声子玻尔兹曼方程的合成迭代加速算法研究

微纳尺度下的热输运机制是当前传热领域的研究热点。由于微纳结构的特征尺寸和声子平均自由程相当，宏观尺度下的傅里叶定律不再适用，而在介观尺度下建模的声子玻尔兹曼方程成为了研究非傅里叶导热的重要理论。声子玻尔兹曼方程定义在高维相空间，因此设计高效的数值方法求解声子玻尔兹曼方程尤为重要。
本文以Callaway 模型为基础，发展了一套能够快速求解声子玻尔兹曼方程的合成迭代加速算法。我们从Callaway 模型方程推导出适用于任意系统尺寸的宏观矩方程，每次迭代同时求解介观玻尔兹曼方程和宏观矩方程，其中介观分布函数为宏观方程提供高阶矩的封闭，宏观方程求解得到的宏观量又反作用于介观方程，引导分布函数朝着稳态迅速演化，从而加快不同尺度下的信息交换，达到快速收敛的目的。介观方程和宏观方程均基于有限体积方法在非结构网格中进行离散求解。通过傅里叶稳定性分析和数值算例验证，发现合成迭代加速算法在求解稳态问题时比常规迭代算法快1~3 个数量级。同时，我们通过Chapman-Enskog 多尺度理论分析和数值计算展示了该算法的渐近保持性质，即相比与常规迭代算法，可以用更少的网格数收敛到准确解，进一步降低计算资源和计算时间的消耗。
基于我们发展的合成迭代加速算法，本文研究了不同声子输运区域内的热传导，发现了频域热反射中不同系统尺寸下加热频率的作用。在有界系统中，随着频率的增加，扩散区域内温度幅值的空间分布始终呈指数下降的形态，而当声子输运处于水动力学区域时，温度幅值分布经历三个阶段的变化：第一阶段保持中间区域线性分布，边界附近出现较大的温度梯度；第二阶段温度幅值在空间上呈现波形分布；第三阶段则表现出与扩散区域相似的规律。对于半无界系统，扩散输运下温度幅值分布随频率变化规律与有界系统类似，而水动力学导热下的温度幅值分布只表现第三阶段的变化规律。本文成果有利于加深对非傅里叶导热现象的理解。并为模拟声子传热问题提供了一个有效的工具，为微纳尺度下的热管理提供指导。

Heat transport mechanism at micro- and nano-scale is a hot research topic. Since the characteristic size of micro- and nano-structures is comparable to the mean free path of phonons, the Fourier law does not work any more, while the phonon Boltzmann equation modeled at the mesoscopic scale has become an important theory for investigating non-Fourier heat conduction. The Boltzmann equation is defined in high-dimensional phase space, so it is crucial to design an efficient numerical method to solve it.
In this work, we have developed a general synthetic iterative scheme (GSIS) for solving the Callaway model equation. The macroscopic moment equations applicable to any system size are derived from the Callaway model. The mesoscopic Boltzmann equation and macroscopic moment equations are solved simultaneously during each iteration. The mesoscopic distribution function provides higher-order moments closure for macroscopic equations, while the macroscopic quantity obtained by solving the moment equations guide the fast evolution of the mesoscopic equation towards the steady state, thus accelerating the information exchange at different scales, and achieving fast convergence. The mesoscopic and macroscopic equations are discretized by finite volume method using unstructured mesh. With Fourier stability analysis and numerical examples verification, we have found that the GSIS is 1~3 order of magnitude faster than the conventional iterative scheme in solving steady-state problems. Meanwhile, the asymptoticpreserving property of this algorithm is demonstrated by Chapman-Enskog multi-scale theoretical analysis and numerical simulation, that is, compared to the conventional iterative scheme, it can converge to the exact solution with fewer grid numbers, further reducing the consumption of computing resources and time.
