Finding Buried Targets Using Acoustic Excitation Zackary R. Kenz - - PowerPoint PPT Presentation

Math Modeling Acoustic Model Simulations Conclusion

Finding Buried Targets Using Acoustic Excitation

Zackary R. Kenz

Advisor: Dr. H.T. Banks In collaboration with Dr. Shuhua Hu, Dr. Grace Kepler, Clay Thompson – NCSU, L3 Communications team led by Dr. Jerrold Levine, and Dr. Richard Albanese Center for Research in Scientific Computation Department of Mathematics North Carolina State University

October 7, 2010

Z.R. Kenz Acoustic/EM Target Detection

Math Modeling Acoustic Model Simulations Conclusion

1

Math Modeling and Differential Equations

2

1D Model Formulation

3

Simulation Setup and Results

4

Conclusion

Z.R. Kenz Acoustic/EM Target Detection

Math Modeling Acoustic Model Simulations Conclusion

Goals for Math Modeling Portion

Introduction to math modeling Introduce differential equations Examine how changing parts of a differential equation can change resulting behavior

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Math Modeling Acoustic Model Simulations Conclusion

Goals for Target Detection Portion

Acoustic Dynamics

Given an impact to the soil, what sort of wave propagation dynamics are expected? How do changes in soil properties affect dynamics?

Electromagnetic Signal Dynamics

Given an arbitrarily moving target in the soil, what will a reflected radar signal look like?

Device Development

Z.R. Kenz Acoustic/EM Target Detection

Math Modeling Acoustic Model Simulations Conclusion

1

Math Modeling and Differential Equations

2

1D Model Formulation

3

Simulation Setup and Results

4

Conclusion

Z.R. Kenz Acoustic/EM Target Detection

Math Modeling Acoustic Model Simulations Conclusion

Math Modeling Overview

Modeling is developing equations to explain some phenomenon, and then using the equations to make predictions or answer questions about the phenomenon. Validating the equations with real life data is of particular interest.

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Math Modeling Acoustic Model Simulations Conclusion

Modeling Cycle

Formalization of properties, relationships and mechanisms which result in a biological or physical model

(ii) The Iterative Modeling Process

Empirical Observations (experiments and data collection)

(i)

Abstraction or Mathematization resulting in a mathematical model

(iii)

Formalization of Uncertainty/Variablity in model and data resulting in a statistical model

(iv)

Model Analysis

(v)

Interpretation and Comparison

(with the real system)

(vi)

Changes in understanding of mechanisms, etc., in the real system.

(vii)

Formation Stage: (i),(ii),(iii),(iv) Solution Stage: (v) Interpretation Stage: (vi), (vii)

Z.R. Kenz Acoustic/EM Target Detection

Math Modeling Acoustic Model Simulations Conclusion

Differential Equations and Modeling

Modeling often uses differential equations to describe relationships between variables There are "ordinary" and "partial" differential equations Question: What are differential equations?

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Math Modeling Acoustic Model Simulations Conclusion

What is an ODE?

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation means the function of interest depends only on one independent variable. Instead of solving for numbers, like solving for x in x2 − 1 = 0, we are solving for a function

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Math Modeling Acoustic Model Simulations Conclusion

ODE Example 1

dy(t) dt = 2 The left hand side is the rate of change of the solution y(t). The equation says that we want the rate of change of the unknown function y(t) to be a constant. What simple function has a constant rate of change (slope)?

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Math Modeling Acoustic Model Simulations Conclusion

Example 1 Solution

dy(t) dt = 2 If we integrate both sides with respect to t, the solution is y(t) = 2t + c

c is a "constant of integration" This could describe the motion of someone walking down the sidewalk at a constant rate.

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Math Modeling Acoustic Model Simulations Conclusion

Adding an Initial Condition

Solution from previous slide: y(t) = 2t + c In a particular problem, we might want the person to start walking at position 5. Mathematically, we write y(0) = 5. Applying that condition to the problem: y(0) = 2 · 0 + c = 5. Solving gives c = 5. The full solution would then be y(t) = 2t + 5

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Math Modeling Acoustic Model Simulations Conclusion

Components of an ODE Problem

A full ODE problem is both the main equation (which includes the derivatives) and also some conditions like our y(0) = 5. The number of conditions depends on the highest number of derivatives you have. Now we’ll look at a slightly more complicated example.

