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