Abstract:
In recent years, the use of phase field models to predict transient phenomena occurring during phase transitions has become increasingly popular. Phase field approaches are especially appropriate to account for physical phenomena that occur on the mesoscopic scale, like twinning, precipitation, thermomechanical micro-stresses, crack and pattern formation, dendritic growth. But also macroscopic effects like hysteresis can be modeled in this way.
In this series of lectures, the general technique of phase field modeling will be introduced, and some important facets of the large body of available mathematical results will be reported, both concerning analysis and numerical approaches. We will also present some "hard" real-world applications, which originated from industrial projects that are currently being investigated by different research teams at the Weierstrass Institute (WIAS) in Berlin.
It is planned to touch upon the following topics during the lectures:

• Phases and phase transitions and order parameters
• Basic thermodynamic concepts
• General scheme of phase field modeling (nonconserved order parameter case)
• The relaxed Stefan problem
• 1D-modeling of Shape Memory Alloys
• Surface hardening of steel
• Nonlocal phase separation and imaging (conserved order parameter case)
• Phase separation in solder alloys

Some References:

• E. Bonetti, P. Colli, W. Dreyer, G. Gilardi, G. Schimperna, J. Sprekels: On a model for phase separation in binary alloys driven by mechanical effects. Physica D 165, 48-65 (2002)
• M. Brokate, J. Sprekels: "Hysteresis and Phase Transitions", Applied Mathematical Sciences Vol. 121, Springer, New York 1996
• P. Colli, P. Krejci, E. Rocca, J. Sprekels: Nonlinear evolution inclusions arising from phase change models. WIAS Preprint No. 974, Berlin 2004 (submitted)
• W. Dreyer, W. H. Müller: A study of the coarsening in tin/lead solders. Internat. J. Solids Structures 37, 3841-3871 (2000)
• H. Gajewski: On a nonlocal model of non-isothermal phase separation. Adv. Math. Sci. Appl. 12, 569-586 (2002)
• H. Gajewski, K. Gärtner: A dissipative discretization scheme for a nonlocal phase segregation model. Z. Angew. Math. Mech. 85, 815-822 (2005)
• H. Gajewski, J. Griepentrog: A descent method for the free energy of multicomponent systems. To appear in: Discr. Cont. Dyn. Syst. 15 (2006)
• D. Hömberg: A mathematical model for induction hardening including mechanical effects. Nonlinear Anal. Real World Appl. 5, 55-90 (2004)
• D. Hömberg, J. Sokolowski: Optimal shape design of inductor coils for surface hardening. SIAM J. Control Optimiz. 42, 1087-1117 (2003)
• D. Hömberg, S. Volkwein:
• Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition. Math. Comp. Model. 37, 1003-1028 (2003)
• D. Hömberg, W. Weiss: PID-control of laser surface hardening of steel. WIAS Preprint No. 876, Berlin 2003
• P. Krejci, E. Rocca, J. Sprekels: Nonlocal temperaure-dependent phase-field models for non-isothermal phase transitions. WIAS Preprint No. 1006, Berlin 2005 (submitted)
• P. Krejci, J. Sprekels: Phase-field models with hysteresis. J. Math. Anal. Appl. 252, 198-219 (2000)
• P. Krejci, J. Sprekels: Nonlocal phase-field models for non-isothermal phase transitions and hysteresis. Adv. Math. Sci. Appl. 14, 593-612 (2004)
• P. Krejci, J. Sprekels, S. Zheng: Asymptotic behaviour for a phase-field system with hysteresis. J. Diff. Eq. 175, 88-107 (2001)
• J. Sprekels, S. Zheng: Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions. J. Math. Anal. Appl. 279, 97-110 (2003)
• A. Visintin: "Models of Phase Transitions", Birkhäuser, Boston 1996

Orario del Corso
Le lezioni svolgeranno presso il Dipartimento di Matematica, Università di Roma "La Sapienza", P. Aldo Moro 2, secondo il seguente orario:

Mer	26,	15.00-17.00	Aula di Consiglio
Giov	27,	11.00-13.00	Aula di Consiglio
Ven	28,	10.00-12.00	Aula C

Le persone interessate sono invitate a registrarsi inviando un mail a falcone@mat.uniroma1.it entro il 15 Aprile.

Tuesday, 28-Mar-2006 11:48:08 CEST