ES 105 Midterm: February 12, 2009

Below you will find the discussion and relevant materials for the ES 105 homework assignment due on January 16, 2009.

Discussion

For our midterm we had to do problem 101: Tide Power in the Bay of Fundy by solving the governing Helmholtz-type equation. We used a Galerkin Method, s.th. the weighting functions are identical to the basis functions to obtain the desired matrix. Please see scanned pdf for Question#1.

Question#2 assumed Dirichlet conditions of z =1 at southwest entrance nodes. The solution to this system of equations for 600+ nodes and 1000+ elements, i.e., the "Status Quo" can bee seen below under Plots. Please consult the Main Fortran Source code for detail on the calculations and implementation.

Question#3 asked usto determine the complex horizontal velocity vector by computing the scalar components (u,v). The values for u and v were computed and their absolute values were plotted as a real vector field at the centroid of the element, i.e., "Vectors @ Element Centroid" can be seen below under Plots. Please consult the written scanned pdf for some details on how u and v were computed and consult the main source code for more explicit details.

Question#4 asked us to determine when R is "effectively infinite" and "effectively zero." This was taken to mean that R is "effectively infinite" s. th. there is a value of R that will lead to no more decrease in the tidal surface elevation at the Minas Basin. Also, R is "effectively zero" s. th. there is a value R > 0 were tidal elevations remain at their max despite a R > 0. See the "Change in R values" image below under Plots.

Question#5 asked us to determine Power over all the generating elements with varying values of R. See the graph "Log(Power) vs R values" below for the end results. Running our code for varying values of R showed that as R increased from 0 --> 1 the sum of Power across all generating elements behaved in an asymptotic manner. Thus, there is an "Ideal R" s.th. one determines a maximum R with a lesser input effort. That is, an "Ideal R" will find a balance were Power is maximized with account of how much effort is needed to maximize that Power. The "Ideal R" was seen to be somehwere between R = 0.1-0.2.

Materials

Source Code

 

Plots

Status Quo

Vectors @ Element Centroid

 

Change in R values

From top to bottom east of the Minas Basin we go down with increasing values of R

Effective Zero ~ 5E-5

Practical Infinite~> 1.0

 

Log(Power) vs R values

As R increase Power approaches a constant value (asymptotes) @ ~ log(Power) = 53,

Thereore an "ideal R" would be one that maximizes the Power with the least input effort (i.e., R)