computationalcosmography

&

theisotropicvectorfielddecompositionmethodology

dr.J.F.(jim)Nystrom

FacultyofInformationTechnology

UniversityofAkureyri

600Akureyri,Iceland

4.mars2005

cosmographydefined∗

Cosmography:(1)ageneraldescriptionoftheworldorofthe

universe.(2)thesciencethatdealswiththeconstitutionofthe

wholeorderofnature.

∗Webster’sNewCollegiateDictionary.

Onthegeometryofspace†

Thegeometryofphysicalspacehadtoberecognizedas

anempiricalproblem;itisthetaskofphysicstosingle

outtheactualspace,i.e.,physicalspace,amongthe

possibletypesofspace.Itcandecidethisquestiononly

byempiricalmeans:buthowshoulditproceed?

†H.Reichenbach,ThePhilosophyofSpace&Time,Dover(1957).

omni-directionalclosestpacking

TheVEisatwelve-around-onearrangement

whichconsistsoffourhexagonalplanes;

eachplanebeingasix-around-onearrangement.

Anisotropicvectormatrixgrid

Vectorequilibrium(VE)cell.IVMgrid.

Contents

¤Introduction

¤computationalcosmographyandemergentcomputation

¤TheIVMCEMtime-domainsolverandtheisotropicvector

fielddecompositionmethodology

¤Questions

computationalcosmographyandemergentcomputation

¤Emergentcomputation

¤Universeasemergentcomputation

¤Onthequestionofinformationinphysicallaw

¤computationalcosmographydefined

¤J.F.Nystrom,“TensionalComputation:FurtherMusingsonthe

ComputationalCosmography,”AppliedMathematicsandComputation120,

211-225(2001).

¤J.F.Nystrom,”OntheOmni-directionalEmergenceofFormin

Computation,”LectureNotesinComputerScience,3305,632-641(2004).

¤J.F.Nystrom,“Statics,dynamics,andpointparticles:Onwardtowards

acomputationalcosmography,”acceptedasaposteratArtificialLifeVII,

Portland,Oregon,USA(2000),butnotpresented.

¤J.F.Nystrom,“MechanismfortheSimulationArgument:Gravityand

MindasTeleologicalPrimeMovers,”tobesubmittedtothePhilosophical

Quarterly.

emergentcomputation‡

“Researchersinseveralfieldshavebeguntoexplorecomputationalmodelsinwhichthebehavioroftheentiresystemisinsomesensemorethanthesumofitsparts.

...Inthesesystemsinterestingglobalbehavioremergefrommanylocalinteractions....

Thepremiseofemergentcomputationisthatinterestingandusefulcomputationalsystemscanbeconstructedbyexploitinginteractionsamongprimitivecomponents,andfurther,thatforsomekindsofproblems(e.g.modelingintelligentbehavior)itmaybetheonlyfeasiblemethod.”

‡S.Forrest,“EmergentComputation:Self-organizing,collective,andco-operativephenomenainnaturalandartificialcomputingnetworks,”inS.Forrest,Ed.,EmergentComputation,MIT/North-Holland,1991.

Universeas

emergentcomputation

Atminimumthefollowinglevelsofemergentbehaviorsare

assumedbythescientificcommunitytoexist:

•Fundamentalparticlesandforcesareassumedtocombinetoformcertainprimitivestableaggregatesystems,which

werefertoasthechemicalelements,H−U.

•Theseaggregatesystemsthencombineincertainwaystoformmoleculesandcompounds,whichisafieldof

emergentbehaviorofconcerntochemists.

•Anotherimportantlevelofemergentbehavioristhecoordinatedinteractionofaggregatesystemsandforces

thatproducethethingsinvolvedinbiology.

•Itcannotbedeniedthattheubiquitouscarbonatomisaveryimportantemergentbehavior,asisthehydrogenbond,

bothkeyparticipantsinthehighestleveloflocalized

emergentbehavioryetfound,thephenomenaoflife.