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Math Modeling Acoustic Model Simulations Conclusion

ODE Example 2

dy(t) dt = 2y(t) This equation says that the rate of change of the solution is proportional to the value of the function at every point t. It’s a little more complicated to get the solution for this case, so we won’t solve it here but just state the answer: y(t) = e2t + c.

Changing the right hand side of the ODE from a constant to the unknown function changed the solution behavior.

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Math Modeling Acoustic Model Simulations Conclusion

ODE Example 3

dz(t) dt = −2z(t) By changing the constant to −2, the solution is now z(t) = e−2t + d.

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Math Modeling Acoustic Model Simulations Conclusion

Example 2 vs Example 3

We’ll set the constants of integration equal to 0 (arbitrary). We can compare the two solutions y(t) = e2t and z(t) = e−2t in the figure below.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 t Function Value y(t) z(t)

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Math Modeling Acoustic Model Simulations Conclusion

A Little More on ODEs

These equations can get more complicated. For example, Higher derivatives can be in the equation Parameters can be in the model. The spring system is a common example: md2y(t) dt2 + c dy(t) dt + ky(t) = 0 m: mass c: damping k: stiffness

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Math Modeling Acoustic Model Simulations Conclusion

Example Solution for Spring System

2 4 6 8 10 −0.5 0.5 1 1.5 2 Time t Postion x(t) Damped Spring System

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Math Modeling Acoustic Model Simulations Conclusion

Brief PDE Discussion

What if we want the solution to depend on multiple independent variables, like the three spatial dimensions? A partial differential equation (PDE) seeks a solution with more than one independent variable. For example, we might want to track the temperature of a room over time. We would then be trying to find a function like T(x, y, z, t).

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Math Modeling Acoustic Model Simulations Conclusion

Brief PDE Discussion

Derivatives are now "partial" derivatives (denoted with ∂), meaning with respect to one variable holding the others constant. For example, if we wanted the rate of change of temperature with respect to time, we’d denote that quantity as ∂T(x, y, z, t) ∂t .

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Math Modeling Acoustic Model Simulations Conclusion

The Wave Equation

An equation that can be used to describe simple wave motion in one spatial dimension is ∂2u(z, t) ∂t2 = c2 ∂2u(z, t) ∂z2 . Upon solving analytically, the constant c represents the speed of the wave. This particular form of the wave equation can be solved by hand, but more complicated forms are solved numerically.

There is a shorthand notation for partial derivatives that turns the wave equation above into utt = c2uzz.

Z.R. Kenz Acoustic/EM Target Detection

Math Modeling Acoustic Model Simulations Conclusion

Moving Forward

With this brief introduction to modeling and differential equations, we can start discussing my research project of detecting buried targets. The project is an example of the modeling cycle.

Z.R. Kenz Acoustic/EM Target Detection

Math Modeling Acoustic Model Simulations Conclusion

1

Math Modeling and Differential Equations

2

1D Model Formulation

3

Simulation Setup and Results

4

Conclusion

Z.R. Kenz Acoustic/EM Target Detection

Math Modeling Acoustic Model Simulations Conclusion

Definitions

Elastic material: returns to its original shape after applying stress forces (rubber band) Viscous material: Resists stresses (tar) Viscoelastic material: Has both properties (rubber)

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Math Modeling Acoustic Model Simulations Conclusion

Problem Situation

Open field where we want to detect buried objects Thump in one area, study wave propagation outward from thumper Likely need a 2D or 3D model to capture full dynamics As a first approximation, we’ll use a 1D model see which features of wave propagation we can capture

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Math Modeling Acoustic Model Simulations Conclusion

1D Problem

One dimensional problem schematic:

air soil soil

.

.

Observations of the wave form at a particular depth will be taken at the z10 position

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Math Modeling Acoustic Model Simulations Conclusion

1D Model: Key Assumptions

Both soil and target are uniform in horizontal (i.e., x and y) directions The column is a continuum Dry soil behaves as a Kelvin-Voigt viscoelastic solid for small-amplitude vibrations1

1B.O. Hardin, The nature of damping in sands, J. Soil Mech. Found. Div.,

91 (1965), 63-97.

