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TheoryNumerical method
ImplementationSimulations
Future developments
Politecnico diTorino
Prof. Marco Di SciuvaProf. Paolo Maggiore
University ofCalifornia,Berkeley
Prof. David Steigmann
Stability and applicationsof the peridynamic method
Candidate Matteo Polleschi
Date July 21, 2010
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Aim of the thesis
Peridynamic method overview
Numerical method stabilization
Qualitative verification
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Aim of the thesis
Peridynamic method overview
Numerical method stabilization
Qualitative verification
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Aim of the thesis
Peridynamic method overview
Numerical method stabilization
Qualitative verification
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (1)
What is peridynamics?
New formulation of continuum mechanics by StewartSilling (Sandia Labs), first published in 2000
Nonlocal, as particles interact at a finite distance
Based upon integral equations, avoiding spatialderivatives =⇒ able to deal with discontinuities(especially fractures)
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (1)
What is peridynamics?
New formulation of continuum mechanics by StewartSilling (Sandia Labs), first published in 2000
Nonlocal, as particles interact at a finite distance
Based upon integral equations, avoiding spatialderivatives =⇒ able to deal with discontinuities(especially fractures)
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (1)
What is peridynamics?
New formulation of continuum mechanics by StewartSilling (Sandia Labs), first published in 2000
Nonlocal, as particles interact at a finite distance
Based upon integral equations, avoiding spatialderivatives =⇒ able to deal with discontinuities(especially fractures)
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (1)
What is peridynamics?
New formulation of continuum mechanics by StewartSilling (Sandia Labs), first published in 2000
Nonlocal, as particles interact at a finite distance
Based upon integral equations, avoiding spatialderivatives =⇒ able to deal with discontinuities(especially fractures)
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (2)
Physical approach:close to molecular dynamics
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (3)
Equation ofmotion
Generic form
ρ(x)u(x, t) =
∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (3)
Equation ofmotion
Generic form
ρ(x)u(x, t) =
∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)
density
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (3)
Equation ofmotion
Generic form
ρ(x)u(x, t) =
∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)
acceleration
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (3)
Equation ofmotion
Generic form
ρ(x)u(x, t) =
∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)
pairwise force function
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (3)
Equation ofmotion
Generic form
ρ(x)u(x, t) =
∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)
pairwise force function
u′ − u relative displacement
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (3)
Equation ofmotion
Generic form
ρ(x)u(x, t) =
∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)
pairwise force function
u′ − u relative displacement
x′ − x relative initial position
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (3)
Equation ofmotion
Generic form
ρ(x)u(x, t) =
∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)
body force density field
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (4)
Horizon Integral is not taken over the entire body.We define a quantity δ,called horizon, such that
if ‖x− x′‖ ≥ δ ⇒ f = 0
δ usually assumed ∼= 3if < 3 ⇒ unnatural crackpathsif > 3 ⇒ wave dispersion,fluid-like behaviour
R
δ
f
x
x'
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
IntroductionEquation of motionHorizonPPF
Theory (5)
Pairwise forcefunction
force/volume2 on a particle at x due to a particle at x′.Completely defines the properties of a material(elasticity, plasticity, yield loads...)
