Computational Mechanics

, Volume 63, Issue 1, pp 99–119 | Cite as

A challenging dam structural analysis: large-scale implicit thermo-mechanical coupled contact simulation on Tianhe-II

  • Rong Tian
  • Mozhen Zhou
  • Jingtao Wang
  • Yang Li
  • Hengbin An
  • Xiaowen Xu
  • Longfei Wen
  • Lixiang Wang
  • Quan Xu
  • Juelin Leng
  • Ran Xu
  • Bingyin Zhang
  • Weijie Liu
  • Zeyao MoEmail author
Original Paper


Due to huge bulk volume and extremely complex geometrical and geological features, it is forbiddingly difficult to perform a dam structural analysis with even moderate geometry fidelity in engineering practices. We present a high resolution of engineering structural analysis of the first ultra-high concrete-faced rockfill dam in China. Mesh resolution is taken to be 20 cm along slab thickness for the bulk volume of 20M \(\hbox {m}^{3}\) of the whole dam. The engineering problem is solved by considering nonlinear behaviors such as joints’ contact nonlinearity, creep deformation, and strong thermo-mechanical coupling, as well as blended continuous-discontinuous approximation, on a mesh model of 1.1 billion dofs using 16K CPU cores of Tianhe-II. The problem to be solved is a challenging non-positive definite, non-symmetric and ill-conditioned matrix problem. The simulation confirms in the first time that the sunlight temperature effect can contribute up to a contact stress increment of maximum 10.9 MPa and explains frequent extrusion damage observed for the dam. As model tests are difficult to perform for high dams, with this first success, we envision that extreme-scale simulation would pose broad impact on the safety evaluation of high dams in future.


Petascale computing Life time safety Ultra high dams Hydraulic engineering Dual mortar method Thermo-mechanical-contact coupling Extended finite element method Preconditioning Linear solver 



This work was partially supported by the National Key R&D Program (Grant #: 2016YFB0201002, 2016YFB0201002), Science Challenge Project (Grant #: TZ2016002, TZ2018002), the National Natural Science Foundation of China (Grant #: 91430218, 91530319, 11472274, 61370067, 51479099), and the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second to the fourth phases) under Grant No. U1501501.

Supplementary material


  1. 1.
    Chen G, Jin D, Mao J et al (2014) Seismic damage and behavior analysis of earth dams during the 2008 Wenchuan earthquake, China. Eng Geol 180:99–129. Google Scholar
  2. 2.
    Luo L, Chen Y, Zhong HT (2013) Application of geomembrane in temporary treatment of extrusion damage of dam face slab. Dam Saf 2(2013):48–51 (in Chinese) Google Scholar
  3. 3.
    Xu ZP, Guo C (2007) Research on the concrete face slab rupture of high CFRD. Water Power 33(9):81–84 (in Chinese) Google Scholar
  4. 4.
    Hao JT, Du ZK (2008) Precaution measures for the spalling failure of the slab joint concrete in high CFRDs. Water Power 34(6):41–44 (in Chinese) Google Scholar
  5. 5.
    Cao KM, Xu JJ (2009) Discussions on critical deflection of face slab and its design improvement for super-high CFRD. Water Power 34(11):98–102 (in Chinese) Google Scholar
  6. 6.
    Ma HQ, Cao KM (2007) Key technology of supper-high CFRD. Eng Sci 9(11):4–10 (in Chinese) MathSciNetGoogle Scholar
  7. 7.
    Li NH, Yang ZY (2012) Technical advances in concrete face rockfill dams in China. Chin J Geotech Eng 34(8):1361–1368 (in Chinese) Google Scholar
  8. 8.
    Cao KM, Zhang ZL (2001) Performance of the Tianshengqiao 1 CFRD. Int J Hydropower Dams 8(5):78–83Google Scholar
  9. 9.
    Zhou MZ, Zhang BY, Zhang ZL et al (2015) Mechanisms and simulation methods for extrusion damage of concrete faces of high concrete-faced rockfill dams. Chin J Geotech Eng 37(8):1426–1432. (in Chinese)Google Scholar
  10. 10.
    Cao KM, Wang YS, Xu JJ et al (2008) Concrete face rockfill dam. China Water Power Press, Beijing, p 147 (in Chinese)Google Scholar
  11. 11.
