Thermal fluid-structure interaction and coupled thermal-stress analysis in a cable stayed bridge exposed to fire

Abstract

In this paper, thermal fluid structure-interaction (TFSI) and coupled thermal-stress analysis are utilized to identify the effects of transient and steady-state heat-transfer on the vortex induced vibration and fatigue of a segmental bridge deck due to fire incidents. Numerical simulations of TFSI models of the deck are dedicated to calculate the lift and drag forces in addition to determining the lock-in regions once using fluid-structure interaction (FSI) models and another using TFSI models. Vorticity and thermal convection fields of three fire scenarios are simulated and analyzed. Simiu and Scanlan benchmark is used to validate the TFSI models, where a good agreement was manifested between the two results. Extended finite element method (XFEM) is adopted to create 3D models of the cable stayed bridge to simulate the fatigue of the deck considering three fire scenarios. Choi and Shin benchmark is used to validate the damaged models of the deck in which a good coincide was seen between them. The results revealed that TFSI models and coupled thermal-stress models are significant in detecting earlier vortex induced vibration and lock-in regions in addition to predicting damages and fatigue of the deck due to fire incidents.

Graphical abstract

Keywords

fire scenario / transient heat transfer / TFSI model / coupled thermal-stress / XFEM

References

[1] Baba S, Kajita T, Ninomiya K. Fire Under a Long San Bridge. 13 th International Association for Bridge and Structural Engineering (IABSE) Congress, Helsinki. 6–10 June 1988

[2] Potgieter I C, Gamble W L. Nonlinear temperature distributions in bridges at different locations in the United States. PCI Journal, 1989, 34(4): 80–103

[3] Bennetts I, Moinuddin K. Evaluation of the impact of potential fire scenarios on structural elements of a cable-stayed bridge. Journal of Fire Protection Engineering, 2009, 19(2): 85–106

[4] Payá-Zaforteza I, Garlock M. A numerical investigation on the fire response of a Steel Girder Bridge. Journal of Constructional Steel Research, 2012, 75: 93–103

[5] Garlock M, Paya-Zaforteza I, Kodur V, Gu L. Fire hazard in bridges: review, assessment and repair strategies. Engineering Structures, 2012, 35: 89–98

[6] Zhou G D,Yi T H. Thermal load in large-scale bridges: a state-of-the-art review. International Journal of Distributed Sensor Networks, 2013, 217983

[7] Peris-Sayol G,Paya-Zaforteza I,Alos-Moya J. Analysis of the Response of a Steel Girder Bridge to Different Tanker Fires Depending on its Structural Boundary Conditions. 8th International Conference on Structures in Fire, Shanghai, China. 11–13 June, 2014

[8] Braxtan N L, Wang Q, Whitney R, Koch G. Numerical Analysis of a Composite Steel Box Girder Bridge in Fire. Applications of Structural Fire Engineering, Dubrovnic, Croatia. 15–16 October 2015

[9] Zhou L, Xia Y, Brownjohn J M W, Koo K Y. Temperature analysis of a long-span suspension bridge based on field monitoring and numerical simulation. Journal of Bridge Engineering, 2016, 21(1): 04015027

[10] Kodur V K, Agrawal A. Critical factors governing the residual response of reinforced concrete beams exposed to fire. Fire Technology, 2016, 52(4): 967–993

[11] Schumacher J. Assessment of Bridge-Structures Under Fired Impact: A Case Study Approach. Dissertation for M.Sc. degree. University of Rhode Island, Kingston, USA. 2016

[12] Choi J. Concurrent Fire Dynamics Models and Thermo-Mechanical Analysis of Steel and Concrete Structures. Dissertation for PhD degree. Georgia Institute of Technology, Georgia, USA. 2008

[13] Giuliani L, Crosti C, Gentili F.  Vulnerability of Bridges to Fire. Proceedings of the 6th International Conference on Bridge Maintenance, Safety and Management (IABMAS), Stresa, Lake Maggiore, Italy. 8–12 July, 2012

