Below you will find the discussion and relevant materials for the ES 105 homework assignment due on March 09, 2009.
DiscussionES105  Winter, 2009 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 12 boundary. Find and save them. ANSWER: In the range of 0.010.001. Nevertheless, this is always dependant on the sensitivity of the instrumentation used to gather information.
ANSWER: k_inf ~100.0 & k=k_inf/10=10.0
Inversion 1. Invert perfect data for the nodal sources. Use Representer method as you did in HW6; compute the bias in U. 1. Voltage (U) 4.Do the MonteCarlo results agree with the direct results?

MaterialsSource Code 
PlotsSolution 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
