Multidisciplinary Undergraduate Research in Computational Mathematics and Nonlinear Dynamics of Biological, Bio-inspired and Engineering Systems

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1 Multidisciplinary Undergraduate Research in Computational Mathematics and Nonlinear Dynamics of Biological, Bio-inspired and Engineering Systems Professor, Mathematical Sciences Director, COMPLETE Center Best Practices for Introducing Undergraduate Students to Computational and Interdisciplinary Research SIAM Annual Meeting July 13, 01

2 UG Students as change agents

3 University & K-1 STEM Collaboration Sarah M. Venuti (Undergraduate Student) Avis Foster (Undergraduate Student) Kris Kappmeyer (K-1 Teacher) Alicia Hamar (High School Student) Stephanie Alley (Undergraduate Student) Andrew Samuelson (Graduate Student) Arterial Wall Blood CSF Transforming Practice Through Undergraduate Researchers, P. Seshaiyer, Council on undergraduate research Quarterly, Fall 01.

4 REU: COMPUTATIONAL MATHEMATICS AND NONLINEAR DYNAMICS OF BIOLOGICAL, BIO- INSPIRED AND ENGINEERING SYSTEMS Summers , , and Stipend $3375; Free on-campus housing and meals; Travel allowance up to $550 and more! Supports 1 students and K-1 teachers each year Sample Research Topics Modeling using deterministic and stochastic differential equations Computational biology, biomechanics and neuroscience Mathematics of Materials Nonlinear dynamics and Micro Air Vehicles

5 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 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?

6 Nature of Student Activities Lecture Series Mentoring Club Guest Colloquium REU Participant Seminars Computational Laboratory Scientific and Social Tours Lesson Study

7 Sample Computational Mathematics Projects A Computational Model for Batten Behavior in Micro Air Vehicles Syeda Khadija F. Zaidi (University of Maryland, College Park) Applying Numerical Methods to Fluid Structure Interactions in Biological Systems Courtney Chancellor (Southern Methodist University) Ordinary Differential Equations in Green Oxidation Processes Angela Dapolite (Clarkson University) Viscoelastic Modeling of Biological Tissue in an Idealized Cerebral Aneurysm Kris Kappmeyer (H-B Woodlawn Highschool) Modeling Evaporation from the pre-lens tear film over a contact lens Amber Xu (Carnegie Mellon University) Optimization of an Antenna Structure for a Photovoltaic Device Emily Forney (Clemson University) Analysis of Spherical Inflation Models for Intracranial Saccular Aneurysm Elasto-dynamics James Halsall (Farmingdale State College, New York)

8 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. Research that happens across traditional mathematics and at the edges of traditional disciplines. Here is the problem, find the mathematics to solve it!

9 Basic Scientific Research Human Health Multidisciplinary Research Impact Areas Energy & Natural resources Defense & National Security Sustainability of the Environment

10 Computational Biomechanics Mathematical Biology Materials Modeling Nonlinear Dynamics Porous Media Mathematics in some Multidisciplinary Research Topics Stochastic Differential Equations Optimization Control Theory Fluid-Structure Interaction Fluid Dynamics

11 Saccular Aneurysms: Rupture

12 Identification of Rupture Potential Characterization of material properties Contact constraints Elasto-dynamics Coupled flow-structure interaction

13 Bifurcating Artery-Aneurysm Model

14 MAV: Membrane Wing Deflection

15 Experimental Challenges in MAV Design Small size High surface-to-volume ratio 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

16 Computational Challenges in MAV Design Structural Dynamics FSI Fluid Dynamics Guidance & Control Propulsion & Power

17 Beam-Fluid Interaction

18 What makes these problems interesting? Fully three-dimensional Heterogeneous/Homogeneous Anisotropic/Isotropic Compressible/Incompressible Non-linearity/Linear Large/Small Deformations Dynamic/Static And more

19 Key Steps in Mechanics Kinematics Forces Balance Relations Constitutive Formulation Boundary/Initial Conditions Boundary/Initial Value Problem

