Below you will find the discussion and relevant materials for the ES 105 homework assignment due on January 16, 2009.
DiscussionFor our midterm we had to do problem 101: Tide Power in the Bay of Fundy by solving the governing Helmholtztype 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 = 0.10.2. MaterialsSource CodeMain Program Fortran source code CSolve Subroutine for LU Decomposition Tecplot FEM Plotting Subroutine Tecplot Vector Plotting Subroutine

PlotsStatus 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 ~ 5E5 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)