    International Commission on Large Dams (ICOLD) (2013) Benchmark workshops on dam safety, Graz, Austria. Accessed 20 Dec 2017
  12. 12.
    Popp A, Seitz A, Gee MW et al (2013) Improved robustness and consistency of 3D contact algorithms based on a dual mortar approach. Comput Methods Appl Mech Eng 264(2013):67–80MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hughes TJR, Taylor RL, Sackman JL et al (1976) A finite element method for a class of contact impact problems. Comput Methods Appl Mech Eng 8(3):249–276zbMATHGoogle Scholar
  14. 14.
    Hallquist JO, Goudreau GL, Benson DJ (1985) Sliding interfaces with contact-impact in large-scale Lagrangian computations. Comput Methods Appl Mech Eng 51(1):107–137MathSciNetzbMATHGoogle Scholar
  15. 15.
    Taylor RL, Papadopoulos P (1991) On a patch test for contact problems in two dimensions. In: Wriggers P, Wagner W (eds) Computational methods in nonlinear mechanics. Springer, Berlin, pp 690–702Google Scholar
  16. 16.
    Crisfield MA (2000) Re-visiting the contact patch test. Int J Numer Meth Eng 48(3):435–449zbMATHGoogle Scholar
  17. 17.
    El-Abbasi N, Bathe KJ (2001) Stability and patch test performance of contact discretizations and a new solution algorithm. Comput Struct 79(16):1473–1486Google Scholar
  18. 18.
    Tan D (2003) Mesh matching and contact patch test. Comput Mech 31(1):135–152MathSciNetzbMATHGoogle Scholar
  19. 19.
    Chen X, Hisada T (2006) Development of finite element contact analysis algorithm passing patch test. Nihon Kikai Gakkai Ronbunshu A Hen (Trans Jpn Soc Mech Eng Part A) 72(713):39–46Google Scholar
  20. 20.
    Kim JH, Lim JH, Lee JH et al (2008) A new computational approach to contact mechanics using variable-node finite elements. Int J Numer Meth Eng 73(13):1966–1988zbMATHGoogle Scholar
  21. 21.
    Kang YS, Kim J, Sohn D et al (2014) A new three-dimensional variable-node finite element and its application for fluid–solid interaction problems. Comput Methods Appl Mech Eng 281(7574):81–105MathSciNetzbMATHGoogle Scholar
  22. 22.
    Zavarise G, Lorenzis LD (2010) A modified node-to-segment algorithm passing the contact patch test. Int J Numer Meth Eng 79(79):379–416zbMATHGoogle Scholar
  23. 23.
    Zhou MZ, Zhang BY, Peng C et al (2016) Three-dimensional numerical analysis of concrete-faced rockfill dam using dual-mortar finite element method with mixed tangential contact constraints. Int J Numer Anal Meth Geomech 40(15):2100–2122. Google Scholar
  24. 24.
    Hüeber S, Wohlmuth BI (2009) Thermo-mechanical contact problems on non-matching meshes. Comput Methods Appl Mech Eng 198(15):1338–1350. zbMATHGoogle Scholar
  25. 25.
    Goodman RE, Taylor RL, Brekke TL (1968) A model for the mechanics of jointed rock. J Soil Mech Found 94(1968):637–660Google Scholar
  26. 26.
    Desai CS, Zaman MM, Lightner JG et al (1984) Thin-layer element for interfaces and joints. Int J Numer Anal Meth Geomech 8(1):19–43Google Scholar
  27. 27.
    Zhang BY, Wang JG, Shi R (2004) Time-dependent deformation in high concrete-faced rockfill dam and separation between concrete face slab and cushion layer. Comput Geotech 31(7):559–573. Google Scholar
  28. 28.
    Bathe KJ, Chaudhary A (1985) A solution method for planar and axisymmetric contact problems. Int J Numer Meth Eng 21(1):65–88zbMATHGoogle Scholar
  29. 29.
    Wriggers P, Simo JC (1985) A note on tangent stiffness for fully nonlinear contact problems. Commun Appl Numer Methods 1(5):199–203zbMATHGoogle Scholar
  30. 30.
    Areias PMA, Sá JMACD, António CAC (2004) Algorithms for the analysis of 3D finite strain contact problems. Int J Numer Meth Eng 61(7):1107–1151MathSciNetzbMATHGoogle Scholar
  31. 31.
