Fast boundary element method based on a 3D pipe model for analyzing cathodic protection
LIU Liqi, WANG Haitao
Key Laboratory of Advanced Reactor Engineering and Safety of the Ministry of Education, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China
Abstract：The boundary element method (BEM) was used to analyze a cathodic protection (CP) system consisting of large pipeline structures. A three-dimensional pipe boundary element model was used to reduce the number of elements on the pipelines as well as the element integral computations. The pipelines were meshed with line elements with the boundary integrals were based on the original shapes. The large-scale CP problem was solved on a common desktop computer using the fast multipole method (FMM) to accelerate the BEM. The nonlinearity introduced by the polarization curve at the cathode was solved iteratively. The numerical results demonstrate that the number of elements can be reduced by one order of magnitude when discretizing pipelines with these line elements compared with triangular elements and that the FMM can solve large CP problems with up to 50 000 dimension of freedoms (DOFs).
刘立祺, 王海涛. 基于三维管道模型的快速边界元法在阴极保护分析中的应用[J]. 清华大学学报（自然科学版）, 2015, 55(9): 1003-1009.
LIU Liqi, WANG Haitao. Fast boundary element method based on a 3D pipe model for analyzing cathodic protection. Journal of Tsinghua University(Science and Technology), 2015, 55(9): 1003-1009.
[1] Stromman R, Arild R. Computerized numerical techniquesapplied in design of offshore cathodic protection systems [J]. Materials Performance, 1981, 20(4): 15-20.
[2] 张鸣镝, 杜元龙, 殷正安, 等. 有限差分法计算海底管道阴极保护时的电位分布 [J].中国腐蚀与防护学报, 1994, 14(1): 77-81.ZHANG Mingdi, DU Yuanlong, YIN Zheng'an, et al. Calculating potential distributions of cathodically protedted subsea pipeline with finite difference method [J]. Journal of Chinese Society for Corrosion and Protection, 1994, 14(1): 77-81.(in Chinese)
[3] Kranc S C, Alberto A S. Detailed modeling of corrosion macrocells on steel reinforcing in concrete [J]. Corrosion Science, 2001, 43(7): 1355-1372.
[4] Hassanein A M, Glass G K, Buenfeld N R. Protection current distribution in reinforced concrete cathodic protection systems [J].Cement and Concrete Composites, 2002, 24(1): 159-167
[5] 邱枫, 徐乃欣. 钢质贮罐底板外侧阴极保护时的电位分布 [J]. 中国腐蚀与防护学报, 1996, 16(1): 29-36.QIU Feng, XU Naixin. Potential distribution on cathodically protected external tank bottom [J]. Journal of Chinese Society for Corrosion and Protection, 1996, 16(1): 29-36. (in Chinese)
[6] 邱枫, 徐乃欣.码头钢管桩阴极保护时的电位分布 [J]. 中国腐蚀与防护学报, 1997, 17(1): 12-18.QIU Feng, XU Naixin. Potential distribution on cathodically protected steel pipe piles [J]. Journal of Chinese Society for Corrosion and Protection, 1997, 17(1): 12-18.(in Chinese)
[7] 邱枫, 徐乃欣. 用带状牺牲阳极对埋地钢管实施阴极保护时的电位和电流分布 [J].中国腐蚀与防护学报, 1997, 17(2): 106-110.QIU Feng, XU Naixin. Potential and current distributions on pipelines cathodically protected with ribbon sacrificial anodes [J]. Journal of Chinese Society for Corrosion and Protection, 1997, 17(2): 106-110.(in Chinese)
[8] Fu J W, Chow J S K. Cathodic protection designs using an intergral equation numercial method [J]. Materials Performance, 1982, 21(10): 9-12.
[9] Degiorgi V G. Evaluation of perfect paint assumptions in modeling of cathodic protection systems [J]. Engineering Analysis with Boundary Elements, 2002, 26(5): 435-445.
[10] Riemer D P, Orazem M E. A mathematical model for the cathodic protection of tank bottoms [J].Corrosion Science, 2005, 47(3): 849-868.
[11] Brichau F, Deconinck J. A numerical model for cathodic protection of buried pipes [J]. Corrosion, 1994, 50(1): 39-49.
[12] Rokhlin V. Rapid solution of integral equations of classical potential theory [J]. Journal of Computational Physics, 1985, 60(2): 187-207.
[13] Greengard L, Rokhlin V. A fast algorithm for particle simulations [J]. Journal of Computational Physics, 1997, 135(2): 280.
[14] Greengard L, Rokhlin V. A new version of the fast multipole method for the Laplace equation in three dimensions [J]. Acta Numerica, 1997, 6(1): 229-269.
[15] Hrycak T, Rokhlin V. An improved fast multipole algorithm for potential fields [J]. SIAM Journal on Scientific Computing, 1998, 19(6): 1804-1826.
[16] Nishimura N, Liu Y J. Thermal analysis of carbon-nanotube composites using a rigid-line inclusion model by the boundary integral equation method [J]. Computational Mechanics, 2004, 35(1): 1-10.
[17] Wang H T, Yao Z H. Application of a new fast multipole bem for simulation of 2D elastic solid with large number of inclusions [J]. Acta Mechanica Sinica, 2004, 20(6): 613-622.
[18] Liu Y, Nishimura N, Otani Y. Large-scale modeling of carbon-nanotube composites by a fast multipole boundary element method [J]. Computational Materials Science, 2005, 34(2): 173-187.
[19] Liu Y J, Nishimura N, Otani Y, et al. A fast boundary element method for the analysis of fiber-reinforced composites based on a rigid-inclusion model [J]. Journal of Applied Mechanics, 2005, 72(1): 115-128.
[20] Wang H T, Yu S Y. Large-scale numerical simulation of mechanical and thermal properties of nuclear graphite using a microstructure-based model [J]. Nuclear Engineering and Design, 2008, 238(12): 3203-3207.
[21] Zhu X, Chen W, Huang Z, et al. Fast multipole boundary element analysis of 2D viscoelastic composites with imperfect interfaces [J]. Science China Technological Sciences, 2010, 53(8): 2160-2171.
[22] Wang H T, Yao Z H. Large-scale thermal analysis of fiber composites using a line-inclusion model by the fast boundary element method [J]. Engineering Analysis with Boundary Elements, 2013, 37(2): 319-326.
[23] Nishimura N. Fast multipole accelerated boundary integral equation methods [J]. Applied Mechanics Reviews, 2002, 55(4): 299-324.
[24] Yoshida K. Applications of Fast Multipole Method to Boundary Integral Equation Method [D]. Kyoto, Japan: Kyoto University, 2001.
[25] 张东东, 杜翠薇, 程学群, 等. 快速多极边界元法在阴极保护求解电位分布中的应用 [C]// 第十二届中国科学技术协会年会, 第二卷.福州: 中国科学技术协会, 2010: 601-608.ZHANG Dongdong, DU Cuiwei, CHEN Xuequn, et al. Fast multipole boundary element methods for cathodic protection potential distribution problems [C]// The Twelfth Annual Conference of CAST, Volume 2. Fuzhou: China Association for Science and Technology, 2010: 601-608. (in Chinese)
2015, Vol. 55 
