Mathematics and the Real World Connecting the Dots

1
Mathematics and the Real World Connecting the
Dots

• by
• Marc Loken
• Minnesota Department of Transportation
• Presented to
• University of St. Thomas
• CAMS Seminar
• February 15, 2006

2
Outline

• The Real World
• Strength Test (BS)
• GPR Test (MS)
• FWD Test (PhD)
• Mathematics Opportunities
• Seeds Program (BS)
• Summer Hires (MS)
• NSF Proposals (PhD)

3
Strength Test (BS)

• Real World Objectives
• Measure the state of stress at failure for a
  geo-material in the laboratory
• Determine the parameters that define a non-linear
  Mohr-Coulomb failure envelope for the material
• Mathematical Tools
• Newtons Method
• Nonlinear regression

4
Strength Test
?3
?2
?1
5
Stress/Strain Curve
?3
?1
6
Mohrs Circle
?3
?1
7
Failure Surface
?f(?)A(?-c)n
8
Slope Equation of Line
m-1/f(??)
?m(?-?m)
?m
9
Nonlinear Equation
?f(?)m(?-?m)
(?,?)
10
Newtons Method/EXCEL
F(??)0f(?)f(?)?-?m ?i1?i-f(?i)/f(?i)
11
Nonlinear System of 3 Eqs. A2n(?i-c)2n-1 ?i
-?m0, i1,33 unknowns (A,n,c)
12
Nonlinear Regression N Eqs. -- 3 Unknowns
SSE ?(r1-r2)2
r1
r2
13
Strength TestCurrent Status

• Theory Done
• Solution Done
• Application -- Done

14
GPR Test (MS)

• Real World Objectives
• Using GPR field test, determine the thickness and
  density of the surface asphalt layer of a roadway
• Using GPR, determine the thickness and moisture
  content of the aggregate base layer
• Mathematical Tools
• Algebra (Create a deterministic system of
  equations)
• Geometry and Optics (Snells Law)
• Numerical Methods (Solve a system of non-linear
  equations and computer implementation)

15
GPR Equipment
16
GPR Geometry
17
GPR Equations
Given
c,h,t0,x0
Measure
t2,t3
Find
d1,v1
18
GPR Images AC GC
19
Numerical Tools
Mathematica
EXCEL
20
GPR Test Results
21
GPR TestCurrent Status

• Theory Done
• Solution
• Improve solution technique
• Connect EXCEL and Mathematica solutions
• Application
• Apply to other GPR geometries
• Automate method
• Apply to other output quantities

22
FWD Test (PhD)

• Real World Objectives
• Using FWD field test, determine the material
  properties (I.e., elastic modulii) that define a
  layered half-space
• Mathematical Tools
• Greens Function (Stokes Problem in 3-D
  Elastodynamics)
• Numerical Methods (Solve a system of non-linear
  equations and computer implementation)

23
FWD Equipment
24
FWD Problem Statement
Given
P(t)

• Determine
• Layer Thickness
• d1,d2,
• Layer Stiffness
• E1,E2,

u1 u2 u3 un
d1,E1
d2,E2
d3,E3
25
Mathematical Tools

• Governing Equations
• Fundamental Solution
• Dynamic Reciprocal Theorem

26
Governing Equations

• Equations of Motion
• ?ij,I ?bj ?uj

• Constitutive Law
• ?ij ? ?kk ?ij 2 ? ?kk

• Kinematics
• ?ij ½(ui,j uj,i)

27
Fundamental Solution
28
Displacement Tensor
29
Stress Tensor
30
Dynamic Reciprocal Theorem
31
Integral Representation
32
Numerical Implementation
33
Numerical Implementation
34
A Note on Analytic Integration
??r -3 du dv arctan(uv/zr) where r2 u2 v2
z2
35
Lambs Problem
36
Lambs Solution
37
FWD Test Status

• Theory Complete
• Solution
• Half-Space Solution Done
• Layered Half-Space Solution Done
• Inverse Problem Needs to be done
• Implementation
• Apply to collected data sets
• Make user-friendly

38
Research Opportunities

• Students
• Seeds Program
• Summer hires
• Faculty
• MnDOT Research Proposals
• NSF Proposals

39
Seeds Program
http//www.dot.state.mn.us/hr/seeds/
40
Proposals

• http//www.dot.state.mn.us.dot.research/tprobstat/
  probstat.html