Based on the GSIS developed by us, we have explored the heat conduction in different phonon transport regimes, and found the effect of heating frequency in the frequency domain thermoreflectance simulation at different system sizes. In bounded systems, the temperature amplitude decreases exponentially in physical space as frequency increasing within diffusion regimes. While in hydrodynamic regimes, the spatial distribution of temperature amplitude goes through three stages: the first stage maintains a linear distribution in the central region, and emerges a big temperature gradient near the boundary; In the second stage, the temperature amplitude exhibits a wave-like distribution in space. The third stage shows a pattern similar to the diffusion regimes. For semi-infinite systems, the variation of temperature amplitude distribution with frequency under diffusion transport is similar to that of bounded systems, while the temperature amplitude distribution under hydrodynamic heat conduction only shows the rule of the third stage. Our results are helpful to build deeper knowledge of the mechanism of non-Fourier heat transfer. They also provide an effective tool for simulating phonon heat transfer, which can guide microand nano-scale thermal management.

[1] 唐祯安, 王立鼎. 关于微尺度理论[J]. 光学精密工程, 2001, 9(6): 493-498.
[2] 段文晖, 张刚. 纳米材料热传导[M]. 科学出版社, 2017.
[3] 华钰超. 微观与宏观结合的声子弹道扩散导热研究[D]. 清华大学, 2018.
[4] ALI S A. Phonon Boltzmann Transport Equation (BTE) Based Modeling of Heat Conduction in Semiconductor Materials at Sub-Micron Scales[D]. The Ohio State University, 2017.
[5] HANAFI M Z M, ISMAIL F S, ROSLI R. Radial plate fins heat sink model design and optimization[C]//2015 10th Asian Control Conference (ASCC). 2015: 1-5.
[6] 罗小平. 微纳导热多尺度模型及低维材料声子水动力学研究[D]. 哈尔滨工业大学, 2021.
[7] LI B, WANG J. Anomalous Heat Conduction and Anomalous Diffusion in One-Dimensional Systems[J]. Physical Review Letters, 2003, 91(4): 044301.
[8] GU X, WEI Y, YIN X, et al. Colloquium: phononic thermal properties of two-dimensional materials[J]. Reviews of Modern Physics, 2018, 90(4): 041002.
[9] XU X, PEREIRA L F C, WANG Y, et al. Length-dependent thermal conductivity in suspended single-layer graphene[J]. Nature Communications, 2014, 5: 3689.
[10] LEE V, WU C H, LOU Z X, et al. Divergent and Ultrahigh Thermal Conductivity in Millimeter-Long Nanotubes[J]. Physical Review Letters, 2017, 118(13): 135901.
[11] GUYER R A, KRUMHANSL J A. Thermal Conductivity, Second Sound, and Phonon Hydrodynamic Phenomena in Nonmetallic Crystals[J]. Physical Review, 1966, 148(2): 778-788.
[12] LEE S, BROIDO D, ESFARJANI K, et al. Hydrodynamic phonon transport in suspended graphene[J]. Nature Communications, 2015, 6(1): 6290.
[13] MASON W P. Phonon Viscosity and Its Effect on Acoustic Wave Attenuation and Dislocation Motion[J]. Journal of the Acoustical Society of America, 1960, 32(4): 458-472.
[14] MARIS, H. J. Phonon Viscosity[J]. Physical Review, 1969, 188(3): 1303-1307.
[15] SMONTARA A, LASJAUNIAS J C, MAYNARD R. Phonon Poiseuille Flow in Quasi-One-Dimensional Single Crystals[J]. Physical Review Letters, 1996, 77(27): 5397-5400.
[16] MACHIDA Y, SUBEDI A, AKIBA K, et al. Observation of Poiseuille flow of phonons in black phosphorus[J]. Science Advances, 2018, 4(6): eaat3374.
[17] CHESTER M. Second Sound in Solids[J]. Physical Review, 1963, 131(5): 2013-2015.
