Theory                      Numerical method                         Implementation                             Simulation...
Theory     Numerical method        Implementation            Simulations   Future developmentsAim of the thesisPeridynamic...
Theory     Numerical method        Implementation            Simulations   Future developmentsAim of the thesisPeridynamic...
Theory     Numerical method        Implementation            Simulations   Future developmentsAim of the thesisPeridynamic...
Theory                          Introduction     Numerical method                          Equation of motion        Imple...
Theory                          Introduction     Numerical method                          Equation of motion        Imple...
Theory                           Introduction      Numerical method                           Equation of motion         I...
Theory                           Introduction      Numerical method                           Equation of motion         I...
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Theory      Numerical method         Implementation             Simulations    Future developmentsFuture developmentsSpati...
Theory      Numerical method         Implementation             Simulations    Future developmentsFuture developmentsSpati...
Theory      Numerical method         Implementation             Simulations    Future developmentsFuture developmentsSpati...
Theory      Numerical method         Implementation             Simulations    Future developmentsFuture developmentsSpati...
Theory     Numerical method        Implementation            Simulations   Future developmentsConclusionsPeridynamic code ...
Theory      Numerical method         Implementation             Simulations    Future developmentsConclusionsPeridynamic c...
Theory      Numerical method         Implementation             Simulations    Future developmentsConclusionsPeridynamic c...
Theory     Numerical method        Implementation            Simulations   Future developmentsThanks for your patience!Got...
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Mixed explicit-implicit peridynamic model

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Thesis defence for the stabilization in time of the peridynamic method. A mixed explicit-implicit code was implemented.

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Mixed explicit-implicit peridynamic model

  1. 1. Theory Numerical method Implementation Simulations Future developmentsPolitecnico di Prof. Marco Di Sciuva Torino Prof. Paolo MaggioreUniversity of Prof. David Steigmann California, Berkeley Stability and applications of the peridynamic method Candidate Matteo Polleschi Date July 21, 2010 Matteo Polleschi Peridynamics: stability and applications
  2. 2. Theory Numerical method Implementation Simulations Future developmentsAim of the thesisPeridynamic method overview Matteo Polleschi Peridynamics: stability and applications
  3. 3. Theory Numerical method Implementation Simulations Future developmentsAim of the thesisPeridynamic method overviewNumerical method stabilization Matteo Polleschi Peridynamics: stability and applications
  4. 4. Theory Numerical method Implementation Simulations Future developmentsAim of the thesisPeridynamic method overviewNumerical method stabilizationQualitative verification Matteo Polleschi Peridynamics: stability and applications
  5. 5. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developmentsTheory (1)What is peridynamics? Matteo Polleschi Peridynamics: stability and applications
  6. 6. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developmentsTheory (1)What is peridynamics?New formulation of continuum mechanics by StewartSilling (Sandia Labs), first published in 2000 Matteo Polleschi Peridynamics: stability and applications
  7. 7. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developmentsTheory (1)What is peridynamics?New formulation of continuum mechanics by StewartSilling (Sandia Labs), first published in 2000Nonlocal, as particles interact at a finite distance Matteo Polleschi Peridynamics: stability and applications
  8. 8. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developmentsTheory (1)What is peridynamics?New formulation of continuum mechanics by StewartSilling (Sandia Labs), first published in 2000Nonlocal, as particles interact at a finite distanceBased upon integral equations, avoiding spatialderivatives =⇒ able to deal with discontinuities(especially fractures) Matteo Polleschi Peridynamics: stability and applications
  9. 9. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developmentsTheory (2)Physical approach:close to molecular dynamics Matteo Polleschi Peridynamics: stability and applications
  10. 10. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developments Theory (3)Equation of Generic form motion ρ(x)¨(x, t) = u f(u − u, x − x)dVx + b(x, t) R Matteo Polleschi Peridynamics: stability and applications
  11. 11. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developments Theory (3)Equation of Generic form motion ρ(x)¨(x, t) = u f(u − u, x − x)dVx + b(x, t) R density Matteo Polleschi Peridynamics: stability and applications
  12. 12. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developments Theory (3)Equation of Generic form motion ρ(x)¨(x, t) = u f(u − u, x − x)dVx + b(x, t) R acceleration Matteo Polleschi Peridynamics: stability and applications