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Math Modeling Acoustic Model Simulations Conclusion

1D Model: Notation

u(z, t) denotes the displacement (units: m) in the z-direction at position z at time t Kelvin-Voigt stress: σ = κ∂u(z,t)

∂z

+ η ∂2u(z,t)

∂t∂z

κ: elastic modulus (i.e. stiffness) in

kg m·s2 = Pa

η: damping coefficient in

kg m·s

ρ: soil density in kg

m3

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Math Modeling Acoustic Model Simulations Conclusion

1D Model: Only Soil

From the equations of motion for a continuum, we obtain ρ ∂2u(z,t)

∂t2

=

∂ ∂z (σ)

=

∂ ∂z

• κ∂u(z,t)

∂z

+ η ∂2u(z,t)

∂t∂z

• Remember from the wave equation slide that the wave speed was the

constant c in utt = c2uzz. For our model above, if we neglect damping we get c =

• κ/ρ.

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Math Modeling Acoustic Model Simulations Conclusion

1D Model: Including Rigid Target

Model the rigid target as a point mass in the column at z10: ρ ∂2u(z,t)

∂t2

=

∂ ∂z

• κ∂u(z,t)

∂z

+ η ∂2u(z,t)

∂t∂z

• , z ∈ (zp0, z10) ∪ (z10, ∞)

M ∂2u(z10,t)

∂t2

= S

• κ∂u(z+

10,t)

∂z

+ η ∂2u(z+

10,t)

∂t∂z

• −S
• κ

∂u(z−

10,t)

∂z

+ η

∂2u(z−

10,t)

∂t∂z

• .

M: mass in kg of target S: surface area in m2 of contact between target and soil under (over) target

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Math Modeling Acoustic Model Simulations Conclusion

1D Model: Additional Assumptions and Conditions

Assume zero displacement and zero velocity initially: u(z, 0) = 0 ∂ ∂z u(z, 0) = 0 At the surface, the normal internal stress is balanced with the applied input force:

• κ∂u(z,t)

∂z

+ η ∂2u(z,t)

∂t∂z

• z=zp0

= −f(t), Computational assumption: the column is finite: u(z00, t) = 0

Z.R. Kenz Acoustic/EM Target Detection

Math Modeling Acoustic Model Simulations Conclusion

1

Math Modeling and Differential Equations

2

1D Model Formulation

3

Simulation Setup and Results

4

Conclusion

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Math Modeling Acoustic Model Simulations Conclusion

Simulation Setup

Surface: zp0 = 0m Observation point/target: z10 = 0.3048m Far boundary: z00 = 50m

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Math Modeling Acoustic Model Simulations Conclusion

Simulation Setup

0.002 0.004 0.006 0.008 0.01 −4 −3 −2 −1 1 2 3 4 x 10

5

t (units: s) f(t) (units:N/m2)

Forcing Function

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Math Modeling Acoustic Model Simulations Conclusion

Questions to Ask

If soil density ρ changes, what happens to the wave form and speed? What happens if the elastic modulus κ changes? What happens when the wave impacts the rigid body target? This is the Model Analysis part of the modeling cycle.

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Math Modeling Acoustic Model Simulations Conclusion

Results, Soil Only

Changes in Soil: Density

0.002 0.004 0.006 0.008 0.01 −2 2 4 6 8 10 x 10

−4

t (units: s) u(0.3048,t) (units: m)

Displacement around z=0.3048m (z=1 ft)

κ=204000000 ρ=1440 κ=204000000 ρ=1800 κ=204000000 ρ=2250

Wave speed heuristic (no damping): v ≈

• κ/ρ

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Math Modeling Acoustic Model Simulations Conclusion

Results, Soil Only

Changes in Soil: Elastic Modulus

0.002 0.004 0.006 0.008 0.01 −2 2 4 6 8 10 12 x 10

−4

t (units: s) u(0.3048,t) (units: m)

Displacement around z=0.3048m (z=1 ft)

