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Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Math: the art in problem solving
Michael Lindstrom
October 22, 201415:00-16:00
email: [email protected]: math.ubc.ca/∼MLRTLM
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Math is everywhere
quantifiable descriptionsglacier motion, spreading of diseases, market inflation,digitally recorded music, ...real world → model → abstractify → analyze → interpretmath is useful
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Outline
applied math technique introselectrodialysis applicationfusion applicationmass spectrometry applicationconcluding remarks
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Modelling Overview
“sketch” relevant physics, biology, etc.assign variables, write equationsinsightful results
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Modelling Example
finding the dimensions of a cylindrical tank with open tophaving a volume of 1000 cm3 with least possible surfacearealet h =height, r =radius; need πr2h = 1000 withminimum value of 2πrh + πr2
cheaper than building thousands of cylinders!
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Modelling Example
finding the dimensions of a cylindrical tank with open tophaving a volume of 1000 cm3 with least possible surfacearealet h =height, r =radius; need πr2h = 1000 withminimum value of 2πrh + πr2
cheaper than building thousands of cylinders!
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Nondimensionalization Overview
make “diorama”remove dimensions, reduce constantskey relationshipsimproved numericsall variables comparable: velocity, pressure, time, etc.
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Nondimensionalization Example
rabbit population P has carrying capacity C satisfiesP ′(T ) = rP(1− P/C) at time T , initial populationP(0) = P0
units: [P] = [C ] = [P0] = rabbits, [r ] = 1/time, [T ] =timenondimensionalize to
p′(t) = p(1− p), p(0) = p0
qualitative behaviour: p0
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Asymptotics Overview
mathematical “impressionism”exploit relative smallness of parameter(s)approximate model problem solution accurately
solution = estimate + correction + smaller correction + ...
analytic insight
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Asymptotics Example
consider 0.0001x3 − x + 1 = 0; let ε = 0.0001� 1asymptotic prediction:
x ∼ 1 + ε+ ...
x ∼ 1√ε− 1
2 + ...
x ∼ −1√ε− 1
2 + ...
Asymptotic Exact1.0001... 1.0001...
99.5... 99.496...-100.5... -100.496...
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Numerical Analysis Overview
“digital snapshot”arbitrarily high precisioncontrol error termsmay require high computational power
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Numerical Analysis Example
evaluate I =∫ 1
0√
x5 + 1dxRiemann sum: let N > 0, h = 1/N and xj = jh so thatI = h
∑Nj=1
√x5
j + 1 + error
theory: |error| ≤ some constantN → 0 as N →∞
N Sum10 0.999
100 1.0661000 1.0738
10000 1.0746100000 1.0747
better methods exist!
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Electrodialysis Motivation
clean water and CO2 sequestration goodmany pollutants ionicelectrodialysis: filter out impure ions from water, makeuseful products with electric field and selective membranes
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Problem
UBC chemical engineers building prototypes, operationalconditions important
Given experimental data of input concentrations, the totalvoltage and the total current drawn across the series ofchannels, what losses can be attributed to each individualchannel and what is taking place in each channel?
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Problem
UBC chemical engineers building prototypes, operationalconditions importantGiven experimental data of input concentrations, the totalvoltage and the total current drawn across the series ofchannels, what losses can be attributed to each individualchannel and what is taking place in each channel?
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Carbonic Acid Channel
given voltage drop V , determine the current I
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Models
model 0: uniform [H+] and [HCO−3 ], current driven byvoltage gradient - wrong species cross membranes
model 1: spatially varying [H+] and [HCO−3 ] withoutreactions - too few ionsmodel 2: with reactions
but ions not produced fast enough #fail
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Models
model 0: uniform [H+] and [HCO−3 ], current driven byvoltage gradient - wrong species cross membranesmodel 1: spatially varying [H+] and [HCO−3 ] withoutreactions - too few ions
model 2: with reactions
but ions not produced fast enough #fail
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Models
model 0: uniform [H+] and [HCO−3 ], current driven byvoltage gradient - wrong species cross membranesmodel 1: spatially varying [H+] and [HCO−3 ] withoutreactions - too few ionsmodel 2: with reactions
but ions not produced fast enough #fail
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Models
model 3
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Models
numerics on asymptotic model more consistent withexperimental datadiscovered device doesn’t operate the way intended!
