SIMULATION METHODS SIMULATION METHODS IN SCIENCE AND IN SCIENCE AND

ENGINEERINGENGINEERING

Dr Dr habhab. . inzinz. . WitoldWitold DzwinelDzwinel, , prof.nprof.n. . AGHAGH

Plan przedmiotu: Metody symulacji w naucePlan przedmiotu: Metody symulacji w nauce

1.1. CeleCele i i srodki modelowania, fizyka a rzeczywistoscsrodki modelowania, fizyka a rzeczywistosc2.2. Nowoczesne problemy symulacji w nauce Nowoczesne problemy symulacji w nauce –– multimulti--resolutionresolution andand crosscross--

scalingscaling3.3. WprowadzenieWprowadzenie do do chaosuchaosu4.4. Metody Monte Metody Monte CarloCarlo –– uklady statyczneuklady statyczne

1.1. Generatory liczb losowychGeneratory liczb losowych2.2. Przypomnienie prostych schematów bazujacych na Przypomnienie prostych schematów bazujacych na zaszas. . MetropolisaMetropolisa3.3. DirectDirect--simulationsimulation Monte Monte CarloCarlo (uklady dynamiczne)(uklady dynamiczne)4.4. Wielkosci fizyczne w metodach MWielkosci fizyczne w metodach M--CC

5.5. Metoda dynamiki molekularnejMetoda dynamiki molekularnej1.1. Elementy kwantowej MD (Elementy kwantowej MD (abab initioinitio CarCar--ParinnelloParinnello))2.2. Podstawy i równaniaPodstawy i równania3.3. Schematy numeryczneSchematy numeryczne4.4. Algorytmy liczenia oddzialywan w modelach Algorytmy liczenia oddzialywan w modelach krótkokrótko--zasiegowychzasiegowych5.5. Algorytm Ewalda dla oddzialywan Algorytm Ewalda dla oddzialywan dalekodaleko--zasiegowychzasiegowych6.6. Fast Fast MultipoleMultipole AlgorithmAlgorithm i obliczenia duzej skalii obliczenia duzej skali7.7. Wielkosci Wielkosci charaktrystycznecharaktrystyczne GreenGreen--KoboKobo relationsrelations etc.etc.8.8. Obliczenia duzej skali Obliczenia duzej skali –– równolegloscrównoleglosc

Plan przedmiotu: Metody symulacji w nauce Plan przedmiotu: Metody symulacji w nauce c.d.c.d.

1.1. Hydrodynamika fluktuacyjna w Hydrodynamika fluktuacyjna w mezoskalimezoskali1.1. DyssypatywnaDyssypatywna dynamika dynamika czastekczastek –– schematy numeryczneschematy numeryczne2.2. Fluid Fluid ParticleParticle ModelModel3.3. GenericGeneric DPDDPD4.4. ChapmanChapman--Enskog’aEnskog’a rozwiniecie, równanie rozwiniecie, równanie FokkeraFokkera--PlancaPlanca--

BoltzmannaBoltzmanna5.5. Symulacje zlozonych cieczySymulacje zlozonych cieczy

2.2. SmoothedSmoothed particleparticle hydrodynamicshydrodynamics3.3. Metody symulacji wykorzystujace metody róznic i elementów Metody symulacji wykorzystujace metody róznic i elementów

skonczonychskonczonych4.4. Wprowadzenie do teorii chaosuWprowadzenie do teorii chaosu5.5. Automaty komórkoweAutomaty komórkowe

1.1. Proste CA, mrówka Proste CA, mrówka LansingtonaLansingtona, , 2.2. Model Model IsingaIsinga3.3. LGALGA4.4. LBGLBG5.5. Zlozone CA Zlozone CA –– modele fizyczne (modele fizyczne (VichniacVichniac model) i geologiczne model) i geologiczne

((anastomosinganastomosing riversrivers))6.6. SOC SOC –– modele trzesien ziemi, plonacego lasu, osypywanie siemodele trzesien ziemi, plonacego lasu, osypywanie sie7.7. Modele Modele PennyPenny starzenia sie, symulacje ewolucji osobniczejstarzenia sie, symulacje ewolucji osobniczej

Plan przedmiotu: Metody symulacji w nauce Plan przedmiotu: Metody symulacji w nauce c.d.c.d.

