Homework 21. Complete Chapter 3, Problem #1 under Project.docx

Homework 2
1. Complete Chapter 3, Problem #1 under “Project: Statistical
Analysis in Inverse Problems
Using Simulated Data” on pages 58–59 of B&T. Use the same
initial conditions as before from
Chapter 2.
2. Consider the logistic population growth model
ẋ = ax− bx2, x(0) = x0.
Let K = a
b
. We will examine the model for the q = (a, b, x0) parameter
vectors
(i) q = (0.5, 0.1, 0.1) ⇒ K = 5 (relatively flat curve),
(ii) q = (0.7, 0.04, 0.1) ⇒ K = 17.5 (moderately sloped curve),
(iii) q = (0.8, 0.01, 0.1) ⇒ K = 80 (relatively steep curve).
Define the regions R0, R1, and R2 as follows:
• R0 is the region where t ∈ [0, 2],
• R1 is the region where t ∈ (2, 12],
• R2 is the region where t ∈ (12, 16].
For each parameter vector q:
(a) Let n = 15. For i = 0, sample n points from region Ri,
distributed uniformly over the interval.
Find the qOLS optimized parameters for 3 different initial

guesses that are far from the true
solution. (You can use the same initial guesses for all regions
and all q parameter vectors).
Calculate J(qOLS) where J is the cost function of the least
squares criterion. Calculate
K
̂ = â
b̂
. Include all results in a table.
For the optimal qOLS with the lowest cost J(qOLS), plot the
solution curve for the true
solution and the estimated solution on the same plot with the
sampled data points. How
do the results compare to the true solution? Determine the
standard errors and confidence
intervals. Are the true parameters contained within the
confidence interval?
Then repeat for i = 1. Then repeat for i = 2.
(b) Repeat problem (a) with n = 50.
(c/d) Repeat problems (a) and (b), but sampling from a uniform
distribution over the entire region
t ∈ [0, 16] instead of a single Ri region.
(e) When sampling from only region Ri, does increasing the
sample size improve the results?
How does this vary for i = 0, 1, 2? What if you sample over all
three regions?
1

MATHEMATICAL
AND EXPERIMENTAL
MODELING OF
PHYSICAL AND
BIOLOGICAL
PROCESSES
TEXTBOOKS in MATHEMATICS
Series Editor: Denny Gulick
PUBLISHED TITLES
COMPLEX VARIABLES: A PHYSICAL APPROACH WITH
APPLICATIONS AND MATLAB®
Steven G. Krantz
INTRODUCTION TO ABSTRACT ALGEBRA
Jonathan D. H. Smith
LINEAR ALBEBRA: A FIRST COURSE WITH
APPLICATIONS
Larry E. Knop
MATHEMATICAL AND EXPERIMENTAL MODELING OF
PHYSICAL AND BIOLOGICAL PROCESSES
H. T. Banks and H. T. Tran
FORTHCOMING TITLES

ENCOUNTERS WITH CHAOS AND FRACTALS
Denny Gulick
MATHEMATICAL
AND EXPERIMENTAL
MODELING OF
PHYSICAL AND
BIOLOGICAL
PROCESSES
H. T. Banks
H. T. Tran
TEXTBOOKS in MATHEMATICS
CRC Press
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Preface
For the past several years, the authors have developed and
taught a two-
semester modeling course sequence based on fundamental
physical and bio-
logical processes: heat flow, wave propagation, fluid and
structural dynamics,
structured population dynamics, and electromagnetism. Among
the specific
topics covered in the courses were thermal imaging and
detection, dynamic
properties (stiffness, damping) of structures such as beams and
plates, acous-
tics and fluid transport, size-structured population dynamics,
electromagnetic
dispersion and optics.
One of the major difficulties (theoretically, computationally,
and technolog-
ically) in mathematical model development is the process of
comparing models
to the field data. Typically, mathematical models contain
parameters and co-
efficients that are not directly measurable in experiments.
Hence, experiments
must be carefully designed in order to provide sufficient data

for model pa-
rameters and/or coefficients to be determined accurately. In this
context, a
major innovative component of the course has been the
exposure of students
to specific laboratory experiments, data collection and analysis.
As usual in
such modeling courses, the pedagogy involves beginning with
first principles
in a physical, chemical or biological process and deriving
quantitative mod-
els (partial differential equations with initial conditions,
boundary conditions,
etc.) in the context of a specific application, which has come
from a “client
discipline” — academic, government laboratory, or industrial
research group,
such as thermal nondestructive damage detection in structures,
active noise
suppression in acoustic chambers, smart material (piezoceramic
sensing and
actuation) structures vibration suppression, or optimizing the
introduction of
mosquitofish into rice fields for the control of mosquitos. The
students then
use the models (with appropriate computational software —
some from MAT-
LAB, some from the routines developed by the instructors
specifically for the
course) to carry out simulations and analyze experimental data.
The students
are exposed to experimental design and data collection through
laboratory de-
mos in certain experiments and through actual hands-on
experience in other
experiments.