  13. 13. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developments Theory (3)Equation of Generic form motion ρ(x)¨(x, t) = u f(u − u, x − x)dVx + b(x, t) R pairwise force function Matteo Polleschi Peridynamics: stability and applications
  14. 14. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developments Theory (3)Equation of Generic form motion ρ(x)¨(x, t) = u f(u − u, x − x)dVx + b(x, t) R pairwise force function u − u relative displacement Matteo Polleschi Peridynamics: stability and applications
  15. 15. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developments Theory (3)Equation of Generic form motion ρ(x)¨(x, t) = u f(u − u, x − x)dVx + b(x, t) R pairwise force function u − u relative displacement x − x relative initial position Matteo Polleschi Peridynamics: stability and applications
  16. 16. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developments Theory (3)Equation of Generic form motion ρ(x)¨(x, t) = u f(u − u, x − x)dVx + b(x, t) R body force density field Matteo Polleschi Peridynamics: stability and applications
  17. 17. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developments Theory (4)Horizon Integral is not taken over the entire body. We define a quantity δ, called horizon, such that R if x−x ≥δ⇒f=0 f x δ usually assumed ∼ 3 = δ x if < 3 ⇒ unnatural crack paths if > 3 ⇒ wave dispersion, fluid-like behaviour Matteo Polleschi Peridynamics: stability and applications
  18. 18. Theory Introduction Numerical method Equation of motion Implementation Horizon Simulations PPF Future developments Theory (5)Pairwise force force/volume 2 on a particle at x due to a particle at x . function Completely defines the properties of a material (elasticity, plasticity, yield loads...) force rupture stretch rupture ⇒ brittle failure Matteo Polleschi Peridynamics: stability and applications
  19. 19. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (1) Previous Dominium discretization ⇒ grid of nodesapproach No elements required ⇒ method is meshless Eq. of motion discretization ¨ ρun = i f(un − un , xp − xi )Vp + bn p i i p Matteo Polleschi Peridynamics: stability and applications
  20. 20. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (1) Previous Dominium discretization ⇒ grid of nodesapproach No elements required ⇒ method is meshless Eq. of motion discretization and linearization ¨ ρun = i C(un − un )(xp − xi )Vp + bn p i i p Matteo Polleschi Peridynamics: stability and applications
  21. 21. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (1) Previous Dominium discretization ⇒ grid of nodesapproach No elements required ⇒ method is meshless Eq. of motion discretization and linearization ¨ ρun = i C(un − un )(xp − xi )Vp + bn p i i p subscript i - node superscript n - time step Matteo Polleschi Peridynamics: stability and applications
  22. 22. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (2)Stability Linearized equation von Neumann stability analysis leads to 2ρ ∆t < p Vp |C(xp − xi )| Drawbacks: Matteo Polleschi Peridynamics: stability and applications
  23. 23. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (2)Stability Linearized equation von Neumann stability analysis leads to 2ρ ∆t < p Vp |C(xp − xi )| Drawbacks: linearization is not always acceptable subject to data entry mistakes not optimal solution Matteo Polleschi Peridynamics: stability and applications
  24. 24. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (3)Mixed method Developed by Professor Zohdi of the University of California, Berkeley, for thermo-chemical multifield problems Matteo Polleschi Peridynamics: stability and applications
  25. 25. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (3)Mixed method Developed by Professor Zohdi of the University of California, Berkeley, for thermo-chemical multifield problems Explicit “external ”time step Matteo Polleschi Peridynamics: stability and applications
  26. 26. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (3)Mixed method Developed by Professor Zohdi of the University of California, Berkeley, for thermo-chemical multifield problems Explicit “external ”time step At each step, implicit ∆t evaluation Matteo Polleschi Peridynamics: stability and applications
  27. 27. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (3)Mixed method Developed by Professor Zohdi of the University of California, Berkeley, for thermo-chemical multifield problems Explicit “external ”time step At each step, implicit ∆t evaluation Error based upon limit on particle movement Matteo Polleschi Peridynamics: stability and applications
  28. 28. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (4)Algorithm Global fixed-point iteration: for all the N nodes, Matteo Polleschi Peridynamics: stability and applications
  29. 29. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (4)Algorithm Global fixed-point iteration: for all the N nodes, compute the new position as ∆t 2 ˙ ui n+1,K ≈ f(ui n+1,K −1 ) + ∆t un + un i i mi Matteo Polleschi Peridynamics: stability and applications
  30. 30. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (4)Algorithm Global fixed-point iteration: for all the N nodes, compute the new position as ∆t 2 ˙ ui n+1,K ≈ f(ui n+1,K −1 ) + ∆t un + un i i mi compute the new (internal cycle) interaction forces (storing them in temporary variables) Matteo Polleschi Peridynamics: stability and applications
  31. 31. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (4)Algorithm Global fixed-point iteration: for all the N nodes, compute the new position as ∆t 2 ˙ ui n+1,K ≈ f(ui n+1,K −1 ) + ∆t un + un i i mi compute the new (internal cycle) interaction forces (storing them in temporary variables) compute the error measures Matteo Polleschi Peridynamics: stability and applications
