Abstract
We present a proof-of-concept particle-based fluid–structure interaction (FSI) computational framework for modeling structural fracture and fragmentation under the impact of extreme hydrodynamic events. The smoothed particle hydrodynamics (SPH) approach is employed to discretize the equations of motion for both the fluid and structural domains. The meshfree nature of the discretization technique accommodates the simulation of scenarios involving extreme structural deformations and material separation, as well as free-surface flows. The framework is supplemented with a phase-field model of brittle fracture that allows for the simulation of crack nucleation, propagation, and branching, which leads to realistic modeling of structural responses during extreme hydrodynamic events. In the end, a novel algorithm for coupling the fluid and solid subproblems is presented. The proposed approach is verified and validated against existing computational methods and experimental results, and in the end, a few challenging problems involving complex fracture patterns and fragmentation are presented.
Graphical abstract









































Similar content being viewed by others
References
Najibi N, Devineni N (2018) Recent trends in the frequency and duration of global floods. Earth Syst Dyn 9(2):757–783
NOAA NCEI (2022) Noaa national centers for environmental information (NCEI) U.S. billion-dollar weather and climate disasters
Monaghan JJ (1994) Simulating free surface flows with SPH. J Comput Phys 110(2):399–406
Chen LF, Zang J, Hillis AJ, Morgan GCJ, Plummer AR (2014) Numerical investigation of wave-structure interaction using OpenFoam. Ocean Eng 88:91–109
Wei Z, Dalrymple RA, Hérault A, Bilotta G, Rustico E, Yeh H (2015) SPH modeling of dynamic impact of tsunami bore on bridge piers. Coast Eng 104:26–42
Akbari H (2017) Simulation of wave overtopping using an improved SPH method. Coast Eng 126:51–68
Sarfaraz M, Pak A (2017) SPH numerical simulation of tsunami wave forces impinged on bridge superstructures. Coast Eng 121:145–157
Arabi MG, Sogut DV, Khosronejad A, Yalciner AC, Farhadzadeh A (2019) A numerical and experimental study of local hydrodynamics due to interactions between a solitary wave and an impervious structure. Coast Eng 147:43–62
Xie Z, Stoesser T (2020) Two-phase flow simulation of breaking solitary waves over surface-piercing and submerged conical structures. Ocean Eng 213:107679
Bagherizadeh E, Zhang Z, Farhadzadeh A, Angelidis D, Ghazian Arabi M, Moghimi S, Khosronejad A (2021) Numerical modelling of solitary wave and structure interactions using level-set and immersed boundary methods by adopting adequate inlet boundary conditions. J Hydraul Res 59(4):559–585
Sogut DV, Sogut E, Farhadzadeh A (2021) Interaction of a solitary wave with an array of macro-roughness elements in the presence of steady currents. Coast Eng 164:103829
Sogut E, Sogut DV, Farhadzadeh A (2021) A comparative study of interaction of random waves with a cluster of structures on a berm with rans and les models. Coast Eng 168:103941
Oñate E, Garcıa J (2001) A finite element method for fluid–structure interaction with surface waves using a finite calculus formulation. Comput Methods Appl Mech Eng 191(6–7):635–660
Wall WA, Genkinger S, Ramm E (2007) A strong coupling partitioned approach for fluid–structure interaction with free surfaces. Comput Fluids 36(1):169–183
Yan J, Korobenko A, Deng X, Bazilevs Y (2016) Computational free-surface fluid–structure interaction with application to floating offshore wind turbines. Comput Fluids 141:155–174
Yan J, Deng X, Korobenko A, Bazilevs Y (2017) Free-surface flow modeling and simulation of horizontal-axis tidal-stream turbines. Comput Fluids 158:157–166
