1029:
which statistical properties are the same at every point in space. In practice, many applications require a more general way of modeling that is neither periodic nor statistically homogeneous. For this end the methods of the homogenization theory have been extended to partial differential equations, which coefficients are neither periodic nor statistically homogeneous (so-called arbitrarily rough coefficients).
1368:
683:
1028:
Classical results of homogenization theory were obtained for media with periodic microstructure modeled by partial differential equations with periodic coefficients. These results were later generalized to spatially homogeneous random media modeled by differential equations with random coefficients
327:
It turns out that the study of these equations is also of great importance in physics and engineering, since equations of this type govern the physics of inhomogeneous or heterogeneous materials. Of course, all matter is inhomogeneous at some scale, but frequently it is convenient to treat it as
993:
As a result of the above, homogenization can therefore be viewed as an extension of the continuum concept to materials which possess microstructure. The analogue of the differential element in the continuum concept (which contains enough atom, or molecular structure to be representative of that
862:
1115:
359:
and therefore they are subjected to loads or forcings which vary on a length scale which is far bigger than the characteristic length scale of the microstructure. In this situation, one can often replace the equation above with an equation of the form
131:
998:" in homogenization and micromechanics. This element contains enough statistical information about the inhomogeneous medium in order to be representative of the material. Therefore averaging over this element gives an effective property such as
454:
284:
721:
416:
1439:
1363:{\displaystyle u_{\epsilon }({\vec {x}})=u({\vec {x}},{\vec {y}})=u_{0}({\vec {x}},{\vec {y}})+\epsilon u_{1}({\vec {x}},{\vec {y}})+\epsilon ^{2}u_{2}({\vec {x}},{\vec {y}})+O(\epsilon ^{3})\,}
1087:
194:
1373:
which generates a hierarchy of problems. The homogenized equation is obtained and the effective coefficients are determined by solving the so-called "cell problems" for the function
909:
962:
56:
988:
1037:
Mathematical homogenization theory dates back to the French, Russian and
Italian schools. The method of asymptotic homogenization proceeds by introducing the fast variable
678:{\displaystyle A_{ij}^{*}=\int _{(0,1)^{n}}A({\vec {y}})\left(\nabla w_{j}({\vec {y}})+{\vec {e}}_{j}\right)\cdot {\vec {e}}_{i}\,dy_{1}\dots dy_{n},\qquad i,j=1,\dots ,n}
322:
1107:
929:
154:
1023:
713:
446:
199:
1672:; Owhadi, H. (November 2010). "Flux Norm Approach to Finite Dimensional Homogenization Approximations with Non-Separated Scales and High Contrast".
448:
is a constant tensor coefficient and is known as the effective property associated with the material in question. It can be explicitly computed as
17:
867:
This process of replacing an equation with a highly oscillatory coefficient with one with a homogeneous (uniform) coefficient is known as
857:{\displaystyle \nabla _{y}\cdot \left(A({\vec {y}})\nabla w_{j}\right)=-\nabla _{y}\cdot \left(A({\vec {y}}){\vec {e}}_{j}\right).}
1997:
366:
1974:
1921:
1548:
1507:
1376:
1883:
1834:
1785:
1607:
1582:
1040:
340:, etc. can be treated as homogeneous materials and associated with these materials are material properties such as
1465:
47:
159:
28:
995:
881:
1992:
1930:
1528:
126:{\displaystyle \nabla \cdot \left(A\left({\frac {\vec {x}}{\epsilon }}\right)\nabla u_{\epsilon }\right)=f}
934:
1644:
967:
289:
1092:
914:
139:
1691:
1001:
691:
424:
1950:
1893:
1844:
8:
1570:
1450:
352:
329:
1695:
1738:
1707:
1681:
1970:
1942:
1917:
1879:
1830:
1781:
1603:
1578:
1544:
1503:
1460:
1753:
1726:
1946:
1909:
1889:
1840:
1773:
1772:. Progress in Nonlinear Differential Equations and Their Applications. Birkhauser.
1748:
1711:
1699:
1536:
1495:
1455:
1966:
1905:
1826:
1669:
279:{\displaystyle A\left({\vec {y}}+{\vec {e}}_{i}\right)=A\left({\vec {y}}\right)}
872:
356:
345:
1913:
1777:
1703:
1540:
1986:
1867:
1822:
341:
1875:
1935:
Averaging of
Processes in Periodic Media (English translation: Kluwer,1989)
1851:
1806:
1499:
1577:. Studies in Mathematics and its Applications. Amsterdam: North-Holland.
