Assume that we have a two-dimensional elastic medium with periodically distributed circular inlays consisting of a stiffer material. If these holes are small then the medium will behave on a macroscopic scale like a homogeneous medium without holes. The elasticity tensor of this medium can be calculated as follows:

- On the representative cell compute a tensor N of rank 3 which correspond to responses of the representative cell to different loads. Boundary conditions for this problem are periodic.
- Compute the effective tensor by evaluating the load functionals on the tensor N.

We used conforming finite elements of order 5 with nonlinear element mappings to approximate the tensor N. Solver for the linear problems was a W(1,1)-multigrid cycle with an overlapping block Gauss-Seidel smoother.

The history of this calculation was as follows (three of the 16 components are shown, due to symmetry properties the remaining ones can be obtained by a suitable permutation of indices):

Cells | Unknowns | Matrix entries | A_{1111} |
A_{1212} |
A_{1221} |
---|---|---|---|---|---|

9 | 1800 | 43060 | 4.1458940638 | 1.3176717343 | 1.2966840277 |

36 | 7200 | 176224 | 4.1412496929 | 1.3139564023 | 1.2979726371 |

144 | 28800 | 705600 | 4.1412384319 | 1.3139473004 | 1.2979716831 |

576 | 115200 | 2822400 | 4.1412383854 | 1.3139472825 | 1.2979716903 |

2304 | 460800 | 11289600 | 4.1412383853 | 1.3139472825 | 1.2979716903 |

Nicolas Neuss, 2017-01-31