“... all models are wrong, but some are useful.” - GeorgeBox
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Models
numerics on asymptotic model more consistent withexperimental datadiscovered device doesn’t operate the way intended!“... all models are wrong, but some are useful.” - GeorgeBox
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Fusion Motivation
fusion energy in sun31H + 2
1H → 42He + 1
0n + 17.6 MeV and other reactionsclean energysustain high particle density and temperaturefusion on earth?magnetized target fusion at General Fusiondesign: compress plasma in imploding shell of lead-lithium,driven by fast high-pressure impulses
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Problem
How much will the plasma compress under such a design?
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Problem
How much will the plasma compress under such a design?
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Simple Dimensionless Model
spherical symmetryhomogeneous plasmanonlinear, coupled system of PDEs: mass and momentumconservationfree boundaries - need to solve for radii within problemnondimensionalization
small initial plasma radiusvery small initial density disturbancevery small impulse timevery, very small initial plasma pressure
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Insights Obtained
wave-like behaviour: sound speed dominatespressure amplitude grows like 1/ralmost all input energy reflected!fusion may be within reach - Lawson condition
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Insights Obtained
minimum radius R∗L :
R∗L ≈C4
s Pplasma,0R7inner,0%
30
πP4maxR4
outer,0T 20
= 4mm
agrees well with numerical methodsParameter Meaning
Cs lead sound speed at atmospheric pressurePplasma,0 initial plasma pressureRinner,0 initial plasma radius%0 lead density at atmospheric pressure
Pmax piston impulse pressureRouter,0 initial lead radius
T0 impulse time scale
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Spectrometry Motivation and Problem
detect chemical species/isotopes with masscharge-to-mass separation in mass spectrometersome current means
magnetic fields costlytime of flight discretequadrupole too specific
PerkinElmer: come up with a design for amass-spectrometer without magnetic fields, withcontinuous measurement and which can detect any mass.
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Spectrometry Motivation and Problem
detect chemical species/isotopes with masscharge-to-mass separation in mass spectrometersome current means
magnetic fields costlytime of flight discretequadrupole too specific
PerkinElmer: come up with a design for amass-spectrometer without magnetic fields, withcontinuous measurement and which can detect any mass.
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Concepts
acceleration in electric field proportional to charge-to-massratio and field strengthpositive charge oscillates between fixed positive charges(one-dimensional)same amplitude and different masses, different periods
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Idea
fire ion beam into tube of chargespredict qualitative trajectories via asymptoticsoscillation wavelength mass-dependentnumerics of ODEs verify distinct masses spatially separate
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Results
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Art
each technique has rolecombination often usefulresults extremely beautiful/satisfying
Math: the artin problem
solving
MichaelLindstrom
Introduction
TechniquesModelling
Nondimensionalization
Asymptotics
Numerical Analysis
ProblemsElectrodialysis
Magnetized TargetFusion
Mass Spectrometry
Summary
Final Remarks
thanks to many collaborators on these projects:- Arman Bonakdarpour, Saad Dara, and David Wilkinson
(UBC Chemical and Biological Engineering Department)- Sandra Barsky and Aaron Froese (General Fusion)- Iain Moyles, Michael Ward, and Brian Wetton (UBC Math
Department)- Samad Bazargan (PerkinElmer)- others
some suggested applied courses:- MATH 215, 316, 400, 401 (differential equations)- MATH 361 (modelling, math-bio)- MATH 210, 405 (numerics)- MATH 450 (asymptotics)- some non-math science courses!
slides posted at: math.ubc.ca/∼MLRTLM/#talks