The method of images in thermoelasticity with an application to wafer heating

Daniel W.M. Veldman*, Rob H.B. Fey, Hans Zwart, Marc M.J. van de Wal, Joris D.B.J. van den Boom, Henk Nijmeijer

*Corresponding author for this work

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Abstract

The well-known method of images relates the solution of the heat equation on (Formula presented.) (typically n = 2 or n = 3) to the solution of the heat equation on certain spatial subdomains Ω of (Formula presented.) By reformulating the method of images in terms of a convolution kernel, two novel extensions are obtained in this paper. First, the method of images is extended from thermal problems to thermoelastic problems, that is, it is demonstrated how the heat-induced deformations on (Formula presented.) can be related to the heat-induced deformations on certain subdomains Ω of (Formula presented.) Secondly, an explicit expression for the convolution kernel for the disk is obtained. This enables the application of the method of images to circular domains to which it could not be applied before. The two obtained extensions lead to a computationally efficient simulation method for repetitive heat loads on a disk. In a representative simulation example of wafer heating, the proposed method is more than ten times faster than a conventional Finite Element approach.

Original languageEnglish
Pages (from-to)970-1010
Number of pages41
JournalJournal of Thermal Stresses
Volume44
Issue number8
Early online date7 Jul 2021
DOIs
Publication statusPublished - 3 Aug 2021

Keywords

  • Analytical solutions
  • boundary conditions
  • disk
  • repetitive heat sources
  • thermal loading

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