[18] NARAYANAMURTI V, DYNES R C. Observation of Second Sound in Bismuth[J]. Physical Review Letters, 1972, 28(22): 1461-1465.
[19] HUBERMAN S, DUNCAN R A, CHEN K, et al. Observation of second sound in graphite at temperatures above 100 K[J]. Science, 2019, 364(6438): 375-379.
[20] 郭洋裕, 王沫然. 声子水动力学: 进展、应用与展望[J]. 中国科学：物理学、力学、天文学, 2017, 47(7): 25.
[21] MCGAUGHEY A, KAVIANY M. Phonon Transport in Molecular Dynamics Simulations: Formulation and Thermal Conductivity Prediction[J]. Advances in Heat Transfer, 2006, 39: 169-255.
[22] HENRY A, CHEN G. High Thermal Conductivity of Single Polyethylene Chains Using Molecular Dynamics Simulations[J]. Physical Review Letters, 2008, 101(23): 235502.
[23] 陈学坤. 准一维杂化纳米结构声子输运的分子动力学研究[D]. 湖南大学, 2016.
[24] GIANNOZZI P, DE GIRONCOLI S, PAVONE P, et al. Ab initio calculation of phonon dispersions in semiconductors[J]. Physical Review B, 1991, 43(9): 7231-7242.
[25] 马金龙. 基于第一性原理研究钎锌矿氮化物及其合金中的声子输运[D]. 华中科技大学,2016.
[26] ALVAREZ P T. Thermal transport in semiconductors: first principles and phonon hydrodynamics[D]. Universitat Autònoma de Barcelona, 2017.
[27] XU Y, WANG J S, DUAN W, et al. Nonequilibrium Green’s function method for phononphonon interactions and ballistic-diffusive thermal transport[J]. Physical Review B, 2008, 78(22): 224303.
[28] WANG J S, WANG J, ZENG N. Nonequilibrium Green’s function approach to mesoscopic thermal transport[J]. Physical Review B, 2006, 74(3): 033408.
[29] GUYER R A, KRUMHANSL J A. Solution of the Linearized Phonon Boltzmann Equation[J]. Physical Review, 1966, 148(2): 766-778.
[30] ALVAREZ F X, JOU D, SELLITTO A. Phonon Hydrodynamics and Phonon-Boundary Scattering in Nanosystems[J]. Journal of Applied Physics, 2009, 105: 014317 - 014317.
[31] CHEN G. Ballistic-Diffusive Heat-Conduction Equations[J]. Physical Review Letters, 2001, 86(11): 2297-2300.
[32] CHEN G. Ballistic-Diffusive Equations for Transient Heat Conduction From Nano to Macro scales[J]. Journal of Heat Transfer, 2001, 124(2): 320-328.
[33] TZOU D Y. A Unified Field Approach for Heat Conduction From Macro- to Micro-Scales[J]. Journal of Heat Transfer, 1995, 117(1): 8-16.
[34] TZOU D Y. The generalized lagging response in small-scale and high-rate heating[J]. International Journal of Heat and Mass Transfer, 1995, 38(17): 3231-3240.
[35] 过增元, 曹炳阳. 基于热质运动概念的普适导热定律[J]. 物理学报, 2008, 57(7): 4273-4281.
[36] 过增元, 朱宏晔. 热质的运动和传递——热子气的守恒方程和傅立叶定律[J]. 工程热物理学报, 2007, 28(1): 86-88.
[37] SIMONCELLI M, MARZARI N, CEPELLOTTI A. Generalization of Fourier’s Law into Viscous Heat Equations[J]. Physical Review X, 2020, 10(1): 011019.
[38] CHAPMAN S, COWLING T G, CERCIGNANI C. The mathematical theory of non-uniform gases[M]. 3rd ed. Cambridge University Press, 1995.
[39] BANACH Z, LARECKI W. Chapman–Enskog method for a phonon gas with finite heat flux [J]. Journal of Physics A: Mathematical and Theoretical, 2008, 41(37): 375502.