Bazilevs Y, Yan J, Deng X, Korobenko A (2019) Computer modeling of wind turbines: 2. Free-surface FSI and fatigue-damage. Arch Comput Methods Eng 26(4):1101–1115
Yan J, Deng X, Fei X, Songzhe X, Zhu Q (2020) Numerical simulations of two back-to-back horizontal axis tidal stream turbines in free-surface flows. J Appl Mech 87(6)
Chen Z-P, Zhang X, Sze KY, Kan L, Qiu X-M (2018) vp material point method for weakly compressible problems. Comput Fluids 176:170–181
Idelsohn SR, Oñate E, Del Pin F, Calvo N (2006) Fluid–structure interaction using the particle finite element method. Comput Methods Appl Mech Eng 195(17–18):2100–2123
Yang X, Liu M, Peng S, Huang C (2016) Numerical modeling of dam-break flow impacting on flexible structures using an improved SPH-EBG method. Coast Eng 108:56–64
Khayyer A, Gotoh H, Falahaty H, Shimizu Y (2018) An enhanced ISPH-SPH coupled method for simulation of incompressible fluid-elastic structure interactions. Comput Phys Commun 232:139–164
Zhan L, Peng C, Zhang B, Wei W (2019) A stabilized TL-WC SPH approach with GPU acceleration for three-dimensional fluid–structure interaction. J Fluids Struct 86:329–353
Sun PN, Le Touzé D, Zhang A-M (2019) Study of a complex fluid–structure dam-breaking benchmark problem using a multi-phase SPH method with APR. Eng Anal Bound Elem 104:240–258
Ng KC, Alexiadis A, Chen H, Sheu TWH (2020) A coupled smoothed particle hydrodynamics-volume compensated particle method (SPH-VCPM) for fluid structure interaction (FSI) modelling. Ocean Eng 218:107923
Sun P-N, Le Touze D, Oger G, Zhang A-M (2021) An accurate FSI-SPH modeling of challenging fluid–structure interaction problems in two and three dimensions. Ocean Eng 221:108552
Khayyer A, Shimizu Y, Gotoh H, Hattori S (2021) Multi-resolution ISPH-SPH for accurate and efficient simulation of hydroelastic fluid–structure interactions in ocean engineering. Ocean Eng 226:108652
O’Connor J, Rogers BD (2021) A fluid–structure interaction model for free-surface flows and flexible structures using smoothed particle hydrodynamics on a GPU. J Fluids Struct 104:103312
Lyu H-G, Sun P-N, Huang X-T, Chen S-H, Zhang A-M (2021) On removing the numerical instability induced by negative pressures in SPH simulations of typical fluid–structure interaction problems in ocean engineering. Appl Ocean Res 117:102938
O’Connor J, Revell A (2019) Dynamic interactions of multiple wall-mounted flexible flaps. J Fluid Mech 870:189–216
Yang Q, Jones V, McCue L (2012) Free-surface flow interactions with deformable structures using an SPH-FEM model. Ocean Eng 55:136–147
McLoone M, Quinlan NJ (2022) Coupling of the meshless finite volume particle method and the finite element method for fluid–structure interaction with thin elastic structures. Eur J Mech B Fluids 92:117–131
Gotoh H, Khayyer A, Shimizu Y (2021) Entirely Lagrangian meshfree computational methods for hydroelastic fluid–structure interactions in ocean engineering-reliability, adaptivity and generality. Appl Ocean Res 115:102822
Khayyer A, Gotoh H, Shimizu Y (2022) On systematic development of FSI solvers in the context of particle methods. J Hydrodyn 34(3):395–407
Cirak F, Deiterding R, Mauch SP (2007) Large-scale fluid–structure interaction simulation of viscoplastic and fracturing thin-shells subjected to shocks and detonations. Comput Struct 85(11–14):1049–1065
Wang KG, Lea P, Farhat C (2015) A computational framework for the simulation of high-speed multi-material fluid–structure interaction problems with dynamic fracture. Int J Numer Methods Eng 104(7):585–623
Moutsanidis G, Kamensky D, Chen JS, Bazilevs Y (2018) Hyperbolic phase field modeling of brittle fracture: Part II-immersed IGA–RKPM coupling for air-blast–structure interaction. J Mech Phys Solids 121:114–132
Behzadinasab M, Moutsanidis G, Trask N, Foster JT, Bazilevs Y (2021) Coupling of IGA and peridynamics for air-blast fluid–structure interaction using an immersed approach. Forces Mech 4:100045