1.1. Symulacje ukladów sypkich i granulatówSymulacje ukladów sypkich i granulatów2.2. Symulacje innych ukladów dyskretnych (ruch pojazdów w Symulacje innych ukladów dyskretnych (ruch pojazdów w

miescie, optymalizacja ruchu na lotnisku itd.)miescie, optymalizacja ruchu na lotnisku itd.)3.3. Systemy Systemy PrusinkiewiczaPrusinkiewicza i i LindemayeraLindemayera4.4. Artificial lifeArtificial life

CrossCross--scales simulations...scales simulations...

Ø Cross-scales simulations –large-scale simulations involving many spatio-temporal scales

• E.Clementi- global ab initio simulations (1983)• P.Lomdahl-Beazley – 3-D NEMD - 109 L-J particles,

penetration dynamics, fracture (1024 CM-5, Los Alamos) (1994)

• F.Abraham– ab initio MD+MD+FEM (fracture dynamics also Holian-Ravello) (1995)

• High-performance computing, ASCI (Options: Red, Blue, Blue Mountain, White) (1995 – NOW)

• Vashishta and Nakano – 3-D long-ranged NEMD, FMM–real electronic structures – 109 particles (1998)

MD MD –– FEMFEM

Molecular Dynamics (MD)

FINER GRID RESOLUTION

INCONSISTENCY?

LARGER NUMBER OF PARTICLES

SPATIAL SCALE [m]

TIME [s]

10 -14

10-12

10-6

1

60

10 -3

10-9 10-6 10-3 1

atoms ⇒ Schwarz procedure ⇒ grid

Finite Differences (FDM) & Finite Elements (FEM)

• chem

ical reactions •

microscopic rheological properties

• mesoscopic flows of colloidal suspension in a capillary

• mesoscopic rheological properties,

• viscoelasticity

• permeability • phase diagram • thermal

conductivity • other global

parameters

ab initio

Molecular Dynamics

• potentials • parameters of chem. reactions

HadjiconstantinouHadjiconstantinou--Patera MD+FEM; Glowinski et. al. DNS;Patera MD+FEM; Glowinski et. al. DNS;

ABRAHAM MODEL OF CRACK PROPAGATIONABRAHAM MODEL OF CRACK PROPAGATION

MultiMulti--grid separation of scalesgrid separation of scales

Fast Fast MultipoleMultipole MethodMethod

... ... and scales crossing modelsand scales crossing models

ØØ Scales crossing models employ Scales crossing models employ mesoscopicmesoscopicobjects representing clusters of atoms and objects representing clusters of atoms and molecules molecules

•• Spatial scaleSpatial scale àà 100 nm 100 nm -- 1010µµm m •• Temporal scaleTemporal scaleàà nanosecondsnanoseconds--milisecondsmiliseconds

ØØ MesoscopicMesoscopic world world –– blend of microblend of micro--macro macro effectseffects

•• Microscopic Microscopic àà thermal fluctuations, chemical reactions, thermal fluctuations, chemical reactions, interatomicinteratomic forces, excluded volume effects forces, excluded volume effects –– e.g., depletion e.g., depletion forces, (also quantum effects)forces, (also quantum effects)

•• Macroscopic Macroscopic àà flows, advection, complex flows flows, advection, complex flows ààhydrodynamic instabilities (Rhydrodynamic instabilities (R--T, RT, R--M, RM, R--B, KelvinB, Kelvin--HelmholtzHelmholtz, , SaffmanSaffman--Taylor) Taylor) àà mixing, vortices formationmixing, vortices formation

OnOn--lattice and offlattice and off--lattice lattice methodsmethods

ØØ OnOn--lattice methods (particles or group of particles lattice methods (particles or group of particles colliding on grid nodes)colliding on grid nodes)ll Cellular automata: CA, LG, LBG Cellular automata: CA, LG, LBG ll Random walker: DLA, RLA, PercolationRandom walker: DLA, RLA, Percolationll Spin glassesSpin glasses

ØØ OffOff--lattice methodslattice methodsll MonteMonte--Carlo: DSMC (direct simulation MC) [Bird 1983]Carlo: DSMC (direct simulation MC) [Bird 1983]ll Moving grid: SPH (smoothed particle hydrodynamics, Moving grid: SPH (smoothed particle hydrodynamics,

[Monaghan 1980] MPS (moving particles semi[Monaghan 1980] MPS (moving particles semi--implicit method)implicit method)ll Fluid particles: DPD [Fluid particles: DPD [HoogerbruggeHoogerbrugge, , KoelmanKoelman],], FPM, TCFPM, TC--DPD DPD