[40] FRYER M J, STRUCHTRUP H. Moment model and boundary conditions for energy transport in the phonon gas[J]. Continuum Mechanics and Thermodynamics, 2014, 26(5): 593-618.
[41] SENDRA L, BEARDO A, TORRES P, et al. Derivation of a hydrodynamic heat equation from the phonon Boltzmann equation for general semiconductors[J]. Physical Review B, 2021, 103(14): L140301.
[42] 张创. 声子输运的介观数值方法及非傅里叶导热机理研究[D]. 华中科技大学, 2021.
[43] OMINI M, SPARAVIGNA A. An iterative approach to the phonon Boltzmann equation in the theory of thermal conductivity[J]. Physica B: Condensed Matter, 1995, 212(2): 101-112.
[44] CHAPUT L. Direct solution to the linearized phonon Boltzmann equation[J]. Physical Review Letters, 2013, 110(26): 265506.
[45] FUGALLO G, LAZZERI M, PAULATTO L, et al. Ab initio variational approach for evaluating lattice thermal conductivity[J]. Physical Review B, 2013, 88(4): 045430.
[46] CHEN G. Nanoscale energy transport and conversion: a parallel treatment of electrons, molecules, phonons, and photons[M]. Oxford university press, 2005.
[47] MURTHY J Y, NARUMANCHI S V J, PASCUAL-GUTIERREZ J A, et al. Review of multiscale simulation in submicron heat transfer[J]. International Journal for Multiscale Computational Engineering, 2005, 3(1): 5-32.
[48] GUO Y, WANG M. Phonon hydrodynamics and its applications in nanoscale heat transport[J]. Physics Reports, 2015, 595: 1-44.
[49] YU C, OUYANG Y, CHEN J. A perspective on the hydrodynamic phonon transport in twodimensional materials[J]. Journal of Applied Physics, 2021, 130(1): 010902.
[50] CALLAWAY J. Model for lattice thermal conductivity at low temperatures[J]. Physical Review, 1959, 113(4): 1046.
[51] MAZUMDER S, MAJUMDAR A. Monte Carlo study of phonon transport in solid thin films including dispersion and polarization[J]. Journal of Heat Transfer, 2001, 123(4): 749-759.
[52] HUA Y C, CAO B Y. An efficient two-step Monte Carlo method for heat conduction in nanostructures[J]. Journal of Computational Physics, 2017, 342: 253-266.
[53] NIE B D, CAO B Y. Thermal wave in phonon hydrodynamic regime by phonon Monte Carlo simulations[J]. Nanoscale and Microscale Thermophysical Engineering, 2020, 24(2): 94-122.
[54] PÉRAUD J P M, HADJICONSTANTINOU N G. Efficient simulation of multidimensional phonon transport using energy-based variance-reduced Monte Carlo formulations[J]. Physical Review B, 2011, 84(20): 205331.
[55] LI R, LEE E, LUO T. Physics-Informed Deep Learning for Solving Coupled Electron and Phonon Boltzmann Transport Equations[J]. Physical Review Applied, 2023, 19(6): 064049.
[56] LI R, WANG J X, LEE E, et al. Physics-Informed Deep Learning for solving phonon Boltzmann transport equation with large temperature non-equilibrium[J]. npj Computational Materials, 2022, 8: 29.
[57] GUO Z, SHU C. Lattice Boltzmann method and its applications in engineering[M]. World Scientific, 2013.
[58] GUO Y, WANG M. Lattice Boltzmann modeling of phonon transport[J]. Journal of Computational Physics, 2016, 315: 1-15.
[59] CHRISTENSEN A, GRAHAM S. Multiscale lattice Boltzmann modeling of phonon transport in crystalline semiconductor materials[J]. Numerical Heat Transfer, Part B: Fundamentals, 2010, 57(2): 89-109.