Our experience with this approach to teaching advanced
mathematics with
a strong laboratory experience has been, not surprisingly,
overwhelmingly
positive. It is one thing to hear lectures on natural modes and
frequencies
(eigenfunctions and eigenvalues) or even to compute them, but
quite another
to go to the laboratory, excite the structure, see the modes, and
take data to
verify your theoretical and computational models.
Indeed, in writing this book, which is based on these
experimentally oriented
modeling courses, the authors aim to provide the reader with a
fundamental
understanding of how mathematics is applied to problems in
science and en-
gineering. Our approach will be through several “case study”
problems that
arise in industrial and scientific research laboratory
applications. For each
case study problem the perception on why a model is needed
and what goals
are to be sought will be discussed. The modeling process begins
with the
examination of assumptions and their translation into
mathematical models.
An important component of the book is the designing of
appropriate exper-
iments that are used to validate the mathematical model’s
development. In
this regard, both hardware and software tools, which are used to

design the
experiments, will be described in sufficient detail so that the
experiments can
be duplicated by the interested reader. Several projects, which
were devel-
oped by the authors in their own teaching of the above-
mentioned modeling
courses, will also be included.
The book is aimed at advanced undergraduate and/or first year
graduate
students. The emphasis of the book is on the application as well
as what
mathematics can tell us about it. The book should serve both to
give the
student an appreciation of the use of mathematics and also to
spark student
interest for deeper study of some of the mathematical and/or
applied topics
involved.
The completion of this text involved considerable assistance
from others.
Foremost, we would like to express our gratitude to many
students, postdoc-
toral fellows and colleagues (university and
industrial/government laboratory
based scientists) over the past decades, who generously
contributed to numer-
ous research efforts on which our modules/projects are based.
Specifically,
we wish to thank Sarah Grove, Nathan Gibson, Scott Beeler,
Brian Lewis,
Cammey Cole, John David, Adam Attarian, Amanda Criner,
Jimena Davis,
Stacey Ernstberger, Sava Dediu, Clay Thompson, Zackary Kenz,

Shuhua Hu
and Nate Wanner among our many young colleagues for their
assistance in
reading various drafts or portions of this book. (Of course, any
remaining
errors, poor explanations, etc., are solely the responsibility of
the authors.)
Finally, the authors wish to acknowledge the unwavering
support of our fami-
lies in our efforts in the development and completion of this
manuscript as well
as other aspects of our professional activities. For their support,
patience and
love, this book is dedicated to Susie, John, Jennifer, Thu, Huy
and Hoang.
H. T. Banks
H. T. Tran
List of Tables
3.1 Estimation using the OLS procedure with CV data for η = 5.
49
3.2 Estimation using the GLS procedure with CV data for η = 5.
49
3.3 Estimation using the OLS procedure with NCV data for η =
5. 49
3.4 Estimation using the GLS procedure with NCV data for η =
5. 49
3.5 χ2(1) values. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1 Range of values of h in Newton cooling. . . . . . . . . . . . . 91
5.2 Type T thermocouples: Coefficients of the approximate
inverse

function giving temperature u as a function of the thermoelec-
tric voltage E in the specified temperature and and voltage
ranges. The function is of the form: u = c0 + c1E + c2E2 +
· · · + c6E6, where E is in microvolts and u is in degrees
Celsius. 97
5.3 Type T thermocouples: Coefficients of the approximate
func-
tion giving the thermoelectric voltage E as a function of tem-
perature u in the specified temperature range. The function is
of the form: E = c0 + c1u + c2u2 + · · · + c8u8, where E is in
microvolts and u is in degrees Celsius. . . . . . . . . . . . . . 97
5.4 Hardware equipment for thermal equipment. . . . . . . . . . 98
5.5 Software tools for thermal equipment. . . . . . . . . . . . . . 99
6.1 Values of E and G for various materials. . . . . . . . . . . . .
112
6.2 Hardware equipment for beam vibration experiment. . . . . .
148
6.3 Software tools for beam vibration experiment. . . . . . . . . .
148
7.1 Beam and patch parameters. . . . . . . . . . . . . . . . . . . 204
8.1 Viscosity values of some gases and liquids at atmospheric
pres-
sure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
9.1 Percent of total catch of selachians. . . . . . . . . . . . . . . . 250

