, Phase diagram of water, 2006.

, lef t) mixed cavitation regime Re = 10 6 , ? v = 0.07, angle of incidence= 1.1 ? , (right) saturation Re = 1.1 × 10 6 , ? v = 0.08, angle of incidence= 5 ? (Briançon-Marjollet 1990)

, R 0 = 1mm, ? = 1. The time is non-dimensionalized time with reference time R 0 of symmetry, 1971.

, , 1987.

, Maximum wall pressure obtained for different stand-off distances, 2012.

, Experimental mass loss rate vs exposure time (Franc, p.10, 2006.

, Schematic of erosion model by Fortes-Patella, 2004.

10 1.10 (lef t) Nearly axi-symmetric pit height profile from AFM (experiments) and FEM (simulation), (right) numerically predicted pressure distribution on surface and pit height, 2006. ,

, 12 1.12 (lef t) Time evolution of pressure from bubble collapse showing liquid jet and bubble ring collapse, (right) zoom near ring collapse pressure peaks for two rigid and two compliant materials R max = 2mm, 11 FEM modelling of material fatigue and mass loss after repetitive impacts (Fivel 2015), 2015.

, Vapor bubble collapse due to imposed pressure difference p liquid p bubble , (top) collapse of an isolated bubble -spherical collapse, (bottom) bubble collapse near a solid wall -non-spherical collapse

, Representation of proposed FSI using CFD-CSM solvers, p.14

15 1.16 Weak scaling of YALES2 on an IBM BlueGene/P computer, 2011. ,

, , vol.20, p.165

, lef t) maximum depth of surface damage produced, R max = 1.45 mm, (right) volume of deformation, Collapse of single cavitation bubble on an aluminum sample, 1998.

, Evolution of velocity field during bubble collapse near a solid wall and development of free vortex ? = 1.1, R max = 340 µm, p.21, 2017.

, Maximum jet velocity and (right) penetration depth from liquid jet into a P AA sample for bubble collapsing near elastic boundaries, 2001.

, Experimental setup for laser generated bubble collapse near solid wall

, Schematic of the experimental setup for laser generated bubble collapse near solid wall

, Left to right, top to bottom: Isolated bubble undergoing a nearly spherical collapse R max = 535 µm, resolution=200000 f ps

, Evolution of bubble radius with time -isolated bubble collapse and rebound R max = 420 µm, resolution=540000 f ps

, Left to right, top to bottom : Liquid micro-jet impacting the solid aluminum surface R max = 400 µm

, Collapse of bubble and resulting "splash" effect , presence of liquid film between the bubble and aluminum surface can be observed during the growth and first collapse, R max = 390µm, ? = 0, vol.87

, splash" effect after the collapse pushes the bubble cavity away from aluminum surface forming a mushroom shaped cavity R max = 430 µm, ? = 1.1, frame size= 1.2 × 1.4 mm 2, p.27

, Collapse of bubble torus and formation of tiny micro-bubbles on aluminum surface R max = 730 µm, ? = 1.4, frame size= 1.9 × 2.2 mm 2

, Elongated bubble torus and liquid micro-jet dissipating into surface micro-bubbles R max = 535 µm, ? = 1.7, frame size= 1.1 × 2 mm 2

, Bubble torus and liquid micro-jet retarded by liquid layer between the surface and collapsing bubble R max = 430 µm, ? = 2.14, frame size= 1.0 × 1.8 mm 2

, distinct surface pits due to liquid micro-jet impact R max = 400 µm, ? = 0.55, (lef t) surface under a high resolution optical microscope after 10 bubble collapses, frame size = 250 × 250 µm 2 , (right) AFM scan of pits, scan size = 110 × 110 µm 2

, 16 Indentation pit of depth 600 nm and pit radius of about 20 µm, scan size = 70 × 70 µm 2

, Indentation pit of depth more than 1 µm and pit radius of about 20 µm, scan size = 110 × 110 µm 2

, no visible surface deformation due to rebounding bubble cavity from "splash" effect R max = 430 µm, ? = 1.1 frame size = 1.5 × 1.5 mm 2

, (right) recovered aluminum sample under optical microscope, frame size = 1.5 × 1.5 mm 2 , (right) AFM scan of top surface pits, scan size= 150 × 150 µm 2, Circular damage pattern after 100 repetitive bubble collapses due to collapsing bubble torus and micro-bubbles R max = 730 µm, ? = 1.4

, 34 2.22 No surface deformation on Al-7075 from detached bubble collapse, frame size = 1.5 × 1.5 mm 2

, Comparison of ? range for observed surface damage in the present experiments and Philipp, 1998.