[60] ALI S A, KOLLU G, MAZUMDER S, et al. Large-scale parallel computation of the phonon Boltzmann Transport Equation[J]. International Journal of Thermal Sciences, 2014, 86: 341- 351.
[61] MAJUMDAR A. Microscale Heat Conduction in Dielectric Thin Films[J]. Journal of Heat Transfer, 1993, 115(1): 7-16.
[62] GUO Y, WANG M. Heat transport in two-dimensional materials by directly solving the phonon Boltzmann equation under Callaway’s dual relaxation model[J]. Physical Review B, 2017, 96(13): 134312.
[63] GUO Z, XU K. Discrete unified gas kinetic scheme for multiscale heat transfer based on the phonon Boltzmann transport equation[J]. International Journal of Heat and Mass Transfer, 2016, 102: 944-958.
[64] LUO X P, YI H L. A discrete unified gas kinetic scheme for phonon Boltzmann transport equation accounting for phonon dispersion and polarization[J]. International Journal of Heat and Mass Transfer, 2017, 114: 970-980.
[65] ZHANG C, GUO Z. Discrete unified gas kinetic scheme for multiscale heat transfer with arbitrary temperature difference[J]. International Journal of Heat and Mass Transfer, 2019, 134: 1127-1136.
[66] ZHANG C, GUO Z, CHEN S. Unified implicit kinetic scheme for steady multiscale heat transfer based on the phonon Boltzmann transport equation[J]. Physical Review E, 2017, 96(6): 063311.
[67] ZHANG C, GUO Z, CHEN S. An implicit kinetic scheme for multiscale heat transfer problem accounting for phonon dispersion and polarization[J]. International Journal of Heat and Mass Transfer, 2019, 130: 1366-1376.
[68] ADAMS M L, LARSEN E W. Fast iterative methods for discrete-ordinates particle transport calculations[J]. Progress in Nuclear Energy, 2002, 40(1): 3-159.
[69] HARTER J R, HOSSEINI S A, PALMER T S, et al. Prediction of thermal conductivity in dielectrics using fast, spectrally-resolved phonon transport simulations[J]. International Journal of Heat and Mass Transfer, 2019, 144: 118595.
[70] LOY J M, MURTHY J Y, SINGH D. A fast hybrid Fourier-Boltzmann transport equation solver for nongray phonon transport[J]. ASME journal of heat and mass transfer, 2013, 135(1): 011008.
[71] LOY J M, MATHUR S R, MURTHY J Y. A coupled ordinates method for convergence acceleration of the phonon Boltzmann transport equation[J]. Journal of Heat Transfer, 2015, 137(1): 012402.
[72] LUO X P, GUO Y Y, WANG M R, et al. Direct simulation of second sound in graphene by solving the phonon Boltzmann equation via a multiscale scheme[J]. Physical Review B, 2019, 100(15): 155401.
[73] CEPELLOTTI A, FUGALLO G, PAULATTO L, et al. Phonon hydrodynamics in twodimensional materials[J]. Nature Communications, 2015, 6(1): 6400.
[74] ZHANG C, CHEN S, GUO Z. Heat vortices of ballistic and hydrodynamic phonon transport in two-dimensional materials[J]. International Journal of Heat and Mass Transfer, 2021, 176: 121282.
[75] ZHANG Z, OUYANG Y, CHENG Y, et al. Size-dependent phononic thermal transport in lowdimensional nanomaterials[J]. Physics Reports, 2020, 860: 1-26.
[76] CHEN G. Non-Fourier phonon heat conduction at the microscale and nanoscale[J]. Nature Reviews Physics, 2021, 3(8): 555-569.
[77] SU W, ZHU L, WU L. Fast convergence and asymptotic preserving of the general synthetic iterative scheme[J]. SIAM Journal on Scientific Computing, 2020, 42(6): B1517-B1540.
