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Boletın De La Sociedad Espa˜Nola De Matemática Aplicada Sema Grupo Bolet¶³n de la Sociedad Espanola~ de Matem¶atica Aplicada S~eMA Grupo Editor J.J. Vald¶es Garc¶³a (U. de Oviedo) E. Fern¶andez Cara (U. de Sevilla) B. Dugnol Alv¶ arez (U. de Oviedo) M. Mateos Alberdi (U. de Oviedo) C.O. Men¶endez P¶erez (U. de Oviedo) P. P¶erez Riera (U. de Oviedo) Comit¶e Cient¶³¯co E. Fern¶andez Cara (U. de Sevilla) A. Bermudez¶ de Castro (U. de Santiago) E. Casas Renter¶³a (U. de Cantabria) J.L. Cruz Soto (U. de C¶ordoba) J.M. Maz¶on Ruiz (U. de Valencia) I. Peral Alonso (U. Aut. de Madrid) J.J. Vald¶es Garc¶³a (U. de Oviedo) J.L. V¶azquez Su¶arez (U. Aut. de Madrid) L. Vega Gonz¶alez (U. del Pa¶³s Vasco) E. Zuazua Iriondo (U. Comp. de Madrid) Responsables de secciones Art¶³culos: E. Fern¶andez Cara (U. de Sevilla) Resumenes¶ de libros: F.J. Sayas Gonz¶alez (U. de Zaragoza) Noticias de S~eMA: R. Pardo San Gil (Secretaria de S~eMA) Congresos y Seminarios: J. Maz¶on Ruiz (U. de Valencia) Matem¶aticas e Industria: M. Lezaun Iturralde (U. del Pa¶³s Vasco) Educaci¶on Matem¶atica R. Rodr¶³guez del R¶³o (U. Comp. de Madrid) P¶agina web de S~eMA http://www.uca.es/sema/ Direcci¶on Editorial: Bolet¶³n de S~eMA. Dpto. de Matem¶aticas. Universidad de Oviedo. Avda. de Calvo Sotelo, s/n. 33007-Oviedo. [email protected] ISSN 1575-9822 Dep¶osito Legal: AS-1442-2002 Imprime: Grupo Bit¶acora. C/ Instituto, 17, Entresuelo. 33201 Gij¶on (Asturias) Diseno~ de portada: Ana Cernea EDITORIAL Estimados amigos: Como ya es habitual, incluimos en este mensaje nuestro m¶as cordial saludo, seguido de unas palabras de introducci¶on a un nuevo numero¶ de nuestro Bolet¶³n. En este caso, junto a las secciones habituales de anuncios, noticias, resumenes¶ de Tesis Doctorales, etc., hemos incluido tres interesantes art¶³culos de investigaci¶on, el trabajo recientemente galardonado con el IV Premio S~eMA de Divulgaci¶on de la Matem¶atica Aplicada y, ¯nalmente, un nuevo art¶³culo de Educaci¶on Matem¶atica. En la secci¶on de Noticias se recogen dos hechos relevantes en el am¶ bito de la Matem¶atica Aplicada que se han producido en los meses de septiembre y octubre. Por una parte, informaremos de la concesi¶on del Premio Nacional de Chile en Ciencias Exactas 2003 al Prof. Carlos Conca, de la Universidad de Chile, viejo conocido y colaborador de muchos socios de S~eMA. Por otro lado, recordaremos que (como probablemente ya sab¶eis) el Premio Nacional de Investigaci¶on 2003 Julio Rey Pastor de Matem¶aticas y Tecnolog¶³as de la Informaci¶on y Comunicaciones ha sido concedido a nuestro companero~ y ex- Presidente Juan Luis V¶azquez, de la Universidad Aut¶onoma de Madrid. Es para nosotros un verdadero placer y orgullo que sean reconocidos de esta forma los m¶eritos de dos investigadores de m¶aximo prestigio en nuestro ¶ambito. Muy en particular, nos congratulamos de lo acaecido en el caso de un ex-Presidente de nuestra sociedad y miembro muy activo de la misma. Las aportaciones cient¶³¯cas de Juan Luis, su decidido trabajo en favor del desarrollo de las Matem¶aticas de calidad y sus actividades de pol¶³tica cient¶³¯ca son dignas de elogio y le hacen merecedor de este prestigioso galard¶on. <Enhorabuena! Una vez m¶as, ha sido para nosotros un placer elaborar este volumen. Por supuesto, seguimos estando a vuestra disposici¶on para cualquier sugerencia que permita mejorar el resultado. Un cordial saludo, Grupo Editor boletin [email protected] 5 Bol. Soc. Esp. Mat. Apl. no26(2003), 7{34 Simulaci¶on num¶erica de las deformaciones termomec¶anicas de una placa de aluminio durante el proceso de colada¤ P. Barral y P. Quintela Departamento de Matem¶atica Aplicada. Universidade de Santiago de Compostela [email protected], [email protected] Resumen En este trabajo se resume la investigaci¶on realizada por las autoras en el campo de la simulaci¶on termomec¶anica de coladas de aluminio. Se propone un modelo matem¶atico para simular las deformaciones de la placa durante su solidi¯caci¶on y se presentan resultados de existencia y regularidad de soluci¶on. A continuaci¶on, se aborda la resoluci¶on num¶erica del problema. Para implementar num¶ericamente la presi¶on metalost¶atica ejercida por el aluminio l¶³quido sobre la placa se utiliza un m¶etodo de dominio ¯cticio, que se justi¯ca mediante un an¶alisis asint¶otico del problema. Para el tratamiento de las no linealidades del problema se aplica el m¶etodo de Bermudez-Moreno¶ combinado con un m¶etodo de Newton generalizado. El problema se discretiza en espacio mediante un m¶etodo de elementos ¯nitos y en tiempo con un esquema impl¶³cito. Finalmente, se presentan resultados num¶ericos para una colada real. Palabras clave: colada, contacto, viscoelasticidad, dominio ¯cticio, desarrollos asint¶oticos, operadores maximales mon¶otonos, elementos ¯nitos. Clasi¯caci¶on por materias AMS: 35, 65, 68, 74. 