OnthequestionofinformationinphysicallawI

“Theoriesofthephysicalworld,whetherornotthey

incorporatethenotionofinformation,dependon

numerically-valuedquantitiesinordertobeabletodescribea

widevarietyofphenomenaaccordingtouniversalphysicallaws

whichassociatesuchquantities:measurablequantitiessuchas

energyandmomentum§”

§C.F.Boyle,PhysicalLawsandInformationContent,inPhysComp’92:Pro-ceedingsoftheWorkshoponPhysicsandComputation,IEEEPress,1992.

OnthequestionofinformationinphysicallawII

“...allinformationisfirstaform,andthemeaningofa

messageisatopologicalrelationbetweentheformofa

messageandtheeigenformsofthereceptor(theformsthat

canprovokeanexcitationofthereceptor)¶.”

¶R.Thom,StructuralStabilityandMorphogenesis:AnOutlineofaGeneralTheoryofModels,W.A.Benjamin,1975.

computationalcosmographydefined

Thenamecomputationalcosmography‖isusedtodescribeacomputationalsystemwhereinphysicalsystembehaviors(and

properties)emergeasby-productsofcomputationsbasedon-

forlackofabetterdescription-theinteractionofvirtual

deformablepolyhedrawhichconformtothegeometry

mechanismsasdescribedinR.B.Fuller’sSynergetics.

‖J.F.Nystrom,“Tensionalcomputation:Furthermusingsonthecomputa-tionalcosmography,”AppliedMathematicsandComputation120,211-225(2001).

InessenceCCisanarchetypeforanewlevelofcellular

automata.

CCisacomputationalmodelthat:∗∗

•AdoptstheviewpointthatUniverseisdiscretelyordered,and

•IsbasedonthepremisethatemergentbehaviorsinUniversearecomputablefrombasic,geometricallygrounded,

fundamentalprinciples.

∗∗J.F.Nystrom,“Statics,dynamics,andpointparticles:Onwardtowardsacomputationalcosmography,”acceptedasaposteratArtificialLifeVII,Portland,Oregon,USA(2000),butnotpresented.

Findingthevectorequilibriuminsidethetetrahedron††

††ImagecourtesyofR.W.Gray.

TheA-module,B-module,andMite(ohmy!)‡‡.

‡‡ThepictureontheleftisfromRichardHawkins’DigitalArchive.ThepictureontherightispartofColorPlate17inR.B.Fuller,Synergetics:ExplorationsintheGeometryofThinking,Macmillan,1975.

TheIVMCEMtime-domainsolver

andthe

isotropicvectorfielddecompositionmethodology

¤IVMgrid(again)andanIVMvectorbasis

¤Afullydiscretesystem

¤Fourhexagonalplanesandanomni-directionalcurloperator

¤Free-spacepropagationofanelectromagneticsolution

¤J.F.NystromandCarrynBellomo,“IsotropicVectorMatrixGridandFace-CenteredCubicLatticeDataStructures,”toappearinLectureNotesinComputerScience.

¤J.F.Nystrom,“High-OrderTime-StableNumericalBoundarySchemefortheTemporallyDependantMaxwellEquationsinTwoDimensions,”JournalofComputationalPhysics178,290-306(2002).

¤J.F.Nystrom,“TheIsotropicVectorFieldDecompositionMethodology,”ACES2002,Monterey,CA,March2002.

¤J.F.Nystrom,“GridConstructionandBoundaryConditionImplementationfortheIsotropicVectorFieldDecompositionMethodology,”ACES2003,Monterey,CA,March2003.

¤J.F.Nystrom,“InSearchofaGeometricalBasisfortheUbiquitousElectromagneticEnergy,”PIERS2002,Boston,MA,July2002.

¤J.F.Nystrom,“Moore’sLawandtheVisualizationofElectromagnetic

Quanta,”PIERS2003,Honolulu,HI,October2003.

Theisotropicvectormatrixgrid(again)

Vectorequilibrium(VE)cell.IVMgrid.