[[EspanolEspanol, 2001], 2001]ll Granular dynamics [Granular dynamics [H.HerrmannH.Herrmann]]

ØØ HeterogeneousHeterogeneousll HardHard--core particles in fluid FEM+MD: e.g., DNS [core particles in fluid FEM+MD: e.g., DNS [GlowinskiGlowinski et. et.

al.], FPD [Tanaka, Araki]al.], FPD [Tanaka, Araki]ll LBG+MD [LBG+MD [Yoemans, Coveney, SlootYoemans, Coveney, Sloot]]

Discrete particles hierarchyDiscrete particles hierarchy

OffOff--lattice lattice àà particlesparticles OOOFFFFFF--- GGGRRRIIIDDD PPPAAARRRTTTIIICCCLLLEEE MMMEEETTTHHHOOODDDSSS

conservative interactions

MD

DDPPDD + dissipative and Brownian forces

FFPP MM + particles rotation, non-central forces

SS PPHH Regularized tensor interactions

AT

OM

S

CL

UST

ER

S OF

AT

OM

S

FL

UID

PA

RT

ICL

ES-V

OL

UM

EL

ESS

MOVING MESH NODES

+ variable mass and particle volume non-isothermal model

TTCC-- DDPP DD FL

UID

PA

RT

ICL

ES O

N V

OR

ON

OI L

AT

TIC

E

Fluid ParticlesFluid Particles

colloidal bead

dissipative particle

BOTTOM-UP APPROACH

MD – particles creating Voronoy clusters

Colloidal bead

Dissipative particle

TOP-DOWN APPROACH

Finite Volume - contiuum description.

• Bottom-up approach, Flekkoy and Coveney, 1999

• Top-down approach, Serrano and Espanol, 2002

( )[ ] ∑∑∑ +

⋅++−++=

lkl

lklklklkl

klkl

klkl

l

lkklk

k

rpLMM

dtd FeeUUeUUgP ~

22η&

( ) ( ) ( ) ( )i

ii

N

ii

i

ii

N

ii

mhWff

mhWff

ρρ,)( ;,)(

11

rrrrrrrr −∇⋅=∇−⋅= ∑∑==

( ) ( ) ( ) , ,,1

21

22 hWmp

dtd

hWpp

mdt

dji

N

jjij

i

iiji

N

j j

j

i

ij

iii

rrvvrrv

−∇−−=−∇⋅

+−= ∑∑

−−

oρ

ερρ

LOGISTIC EQUATIONLOGISTIC EQUATION

CHAOTIC BEHAVIORCHAOTIC BEHAVIOR

FORMFORM

)1(1 nnn xxrx −⋅⋅=+

BIFFURCATIONBIFFURCATION

r

x∞

…… at first it looks like thisat first it looks like this

and then it comes into chaosand then it comes into chaos

REMARKS 1REMARKS 1

REMARKS 2REMARKS 2

Scaling relationsScaling relations

Logistic equation 1Logistic equation 1

Logistic equation 2Logistic equation 2

Logistic equation 3Logistic equation 3

ATTRACTORSATTRACTORS

LorentzLorentz systemsystem

LAPUNOV EXPONENTSLAPUNOV EXPONENTS

LapunovLapunov exponent and chaosexponent and chaos

Characteristics of chaosCharacteristics of chaos

Lotka Volterra modelLotka Volterra model

EQUATIONEQUATIONLotka-Volterra model is the simplest model of predator-prey interactions. Themodel was developed independently by Lotka (1925) and Volterra (1926):

It has two variables (P, H) and several parameters:H = density of preyP = density of predatorsr = intrinsic rate of prey population increasea = predation rate coefficientb = reproduction rate of predators per 1 prey eatenm = predator mortality rate

SOLUTIONSOLUTION

COMMENTCOMMENT

ØØ The model of The model of LotkaLotka and and VolterraVolterra is not very is not very realistic. realistic.

ØØ It does not consider any competition among prey It does not consider any competition among prey or predators. As a result, prey population may or predators. As a result, prey population may grow infinitely without any resource limits. grow infinitely without any resource limits.

ØØ Predators have no saturation: their consumption Predators have no saturation: their consumption rate is unlimited. rate is unlimited.

ØØ The rate of prey consumption is proportional to The rate of prey consumption is proportional to prey density. Thus, it is not surprising that model prey density. Thus, it is not surprising that model behavior is unnatural showing no asymptotic behavior is unnatural showing no asymptotic stability. stability.