[14] Wright W, Lattimer B, Woodworth M,Nahid M, Sotelino E. Highway Bridge Fire Hazard Assessment, Blacksburg, VA, USA: Virginia Polytechnic Institute and State University. 2013

[15] Dotreppe J C, Majkut S, Franssen J M. Failure of a tied-arch bridge submitted to a severe localized fire. Structures and Extreme Events, IABSE Symposium, Budapest, Hungary, September 2006. 92, 272–273

[16] Kodur V, Aziz E, Dwaikat M. Evaluating fire resistance of steel girders in bridges. Journal of Bridge Engineering, 2013, 18(7): 633–643

[17] Zhuang X, Huang R, LiangC, Rabczuk T. A coupled thermo-hydro-mechanical model of jointed hard rock for compressed air energy storage. Mathematical Problems in Engineering, 2014, 2014: 179169

[18] Rabizadeh E, Bagherzadeh A S, Rabczuk T. Goal-oriented error estimation and adaptive mesh refinement in dynamic coupled thermoelasticity. Computers & Structures, 2016, 173: 187–211

[19] Amiri F, Millan D, Arroyo M, Silani M, Rabczuk T. Fourth order phase-field model for local max-ent approximants applied to crack propagation. Computer Methods in Applied Mechanics and Engineering, 2016, 312(C): 254–275

[20] Amiri F, Millán D, Shen Y, Rabczuk T, Arroyo M. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 2014, 69: 102–109

[21] Rabczuk T, Belytschko T,Xiao S P. Stable particle methods based on Lagrangian kernels. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12–14): 1035–1063

[22] Rabczuk T, Belytschko T. Adaptivity for structured meshfree particle methods in 2D and 3D. International Journal for Numerical Methods in Engineering, 2005, 63(11): 1559–1582

[23] Vu-Bac N, Nguyen-Xuan H, Chen L, Lee C K, Zi G, Zhuang X, Liu G R, Rabczuk T. A phantom-node method with edge-based strain smoothing for linear elastic fracture mechanics. Journal of Applied Mathematics, 2013, 2013: 978026

[24] Chau-Dinh T, Zi G, Lee P S, RabczukT, Song J H. Phantom-node method for shell models with arbitrary cracks. Computers & Structures, 2012, 92–93: 242–256

[25] Rabczuk T, Zi G, Gerstenberger A, Wall W A. A new crack tip element for the phantom node method with arbitrary cohesive cracks. International Journal for Numerical Methods in Engineering, 2008, 75(5): 577–599

[26] Rabczuk T, Belytschko T. Application of particle methods to static fracture of reinforced concrete structures. International Journal of Fracture, 2006, 137(1–4): 19–49

[27] Rabczuk T, Areias P M A. A meshfree thin shell for arbitrary evolving cracks based on an external enrichment. CMES-Computer Modeling in Engineering and Sciences, 2000, 1(1): 11–26

[28] Rabczuk T, Bordas S, Zi G. A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics. Computational Mechanics, 2007, 40(3): 473–495

[29] Zi G, Rabczuk T, Wall W A. Extended meshfree methods without branch enrichment for cohesive cracks. Computational Mechanics, 2007, 40(2): 367–382

[30] Rabczuk T, Zi G. A meshfree method based on the local partition of unity for cohesive cracks. Computational Mechanics, 2007, 39(6): 743–760

[31] Rabczuk T, Areias P M A, Belytschko T. A meshfree thin shell method for nonlinear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548

[32] Bordas S, Rabczuk T, Zi G. Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree methods without asymptotic enrichment. Engineering Fracture Mechanics, 2008, 75(5): 943–960

[33] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A geometrically non-linear three dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 2008, 75(16): 4740–4758

[34] Rabczuk T, Bordas S, Zi G. On three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 2010, 88(23–24): 1391–1411

[35] Amiri F, Anitescu C, Arroyo M, Bordas S, Rabczuk T. XLME interpolants, a seamless bridge between XFEM and enriched meshless methods. Computational Mechanics, 2014, 53(1): 45– 57