Ni R, Li J, Zhang X, Zhou X, Cui X (2022) An immersed boundary-material point method for shock-structure interaction and dynamic fracture. J Comput Phys 470:111558
Rahimi MN, Kolukisa DC, Yildiz M, Ozbulut M, Kefal A (2022) A generalized hybrid smoothed particle hydrodynamics-peridynamics algorithm with a novel Lagrangian mapping for solution and failure analysis of fluid–structure interaction problems. Comput Methods Appl Mech Eng 389:114370
Yao X, Huang D (2022) Coupled PD-SPH modeling for fluid–structure interaction problems with large deformation and fracturing. Comput Struct 270:106847
Sun W-K, Zhang L-W, Liew KM (2022) A coupled SPH-PD model for fluid–structure interaction in an irregular channel flow considering the structural failure. Comput Methods Appl Mech Eng 401:115573
Ke W, Yang D, Wright N (2016) A coupled SPH-DEM model for fluid–structure interaction problems with free-surface flow and structural failure. Comput Struct 177:141–161
Ng KC, Low WC, Chen H, Tafuni A, Nakayama A (2022) A three-dimensional fluid–structure interaction model based on SPH and lattice-spring method for simulating complex hydroelastic problems. Ocean Eng 260:112026
Yang F, Xin G, Xia X, Zhang Q (2022) A peridynamics-immersed boundary-lattice Boltzmann method for fluid–structure interaction analysis. Ocean Eng 264:112528
Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217:77–95
Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc 181(3):375–389
Libersky LD, Petschek AG, Carney TC, Hipp JR, Allahdadi FA (1993) High strain Lagrangian hydrodynamics: a three-dimensional SPH code for dynamic material response. J Comput Phys 109(1):67–75
Monaghan JJ, Kos A (1999) Solitary waves on a cretan beach. J Waterw Port Coast Ocean Eng 125(3):145–155
Vignjevic R, Reveles JR, Campbell J (2006) SPH in a total Lagrangian formalism. CMC-Tech Science Press- 4(3):181
Liu MB, Liu GR (2010) Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch Comput Methods Eng 17(1):25–76
Vacondio R, Rogers BD, Stansby PK, Mignosa P, Feldman J (2013) Variable resolution for SPH: a dynamic particle coalescing and splitting scheme. Comput Methods Appl Mech Eng 256:132–148
Crespo AJC, Domínguez JM, Rogers BD, Gómez-Gesteira M, Longshaw S, Canelas RJFB, Vacondio R, Barreiro A, García-Feal O (2015) Dualsphysics: Open-source parallel CFD solver based on smoothed particle hydrodynamics (SPH). Comput Phys Commun 187:204–216
Fourtakas G, Stansby PK, Rogers BD, Lind SJ, Yan S, Ma Q (2018) On the coupling of incompressible SPH with a finite element potential flow solver for nonlinear free-surface flows. Int J Offsh Polar Eng 28(03):248–254
Fourtakas G, Rogers BD, Nasar AMA (2021) Towards pseudo-spectral incompressible smoothed particle hydrodynamics (ISPH). Comput Phys Commun 266:108028
King JRC, Lind SJ, Rogers BD, Stansby PK, Vacondio R (2022) Large eddy simulations of bubbly flows and breaking waves with smoothed particle hydrodynamics. arXiv:2206.01641
Feng R, Fourtakas G, Rogers BD, Lombardi D (2022) Two-phase fully-coupled smoothed particle hydrodynamics (SPH) model for unsaturated soils and its application to rainfall-induced slope collapse. Comput Geotech 151:104964
Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199(45–48):2765–2778
Borden MJ, Hughes TJR, Landis CM, Verhoosel CV (2014) A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework. Comput Methods Appl Mech Eng 273:100–118
Ambati M, Gerasimov T, De Lorenzis L (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55(2):383–405
Kakouris EG, Triantafyllou SP (2017) Phase-field material point method for brittle fracture. Int J Numer Methods Eng 112(12):1750–1776
Kamensky D, Moutsanidis G, Bazilevs Y (2018) Hyperbolic phase field modeling of brittle fracture: part I-theory and simulations. J Mech Phys Solids 121:81–98
