摘要
区域分解算法作为求解大规模科学与工程问题的一种有效计算手段,已经在地球物理电磁法领域取得了一定的应用,但影响区域分解算法计算效率的因素复杂,前人的研究缺乏对计算效率影响因素的系统讨论和研究。将目前应用广泛的平衡区域分解算法引入到直流电阻率三维正演中,首先对三维模型进行有限差分离散得到线性方程组,然后将求解区域分解为多个不重叠的子域,使用Schur补偿算法将线性方程解耦为子域和共享边界的方程,最后对边界方程进行平衡预处理,并对子域和共享边界的方程进行迭代求解,实现了直流电阻率三维正演,通过与2层水平介质模型的解析解对比验证了算法的准确性和可行性。着重对影响平衡区域分解算法计算效率的因素进行了讨论分析,结果表明子域数目、子域问题和边界方程的解法以及网格大小都会对计算效率产生不同程度的影响。平衡区域分解算法的计算速度随子域数目先减小后增大,随网格增大呈指数增加。采用的3种子域问题和边界方程的解法中,预处理共轭梯度法效率最高,稳定双共轭梯度法次之,最速下降法效率最低。
As a powerful means to solve large-scale scientific and engineering problems, domain decomposition method has been applied in the field of geophysical electromagnetic to some extent. However, the factors affecting the computational efficiency of domain decomposition method are complex, and previous studies lack of discussion on the factors affecting the computational efficiency. In this paper, the widely used balanced domain decomposition method was introduced into the three-dimensional forward modeling of DC resistivity. Firstly, the three-dimensional model was discretized by finite difference method to obtain a set of linear equations. Then, the solution domain was decomposed into several non-overlapping sub-domains. The Schur compensation algorithm was used to decouple the linear equations into sub-domains and shared boundary equations. Finally, the boundary equation was balanced preconditioned, the equations of sub-domains and shared boundary were solved iteratively, and the three-dimensional forward modeling of DC resistivity is realized. The accuracy and feasibility of the proposed algorithm were verified by comparing with the analytical solutions of the two-layer horizontal medium model. In this paper, the factors affecting the computational efficiency of the balanced domain decomposition method were discussed and analyzed. The results show that the number of subdomains, the solution of subdomain and boundary problems, and the mesh sizes will have different effects on the computational efficiency. The computational speed of the balanced domain decomposition method decreases first and then increases with the number of sub-domains, and increases exponentially with the increase of meshes. Among three kinds of iteration methods for subdomain and boundary problems adopted in this paper, PCG has the highest efficiency, BICG takes the second place, Steepest Descent method has the lowest efficiency.
引文
[1] 唐声海.直流电阻率测深在川西高原康定白土坎滑坡勘探中的应用[J].中国地质灾害与防治学报,1999(1):100-105.
[2] 何中江.直流电阻率法在寻找地下水中的综合应用[J].价值工程,2015(2):47-49.
[3] 郝晋荣,李毛飞,廖昱翔,等.直流电阻率法高阻管道漏点位置的探测与分析[J].地球物理学进展,2018,33(2):803-807.
[4] 吕涛.区域分解算法:偏微分方程数值解法新技术[M].北京:科学出版社,1992.
[5] Dawson C, Du Q, Dupont T. A finite difference domain decomposition algorithm for numerical solution of the heat equation[J]. Mathematics of Computation, 1989, 57(195): 63-71.
[6] 顾彩梅.热传导方程的有限差分区域分解算法[D].济南:山东大学, 2006.
[7] Tallec P L. Domain decomposition methods in computational mechanic[J]. Comput Mech Adv, 1994, 111(1): 227-237.
[8] 刘佩, 姚谦峰. 结构动力可靠度计算的区域分解法和重要抽样法[J]. 振动与冲击, 2008, 27(12): 52-55.
[9] Stupfel B. A fast-domain decomposition method for the solution of electromagnetic scattering by large objects[J]. IEEE Transactions on Antennas & Propagation, 1996, 44(10): 1375-1385.
[10] 厉天威,阮江军,张宇,等.基于区域分解法的电磁场并行计算研究[J].高电压技术, 2007, 33(5): 58-61.
[11] Xiong Z. Domain decomposition for 3D electromagnetic modeling[J]. Earth Planets & Space,1999,51(10):1013-1018.
[12] Commer M, Newman G A, Carazzone J J, et al. Massively parallel electrical-conductivity imaging of hydrocarbons using the IBM Blue Gene/L supercomputer[J]. Ibm Journal of Research & Development, 2008, 52(1/2):93-103.
[13] Silva N V D, Macgregor L, Morgan J, et al. A domain decomposition method for 3-D controlled source electromagnetics[C]//Seg Technical Program Expanded, 2010: 660-664.
[14] Zyserman F I, Guarracino L, Santos J E. A hybridized mixed finite element domain decomposed method for two dimensional magnetotelluric modelling[J]. Earth Planets & Space, 1999, 51(4): 297-306.
[15] Rungarunwan T, Siripunvaraporn W. An efficient modified hierarchical domain decomposition for two-dimensional magnetotelluric forward modelling[J]. Geophysical Journal International, 2010, 183(2):634-644.
[16] Ren Z Y, Kalscheuer T, Greenhalgh S, et al. A finite-element-based domain-decomposition approach for plane wave 3D electromagnetic modeling[J]. Geophysics, 2014; 79(6): E255-E268.
[17] Bihlo A, Farquharson C G, Haynes R D, et al. Probabilistic domain decomposition for the solution of the two-dimensional magnetotelluric problem[J]. Computational Geosciences, 2017, 21(1): 117-129.
[18] 李丹.基于区域分解算法的大地电磁二维正演研究[D].南昌:东华理工大学, 2017.
[19] Mandel J, Dohrmann C R. Convergence of a balancing domain decomposition by constraints and energy minimization[J]. Numerical Linear Algebra with Applications, 2003, 10(7):639-659.
[20] Dey A, Morrison H F. Resistivity modeling for arbitrarily shaped three-dimensional structures[J]. Geophysical Prospecting, 1979, 27(1):106-136.