List of Figures
1.1 Schematic diagram of the iterative modeling process. . . . . 3
2.1 Spring-mass system (with the mass in equilibrium position).
. 8
2.2 Graph of the simple harmonic motion, y(t) = R cos(ωt−φ). .
9
2.3 Spring-mass system (with “massless” paddles attached to
the
body). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Plot of y(t) = Re−ct/2m cos(νt−δ). . . . . . . . . . . . . . . . 11
3.1 Plot of the pdf p(x) of a uniform distribution. . . . . . . . . . 23
3.2 The pdf graph of a Gaussian distributed random variable. . .
24
3.3 The pdf graph of a chi-square distribution for various
degrees
of freedom k. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Original and truncated logistic curve with K = 17.5, r = .7
and z0 = .1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5 Residual vs. time plots: Original and truncated logistic
curve
for q̂ CVOLS with η = 5. . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 Residual vs. model plots: Original and truncated logistic
curve
for q̂ CVOLS with η = 5. . . . . . . . . . . . . . . . . . . . . . . . . 50
curve
for q̂ NCVOLS with η = 5. . . . . . . . . . . . . . . . . . . . . . . . 51

3.8 Residual vs. model plots: Original and truncated logistic
curve
for q̂ NCVOLS with η = 5. . . . . . . . . . . . . . . . . . . . . . . . 51
curve
for q̂ CVGLS with η = 5. . . . . . . . . . . . . . . . . . . . . . . . . 52
3.10 Modified residual vs. model plots: Original and truncated
lo-
gistic curve for q̂ CVGLS with η = 5. . . . . . . . . . . . . . . . . . 52
3.11 Modified residual vs. time plots: Original and truncated
logis-
tic curve for q̂ NCVGLS with η = 5. . . . . . . . . . . . . . . . . . . 53
3.12 Modified residual vs. model plots: Original and truncated
lo-
gistic curve for q̂ NCVGLS with η = 5. . . . . . . . . . . . . . . . . .
53
3.13 Example of U ∼ χ2(4) density. . . . . . . . . . . . . . . . . . 56
3.14 Beam excitation. . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1 Two chamber compartments separated by a membrane. . . .
67
4.2 Binary molecules movement. . . . . . . . . . . . . . . . . . . . 71
4.3 Moving fluid through a pipe. . . . . . . . . . . . . . . . . . . 72
4.4 Incremental volume element. . . . . . . . . . . . . . . . . . . 73
4.5 Plug flow model. . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1 Diagram of SMC-adhesive-SMC joint. . . . . . . . . . . . . . 82
5.2 A schematic diagram of the NDE method for the detection

of
structural flaws. The sensor measures the surface temperature,
and the measured temperature is different for the smooth versus
the corroded surface. . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 Transient conduction in one-dimensional cylindrical rod. . .
. 85
5.4 (a) A general three-dimensional region. (b) An infinitesimal
volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.5 Hardware connections used to validate the one-dimensional
heat
equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.6 Heat experiment as set up in our own laboratory. . . . . . . .
98
6.1 2-D fluid/structure interaction system. . . . . . . . . . . . . . 104
6.2 Prismatic bar deformation due to tensile forces. . . . . . . . .
105
6.3 Normal stresses on the prismatic bar. . . . . . . . . . . . . . . 105
6.4 Stress-strain diagram for a typical structural steel in
tension. 107
6.5 Necking of a prismatic bar in tension. . . . . . . . . . . . . . 107
6.6 Bolt subjected to bearing stresses in a bolted connection. . .
109
6.7 Shearing stresses exerted on the bolt by the prismatic bar
and
the clevis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.8 Shear stress acts on a rectangular cube. . . . . . . . . . . . . 111
6.9 Shear stresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.10 Shear strains on the front side of the rhombus. . . . . . . . .
112
6.11 A cantilever beam. . . . . . . . . . . . . . . . . . . . . . . . . 113