    Sabetamal H, Nazem M, Sloan SW et al (2016) Frictionless contact formulation for dynamic analysis of nonlinear saturated porous media based on the mortar method. Int J Numer Anal Methods Geomech 40(1):25–61Google Scholar
  32. 32.
    Kartal ME, Bayraktar A, Başağa HB (2012) Nonlinear finite element reliability analysis of concrete-faced rockfill (CFR) dams under static effects. Appl Math Model 36(11):5229–5248zbMATHGoogle Scholar
  33. 33.
    Park KC, Felippa CA, Rebel G (2002) A simple algorithm for localized construction of non-matching structural interfaces. Int J Numer Meth Eng 53(9):2117–2142zbMATHGoogle Scholar
  34. 34.
    Simo JC, Wriggers P, Taylor RL (1986) A perturbed Lagrangian formulation for the finite element solution of contact problems. Comput Methods Appl Mech Eng 50(2):163–180MathSciNetzbMATHGoogle Scholar
  35. 35.
    Papadopoulos P, Taylor RL (1992) A mixed formulation for the finite element solution of contact problems. Comput Methods Appl Mech Eng 94(3):373–389zbMATHGoogle Scholar
  36. 36.
    Zavarise G, Wriggers P (1998) A segment-to-segment contact strategy. Math Comput Modell Int J 28(4–8):497–515zbMATHGoogle Scholar
  37. 37.
    Wriggers P, Zavarise G (2008) A formulation for frictionless contact problems using a weak form introduced by Nitsche. Comput Mech 41(3):407–420zbMATHGoogle Scholar
  38. 38.
    Konyukhov A, Schweizerhof K (2013) Surface-to-surface contact-various aspects for implementations within the finite element method. Computational contact mechanics. Springer, Berlin, pp 209–291Google Scholar
  39. 39.
    Bernardi C, Maday Y, Patera AT (1994) A new nonconforming approach to domain decomposition: the mortar element method. In: Brezis H, Lions J-L (eds) Collége de France seminar XI. Pitman, New York City, pp 13–51Google Scholar
  40. 40.
    Belgacem FB, Hild P, Laborde P (1997) Approximation of the unilateral contact problem by the mortar finite element method. Comptes Rendus De L’Academie Des Sciences 324(1):123–127MathSciNetzbMATHGoogle Scholar
  41. 41.
    Hild P (2000) Numerical implementation of two nonconforming finite element methods for unilateral contact. Comput Methods Appl Mech Eng 184(1):99–123MathSciNetzbMATHGoogle Scholar
  42. 42.
    Mcdevitt TW, Laursen TA (2000) A mortar-finite element formulation for frictional contact problems. Int J Numer Meth Eng 48(10):1525–1547MathSciNetzbMATHGoogle Scholar
  43. 43.
    Yang B, Laursen TA, Meng X (2005) Two dimensional mortar contact methods for large deformation frictional sliding. Int J Numer Meth Eng 62(9):1183–1225MathSciNetzbMATHGoogle Scholar
  44. 44.
    Puso MA, Laursen TA, Solberg J (2008) A segment-to-segment mortar contact method for quadratic elements and large deformations. Comput Methods Appl Mech Eng 197(6–8):555–566MathSciNetzbMATHGoogle Scholar
  45. 45.
    Hesch C, Betsch P (2009) A mortar method for energy-momentum conserving schemes in frictionless dynamic contact problems. Int J Numer Meth Eng 77(10):1468–1500MathSciNetzbMATHGoogle Scholar
  46. 46.
    Tur M, Fuenmayor FJ, Wriggers P (2009) A mortar-based frictional contact formulation for large deformations using Lagrange multipliers. Comput Methods Appl Mech Eng 198(198):2860–2873MathSciNetzbMATHGoogle Scholar
  47. 47.
    Weißenfels C, Wriggers P (2015) Methods to project plasticity models onto the contact surface applied to soil structure interactions. Comput Geotech 65(2015):187–198Google Scholar
  48. 48.
    Puso MA, Laursen TA (2004) A mortar segment-to-segment frictional contact method for large deformations. Comput Methods Appl Mech Eng 193(45–47):4891–4913MathSciNetzbMATHGoogle Scholar
  49. 49.
    Wohlmuth BI (2000) A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J Numer Anal 38(3):989–1012MathSciNetzbMATHGoogle Scholar
  50. 50.
