scholarly journals Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and $ 1-d $ heat diffusion equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Abdallah Benabdallah ◽  
Mohsen Dlala
Keyword(s):  
Mixed Boundary ◽  
Nonlinear Ordinary Differential Equation ◽  
Coupled Systems ◽  
Heat Diffusion ◽  
Exponential Stabilization ◽  
Heat Diffusion Equation ◽  
Infinite Dimensional ◽  
Boundary State ◽  
Linear Ode ◽  
Nonlinear Ode

<p style='text-indent:20px;'>In this paper, we solve the problem of rapid exponential stabilization for coupled nonlinear ordinary differential equation (ODE) and <inline-formula><tex-math id="M2">\begin{document}$ 1-d $\end{document}</tex-math></inline-formula> unstable linear heat diffusion. The control acts at a boundary of the heat domain and the heat equation enters in the ODE by Dirichlet connection. We show that the infinite dimensional backstepping transformation introduced recently for stabilization of coupled linear ODE-PDE can deal with a nonlinear ODE and obtain a global stabilization result. Our result is innovative and no similar result can be found in the literature as it combines the three following factors, i) nonlinear term in the ODE subsystem, ii) unstable PDE subsystem and iii) mixed boundary condition. Not only this, the techniques used in this work can provide answers to fundamental questions, such as the stabilization of coupled systems where both subsystems may contain nonlinear terms.</p>

Mapping Thickness Dependent Thermal Conductivity of GaN

2016 ◽  
Vol 138 (2) ◽  
Author(s):  
Elbara Ziade ◽  
Jia Yang ◽  
Gordie Brummer ◽  
Denis Nothern ◽  
Theodore Moustaks ◽  
...  
Keyword(s):  
Thermal Conductivity ◽  
Pump Beam ◽  
Continuous Wave ◽  
High Electron Mobility Transistors ◽  
Heat Diffusion ◽  
Phase Lag ◽  
High Electron ◽  
Heat Diffusion Equation ◽  
Phase Images ◽  
Gan Film

Frequency domain thermoreflectance (FDTR) is used to create quantitative maps of thermal conductivity and thickness for a thinning gallium nitride (GaN) film on silicon carbide (SiC). GaN was grown by molecular beam epitaxy on a 4H-SiC substrate with a gradient in the film thickness found near the edge of the chip. The sample was then coated with a 5 nm nickel adhesion layer and a 85 nm gold transducer layer for the FDTR measurement. A piezo stage raster scans the sample to create phase images at different frequencies. For each pixel, a periodically modulated continuous-wave laser (the red pump beam) is focused to a Gaussian spot, less than 2 um in diameter, to locally heat the sample, while a second beam (the green probe beam) monitors the surface temperature through a proportional change in the reflectivity of gold. The pump beam is modulated simultaneously at six frequencies and the thermal conductivity and thickness of the GaN film are extracted by minimizing the error between the measured probe phase lag at each frequency and an analytical solution to the heat diffusion equation in a multilayer stack of materials. A scanning electron microscope image verifies the thinning GaN. We mark the imaged area with a red box. A schematic of the GaN sample in our measurement system is shown in the top right corner, along with the two fitting properties highlighted with a red box. We show the six phase images and the two obtained property maps: thickness and thermal conductivity of the GaN. Our results indicate a thickness dependent thermal conductivity of GaN, which has implications of thermal management in GaN-based high electron mobility transistors.

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.

scholarly journals Computational and Mathematical Model with Phase Change and Metal Addition Applied to GMAW

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Alfredo dos Santos Maia Neto ◽  
Marcelo Gonçalves de Souza ◽  
Edson Alves Figueira Júnior ◽  
Valério Luiz Borges ◽  
Solidônio Rodrigues de Carvalho
Keyword(s):  
Mathematical Model ◽  
Phase Change ◽  
Thermal Model ◽  
Complex Geometry ◽  
Golden Section ◽  
Simulated Data ◽  
Heat Diffusion ◽  
Data Logger ◽  
Heat Diffusion Equation ◽  
Metal Addition

This work presents a 3D computational/mathematical model to solve the heat diffusion equation with phase change, considering metal addition, complex geometry, and thermal properties varying with temperature. The finite volume method was used and the computational code was implemented in C++, using a Borland compiler. Experimental tests considering workpieces of stainless steel AISI 304 were carried out for validation of the thermal model. Inverse techniques based on Golden Section method were used to estimate the heat transfer rate to the workpieces. Experimental temperatures were measured using thermocouples type J—in a total of 07 (seven)—all connected to the welded workpiece and the Agilent 34970A data logger. The workpieces were chamfered in a 45° V-groove in which liquid metal was added on only one weld pass. An innovation presented in this work when compared to other works in scientific literature was the geometry of the weld pool. The good relation between experimental and simulated data confirmed the quality and robustness of the thermal model proposed in this work.