  32. 32. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (4)Algorithm Global fixed-point iteration: for all the N nodes, compute the new position as ∆t 2 ˙ ui n+1,K ≈ f(ui n+1,K −1 ) + ∆t un + un i i mi compute the new (internal cycle) interaction forces (storing them in temporary variables) compute the error measures if tolerance met Matteo Polleschi Peridynamics: stability and applications
  33. 33. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (4)Algorithm Global fixed-point iteration: for all the N nodes, compute the new position as ∆t 2 ˙ ui n+1,K ≈ f(ui n+1,K −1 ) + ∆t un + un i i mi compute the new (internal cycle) interaction forces (storing them in temporary variables) compute the error measures if tolerance met increment time t = t + ∆t and start from the beginning Matteo Polleschi Peridynamics: stability and applications
  34. 34. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (4)Algorithm Global fixed-point iteration: for all the N nodes, compute the new position as ∆t 2 ˙ ui n+1,K ≈ f(ui n+1,K −1 ) + ∆t un + un i i mi compute the new (internal cycle) interaction forces (storing them in temporary variables) compute the error measures if tolerance met increment time t = t + ∆t and start from the beginning construct new time step ∆t = ΦK ∆t Matteo Polleschi Peridynamics: stability and applications
  35. 35. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (4)Algorithm Global fixed-point iteration: for all the N nodes, compute the new position as ∆t 2 ˙ ui n+1,K ≈ f(ui n+1,K −1 ) + ∆t un + un i i mi compute the new (internal cycle) interaction forces (storing them in temporary variables) compute the error measures if tolerance not met Matteo Polleschi Peridynamics: stability and applications
  36. 36. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (4)Algorithm Global fixed-point iteration: for all the N nodes, compute the new position as ∆t 2 ˙ ui n+1,K ≈ f(ui n+1,K −1 ) + ∆t un + un i i mi compute the new (internal cycle) interaction forces (storing them in temporary variables) compute the error measures if tolerance not met construct new time step ∆t = ΦK ∆t Matteo Polleschi Peridynamics: stability and applications
  37. 37. Theory Previous approach Numerical method Explicit stability Implementation Mixed method Simulations Algorithm Future developments Numerical method (4)Algorithm Global fixed-point iteration: for all the N nodes, compute the new position as ∆t 2 ˙ ui n+1,K ≈ f(ui n+1,K −1 ) + ∆t un + un i i mi compute the new (internal cycle) interaction forces (storing them in temporary variables) compute the error measures if tolerance not met construct new time step ∆t = ΦK ∆t restart from time t Matteo Polleschi Peridynamics: stability and applications
  38. 38. Theory Numerical method Implementation Simulations Future developments Implementation Pre-processor Geometry and Mesh: Salom` e Constraints, loads and initial velocities: Impact Solver C++ solver built from scratch Parallelization by use of OpenMP (shared memory) External libraries: Armadillo (linear algebra), VTK (visualization)Post-processor Real-time visualization: VisIt Picture production: Gmsh Matteo Polleschi Peridynamics: stability and applications
  39. 39. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (1) Membrane dampedobscillations Matteo Polleschi Peridynamics: stability and applications
  40. 40. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (1) Membrane dampedobscillations Matteo Polleschi Peridynamics: stability and applications
  41. 41. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (1) Membrane dampedobscillations Matteo Polleschi Peridynamics: stability and applications
  42. 42. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (1) Membrane dampedobscillations Matteo Polleschi Peridynamics: stability and applications
  43. 43. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (1) Membrane dampedobscillations Matteo Polleschi Peridynamics: stability and applications
  44. 44. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (1) Membrane dampedobscillations Matteo Polleschi Peridynamics: stability and applications
  45. 45. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (1) Membrane dampedobscillations Matteo Polleschi Peridynamics: stability and applications
  46. 46. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (1) Membrane dampedobscillations Matteo Polleschi Peridynamics: stability and applications
  47. 47. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (1) Membrane dampedobscillations Matteo Polleschi Peridynamics: stability and applications
  48. 48. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (1) Time stepsover execution time Matteo Polleschi Peridynamics: stability and applications
  49. 49. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (2)Plate with holebrittle fracture Matteo Polleschi Peridynamics: stability and applications
  50. 50. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (2)Plate with holebrittle fracture Matteo Polleschi Peridynamics: stability and applications
  51. 51. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (2)Plate with holebrittle fracture Matteo Polleschi Peridynamics: stability and applications
  52. 52. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (2)Plate with holebrittle fracture Matteo Polleschi Peridynamics: stability and applications