PROSTE ALGORYTMYPROSTE ALGORYTMYZLOZONY SWIATZLOZONY SWIAT

Prof. dr hab. inz. Witold Dzwinel

Katedra Informatyki AGH

RANDOM NUMBER RANDOM NUMBER GENERATORGENERATOR

IBM generator

75= 16807

RANDU

65539=216+3

INTEGRALS INTEGRALS –– MONTE CARLOMONTE CARLO

∫∫=circleS

dS

W POSZUKIWANIU OPTIMUMW POSZUKIWANIU OPTIMUM

SIMULATED ANNEALING SIMULATED ANNEALING ––SUPERSUPER--ALGORYTM XX WIEKUALGORYTM XX WIEKU

ALGORYTM GENETYCZNYALGORYTM GENETYCZNY

TSP PROBLEM TSP PROBLEM

DYNAMIKA MOLEKULARNA DYNAMIKA MOLEKULARNA PROBLEM NPROBLEM N--CIALCIAL

Molecular Dynamics Molecular Dynamics BasicsBasics

Aaron Aaron WemhoffWemhoffMultiphase Transport LaboratoryMultiphase Transport Laboratory

MicroscaleMicroscale Heat Transfer LaboratoryHeat Transfer LaboratoryDepartment of Mechanical EngineeringDepartment of Mechanical Engineering

University of California, BerkeleyUniversity of California, Berkeley

About Molecular DynamicsAbout Molecular Dynamics• Molecule behavior based on basic Newtonian mechanics of

monatomic condensed gases

• Allows for “bottom-up” approach to predicting system phenomena at the nanoscale

• Newton → Idea

• Computing Age → Implementation

• Computationally intensive

• Advantage: simple theoretical background

• Disadvantage: limited system size

Why Monatomic Gases?Why Monatomic Gases?• Monatomic Gases → van der Waals liquids and solids

• The van der Waals forces are weak…

→ Forces are sufficient at low temperatures (small kBT).

• It won’t apply at room temperatures…→ Same phenomena as common solids at room temperature b/c of the ratio of interatomic to thermal energies

• Hydrogen bonding and other polar interactions are not included…

→ Can still observe basic phenomena at the nanoscale; more complex situations can be considered through complex intermolecular potential functions.

• Newton:dtvd

mF ii

jij

rr=∑ ij

ij

ijij r

rF ˆ

∂∂

−=φr

i

j

Fij

vi(t)

Force

Mass

Acceleration

Potential Energy

Unit vector from i to j

KineticsKinetics

• Newton:dtvd

mF ii

jij

rr=∑ ij

ij

ijij r

rF ˆ

∂∂

−=φr

• Updated Position:

( ) ( )i

iiii m

tFttvttrttr

)(21

)()()( 2

rrrr

∆+∆+=∆+

new position

old positionold velocity

old force

KineticsKinetics

i

Fij

vi

i

(t)

(t + ∆t)

j

• Newton:dtvd

mF ii

jij

rr=∑ ij

ij

ijij r

rF ˆ

∂∂

−=φr

• Updated Position:

( ) ( )i

iiii m

tFttvttrttr

)(21

)()()( 2

rrrr

∆+∆+=∆+

• Updated Velocity:

( )i

iii m

tFttv

ttv

)(21

)(2

rrr

∆+=

∆

+ ( )i

iii m

ttFtttvttv )(21

2)( ∆+∆+

∆+=∆+

rr

first half of time step second half of time step

KineticsKinetics

i

Fij

vi

i

(t)

Fij

vi

(t + ∆t)

j

• Newton:dtvd

mF ii

jij

rr=∑ ij

ij

ijij r

rF ˆ

∂∂

−=φr

• Updated Position:

( ) ( )i

iiii m

tFttvttrttr

)(21

)()()( 2

rrrr

∆+∆+=∆+

• Updated Velocity:

( )i

iii m

tFttv

ttv

)(21

)(2

rrr

∆+=

∆

+ ( )i

iii m

ttFtttvttv )(21

2)( ∆+∆+

∆+=∆+

rr

• System Temperature:

TNkmv B

N

ii 2

321

1

2 =∑=

∑=

=N

iii

B

vmNk

T1

2

31

system kinetic energy equipartition theorem

KineticsKinetics

i

Fij

vi

i

(t)

Fij

vi

(t + ∆t)

j

Carey, V.P., Statistical Thermodynamics and Microscale Thermophysics, New York: Cambridge University Press, 1999.