[36] Areias P, Rabczuk T, Msekh M. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312(C): 322–350

[37] Areias P, Reinoso J, Camanho P, Rabczuk T. A constitutive-based element-by-element crack propagation algorithm with local mesh refinement. Computational Mechanics, 2015, 56(2): 291–315

[38] Areias P, Rabczuk T, Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137

[39] Areias P, Rabczuk T, Camanho P P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947

[40] Areias P, Rabczuk T. Finite strain fracture of plates and shells with configurational forces and edge rotation. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122

[41] Areias P, Rabczuk T, de Sá J C. A novel two-stage discrete crack method based on the screened Poisson equation and local mesh refinement. Computational Mechanics, 2016, 58(6): 1003–1018

[42] Areias P, Msekh M A, Rabczuk T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143

[43] Rabczuk T, Belytschko T. Cracking particles: a simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[44] Rabczuk T, Areias P M A. A new approach for modelling slip lines in geological materials with cohesive models. International Journal for Numerical and Analytical Methods in Engineering, 2006, 30(11): 1159–1172

[45] Rabczuk T, Belytschko T. A three dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29–30): 2777–2799

[46] Rabczuk T, Areias P M A, Belytschko T. A simplified meshfree method for shear bands with cohesive surfaces. International Journal for Numerical Methods in Engineering, 2007, 69(5): 993–1021

[47] Rabczuk T, Samaniego E. Discontinuous modelling of shear bands using adaptive meshfree methods. Computer Methods in Applied Mechanics and Engineering, 2008, 197(6–8): 641–658

[48] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A simple and robust three dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37–40): 2437–2455

[49] Areias P M A, Rabczuk T, Camanho P P. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63

[50] ThaiT Q, Rabczuk T, Bazilevs Y, Meschke G. A higher-order stress-based gradient-enhanced damage model based on isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2016, 304: 584–604

[51] Ghorashi S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based XIGA for fracture analysis of orthotropic media. Computers & Structures, 2015, 147: 138–146

[52] Jia Y, Anitescu C, Ghorashi S, Rabczuk T. Extended isogeometric analysis for material interface problems. IMA Journal of Applied Mathematics, 2015, 80(3): 608–633

[53] Nguyen-Thanh N, Valizadeh N, Nguyen M N, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T. An extended isogeometric thin shell analysis based on Kirchhoff-Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291

[54] Budarapu P, Gracie R, Bordas S, Rabczuk T. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 2014, 53(6): 1129–1148

[55] Budarapu P, Gracie R, Yang S W , Zhuang X , Rabczuk T. Efficient coarse graining in multiscale modeling of fracture. Theoretical and Applied Fracture Mechanics, 2014, 69: 126–143

[56] Silani M, Talebi H, Hamouda A S, Rabczuk T. Nonlocal damage modelling in clay/epoxy nanocomposites using a multiscale approach. Journal of Computational Science, 2016, 15: 18–23

[57] Talebi H, Silani M, Rabczuk T. Concurrent multiscale modelling of three dimensional crack and dislocation propagation. Advances in Engineering Software, 2015, 80: 82–92

[58] Silani M, Ziaei-Rad S, Talebi H, Rabczuk T. A semi-concurrent multiscale approach for modeling damage in nanocomposites. Theoretical and Applied Fracture Mechanics, 2014, 74: 30–38

[59] Talebi H, Silani M, Bordas S, Kerfriden P, Rabczuk T. A computational library for multiscale modelling of material failure. Computational Mechanics, 2014, 53(5): 1047–1071

[60] Talebi H, Silani M, Bordas S P A, Kerfriden P, Rabczuk T. Molecular dynamics/XFEM coupling by a three-dimensional extended bridging domain with applications to dynamic brittle fracture. International Journal for Multiscale Computational Engineering, 2013, 11(6): 527–541

