potential driven flows through bifurcating networks aerospace and mechanical engineering graduate...
DESCRIPTION
Background / Motivation self-similarity Natural physical examples Broccoli Artificially occurring examples Artificial terrain Non-physical examples Data series Music "When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar." - Mandelbrot From Hofstadter's classic, Godel, Escher, Bach: "A fugue is like a canon, in that it is usually based on one theme which gets played in different voices and different keys, and occasionally at different speeds or upside down or backwards."TRANSCRIPT
Potential Driven Flows Through Bifurcating Networks
Aerospace and Mechanical Engineering Graduate Student Conference 200619 October, 2006
Jason MayesAdvisor: Dr. Mihir Sen
Outline Background/Motivation Objectives A self-similar model Simplifying the model Forms of similarity Examples Conclusions / Future work
Background / Motivationself-similarity
Natural physical examples Broccoli
Artificially occurring examples Artificial terrain
Non-physical examples Data series
Music
"When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar." - Mandelbrot
From Hofstadter's classic, Godel, Escher, Bach: "A fugue is like a canon, in that it is usually based on one theme which gets played in different voices and different keys, and occasionally at different speeds or upside down or backwards."
Background / Motivationself-similar systems
Self-similarity in engineering many systems can be considered self-similar
over several scales Large scale problems
as systems grow, solutions become more difficult
can we simplify?
Objectivessimplification and reduction
Take advantage of similarity to simplify analysis
Use known structure / behavior to simplify Using structure Extending behavior from one scale to another
Reduce large equation sets
A self-similar modela model for analysis
The bifurcating tree geometry Geometry seen in a wide variety of
applications Potential-driven flow or transfer
ex. heat, fluid, energy, ect. Conservation at bifurcation points
q
q1
q2q = q1+q2
A self-similar modelpotential-driven transfer
Assumptions Transfer governed by a linear operator i.e., for each branch: pLq
A self-similar modelsimplification
Goal is to reduce or simplify the system to the single equation
Generation two (N=2) Network
A self-similar modelsimplification
Apply recursively to eliminate q1,1 and q2,2
A self-similar modela simple result
Result of simplification for N=2 network
For the more general N-generation network
For integro-differential operators Process repeated in the Laplace domain Inverses become simple algebraic inverses Can be written as a continued fraction
Forms of similaritywithin and between
Similarity can be used to further simplify Two forms of similarity:
Similarity ‘within’ a generation Symmetric networks: the operators within a generation are identical Asymmetric networks: the operators within a generation are not identical
Similarity ‘between’ generations Generation dependent operators depend on the generation in which the
operator occurs and change between successive generations Generation independent operators do not change between generations
Four possible combinations Symmetric with generation independent operators Asymmetric with generation independent operators Symmetric with generation dependent operators Asymmetric with generation independent operators
Examplestree of resistors
Symmetric with generation independent operators
Examplestree of resistors
Tree constructed of identical resistors Current i(t) through each branch is driven by the potential
difference v(t) across the branch Each branch governed by
For an N generational tree of resistors
For an infinite tree of resistors
)()( tvtRi )(
)(tvp
tiqRL
)()(211 tvtiR
N
)()( tvtRi
Examplesfractional order visco-elasticity models
Asymmetric with generation independent operators
Examplesfractional order visco-elasticity models
Tree is composed of springs and dashpots Branches governed by linear operators Result is a fractional-order visco-elasticity
modelkLk a
dtdL
dtd
a
Exampleslaminar pipe flows in branching networks
Symmetric with generation dependent operators
Exampleslaminar pipe flows in branching networks
Each pipe governed by
In Laplace domain
In the time domain
Conclusions / Future Work Known behavior on one scale can be extended
to better understand behavior on another Similarity or structure can be used to help
simplify For equation sets: patterns or structures can
be used to simplify Future work:
Analyze other self-similar geometries Study probabilistically self-similar geometries
Questions?