Svolos L, Bronkhorst CA, Waisman H (2020) Thermal-conductivity degradation across cracks in coupled thermo-mechanical systems modeled by the phase-field fracture method. J Mech Phys Solids 137:103861
Kristensen PK, Niordson CF, Martínez-Paneda E (2020) Applications of phase field fracture in modelling hydrogen assisted failures. Theor Appl Fract Mech 110:102837
Cui C, Ma R, Martínez-Pañeda E (2021) A phase field formulation for dissolution-driven stress corrosion cracking. J Mech Phys Solids 147:104254
Svolos L, Mourad HM, Bronkhorst CA, Waisman H (2021) Anisotropic thermal-conductivity degradation in the phase-field method accounting for crack directionality. Eng Fract Mech 245:107554
Svolos L, Mourad HM, Manzini G, Garikipati K (2022) A fourth-order phase-field fracture model: formulation and numerical solution using a continuous/discontinuous galerkin method. J Mech Phys Solids 165:104910
Rahimi MN, Moutsanidis G (2022) A smoothed particle hydrodynamics approach for phase field modeling of brittle fracture. Comput Methods Appl Mech Eng 398:115191
Dalrymple RA, Rogers BD (2006) Numerical modeling of water waves with the SPH method. Coast Eng 53(2–3):141–147
Smagorinsky J (1963) General circulation experiments with the primitive equations: I. The basic experiment. Mon Weather Rev 91(3):99–164
Griffith AA (1921) Vi. The phenomena of rupture and flow in solids. Philos Trans R Soc Lond Ser A 221(582–593):163–198
Bourdin B, Francfort GA, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826
Kuhn C, Müller R (2008) A phase field model for fracture. In PAMM: Proceedings in applied mathematics and mechanics, vol 8. Wiley Online Library. p 10223–10224
Rahimi MN, Moutsanidis G (2022) Modeling dynamic brittle fracture in functionally graded materials using hyperbolic phase field and smoothed particle hydrodynamics. Comput Methods Appl Mech Eng 401:115642
Monaghan JJ (2005) Smoothed particle hydrodynamics. Rep Prog Phys 68(8):1703–1759
Bonet J, Lok TSL (1999) Variational and momentum preservation aspects of Smooth particle hydrodynamic formulations. Comput Methods Appl Mech Eng 180(1–2):97–115
Tofighi N, Ozbulut M, Rahmat A, Feng JJ, Yildiz M (2015) An incompressible smoothed particle hydrodynamics method for the motion of rigid bodies in fluids. J Comput Phys 297:207–220
Vincent S, Brändle JC, de Motta A, Sarthou J-LE, Simonin O, Climent E (2014) A Lagrangian VOF tensorial penalty method for the DNS of resolved particle-laden flows. J Comput Phys 256:582–614
Shadloo MS, Zainali A, Sadek SH, Yildiz M (2011) Improved incompressible smoothed particle hydrodynamics method for simulating flow around bluff bodies. Comput Methods Appl Mech Eng 200(9):1008–1020
Ozbulut M, Tofighi N, Goren O, Yildiz M (2017) Investigation of wave characteristics in oscillatory motion of partially filled rectangular tanks. J Fluids Eng 140(4):12
Khayyer A, Shimizu Y, Gotoh T, Gotoh H (2023) Enhanced resolution of the continuity equation in explicit weakly compressible SPH simulations of incompressible free-surface fluid flows. Appl Math Model 116:84–121
Belytschko T, Guo Y, Kam Liu W, Ping Xiao S (2000) A unified stability analysis of meshless particle methods. Int J Numer Methods Eng 48(9):1359–1400
Swegle JW, Hicks DL, Attaway SW (1995) Smoothed particle hydrodynamics stability analysis. J Comput Phys 116(1):123–134
Islam MRI, Peng C (2019) A total Lagrangian SPH method for modelling damage and failure in solids. Int J Mech Sci 157:498–511
Wang L, Fei X, Yang Y (2021) An improved total Lagrangian SPH method for modeling solid deformation and damage. Eng Anal Bound Elem 133:286–302
Monaghan JJ, Gingold RA (1983) Shock simulation by the particle method SPH. J Comput Phys 52(2):374–389
