Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
Padmanabhan Seshaiyer
Professor, Mathematical Sciences
Director, COMPLETE Center
George Mason University
Email: [email protected]
Multidisciplinary Undergraduate Research in Computational Mathematics and Nonlinear Dynamics of
Biological, Bio-inspired and Engineering Systems
Best Practices for Introducing Undergraduate Students to
Computational and Interdisciplinary Research
SIAM Annual Meeting
July 13, 2012
UG Students as change agents
Padmanabhan Seshaiyer
George Mason University
University & K-12 STEM Collaboration
Sarah M. Venuti (Undergraduate Student)Avis Foster (Undergraduate Student)Kris Kappmeyer (K-12 Teacher)Alicia Hamar (High School Student)Stephanie Alley (Undergraduate Student)Andrew Samuelson (Graduate Student)
Padmanabhan Seshaiyer
George Mason University
BloodArterial Wall
CSF
Transforming Practice Through Undergraduate Researchers, P. Seshaiyer, Council on undergraduate research Quarterly, Fall 2012.
REU: COMPUTATIONAL MATHEMATICS AND NONLINEAR DYNAMICS OF BIOLOGICAL, BIO-
INSPIRED AND ENGINEERING SYSTEMS
• Summers 2012-2013, 2011-2012, 2009-2010 and 2007-2008
• Stipend $3375; Free on-campus housing and meals; Travel allowance up to $550 and more!and more!
• Supports 12 students and 2 K-12 teachers each year
• http://math.gmu.edu/reu
Sample Research Topics
Modeling using deterministic and stochastic differential equations
Computational biology, biomechanics and neuroscience
Mathematics of Materials
Nonlinear dynamics and Micro Air Vehicles
Selected Quotes from participants
"It has inspired me to push the limits, to keep asking questions and keep believing we can solve any problem.”
“I have been exposed to many applications that I
Padmanabhan Seshaiyer
George Mason University
“I have been exposed to many applications that I did not know existed before.”
“I used to think doing research in math was purely proof based. Now I see that research in math has endless possibilities.”
“The research program has greatly helped me to answer the question that my students always seem to have…when am I ever going to use this?”
Nature of Student Activities
• Lecture Series
• Mentoring Club
• Guest Colloquium
• REU Participant Seminars
Padmanabhan Seshaiyer
George Mason University
• REU Participant Seminars
• Computational Laboratory
• Scientific and Social Tours
• Lesson Study
A Computational Model for Batten Behavior in Micro Air VehiclesSyeda Khadija F. Zaidi (University of Maryland, College Park)
Applying Numerical Methods to Fluid Structure Interactions in Biological SystemsCourtney Chancellor (Southern Methodist University)
Ordinary Differential Equations in Green Oxidation Processes
Sample Computational Mathematics Projects
Padmanabhan Seshaiyer
George Mason University
Ordinary Differential Equations in Green Oxidation ProcessesAngela Dapolite (Clarkson University)
Viscoelastic Modeling of Biological Tissue in an Idealized Cerebral AneurysmKris Kappmeyer (H-B Woodlawn Highschool)
Modeling Evaporation from the pre-lens tear film over a contact lensAmber Xu (Carnegie Mellon University)
Optimization of an Antenna Structure for a Photovoltaic DeviceEmily Forney (Clemson University)
Analysis of Spherical Inflation Models for Intracranial Saccular Aneurysm Elasto-dynamicsJames Halsall (Farmingdale State College, New York)
Multidisciplinary Research in Mathematics
• As concluded by the National Research Council: Undergraduate education will not change in a permanent way through the efforts of “Lone Rangers.” Change requires ongoing interaction among communities of people and institutions that will reinforce and drive reform.
Padmanabhan Seshaiyer
George Mason University
reform.
• Research that happens across traditional mathematics and at the edges of traditional disciplines.
• Here is the problem,
find the mathematics to solve it!