κ=102000000 ρ=1800 κ=204000000 ρ=1800 κ=408000000 ρ=1800

Heuristic: v ≈

• κ/ρ

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Math Modeling Acoustic Model Simulations Conclusion

Results, Soil Only

Changes in Soil: Density and Elastic Modulus

0.002 0.004 0.006 0.008 0.01 −2 2 4 6 8 10 12 14 x 10

−4

t (units: s) u(0.3048,t) (units: m)

Displacement around z=0.3048m (z=1 ft)

κ=102000000 ρ=1440 κ=204000000 ρ=1800 κ=408000000 ρ=2250

Heuristic: v ≈

• κ/ρ

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Math Modeling Acoustic Model Simulations Conclusion

Results, Soil Only

0.002 0.004 0.006 0.008 0.01 −2 2 4 6 8 10 12 x 10

−4

t (units: s) u(0.3048,t) (units: m)

Displacement around z=0.3048m (z=1 ft)

κ=163200000 ρ=1440 κ=204000000 ρ=1800 κ=255000000 ρ=2250

Heuristic: v ≈

• κ/ρ

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Math Modeling Acoustic Model Simulations Conclusion

Results, With Target: Wave Form Passing by Target

0.5 1 1.5 2 −2 2 4 6 8 10 12 14 x 10

−5

Wave form in z−domain, at time t=0.00067831

Distance under ground, units: m Displacement, units: m

Dashed line represents location of target in column

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Math Modeling Acoustic Model Simulations Conclusion

Results, With Target: Wave Form Passing by Target

0.5 1 1.5 2 2 4 6 8 x 10

−4

Wave form in z−domain, at time t=0.0020486

Distance under ground, units: m Displacement, units: m

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Math Modeling Acoustic Model Simulations Conclusion

Results, With Target: Wave Form Passing by Target

0.5 1 1.5 2 2 4 6 8 x 10

−4

Wave form in z−domain, at time t=0.0054744

Distance under ground, units: m Displacement, units: m

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Math Modeling Acoustic Model Simulations Conclusion

Results, With Target: Wave Form Passing by Target

0.5 1 1.5 2 1 2 3 4 5 6 7 8 x 10

−4

Wave form in z−domain, at time t=0.0068447

Distance under ground, units: m Displacement, units: m

Z.R. Kenz Acoustic/EM Target Detection

Math Modeling Acoustic Model Simulations Conclusion

1

Math Modeling and Differential Equations

2

1D Model Formulation

3

Simulation Setup and Results

4

Conclusion

Z.R. Kenz Acoustic/EM Target Detection

Math Modeling Acoustic Model Simulations Conclusion

Discussion

Model successes: The 1D model can capture basic wave dynamics in the soil Models results of changes in soil properties the way we would expect Model shows what we would expect when wave impacts a rigid body Modeled effects that were seen in field tests

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Math Modeling Acoustic Model Simulations Conclusion

Discussion

Model drawbacks: Missing any dynamics outside soil column Assumes everything is uniform, clearly not true in practice Only models one wave type - in reality, multiple types result from a single impact Doesn’t model effects of waves hitting target at an angle – Here target is modeled as a point, real target is 3D

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Math Modeling Acoustic Model Simulations Conclusion

Future Work

More complicated equations - need more spatial independent variables since real life is three-dimentional Further studies on how well we can estimate model parameters like density using real life data Couple with electromagnetic detection portion Discrimanants for target detection

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Math Modeling Acoustic Model Simulations Conclusion

Acknowledgements

Project supported by Air Force Office of Scientific Research, grant FA9550-09-1-0226 My efforts were supported by a Department of Education GAANN Fellowship

Z.R. Kenz Acoustic/EM Target Detection

Math Modeling Acoustic Model Simulations Conclusion

Questions?

For further details on elasticity theory and this computational example, see: "A Brief Review of Elasticity and Viscoelasticty", CRSC Technical Report CRSC-TR10-08, May 2010, www.ncsu.edu/crsc/reports/reports10.html

• r "A Brief Review of Elasticity and Viscoelasticity for Solids," Advances in

Applied Mathematics, to appear

Z.R. Kenz Acoustic/EM Target Detection