dynamic simulation of particle-filled hollow spheres · 2017. 2. 6. · numerical results...

Dynamic Simulation ofParticle-Filled Hollow Spheres

Tobias Steinle, Andrea Walther, Jadran Vrabec

Universitat PaderbornInstitut fur Mathematik

GAMM 2012, Darmstadt

T. Steinle, A. Walther, J. Vrabec 1 / 16 Particle-Filled Hollow Spheres

Table of contents

Motivation

Mathematical ModellingMolecular DynamicsTime IntegrationPotentials

Numerical Results

Current Cooperation: Fraunhofer Institute for ManufacturingTechnology and Advanced Materials, Dresden (Ulrike Jehring)

T. Steinle, A. Walther, J. Vrabec 2 / 16 Particle-Filled Hollow Spheres March 27th, 2012

Motivation

Motivation

I vibration can cause many problems, e.g., noise and wearI development of light-weight material at

Fraunhofer Institute for Manufacturing Technology andAdvanced Materials (Ulrike Jehring)

I research on hollow sphere structures


Motivation

Motivation


Motivation

Advantages of hollow sphere structures

I easily adaptable to different shapesI solvent resistance, thermal resistance, noise reduction

AdditionallyI hollow spheres with particlesI yields high dampening-to-weight-ratio


Motivation


I easily adaptable to different shapesI solvent resistance, thermal resistance, noise reduction

AdditionallyI hollow spheres with particles

I yields high dampening-to-weight-ratio


Motivation



Mathematical Modelling

Simulation of a sphere

Two basic possibilities

Collision Detection

computation of the next collisionfollowing paths of particles

2D: diploma thesis (Denise Holfeld)high complexity



Simulation of a sphere

Two basic possibilities

Collision Detection Time Integration

computation of the next collision well established methodsfollowing paths of particles available for 3D case

2D: diploma thesis (Denise Holfeld)high complexity



Molecular Dynamics

Molecular Dynamics

I discrete element method

I millions of moleculesI equally distributed particlesI periodic boundariesI cuboid simulation regionI potential-basedI basic program available Thermodynamics and Energy

Technology (Jadran Vrabec)



Molecular Dynamics

Molecular Dynamics

I discrete element methodI millions of molecules

I equally distributed particlesI periodic boundariesI cuboid simulation regionI potential-basedI basic program available Thermodynamics and Energy




Molecular Dynamics

Molecular Dynamics

I discrete element methodI millions of moleculesI equally distributed particles

I periodic boundariesI cuboid simulation regionI potential-basedI basic program available Thermodynamics and Energy




Molecular Dynamics

Molecular Dynamics

I discrete element methodI millions of moleculesI equally distributed particlesI periodic boundaries

I cuboid simulation regionI potential-basedI basic program available Thermodynamics and Energy




Molecular Dynamics

Molecular Dynamics

I discrete element methodI millions of moleculesI equally distributed particlesI periodic boundariesI cuboid simulation region

I potential-basedI basic program available Thermodynamics and Energy




Molecular Dynamics

Molecular Dynamics

I discrete element methodI millions of moleculesI equally distributed particlesI periodic boundariesI cuboid simulation regionI potential-based

I basic program available Thermodynamics and EnergyTechnology (Jadran Vrabec)



Molecular Dynamics

Molecular Dynamics

I discrete element methodI millions of moleculesI equally distributed particlesI periodic boundariesI cuboid simulation regionI potential-basedI basic program available Thermodynamics and Energy




Molecular Dynamics

Adaptations for filled spheresI sphere with reflective boundary conditions

I gravityI frictionI deformation and movement of the boundaryI different particle shapes

particles as aggregation of spheres



Molecular Dynamics

Adaptations for filled spheresI sphere with reflective boundary conditionsI gravity

I frictionI deformation and movement of the boundaryI different particle shapes




Molecular Dynamics

Adaptations for filled spheresI sphere with reflective boundary conditionsI gravityI friction

I deformation and movement of the boundaryI different particle shapes




Molecular Dynamics

Adaptations for filled spheresI sphere with reflective boundary conditionsI gravityI frictionI deformation and movement of the boundary

I different particle shapesparticles as aggregation of spheres



Molecular Dynamics

Adaptations for filled spheresI sphere with reflective boundary conditionsI gravityI frictionI deformation and movement of the boundaryI different particle shapes




Time Integration

Translation

I problem is derived from the equation of motion

x = v =Fm

I formulation as system of first order ODEs

v =Fm

x = v



Time Integration

The Leapfrog-Algorithm

I positions and velocities are calculated alternatingly

vn+ 1

2i = v

n− 12

i +dtmi

F ni

xn+1i = xn

i + dtvn+ 1

2i

I F is based on a potential

Fij =∂Vij

∂rij



Potentials

Lennard-Jones Potential

I pairwise-potentialI potential has two parts, one attracting and one rejecting

V (r) = −4ε(( r

σ

)12−( rσ

)6)

I usually cut off at a distance rc

I for reflections: hull potential



Potentials

Lennard-Jones potentialstandard new

0.8 1 1.2 1.4 1.6 1.8−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

r

V(r

)

0.8 1 1.2 1.4 1.6 1.8−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

r

V(r

)

−ε

rm

=21/6σσ rc r

c



Potentials

Reflections at the boundary

With the current implementation, there are two possibilitiesT. Steinle, A. Walther, J. Vrabec 12 / 16 Particle-Filled Hollow Spheres March 27th, 2012


Potentials


With the current implementation, there are two possibilities

hull potential

ghost particle on the boundarypotential forcespseudo friction



Potentials


With the current implementation, there are two possibilities

hull potential conservation of momentum

ghost particle on the boundary elastic or inelastic collisionspotential forces (coeff. of restitutionpseudo friction from experiments)



Potentials

InitializationI sphere shapedI include experimental dataI pseudo friction on contact



Potentials

Movement of the simulation volumeI pulsing surfaceI hoppingI deformation


Numerical Results

Examples


Numerical Results

Conclusions and Outlook

I MD can be used for fast particle calculationsI adapted MD towards our application

I fitting with experiments (falling sphere)I adaptive linked cell algorithmI coupling of spheresI modeling of friction

Thank you for your attention!


Numerical Results

Conclusions and Outlook

I MD can be used for fast particle calculationsI adapted MD towards our applicationI fitting with experiments (falling sphere)I adaptive linked cell algorithmI coupling of spheresI modeling of friction

Thank you for your attention!


Numerical Results

RotationI rotation is derived from the equation of rigid body rotational

motion, e.g. angular momentum

jn+ 1

2i = j

n− 12

i + tni

I angular velocity is related to the angular momentum by theinertia tensor

ω = I−1j

I quaternions useful for the orientation

qn+1i = qn

i +dt2

Q(qn+ 1

2i )ω

n+ 12

i where ω = (0, ω)T

I Fincham’s rotational quaternion algorithm


Numerical Results

Complexitycurrent status: up to 20000 particlesquadratic complexity

goal: 200000 particles


Numerical Results

Linked Cell algorithmI LCA linear in particle numberI LCA divides simulation volume into cellsI reduction in the number of possible contact particles


dynamic simulation of particle-filled hollow spheres · 2017. 2. 6. · numerical results...

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