• No long range interactions• Interact via elastic collisions

“Hard Sphere” Potential

Gases• Electric dipole ~ r -3

• 2 dipoles ~ r -6

• Repulsive nuclear forces ~ r -12

• Total Potential = (Attractive) + (Repulsive)“Lennard-Jones 6-12 Potential”

Dense Gases, Liquids and SolidsIntermolecular PotentialsIntermolecular Potentials

• Large systems are composed of a grid of equivalent smaller systems

∞

∞

∞∞

v

Periodic Boundary Periodic Boundary ConditionsConditions

• Large systems are composed of a grid of equivalent smaller systems

• When molecules pass through one boundary another molecule enters in the opposite boundary

∞

∞

∞∞

Periodic Boundary Periodic Boundary ConditionsConditions

• Thin liquid film mixture density profile

Matsumoto, M., Takaoka, Y. and Kataoka, Y., 1993, Liquid-Vapor Interface of Water-Methanol Mixture, J. Chem. Physics, Vol. 98, pp. 1464-1472.

Tarek, M., Tobias, D.J. and Klein, M.L., 1996, Molecular Dynamics of an Ethanol-Water Solution, Physica A, Vol. 231, pp. 117-122.

ExamplesExamples

Weng, J.G., Park, S.H., Lukes, J.R., and Tien, C.L., 2000, Molecular dynamics investigation of thickness effect on liquid films, Journal of Chemistry and Physics, Vol. 113, pp. 5917-5923.

• Thin liquid film stress distribution

ExamplesExamples

• Molecular transport in droplets

Maruyama, S., Matsumoto, S., and Ogita, A., 1994, Surface Phenomena of Molecular Clusters by Molecular Dynamics Method, Thermal Science and Engineering, Vol. 2, No. 1.

• Bubble nucleation on solid surfaces

Maruyama, S, and Kimura, T., 2000, A Molecular Dynamics Simulation of a Bubble Nucleation on Solid Surface, Heat and Technology, Vol. 18, pp. 69-73.

ExamplesExamples

• Rayleigh-Taylor instability generation

Dzwinel, W., Alda, W., Pogoda, M., and Yuen, D.A., 2000, Turbulent mixing in the microscale : a 2D molecular dynamics simulation, Physica D, Vol. 137, pp. 157-171.

ExamplesExamples

More about MDMore about MD

By By WitoldWitold DzwinelDzwinel

• Newton:dtvd

mF ii

jij

rr=∑ ij

ij

ijij r

rF ˆ

∂∂

−=φr

• Updated Position:

( ) ( )i

iiii m

tFttvttrttr

)(21

)()()( 2

rrrr

∆+∆+=∆+

• Updated Velocity:

( )i

iii m

tFttv

ttv

)(21

)(2

rrr

∆+=

∆

+ ( )i

iii m

ttFtttvttv )(21

2)( ∆+∆+

∆+=∆+

rr

• System Temperature:

TNkmv B

N

ii 2

321

1

2 =∑=

∑=

=N

iii

B

vmNk

T1

2

31

system kinetic energy equipartition theorem

KineticsKinetics

i

Fij

vi

i

(t)

Fij

vi

(t + ∆t)

j

More about forcesMore about forces

Unbounded forcesUnbounded forces

Other short range forces in MDOther short range forces in MD

LongLong--range nonrange non--bounded bounded

Electrostatic on periodic gridElectrostatic on periodic grid

EWALD SUMMATIONEWALD SUMMATION

How to speedHow to speed--up up EwaldEwald summationsummation

SUMMARYSUMMARY

MultiMulti--Grid separation of scalesGrid separation of scales

3D potential3D potential

Octal treesOctal trees

Cost and accuracyCost and accuracy

Morton particles sorting for parallel Morton particles sorting for parallel processingprocessing

Simulating Dynamical Features of Escape Simulating Dynamical Features of Escape PanicPanic

Dirk Helbing, Illes Farkas, and Dirk Helbing, Illes Farkas, and Tamas VicsekTamas Vicsek

Presentation byPresentation byAndrew GoodmanAndrew Goodman

The ProblemThe Problem

ØØCrowd stampedes can be deadlyCrowd stampedes can be deadlyØØ People act in uncoordinated and People act in uncoordinated and

dangerous ways when panickingdangerous ways when panickingØØ It is difficult to obtain real data on crowd It is difficult to obtain real data on crowd

panicspanics

The SolutionThe Solution

ØØModel people as selfModel people as self--driven particlesdriven particlesØØModel physical and socioModel physical and socio--psychological psychological

influences on people’s movement as influences on people’s movement as forcesforces

