balance laws traffic flow behavior greenshields model...
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Continuum approach Motion descriptionBalance lawsTraffic flow behaviorGreenshields modelLinear traffic waves Method of characteristics
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If we consider an artery with obstacles thesituation can be compared to a highwaypopulated by travelling vehicles where inthis case those cars are cells of the blood.
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𝐁𝐁𝐁𝐁𝐁𝐁𝐁𝐁𝐁𝐁 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐁𝐁
µmµm
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𝟏𝟏𝟏𝟏−𝟒𝟒 𝟏𝟏𝟏𝟏−𝟔𝟔𝟏𝟏𝟏𝟏−𝟑𝟑𝟏𝟏𝟏𝟏𝟏𝟏𝒎𝒎
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Displacement
=0
Control Volume
Trajectory
Flow direction
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1-D conservation equation
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Traffic flow behavior
The flowMacroscopic)
Individual cars(Microscopic)
The observed behavior of cars involvesdriving speeds
driving lane
FactorsTraffic conditionsCars characteristics
Road conditionsExternal agents
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𝒒𝒒
𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎
𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎
𝒒𝒒𝒎𝒎𝒎𝒎𝒎𝒎
𝒗𝒗
𝝆𝝆 𝝆𝝆
Constitutive law 𝒗𝒗 = 𝒇𝒇 𝝆𝝆
𝒗𝒗 𝝆𝝆 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 − 𝝆𝝆𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎
𝒒𝒒 𝝆𝝆 = 𝝆𝝆 𝒗𝒗 𝝆𝝆
Empirical dataGreenshields model
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𝝏𝝏𝝆𝝆𝝏𝝏𝝏𝝏
+ 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 −𝟐𝟐𝝆𝝆𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎
𝝏𝝏𝝆𝝆𝝏𝝏𝒎𝒎
= 𝟏𝟏
𝝏𝝏𝝆𝝆𝝏𝝏𝝏𝝏
+ 𝒒𝒒𝒒(𝝆𝝆)𝝏𝝏𝝆𝝆𝝏𝝏𝒎𝒎
= 𝟏𝟏
𝒒𝒒 𝝆𝝆 = 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝝆𝝆 − 𝝆𝝆𝟐𝟐
𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎𝒗𝒗 𝝆𝝆 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 − 𝝆𝝆
𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎𝒒𝒒 𝝆𝝆 = 𝝆𝝆 𝒗𝒗 𝝆𝝆
𝒒𝒒′ 𝝆𝝆 = 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 −𝟐𝟐𝝆𝝆𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎
Quasi linearPDE
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𝒒𝒒′ 𝝆𝝆𝟏𝟏 + 𝜹𝜹𝝆𝝆 = 𝒒𝒒′ 𝝆𝝆𝟏𝟏 + 𝒒𝒒′′ 𝝆𝝆𝟏𝟏 𝜹𝜹𝝆𝝆 + …
Linearized form
Solution form𝒒𝒒′ 𝝆𝝆𝟏𝟏 = 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 − 𝟐𝟐𝝆𝝆𝟏𝟏
𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎= 𝒄𝒄
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𝒅𝒅𝒅𝒅𝒅𝒅𝝏𝝏𝒎𝒎𝒅𝒅𝒄𝒄𝒅𝒅 𝒎𝒎
𝟏𝟏𝒄𝒄
𝟏𝟏
𝒅𝒅𝒎𝒎𝒅𝒅𝝏𝝏 = 𝒄𝒄
𝝏𝝏𝒅𝒅𝒎𝒎𝒅𝒅𝝏𝝏
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Thank you for your attention