20 Solution Methodology Physical System Mathematical Model

21 Predator-Prey Problem dy dt α Y α 1 XY dy dt α y( t) dy K y dt y( t) y e 0 Kt dx dt + α 3 X α 4 XY Y(0) Y 0 X(0) X 0

22 Displacement of a Linear Elastic Bar f(x) Xa u(x) Xb du σ α dx dσ f dx (x) σ K du dx d dx du K f dx u( a) 0 u( b) 0 ( x)

23 Solution Methodology Physical System Mathematical Model Compare Analytical Solution

24 Finding an Analytical Solution d dx BVP du K f dx u( a) 0 u( b) 0 ( x) dy dt dx dt α Y α 1 IVP α 3 X + α 4 XY XY Y(0) Y 0 X(0) X 0

25 Solution Methodology Physical System Mathematical Model Analytical Solution Compare Numerical Solution Compare Experimental Data Compare

26 Experimental Data BVP or IVP Finite Element Methods Finite Difference Methods... Shooting Methods Runge Kutta Methods Solution to a System of Equations Direct Methods Error Analysis Indirect Methods Solution to BVP/IVP

27 Fluid Flow-structure interaction (FSI) Γ Solid Fluid Solid 0. ). ( ) (0, : + + Ω u f p u u u t u T Fluid f f ν ρ r Γ ] ) ( 0.5 [ ~ ~ ~ ) ( ~ ~. ) (0, : T s s w w tr b t w T Solid + + Ω ε ε µ ε λ σ σ ρ Γ

28 Beam Flow Interaction

29 Modeling Arterial Wall-Flow Interaction Arterial Wall Blood CSF Mathematical Model Modeling, Analysis and Computation of Fluid Structure Interaction Models for Biological Systems S. Minerva Venuti and P. Seshaiyer (Mentor), SIAM Undergraduate Research Online (010)

30 One-dimensional Model Inner wall Arterial Wall Outer wall u ( x, t ) Blood k m Spinal Fluid x-direction X b X 0 sin( ωt) rho 1000 kg/m 3 density of spinal fluid mu 0 Stokes viscosity of spinal fluid c 1500 m/s velocity of spinal fluid m 1.0 kg mass of the arterial wall a 0.01 m area of the outer arterial wall k 8000 N/m spring constant omega 1 rad/s frequency of heart beat X amplitude of heart beat X b displacement imposed by the pressure exerted against the arterial wall by the blood

31 Coupled FSI System t u c x u 0 < x < L ρ c u a x + m u( 0, t) + ku(0, t) kx 0 sin( ω t) u u ( L, t) c ( L, t) for x x L 0 for x 0 x u(x,0) u (,0) 0

32 Analytical Solution t D t C Be Ae t u t r t r sin cos ) 0, ( ) / 4( ) / ( / 1 m k m ac m ac r ρ ρ r ) / 4( ) / ( / m k m ac m c a + ρ ρ,, ) )*( ( ) / ( ω ω + r r r X m k A, ) )*( ( ) / ( 1 0 ω ω + r r r X m k B 4 0 ) / ( ) / ( k m k m m c a a c m X m k C + + ω ω ω ρ ω ρ and 4 0 ) / ( ) ( ) / ( k m k m m c a k m m X m k D + + ω ω ω ρ ω ω.

33 Displacement at x0

34 Variations on parameter k - stiffness K4000 K16000 K8000 K3000

35 Variations of rho - density Rho 500 Rho 000 Rho 1000 Rho 4000

36 How does new research problems evolve? Artery wall F spring u F blood kx 0 cosωt γ fluid F damping mu tt (0,t) v ρ t + ρ v v x + P x + v µ x F

37 What type of transformative research and training can one do? Modify Key Assumptions Build Realistic Geometry Optimize Mathematical Techniques Enhance Mathematical Software Match Experimental Data Refine Mathematical Model Perform Parameter Estimation Studies

38 My Contacts! Mathematical Sciences Web: