系统级封装应用中时谐Maxwell方程大规模计算的求解算法:现状与挑战

doi:10.19596/j.cnki.1001-246x.8231

摘要/Abstract

摘要： 系统级封装（SiP）是当前电子学系统设计的主流技术途径，数值模拟是进行系统级封装（SiP）设计的主要手段。由于系统级封装应用特有的复杂性，现有的求解时谐Maxwell方程离散系统的算法面临很大的挑战，成为制约该类应用大规模数值模拟效率的瓶颈。本文综述系统级封装应用时谐Maxwell方程解法器求解算法，针对典型实际模型，评估现有算法的现状和面临的挑战，分析应用特征对算法计算能力的影响，并在现有算法的基础上提出一种可行的预条件算法策略。

关键词: 系统级封装(SiP), 数值模拟, 时谐Maxwell方程, 线性解法器, 预条件算法

Abstract: System in package (SiP) is mainstream technology in the design of electronics system. Numerical simulation plays an important role in SiP applications. However, due to the specific complexity of SiP applications, existing algorithms for linear systems arising from time-harmonic Maxwell's equations are faced with great challenges, which become a bottleneck restricting efficiency of large-scale numerical simulations. In this paper, we review algorithms for time-harmonic Maxwell's equations in SiP applications. Based on capability assessment of existing algorithms for realistic SiP models, we propose a preconditioning strategy, and show its feasibility and efficiency. Furthermore, we analyze impact of such applications on performance behavior of current algorithms and the challenges we faced with.

Key words: system in package(SiP), numerical simulation, time-harmonic Maxwell's equations, linear solver, preconditioning algorithm

中图分类号:

O246

胡少亮, 徐小文, 郑宇腾, 赵振国, 王卫杰, 徐然, 安恒斌, 莫则尧. 系统级封装应用中时谐Maxwell方程大规模计算的求解算法:现状与挑战[J]. 计算物理, 2021, 38(2): 131-145.

HU Shaoliang, XU Xiaowen, ZHENG Yuteng, ZHAO Zhenguo, WANG Weijie, XU Ran, AN Hengbin, MO Zeyao. Algorithms in Linear Solver for Large-scale Time-harmonic Maxwell's Equations in SiP Applications: State-of-the-art and Challenges[J]. Chinese Journal of Computational Physics, 2021, 38(2): 131-145.