Influence of Hydrodynamic Journal Bearings With Multiple Slip Zones on Rotordynamic Behavior

2017 ◽  
Vol 139 (6) ◽  
Author(s):  
A. Bhattacharya ◽  
J. K. Dutt ◽  
R. K. Pandey
Keyword(s):  
Slip Length ◽  
Stability Limit ◽  
Substantial Improvement ◽  
Journal Bearings ◽  
Heat Diffusion ◽  
Small Range ◽  
Heat Diffusion Equation ◽  
Multiple Slip ◽  
Lubricating Film ◽  
Marginal Improvement

This paper mainly reports stability investigations of rotors supported on fluid film journal bearings possessing multilocational slip-no-slip zones at the bush–film interface. The coupled solution of governing equations (Reynolds equation, energy equation, heat diffusion equation, lubricant rheological relation, and thermal boundary conditions) has been used to find pressure distributions in the lubricating film followed by evaluation of bearing coefficients. These coefficients have been used to determine stability limit speed (SLS) of the system and its robustness for both short (nearly inflexible) and long (flexible) rotors. Numerical simulations show that the pattern of pressure distribution with multiple slip-no-slip zones is similar to that obtained for multilobe bearings, resulting in substantial improvement of rotor–bearing stability irrespective of eccentricity ratio. A reduction in friction force (up to Sommerfeld number 1.8) and an increase in SLS and robustness compared to conventional bearings are observed when used with short rotors. Typically, up to six pairs of slip-no-slip zones improve SLS of the rotor–shaft system and robustness for short rotors, although more pairs deteriorate both. However, for long rotors, where dynamic rotor forces also act, these bearings provide marginal improvement in stability and robustness only for a small range of slip length.

scholarly journals A Numerical Study of Diffusion of Nanoparticles in a Viscous Medium During Solidification

2017 ◽  
Author(s):  
Kazi M. Rahman ◽  
M. Ruhul Amin ◽  
Ahsan Mian
Keyword(s):  
Fluid Flow ◽  
Metal Matrix ◽  
Marangoni Number ◽  
Particle Distribution ◽  
Heat Diffusion ◽  
Solidification Time ◽  
Original Position ◽  
Heat Diffusion Equation ◽  
Metal Matrix Nanocomposites ◽  
Liquid Titanium

In the field of additive manufacturing process, laser cladding is widely considered due to its cost effectiveness, small localized heat generation and full fusion to metals. Introducing nanoparticles with cladding metals produces metal matrix nanocomposites which in turn improves the material characteristics of the clad layer. The strength of the laser cladded reinforced metal matrix composite are dependent on the location and concentration of the nanoparticles infused in metals. The governing equations that control the fluid flow are standard incompressible Navier-Stokes and heat diffusion equation whereas the Euler-Lagrange approach has been considered for particle tracking. The mathematical formulation for solidification is adopted based on enthalpy porosity method. Liquid titanium has been considered as the initial condition where particle distribution has been assumed uniform throughout the geometry. During the solidification process of liquid titanium, particle flow and distribution has been observed until the entire geometry solidified. A numerical model implemented in a commercial software based on control volume method has been developed that allows to simulate the fluid flow during solidification as well as tracking nanoparticles during this process. The influence of the free surface of the melt pool has a high importance on the fluid flow as well as the influence of pure natural convection. Thus both buoyancy and Marangoni convection have been considered in terms of fluid flow in the molten region. A detailed parametric study has been conducted by changing the Marangoni number, convection heat transfer coefficient, constant temperature below the melting point of titanium and insulated boundary conditions to analyze the behavior of the nanoparticle movement. With the change in Marangoni number and solidification time, a significant change in particle distribution has been observed. The influence of increase in Marangoni number results in a higher concentration of nanoparticles in some portions of the geometry and lack of nanoparticles in rest of the geometry. The high concentration of nanoparticles decrease with a decrease in Marangoni number. Furthermore, an increase in the rate of solidification time limits the nanoparticle movement from its original position which results in different distribution patterns with respect to the solidification time.

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.

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