Final Problem Solving Lab Example
The following provides an example for the final version of the Problem Solving Lab Assignments in EGR 312. The problem explained and described below refers to the "falling parachutist" problem that is described in the course text book (Numerical Methods for Engineers 8th Edition - Chapra and Canale) and in class. Please use this as a guide when completing your final problem solving lab as it includes all key elements that you are expected to complete as part of your submission. Your final assignment should be in the form of a webpage. Remember that you are encouraged to present your results in a way that seems most appropriate to your problem and you should not exactly replicate the format and content you see here. The content below was developed by Dr. Di Vittorio, Dr. Lauren Lowman (co-instructor), and Nick Corak (EGR 312 TA). An example of an Initial Problem Solving Lab is also provided.
Figure 2: Mathematical Model
Figure 3: Explanation of Numerical Approach
The following pseudocode was used to implement the simulation in MATLAB. The MATLAB code is included at the end of this webpage.
Figure 4: Sample Calculations for Numerical Approach
Figure 6: Sample Calculations for Analytical Approach
Figure 5: Analytical Derivation of Velocity
Figure 7: Analytical and Numerical Results
Figure 8: Error for Each Numerical Simulation
The results in Figure 7 show that the numerical simulations that used Euler's method overestimate the true velocity. However, as the step size is reduced, the numerical simulations more closely approximate the true values. In addition, all of the simulations eventually reach the terminal velocity. Therefore, if we are only concerned with achieving the terminal velocity, then a large step size is sufficient as long as the velocity converges (stops changing). However, if we want to know how the velocity changes over time before reaching terminal velocity, then a smaller step size will greatly reduce the numerical error.
Figure 8 displays the true percent relative error over time for each time step. A step size of 5 seconds contains errors greater than 50% at the beginning of the simulation. If the step size is reduced by a factor of 10 (0.5s) then the maximum error is less than 5%. Both Figures 7 and 8 show that the velocity simulation converges around t = 25 seconds. Therefore, the parachutist will reach terminal velocity about 25 seconds after jumping from the plane.
When truncation error is reduced from decreasing the step size, the round-off error increases because more computations are performed. However, round-off error appears to be minor in this simulation, as the error is still drastically reduced for a step size of 0.5 seconds. The error could be further reduced for a smaller step size, but considering the simplicity of this model and processes that we have likely neglected the error reduction is likely small compared to overall model uncertainties. For instance, there could be wind currents that exert a horizontal force on the parachutist. The parachutists would also likely move and adjust their position, which would create a non-constant drag coefficient.
Regardless of the larger model uncertainties, this simple model is still helpful for answering questions such as the following: What is the maximum velocity for which we should design the parachute dimensions and material properties? If we are given an initial elevation, when should the parachutist open the chute to ensure they have enough time to safely land? What is the equivalent "wind speed" that the parachutist feels while falling? Numerical simulations of complex systems allow us to perform a variety of quick analyses at a low cost to develop safer engineering designs.