, Schematic of CPS algorithm-estimation of sound speed and pressure from equation of state (in green)

Schematic of CCS algorithm-estimation of intermediate sound speed and corrected pressure from equation of state (in yellow), p.48 ,

, Subtri of a node pair for a triangular element (b) CV of a node in a mesh (c) Subtri of an edge-mesh face in a hexahedron (d) Exterior triangles of all subtetrahedra showing the contribution node CV for hexahedral element, Control volume representation for spatial discretization: (a), 2011.

, Waves entering and leaving the computational domain through an inlet plane at x 1 = 0 and an outlet plane x 1 = L for a subsonic flow (Poinsot 1992)

, Density vs pressure evolution in the cavitation model, p.56

, Speed of sound vs density evolution in the cavitation model, p.57

, Dynamic viscosity vs density evolution in the cavitation model, p.58

, 8 1D representation of bubble radius initialization

, 59 3.10 1D Lagrangian, Eulerian and ALE mesh node motion with associated material particle motion, 1D representation of bubble radius initialization for different mesh spacing of ?x=10, vol.5, p.61, 2004.

One to one transformation between the material domain, spatial domain and the referential domain for ALE (Donea 2004), p.62 ,

, Finite element mesh with elements, p.67, 2004.

, 2 1D finite element (a) Linear (b) Quadratic (c) Linear approximation (d) Quadratic approximation of field variable, p.68, 2004.

, Elastic and plastic response to an applied load, p.68, 2004.

, Al-7075 showing the elastic and plastic regime separated by the yield strength ? y , (right) elastic regime for Al-7075, Stress-strain curve at strain rate 1.0 s ?1

, Stress-strain curves for Al-7075, St A-2025 and NAB at strain rate 1.0 s ?1, p.70

Accumulated plastic strain P ?p (Di Paola 2017a), p.70 ,

, Mechanical calculation with PASAPAS procedure (Di Paola 2017b), p.73

, Boundary conditions and mesh used for FEM simulations

,

,

, Shock tube test case representation

, 2 Numerical oscillations from centred difference scheme near discontinuities in CPS solver, without the use of any filtering

, Effect of pressure and density filtering in CPS with coarse mesh ?x = 1mm, vol.83

, Effect of pressure and density filtering in CPS with fine mesh ?x = 100µm, vol.84

, Numerical oscillations from centred difference scheme near discontinuities in CPS and CCS solver

, Effect of pressure and density filtering in CCS with coarse mesh ?x = 1mm, p.85

, Effect of pressure and density filtering in CCS with fine mesh ?x = 100µm, p.85

, Effect of isolated filtering of density in CCS coarse mesh ?x = 1 mm, vol.86

, Effect of isolated filtering of density in CCS fine mesh ?x = 100 µm. . . 86 5.10 Full vs 1/4 th domain for 2D Rayleigh collapse, vol.87

, 2D Rayleigh collapse comparison for R b = 17R 0 (t rayleigh = 8.75 µs) & R b = 25R 0 (t rayleigh = 9.3 µs). Initial bubble radius R 0 = 500 µm, p.88

, Full computational domain for 3D Rayleigh-Plesset comparison, 89 5.13 1/8 th symmetric computational domain for 3D Rayleigh-Plesset compar

,

, Vapor bubble iso-surface of ? = 0.5, R 0 = 500 µm, p.90

, 3D Rayleigh Plesset validation R 0 = 500 µm, p.91

, Temporal evolution of 2D bubble collapse R max = 100µm, p ? = 100M P a, t = 0 & 250ns -stationary vs moving domain

, Temporal evolution of 2D bubble collapse R max = 100µm, p ? = 100M P a, t = 380 & 500ns -stationary vs moving domain

, Radius evolution during 2D bubble collapse R max = 100 µm, p ? = 100 M P a -stationary vs moving domain

, Bubble position for different stand-off ? = 0.5, 0.8, 0.9, 1.4, solid wall at the bottom of the frame

, Time step ?t evolution for 2D bubble collapse computation,? = 0.9

, Temporal evolution of density field during liquid pressure-induced attached bubble collapse near solid wall, ? = 0, vol.9

, frame size = 1.5 × 1.5 mm 2

, Location of probe points on the solid wall-points F 00(x, y) = (0, 0) and F 10(x, y) = (0.001, 0)

