last update January 17-th 2007
EIGENVALUE 1 = -2.817 COLOR SECTION=EIGENVECTOR1(X,Y,0,0,0,0)
EIGENVALUE 1 = -2.817 PROFILE Z=EIGENVECTOR1(X,Y,0,0,0,0)
EIGENVALUE 1 = -2.817 PROFILE Z=EIGENVECTOR1(X,Y,0,u=0,0,0)
EIGENVALUE 1 = -2.817 PROFILE Y=EIGENVECTOR1(X,0,Z,0,0,0)
EIGENVALUE 1 = -2.817 PROFILE X=EIGENVECTOR1(0,Y,Z,0,0,0)
EIGENVALUE 2 = -2.050 COLOR SECTION=EIGENVECTOR1(X,Y,0,0,0)
EIGENVALUE 2 = -2.050 COLOR SECTION=EIGENVECTOR1(X,0,Z,0,0,0)
EIGENVALUE 2 = -2.050 PROFILE Z=EIGENVECTOR1(0,Y,Z,0,0,0)
EIGENVALUE 2 = -2.050 PROFILE Z=EIGENVECTOR1(0,Y,Z,0,0,0)
EIGENVALUE 2 = -2.050 PROFILE Y=EIGENVECTOR1(X,0,Z,0,0,0)
EIGENVALUE 2 = -2.050 PROFILE Z=EIGENVECTOR1(0,Y,0,U,0,0)
EIGENVALUE 2 = -2.050 PROFILE Y=EIGENVECTOR1(X,0,0,U,0,0)
EIGENVALUE 2 = -2.050 PROFILE X=EIGENVECTOR1(X,Y,0,0,0,0)
EIGENVALUE 3 = -1.830 COLOR SECTION=EIGENVECTOR1(X,Y,0,0,0)
BR>EIGENVALUE 3 = -1.830 PROFILE Z=EIGENVECTOR1(X,Y,0,0,0,0)
EIGENVALUE 3 = -1.830 PROFILE Z=EIGENVECTOR1(0,Y,0,U,0,0)
EIGENVALUE 3 = -1.830 PROFILE Y=EIGENVECTOR1(X,0,0,U,0,0)
EIGENVALUE 4 = -1.830 COLOR SECTION=EIGENVECTOR4(X,Y,0,0,0,0)
EIGENVALUE 4 = -1.830 COLOR SECTION=EIGENVECTOR4(X,0,Z,0,0,0) et (X,Y,0,0,0,0)
EIGENVALUE 4 = -1.830 PROFILE Z=EIGENVECTOR4(X,Y,0,0,0,0)
EIGENVALUE 4 = -1.830 PROFILE Y=EIGENVECTOR4(X,0,Z,0,0,0)
EIGENVALUE 4 = -1.830 PROFILE X=EIGENVECTOR4(0,Y,Z,0,0,0)
EIGENVALUE 4 = -1.830 ISOSURFACES of EIGENVECTOR4(X,Y,Z,0,0,0)
EIGENVALUE 4 = -1.830 ISOSURFACES of EIGENVECTOR4(X,Y,0,U,0,0)
EIGENVALUE 10 = -1.098 PROFILE Z=EIGENVECTOR10(X,Y,0,0,0,0)
EIGENVALUE 13 = -1.055 PROFILE Z=EIGENVECTOR13(X,Y,0,U=0,0,0)
EIGENVALUE 13 = -1.055 COLOR SECTION=EIGENVECTOR13(X,Y,0,U=0,0,0)
EIGENVALUE 13 = -1.055 ISOSURFACES of EIGENVECTOR13(X,Y,0,U,0,0)
EIGENVALUE 14 = -1.037 PROFILE Z=EIGENVECTOR14(X,Y,0,0,0,0)
EIGENVALUE 14 = -1.037 PROFILE Z=EIGENVECTOR14(X,Y,0)
EIGENVALUE 14 = -1.037 COLOR SECTION=EIGENVECTOR14(X,Y,0)
EIGENVALUE 14 = -1.037 ISOSURFACES of EIGENVECTOR14(X,Y,Z)
Page written by Alain Perronnet
Last update January 17-th 2007