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A core Course on Modeling. Introduction to Modeling 0LAB0 0LBB0 0LCB0 0LDB0 [email protected]e.nl [email protected] P.14. Structural and quantitative confidence in the lantern model. Contents Scaffolding Correct qualitative behavior V arying cat.-I quantities

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a core course on modeling

A core Course on Modeling

Introductionto Modeling

0LAB0 0LBB0 0LCB0 0LDB0

[email protected]

[email protected]

P.14

slide2

Structuralandquantitativeconfidence in the lantern model.

  • Contents
  • Scaffolding
  • Correct qualitativebehavior
  • Varying cat.-I quantities
  • Plottinggraphsusing the analysis tab
  • Correct asymptoticbehavior
  • Correct singularities
  • Convergence
slide3

Scaffolding.

  • Build extra features in the model, not neededfor the calculation of the cat.-II quantities, but toinspect the validity of the model.
  • Lanterns:
  • brightnessdistribution over the road
  • differencebetween max and min intensity

try!

try!

slide4

Scaffolding.

  • Build extra features in the model, not neededfor the calculation of the cat.-II quantities, but toinspect the validity of the model.
  • Lanterns:
  • brightnessdistribution over the road
  • differencebetween max and min intensity

try!

try!

slide5

Scaffolding.

  • Build extra features in the model, not neededfor the calculation of the cat.-II quantities, but toinspect the validity of the model.
  • Lanterns:
  • brightnessdistribution over the road
  • differencebetween max and min intensity

try!

try!

slide6

Correct qualitativebehavior.

  • Varying cat.-I quantities. Examples:

dL=25 m; h=5.5 m; p  2.4 kW

dL=25 m; h=12 m; p  2.4 kW: higherlanterns, solessfluctuations

dL=15 m; h=12 m; p  2.4 kW: lanterns closer together: verylittlefluctuations

dL=15 m; h=3 m; p  1 kW: shorterlanterns, soless energy/m

( 1/15 < 2.4/25) – but more fluctuations

slide7

Correct qualitativebehavior.

  • Plottinggraphsusing the analysis-tab. Examples:

Both maxIntandminInt are proportionalto P.

Sotheirdifferencealso

shouldbeproportionalto p.

Tallerlanternscause more blending lessfluctuations

Lanternsfurther apart givelargerfluctuations, solargerdifferencebetweenmaxIntandminInt.

slide8

Correct asymptoticbehavior.

  • If the street is long enough (100m) comparedto the height of the lanterns (2 ... 32 m) we expecttosee the boundaryeffectsseparatedfrom the regularbehavior:

h=2 m

h=4 m

hardlyany flat region

flat region?

h=8 m

flat region

flat region

flat region

h=16 m

h=32 m

slide9

Correct singularities.

  • IfdL is very large comparedto h, we canignoreaddingcontributions of multiple lanterns.
  • The maximum intensitycanthenbecalculated as p/h2:
  • p = 8069.5 W
  • h = 9.9 m
  • p/h2 = 82.3 W/m2
  • ACCEL gives 84 W/m2.
  • OK because of
  • contribution of
  • otherlanterns.
slide10

Correct convergence.

  • Suppose we increase performance bytakingfewerlanternsinto account.
  • allowsto set # lanternsto taken into account.

Thisversion of the model

slide11

Correct convergence.

  • Suppose we increase performance bytakingfewerlanternsinto account.
  • How few lanterns is enoughforreliableresult depends on regime

dL=25 m; nr. neighbours = 0

dL=25 m; nr. neighbours = 2

dL=25 m; nr. neighbours = 3  no noticeabledifference, soleftand right 2 lanterns is enough.

slide12

Correct convergence.

  • Suppose we increase performance bytakingfewerlanternsinto account.
  • How few lanterns is enoughforreliableresult depends on regime

dL=1 m; nr. neighbours = 0

dL=1 m; nr. neighbours = 5

dL=1 m; nr. neighbours = 10

dL=1 m; nr. neighbours = 20: graphstill looks not smooth, we presumablyneed more lanterns.

slide13

Summary:

  • Scaffoldingmaybeusefulto help ascertain the convincingness of the model;
  • Qualitativebehavor: think of configurationsthatcaneasilybeunderstood;
  • Asymptoticbehavior: can we separate ‘boundaryeffects’ thatcanbe ‘reasonedaway’;
  • Singularbehavior: are there cases where we can do the math even without a computer;
  • Convergence: do we take enough terms, steps, ... .
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