Como dito no cap´ıtulo introdut´orio e aqui reafirmado, o presente texto n˜ao tem a pretens˜ao de prover uma vis˜ao aprofundada nem muito menos que esgote o tema. As referˆencias citadas ao longo do texto ( e aquelas citadas no interior destes) podem servir de ponto de partida para aqueles que desejarem uma vis˜ao mais detalhada desta desafiante ´area da modelagem multiescala.
4.6. OBSERVAC¸ ˜OES FINAIS 47 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1
Fig. 4.19: Gr´aficos de u e de sua aproxima¸c˜ao por elementos finitos, com = 1/2
4.6. OBSERVAC¸ ˜OES FINAIS 48 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 Solu¸c˜ao Exata
Solu¸c˜ao por elementos finitos multiescala
Fig. 4.20: Gr´aficos de u e de sua aproxima¸c˜ao por elementos finitos multiescala,
4.6. OBSERVAC¸ ˜OES FINAIS 49
curso despertar o interesse para a ´area de modelagem multiescala voltada para aplica¸c˜oes que envolvem novos materiais aplicados a sistemas de engenharia. O que deve ter ficado claro ao longo desta exposi¸c˜ao ´e que se est´a diante de uma ´area onde a multidisciplinaridade se faz importante a cada passo. Em particular, face ao contexto em que este texto vem sendo explorado, torna-se fundamental destacar a importˆancia da participa¸c˜ao da matem´atica aplicada. O cap´ıtulo 4, no qual t´ecnicas s˜ao detalhadas, deixa isso muito claro.
Desta forma os autores esperam estar dando sua contribui¸c˜ao no sentido de motivar a forma¸c˜ao de n´ucleos e grupos multidisciplinares para atuar na ´area que, sem sombra de d´uvida, jogar´a um papel cada vez mais importante no mundo da alta tecnologia contemporˆanea.
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´Indice
Coercividade 35 Continuidade 35 Deforma¸c˜oes 11,13
Desigualdade de Poincar´e 35 Espa¸cos de Fun¸c˜oes
L224
H1 28
El´astico 13
Elementos Finitos 28–46
Elementos Finitos Lineares 28–36, 44– 46
Elementos Finitos Multiescala 36– 44, 46
Malha 28 Equil´ıbrio 7 Estimativas
de regularidade 36
de erro de modelagem por homo- geneiza¸c˜ao 24
de erro de modelagem por ele- mentos finitos lineares 36 de erro de modelagem por ele-
mentos finitos multiescala 40, 42 Formas Fraca 28
Formas Bilineares 30
Fun¸c˜oes Lineares por Partes 28 Homogeneiza¸c˜ao 24, 24, 42, 43
Lema de Cea 35, 39, 40 42 Multiescala 15, 22, 36 Modelo Unidimensional 21 Residual Free Bubbles 44