参考文献

[1] SHAO Y, PENG Z, LEE J, et al. High speed interconnects of multi-layer PCB analysis by using non-conformal domain decomposition method[C]. International Symposium on Electromagnetic Compatibility, 2010:637-642.
[2] SHAO Y, PENG Z, LEE J, et al. Full-wave real-life 3-D package signal integrity analysis using nonconformal domain decomposition method[J]. IEEE Transactions on Microwave Theory and Techniques, 2011, 59(2):230-241.
[3] SHAO Y, PENG Z, LEE J, et al. Signal integrity analysis of high-speed interconnects by using nonconformal domain decomposition method[J]. IEEE Transactions on Components, Packaging and Manufacturing Technology, 2012, 2(1):122-130.
[4] MEINDL J D. Beyond Moore's law:The interconnect era[J]. Computing in Science and Engineering, 2003, 5(1):20-24.
[5] 詹文浩, 戴国华. 手机射频前端发展状况及技术分析[J]. 移动通信, 2017, 041(007):5-9.
[6] BUCCELLA C, FELIZIANI M, MANZI G, et al. Three-dimensional FEM approach to model twisted wire pair cables[J]. IEEE Conference on Electromagnetic Field Computation, 2006, 43(4):1373-1376.
[7] HOLLAUS K, BIRO O, CALDERA P, et al. Simulation of crosstalk on printed circuit boards by FDTD, FEM, and a circuit model[J]. IEEE Transactions on Magnetics, 2008, 44(6):1486-1489.
[8] ZHU X, WU W, ZHANG G, CAI L. Parallel simulation and analysis of large EMP bounded wave simulator with horizontal polarization[J]. Chinese Journal of Computational Physic, 2019, 36(6):691-698.
[9] ZHU X, CHEN Z, WU W, et al. Simulation of large vertically polarized EMP radiating wave simulator with discrete resistors using parallel FDTD method[J]. Chinese Journal of Computational Physic, 2019, 36(3):349-356.
[10] YANG Y. A magnetic field divergence cleaning method in MHD numerical simulations[J]. Chinese Journal of Computational Physics, 2018, 35(4):437-442.
[11] MAO J, XIANG K, WANG Y, WANG H. Numerical simulation of magnetohydrodynamic duct flow with sudden expansion[J]. Chinese Journal of Computational Physic, 2018, 35(5):597-605.
[12] CHEN J, CHEN Z, CUI T, et al. An adaptive finite element method for the eddy current model with circuit/field couplings[J]. SIAM Journal on Scientific Computing, 2010, 32(2):1020-1042.
[13] LEE S, VOUVAKIS M N, LEE J, et al. A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays[J]. Journal of Computational Physics, 2005, 203(1):1-21.
[14] PENG Z, LEE J F. Non-conformal domain decomposition method with mixed true second order transmission condition for solving large finite antenna arrays[J]. IEEE Transactions on Antennas and Propagation, 2011, 59(5):1638-1651.
[15] SHAO Y, PENG Z, LIM K H, et al. Non-conformal domain decomposition methods for time-harmonic Maxwell equations[J]. Proceedings of the Royal Society A:Mathematical, Physical and Engineering Sciences, 2012, 468(2145):2433-2460.
[16] WANG W J, XU R, LI H Y, et al. Massively parallel simulation of large-scale electromagnetic problems using one high-performance computing scheme and domain decomposition method[J]. IEEE Transactions on Electromagnetic Compatibility, 2017, 59(5):1523-1531.
[17] WANG W J, YE Z B, ZHOU H J. Novel strategies based on domain decomposition method for electromagnetic problem analysis[C]. Cross Strait Quad-Regional Radio Science and Wireless Technology Conference (CSQRWC), 2018. IEEE, 2018:1-3.
[18] ENGQUIST B, YING L. Sweeping preconditioner for the Helmholtz equation:Moving perfectly matched layers[J]. Multiscale Modeling & Simulation, 2011, 9(2):686-710.
[19] LIU F, YING L. Sparsifying preconditioner for the time-harmonic Maxwell's equations[J]. Journal of Computational Physics, 2019, 376:913-923.
[20] POULSON J, ENGQUIST B, LI S, et al. A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations[J]. SIAM Journal on Scientific Computing, 2013, 35(3):C194-C212.
[21] CHEN Z, XIANG X. A source transfer domain decomposition method for Helmholtz equations in unbounded domain[J]. SIAM Journal on Numerical Analysis, 2013, 51(4):2331-2356.
[22] CHEN Z, XIANG X. A source transfer domain decomposition method for Helmholtz equations in unbounded domain part II:Extensions[J]. Numerical Mathematics-Theory Methods and Applications, 2013, 6(3):538-555.
[23] RAWAT V, LEE J. Nonoverlapping domain decomposition with second order transmission condition for the time-harmonic Maxwell's equations[J]. SIAM Journal on Scientific Computing, 2010, 32(6):3584-3603.
[24] PENG Z, LEE J F. A scalable nonoverlapping and nonconformal domain decomposition method for solving time-harmonic Maxwell equations in R³[J]. SIAM Journal on Scientific Computing, 2012, 34(3):A1266-A1295.
[25] BONAZZOLI M, DOLEAN V, GRAHAM I G, et al. A two-level domain-decomposition preconditioner for the time-harmonic Maxwell's equations[C]. Computational Science and Engineering, 2018:149-157.
