Induction hardening of steel with restrained Joule heating and nonlinear law for magnetic induction field: Solvability

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• 作者：Jaroslav Chovan ^{jaroslav.chovan@ugent.be" class="auth_mail" title="E-mail the corresponding author} ; Mariá ; n Slodička ; ^{marian.slodicka@ugent.be" class="auth_mail" title="E-mail the corresponding author} ;
• 关键词：35K61 ; 35Q61 ; 35Q79 ; 65M12
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：311
• 期：Complete
• 页码：630-644
• 全文大小：478 K

文摘

We study a coupled system of Maxwell’s equations with nonlinear heat equation. We start with derivation of a mathematical model assuming a nonlinear dependency of magnetic field source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716303764&_mathId=si1.gif&_user=111111111&_pii=S0377042716303764&_rdoc=1&_issn=03770427&md5=9e365f4c208845d1acf1bdf282c8b44f">source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0377042716303764-si1.gif">$H$ on magnetic induction source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716303764&_mathId=si2.gif&_user=111111111&_pii=S0377042716303764&_rdoc=1&_issn=03770427&md5=959c62bc50785d894c438503abfaab46">source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0377042716303764-si2.gif">$B$. Moreover the electric conductivity method=retrieve&_eid=1-s2.0-S0377042716303764&_mathId=si3.gif&_user=111111111&_pii=S0377042716303764&_rdoc=1&_issn=03770427&md5=ba1f87e62c9230cfc22b106cbd9c4282" title="Click to view the MathML source">σ$\sigma$ is supposed to be temperature reliant. The coupling between the thermal and electromagnetic equations is provided via the Joule heating term and method=retrieve&_eid=1-s2.0-S0377042716303764&_mathId=si3.gif&_user=111111111&_pii=S0377042716303764&_rdoc=1&_issn=03770427&md5=ba1f87e62c9230cfc22b106cbd9c4282" title="Click to view the MathML source">σ$\sigma$. Next we use time discretization based on the Rothe’s method to provide energy estimates for discretized system. We prove the existence of a weak solution source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716303764&_mathId=si5.gif&_user=111111111&_pii=S0377042716303764&_rdoc=1&_issn=03770427&md5=137857137078c91de3e6134e6b4103d4">source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0377042716303764-si5.gif">$\{u,B\}$ to this coupled system with controlled Joule heating term.