Discussion

ES105 - Winter, 2009
Problem Set 7
Due: Monday, 9 March, 11:15 AM Read: Text, Chapter 18 (Statistical Interpolation).

Find the Crack

(Note: All (x,y) units in this problem are stated in the units of the .nod file.)

Continuation of Homeworks 5 and 6. The node and element files are the same, Problem 44. BUT -- there is a crack in the plate! And it leaks current.

The crack is located between zones 1 and 2 of the element file; it runs from top to bottom, without interruption. It is imperfectly grounded, resulting in a leak of current out of the system. It is to be modeled as a set of point sinks of current, proportional to the local voltage.

div (t grad U) = Sum_m ( k U delta_m )

where U is the voltage, delta_m is the Dirac delta function concentrated at node m along the crack. The coefficient k is uniform along the crack. Everything else from Problem 44 is unchanged -- the ground nodes, the point source, and the boundary efflux. The current leak along the crack is new.

Truth

1. The boundary element file hw44.bel shows the boundaries among 3 zones: 0 (outside), 1 and 2. All the 'crack' nodes are on the 1-2 boundary. Find and save them.
2. Formulate a new forward model that includes the crack sources. (There is a brief note on the formulation here.)
3. Simulate truth for a range of the parameter k.
1. Show that your program recovers previous (uncracked) results when k is small. How small?

ANSWER: In the range of 0.01-0.001. Nevertheless, this is always dependant on the sensitivity of the instrumentation used to gather information.

2. Demonstrate the effect of increasing k; how big is 'infinite', and do the results (voltage and current) agree with intuition? Call this value k_inf

ANSWER: k_inf ~100.0 & k=k_inf/10=10.0

3. The truth is, k is 1/10 of k_inf. Find the true voltage and current.
4. Measure the truth as in HW6, with the same measurement apparatus.

Inversion
Find the sources as in HW6. Use the unchanged inverse model from HW6. Specifically,

1. Invert perfect data for the nodal sources. Use Representer method as you did in HW6; compute the bias in U.
2. Compute the inverse noise.
3. Generate and invert a MonteCarlo ensemble of data vectors. From this ensemble, compute bias and noise (imprecision) maps of

1. Voltage (U)
2. Source distriution

4.Do the MonteCarlo results agree with the direct results?

Plots

Solution using SOLVE @ k=0

Difference Map between:

SOLVE @ k=0 & SOLVE @ k=0.01

i.e., k is "practically" zero depending on instrument sensitivity

Difference Map between:

SOLVE @ k=0 & SOLVE @ k=0.001

i.e., k is "practically" zero depending on instrument sensitivity

Solution using SOLVE @ k=10.0 i.e., k=k_inf/10

Difference Map between:

SOLVE @ k=0 & SOLVE @ k=10.0

i.e. k=k_inf/10

Difference Map between:

SOLVE @ k=0 & SOLVE @ k=100.0

i.e., k is "practically" infinite

Solution using REPRESENTER's to INVERT DATA (@ k=10)

Difference Map between Solutions using SOLVE.F and REPRESENTERs

RHS using REPRESENTER's to INVERT DATA

INVERSE NOISE using REPRESENTER's to INVERT DATA

Monte Carlo Simulation

SOLUTION

k=10 and 5000 iterations

RHS (SOURCES)

k=10 and 5000 iterations

BIAS

k=10 and 5000 iterations