IVMvectorbasis

e1

e2

e3

e4

e5

e6

IVMbasisvectorse1-e6withinthebasicVEcell.

e1=ax(√32)−ay(

12),

e2=−ax(√32)−ay(

12),

e3=−ax(1

2√3)−ay(

12)+az(

√2√3),

e4=ax(1√3)+az(

√2√3),

e5=−ax(1

2√3)+ay(

12)+az(

√2√3),

e6=ay.

BasisvectorswithinchosenCartesianbasis.

Afullydiscretesystem

¤Forourpresentpurposes,IusetheMaxwellequations

writtenintermsoftheelectricfieldEandthemagneticflux

densityB(shownhereinSIunitsforfreespacepropagation):

∂E

∂t=c

2∇×B,

∂B

∂t=−∇×E,

wherecisthespeedofelectromagneticpropagation.

Furthermore,c2=1/(µ0²0),whereµ0isthepermeabilityof

freespaceand²0isthepermittivityoffreespace.

¤NowemployaRunge-Kuttamethodoforderfour,andthus

evolveallthefieldcomponents(intime)simultaneously.

¤Forexample,letthevalueofE[0]afterntimestepsbewrittenas

En[0]=E

n1[0]e1+E

n2[0]e2+E

n3[0]e3+E

n4[0]e4+E

n5[0]e5+E

n6[0]e6.

¤IdenotethethefirststagevaluefortheRunge-Kutta

methodoforderfour(tobeusedinthecalculationofEn+11[0])

asEk11;thevalueforthesecondstageasE

k21;thevalueforthe

thirdstageasEk31;andthevalueofthefourthstageasE

k41.If

weletδtdenotethetimestepofthesimulation,then,for

example,En+11[0]iscalculatedthus:

En+11[0]=E

n1[0]+

δt

6

(

Ek11[0]+2E

k21[0]+2E

k31[0]+E

k41[0]

)

.

ThefourhexagonalplanesoftheVE

e1

e2

e3

e4

e5

e6

(Herethea-planeisshowninmagenta,theb-planeisshowninred,the

c-planeisshowningreen,andthed-planeiscoloredaqua.)

a-plane:[+1][-2][+6][-1][+2][-6],b-plane:[+1][+4][+5][-1][-4][-5],c-plane:[-2][+4][+3][+2][-4][-3],d-plane:[-5][-3][+6][+5][+3][-6].

CalculationoftheVE-basedcurl

¤GivenT,wewanttosolvetheequation

S=∇×T.

¤BothSandTarewrittenintheIVMbasis:

S=S1e1+S2e2+S3e3+S4e4+S5e5+S6e6,

T=T1e1+T2e2+T3e3+T4e4+T5e5+T6e6.

¤Startwiththedefinitionofthecurl,

S=limdA→0

(1/dA)[n∮

T·dl],

whichistobeevaluatedoneachofthefourhexagonalplanes

oftheVE.

¤Forexample,aroundthea-plane,Itakethedotproductof

theresultingcontourwiththeunitvectornormaltothe

a-plane,aa(=azinthiscase);thus

(S3e3+S4e4+S5e5)·aa=1

dA[∮

T·dl]a.

¤Aroundtheb-plane;

(S2e2+S3e3−S6e6)·ab=1

dA[∮

T·dl]b.

¤Notethat:

e3·aa=e4·aa=e5·aa=√2

√3

,

e2·ab=e3·ab=e6·ab=√2

√3

.

¤Integratingaroundallfourofthehexagonalplanesbegetsfourequations(oneforeachoftheplanesvariables:a′,b′,c′andd′):

(S3+S4+S5)=a′,wherea′=√3

√2dA

[∮

T·dl]a,

(S2+S3−S6)=b′,whereb′=√3

√2dA

[∮

T·dl]b,

(S1−S5−S6)=c′,wherec′=√3

√2dA

[∮

T·dl]c,

(S1−S2+S4)=d′,whered′=√3

√2dA

[∮

T·dl]d.

¤Evidently

a′−b′+c′−d′=0.

