Numerical solution of coupled problems using code Agros2D
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Numerical solution of coupled problems using code Agros2D ◦ Pavel Karban · František Mach · Pavel Kus · David Pánek · Ivo Doležel
Received: 5 November 2012 / Accepted: 9 January 2013 / Published online: 26 January 2013 © Springer-Verlag Wien 2013
Abstract New code Agros2D for 2D numerical solution of coupled problems is presented. This code is based on the fully adaptive higher-order finite element method and works with library Hermes2D containing the most advanced numerical algorithms for the numerical processing of systems of second-order partial differential equations. It is characterized by several quite unique features such as work with hanging nodes of any level, multimesh technology (every physical field can be calculated on a different mesh generally varying in time) and a possibility of combining triangular, quadrilateral and curved elements. The power of the code is illustrated by three typical coupled problems. Keywords Higher-order finite element method · Coupled problems · Monolithic solution · hp-adaptivity · Hanging nodes Mathematics Subject Classification
65M60 · 68N01
◦ P. Karban (B) · F. Mach · P. Kus · D. Pánek · I. Doležel Department of Theory of Electrical Engineering, University of West Bohemia, Pilsen, Czech Republic e-mail: [email protected]
F. Mach e-mail: [email protected] ◦ P. Kus e-mail: [email protected]
D. Pánek e-mail: [email protected] I. Doležel e-mail: [email protected]
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1 Introduction Numerous physical phenomena are characterized by an interaction of several physical fields, influencing mutually one another. From a wide spectrum we can mention, for example, thermal and force effects of electromagnetic fields, changes in solid and liquid structures subject to varying temperature or pressure, or behavior of plasma in different external conditions. Let us illustrate this category of phenomena by a typical technical example–induction heating of a steel cylinder (a heat-treatment technology widely used in many industrial applications) in an inductor carrying time variable, mostly periodic current, see Fig. 1. The current passing through the inductor produces in its vicinity time-variable magnetic field that generates induced currents in the cylinder. These currents (whose density Ji (t) lag by almost 180 ◦ behind the field current density J(t)) produce in it the Joule losses that are transformed into heat. The temperature of the cylinder starts growing. We can say that the primary harmonic magnetic field in the inductor produces the secondary temperature field in the heated billet. On the other hand, it is well known that the physical properties of heated materials vary with their temperature. This also holds for electrical conductivity and magnetic permeability of steel that start decreasing. Thus, the secondary temperature field in the billet affects the primary magnetic field in the inductor. It is obvious that both physical fields influence one another. When analyzing the process in more details, we can add, moreover, that heating of the cylinder s
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