ØØ Simulate crowd panics and see what Simulate crowd panics and see what happenshappens

Acceleration of Simulated Acceleration of Simulated PeoplePeople

ØØ vvii00(t) = desired speed(t) = desired speed

ØØ eeii00(t) = desired direction(t) = desired direction

ØØ vvii(t) = actual velocity(t) = actual velocityØØ tt i i = characteristic time= characteristic timeØØ mmii = mass= mass

Forces from Other PeopleForces from Other People

ØØ Force from other people’s bodies being in Force from other people’s bodies being in the waythe way

ØØ Force of friction preventing people from Force of friction preventing people from slidingsliding

ØØ Psychological “force” of tendency to avoid Psychological “force” of tendency to avoid each othereach other

ØØ Sum of forces of person j on person i is fSum of forces of person j on person i is f ijij

Total Force of Other PeopleTotal Force of Other People

ØØ AAiiexp[(rexp[(rijij –– ddijij)/B)/Bii]]nnijij is psychological “force”is psychological “force”•• rrijij is the sum of the people’s radiiis the sum of the people’s radii•• ddijij is the distance between their centers of massis the distance between their centers of mass•• nnij ij is the normalized vector from j to iis the normalized vector from j to i•• AAii and Band B ii are constantsare constants

Physical ForcesPhysical Forces

ØØ kg(rkg(rijij –– ddijij))nnijij is the force from other is the force from other bodiesbodies

ØØ ??gg(r(rijij –– ddijij))?? vvttijijttijij is the force of sliding is the force of sliding

frictionfrictionØØ g(x) is 0 if the people don’t touch and x g(x) is 0 if the people don’t touch and x

if they do touchif they do touchØØ ttijij is the tangential directionis the tangential directionØØ ?? vv tt

ijij is the tangential velocity differenceis the tangential velocity differenceØØ k and k and ?? are constantsare constants

Forces from WallsForces from Walls

ØØ Forces from walls are calculated in basically Forces from walls are calculated in basically the same way as forces from other peoplethe same way as forces from other people

Values Used for Constants and Values Used for Constants and ParametersParameters

ØØ Values chosen to match flows of people Values chosen to match flows of people through an opening under nonthrough an opening under non--panic panic conditionsconditions

ØØ People are modeled as the same except People are modeled as the same except for their radiusfor their radius

ØØ Insufficient data on actual panic situations Insufficient data on actual panic situations to analyze the algorithm quantitativelyto analyze the algorithm quantitatively

Simulation of CloggingSimulation of Clogging

Simulation of CloggingSimulation of Clogging

ØØ As desired speed increases beyond 1.5m As desired speed increases beyond 1.5m ss--11, it takes more time for people to leave, it takes more time for people to leave

ØØ As desired speed increases, the outflow of As desired speed increases, the outflow of people becomes irregularpeople becomes irregular

ØØ Arch shaped clogging occurs around the Arch shaped clogging occurs around the doorway doorway

Widening Can Create CrowdingWidening Can Create Crowding

Mass BehaviorMass Behavior

ØØ Panicking people tend to exhibit herding Panicking people tend to exhibit herding behaviorbehavior

ØØ Herding simulated using “panic parameter” pHerding simulated using “panic parameter” p

Effects of HerdingEffects of Herding

Injured People Block ExitInjured People Block Exit

A Column Can Increase OutflowA Column Can Increase Outflow

FindingsFindings

ØØ Bottlenecks cause cloggingBottlenecks cause cloggingØØ Asymmetrically placed columns around Asymmetrically placed columns around

exits can reduce clogging and prevent exits can reduce clogging and prevent build up of fatal pressuresbuild up of fatal pressures

ØØ A mixture of herding and individual A mixture of herding and individual behavior is idealbehavior is ideal

Some QuestionsSome Questions

ØØ Are parameters based on nonAre parameters based on non--panic panic situations correct for panic situations?situations correct for panic situations?

ØØHow can we get quantitative data about How can we get quantitative data about panic situations to test simulations?panic situations to test simulations?

ØØWhat happens when injured people are What happens when injured people are allowed to fall over (and possibly be allowed to fall over (and possibly be trampled)?trampled)?