Solution

¤AsolutionshowinghowtocalculateS[0]cannowbewritten:

S1=(c′+d′)/4,S2=(b′−d′)/4,S3=(a′+b′)/4,S4=(a′+d′)/4,S5=(a′−c′)/4,S6=(−b′−c′)/4,

whichshowsthateachvectorcomponentofSisdependentonthecontoursaroundtwoseparatehexagonalplanes.Theminussignsensurearight-handedsensetoeachcontour-asitrelatestoeachparticularcomponentofS.

Now,usingthesolutionforS1,wegetbacktothestagesoftheRK4integratorandlookatthesolutionforE1:

Ek11[0]=

c′(Bn)+d′(Bn)

4µ0²0,

Ek21[0]=

c′(Bn+δt2Bk1)+d′(Bn+δt

2Bk1)

4µ0²0,

Ek31[0]=

c′(Bn+δt2Bk2)+d′(Bn+δt

2Bk2)

4µ0²0,

Ek41[0]=

c′(Bn+δtBk3)+d′(Bn+δtBk3)

4µ0²0.

Thisnowcompletelyshowshowtheintegratorstagesforthe

timeevolutionofE1[0]arecalculated.

Visualizingthevectorcurl

e3

Theaandbplanes

interactandbegetthe

e3componentofacurl.

e5

Theaandcplanes

interactandbegetthe

e5componentofacurl.

e1

e2

e4

e6

Contourintegralevaluation

¤TheareaenclosedbyeachcircularcontourisdA=πh2

(wherehisthegridspacing).

¤Eachintegralisimplementeddiscretelybyevaluatingthe

integralatsixlocationsonthehexagonalplane.Forexample,

onthea-plane:

[∮

T·dl]a=hπ

3T·dla[+1]+h

π

3T·dla[−2]+h

π

3T·dla[+6]+

hπ

3T·dla[−1]+h

π

3T·dla[+2]+h

π

3T·dla[−6],

where,forexample,

dla[+1]=−e1×aa

isthevectortangenttothecontourat[+1],and

aa=

(

e2×e1|e2×e1|

)

istheunitnormaltothea-plane.

¤Thus,thespatiallydiscreteequationfortheplanevariablea′

isgivenintermsofcomponentsofTatthevertices[+1],[-1],

[-2],[+2],[+6],and[-6]as

a′=1

h√6

{[√3

2(T6−T2)+

1

2√3(T4+T5)−

1√3

T3

]

[+1]

+

[√3

2(T2−T6)−

1

2√3(T4+T5)+

1√3

T3

]

[−1]

+

[√3

2(T6−T1)−

1

2√3(T3+T4)+

1√3

T5

]

[−2]

+

[√3

2(T1−T6)+

1

2√3(T3+T4)−

1√3

T5

]

[+2]

+

[√3

2(T2−T1)+

1

2√3(T3+T5)−

1√3

T4

]

[+6]

+

[√3

2(T1−T2)−

1

2√3(T3+T5)+

1√3

T4

]

[−6]}

.

Free-spacepropagation

¤HereImodelthepropagationofatransverse

electromagneticsolutioninthetime-domain.Iseeda

100-frequencyIVMgridwiththefollowingsolution:

E=azsin(ky),andB=axsin(ky)

c,

wherek=2πf/cisthewavenumber,f=3GHzisthe

frequency,andcisthespeedofelectromagneticpropagation.

Examplepropagation

−10−8−6−4−20246810

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Grid points near the origin. (In [−6] and [+6] directions.)

Ez value

n=0n=44n=88

CFL=1andPPW=11.

Electricfieldastravelingtetrahedron

Somefutureresearchdirections

¤IVMCEMjournalpublicationandwaveguidesimulation

¤DataminingofIVMCEMtime-domainsimulationdata

¤ClusterandGridimplementationofIVMCEMcodes

¤IVMcellularautomataandlatticeBoltzmann

¤IVMcellpopulationdynamicsandwoundhealing(Carryn)

¤Graphics,visualizationandVRimplementations

¤CC!

ABIGSimulation

Bladesofgrassliterallyeatelectromagneticenergy.AsimulationinvolvingthecompletephotosynthesisprocessofabladeofgrasswouldbeaBIGsimulationindeed.(Andinvolvemorethanjustacomputationalelectromagneticsolver.)