[61] Amani J, Oterkus E, Areias P M A, Zi G, Nguyen-Thoi T, Rabczuk T. A non-ordinary state-based peridynamics formulation for thermoplastic fracture. International Journal of Impact Engineering, 2016, 87: 83–94

[62] Ren H, Zhuang X, Rabczuk T. Dual-horizon peridynamics: a stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782

[63] Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-Horizon Peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476

[64] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

[65] Vu-Bac N, Rafiee R, Zhuang X, Lahmer T, Rabczuk T. Uncertainty quantification for multiscale modeling of polymer nanocomposites with correlated parameters. Composites. Part B, Engineering, 2015, 68: 446–464

[66] Nanthakumar S, Valizadeh N, Park H, Rabczuk T. Surface effects on shape and topology optimization of nanostructures. Computational Mechanics, 2015, 56(1): 97–112

[67] Ghasemi H, Park H S, Rabczuk T. A level-set based IGA formulation for topology optimization of exoelectric materials. Computer Methods in Applied Mechanics and Engineering, 2017, 313: 239–258

[68] Anitescu C, Jia Y, Zhang Y, Rabczuk T. An isogeometric collocation method using superconvergent points. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 1073–1097

[69] Chan C L, Anitescu C, Rabczuk T. Volumetric parametrization from a level set boundary representation with a PHT-splines. Computer Aided Design, 2017, 82: 29–41

[70] Nguyen V P, Anitescu C, Bordas S P A, Rabczuk T. Isogeometric analysis: an overview and computer implementation aspects. Mathematics and Computers in Simulations, 2015, 117, art. no. 4190, 89–116

[71] Valizadeh N, Bazilevs Y, Chen J S, Rabczuk T. A coupled IGA-meshfree discretization of arbitrary order of accuracy and without global geometry parameterization. Computer Methods in Applied Mechanics and Engineering, 2015, 293: 20–37

[72] Nanthakumar S S, Lahmer T, Rabczuk T. Detection of flaws in piezoelectric structures using extended FEM. International Journal for Numerical Methods in Engineering, 2013, 96(6): 373–389

[73] Vu-Bac N, Duong T X, Lahmer T, Zhuang X, Sauer R A, Park H S, Rabczuk T. A NURBS-based inverse analysis for reconstruction of nonlinear deformations in thin shell structures. Computer Methods in Applied Mechanics and Engineering, 2017, http://doi.org/10.1016/j.cma.2017.09.034

[74] Anitescu C, Hossain N, Rabczuk T. Recovery-based error estimation and adaptivity using high-order splines over hierarchical T-meshes. Computer Methods in Applied Mechanics and Engineering, 2018, 324: 638–662

[75] Msekh M A, Nguyen-Cuong H, Zi G, Areias P, Zhuang X, Rabczuk T. Fracture properties prediction of clay/epoxy nanocomposites with interphase zones using a phase field model. Engineering Fracture Mechanics.

[76] Rabczuk T, Ren H. A peridynamics formulation for quasi-static fracture and contact in rock. Engineering Geology, 2017, 225: 42–48

[77] Hamdia K, Silani M, Zhuang X, He P, Rabczuk T. Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions. International Journal of Fracture, 2017, 206(2): 215–227

[78] Areias P, Rabczuk T. Steiner-point free edge cutting of tetrahedral meshes with applications in fracture. Finite Elements in Analysis and Design, 2017, 132: 27–41

[79] Rabczuk T, Gracie R, Song J H, Belytschko T. Immersed particle method for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 2010, 81(1): 48–71

[80] Carstensen J V. Material Modelling of Reinforced Concrete at Elevated Temperatures. Dissertation for M.Sc degree. Department of Civil Engineering, Technical University of, Denmark, Lyngby, Denmark. 2011

[81] Sangluaia C, Haridharan M K, Natarajan C, Rajaraman A. Behaviour of reinforced concrete slab subjected to fire. International Journal of Computational Engineering Research, 2013, 3(1): 195–206

[82] Babu N R C, Haridharan M K, Natarajan C. Temperature distribution in concrete slabs exposed to elevated temperature. International Journal of Engineering Science Invention, 2014, 3(3): 35–43

