Intro Multi

7/18/2019 Intro Multi

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An Introduction to

Multiscale Modeling

Scientifc Computing

and Numerical AnalysisSeminar

CAAM 699


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Outline

Multiscale Nature o Matter Physical Scales

Temporal Scales

Dierent !a"s or Dierent Scales

Computational Di#culties

$omogeneous %lastic String &nhomogeneous %lastic String

O'er'ie" o Seminar Topics


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Physical Scales

Discrete Natureo Matter

MultiplePhysical(Spatial) scales%*ist

%*ample+ ,i'er Physical Scale+

-m . /01mhttp+22a-3"ater3usgs3go'2yu-on2inde*3php


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Physical Scales

4ater Cluster Physical Scale+

5 nm . 5 * /069m

http+22eyeothefsh3org2lea-yleush-e2

http+22"""37tinternet3com28martin3chaplin2clusters3html

4ater Drops

Physical Scale+ mm . /01m


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Physical Scales

4ater Molecule

Physical Scale+

03:; nm . 3:; * /06/0

m

http+22commons3"i-imedia3org2"i-i2


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Temporal Scales

Multiple Time Scales inMatter

Time Scale o &nterestDepends on Phenomenon o&nterest


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Dierent Scales> Dierent!a"s

?o'erning %@uations dierent ordierent scales

%*ample+ Modeling a Su7atomic Description o


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Model Choice

Could represent ri'er as discrete uidparticles> and utilie moleculardynamics to model its o"

More details included in the model>the more accurate your model "illli-ely 7e

4hatBs the pro7lemEEE?ood !uc- trying to do this

computationallyFFF


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Computational Di#culties

Num7er o elements

Smaller Spatial Scale may "arrant asmaller time scale in order to -eepnumerical methods sta7le %*ample+ C

PD%s


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Model Choice

Galance detailand computationalcomplexity

Choice oten made to model a material as acontinuum

?oal is to then fnd a constitutive law thatcan e*plain ho" the material 7eha'es

& the material is homogeneous> thecontinuum assumption is typically

accepta7le and constituti'e la"s can 7eound

Heterogeneous materials are moredifcult to model, and motivate the

need or multiscale models


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$omogeneous %lastic String

Discrete Scale+ MassSpring system

point masses o mass

Springs 7et"een each mass ha'e springconstant

&n ero strain state> springs are length

Deri'e %@uation or !ongitudinal Motion

k

N m

h


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!et 7e the displacement omass rom its ero strain state attime

%@uation o motion or mass can7e "ritten using Ne"tonBs !a"+

The can 7e "ritten as

)(tuj jt

j

maF=

ma )(tum j


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%lasticity Modulus

!inear Mass Density

Ta-e !imit as

[ ]2

11 )()(2)()(

h

tututuEtu

jjj

j

+ +=

khE=

h

m=

0h


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/D 4a'e %@uation

Continuum!e'el model> limit o themicroscopic (discrete) model

4a'e speed determined 7y

does NOT depend on location in thestring

$yper7olic PD%

easy to simulate

),(),( txuEtxu xxtt

=

,E


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&nhomogeneous %lasticString

Discrete Model> MassSpring system

Num7er o springs 7et"een eachpoint mass can 'ary


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point masses o mass

Springs 7et"een each mass ha'espring constant

&n ero strain state> springs are length

displacement o mass

. num7er o springs 7et"eenmass and at time

k

N m

h

)(tuj j

)(1 tnj+ j

1+j t


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%@uation o Motion or mass

[ ] [ ])()()()()()()( 111 tututkntututkntum jjjjjjj ++ =j

[ ] [ ]( ))()()()()()()( 111 tututntututnktum jjjjjjj ++ =

[ ] [ ]( )2

1112 )()()()()()(

)(h

tututntututn

m

khtu

jjjjjj

j

++ =


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%@uation o Motion or mass

Ta-e !imit as

j

= ++

h

tututn

h

tututn

mh

khtu

jj

j

jj

jj

)()()(

)()()()(

11

1

2

0h

( )xxtt

txutxnE

txu ),(),(),(

=


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4a'e e@uation "ith locally'arying "a'e speed+

To sol'e this "a'e e@uationyou need to -no" 7utthis is a microscopic @uantityF(!ocal density o springs)

Micro @uantity needed in acontinuum e@uation

),( txnE

),( txn


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Put another "ay in the orm o aconstituti'e la" (relation 7et"eenstress and strain)

Di'iding 7y h and ta-ing the limit h 0

== AFmutt)(

+= )2

(()

2

(( h

xh

xAuAh tt

xttu )( =

& h %l ti


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%@uating these t"o @uantities gi'es+

Htiliing+

%lastic properties o the spring 'ary

spatially

xttu )(= ( )xxtt txutxnEtxu ),(),(),(

=

( )xxx

txutxnE ),(),()( =

),()( txEn=xu=


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Practical %*ample

,upturing String+

Assume springs 7rea- i thesegment length or somedistance

Microscopic Model+

Continuum Model+

ppduu ii >+1 d

= ++

h

tututn

h

tututn

mh

khtu

jj

j

jj

jj

)()()(

)()()()(

11

1

2

( )xxtt

txutxnE

txu ),(),(),(

=


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Practical %*ample

Gegin "ith string in ero strain stateattached at one end to "all

This string is stretched at the otherend 7y a constant strain rate

// point masses> /00 springs7et"een each pair o masses

4hen distance 7et"een massese*ceeds then springs 7rea-p

pd


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Rupturing tring


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Rupturing tring

Displacement

%ng@uist G> JMultiscaleModeling and ComputationK> Noticeso the AMS> Lol 50+9> p3 /06/0:0

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