Trefftz Functions Applied to Direct and Inverse Non-Fourier Heat Conduction Problems

2014 ◽  
Vol 136 (9) ◽  
Author(s):  
Krzysztof Grysa ◽  
Artur Maciag ◽  
Justyna Adamczyk-Krasa
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Inverse Problems ◽  
Approximate Method ◽  
Heat Conduction Equation ◽  
Input Data ◽  
Direct And Inverse Problems ◽  
Ill Posed ◽  
Fourier Heat Conduction ◽  
Boundary Inverse Problems

The paper presents a new approximate method of solving non-Fourier heat conduction problems. The approach described here is suitable for solving both direct and inverse problems. The way of generating Trefftz functions for non-Fourier heat conduction equation has been shown. Obtained functions have been used for solving direct and boundary inverse problems (identification of boundary condition). As a rule, inverse problems are ill-posed. Therefore, each method of solving these problems has to be checked according to disturbance of the input data. Presented examples confirm high usability of the presented approach for solving direct and inverse non-Fourier heat conduction problems.

Assessment of Models for Extracting Thermal Conductivity From Frequency Domain Thermoreflectance Experiments

2020 ◽  
Author(s):  
Siddharth Saurav ◽  
Sandip Mazumder
Keyword(s):  
Thermal Conductivity ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Frequency Domain ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Hyperbolic Heat Conduction ◽  
Phase Lag ◽  
Computational Domain ◽  
Fourier Heat Conduction

Abstract The Fourier heat conduction and the hyperbolic heat conduction equations were solved numerically to simulate a frequency-domain thermoreflectance (FDTR) experimental setup. Numerical solutions enable use of realistic boundary conditions, such as convective cooling from the various surfaces of the substrate and transducer. The equations were solved in time domain and the phase lag between the temperature at the center of the transducer and the modulated pump laser signal were computed for a modulation frequency range of 200 kHz to 200 MHz. It was found that the numerical predictions fit the experimentally measured phase lag better than analytical frequency-domain solutions of the Fourier heat equation based on Hankel transforms. The effects of boundary conditions were investigated and it was found that if the substrate (computational domain) is sufficiently large, the far-field boundary conditions have no effect on the computed phase lag. The interface conductance between the transducer and the substrate was also treated as a parameter, and was found to have some effect on the predicted thermal conductivity, but only in certain regimes. The hyperbolic heat conduction equation yielded identical results as the Fourier heat conduction equation for the particular case studied. The thermal conductivity value (best fit) for the silicon substrate considered in this study was found to be 108 W/m/K, which is slightly different from previously reported values for the same experimental data.

scholarly journals Analytical Solution of Heat Conduction in a Symmetrical Cylinder Using the Solution Structure Theorem and Superposition Technique

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1522 ◽  
Author(s):  
Rasool Kalbasi ◽  
Seyed Mohammadhadi Alaeddin ◽  
Mohammad Akbari ◽  
Masoud Afrand
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Solution Structure ◽  
Structure Theorem ◽  
Initial Conditions ◽  
Hyperbolic Heat Conduction ◽  
Fourier Heat Conduction ◽  
Superposition Technique ◽  
Complex Origin

In this paper, non-Fourier heat conduction in a cylinder with non-homogeneous boundary conditions is analytically studied. A superposition approach combining with the solution structure theorems is used to get a solution for equation of hyperbolic heat conduction. In this solution, a complex origin problem is divided into, different, easier subproblems which can actually be integrated to take the solution of the first problem. The first problem is split into three sub-problems by setting the term of heat generation, the initial conditions, and the boundary condition with specified value in each sub-problem. This method provides a precise and convenient solution to the equation of non-Fourier heat conduction. The results show that at low times (t = 0.1) up to about r = 0.4, the contribution of T1 and T3 dominate compared to T2 contributing little to the overall temperature. But at r > 0.4, all three temperature components will have the same role and less impact on the overall temperature (T).

scholarly journals Non-linear unsteady inverse boundary problem for heat conduction equation

2017 ◽  
Vol 38 (2) ◽  
pp. 81-100 ◽  
Author(s):  
Magda Joachimiak ◽  
Michał Ciałkowski
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Linear Problem ◽  
Conduction Equation ◽  
Direct And Inverse Problems ◽  
Heat Conduction Coefficient ◽  
Nitrification Process ◽  
Non Linear ◽  
Unsteady Heat Conduction ◽  
Inverse Boundary Problem