6.12 A simple beam. . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.13 A cantilever beam with a tip mass at the free end and is
sub-
jected to a distributed force f. . . . . . . . . . . . . . . . . . 114
6.14 Shearing forces and moments on a cantilever beam with a
tip
mass at the free end. . . . . . . . . . . . . . . . . . . . . . . . 115
6.15 Force balance on an incremental element of the beam. . . . .
116
6.16 Local deformation of a segment of the beam due to
bending. 118
6.17 Stress and strain as functions of distances from the neutral
axis
at the point x (or e) on the neutral axis. . . . . . . . . . . . . 120
6.18 Segment of a beam with a rectangular cross-sectional area.
. 122
6.19 Pinned end support. . . . . . . . . . . . . . . . . . . . . . . . 124
6.20 Frictionless roller end support. . . . . . . . . . . . . . . . . . 125
6.21 Cantilever beam with a tip mass. . . . . . . . . . . . . . . . . 125
6.22 Local deformation of the cantilever beam with tip mass. . .
. 126
6.23 Force balance at the tip mass. . . . . . . . . . . . . . . . . . . 127
6.24 Deformation of the beam due to the rotation of the beam
cross
section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.25 Moment balance at the tip mass. . . . . . . . . . . . . . . . . 129
6.26 Hat basis functions. . . . . . . . . . . . . . . . . . . . . . . . 144
6.27 Hardware used for modal analysis and model validation of
the

cantilever beam model. . . . . . . . . . . . . . . . . . . . . . . 149
7.1 A spring-mass-dashpot platform system. . . . . . . . . . . . .
164
7.2 State vector x(t) for t1 = 1 second. . . . . . . . . . . . . . . . 165
7.3 State vector x(t) for t1 = .5 second. . . . . . . . . . . . . . . . 166
7.4 Control u(t) for t1 = 1 second (solid line) and t1 = .5 second
(dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.5 A pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.6 A closed-loop or feedback control system. . . . . . . . . . . .
178
7.7 An open-loop control system. . . . . . . . . . . . . . . . . . . 179
7.8 Dynamic output compensator. . . . . . . . . . . . . . . . . . 192
7.9 The uncontrolled system (u ≡ 0.) . . . . . . . . . . . . . . . . 194
7.10 The state vector, x(t), of the closed-loop system with K =
(−2 − 3 − 3) and G = ( 14 8 − 4)T . . . . . . . . . . . . . . 195
7.11 The estimator error, e(t), of the closed-loop system with K
=
(−2 − 3 − 3) and G = ( 14 8 − 4)T . . . . . . . . . . . . . . 195
7.12 The state estimator, x̂ (t), with K = (−2 − 3 − 3) and G =
( 14 8 − 4)T . The label xe1(t) denotes x̂ 1(t), etc. . . . . . . 196
7.13 The state vector, x(t), with K = (−68 − 48 − 12) and G =
(110 95 20)T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.14 The estimator error, e(t), with K = (−68 − 48 − 12) and
G = (110 95 20)T . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.15 The state estimator, x̂ (t), with K = (−68 − 48 − 12) and
G = (110 95 20)T . The label xe1(t) denotes x̂ 1(t), etc. . . . 198
7.16 Cantilever beam with piezoceramic patches. . . . . . . . . . .
200

7.17 Experimental beam with piezoceramic patches. . . . . . . . .
205
7.18 Experimental setup and implementation of online
component
of the Real-Time Control Algorithm. . . . . . . . . . . . . . . 206
7.19 Uncontrolled and controlled displacements at xob =
0.11075m. 207
7.20 Control voltages. . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.21 The inverted pendulum. . . . . . . . . . . . . . . . . . . . . . 208
7.22 Free body diagram of the inverted pendulum. . . . . . . . . .
208
8.1 A fluid initially at rest between two parallel plates. . . . . . .
217
8.2 Transient velocity profile of a fluid between two parallel
plates. 218
8.3 Fluid shear in steady-state between two parallel plates. . . . .
219
8.4 A fluid element fixed in space through which a fluid is
flowing. 221
8.5 A fluid element of volume ∆x∆y∆z fixed in space through
which the x-component of the momentum is transported. . . 223
8.6 Hardware used for studying various types of boundary
condi-
tions associated with the one-dimensional wave equation. . . 239
8.7 Hewlett-Packard dynamic signal analyzer. . . . . . . . . . . .
239
9.1 Graphs of the population p(t). . . . . . . . . . . . . . . . . . 247
9.2 Graph of the solution to the logistic model. . . . . . . . . . .