[78] LIU J, ZHANG C, YUAN H, et al. A fast-converging scheme for the phonon Boltzmann equation with dual relaxation times[J]. Journal of Computational Physics, 2022, 467: 111436.
[79] SHANG M Y, ZHANG C, GUO Z, et al. Heat vortex in hydrodynamic phonon transport of two-dimensional materials[J]. Scientific Reports, 2020, 10(1): 8272.
[80] FEDORENKO R. The speed of convergence of one iterative process[J]. USSR Computational Mathematics and Mathematical Physics, 1964, 4(3): 227-235.
[81] 朱亚军. 统一气体动理学格式加速算法研究[D]. 西北工业大学, 2020.
[82] MOUKALLED F, MANGANI L, DARWISH M. The Finite Volume Method in Computational Fluid Dynamics[M]. Springer International Publishing, 2016.
[83] MAZUMDER S. Numerical Methods for Partial Differential Equations[M]. Academic Press, 2016.
[84] SEVILLA R, HUERTA A. HDG-NEFEM with degree adaptivity for Stokes flows[J]. Journal of Scientific Computing, 2018, 77: 1953-1980.
[85] 陶文铨. 数值传热学[M]. 西安交通大学出版社, 2001.
[86] YANG R, YUE S, LIAO B. Hydrodynamic phonon transport perpendicular to diffuse-gray boundaries[J]. Nanoscale and Microscale Thermophysical Engineering, 2019, 23(1): 25-35.
[87] ZHANG C, CHEN S, GUO Z, et al. A fast synthetic iterative scheme for the stationary phonon Boltzmann transport equation[J]. International Journal of Heat and Mass Transfer, 2021, 174: 121308.
[88] ZHU J, TANG D, WANG W, et al. Ultrafast thermoreflectance techniques for measuring thermal conductivity and interface thermal conductance of thin films[J]. Journal of Applied Physics, 2010, 108(9): 094315.
[89] KOH Y K, SINGER S L, KIM W, et al. Comparison of the 3ω method and time-domain thermoreflectance for measurements of the cross-plane thermal conductivity of epitaxial semiconductors[J]. Journal of Applied Physics, 2009, 105(5): 2590.
[90] KOH Y K, CAO Y, CAHILL D G, et al. Heat transport mechanisms in superlattices[J]. Advanced Functional Materials, 2009, 19(4): 610-615.
[91] SCHMIDT A J, CHEN X, CHEN G. Pulse accumulation, radial heat conduction, andanisotropic thermal conductivity in pump-probe transient thermoreflectance[J]. Review of Scientific Instruments, 2008, 79(11): 114902.
[92] COSTESCU R M, WALL M A, CAHILL D G. Thermal conductance of epitaxial interfaces[J]. Physical Review B, 2003, 67(5): 054302.
[93] KOH Y K, CAHILL D G. Frequency dependence of the thermal conductivity of semiconductor alloys[J]. Physical Review B, 2007, 76(7): 075207.
[94] REGNER K T, MAJUMDAR S, MALEN J A. Instrumentation of broadband frequency domain thermoreflectance for measuring thermal conductivity accumulation functions[J]. Review of Scientific Instruments, 2013, 84(6): 064901.
[95] REGNER K T, SELLAN D P, SU Z, et al. Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain thermoreflectance[J]. Nature communications, 2013, 4(1): 1-7.
[96] LIU J, WU L. Fast-Converging and Asymptotic-Preserving Simulation of Frequency Domain Thermoreflectance[J]. Communications in Computational Physics, 2023, 34(1): 65-93.
[97] YANG F, DAMES C. Heating-frequency-dependent thermal conductivity: An analytical solution from diffusive to ballistic regime and its relevance to phonon scattering measurements[J]. Physical Review B, 2015, 91(16): 165311.
[98] HUBERMAN S, ZHANG C, HAIBEH J A. On the question of second sound in germanium: A theoretical viewpoint[A]. 2023. arXiv: 2206.02769.