    Flemisch B, Puso MA, Wohlmuth BI (2005) A new dual mortar method for curved interfaces: 2D elasticity. Int J Numer Meth Eng 63(6):813–832MathSciNetzbMATHGoogle Scholar
  51. 51.
    Lamichhane BP, Stevenson RP, Wohlmuth BI (2005) Higher order mortar finite element methods in 3D with dual Lagrange multiplier bases. Numer Math 102(1):93–121MathSciNetzbMATHGoogle Scholar
  52. 52.
    Flemisch B, Wohlmuth BI (2007) Stable Lagrange multipliers for quadrilateral meshes of curved interfaces in 3D. Comput Methods Appl Mech Eng 196(8):1589–1602MathSciNetzbMATHGoogle Scholar
  53. 53.
    Hartmann S, Ramm E (2008) A mortar based contact formulation for non-linear dynamics using dual Lagrange multipliers. Finite Elem Anal Des 44(5):245–258MathSciNetGoogle Scholar
  54. 54.
    Popp A, Gitterle M, Gee MW et al (2010) A dual mortar approach for 3D finite deformation contact with consistent linearization. Int J Numer Meth Eng 83(11):1428–1465MathSciNetzbMATHGoogle Scholar
  55. 55.
    Doca T, Pires FMA, Sa JMACD (2014) A frictional mortar contact approach for the analysis of large inelastic deformation problems. Int J Solids Struct 51(9):1697–1715Google Scholar
  56. 56.
    Popp A, Wall WA (2014) Dual mortar methods for computational contact mechanics—overview and recent developments. GAMM Mitteilungen 37(1):66–84MathSciNetzbMATHGoogle Scholar
  57. 57.
    Sitzmann S, Willner K, Wohlmuth BI (2015) A dual Lagrange method for contact problems with regularized frictional contact conditions: modelling micro slip. Comput Methods Appl Mech Eng 285(3):468–487MathSciNetzbMATHGoogle Scholar
  58. 58.
    Wohlmuth BI, Krause RH (2003) Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems. SIAM J Sci Comput 25(1):324–347MathSciNetzbMATHGoogle Scholar
  59. 59.
    Cichosz T, Bischoff M (2011) Consistent treatment of boundaries with mortar contact formulations using dual Lagrange multipliers. Comput Methods Appl Mech Eng 200(9):1317–1332MathSciNetzbMATHGoogle Scholar
  60. 60.
    Hüeber S, Wohlmuth BI (2005) A primal-dual active set strategy for non-linear multibody contact problems. Comput Methods Appl Mech Eng 194(27–29):3147–3166MathSciNetzbMATHGoogle Scholar
  61. 61.
    Brunssen S, Schmid F, Schäfer M et al (2006) A fast and robust iterative solver for nonlinear contact problems using a primal-dual active set strategy and algebraic multigrid. Int J Numer Meth Eng 69(3):524–543MathSciNetzbMATHGoogle Scholar
  62. 62.
    Eber S, Stadler G, Wohlmuth BI (2008) A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction. SIAM J Sci Comput 30(2):572–596MathSciNetzbMATHGoogle Scholar
  63. 63.
    Popp A, Gee MW, Wall WA (2010) A finite deformation mortar contact formulation using a primal-dual active set strategy. Int J Numer Meth Eng 79(11):1354–1391MathSciNetzbMATHGoogle Scholar
  64. 64.
    Gitterle M, Popp A, Gee MW et al (2010) Finite deformation frictional mortar contact using a semi-smooth Newton method with consistent linearization. Int J Numer Meth Eng 84(5):543–571MathSciNetzbMATHGoogle Scholar
  65. 65.
    Hartmann S, Brunssen S, Ramm E et al (2007) Unilateral non-linear dynamic contact of thin-walled structures using a primal-dual active set strategy. Int J Numer Meth Eng 70(8):883–912MathSciNetzbMATHGoogle Scholar
  66. 66.
    Yagawa G, Soneda N, Yoshimura S (1991) A large-scale finite element analysis using domain decomposition method on a parallel computer. Comput Struct 1991:615–625zbMATHGoogle Scholar
  67. 67.
    Yagawa G, Yoshinoka A, Yoshimura S, Soneda N (1993) A parallel finite element method with a supercomputer network. Comput Struct 47(3):407–418zbMATHGoogle Scholar
  68. 68.