[26] BONAZZOLI M, DOLEAN V, GRAHAM I G, et al. Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption[J]. Mathematics of Computation, 2019, 88(320):2559-2604.
[27] COCQUET P H, GANDER M J. How large a shift is needed in the shifted Helmholtz preconditioner for its effective inversion by multigrid[J]. SIAM Journal on Scientific Computing, 2017, 39(2):A438-A478.
[28] GANDER M J, GRAHAMI G, SPENCEE A, et al. Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning:What is the largest shift for which wavenumber-independent convergence is guaranteed?[J]. Numerische Mathematik, 2015, 131(3):567-614.
[29] HU Q, YUAN L. A plane-wave least-squares method for time-harmonic Maxwell's equations in absorbing media[J]. SIAM Journal on Scientific Computing, 2014, 36(4):A1937-A1959
[30] HU Q, ZHANG H. Substructuring preconditioners for the systems arising from plane wave discretization of Helmholtz equations[J]. SIAM Journal on Scientific Computing, 2016, 38(4):A2232-A2261.
[31] HU Q, LI X. Novel multilevel preconditioners for the systems arising from plane wave discretization of Helmholtz equations with large wave numbers[J]. SIAM Journal on Scientific Computing, 2017, 39(4):A1675-A1709.
[32] HU Q, LI X. Efficient multilevel preconditioners for three-dimensional plane wave Helmholtz systems with large wave numbers[J]. Multiscale Modeling & Simulation, 2017, 15(3):1242-1266.
[33] PENG J, WANG J, SHU S, et al. Adaptive BDDC algorithms for the system arising from plane wave discretization of Helmholtz equations[J]. International Journal for Numerical Methods in Engineering, 2018, 116:683-707.
[34] PENG Z, RAWAT V, LEE J, et al. One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems[J]. Journal of Computational Physics, 2010, 229(4):1181-1197.
[35] PENG Z, LEE J. Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics[J]. Journal of Computational Physics, 2010, 229(16):5615-5629.
[36] EI MRABET O. High frequency structure simulator (HFSS) tutorial[J]. IETR, UMR CNRS, 2006, 6164:2005-2006.
[37] AGUILAR A G, VAN TONDER J, JAKOBUS U, et al. Overview of recent advances in the electromagnetic field solver FEKO[C]. 9th European Conference on Antennas and Propagation (EuCAP), 2015. IEEE, 2015:1-5.
[38] DICKINSON E J, EKSTROM H, FONTES E, et al. COMSOL multiphysics:Finite element software for electrochemical analysis:A mini-review[J]. Electrochemistry Communications, 2014:71-74.
[39] NEDELEC J C. Mixed finite elements in R³[J]. Numerische Mathematik, 1980, 35(3):315-341.
[40] XU X, MO Z. Algebraic interface-based coarsening AMG preconditioner for multi-scale sparse matrices with applications to radiation hydrodynamics computation[J]. Numerical Linear Algebra with Applications, 2017, 24(2):e2078.
[41] 赵振国, 李光荣, 童杰, 等. 封装结构强电磁脉冲多物理效应并行计算程序研制[J]. 强激光与粒子束, 2018, 030(008):29-33.
[42] LIU Q, MO Z, ZHANG A, et al. JAUMIN:A programming framework for large-scale numerical simulation on unstructured meshes[J]. CCF Trans HPC, 2019, 1(1):35-48.
[43] TIAN R, ZHOU M, WANG J, et al. A challenging dam structural analysis:Large-scale implicit thermo-mechanical coupled contact simulation on Tianhe-II[J]. Computational Mechanics, 2019, 63(1):99-119.
[44] BALAY S, ABHYANKAR S, ADAMS M, et al. PETSc users manual revision 3.8[R]. Argonne National Lab (ANL), Argonne, IL(United States),2017.
[45] AMESTOY P R, DUFF I S, LECELLENT J, et al. MUMPS:A general purpose distributed memory sparse solver[C]. Parallel Computing, 2000:121-130.
[46] FALGOUT R D, YANG U M. Hypre:A library of high performance preconditioners[C]. International Conference on Computational Science, 2002:632-641.
[47] XU X, MO Z, YUE X, et al. α Setup-AMG:An adaptive-setup-based parallel AMG solver for sequence of sparse linear systems[J/OL]. CCF Trans HPC, 2020. https://doi.org/10.1007/s42514-020-00033-w.
[48] CAI X, SARKIS M. A restricted additive Schwarz preconditioner for general sparse linear systems[J]. SIAM Journal on Scientific Computing, 1999, 21(2):792-797.
[49] HIPTMAIR R, XU J. Nodal auxiliary space preconditioning in H(curl) and H(div) spaces[J]. SIAM Journal on Numerical Analysis, 2007, 45(6):2483-2509.
[50] CHEN Z, WANG L, ZHENG W, et al. An adaptive multilevel method for time-harmonic Maxwell equations with singularities[J]. SIAM Journal on Scientific Computing, 2007, 29(1):118-138.
[51] HU Q, LIU C, SHU S, et al. An effective preconditioner for a PML system for electromagnetic scattering problem[J]. ESAIM:Mathematical Modelling and Numerical Analysis, 2015, 49(3):839-854.
[52] KOLEV T V, VASSILEVSKI P S. Parallel auxiliary space AMG for H(curl) problems[J]. Journal of Computational Mathematics, 2009, 27(5):604-623.