35:
1818:
328:
homogeneous. A good example is the continuum concept which is used in
1859:
1743:
1686:
1811:
Homogenization of differential operators and integral functionals
39:
1602:. Modern Mechanics and Mathematics. Chapman and Hall/CRC Press.
1962:
1938:
1863:
1814:
1645:"Boundary Value Problems with Rapidly Oscillating Coefficients"
337:
333:
1961:, Oxford Lecture Series in Mathematics and Its Applications,
1871:
1858:, Studies in Mathematics and its Applications, vol. 26,
1494:. Lecture Notes in Physics. Vol. 127. Springer Verlag.
1623:
Kozlov, S.M. (1979). "Homogenization of Random
Operators".
1568:
1633:(English transl.: Math. USSR, Sb. 37:2, 1980, pp. 167-180)
1032:
871:. This subject is inextricably linked with the subject of
878:
In homogenization one equation is replaced by another if
411:{\displaystyle \nabla \cdot \left(A^{*}\nabla u\right)=f}
1535:. Mathematics and its Applications. Dordrecht: Kluwer.
1904:, Interdisciplinary Applied Mathematics, vol. 6,
1899:
1856:
Mathematical problems in elasticity and homogenization
1850:
1434:{\displaystyle u_{1}({\vec {x}},{\vec {x}}/\epsilon )}
1533:
Homogenisation: Averaging
Processes in Periodic Media
1379:
1118:
1095:
1043:
1004:
970:
937:
917:
884:
724:
694:
457:
427:
369:
292:
202:
162:
142:
59:
1642:
1956:
1804:
1600:
Microstructural randomness and scaling in materials
1929:
1527:
1433:
1362:
1101:
1081:
1017:
982:
956:
923:
903:
856:
707:
677:
440:
410:
316:
278:
188:
148:
125:
1724:
1597:
1984:
1489:
1082:{\displaystyle {\vec {y}}={\vec {x}}/\epsilon }
50:with rapidly oscillating coefficients, such as
1727:"Localization of elliptic multiscale problems"
1668:
351:Frequently, inhomogeneous materials (such as
1643:Papanicolaou, G. C.; Varadhan, S.R. (1981).
1674:Archive for Rational Mechanics and Analysis
1575:Asymptotic Analysis for Periodic Structures
332:. Under this assumption, materials such as
1492:Non-homogeneous media and vibration theory
1752:
1742:
1685:
1359:
614:
1854:; Shamaev, A.S.; Yosifian, G.A. (1991),
1767:
1652:Seria Colloq. Math. Society Janos Bolyai
189:{\displaystyle A\left({\vec {y}}\right)}
1033:The method of asymptotic homogenization
14:
1985:
1957:Braides, A.; Defranceschi, A. (1998),
1622:
904:{\displaystyle u_{\epsilon }\approx u}
1564:
1562:
1560:
1523:
1521:
1519:
1485:
1483:
1481:
1959:Homogenization of Multiple Integrals
1725:Målqvist, A.; Peterseim, D. (2014).
24:
1557:
1516:
1478:
957:{\displaystyle u_{\epsilon }\to u}
789:
764:
726:
534:
391:
370:
99:
60:
25:
2009:
1089:and posing a formal expansion in
1770:An Introduction to Γ-Convergence
156:is a very small parameter and
1902:Homogenization and Porous Media
1900:Hornung, Ulrich (Ed.). (1997),
1754:10.1090/S0025-5718-2014-02868-8
1466:Effective medium approximations
647:
1998:Partial differential equations
1761:
1718:
1662:
1636:
1616:
1598:Ostoja-Starzewski, M. (2007).
1591:
1428:
1414:
1399:
1390:
1356:
1343:
1334:
1328:
1313:
1304:
1278:
1272:
1257:
1248:
1229:
1223:
1208:
1199:
1183:
1177:
1162:
1153:
1144:
1138:
1129:
1065:
1050:
983:{\displaystyle \epsilon \to 0}
974:
948:
834:
824:
818:
809:
761:
755:
746:
602:
575:
562:
556:
547:
526:
520:
511:
497:
484:
266:
233:
217:
176:
85:
48:partial differential equations
29:Homogenization of a polynomial
13:
1:
1798:
1490:Sanchez-Palencia, E. (1980).
996:Representative Volume Element
196:is a 1-periodic coefficient:
994:material), is known as the "
964:in some appropriate norm as
317:{\displaystyle i=1,\dots ,n}
18:Homogenization (mathematics)
7:
1933:; Panasenko, G. P. (1984),
1573:; Papanicolaou, G. (1978).