    Iizuka M, Nakamura H, Garatani K, Nakajima K, Okuda H, Yagawa G (1999) GeoFEM: high performance parallel FEM for geophysical applications. In: High performance computing, second international symposium, ISHPC 99. Lecture notes in computer science, vol 1615, pp 292–303Google Scholar
  69. 69.
    Papadrakakis M, Bitzarakis S (1996) Domain decomposition PCG methods for serial and parallel processing. Adv Eng Softw 25:291–307zbMATHGoogle Scholar
  70. 70.
    Bitzarakis S, Papadrakakis M, Katsopulos A (1997) Parallel solution techniques in computational structural mechanics. Comput Methods Appl Mech Eng 148(1–2):75–105MathSciNetzbMATHGoogle Scholar
  71. 71.
    Farhat C, Crivelli L, Roux FX (1994) A transient FETI methodology for large-scale parallel implicit computations in structural mechanics. Int J Numer Meth Eng 37:1945–1975MathSciNetzbMATHGoogle Scholar
  72. 72.
    Farhat C, Mandel J, Roux FX (1994) Optimal convergence properties of the FEIT domain decomposition method. Comput Methods Appl Mech Eng 115:367–388Google Scholar
  73. 73.
    Yoshimura S, Yamada T, Kawai K, Miyamura T, Ogino M, Shioya R (2015) Petascale coupled simulations of real world’s complex structures. IACM Exp 36:9–13Google Scholar
  74. 74.
    Yoshimura S, Shioya R, Noguchi H, Miyamura T (2002) Advanced general-purpose computational mechanics system for large-scale analysis and design. J Comput Appl Math 49:279–296zbMATHGoogle Scholar
  75. 75.
    Danielson K, Hao S, Liu WK, Uras A, Li SF (2000) Parallel computational of meshless methods for explicit dynamic analysis. Int J Numer Meth Eng 47:1367–1379zbMATHGoogle Scholar
  76. 76.
    Liu WK (1987) Parallel computations for mixed-time integrations. In: Lewis RW, Hinton E, Bettess P, Schrefler BA (eds) Numerical methods for transient and coupled system. Wiley, London, pp 261–277Google Scholar
  77. 77.
    Liu GR (2011) On future computational methods for exascale computer. IACM Exp 30:8–10Google Scholar
  78. 78. Accessed 20 Dec 2017
  79. 79.
    Chen HQ, Ma HF, Tu J, Cheng GQ, Tang JZ (2008) Parallel computation of seismic analysis of high arch dam. Earthq Eng Eng Vib 7:1–11Google Scholar
  80. 80.
    Zhong H, Lin G (2010) Research on parallel computing of damage prediction of high arch dams subjected to earthquakes. Chin J Comput Mech 27:218–224 (in Chinese) Google Scholar
  81. 81.
    Xu XW, Mo ZY (2017) Algebraic interface based coarsening AMG preconditioner for multi-scale sparse matrices with applications to radiation hydrodynamics computation. Numer Linear Algebra Appl 24:2. MathSciNetzbMATHGoogle Scholar
  82. 82.
    Wen LF, Tian R (2016) Improved XFEM: accurate and robust dynamic crack growth simulation. Comput Methods Appl Mech Eng 308:256–285MathSciNetGoogle Scholar
  83. 83.
    Tian R, Yagawa G, Terasaka H (2006) Linear dependence problems of partition of unity based generalized FEMs. Comput Methods Appl Mech Eng 195:4768–4782MathSciNetzbMATHGoogle Scholar
  84. 84.
    Tian R (2013) Extra-dof-free and linearly independent enrichments in GFEM. Comput Methods Appl Mech Eng 266(6):1–22zbMATHGoogle Scholar
  85. 85.
    Tian R, To AC, Liu WK (2011) Conforming local meshfree method. Int J Numer Meth Eng 86(3):335–357zbMATHGoogle Scholar
  86. 86.
    Shi GM, He YB, Wu RA, Mo J, Li YC, Zhang YL (2010) Object-oriented finite element parallel computation framework PANDA. Comput Aided Eng 19(4):8–14 (in Chinese) Google Scholar
  87. 87.
    Xu JG, Shi ZJ, Hao ZM, He YB, Li YF (2010) Design and verification of a nonlinear statics FEM parallel computing code based on PANDA framework. Chin J Solid Mech 31(special issue):294–298 (in Chinese) Google Scholar
  88. 88.
