Numerical analysis of a transient non-linear axisymmetric eddy current model. (October 2015)

• Record Type: Journal Article
• Authors: Bermúdez, Alfredo; Gómez, Dolores; Rodríguez, Rodolfo; Venegas, Pablo
• Is Part Of: Computers & mathematics with applications. Volume 70:issue 8(2015)
• Page Start: 1984
• Page End: 2005
• Publication Date: 2015-10
• Subjects: Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2015.08.017
• Languages: English
• ISSNs: 0898-1221

Abstract

<abstract xml:lang="en" abstract-type="author" id="a000005"> <title id="st000005">Abstract</title> <sec> <p id="sp000030">This paper deals with the numerical solution of an axisymmetric transient eddy current problem in a conductive non-linear magnetic media. This means that the relation between the magnetic field and the magnetic induction (i.e., the so-called <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2svh1fx6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si29.gif" display="inline" overflow="scroll" id="d13e576" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi><mml:mtext>–</mml:mtext><mml:mi>H</mml:mi></mml:math></alternatives></inline-formula> curve) is non-linear. We analyze a weak formulation of the resulting problem in the axisymmetric case, with the source term given by means of a non-homogeneous Dirichlet boundary condition. For its numerical approximation, we propose a fully discrete scheme based on a finite element method combined with a backward Euler time discretization. We establish its well-posedness and derive error estimates in appropriate norms for the proposed scheme. In particular, we obtain an <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2svgr1xp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si30.gif" display="inline" overflow="scroll" id="d13e584" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>L</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula> rate of convergence of order <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2svhbcgb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si31.gif" display="inline" overflow="scroll" id="d13e594" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>h</mml:mi><mml:mo>+</mml:mo><mml:mi>Δ</mml:mi><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> without assuming any additional regularity of the solution. Moreover, under appropriate smoothness assumptions, we also prove an <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2svgr1xp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si30.gif" display="inline" overflow="scroll" id="d13e611" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>L</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>-like rate of convergence of order <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgj2svh00bm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si33.gif" display="inline" overflow="scroll" id="d13e621" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>Δ</mml:mi><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. Finally, some numerical results, which confirm the theoretically predicted behavior of the method, are reported.</p> </sec> </abstract>