Stochastic Analysis of Heat Conduction and Thermal Stresses in Solids: A Review

Chiba, Ryoichi

doi:10.5772/50994

Cited by 12 publications

(11 citation statements)

References 124 publications

(117 reference statements)

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“…Therefore, it is reasonable to use uniformly random variables for the mathematical representation of uncertainty in α . As for κ , the uniform ellipticity condition is physically quite reasonable to assume (, p. 252]).…”

Section: Problem Statementmentioning

confidence: 99%

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Robust optimal Robin boundary control for the transient heat equation with random input data

Martínez-Frutos

Kessler

Münch

et al. 2016

Numerical Meth Engineering

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Summary The problem of robust optimal Robin boundary control for a parabolic partial differential equation with uncertain input data is considered. As a measure of robustness, the variance of the random system response is included in two different cost functionals. Uncertainties in both the underlying state equation and the control variable are quantified through random fields. The paper is mainly concerned with the numerical resolution of the problem. To this end, a gradient‐based method is proposed considering different functional costs to achieve the robustness of the system. An adaptive anisotropic sparse grid stochastic collocation method is used for the numerical resolution of the associated state and adjoint state equations. The different functional costs are analysed in terms of computational efficiency and its capability to provide robust solutions. Two numerical experiments illustrate the performance of the algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

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Section: Problem Statementmentioning

confidence: 99%

“…We are now in a position to prove the well-posedness of the state law (1) in this functional framework. (1).…”

Section: Remarkmentioning

confidence: 99%

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Robust optimal Robin boundary control for the transient heat equation with random input data

Martínez-Frutos

Kessler

Münch

et al. 2016

Numerical Meth Engineering

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“…Thus, when time functions included as a random internal heat supply or/and the thermal boundary conditions are equipped with sectional random heat supply, then the mathematical complicity further increases. Therefore, the analyses treated in the framework of the theory of a stationary random process ought to be extended to nonstationary random processes was explained in the highly cited review paper [1]. A short history of the research work associated with random pulse and application of probabilistic methods are discussed in detail as a prerequisite.…”

Section: Introductionmentioning

confidence: 99%

Transient temperature distribution and thermal stresses in a thick plate subjected to random pulses

Bhoyar¹,

Varghese²,

Khalsa³

2019

International Journal of Thermodynamics

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This paper deals with the determination of second-order statistics (the mean of each component and correlations between components) of the quasi-static temperature distribution and thermal stresses on a thick plate under thermal load. The heated thick circular plate with random internal heat generation acting at random instants of time, in addition to sectional heat supply in the form of random impulses on the upper face, while the lower face is kept at zero temperature, and the fixed circular edge thermally insulated in considered for investigation. A type of Laplace transform method is employed to obtain the analytical solutions for the temperature and stresses statistics. The effects of the statistics of the random temperature distribution and its associated stresses are signified graphically that highlight the developments proposed in this paper.

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“…To estimate the effects quantitatively, probabilistic methods have been applied to heat transfer problems. Existing literature about the application of the methods to heat conduction problems is summarised in review articles [2][3][4], and therefore we do not enter further into the subject here. On the other hand, focusing on the application to convective heat transfer problems, one can find early studies of stochastic forced convection with heat source and initial and boundary conditions being white noises [5,6].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Analysis of Natural Convection in Vertical Channels with Random Wall Temperature

Chiba

2017

Mathematical Problems in Engineering

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This study attempts to derive the statistics of temperature and velocity fields of laminar natural convection in a heated vertical channel with random wall temperature. The wall temperature is expressed as a random function with respect to time, or a random process. First, analytical solutions of the transient temperature and flow velocity fields for an arbitrary temporal variation in the channel wall temperature are obtained by the integral transform and convolution theorem. Second, the autocorrelations of the temperature and velocity are formed from the solutions, assuming a stationarity in time. The mean square values of temperature and velocity are computed under the condition that the fluctuation in the channel wall temperature can be considered as white noise or a stationary Markov process. Numerical results demonstrate that a decrease in the Prandtl number or an increase in the correlation time of the random process increases the level of mean square velocity but does not change its spatial distribution tendency, which is a bell-shaped profile with a peak at a certain horizontal distance from the channel wall. The peak position is not substantially affected by the Prandtl number or the correlation time.

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Stochastic Analysis of Heat Conduction and Thermal Stresses in Solids: A Review

Cited by 12 publications

References 124 publications

Robust optimal Robin boundary control for the transient heat equation with random input data

Robust optimal Robin boundary control for the transient heat equation with random input data

Transient temperature distribution and thermal stresses in a thick plate subjected to random pulses

Stochastic Analysis of Natural Convection in Vertical Channels with Random Wall Temperature

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