, Temporal evolution of pressure at point F 00 and maximum pressure p max?wall , ? = 0, vol.9

, Maximum pressure at the solid wall p max?wall and its location on the wall as a function of distance from the axis of symmetry i.e point F 00, ? = 0, vol.9

, Contour showing -(lef t) numerical Schlieren, (center) pressure and (right) velocity field during liquid jet impact, ? = 0, vol.9

, frame size = 500 × 500 µm 2

, R max = 495 µm, p = 100 M P a, case P 0.9, frame size = 500 × 500 µm 2, Contour showing -(lef t) numerical Schlieren, (center) pressure and (right) velocity field on remaining bubble collapse and shock wave superimposition ? = 0, vol.9

, Pressure plots on the solid wall between points F 00 and F 10 at different time instants ? = 0.9

, and (bottom)? = 0.8, (lef t) entire simulation time, (right) zoomed on the dynamical pressure peaks, and p max?wall for (top)? = 0.5, p.0

, Collapsing bubble shape (top) 2D numerical simulation ? = 0.5, R max = 449 µm, p = 100 M P a, (bottom) experimental bubble collapse in atmospheric condition ? = 0, vol.55

, Contour showing -(lef t) numerical Schlieren, (center) pressure and (right) velocity field on remaining bubble collapse and shock wave superimposition ? = 0.8, R max = 487 µm, p ? = 100 M P a, frame size = 500 × 500 µm, p.105

, Contour showing -(lef t) numerical Schlieren, (center) pressure and (right) velocity field on remaining bubble collapse and shock wave superimposition ? = 0.5, R max = 449 µm, p ? = 100 M P a

, Plot showing (lef t) spatial and (right) temporal convergence of the numerical solution at p F 00 , ? = 0.8

, Reduced computational domain for 2D bubble collapse -domain boundary 10R 0 × 5R 0 , domain size = 5 × 2.5 mm 2

, Plot showing agreement of results for domain size 40R 0 and reduced computational domain size 10R 0 ×5R 0 , ? = 0.9

, Temporal evolution of density field during liquid pressure-induced detached bubble collapse near solid wall -appearance of expansion wave driven secondary cavitation near the solid wall, ? = 1.4, R max = 500µm

, pressure contour on each frame showing temporal evolution of a shock-induced 2D bubble collapse, ? = 0, R max = 495 µm, p = 0.1 M P a, p shock = 50 M P a, case SH0.9, vol.9

, 20 2D attached bubble: pressure peaks at p F 00 and p max?wall on the solid wall for shock-induced collapse, ? = 0.9, case SH0.9, p.110

, 21 2D attached bubble: pressure plots on the solid wall between points F 00 and F 10 at different time instants, ? = 0.9, case SH0.9, p.111

, Velocity vectors showing the flow field during the final stages of collapse near the solid wall, ? = 0, vol.9

, frame size = 500 × 500 µm 2, vol.9

, 112 6.24 2D detached bubble: comparison of bubble shapes in (top) numerical simulation ? = 1.4, R max = 500 µm, Numerical Schlieren showing the sequence of events during the remaining bubble collapse-emission of primary and secondary shock, ? = 0, vol.9

, 25 2D detached bubble: final stages of collapse showing shock propagation near the solid wall ? = 1.4, R max = 500 µm, p = 0.1 M P a

, 4, frame size = 1 × 2.5 mm 2

, 26 2D detached bubble: pressure peaks at p F 00 and p max?wall on the solid wall for shock

116 6.28 3D attached bubble: evolution of velocity on the iso-volume of attached bubble and pressure contour on solid wall, ? = 0, th cut section of bubble iso-volume attached on the solid wall, ? = 0.9, vol.9 ,

, 50M P a, frame size = 2 × 2 mm 2

, 118 6.30 3D detached bubble: evolution of velocity on the iso-volume of attached bubble and pressure contour on solid wall, 29 3D attached bubble: pressure peaks at p F 00 and p max?wall on the solid wall for shock-induced collapse ? = 0.9

, 50M P a, frame size = 2 × 2 mm 2

, 31 3D detached bubble: pressure peaks at p F 00 and p max?wall on the solid wall for shock-induced collapse, ? = 1.4

, Top view and front view showing the initial setup of planar 3D bubble cloud, frame size = 4 × 4 mm 2

, Temporal evolution of bubble shapes and pressure on the solid wall from 3D collapsing cloud, p = 0.1 M P a, p shock = 50M P a