1444:
10:
2014:
1731:Mathematics of Computation
688:from 1-periodic functions
26:
1914:10.1007/978-1-4612-1920-0
1778:10.1007/978-1-4612-0327-8
1704:10.1007/s00205-010-0302-1
1541:10.1007/978-94-009-2247-1
1102:{\displaystyle \epsilon }
924:{\displaystyle \epsilon }
149:{\displaystyle \epsilon }
1531:; Panasenko, G. (1989).
1471:
46:is a method of studying
27:Not to be confused with
1809:; Zhikov, V.V. (1994),
1435:
1364:
1103:
1083:
1019:
984:
958:
925:
905:
875:for this very reason.
858:
709:
679:
442:
412:
318:
280:
190:
150:
127:
1768:Dal Maso, G. (1993).
1658:. Amsterdam: 835–873.
1500:10.1007/3-540-10000-8
1436:
1365:
1104:
1084:
1020:
1018:{\displaystyle A^{*}}
985:
959:
926:
906:
859:
710:
708:{\displaystyle w_{j}}
680:
443:
441:{\displaystyle A^{*}}
413:
319:
281:
191:
151:
128:
1377:
1116:
1093:
1041:
1002:
968:
935:
915:
882:
722:
692:
455:
425:
367:
290:
200:
160:
140:
57:
1993:Asymptotic analysis
1696:2010ArRMA.198..677B
1451:Asymptotic analysis
475:
353:composite materials
330:continuum mechanics
1737:(290): 2583–2603.
1431:
1360:
1099:
1079:
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980:
954:
921:
901:
854:
705:
675:
458:
438:
408:
314:
276:
186:
146:
123:
1976:978-0-198-50246-3
1923:978-1-4612-7339-4
1550:978-94-010-7506-0
1509:978-3-540-10000-3
1461:Mosco convergence
1417:
1402:
1331:
1316:
1275:
1260:
1226:
1211:
1180:
1165:
1141:
1068:
1053:
911:for small enough
837:
821:
758:
605:
578:
559:
523:
269:
236:
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179:
93:
88:
16:(Redirected from
2005:
1979:
1953:
1931:Bakhvalov, N. S.
1926:
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1569:Bensoussan, A.;
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79:
21:
2013:
2012:
2008:
2007:
2006:
2004:
2003:
2002:
1983:
1982:
1977:
1967:Clarendon Press
1924:
1906:Springer-Verlag
1886:
1837:
1827:Springer-Verlag
1801:
1796:
1795:
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1766:
1762:
1723:
1719:
1667:
1663:
1647:
1641:
1637:
1631:(151): 188–202.
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1610:
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55:
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32:
23:
22:
15:
12:
11:
5:
2011:
2001:
2000:
1995:
1981:
1980:
1975:
1954:
1927:
1922:
1897:
1884:
1848:
1835:
1805:Kozlov, S.M.;
1800:
1797:
1794:
1793:
1786:
1760:
1717:
1680:(2): 677–721.
1661:
1635:
1615:
1608:
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873:micromechanics
869:homogenization
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836:
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826:
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346:elastic moduli
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44:homogenization
9:
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1885:0-444-88441-6
1881:
1877:
1876:North-Holland
1873:
1869:
1868:New York City
1865:
1861:
1857:
1853:
1852:Oleinik, O.A.
1849:
1846:
1842:
1838:
1836:3-540-54809-2
1832:
1828:
1824:
1823:New York City
1820:
1816:
1812:
1808:
1807:Oleinik, O.A.
1803:
1802:
1789:
1787:9780817636791
1783:
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1609:9781584884170
1605:
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1584:0-444-85172-0
1580:
1576:
1572:
1565:
1563:
1561:
1552:
1546:
1542:
1538:
1534:
1530:
1529:Bakhvalov, N.
