Mixed explicit-implicit peridynamic model

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



Future developments

Aim of the thesis

Peridynamic method overview

Numerical method stabilization

Qualitative verification




Future developments

Aim of the thesis







Future developments

Aim of the thesis







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)




Future developments


Theory (1)








Future developments


Theory (1)








Future developments


Theory (1)








Future developments


Theory (2)

Physical approach:close to molecular dynamics




Future developments


Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)




Future developments


Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)

density




Future developments


Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)

acceleration




Future developments


Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)

pairwise force function




Future developments


Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)


u′ − u relative displacement




Future developments


Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)


u′ − u relative displacement

x′ − x relative initial position




Future developments


Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)

body force density field




Future developments


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'




Future developments


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




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




Future developments



Previousapproach

Dominium discretization ⇒ grid of nodesNo elements required ⇒ method is meshlessEq. of motion discretizationand linearization

ρuni =∑p

C(unp − uni )(xp − xi)Vp + bni




Future developments



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




Future developments



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




Future developments



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




Future developments



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




Future developments










Future developments










Future developments










Future developments



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




Future developments






uin+1,K ≈ ∆t2




if tolerance met







Future developments






uin+1,K ≈ ∆t2




if tolerance met







Future developments






uin+1,K ≈ ∆t2




if tolerance met







Future developments






uin+1,K ≈ ∆t2




if tolerance met







Future developments






uin+1,K ≈ ∆t2




if tolerance met

increment time t = t + ∆t and start from the beginning

construct new time step ∆t = ΦK∆t






Future developments






uin+1,K ≈ ∆t2




if tolerance met







Future developments






uin+1,K ≈ ∆t2




if tolerance met







Future developments






uin+1,K ≈ ∆t2




if tolerance met



construct new time step ∆t = ΦK∆t

restart from time t




Future developments






uin+1,K ≈ ∆t2




if tolerance met







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




Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (1)

Membranedamped

obscillations




Future developments


Simulations (1)

Membranedamped

obscillations




Future developments


Simulations (1)

Membranedamped

obscillations




Future developments


Simulations (1)

Membranedamped

obscillations




Future developments


Simulations (1)

Membranedamped

obscillations




Future developments


Simulations (1)

Membranedamped

obscillations




Future developments


Simulations (1)

Membranedamped

obscillations




Future developments


Simulations (1)

Membranedamped

obscillations




Future developments


Simulations (1)

Membranedamped

obscillations




Future developments


Simulations (1)

Time stepsover execution

time




Future developments


Simulations (2)

Plate with holebrittle fracture




Future developments


Simulations (2)





Future developments


Simulations (2)





Future developments


Simulations (2)





Future developments


Simulations (2)





Future developments


Simulations (2)





Future developments


Simulations (2)





Future developments


Simulations (2)





Future developments


Simulations (2)





Future developments


Simulations (3)

Impact




Future developments


Simulations (3)

Impact




Future developments


Simulations (3)

Impact




Future developments


Simulations (3)

Impact




Future developments


Simulations (3)

Impact




Future developments


Simulations (3)

Impact




Future developments


Simulations (3)

Impact




Future developments


Simulations (3)

Impact




Future developments


Simulations (3)

Impact




Future developments


Simulations (4)

Specimentraction




Future developments


Simulations (4)

Specimentraction




Future developments


Simulations (4)

Specimentraction




Future developments


Simulations (4)

Specimentraction




Future developments


Simulations (4)

Specimentraction




Future developments


Simulations (4)

Specimentraction




Future developments


Simulations (4)

Specimentraction




Future developments


Simulations (4)

Specimentraction




Future developments


Simulations (4)

Specimentraction




Future developments


Simulations (4)

Specimentraction




Future developments


Simulations (4)

Specimentraction




Future developments


Simulations (4)

Specimentraction




Future developments


Simulations (4)

Specimentraction




Future developments

Future developments

Spatial discretization

Complete range of material behaviour

Fatigue (variable loads)

Maintenance support by simulations




Future developments

Future developments








Future developments

Future developments








Future developments

Future developments








Future developments

Conclusions

Peridynamic code from scratch

Stability

Results coherent with brittle fracture




Future developments

Conclusions


Stability





Future developments

Conclusions


Stability





Future developments

Thanks for your patience!Got any question?


Technology

Mixed explicit-implicit peridynamic model