[83] Palmklint E, Svensson M. Thermal and Mechanical Response of Composite Slabs Exposed to Travelling Fire. Dissertation for M.Sc. degree. Department of Civil, Environmental and Natural Resources Engineering, Lulea University of Technology, Lulea, Sweden. 2016

[84] Nakne J K D. FE-Analysis of a Beam-Column Connection in Composite Structures exposed to Fire. Dissertation for M.Sc. degree. Department of Civil and Environmental Engineering, Chalmers University of Technology. Chalmers, Sweden, 2011

[85] Allam S M, Elbakry H M F, Rabeai A G. Behavior of one-way reinforced concrete slabs subjected to fire. Alexandria Engineering Journal, 2013, 52(4): 749–761

[86] Patade H K, Chakrabarti M A. Thermal stress analysis of beam subjected to fire. Journal of Engineering Research and Applications, 2013, 3(5): 420–424

[87] Ab Kadir M A. Fire Resistance of Earthquake Damaged Reinforced Concrete Frames. Dissertation for PhD degree. The University of Edinburgh, Edinburgh, UK. 2013

[88] Elbadry M M, Ghali A. Temperature variations in concrete bridges. Journal of Structural Engineering, 1983, 109(10): 2355–2374

[89] Lienhard J, Lienhard J. A Heat Transfer Textbook (3rd ed). Cambridge: Phlogiston Press. 2003

[90] Choi J, Haj-Ali R, Kim H S. Integrated fire dynamic and thermo-mechanical modeling of a bridge under fire. Structural Engineering and Mechanics, 2012, 42(5): 815–829

[91] Pironkov P. Numerical Simulation of Thermal Fluid-Structure-Interaction. Dissertation for PhD degree. Department of Mechanical Engineering, Technische Universitat Darmstadt, Darmstadt, Germany. 2010

[92] Grilli M, Hickel S, Adams N A. An innovative approach to thermo-fluid-structure interaction based on an immersed interface method and a monolithic thermo-structure interaction algorithm. 42nd AIAA Fluid Dynamics Conference and Exhibit. Louisiana, USA. 25–28 June, 2012

[93] Yin L, Jiang J, Chen L. A monolithic solution procedure for a thermal fluid-structure interaction system of thermal shock. Procedia Engineering, 2012, 31: 1131–1139

[94] Gleim T, Birken P, Weiland M, Kuhl D, Meister A, Wunsch O. Thermal fluid-structure-interaction–Experimental and numerical analysis. Proceedings of the International Conference on Numerical Analysis and Applied Mathematics (ICNAAM), Rhodes, Greece. 22–28 September, 2014

[95] Birken P, Gleim T, Kuhl D, Meister A. Fast solvers for unsteady thermal fluid structure interaction. International Journal for Numerical Methods in Fluids, 2015, 79(1): 16–29

[96] Talebi E, Tahir M, Zahmatkesh F, Kueh A B H. Fire response of a 3D multi-storey building with buckling restrained braces. Latin American Journal of Solids and Structures, 2015, 12(11): 2118–2142

[97] Stratford T, Dhakal R P. Spalling of Concrete: Implications for Structural Performance in Fire. 20th Australasian Conference on Mechanics of Structure and Materials, Toowoomba, Australia. 2–5 December, 2008

[98] Bo S. Finite Element Simulation of Fire Induced Spalling in High Strength Concrete Slabs. Dissertation for M.Sc. degree. Department of Civil and Environmental Engineering, Lehigh University, Pennsylvania, USA. 2011

[99] Ervine A. Damaged Reinforced Concrete Structures in Fire. Dissertation for PhD degree. Department of Civil Engineering, University of Edinburgh, Edinburgh, UK. 2012

[100] Chung C H, Im C R, Park J. Structural test and analysis of RC slab after fire loading. Nuclear Engineering and Technology, 2013, 45(2): 223–236

