HTML Injection Attacks: Impact and Mitigation Strategies
Mixed explicit-implicit peridynamic model
1. Theory
Numerical method
Implementation
Simulations
Future developments
Politecnico di Prof. Marco Di Sciuva
Torino Prof. Paolo Maggiore
University 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. Theory
Numerical method
Implementation
Simulations
Future developments
Aim of the thesis
Peridynamic method overview
Matteo Polleschi Peridynamics: stability and applications
3. Theory
Numerical method
Implementation
Simulations
Future developments
Aim of the thesis
Peridynamic method overview
Numerical method stabilization
Matteo Polleschi Peridynamics: stability and applications
4. Theory
Numerical method
Implementation
Simulations
Future developments
Aim of the thesis
Peridynamic method overview
Numerical method stabilization
Qualitative verification
Matteo Polleschi Peridynamics: stability and applications
5. Theory
Introduction
Numerical method
Equation of motion
Implementation
Horizon
Simulations
PPF
Future developments
Theory (1)
What is peridynamics?
Matteo Polleschi Peridynamics: stability and applications
6. Theory
Introduction
Numerical method
Equation of motion
Implementation
Horizon
Simulations
PPF
Future developments
Theory (1)
What is peridynamics?
New formulation of continuum mechanics by Stewart
Silling (Sandia Labs), first published in 2000
Matteo Polleschi Peridynamics: stability and applications
7. Theory
Introduction
Numerical method
Equation of motion
Implementation
Horizon
Simulations
PPF
Future developments
Theory (1)
What is peridynamics?
New formulation of continuum mechanics by Stewart
Silling (Sandia Labs), first published in 2000
Nonlocal, as particles interact at a finite distance
Matteo Polleschi Peridynamics: stability and applications
8. Theory
Introduction
Numerical method
Equation of motion
Implementation
Horizon
Simulations
PPF
Future developments
Theory (1)
What is peridynamics?
New formulation of continuum mechanics by Stewart
Silling (Sandia Labs), first published in 2000
Nonlocal, as particles interact at a finite distance
Based upon integral equations, avoiding spatial
derivatives =⇒ able to deal with discontinuities
(especially fractures)
Matteo Polleschi Peridynamics: stability and applications
9. Theory
Introduction
Numerical method
Equation of motion
Implementation
Horizon
Simulations
PPF
Future developments
Theory (2)
Physical approach:
close to molecular dynamics
Matteo Polleschi Peridynamics: stability and applications
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. 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. 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. 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. 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. 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. 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. 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. 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. Theory
Previous approach
Numerical method
Explicit stability
Implementation
Mixed method
Simulations
Algorithm
Future developments
Numerical method (1)
Previous Dominium discretization ⇒ grid of nodes
approach 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. Theory
Previous approach
Numerical method
Explicit stability
Implementation
Mixed method
Simulations
Algorithm
Future developments
Numerical method (1)
Previous Dominium discretization ⇒ grid of nodes
approach 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. Theory
Previous approach
Numerical method
Explicit stability
Implementation
Mixed method
Simulations
Algorithm
Future developments
Numerical method (1)
Previous Dominium discretization ⇒ grid of nodes
approach 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (1)
Membrane
damped
obscillations
Matteo Polleschi Peridynamics: stability and applications
40. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (1)
Membrane
damped
obscillations
Matteo Polleschi Peridynamics: stability and applications
41. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (1)
Membrane
damped
obscillations
Matteo Polleschi Peridynamics: stability and applications
42. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (1)
Membrane
damped
obscillations
Matteo Polleschi Peridynamics: stability and applications
43. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (1)
Membrane
damped
obscillations
Matteo Polleschi Peridynamics: stability and applications
44. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (1)
Membrane
damped
obscillations
Matteo Polleschi Peridynamics: stability and applications
45. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (1)
Membrane
damped
obscillations
Matteo Polleschi Peridynamics: stability and applications
46. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (1)
Membrane
damped
obscillations
Matteo Polleschi Peridynamics: stability and applications
47. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (1)
Membrane
damped
obscillations
Matteo Polleschi Peridynamics: stability and applications
48. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (1)
Time steps
over execution
time
Matteo Polleschi Peridynamics: stability and applications
49. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (2)
Plate with hole
brittle fracture
Matteo Polleschi Peridynamics: stability and applications
50. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (2)
Plate with hole
brittle fracture
Matteo Polleschi Peridynamics: stability and applications
51. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (2)
Plate with hole
brittle fracture
Matteo Polleschi Peridynamics: stability and applications
52. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (2)
Plate with hole
brittle fracture
Matteo Polleschi Peridynamics: stability and applications
53. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (2)
Plate with hole
brittle fracture
Matteo Polleschi Peridynamics: stability and applications
54. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (2)
Plate with hole
brittle fracture
Matteo Polleschi Peridynamics: stability and applications
55. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (2)
Plate with hole
brittle fracture
Matteo Polleschi Peridynamics: stability and applications
56. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (2)
Plate with hole
brittle fracture
Matteo Polleschi Peridynamics: stability and applications
57. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (2)
Plate with hole
brittle fracture
Matteo Polleschi Peridynamics: stability and applications
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. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (3)
Impact
Matteo Polleschi Peridynamics: stability and applications
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. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (3)
Impact
Matteo Polleschi Peridynamics: stability and applications
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. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (3)
Impact
Matteo Polleschi Peridynamics: stability and applications
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. Theory
Membrane
Numerical method
Plate with hole
Implementation
Impact
Simulations
Specimen traction
Future developments
Simulations (3)
Impact
Matteo Polleschi Peridynamics: stability and applications
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Theory
Numerical method
Implementation
Simulations
Future developments
Future developments
Spatial discretization
Matteo Polleschi Peridynamics: stability and applications
81. Theory
Numerical method
Implementation
Simulations
Future developments
Future developments
Spatial discretization
Complete range of material behaviour
Matteo Polleschi Peridynamics: stability and applications
82. Theory
Numerical method
Implementation
Simulations
Future developments
Future developments
Spatial discretization
Complete range of material behaviour
Fatigue (variable loads)
Matteo Polleschi Peridynamics: stability and applications
83. Theory
Numerical method
Implementation
Simulations
Future developments
Future developments
Spatial discretization
Complete range of material behaviour
Fatigue (variable loads)
Maintenance support by simulations
Matteo Polleschi Peridynamics: stability and applications
84. Theory
Numerical method
Implementation
Simulations
Future developments
Conclusions
Peridynamic code from scratch
Matteo Polleschi Peridynamics: stability and applications
85. Theory
Numerical method
Implementation
Simulations
Future developments
Conclusions
Peridynamic code from scratch
Stability
Matteo Polleschi Peridynamics: stability and applications
86. Theory
Numerical method
Implementation
Simulations
Future developments
Conclusions
Peridynamic code from scratch
Stability
Results coherent with brittle fracture
Matteo Polleschi Peridynamics: stability and applications
87. Theory
Numerical method
Implementation
Simulations
Future developments
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Matteo Polleschi Peridynamics: stability and applications
