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1. On the Determination of Thermal Conductivity From Frequency Domain Thermoreflectance Experiments

   https://doi.org/10.1115/1.4052655  

   Saurav, Siddharth; Mazumder, Sandip (January 2022, Journal of Heat Transfer)

   Abstract The Fourier and the hyperbolic heat conduction equations were solved numerically to simulate a frequency-domain thermoreflectance (FDTR) experiment. Numerical solutions enable isolation of pump and probe laser spot size effects and use of realistic boundary conditions. The equations were solved in time domain and the phase lag between the temperature of the transducer (averaged over the probe laser spot) and the modulated pump laser signal was computed for a modulation frequency range of 200 kHz–200 MHz. Numerical calculations showed that extracted values of the thermal conductivity are sensitive to both the pump and probe laser spot sizes, while analytical solutions (based on Hankel transform) cannot isolate the two effects. However, for the same effective (combined) spot size, the two solutions are found to be in excellent agreement. If the substrate (computational domain) is sufficiently large, the far-field boundary conditions were found to have no effect on the computed phase lag. The interface conductance between the transducer and the substrate was found to have some effect on the extracted thermal conductivity. The hyperbolic heat conduction equation yielded almost the same results as the Fourier heat conduction equation for the particular case studied. The numerically extracted thermal conductivity value (best fit) for the silicon substrate considered in this study was found to be about 82–108 W/m/K, depending on the pump and probe laser spot sizes used. 

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2. Anisotropic Fourier Heat Conduction and phonon Boltzmann transport equation based simulation of time domain thermo-reflectance experiments

   https://doi.org/10.1016/j.ijheatmasstransfer.2024.125698  

   Saurav, Siddharth; Mazumder, Sandip (August 2024, International Journal of Heat and Mass Transfer)

   The anisotropic Fourier Heat Conduction Equation (FHCE) and the multidimensional phonon Boltzmann Transport Equation (BTE) were solved numerically in cylindrical coordinates and in time domain to simulate a Time Domain Thermo-Reflectance (TDTR) experimental silicon/aluminum substrate/transducer setup. The out-of-phase response of the probe laser was predicted at various beam offset distances for a pump laser pulse frequency of 80 MHz and modulation frequency of 10 MHz and compared against experimental measurements for a silicon substrate. The isotropic FHCE was also solved for comparison. Results show that the isotropic FHCE with bulk thermal conductivity of 145 W/m/K significantly underpredicts the out-of-phase temperature difference, particularly at smaller beam offsets. With an isotropic thermal conductivity of 105 W/m/K, the computed results match experimental data at smaller beam offsets well, but overpredicts the experimental data at larger beam offsets. An almost-perfect match is obtained by using an anisotropic thermal conductivity wherein the radial (in-plane) thermal conductivity is set to 85 W/m/K and the axial (through-plane) conductivity is set to 130 W/m/K. The multidimensional frequency and polarization dependent phonon BTE is next solved. The BTE results for the out-of-phase temperature difference match experimental observations well at small and intermediate beam offsets, but overpredicts the experimental data at larger beam offsets. FHCE results are fitted to the BTE predictions, and the extracted (best fit) thermal conductivity is found to be 110 W/m/K. 

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3. Phonon Boltzmann Transport Equation Based Simulation of Frequency Domain Thermoreflectance Experiments

   https://doi.org/10.1115/IMECE2022-95630  

   Saurav, Siddharth; Mazumder, Sandip (October 2022, Proceedings of the International Mechanical Engineering Congress and Exposition, IMECE2022)

   Abstract The Boltzmann Transport equation (BTE) was solved numerically in cylindrical coordinates and in time domain to simulate a Frequency Domain Thermo-Reflectance (FDTR) experiment. First, a parallel phonon BTE solver that accounts for all phonon modes, frequencies, and polarizations was developed and tested. The solver employs the finite-volume method (FVM) for discretization of physical space, and the finite-angle method (FAM) for discretization of angular space. The solution was advanced in time using explicit time marching. The simulations were carried out in time domain and band-based parallelization of the BTE solver was implemented. The phase lag between the temperature averaged over the probed region of the transducer and the modulated laser pump signal was extracted for a pump laser modulation frequency ranging from 20–200 MHz. It was found that with the relaxation time scales used in the present study, the computed phase lag is underpredicted when compared to experimental data, especially at smaller modulation frequencies. The challenges in solving the BTE for such applications are highlighted. 

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4. Conformal Topology Optimization of Heat conduction problems on Manifolds using an Extended Level Set Method (X-LSM)

   Xu, Xiaoqiang; Chen, Shikui; Gu, Xianfeng David; Wang, Michael Yu (August 2021, ASME 2021 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference)

   null (Ed.)

   In this paper, the authors propose a new dimension reduction method for level-set-based topology optimization of conforming thermal structures on free-form surfaces. Both the Hamilton-Jacobi equation and the Laplace equation, which are the two governing PDEs for boundary evolution and thermal conduction, are transformed from the 3D manifold to the 2D rectangular domain using conformal parameterization. The new method can significantly simplify the computation of topology optimization on a manifold without loss of accuracy. This is achieved due to the fact that the covariant derivatives on the manifold can be represented by the Euclidean gradient operators multiplied by a scalar with the conformal mapping. The original governing equations defined on the 3D manifold can now be properly modified and solved on a 2D domain. The objective function, constraint, and velocity field are also equivalently computed with the FEA on the 2D parameter domain with the properly modified form. In this sense, we are solving a 3D topology optimization problem equivalently on the 2D parameter domain. This reduction in dimension can greatly reduce the computing cost and complexity of the algorithm. The proposed concept is proved through two examples of heat conduction on manifolds. 

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5. Eshelby’s inclusion and inhomogeneity problems under harmonic heat transfer

   https://doi.org/10.1098/rspa.2024.0824  

   Wu, Chunlin; Wang, Tengxiang; Yin, Huiming (July 2025, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Eshelby’s equivalent inclusion method (EIM) has been formulated to solve harmonic heat transfer problems of an infinite or semi-infinite domain containing an inclusion or inhomogeneity. For the inclusion problem, the heat equation is reduced to a modified Helmholtz’s equation in the frequency domain through the Fourier transform, and the harmonic Eshelby’s tensor is derived from the domain integrals of the corresponding Green’s function in the form of Helmholtz’s potential. Using the convolution property of the Fourier space, Helmholtz’s potential with polynomial-form source densities is integrated over an ellipsoidal inclusion, which is reduced to a one-dimensional integral for spheroids and an explicit, exact expression for spheres. The material mismatch in the inhomogeneity problem is simulated by continuously distributed eigen-fields, namely, the eigen-temperature-gradient (ETG) and eigen-heat-source (EHS) for thermal conductivity and specific heat, respectively. The proposed EIM formulation is verified by the conventional boundary integral method with the harmonic Green’s function and multi-domain interfacial continuity, and the accuracy and efficacy of the solution are discussed under different material and load settings. 

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