[101] Mueller K A, Kurama Y C. (2014). Through-thickness thermal behavior of two RC bearing walls under fire. Structures Congress, 2014, 1159–1169

[102] Klingsch E W H. Explosive Spalling of Concrete in Fire. Dissertation for PhD degree. Institute of Structural Engineering, ETH Zurich, Zurch, Switzerland. 2014

[103] Khoury G A. Effect of fire on concrete and concrete structures. Progress in Structural Engineering and Materials, 2000, 2(4): 429–447

[104] Akhtaruzzaman A A, Sullivan P J. Explosive spalling of concrete exposed to high temperature. Concrete Structures and Technology Research Report. London, Imperial. 1970

[105] Choi E G, Shin Y S. The structural behavior and simplified thermal analysis of normal-strength and high-strength concrete beams under fire. Engineering Structures, 2011, 33(4): 1123–1132

[106] Alos-Moya J, Paya-Zaforteza I, Garlock M, Loma-Ossorio E, Schiffner D, Hospitaler A. Analysis of a bridge failure due to fire using computational fluid dynamics and finite element models. Engineering Structures, 2014, 68: 96–110

[107] Braxtan N L, Whitney R, Wang Q, Koch G. Preliminary investigation of composite steel box girder bridges in fire. Bridge Structures (Abingdon), 2015, 11(3): 105–114

[108] Nariman N A. Control efficiency optimization and Sobol’s sensitivity indices of MTMDs design parameters for buffeting and flutter vibrations in a cable stayed bridge. Frontiers of Structural and Civil Engineering, 2017, 11(1): 66–89

[109] Nariman N A. Influence of fluid-structure interaction on vortex induced vibration and lock-in phenomena in long span bridges. Frontiers of Structural and Civil Engineering, 2016, 10(4): 363–384

[110] Nariman N A. A novel structural modification to eliminate the early coupling between bending and torsional mode shapes in a cable stayed bridge. Frontiers of Structural and Civil Engineering, 2017, 11(2): 131–142

[111] Nariman N A. Kinetic Energy Based Model Assessment and Sensitivity Analysis of Vortex Induced Vibration of Segmental Bridge Decks. Frontiers of Structural and Civil Engineering, 2017, https://doi.org/10.1007/s11709-017-0435-5

[112] Nariman N A. Aerodynamic stability parameters optimization and global sensitivity analysis for a cable stayed bridge. KSCE Journal of Civil Engineering, 2016, https://doi.org/10.1007/s12205-016-0962-y

[113] Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 1999, 45(5): 601–620

[114] Dolbow J. An extended finite element method with discontinuous enrichment for applied mechanics. Dissertation for PhD degree. Northwestern University, Illinois, USA. 1999

[115] Dolbow J, Moës N, Belytschko T. Discontinuous enrichment infinite elements with a partition of unity method. Finite Elements in Analysis and Design, 2000, 36(3–4): 235–260

[116] Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 1999, 46: 131–150

[117] Sukumar N, Moës N, Moran B, Belytschko T. Extended finite element method for three dimensional crack modelling. International Journal for Numerical Methods in Engineering, 2000, 48: 1549–1570

[118] Belytschko T, Moës N, Usui S, Parimi C. Arbitrary discontinuities in finite elements. International Journal for Numerical Methods in Engineering, 2001, 50: 993–1013

[119] Stolarska M, Chopp D L, Moës N, Belytschko T. Modelling crack growth by level sets in the extended finite element method. International Journal for Numerical Methods in Engineering, 2001, 51(8): 943–960

[120] Osher S, Sethian J. Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics, 1988, 79(1): 12–49

[121] Ventura G, Budyn E, Belytschko T. Vector level sets for description of propagating crack in finite elements. International Journal for Numerical Methods in Engineering, 2003, 58(10): 1571–1592

[122] McNary M J. Implementation of the Extended Finite Element Method XFEM in the ABAQUS software Package. Dissertation for MSc degree. Department of mechanical engineering, Georgia Institute of Technology, Georgia, USA. 2009

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