    Liu QK, Zhao WB, Cheng J, et al (2016) A programming framework for large scale numerical simulations on unstructured mesh. In: Proceedings of the 2nd IEEE international conference on high performance and smart computing (IEEE HPSC), New YorkGoogle Scholar
  89. 89.
    Mo ZY, Zhang AQ, Cao XL, Liu QK, Xu XW, An HB, Pei WB, Zhu XP (2000) JASMIN: a parallel software infrastructure for scientific computing. Front Comput Sci China 4(4):480–488Google Scholar
  90. 90.
  91. 91.
  92. 92.
    Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Meth Eng 45(5):601–620zbMATHGoogle Scholar
  93. 93.
    Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Meth Eng 46(1):131–150zbMATHGoogle Scholar
  94. 94.
    Tian R, Wen LF (2015) Improved XFEM–an extra-dof free, well-conditioning, and interpolating XFEM. Comput Methods Appl Mech Eng 285(3):639–658MathSciNetzbMATHGoogle Scholar
  95. 95.
    Sulsky D, Chen Z, Schreyer HL (1994) A particle method for history-dependent materials. Comput Methods Appl Mech Eng 118:179–196MathSciNetzbMATHGoogle Scholar
  96. 96.
    Liu MB, Liu GR (2010) Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch Comput Methods Eng 17(1):25–76MathSciNetzbMATHGoogle Scholar
  97. 97.
    Tian R (2014) Simulation at extreme-scale: co-design thinking and practices. Arch Comput Methods Eng 21(1):39–58Google Scholar
  98. 98.
    Wang YR, Li LS, Tian R (2017) Large-scale parallelization of smoothed particle hydrodynamics method on heterogeneous cluster. In: 46th international conference on parallel processing (ICPP2017), Bristol, UK, 14–17 AugGoogle Scholar
  99. 99.
    Wang YR, Li LS, Wang JT, Tian R (2016) GPU acceleration of smoothed particle hydrodynamics for the Navier–Stokes equations. In: 24th Eruomicro international conference on parallel, distributed, and network-based processing (PDP2016), Greece, pp 478–485.
  100. 100.
    Li LS, Wang YR, Ma ZT, Tian R (2014) petaPar: a scalable Petascale framework for meshfree/particle simulation. In: Proceedings of the 2014 IEEE international symposium on parallel and distributed processing with applications (ISPA’14), pp 50–57Google Scholar
  101. 101.
    Balay S, Abhyankar S, Adams M, et al (2014) PETSc users manual (revision 3.5). Argonne National Laboratory, ANL-95/11Google Scholar
  102. 102.
    Berger M, Bokhari S (1987) A partitioning strategy for nonuniform problems on multiprocessors. IEEE Trans Comput 36:570–580Google Scholar
  103. 103.
    Karypis G, Kumar V (2006) A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J Sci Comput 20(1):359–392MathSciNetzbMATHGoogle Scholar
  104. 104.
    Mazars J (1985) A description of microscale and macroscale damage of concrete structures. Eng Fract Mech 107(1):83–89Google Scholar
  105. 105.
    Duncan JM, Byrne P, Wong KS et al (1981) Strength, stress–strain and bulk modulus parameters for finite element analysis of stress and movements in soil masses. J Consult Clin Psychol 49(4):554–567Google Scholar
  106. 106.
    Yu YZ, Zhang BY, Yuan HN (2007) An intelligent displacement back-analysis method for earth-rockfill dams. Comput Geotech 34(6):423–434. Google Scholar
  107. 107.
    Li GY, Mi ZK, Fu H (2004) Experimental studies on rheological behaviors for rockfills in concrete faced rockfill dam. Rock Soil Mech 25(11):1712–1716Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Rong Tian
    • 1
  • Mozhen Zhou
    • 1
  • Jingtao Wang
    • 1
  • Yang Li
    • 1
  • Hengbin An
    • 1
  • Xiaowen Xu
    • 1
  • Longfei Wen
    • 1
  • Lixiang Wang
    • 1
  • Quan Xu
    • 1
  • Juelin Leng
    • 1
  • Ran Xu
    • 1
  • Bingyin Zhang
    • 2
  • Weijie Liu
    • 1
  • Zeyao Mo
    • 1
    Email author
  1. 1.Software Center for High Performance Numerical SimulationIAPCMBeijingChina
  2. 2.State Key Laboratory of Hydroscience and EngineeringTsinghua UniversityBeijingChina

Personalised recommendations