Random 2D Composites and the Generalized Method of Schwarz
Two-phase composites with nonoverlapping inclusions randomly embedded in matrix are investigated. A straightforward approach is applied to estimate the effective properties of random 2D composites. First, deterministic boundary value problems are solved for all locations of inclusions, that is, for...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2015/535128 |
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| _version_ | 1832565935204990976 |
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| author | Vladimir Mityushev |
| author_facet | Vladimir Mityushev |
| author_sort | Vladimir Mityushev |
| collection | DOAJ |
| description | Two-phase composites with nonoverlapping inclusions randomly embedded in matrix are investigated. A straightforward approach is applied to estimate the effective properties of random 2D composites. First, deterministic boundary value problems are solved for all locations of inclusions, that is, for all events of the considered probabilistic space C by the generalized method of Schwarz. Second, the effective properties are calculated in analytical form and averaged over C. This method is related to the traditional method based on the average probabilistic values involving the n-point correlation functions. However, we avoid computation of the correlation functions and compute their weighted moments of high orders by an indirect method which does not address the correlation functions. The effective properties are exactly expressed through these moments. It is proved that the generalized method of Schwarz converges for an arbitrary multiply connected doubly periodic domain and for an arbitrary contrast parameter. The proposed method yields an algorithm which can be applied with symbolic computations. The Torquato-Milton parameter ζ1 is exactly written for circular inclusions. |
| format | Article |
| id | doaj-art-833deebf227f4c3fa371d175d52ffee9 |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-833deebf227f4c3fa371d175d52ffee92025-02-03T01:06:14ZengWileyAdvances in Mathematical Physics1687-91201687-91392015-01-01201510.1155/2015/535128535128Random 2D Composites and the Generalized Method of SchwarzVladimir Mityushev0Pedagogical University, ul. Podchorazych 2, 30-084 Krakow, PolandTwo-phase composites with nonoverlapping inclusions randomly embedded in matrix are investigated. A straightforward approach is applied to estimate the effective properties of random 2D composites. First, deterministic boundary value problems are solved for all locations of inclusions, that is, for all events of the considered probabilistic space C by the generalized method of Schwarz. Second, the effective properties are calculated in analytical form and averaged over C. This method is related to the traditional method based on the average probabilistic values involving the n-point correlation functions. However, we avoid computation of the correlation functions and compute their weighted moments of high orders by an indirect method which does not address the correlation functions. The effective properties are exactly expressed through these moments. It is proved that the generalized method of Schwarz converges for an arbitrary multiply connected doubly periodic domain and for an arbitrary contrast parameter. The proposed method yields an algorithm which can be applied with symbolic computations. The Torquato-Milton parameter ζ1 is exactly written for circular inclusions.http://dx.doi.org/10.1155/2015/535128 |
| spellingShingle | Vladimir Mityushev Random 2D Composites and the Generalized Method of Schwarz Advances in Mathematical Physics |
| title | Random 2D Composites and the Generalized Method of Schwarz |
| title_full | Random 2D Composites and the Generalized Method of Schwarz |
| title_fullStr | Random 2D Composites and the Generalized Method of Schwarz |
| title_full_unstemmed | Random 2D Composites and the Generalized Method of Schwarz |
| title_short | Random 2D Composites and the Generalized Method of Schwarz |
| title_sort | random 2d composites and the generalized method of schwarz |
| url | http://dx.doi.org/10.1155/2015/535128 |
| work_keys_str_mv | AT vladimirmityushev random2dcompositesandthegeneralizedmethodofschwarz |