Transcript Slide 1

P. Venkataraman
Mechanical Engineering
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
Todays Presentation
1.
2.
3.
4.
Introduction to Lumped Parameter Inverse BVP
The Solution Procedure
The Example
Conclusion
Mechanical Engineering
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
Lumped Parameter Inverse BVP
An Example :
y1  b1 y12  b2 y1 y1  b3  0;
y2  b4 y1 y2  b5  0;
Fluid flow in a long vertical channel with fluid
injection
y3  b6 y1 y3  0;
Forward Problem:
Given:
y1 (1)  1;
Find:
y1 (0)  0;
y1 (0)  0;
y1 (1)  0;
y2 (0)  0;
y2 (1)  0;
y3 (0)  0;
y3 (1)  1;
b1  100; b2  100; b3  276;
and b  100; b  1; b  70;
4
5
6
y1  x  , y2  x  , y3  x 
Inverse Problem:
Given:
y1  x  , y2  x  , y3  x 
Find:
b1 ,b2 ,b3 ,b4 ,b5 ,b6
and
y1 (0)  0;
y1 (1)  1;
y1 (0)  0;
y1 (1)  0;
y2 (0)  0;
y2 (1)  0;
y3 (0)  0;
y3 (1)  1;
Mechanical Engineering
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
3
Lumped Parameter Inverse BVP
Forward Problem:
Given: y1 (0)  0; y1 (0)  0; y2 (0)  0; y3 (0)  0; and
y1 (1)  1; y1 (1)  0;
y1  x  , y2  x  , y3  x 
•
•
•
•
•
y2 (1)  0;
y3 (1)  1;
b1  100; b2  100; b3  276;
b4  100; b5  1; b6  70;
Find:
• is well posed (solution exist and unique)
• assumes perfect measurement of parameters and boundary conditions
• error in the solution will vanish as the perturbation in the parameters tends
to zero
Inverse Problem:
y1 (0)  0; y1 (0)  0; y2 (0)  0; y3 (0)  0;
Given: y1  x  , y2  x  , y3  x  and
y1 (1)  1; y1 (1)  0; y2 (1)  0; y3 (1)  1;
b1 ,b2 ,b3 ,b4 ,b5 ,b6
Find:
Inverse problems are considered naturally unstable, ill-posed, not unique
cannot be satisfactorily solved mathematically
no valid inverse problem based on smooth or perfect data
all current methods use some sort of regularization (artificial objective
function for minimization)
inverse problem cannot be satisfactorily solved without partial information
Mechanical Engineering
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
4
Lumped Parameter Inverse BVP
The solution of inverse BVP in this paper is robust:
is natural
based on derivative information
procedure can be adapted fro the forward problem too
does not require regularization
does not require dimensional control
does not require partial information
Mechanical Engineering
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
5
The Solution Procedure
Some Assumptions :
• we simulate discrete non smooth data to represent
measurement error from smooth data of the forward problem
• each data stream is connected to a differential equation
• the solution of the inverse BVP is the value of the lumped
parameters
• the solution of the inverse BVP is also the trajectory based on
the parameters
• quality of solution is determined by closeness of trajectory
determined using the value for parameters to the original
smooth trajectory
• final trajectory is obtained using numerical integration
(collocation)
• for comparison we assume that the boundary conditions are not
perturbed
Mechanical Engineering
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
6
The Solution Procedure
Step 1: Data Smoothing using a recursive Bezier filter
Bezier filter determines the best order that minimizes the sum of least
squared error (LSE) and the sum of the absolute error (LAE)
Step 2: The first optimization procedure
Obtain the first estimate for the lumped parameters by the minimization
of the sum of the residuals over a set of available data points
Step 3: The second optimization procedure
Obtain the second estimate for the lumped parameters by minimizing
the sum of the error between original data and data obtained through
numerical integration (collocation) over a reduced region
Step 4: Final numerical integration to generate the trajectory based on the
solution in Step 3
Boundary conditions are assumed perfect to compare the trajectory
Trajectory is compared to underlying smooth trajectory
Mechanical Engineering
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
7
The Solution Procedure
MATLAB was used for all calculations
A combination of symbolic and numerical processing was used to
postpone round-off errors
The following MATLAB functions was used in the implementation
• fminunc : unconstrained function minimization
• matlabFunction : conversion of symbolic objects
• bvp4c :
No special programming techniques were necessary
Computations were performed using a standard laptop
Mechanical Engineering
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
8
The Example
Example of fluid flow in a long vertical channel with fluid injection on one side
 
2


y1  R  y1  y1 y1   RA  0;


y   Ry y   1  0;
2
1 2
y3  Pe y1 y3  0;
y1 (0)  0;
y1 (1)  1;
y1 (0)  0;
y1 (1)  0;
y2 (0)  0;
y2 (1)  0;
y3 (0)  0;
y3 (1)  1;
R is the Reynolds number and Pe is the Peclet number. A is an unknown
parameter which is determined through the extra boundary condition. The
example is defined for a Reynolds number of 100 for which the value for A is
2.76.
Mechanical Engineering
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
9
The Example – Solution to Forward Problem
Solution to forward problem
using Bezier function –
order 20
Mechanical Engineering
10
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
The Example – Inverse Problem with Smooth Data
The Bezier function technique with 18th order functions
IG : Initial Guess
Opt1 : First Optimization
Opt2 : Second Optimization
ES : Expected Solution
Mechanical Engineering
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
11
The Example – Perturbation in Boundary Conditions
Most solution to
inverse BVP do not
consider change in
BC
This work
accommodates
changes in BC as it
works with a clipped
region
Mechanical Engineering
12
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
The Example – Inverse Problem with Non Smooth Data
10% perturbation – 31 points
Variable y1
Mechanical Engineering
Variable y2
13
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
The Example – Inverse Problem with Non Smooth Data
10% perturbation – 31 points
Variable y3
Mechanical Engineering
14
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
The Example – Inverse Problem with Non Smooth Data
20% perturbation – 31 points
Variable y1
Mechanical Engineering
Variable y2
15
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
The Example – Inverse Problem with Non Smooth Data
20% perturbation – 31 points
Variable y3
Mechanical Engineering
16
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem
Conclusions
This paper presents a robust method for inverse lumped parameter BVP
The method is based on describing the data using Bezier functions
The method involves three sequential applications of unconstrained
optimization
The method does not require regularization
The method does not require dimensional control
The method does not require additional information on the nature of the
problem or solution
The method accommodates the perturbation of the boundary conditions
Mechanical Engineering
17
P. Venkataraman Rochester Institute of Technology
32nd CIE, Chicago IL, Aug 2012
P. Venkataraman
DETC2012
– 70343: A Robust Technique for Lumped Parameter Inverse Boundary Value Problem