COMPARISON OF NINE THREE-DIMENSIONAL NUMERICAL METHODS FOR THE SOLUTION OF THE HEAT DIFFUSION EQUATION

1985 ◽  
Vol 8 (3) ◽  
pp. 281-298 ◽  
Author(s):  
Jules Thibault
Keyword(s):  
Numerical Methods ◽  
Diffusion Equation ◽  
Three Dimensional ◽  
Heat Diffusion ◽  
Heat Diffusion Equation

scholarly journals The Effect of Multigrid Parameters in a 3D Heat Diffusion Equation

2018 ◽  
Vol 23 (1) ◽  
pp. 213-221 ◽  
Author(s):  
F. De Oliveira ◽  
S.R. Franco ◽  
M.A. Villela Pinto
Keyword(s):  
Diffusion Equation ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Heat Diffusion ◽  
Algebraic Equations ◽  
Central Difference ◽  
Heat Diffusion Equation ◽  
Cpu Time ◽  
Correction Scheme ◽  
Geometric Multigrid Method

AbstractThe aim of this paper is to reduce the necessary CPU time to solve the three-dimensional heat diffusion equation using Dirichlet boundary conditions. The finite difference method (FDM) is used to discretize the differential equations with a second-order accuracy central difference scheme (CDS). The algebraic equations systems are solved using the lexicographical and red-black Gauss-Seidel methods, associated with the geometric multigrid method with a correction scheme (CS) and V-cycle. Comparisons are made between two types of restriction: injection and full weighting. The used prolongation process is the trilinear interpolation. This work is concerned with the study of the influence of the smoothing value (v), number of mesh levels (L) and number of unknowns (N) on the CPU time, as well as the analysis of algorithm complexity.

Fast Fluid Thermodynamics Simulation by Solving Heat Diffusion Equation

2021 ◽  
Vol 11 (04) ◽  
pp. 1-11
Author(s):  
Wanwan Li
Keyword(s):  
Diffusion Equation ◽  
Real Time ◽  
Stokes Equations ◽  
Fluid Simulation ◽  
Heat Diffusion ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Heat Diffusion Equation ◽  
Acceleration Techniques ◽  
Speed Up

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.

Computer simulation of target link explosion in laser programmable redundancy for silicon memory

1986 ◽  
Vol 1 (2) ◽  
pp. 368-381 ◽  
Author(s):  
L.M. Scarfone ◽  
J.D. Chlipala
Keyword(s):  
Thermal Evolution ◽  
Three Dimensional ◽  
Heat Diffusion ◽  
Dielectric Films ◽  
Threshold Temperature ◽  
Heat Diffusion Equation ◽  
Heating Effects ◽  
Transparent Dielectric ◽  
Difference Form ◽  
Pulse Energies

Pulses of Q-switched Nd-YAG radiation have been used to remove polysilicon target links during the implementation of laser programmable redundancy in the fabrication of silicon memory. The link is encapsulated by transparent dielectric films that give rise to important optical interference effects modifying the laser flux absorbed by the link and the silicon substrate. Estimates of these effects are made on the basis of classical plane-wave procedures. Thermal evolution of the composite structure is described in terms of a finite-difference form of the three-dimensional heat diffusion equation with a heat generation rate having a Gaussian spatial distribution of intensity and temporal shapes characteristic of commercial lasers. Temperature-dependent thermal diffusivity and melting of the polysilicon link are included in the computer modeling. The calculations account for the discontinuous change in the link absorption coefficient at the transition temperature. A threshold temperature and corresponding pressure, sufficiently high to rupture the dielectric above the link and initiate the removal process, are estimated by treating the molten link as a hard-sphere fluid. Numerical results are presented in the form of three-dimensional temperature distributions for 1.06 and 0.53 μm radiation with pulse energies 3.5 and 0.15μJ, respectively. Similarities and differences between heating effects produced by long (190 ns FWHM/740 ns duration) and short (35 ns FWHM/220 ns duration) pulses are pointed out.

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

2010 ◽  
Vol 14 (2) ◽  
pp. 291-316 ◽  
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.

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

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.

Group Transformations and the Nonlinear Heat Diffusion Equation

1981 ◽  
Vol 40 (2) ◽  
pp. 191-207 ◽  
Author(s):  
A. Munier ◽  
J. R. Burgan ◽  
J. Gutierrez ◽  
E. Fijalkow ◽  
M. R. Feix
Keyword(s):  
Diffusion Equation ◽  
Heat Diffusion ◽  
Heat Diffusion Equation

Modeling of Titanium Oxide Layer Growth Produced by Fiber Laser Beam

2013 ◽  
Vol 336 ◽  
pp. 11-18 ◽  
Author(s):  
Farida Hamadi ◽  
El Hachemi Amara ◽  
Djamila Bennaceur-Doumaz ◽  
R. Boutaka ◽  
H. Kellou ◽  
...  
Keyword(s):  
Temperature Field ◽  
Laser Beam ◽  
Film Growth ◽  
Three Dimensional ◽  
Layer Growth ◽  
Heat Diffusion ◽  
Metallic Surface ◽  
Heat Diffusion Equation ◽  
Computational Procedures ◽  
Volume Method

In this paper, we study the oxidation process during the heating of a titanium metallic surface by a Nd-YAG fiber pulsed laser beam under air environment. For this, we adopted an approach that considers a three-dimensional heat diffusion model coupled with an oxidation parabolic law (oxidation kinetics). The heat diffusion equation solved numerically, gives the temperature field. The oxide film growth is simulated by implementing a dynamic mesh technique. We developed computational procedures UDFs (User Defined Function) running interactively with the Fluent fluid dynamics software [ that implements the finite volume method. These UDFs are developed to insert the oxidation law, the temperature field, the specific boundary conditions and the mesh deformation into the calculation.

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