248
9.3 Orbital solutions of the predator/prey model. . . . . . . . . .
251
9.4 Total population from size a to b at time t0. . . . . . . . . . .
253
9.5 Size trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . 254
9.6 Growth characteristic of the conservation equation. . . . . . .
256
9.7
Solution
to equation (9.10) along the characteristic curve for
g(t,x) ≡ a and µ = 0. . . . . . . . . . . . . . . . . . . . . . . 258
9.8 Characteristic curve. . . . . . . . . . . . . . . . . . . . . . . . 260
9.9 Regions in the (t,x) plane defining the solution. . . . . . . . .
261
9.10 Mosquitofish data. . . . . . . . . . . . . . . . . . . . . . . . . 269
Contents
1 Introduction: The Iterative Modeling Process 1
2 Modeling and Inverse Problems 7

2.1 Mechanical Vibrations . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . 11
References 15
3 Mathematical and Statistical Aspects of Inverse Problems 17
3.1 Probability and Statistics Overview . . . . . . . . . . . . . . 18
3.1.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.2 Random Variables . . . . . . . . . . . . . . . . . . . . 20
3.1.3 Statistical Averages of Random Variables . . . . . . . 21
3.1.4 Special Probability Distributions . . . . . . . . . . . . 22
3.2 Parameter Estimation or Inverse Problems . . . . . . . . . . 29
3.2.1 The Mathematical Model . . . . . . . . . . . . . . . . 29
3.2.2 The Statistical Model . . . . . . . . . . . . . . . . . . 30
3.2.3 Known Error Processes: Maximum Likelihood Estima-
tors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.3.1 Normally Distributed Errors . . . . . . . . . 31
3.2.4 Unspecified Error Distributions and Asymptotic Theory 33
3.2.5 Ordinary Least Squares (OLS) . . . . . . . . . . . . . 34
3.2.6 Numerical Implementation of the Vector OLS Proce-

dure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.7 Generalized Least Squares (GLS) . . . . . . . . . . . . 38
3.2.8 GLS Motivation . . . . . . . . . . . . . . . . . . . . . 39
3.2.9 Numerical Implementation of the GLS Procedure . . . 40
3.3 Computation of Σ̂n, Standard Errors and Confidence
Intervals 41
3.4 Investigation of Statistical Assumptions . . . . . . . . . . . . 45
3.4.1 Residual Plots . . . . . . . . . . . . . . . . . . . . . . 46
3.4.2 An Example Using Residual Plots . . . . . . . . . . . 47
3.5 Statistically Based Model Comparison Techniques . . . . . .
51
3.5.1 RSS Based Statistical Tests . . . . . . . . . . . . . . . 54
3.5.1.1 P-Values . . . . . . . . . . . . . . . . . . . . 56
3.5.1.2 Alternative Statement . . . . . . . . . . . . . 57
3.5.2 Application: Cat-Brain Diffusion/Convection Problem 57
References 63

4 Mass Balance and Mass Transport 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Compartmental Concepts . . . . . . . . . . . . . . . . . . . . 65
4.3 Compartment Modeling . . . . . . . . . . . . . . . . . . . . . 67
4.4 General Mass Transport Equations . . . . . . . . . . . . . . 71
4.4.1 Mass Flux Law in a Stationary (Non-Moving) Fluid . 73
4.4.2 Mass Flux in a Moving Fluid . . . . . . . . . . . . . . 75
References 79
5 Heat Conduction 81
5.1 Motivating Problems . . . . . . . . . . . . . . . . . . . . . . 81
5.1.1 Radio-Frequency Bonding of Adhesives . . . . . . . . 81
5.1.2 Thermal Testing of Structures . . . . . . . . . . . . . 82
5.2 Mathematical Modeling of Heat Transfer . . . . . . . . . . . 83
5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 83
5.2.2 Fourier’s Law of Heat Conduction . . . . . . . . . . . 84
5.2.3 Heat Equation . . . . . . . . . . . . . . . . . . . . . . 85
5.2.4 Boundary Conditions and Initial Conditions . . . . . . 89
5.2.5 Properties of

Homework 21. Complete Chapter 3, Problem #1 under Project.docx

Homework 21. Complete Chapter 3, Problem #1 under Project.docx

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