HumanHealth
BasicScientificResearch
Energy &Natural resourcesMultidisciplinary
Research
Padmanabhan Seshaiyer
George Mason University
Health
Defense &National Security
Sustainabilityof the
Environment
ResearchImpactAreas
ComputationalBiomechanics
MathematicalBiology
MaterialsModeling
NonlinearDynamics
StochasticDifferential Porous
Mathematics in someMultidisciplinary
Padmanabhan Seshaiyer
George Mason University
Differential Equations
FluidDynamics
Fluid-StructureInteraction
ControlTheory
Optimization
PorousMedia
MultidisciplinaryResearch Topics
Saccular Aneurysms: Rupture
Padmanabhan Seshaiyer
George Mason University
Identification of Rupture Potential
Characterization of material properties
Padmanabhan Seshaiyer
George Mason University
Contact constraints
Elasto-dynamics
Coupled flow-structure interaction
Bifurcating Artery-Aneurysm Model
Padmanabhan Seshaiyer
George Mason University
MAV: Membrane Wing Deflection
Padmanabhan Seshaiyer
George Mason University
Experimental Challenges in MAV Design
Small size
High surface-to-volume ratio
Constrained weight and volume limitations
Padmanabhan Seshaiyer
George Mason University
Constrained weight and volume limitations
Low Reynolds number regime
Low aspect ratio fixed to rotary to flapping wings
Longer flight time
Better range-payload performance
Computational Challenges in MAV Design
Structural Dynamics
Fluid Dynamics
FSI
Padmanabhan Seshaiyer
George Mason University
Guidance & Control
Propulsion &Power
Beam-Fluid Interaction
Padmanabhan Seshaiyer
George Mason University
What makes these problems interesting?
Fully three-dimensional
Heterogeneous/Homogeneous
Anisotropic/Isotropic
Padmanabhan Seshaiyer
George Mason University
Anisotropic/Isotropic
Compressible/Incompressible
Non-linearity/Linear
Large/Small Deformations
Dynamic/Static
And more ………
Key Steps in Mechanics
Kinematics
Forces
Balance Relations
Padmanabhan Seshaiyer
George Mason University
Balance Relations
Constitutive Formulation
Boundary/Initial Conditions
Boundary/Initial Value Problem
Solution Methodology
Physical System
Padmanabhan Seshaiyer
George Mason University
Mathematical Model
Predator-Prey Problem
XYYdt
dY= 1α
2α__
dX
Padmanabhan Seshaiyer
George Mason University
Kteyty
yKdt
dyty
dt
dy
0)(
)(
=⇒
=⇒α3
α XY4α__ +
Y(0) = Y0
X(0) = X0
Xdt
dX=
Displacement of a Linear Elastic Bar
X=a X=b
f(x)
u(x)
Padmanabhan Seshaiyer
George Mason University
)(xfdx
d
dx
duK
dx
du
−=
=⇒
σ
σασ
0)(
0)(
)(
=
=
=
−
bu
au
xfdx
duK
dx
d
Solution Methodology
Physical System
Padmanabhan Seshaiyer
George Mason University
Mathematical Model
Analytical Solution
Compare
Finding an Analytical Solution
)(=
− xf
dx
duK
dx
dXYY
dt
dY= 1α
2α__
BVP IVP
Padmanabhan Seshaiyer
George Mason University
0)(
0)(
=
=
bu
au
dxdx
3α
dt
XY4α__ +
Y(0) = Y0
X(0) = X0
Xdt
dX=
Solution Methodology
Physical System
Mathematical Model
Padmanabhan Seshaiyer
George Mason University
Mathematical Model
Numerical SolutionAnalytical Solution
Experimental DataCompare Compare
Compare
BVP or IVP
Finite Element Methods
Finite Difference Methods
Shooting Methods
RungeKutta
Methods
….............