AbstractDirect and inverse problems for unsteady heat conduction equation for a cylinder were solved in this paper. Changes of heat conduction coefficient and specific heat depending on the temperature were taken into consideration. To solve the non-linear problem, the Kirchhoff’s substitution was applied. Solution was written as a linear combination of Chebyshev polynomials. Sensitivity of the solution to the inverse problem with respect to the error in temperature measurement and thermocouple installation error was analysed. Temperature distribution on the boundary of the cylinder, being the numerical example presented in the paper, is similar to that obtained during heating in the nitrification process.

scholarly journals Reconstruction of basal properties in ice sheets using iterative inverse methods

2012 ◽  
Vol 58 (210) ◽  
pp. 795-808 ◽  
Author(s):  
Marijke Habermann ◽  
David Maxwell ◽  
Martin Truffer
Keyword(s):  
Inverse Problems ◽  
Input Data ◽  
Large Scale ◽  
Gradient Methods ◽  
Inverse Methods ◽  
Model Parameters ◽  
Discrepancy Principle ◽  
Regularization Techniques ◽  
Ill Posed ◽  
Nonlinear Conjugate Gradient Methods

AbstractInverse problems are used to estimate model parameters from observations. Many inverse problems are ill-posed because they lack stability, meaning it is not possible to find solutions that are stable with respect to small changes in input data. Regularization techniques are necessary to stabilize the problem. For nonlinear inverse problems, iterative inverse methods can be used as a regularization method. These methods start with an initial estimate of the model parameters, update the parameters to match observation in an iterative process that adjusts large-scale spatial features first, and use a stopping criterion to prevent the overfitting of data. This criterion determines the smoothness of the solution and thus the degree of regularization. Here, iterative inverse methods are implemented for the specific problem of reconstructing basal stickiness of an ice sheet by using the shallow-shelf approximation as a forward model and synthetically derived surface velocities as input data. The incomplete Gauss-Newton (IGN) method is introduced and compared to the commonly used steepest descent and nonlinear conjugate gradient methods. Two different stopping criteria, the discrepancy principle and a recent- improvement threshold, are compared. The IGN method is favored because it is rapidly converging, and it incorporates the discrepancy principle, which leads to optimally resolved solutions.

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

2008 ◽  
Vol 25 (2) ◽  
pp. 205-210 ◽  
Author(s):  
Youtong Zhang ◽  
Changsong Zheng ◽  
Yongfeng Liu ◽  
Liang Shao ◽  
Chenhua Gou
Keyword(s):  
Heat Conduction ◽  
Exact Solutions ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Fourier Heat Conduction

Mesoscopic Heat Conduction and Onset of Periodicity

2003 ◽  
Author(s):  
Kal Renganathan Sharma
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Time Domain ◽  
Heat Conduction Equation ◽  
Higher Order ◽  
Heat Propagation ◽  
Conduction Equation ◽  
Transient Temperature ◽  
Higher Order Terms ◽  
Fourier Heat Conduction

Mesoscopic approach deals with study that considers temporal fluctuations which is often averaged out in a macroscopic approach without going into the molecular or microscopic approach. Transient heat conduction cannot be fully described by Fourier representation. The non-Fourier effects or finite speed of heat propagation effect is accounted for by some investigators using the Cattaneo and Vernotte non-Fourier heat conduction equation: q=−k∂T/∂x−τr∂q/∂t(1) A generalized expression to account for the non-Fourier or thermal inertia effects suggested by Sharma (5) as: q=−k∂T/∂x−τr∂q/∂t−τr2/2!∂2q/∂t2−τr3/3!∂3q/∂t3−…(2) This was obtained by a Taylor series expansion in time domain. Manifestation of higher order terms in the modified Fourier’w law as periodicity in the time domain is considered in this study. When a CWT is maintained at one end of a medium of length L where L is the distance from the isothermal wall beyond which there is no appreciable temperature change from the initial condition during the duration of the study the transient temperature profile is obtained by the method of Laplace transforms. The space averaged heat flux is obtained and upon inversion from Laplace domain found to be a constant for the the case obeying Fourier’s law; 1 − exp(−τ) using the Cattaneo and Vernotte non-Fourier heat conduction equation, and upon introduction of the second derivative in time of the heat flux the expression becomes, 1 − exp(−τ)(Sin(τ) + Cos(τ)). Thus the periodicity in time domain is lost when the higher order terms in the generalized Fourier expression is neglected.

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