stretch
force
rupture
rupture
⇒ brittle failure
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (1)
Previousapproach
Dominium discretization ⇒ grid of nodesNo elements required ⇒ method is meshlessEq. of motion discretization
ρuni =∑p
f(unp − uni , xp − xi)Vp + bni
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (1)
Previousapproach
Dominium discretization ⇒ grid of nodesNo elements required ⇒ method is meshlessEq. of motion discretizationand linearization
ρuni =∑p
C(unp − uni )(xp − xi)Vp + bni
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (1)
Previousapproach
Dominium discretization ⇒ grid of nodesNo elements required ⇒ method is meshlessEq. of motion discretizationand linearization
ρuni =∑p
C(unp − uni )(xp − xi)Vp + bni
subscript i - nodesuperscript n - time step
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (2)
Stability Linearized equation von Neumann stability analysis leadsto
∆t <
√2ρ∑
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
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (2)
Stability Linearized equation von Neumann stability analysis leadsto
∆t <
√2ρ∑
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
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (3)
Mixed method Developed by Professor Zohdi of the University ofCalifornia, Berkeley, for thermo-chemical multifieldproblems
Explicit “external ”time step
At each step, implicit ∆t evaluation
Error based upon limit on particle movement
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (3)
Mixed method Developed by Professor Zohdi of the University ofCalifornia, Berkeley, for thermo-chemical multifieldproblems
Explicit “external ”time step
At each step, implicit ∆t evaluation
Error based upon limit on particle movement
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (3)
Mixed method Developed by Professor Zohdi of the University ofCalifornia, Berkeley, for thermo-chemical multifieldproblems
Explicit “external ”time step
At each step, implicit ∆t evaluation
Error based upon limit on particle movement
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (3)
Mixed method Developed by Professor Zohdi of the University ofCalifornia, Berkeley, for thermo-chemical multifieldproblems
Explicit “external ”time step
At each step, implicit ∆t evaluation
Error based upon limit on particle movement
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (4)
Algorithm Global fixed-point iteration:
for all the N nodes,
compute the new position as
uin+1,K ≈ ∆t2
mif(uin+1,K−1) + ∆tuni + uni
compute the new (internal cycle) interaction forces (storingthem in temporary variables)
compute the error measures
if tolerance met
increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t
if tolerance not met
construct new time step ∆t = ΦK∆trestart from time t
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (4)
Algorithm Global fixed-point iteration:
for all the N nodes,
compute the new position as
uin+1,K ≈ ∆t2
mif(uin+1,K−1) + ∆tuni + uni
compute the new (internal cycle) interaction forces (storingthem in temporary variables)
compute the error measures
if tolerance met
increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t
if tolerance not met
construct new time step ∆t = ΦK∆trestart from time t
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (4)
Algorithm Global fixed-point iteration:
for all the N nodes,
compute the new position as
uin+1,K ≈ ∆t2
mif(uin+1,K−1) + ∆tuni + uni
compute the new (internal cycle) interaction forces (storingthem in temporary variables)
compute the error measures
if tolerance met
increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t
if tolerance not met
construct new time step ∆t = ΦK∆trestart from time t
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (4)
Algorithm Global fixed-point iteration:
for all the N nodes,
compute the new position as
uin+1,K ≈ ∆t2
mif(uin+1,K−1) + ∆tuni + uni
compute the new (internal cycle) interaction forces (storingthem in temporary variables)
compute the error measures
if tolerance met
increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t
if tolerance not met
construct new time step ∆t = ΦK∆trestart from time t
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (4)
Algorithm Global fixed-point iteration:
for all the N nodes,
compute the new position as
uin+1,K ≈ ∆t2
mif(uin+1,K−1) + ∆tuni + uni
compute the new (internal cycle) interaction forces (storingthem in temporary variables)
compute the error measures
if tolerance met
increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t
if tolerance not met
construct new time step ∆t = ΦK∆trestart from time t
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (4)
Algorithm Global fixed-point iteration:
for all the N nodes,
compute the new position as
uin+1,K ≈ ∆t2
mif(uin+1,K−1) + ∆tuni + uni
compute the new (internal cycle) interaction forces (storingthem 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
if tolerance not met
construct new time step ∆t = ΦK∆trestart from time t
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (4)
Algorithm Global fixed-point iteration:
for all the N nodes,
compute the new position as
uin+1,K ≈ ∆t2
mif(uin+1,K−1) + ∆tuni + uni
compute the new (internal cycle) interaction forces (storingthem in temporary variables)
compute the error measures
if tolerance met
increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t
if tolerance not met
construct new time step ∆t = ΦK∆trestart from time t
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (4)
Algorithm Global fixed-point iteration:
for all the N nodes,
compute the new position as
uin+1,K ≈ ∆t2
mif(uin+1,K−1) + ∆tuni + uni
compute the new (internal cycle) interaction forces (storingthem in temporary variables)
compute the error measures
if tolerance met
increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t
if tolerance not met
construct new time step ∆t = ΦK∆trestart from time t
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (4)