Experimental Data
Padmanabhan Seshaiyer
George Mason University
Solution to a System of Equations
Direct Methods
Indirect Methods
Error Analysis
Solution to BVP/IVP
Fluid
Flow-structure interaction (FSI)
Γ
Solid FluidSolid
Padmanabhan Seshaiyer
George Mason University
0.
).(
),0(:
=∇
=∇+∇+∆−∂
∂
×Ω
u
fpuuut
u
TFluid
f
f
νρr
Γ
])([5.0~
~2)~(~
~.
),0(:
2
2
T
s
s
ww
tr
bt
w
TSolid
∇+∇=
+=
=∇−∂
∂
×Ω
ε
εµελσ
σρ
Γ
Beam – Flow Interaction
Modeling Arterial Wall-Flow Interaction
BloodArterial Wall
CSF
Padmanabhan Seshaiyer
George Mason University
MathematicalModel
Modeling, Analysis and Computation of Fluid Structure Interaction Models for Biological SystemsS. Minerva Venuti and P. Seshaiyer (Mentor), SIAM Undergraduate Research Online (2010)
Inner wall Outer wall
Arterial Wall
km Spinal Fluid ∞
),( txu
Blood
One-dimensional Model
Padmanabhan Seshaiyer
George Mason University
x-direction
)sin(0
tXX b ω=
rho 1000 kg/m 3
density of spinal fluid
mu 0 Stokes viscosity of spinal fluidc 1500 m/s velocity of spinal fluidm 1.0 kg mass of the arterial walla 0.01 m2 area of the outer arterial wallk 8000 N/m spring constant
omega 1 rad/s frequency of heart beatX0 0.01 amplitude of heart beatXb displacement imposed by the pressure
exerted against the arterial wall by the blood
Coupled FSI System
2
22
2
2
x
uc
t
u
∂
∂=
∂
∂Lx <<0
Padmanabhan Seshaiyer
George Mason University
0for 0)sin(),0(),0( 0
2 ==−++∂
∂−
••
xtkXtkutumax
uc ωρ
LxtLx
uctLu =
∂
∂−=
•
for ),(),(
=)0,(xu 0)0,( =•
xu
Analytical Solution
tDtCBeAetutrtr
sincos),0( 21 +++=
2
)/(4)/(/2
1
mkmcamcar
−−−=
ρρ=2r 2
)/(4)/(/2
mkmcamca −+−
ρρ,
Padmanabhan Seshaiyer
George Mason University
,)(*)(
)/(22
121
0
ω
ω
+−=
rrr
XmkA ,
)(*)(
)/(22
212
0
ω
ω
+−=
rrr
XmkB
222422222
0
)/(2
)/(
kmkmmca
mcaXmkC
+−+−=
ωωωρ
ρωand
222422222
2
0
)/(2
)()/(
kmkmmca
kmmXmkD
+−+
−−=
ωωωρ
ωω
.
Displacement at x=0
Padmanabhan Seshaiyer
George Mason University
Variations on parameter k - stiffness
K=4000K=4000 K=16000
K=8000 K=32000
Variations of rho - density
Rho = 500 Rho = 2000
Rho = 1000 Rho = 4000
Rho = 500 Rho = 2000
How does new research problems evolve?
uF spring
Artery wall
Padmanabhan Seshaiyer
George Mason University
γ
fluid
F damping
F blood = kX0cosωt
mutt(0,t)
Fx
v
x
P
x
vv
t
v=
∂
∂+
∂
∂+
∂
∂+
∂
∂2
2
µρρ
What type of transformative research and training can one do?
Modify Key Assumptions
Build Realistic Geometry
Padmanabhan Seshaiyer
George Mason University
Optimize Mathematical Techniques
Enhance Mathematical Software
Match Experimental Data
Refine Mathematical Model
Perform Parameter Estimation Studies
My Contacts!
Padmanabhan Seshaiyer
Mathematical SciencesMathematical Sciences
George Mason University
Email: [email protected]
Web: http://math.gmu.edu/~pseshaiy
http://math.gmu.edu/~pseshaiy/outreach.html