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# Solving Fully Three-Dimensional Microscale Dual Phase Lag Problem Using Mixed-Collocation, Finite Difference Discretization

[+] Author and Article Information
Alaeddin Malek, Zahra Kalateh Bojdi, Parisa Nuri Niled Golbarg

Department of Applied Mathematics, Faculty of Mathematical Sciences,  Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iranmala@modares.ac.ir

J. Heat Transfer 134(9), 094504 (Jul 02, 2012) (6 pages) doi:10.1115/1.4006271 History: Received July 24, 2011; Revised February 19, 2012; Published July 02, 2012; Online July 02, 2012

## Abstract

In the present work, we investigate laser heating of nanoscale thin-films irradiated in three dimensions using the dual phase lag (DPL) model. A numerical solution based on mixed-collocation, finite difference method has been employed to solve the DPL heat conduction equation. Direct substitution in the model transforms the differential equation into a linear system of equations in which related system is solved directly without preconditioning. Consistency, stability, and convergence of the proposed method based on a mixed-collocation, finite difference approximation are proved, and numerical results are presented. The general form of matrices and their corresponding eigenvalues are presented.

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## Figures

Figure 4

Temperature distribution at t=  0.5 ps for N1  = N2  = N3  =  16, at Δt=  0.5 fs

Figure 3

Temperature distribution at t=  0.25 ps for N1  = N2  = N3  =  16, at Δt=  0.5 fs

Figure 2

(a) Graph of temperature and (b) normalized temperature change at x= y=  250 nm, and z=  0 using three different meshes and a time increment of 0.5 fs

Figure 1

Eigenvalue of matrix PN at Δt=  0.5 fs for (a) N1  = N2  = N3  =  10 and (b) N1  = N2  = N3  =  12

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