## Solved problems

### GEOC

*Modelling of microstructure of geocomposites*.
The micro finite element analysis, in which FEM is adapted for the derivation of material
behaviour from its complex microstructure. This modelling helps in the assessment and
optimization of the grouting technology. The goal may be to find out some physical parameters
of materials, which have complicated heterogeneous inner structure, but which can be considered
as piecewise homogeneous from a certain scale.

Studied geocomposite materials are produced by injection of polyurethane resin into coal environment, e.g. to reinforce coal pillars. Due to permeable and fractured coal, the geocomposites have complex microstructure.

Model captures the microstructure of a cubic geocomposite sample 75mm in size, which has been scanned with a special industrial X-ray computer tomograph producing a set of images corresponding to cuts through the sample.

*Discretization*.
A domain is discretized by a uniform grid of 231 × 231 × 37 grid voxels, resulting in a linear
system of 6 135 936 DOF. The model assumes homogeneous material in voxels, which is assigned
according to the CT scan values. The homogenized material properties of the cube are computed
making use of numerical upscaling via stress and strain driven tests.

### KBS

*Large-scale geoenvironmental model*.
Model of the prototype repository at Aspo following the KBS3 concept of storage of the
spent nuclear fuel and considering thermo-mechanical behavior of 3D domain in the time
period of 50 or 100 years.

The studied domain includes 65 m long deposition tunnel located 450 meters bellow surface, backfilled. There are two sections with 4 and 2 deposition holes (1.75 m diameter and 8 m deep), where cannisters with the spent nuclear fuel are stored. The used buffer material is bentonite.

*Discretization*.
The domain is discretized in space by a regular mesh with 391 x 63 x 105 nodes. The bricks
are further decomposed to six tetrahedra. In time, the finite differences are applied together
with implicit Euler scheme and adaptive time stepping.

The thermal part is influenced mainly by radioactive waste as the heat source and by heat conduction in rock, buffer and backfill. It results in a sequence of linear systems, each with 2 586 465 DOF.

Moreover in predermined time points, the solution of a linear system with 7 759 395 DOF is computed. It corresponds to the mechanical part, which is influenced by initial stress loading, tunnel excavation and heat load from the nuclear waste.

*Times of parallel computing*.
Computations performed on Natan and Thea. For the mechanical part only, in time of 2 years
after the heat loading by the nuclear waste:

- DD: Solution of the FEM system by two-level overlapping Schwarz method with minimal overlap, replacing of subdomain problems by incomplete factorization, creating of coarse grid problem by aggregation (two choices of aggregation 6 x 6 x 6 and 12 x 6 x 9), approximate solution of the coarse grid problem by inner PCG iterations with relative accuracy eps=0.1. The FEM system is solved from the zero initial guess up to relative residual accuracy eps=0.0001.
- DiD: Displacement decomposition preconditioning, subproblems replaced by incomplete factorization (IF) or solved by inner PCG iterations (II) with inner accuracy eps=0.1. The FEM system is again solved from the zero initial guess up to relative residual accuracy eps=0.0001.

# C | Simba (OpenMP) | Simba (MPICH) | Ra (SCALI) | ||||||
---|---|---|---|---|---|---|---|---|---|

# It | T [s] | S | # It | T [s] | S | # It | T [s] | S | |

1 | 1341 | 6292 | 1344 | 5931.4 | 1344 | 1144.0 | |||

2 | 1421 | 4101 | 1.63 | 1424 | 3169.1 | 1.87 | 1421 | 643.2 | 1.78 |

4 | 1425 | 2082 | 3.44 | 1428 | 1577.0 | 3.76 | 1426 | 314.3 | 3.64 |

8 | 1514 | 1120 | 6.34 | 1514 | 833.2 | 7.12 | 1514 | 909.4 | |

12 | 1578 | 872 | 8.48 | 1581 | 596.2 | 9.95 | 1579 | 1192.6 | |

16 | 1614 | 751 | 10.09 | 1618 | 482.9 | 12.28 | 1616 | 1355.9 | |

20 | 1691 | 471.1 | |||||||

24 | 1710 | 554.3 |

Computations performed on Simba and Ra. For the thermal part only, in time interval of 100 years after the heat loading by the nuclear waste. The adaptive choice of the time step is done and the total number of time steps is 47. The FEM system is solved up to relative residual accuracy eps=0.000001 and with the initial guess taken from the previous time step.

### DR

*Large-scale geotechnical model*.
Computations of stress changes due to uranium ore mining in the depth up to 1200
meters at the mine GEAM Dolni Rozinka (DR) in the Bohemian - Moravian Highlands
(the Czech Republic). The computed stress changes are used for comparison of
different mining methods in relation to a risk of dangerous rockbursts.

The whole modelling simulates four stages of mining, represented by a four-step sequence of problems with different material distributions. The last situation is used as a benchmark problem for comparison of different linear system solvers.

*Short description*.
Elasticity problem in a domain of 1430 x 550 x 600 meters located 800 meters bellow
the surface, loading by weight of overburden on the top domain face, symmetry plane
boundary conditions on the other faces, weight of rocks, initial stress.

*Discretization*.
Regular grid of 124 x 137 x 76 nodes, each brick is divided into six tetrahedra by
Kuhn's type division. Linear tetrahedral finite elements results in a linear system
with 3 873 264 DOF.

*Times of parallel computing*.
Solution of the FEM system by two-level overlapping Schwarz method with minimal overlap,
replacing of subdomain problems by incomplete factorization, creating of coarse grid
problem by aggregation, approximate solution of the coarse grid problem by inner PCG
iterations with relative accuracy eps=0.1. The FEM system is solved from the zero initial
guess up to relative residual accuracy eps=0.0001.