, bubble radius R max = 500µm, (bottom) solid domain, size = 1×2.5mm 2 , (top) fluid domain, size = 5 × 2.5 mm 2, FSI coupling domain and interface between the fluid and solid, max

, Stress-strain curves for Al-7075, St A-2025 and NAB at strain rate 1.0 s ?1, p.124

, S10(x, y) = (0.001, 0), Location of probe points in the solid domain, S00(x, y) = (0, 0)

, Solid wall interface profile at (a) after liquid jet impact, (b) after entire simulation time t = 6 µs, case P 0.9

, Temporal evolution of surface displacement at (a) S00, (b) S10, (c) relative displacement, case P 0.9

, Solid wall interface profile at (a) after liquid jet impact, (b) after entire simulation time t = 6 µs, case SH0.9

, Temporal evolution of surface displacement at (a) S00, (b) S10,(c) relative displacement, case SH0.9

, Temporal convergence of relative displacement (a) interface profile at 6µs and 11 µs, (b) total displacement at S00, (c) relative displacement at S00 for total time t = 11 µs, case SH0

, remaining bubble collapse (2D equivalent of 3D bubble torus at t = 4.420 µs) and subsequent shock wave superimposition at bubble symmetry axis (t = 4.470 µs) for St A-2205, R max = 495 µm, case SH0.9, frame size = 1 × 1 mm 2, von Mises stress ? V M at different time instants during liquid jet impact (t = 4.295 µs)

, von Mises stress ? V M contour showing propagation of stress waves in the solid for St A-2205

, Accumulated plastic strain P ?p for the three considered material at the end of simulation time t = 6 µs, R max = 495 µm, case SH0.9, frame size= 1 × 1 mm 2

, Solid wall interface position (a) after water-hammer shock impact at symmetry axis, (b) after entire simulation time t = 6 µs, p.130

, Interface position at different time instant showing propagation of waterhammer shock on St A-2205

, von Mises stress ? V M contour showing propagation of stress waves in the solid for St A-2205

, Temporal evolution pressure, sound speed on the wall and fluid velocity u x at ?y = 5 µm (wall parallel i.e. along x-direction from viscous fluid calculations) near solid wall for detached bubble collapse

, Accumulated plastic strain P ?p for the considered material at t = 6 µs, R max = 500 µm, case SH1.4, frame size= 1 × 1 mm 2, p.132

, Temporal evolution of displacement at (a) S00, (b) S10,(c) relative displacement

, Pressure dampening in step-wise coupled FSI showing convergence of pressure on St A-2205 in four steps, case SH0.9

, Maximum pressure p max?wall evolution on rigid wall and on the deformable materials with two-way FSI, case SH0.9, p.136

, on maximum pressure p max?wall evolution on rigid wall and on the deformable materials with two-way FSI, case SH0.9, p.136

, Numerical Schlieren and accumulated plastic strain P ?p in two-way FSI for St A-2205 at different time instants showing the dynamical features of bubble collapse and corresponding generation of plasticity, R max = 495 µm, case SH0.9, frame size=1 × 1 mm 2

, Comparison of accumulated plasticity P ?p contour for one-way and twoway coupled FSI at t = 6 µs, R max = 495 µm, case SH0.9, frame size=250 × 250 µm 2

, Comparison of solid interface profile for one-way and two-way coupled FSI, case SH0

, Surface damage as a function of distance from the wall, p.141, 2019.

Computational domain for full 2D Rayleigh collapse case, R max = 8.5mm, N cells = 205648 ,

, Cell size transition between the uniform quad-elements and surrounding tri-elements, growth ratio= 1.05

, Full domain for 3D Rayleigh-Plesset validation, domain size 10 × 10 × 10 mm 3 , N cells = 14771173

, th symmetrical domain for 3D Rayleigh-Plesset validation, domain size 20 × 20 × 20 mm 3 , N cells = 6117566

Computational domain for 2D bubble collapse near a solid wall -domain 40R 0 , size 20 × 20 mm 2 , N cells = 176058, max. skewness=0.64, p.159 ,

Cell size transition between the two meshed domains -domain ,

, 2D bubble collapse near a solid wall with cartesian mesh -boundary 40R 0 , domain size 5 × 2.5 mm 2 , computational cells 500 × 500, N cells = 250000, max. skewness=0 -domain 10R 0

, Computational domain for 3D bubble collapse near solid wall with cartesian mesh -domain size 2 × 2.5 × 2 mm 3 , computational cells 200 × 250 × 200, N cells = 10000000

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