Rahimi MN, Moutsanidis G (2022) A coupled total Lagrangian SPH-phase-field framework for modeling dynamic brittle fracture. In: Giuseppe B (ed) 2022 International SPHERIC work., number June. Catania, Italy. Istituto Nazionale di Geofisica e Vulcanologi. p 71–76
Shimizu Y, Khayyer A, Gotoh H (2022) An implicit SPH-based structure model for accurate fluid–structure interaction simulations with hourglass control scheme. Eur J Mech B Fluids 96:122–145
Khayyer A, Shimizu Y, Gotoh H, Nagashima K (2021) A coupled incompressible SPH-Hamiltonian SPH solver for hydroelastic FSI corresponding to composite structures. Appl Math Model 94:242–271
Diehl P, Lipton R, Wick T, Tyagi M (2022) A comparative review of peridynamics and phase-field models for engineering fracture mechanics. Comput Mech 69:1259–1293
Landau LD, Lifšic EM, Lifshitz EM, Kosevich AM, Pitaevskii LP (1986) Theory of elasticity, vol 7. Elsevier, Amsterdam
Gray JP, Monaghan JJ, Swift RP (2001) SPH elastic dynamics. Comput Methods Appl Mech Eng 190(49):6641–6662
Monaghan JJ, Rafiee A (2013) A simple SPH algorithm for multi-fluid flow with high density ratios. Int J Numer Methods Fluids 71(5):537–561
Antuono M, Marrone S, Colagrossi A, Bouscasse B (2015) Energy balance in the \(\delta\)-SPH scheme. Comput Methods Appl Mech Eng 289:209–226
Khayyer A, Gotoh H, Shimizu Y (2017) Comparative study on accuracy and conservation properties of two particle regularization schemes and proposal of an optimized particle shifting scheme in ISPH context. J Comput Phys 332:236–256
Lobovský L, Botia-Vera E, Castellana F, Mas-Soler J, Souto-Iglesias A (2014) Experimental investigation of dynamic pressure loads during dam break. J Fluids Struct 48:407–434
Antuono M, Colagrossi A, Marrone S, Molteni D (2010) Free-surface flows solved by means of SPH schemes with numerical diffusive terms. Comput Phys Commun 181(3):532–549
Ting TS, Prakash M, Cleary PW, Thompson MC (2006) Simulation of high ||Reynolds number flow over a backward facing step using SPH. In: Stacey A, Blyth B, Shepherd J, Roberts AJ (eds) Proceedings of the 7th biennial engineering mathematics and applications conference, EMAC-2005, volume 47 of ANZIAM J. p C292–C309
Kim J, Kline SJ, Johnston JP (1980) Investigation of a reattaching turbulent shear layer: flow over a backward-facing step. J Fluids Eng 102(3):302–308
Fourey G, Hermange C, Le Touzé D, Oger G (2017) An efficient FSI coupling strategy between smoothed particle hydrodynamics and finite element methods. Comput Phys Commun 217:66–81
Stephen T, Sergius W-K et al (1959) Theory of plates and shells, vol 2. McGraw-hill, New York
Li Z, Leduc J, Nunez-Ramirez J, Combescure A, Marongiu JC (2015) A non-intrusive partitioned approach to couple smoothed particle hydrodynamics and finite element methods for transient fluid–structure interaction problems with large interface motion. Comput Mech 55(4):697–718
Rafiee A, Thiagarajan KP (2009) An SPH projection method for simulating fluid-hypoelastic structure interaction. Comput Methods Appl Mech Eng 198(33):2785–2795
Marti J, Idelsohn S, Limache A, Calvo N, D’Elía J (2006) A fully coupled particle method for quasi incompressible fluid-hypoelastic structure interactions. Mecánica Computacional 809–828
Acknowledgements
The authors would like to thank Stony Brook Research Computing and Cyberinfrastructure, and the Institute for Advanced Computational Science at Stony Brook University for access to the high-performance SeaWulf computing system, which was made possible by a $1.4 M National Science Foundation grant (#1531492).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Rahimi, M.N., Moutsanidis, G. An SPH-based FSI framework for phase-field modeling of brittle fracture under extreme hydrodynamic events. Engineering with Computers 39, 2365–2399 (2023). https://doi.org/10.1007/s00366-023-01857-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00366-023-01857-0