The Effectiveness of the Finite Differences Method on Physical and Medical Images Based on a Heat Diffusion Equation

In mechanical engineering educations, simulating fluid thermodynamics is rather helpful for students to understand the fluid’s natural behaviors. However, rendering both high-quality and realtime simulations for fluid dynamics are rather challenging tasks due to their intensive computations. So, in order to speed up the simulations, we have taken advantage of GPU acceleration techniques to simulate interactive fluid thermodynamics in real-time. In this paper, we present an elegant, basic, but practical OpenGL/SL framework for fluid simulation with a heat map rendering. By solving Navier-Stokes equations coupled with the heat diffusion equation, we validate our framework through some real-case studies of the smoke-like fluid rendering such as their interactions with moving obstacles and their heat diffusion effects. As shown in Fig. 1, a group of experimental results demonstrates that our GPU-accelerated solver of Navier-Stokes equations with heat transfer could give the observers impressive real-time and realistic rendering results.

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Heat-balance integral to fractional (half-time) heat diffusion sub-model

Thermal Science ◽

10.2298/tsci1002291h ◽

2010 ◽

Vol 14 (2) ◽

pp. 291-316 ◽

Cited By ~ 29

Author(s):

Jordan Hristov

Keyword(s):

Diffusion Equation ◽

Heat Balance ◽

Integral Method ◽

Time Derivative ◽

Heat Diffusion ◽

Half Time ◽

Heat Diffusion Equation ◽

Parabolic Profile ◽

Heat Balance Integral Method ◽

The Heat Balance

The fractional (half-time) sub-model of the heat diffusion equation, known as Dirac-like evolution diffusion equation has been solved by the heat-balance integral method and a parabolic profile with unspecified exponent. The fractional heat-balance integral method has been tested with two classic examples: fixed temperature and fixed flux at the boundary. The heat-balance technique allows easily the convolution integral of the fractional half-time derivative to be solved as a convolution of the time-independent approximating function. The fractional sub-model provides an artificial boundary condition at the boundary that closes the set of the equations required to express all parameters of the approximating profile as function of the thermal layer depth. This allows the exponent of the parabolic profile to be defined by a straightforward manner. The elegant solution performed by the fractional heat-balance integral method has been analyzed and the main efforts have been oriented towards the evaluation of fractional (half-time) derivatives by use of approximate profile across the penetration layer.

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Sub-Continuum Simulations of Heat Conduction in Silicon-on-Insulator Transistors

10.1115/imece1999-1061 ◽

1999 ◽

Author(s):

Per G. Sverdrup ◽

Y. Sungtaek Ju ◽

Kenneth E. Goodson

Keyword(s):

Diffusion Equation ◽

Free Path ◽

Temperature Rise ◽

Mean Free Path ◽

Channel Length ◽

Silicon On Insulator ◽

Heat Diffusion ◽

Boltzmann Transport ◽

Heat Diffusion Equation ◽

Silicon Devices

Abstract The temperature rise in compact silicon devices is predicted at present by solving the heat diffusion equation based on Fourier’s law. The validity of this approach needs to be carefully examined for semiconductor devices in which the region of strongest electronphonon coupling is narrower than the phonon mean free path, Λ, and for devices in which Λ is comparable to or exceeds the dimensions of the device. Previous research estimated the effective phonon mean free path in silicon near room temperature to be near 300 nm, which is already comparable with the minimum feature size of current generation transistors. This work numerically integrates the phonon Boltzmann transport equation (BTE) within a two-dimensional Silicon-on-Insulator (SOI) transistor. The BTE is coupled with the classical heat diffusion equation, which is solved in the silicon dioxide layer beneath a transistor with a channel length of 400 nm. The sub-continuum simulations yield a peak temperature rise that is 159 percent larger than predictions using only the classical heat diffusion equation. This work will facilitate the development of simpler calculation strategies, which are appropriate for commercial device simulators.

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