Algorithm Global fixed-point iteration:
for all the N nodes,
compute the new position as
uin+1,K ≈ ∆t2
mif(uin+1,K−1) + ∆tuni + uni
compute the new (internal cycle) interaction forces (storingthem in temporary variables)
compute the error measures
if tolerance met
increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t
if tolerance not met
construct new time step ∆t = ΦK∆t
restart from time t
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Previous approachExplicit stabilityMixed methodAlgorithm
Numerical method (4)
Algorithm Global fixed-point iteration:
for all the N nodes,
compute the new position as
uin+1,K ≈ ∆t2
mif(uin+1,K−1) + ∆tuni + uni
compute the new (internal cycle) interaction forces (storingthem in temporary variables)
compute the error measures
if tolerance met
increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t
if tolerance not met
construct new time step ∆t = ΦK∆trestart from time t
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Implementation
Pre-processor Geometry and Mesh: SalomeConstraints, loads and initial velocities: Impact
Solver C++ solver built from scratchParallelization by use of OpenMP (shared memory)External libraries: Armadillo (linear algebra), VTK(visualization)
Post-processor Real-time visualization: VisItPicture production: Gmsh
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (1)
Membranedamped
obscillations
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (1)
Membranedamped
obscillations
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (1)
Membranedamped
obscillations
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (1)
Membranedamped
obscillations
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (1)
Membranedamped
obscillations
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (1)
Membranedamped
obscillations
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (1)
Membranedamped
obscillations
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (1)
Membranedamped
obscillations
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (1)
Membranedamped
obscillations
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (1)
Time stepsover execution
time
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (2)
Plate with holebrittle fracture
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (2)
Plate with holebrittle fracture
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (2)
Plate with holebrittle fracture
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (2)
Plate with holebrittle fracture
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (2)
Plate with holebrittle fracture
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (2)
Plate with holebrittle fracture
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (2)
Plate with holebrittle fracture
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (2)
Plate with holebrittle fracture
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (2)
Plate with holebrittle fracture
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (3)
Impact
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (3)
Impact
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (3)
Impact
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (3)
Impact
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (3)
Impact
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (3)
Impact
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (3)
Impact
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (3)
Impact
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (3)
Impact
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (4)
Specimentraction
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (4)
Specimentraction
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (4)
Specimentraction
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (4)
Specimentraction
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (4)
Specimentraction
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (4)
Specimentraction
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (4)
Specimentraction
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (4)
Specimentraction
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (4)
Specimentraction
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (4)
Specimentraction
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (4)
Specimentraction
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (4)
Specimentraction
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
MembranePlate with holeImpactSpecimen traction
Simulations (4)
Specimentraction
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Future developments
Spatial discretization
Complete range of material behaviour
Fatigue (variable loads)
Maintenance support by simulations
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Future developments
Spatial discretization
Complete range of material behaviour
Fatigue (variable loads)
Maintenance support by simulations
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Future developments
Spatial discretization
Complete range of material behaviour
Fatigue (variable loads)
Maintenance support by simulations
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Future developments
Spatial discretization
Complete range of material behaviour
Fatigue (variable loads)
Maintenance support by simulations
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Conclusions
Peridynamic code from scratch
Stability
Results coherent with brittle fracture
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Conclusions
Peridynamic code from scratch
Stability
Results coherent with brittle fracture
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Conclusions
Peridynamic code from scratch
Stability
Results coherent with brittle fracture
Matteo Polleschi Peridynamics: stability and applications
TheoryNumerical method
ImplementationSimulations
Future developments
Thanks for your patience!Got any question?
Matteo Polleschi Peridynamics: stability and applications