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Published in:
Micro-Electro-Mechanical-Systems (MEMS) Read chapter Read first chapter

2018 | OriginalPaper | Chapter

4. Resistance Spot Welding

Authors : Nikos I. Kavallaris, Takashi Suzuki

Published in: Non-Local Partial Differential Equations for Engineering and Biology

Publisher: Springer International Publishing

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Abstract

In the current chapter we debate a joining process called resistance spot welding which finds many applications ranging from the automobile industry to robotics. In the first part of the chapter we present the construction of a non-local mathematical model illustrating the phase transition occurs during this joining process. Since the derived model consists of a degenerate non-local parabolic equation its analytical study is rather hard due to many arising technicalities and so we appeal to a numerical approach. We then consider a time discretization scheme for solving the resulting non-local moving boundary problem. The scheme consists of solving at each time step a linear elliptic partial differential equation and then making a correction which takes into account the nonlinearity. The stability and error estimates of the developed scheme are also investigated. Finally, some numerical experiments are presented which verify the efficiency of the developed numerical algorithm, as well as demonstrate the emergent interfaces due to the degeneracy of the problem.

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previous chapter Linear Friction Welding
next chapter Gierer–Meinhardt System
Literature
1.
Antontsev, S.N., Chipot, M.: The analysis of blow-up for the thermistor problem. Sib. Math. J. 38, 827–841 (1997)CrossRef
2.
Bebernes, J., Talaga, P.: Nonlocal problems modelling shear banding. Commun. Appl. Nonlinear Anal. 3, 79–103 (1996)MathSciNetMATH
3.
Berger, A.E., Brezis, H., Rogers, J.C.W.: A numerical method for solving the problem \(u_t - \Delta f(u) = 0\). RAIRO Anal. Numer. 13, 297–312 (1979)
4.
Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Applied Mathematical Sciences. Springer, New York (1996)
5.
6.
Hömberg, D., Rocca, E.: A model for resistance welding including phase transitions and Joule heating. Math. Methods Appl. Sci. 34, 2077–2088 (2011)MathSciNetCrossRefMATH
7.
Kavallaris, N.I.: Asymptotic behaviour and blow-up for a nonlinear diffusion problem with a non-local source term. Proc. Edinb. Math. Soc. 47, 375–395 (2004)MathSciNetCrossRefMATH
8.
9.
Kavallaris, N.I., Yan, Y.: A time discretization scheme for a nonlocal degenerate problem modelling resistance spot welding. Math. Model. Nat. Phenom. 10(6), 90–112 (2015)MathSciNetCrossRefMATH
10.
Kavallaris, N.I., Lacey, A.A., Nikolopoulos, C.V., Voong, C.: Behaviour of a non-local equation modelling linear friction welding. IMA J. Appl. Math. 72, 597–616 (2007)MathSciNetCrossRefMATH
11.
Lacey, A.A.: Thermal runaway in a nonlocal problem modelling Ohmic heating, part I, model derivation and some special cases. Eur. J. Appl. Math. 6, 127–144 (1995)MATH
12.
Latos, E.A., Tzanetis, D.E.: Existence and blow-up of solutions for a non-local filtration and porous medium problem. Proc. Edinb. Math. Soc. 53, 195–209 (2010)MathSciNetCrossRefMATH
13.
Latos, E.A., Tzanetis, D.E.: Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source. Nonlinear Differ. Equ. Appl. 17, 137–151 (2010)MathSciNetCrossRefMATH
14.
Nochetto, R.H., Verdi, C.: Approximation of degenerate parabolic problems using numerical integration. SIAM J. Numer. Anal. 25, 784–814 (1988)MathSciNetCrossRefMATH
15.
Nochetto, R.H., Verdi, C.: An efficient linear scheme to approximate parabolic free boundary problems: error estimates and implementation. Math. Comput. 51, 27–53 (1988)MathSciNetCrossRefMATH
16.
Saunders, N., Li, X., Miodownik, A.P., Schillé, J-Ph.: Modelling of the thermo-physical and physical properties relevant to solidification. In: Stefanescu, D., Warren, J.A., Jolly, M.R., Krane, M.J.M. (eds.) Advanced Solidification Processes X, p. 669. TMS, Warrendale (2003)
17.
Sidi Ammi, M.R., Mul, O.: Error estimates for the Chernoff scheme to approximate a nonlocal problem. Proc. Est. Acad. Sci. Phys. Math. 56, 359–372 (2007)MathSciNetMATH
18.
Steinbach, I., Apel, M.: Multi phase field model for solid state transformation with elastic strain. Phys. D 217, 153–160 (2006)CrossRefMATH
19.
Tzanetis, D.E.: Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating. Electron. J. Differ. Equ. 11, 1–26 (2002)MathSciNetMATH
20.
Tzanetis, D.E., Vlamos, P.M.: A nonlocal problem modelling Ohmic heating with variable thermal conductivity. Nonlinear Anal. RWA 2, 443–454 (2001)MathSciNetCrossRefMATH
21.
Tzanetis, D.E., Vlamos, P.M.: Some interesting special cases of a non-local problem modelling Ohmic heating with variaable thermal conductivity. Proc. Edinb. Math. Soc. 44, 585–595 (2001)MathSciNetCrossRefMATH
22.
Vazquez, J.L.: The Porous Medium Equation: Mathematical Theory. Oxford Science Publications, Oxford (2007)MATH
23.
24.
Zel’dovich, Ya.B., Raizer, Yu.P.: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, vol. II. Academic Press, New York (1966)
Metadata
Title
Resistance Spot Welding
Authors
Nikos I. Kavallaris
Takashi Suzuki
Copyright Year
2018
Book
Non-Local Partial Differential Equations for Engineering and Biology

Print ISBN: 978-3-319-67942-6

Electronic ISBN: 978-3-319-67944-0

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-67944-0_4

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