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Fully discrete T ‐ ψ finite element method to solve a nonlinear induction hardening problem
Author(s) -
Kang Tong,
Wang Ran,
Zhang Huai
Publication year - 2021
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22540
Subject(s) - finite element method , mathematics , nonlinear system , mathematical analysis , maxwell's equations , electromagnetic induction , scalar (mathematics) , magnetic potential , induction hardening , mixed finite element method , physics , geometry , electromagnetic coil , residual stress , materials science , quantum mechanics , composite material , thermodynamics
We study an induction hardening model described by Maxwell's equations coupled with a heat equation. The magnetic induction field is assumed a nonlinear constitutional relation and the electric conductivity is temperature‐dependent. The T ‐ ψ method is to transform Maxwell's equations to the vector–scalar potential formulations and to solve the potentials by means of the finite element method. In this article, we present a fully discrete T ‐ ψ finite element scheme for this nonlinear coupled problem and discuss its solvability. We prove that the discrete solution converges to a weak solution of the continuous problem. Finally, we conclude with two numerical experiments for the coupled system.