### Abstract

We present in this paper a method for determining the convergence characteristics of the Neumann iterative solution of a discrete version of a second-type Fredholm equation. Implemented as the so-called 'equivalent inclusion problem' within the context of mechanical stress-strain analysis, it allows the modelling of elastically highly heterogeneous bodies with the aid of discrete Fourier transforms. A method is developed with which we can quantify, pre-analysis (i.e. at iteration zero), the convergence behaviour of the Neumann scheme depending on the choice of an auxiliary stiffness tensor, specifically for the linear elastic case. It is shown that a careful choice of this tensor results in both guaranteed convergence and a smaller convergence radius for the solution. Furthermore, there is some indication that, as the convergence radius decreases, the scheme may converge to a solution at a faster rate, translating into an increase in computational efficiency.

Original language | English |
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Pages (from-to) | 1967-1987 |

Number of pages | 21 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 458 |

Issue number | 2024 |

DOIs | |

Publication status | Published - 8 Aug 2002 |

### Keywords

- Composites
- Equivalent inclusion technique
- Fixed-point theorem
- Linear elasticity
- Neumann iteration

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## Cite this

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*,

*458*(2024), 1967-1987. https://doi.org/10.1098/rspa.2001.0955