  53. 53. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (2)Plate with holebrittle fracture Matteo Polleschi Peridynamics: stability and applications
  54. 54. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (2)Plate with holebrittle fracture Matteo Polleschi Peridynamics: stability and applications
  55. 55. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (2)Plate with holebrittle fracture Matteo Polleschi Peridynamics: stability and applications
  56. 56. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (2)Plate with holebrittle fracture Matteo Polleschi Peridynamics: stability and applications
  57. 57. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (2)Plate with holebrittle fracture Matteo Polleschi Peridynamics: stability and applications
  58. 58. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (3)Impact Matteo Polleschi Peridynamics: stability and applications
  59. 59. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (3)Impact Matteo Polleschi Peridynamics: stability and applications
  60. 60. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (3)Impact Matteo Polleschi Peridynamics: stability and applications
  61. 61. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (3)Impact Matteo Polleschi Peridynamics: stability and applications
  62. 62. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (3)Impact Matteo Polleschi Peridynamics: stability and applications
  63. 63. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (3)Impact Matteo Polleschi Peridynamics: stability and applications
  64. 64. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (3)Impact Matteo Polleschi Peridynamics: stability and applications
  65. 65. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (3)Impact Matteo Polleschi Peridynamics: stability and applications
  66. 66. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (3)Impact Matteo Polleschi Peridynamics: stability and applications
  67. 67. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (4)Specimen traction Matteo Polleschi Peridynamics: stability and applications
  68. 68. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (4)Specimen traction Matteo Polleschi Peridynamics: stability and applications
  69. 69. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (4)Specimen traction Matteo Polleschi Peridynamics: stability and applications
  70. 70. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (4)Specimen traction Matteo Polleschi Peridynamics: stability and applications
  71. 71. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (4)Specimen traction Matteo Polleschi Peridynamics: stability and applications
  72. 72. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (4)Specimen traction Matteo Polleschi Peridynamics: stability and applications
  73. 73. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (4)Specimen traction Matteo Polleschi Peridynamics: stability and applications
  74. 74. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (4)Specimen traction Matteo Polleschi Peridynamics: stability and applications
  75. 75. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (4)Specimen traction Matteo Polleschi Peridynamics: stability and applications
  76. 76. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (4)Specimen traction Matteo Polleschi Peridynamics: stability and applications
  77. 77. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (4)Specimen traction Matteo Polleschi Peridynamics: stability and applications
  78. 78. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (4)Specimen traction Matteo Polleschi Peridynamics: stability and applications
  79. 79. Theory Membrane Numerical method Plate with hole Implementation Impact Simulations Specimen traction Future developments Simulations (4)Specimen traction Matteo Polleschi Peridynamics: stability and applications
  80. 80. Theory Numerical method Implementation Simulations Future developmentsFuture developmentsSpatial discretization Matteo Polleschi Peridynamics: stability and applications
  81. 81. Theory Numerical method Implementation Simulations Future developmentsFuture developmentsSpatial discretizationComplete range of material behaviour Matteo Polleschi Peridynamics: stability and applications
  82. 82. Theory Numerical method Implementation Simulations Future developmentsFuture developmentsSpatial discretizationComplete range of material behaviourFatigue (variable loads) Matteo Polleschi Peridynamics: stability and applications
  83. 83. Theory Numerical method Implementation Simulations Future developmentsFuture developmentsSpatial discretizationComplete range of material behaviourFatigue (variable loads)Maintenance support by simulations Matteo Polleschi Peridynamics: stability and applications
  84. 84. Theory Numerical method Implementation Simulations Future developmentsConclusionsPeridynamic code from scratch Matteo Polleschi Peridynamics: stability and applications
  85. 85. Theory Numerical method Implementation Simulations Future developmentsConclusionsPeridynamic code from scratchStability Matteo Polleschi Peridynamics: stability and applications
  86. 86. Theory Numerical method Implementation Simulations Future developmentsConclusionsPeridynamic code from scratchStabilityResults coherent with brittle fracture Matteo Polleschi Peridynamics: stability and applications
  87. 87. Theory Numerical method Implementation Simulations Future developmentsThanks for your patience!Got any question? Matteo Polleschi Peridynamics: stability and applications

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