1524:
1522:
1520:
1511:
1505:
1501:
1497:
1493:
1486:
1484:
1482:
1477:
1467:
1464:
1462:
1459:
1457:
1456:Γ-convergence
1454:
1452:
1449:
1448:
1442:
1425:
1421:
1411:
1405:
1396:
1385:
1381:
1351:
1347:
1340:
1337:
1325:
1319:
1310:
1299:
1295:
1289:
1285:
1281:
1269:
1263:
1254:
1243:
1239:
1235:
1232:
1220:
1214:
1205:
1194:
1190:
1186:
1174:
1168:
1159:
1150:
1147:
1135:
1124:
1120:
1112:
1111:
1110:
1096:
1076:
1072:
1062:
1056:
1047:
1030:
1026:
1010:
1006:
997:
991:
977:
971:
951:
943:
939:
918:
898:
895:
890:
886:
876:
874:
870:
851:
847:
841:
831:
815:
806:
802:
798:
793:
785:
782:
778:
772:
768:
752:
743:
739:
735:
730:
718:
717:
716:
700:
696:
672:
669:
666:
663:
660:
657:
654:
651:
648:
644:
639:
635:
631:
628:
623:
619:
615:
609:
599:
592:
588:
582:
572:
565:
553:
542:
538:
530:
517:
508:
501:
493:
490:
487:
480:
476:
471:
466:
463:
459:
451:
450:
449:
433:
429:
405:
402:
398:
394:
386:
382:
377:
373:
363:
362:
361:
358:
354:
349:
347:
343:
342:shear modulus
339:
335:
331:
325:
311:
308:
305:
302:
299:
296:
293:
272:
263:
257:
253:
250:
246:
240:
230:
223:
214:
207:
203:
182:
173:
167:
163:
143:
120:
117:
113:
107:
103:
95:
90:
82:
75:
71:
67:
63:
53:
52:
51:
49:
45:
41:
37:
30:
19:
1958:
1934:
1901:
1855:
1810:
1769:
1763:
1734:
1730:
1720:
1677:
1673:
1670:Berlyand, L.
1664:
1655:
1651:
1638:
1628:
1625:Mat. Sbornik
1624:
1618:
1599:
1593:
1574:
1532:
1491:
1372:
1036:
1027:
992:
877:
868:
866:
715:satisfying:
687:
420:
350:
326:
135:
43:
33:
1571:Lions, J.L.
931:, provided
36:mathematics
1987:Categories
1951:0607.73009
1894:0768.73003
1845:0838.35001
1819:Heidelberg
1799:References
355:) possess
1860:Amsterdam
1744:1110.0692
1687:0901.1463
1426:ϵ
1415:→
1400:→
1348:ϵ
1329:→
1314:→
1286:ϵ
1273:→
1258:→
1236:ϵ
1224:→
1209:→
1178:→
1163:→
1139:→
1125:ϵ
1097:ϵ
1077:ϵ
1066:→
1051:→
1011:∗
975:→
972:ϵ
949:→
944:ϵ
919:ϵ
896:≈
891:ϵ
835:→
819:→
799:⋅
790:∇
786:−
765:∇
756:→
736:⋅
727:∇
667:…
629:…
603:→
593:⋅
576:→
557:→
535:∇
521:→
481:∫
472:∗
434:∗
392:∇
387:∗
374:⋅
371:∇
306:…
267:→
234:→
218:→
177:→
144:ϵ
108:ϵ
100:∇
91:ϵ
86:→
64:⋅
61:∇
1445:See also
1712:1337370
1692:Bibcode
1025:above.
348:, etc.
40:physics
1973:
1963:Oxford
1949:
1939:Moscow
1920:
1892:
1882:
1864:London
1843:
1833:
1815:Berlin
1784:
1710:
1606:
1581:
1547:
1506:
421:where
338:solids
334:fluids
136:where
1943:Nauka
1872:Tokyo
1739:arXiv
1708:S2CID
1682:arXiv
1648:(PDF)
1472:Notes
1971:ISBN
1918:ISBN
1880:ISBN
1831:ISBN
1782:ISBN
1604:ISBN
1579:ISBN
1545:ISBN
1504:ISBN
38:and
1947:Zbl
1910:doi
1890:Zbl
1841:Zbl
1774:doi
1749:doi
1700:doi
1678:198
1629:109
1537:doi
1496:doi
286:,
34:In
1989::
1969:,
1965::
1945:,
1941::
1937:,
1916:,
1908:,
1888:,
1878:,
1874::
1870:-
1866:-
1862:-
1839:,
1829:,
1825::
1813:,
1780:.
1747:.
1735:83
1733:.
1729:.
1706:.
1698:.
1690:.
1676:.
1656:27
1654:.
1650:.
1627:.
1559:^
1543:.
1518:^
1502:.
1480:^
1441:.
1109::
990:.
344:,
336:,
324:.
42:,
1912::
1821:-
1817:-
1790:.
1776::
1757:.
1751::
1741::
1714:.
1702::
1694::
1684::
1612:.
1587:.
1553:.
1539::
1512:.
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1200:(
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1175:y
1169:,
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978:0
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