Numerical Solution for a Transient Temperature Distribution on a Finite Domain Due to a Dithering or Rotating Laser Beam

Numerical Solution for a Transient Temperature Distribution on a Finite Domain Due to a Dithering or Rotating Laser Beam

Tsuwei Tan (Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, USA) and Hong Zhou (Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, USA)
DOI: 10.4018/ijoris.2013100102

Abstract

The temperature distribution due to a rotating or dithering Gaussian laser beam on a finite body is obtained numerically. The authors apply various techniques to solve the nonhomogeneous heat equation in different spatial dimensions. The authors’ approach includes the Crank-Nicolson method, the Fast Fourier Transform (FFT) method and the commercial software COMSOL. It is found that the maximum temperature rise decreases as the frequency of the rotating or dithering laser beam increases and the temperature rise induced by a rotating beam is smaller than the one induced by a dithering beam. The authors’ numerical results also provide the asymptotic behavior of the maximum temperature rise as a function of the frequency of a rotating or dithering laser beam.
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1-D Mathematical Modeling

Consider a laser beam hitting a 1-D rod with finite length ijoris.2013100102.m01. The beam moves along the rod which is insulated at the two endpoints. Mathematically, the temperature distribution of the rod can be modeled by the nonhomogeneous heat equation:

ijoris.2013100102.m02
(1) where ijoris.2013100102.m03 denotes the temperature rise of the rod at position ijoris.2013100102.m04 and time ijoris.2013100102.m05, ijoris.2013100102.m06 is the thermal diffusivity of the rod, ijoris.2013100102.m07 is the thermal conductivity, and ijoris.2013100102.m08 is the energy distribution of the moving laser beam. In the case of a dithering laser beam shown in Figure 1 (1), ijoris.2013100102.m09 can be expressed as
ijoris.2013100102.m10
(2)
Figure 1.

(1) A dithering laser beam on a 1-D rod. (2) 1-D temperature distribution along the rod from various numerical methods. (3) 1D maximum temperature rise of steel AISI 4340 versus frequency of the dithering laser beam. (4) The curve in (3) is well approximated by the function ijoris.2013100102.m46.

ijoris.2013100102.f01
where ijoris.2013100102.m11 is the position of the dithering Gaussian beam, ijoris.2013100102.m12 is the initial position of the laser beam, ijoris.2013100102.m13 is the intensity of the laser beam, ijoris.2013100102.m14 is the effective radius of the laser beam, and ijoris.2013100102.m15is a constant used for the Gaussian model. The initial condition for ijoris.2013100102.m16 is zero which assumes that the rod has the same temperature as the ambient initially. The boundary conditions impose that the rod is insulated at the two ends:

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