1 El proceso de producci¶on de aluminio El proceso de producci¶on de aluminio, esquematizado en la Figura 1, es muy complejo y se puede dividir en las siguientes etapas: ¤ Investigaci¶on ¯nanciada por: ALCOA-INESPAL S.A. (A Coruna),~ CICYT (proyecto MAT96-0715), CYCIT-FEDER (proyecto DPI2001-29098), IGAPE y Xunta de Galicia (proyectos PGIDIT02PXIC20701PN, PGIDIT00PXI20701PR). Fecha de recepci¶on: 24 de marzo de 2003 7 8 P. Barral y P. Quintela Figura 1: Esquema del proceso de producci¶on de aluminio Extracci¶on de bauxita La bauxita se extrae directamente de distintos yacimientos y se transporta hasta la estaci¶on de trituraci¶on. All¶³ se traslada hasta el molino que tiene como funci¶on reducir el material a una granulometr¶³a menor de 100 mm para su transporte y mejor manejo. Obtenci¶on de alumina¶ El proceso de obtenci¶on de alumina¶ de grado metalurgico¶ a partir de la bauxita consiste en la reducci¶on del tamano~ de part¶³culas de bauxita, la extracci¶on de la alumina¶ contenida en ella por medio de la digesti¶on con soda c¶austica y la separaci¶on de las impurezas que acompanan~ a la alumina¶ a ¯n de prevenir la contaminaci¶on del producto ¯nal. Despu¶es de varios ¯ltrados, el l¶³quido, rico en alumina,¶ se somete a una fase de enfriamiento que lo prepara para la fase de precipitaci¶on, donde se obtienen los cristales de alumina¶ hidratada. El proceso concluye en los calcinadores, constituidos por grandes hornos que eliminan la humedad de la alumina¶ hidratada para obtener la alumina¶ de grado metalurgico,¶ producto ¯nal dispuesto para ser utilizado en las cubas electrol¶³ticas. Electrolisis La planta de reducci¶on o de cubas electrol¶³ticas es el coraz¶on del proceso de producci¶on del aluminio. All¶³ se disuelve la alumina¶ en un medio electrol¶³tico de criolita fundida a 950oC aproximadamente, descomponi¶endola en sus dos elementos b¶asicos: ox¶³geno y aluminio. El Simulaci¶on num¶erica de un proceso de colada 9 Figura 2: El aluminio l¶³quido viaja desde el horno hasta la mesa de colada, donde solidi¯ca. El lingote obtenido sufre posteriormente un proceso de extrusi¶on, tre¯lado o laminaci¶on. ox¶³geno es atra¶³do por los ¶anodos hacia la parte superior de la celda, quemado y convertido en di¶oxido de carbono en el ano¶ do. El aluminio, a su vez, se precipita hacia el fondo y se extrae fundido por succi¶on hacia el crisol, para ser enviado a la planta de fundici¶on. Colada En la planta de fundici¶on se preparan las aleaciones de aluminio atendiendo a criterios comerciales. El metal, procedente de las cubas de electrolisis, se vac¶³a en los hornos de retenci¶on donde se le anaden~ aditivos como titanio, magnesio, cobre o hierro, para preparar las distintas aleaciones. El metal l¶³quido es sometido a diversos an¶alisis y controles de calidad dentro de los hornos, para luego ser transferido a la mesa de colada. En el proceso de colada se vierte el metal l¶³quido a trav¶es de canales a los diferentes moldes que son enfriados por agua (ver Figura 5). El producto ¯nal es aluminio primario en forma de lingote para tre¯lado, cilindros para extrusi¶on y placas para laminaci¶on. Tre¯lado, extrusi¶on y laminaci¶on Finalmente, mediante tre¯lado, extrusi¶on o laminaci¶on, se obtienen bobinas de alambre, barras o l¶aminas, que son enviadas a los clientes, quienes las transformar¶an en recipientes, ventanas, piezas de autom¶oviles, etc. (ver Figura 2). 10 P. Barral y P. Quintela 2 Proceso de colada En este trabajo nos centramos en el proceso de colada. Como ya hemos mencio- nado, al llegar a la mesa de colada el aluminio se vierte en el molde o lingotera, donde un sistema de refrigeraci¶on por agua lo hace solidi¯car. Se forma as¶³ una corteza, correspondiente al aluminio ya solidi¯cado, que sirve de recept¶aculo al aun¶ l¶³quido. Al mismo tiempo, una plataforma m¶ovil, situada en la parte infe- rior y llamada falso fondo, va descendiendo de forma continua a una velocidad aproximada de 1mm/s, mientras sobre la piscina de metal sigue llegando nuevo aluminio procedente del horno. Para m¶as informaci¶on sobre el proceso industrial de solidi¯caci¶on del aluminio se puede consultar D¶³az [15] o Drezet-Plata [17]. Existen dos tipos de tecnolog¶³as para la colada de aluminio: la colada cl¶asica, en la cual el aluminio l¶³quido se vierte en un molde f¶³sico y la electromagn¶etica, en la que el aluminio se con¯na mediante un campo magn¶etico.
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