*This section is a synopsis of a portion of a paper written by Zafer Gurdal of Virginia Tech.
As part of the MAD (Multidisciplinary Analysis and Design) Center activities, we decided to incorporate concepts, methodologies, and the applications of multidisciplinary design optimization into the curriculum as a means of introducing more formal design methods. Rather than developing an entirely new course, we decided to modify a course which was part of the current curriculum along the lines of multidisciplinary optimization. The senior level course "Engineering Design Optimization" (AOE/ESM 4084), which is a cross-listed course between the Department of Aerospace and Ocean Engineering, and the Department of Engineering Science and Mechanics at Virginia Tech was ideally suited for that purpose.
AOE/ESM 4084 was originally designed to introduce the use of the methods of mathematical programming for engineering design optimization. The methods taught in this course included linear programming, unconstrained nonlinear programming such as the steepest descent, conjugate gradient, and Newton's methods, and constrained nonlinear programming such as the penalty function approach, sequential linear programming, and sequential quadratic programming approaches. Applications of these methods to minimum weight design of structures, machine design, and appropriate design problems from other engineering disciplines were demonstrated.
Most of the material in the original course was needed for the MDO course. In addition, new material on MDO specific topics such as the introduction to MDO formulations and the Global Sensitivity Equation method was needed. Also, instead of the small, mostly textbook examples, typically used in the course, we needed to have the students solve more complex problems that had significant MDO content. The problem of squeezing more material and more complex problems into the same time period was solved by extensive use of Mathematica TM. With Mathematica TM available to the students, solving complex problems became much easier, and as explained below, Mathematica TM also enabled us to present optimization algorithms better and faster to students.
We had to carefully consider the introduction of the MDO problems. Including design examples from different disciplines into a senior level course is difficult. Such examples require the students to have at least a rudimentary knowledge of the analysis techniques used in those disciplines. The difficulty is mostly associated with the complexity of the analyses needed in realistic multidisciplinary problems. The existing course was already crammed with too much material.
A solution to the problem was found using two new approaches. First, we introduced the use of the PC based package Mathematica TM into the course as mentioned above. Second, rather than introducing multiple examples, we decided to include a single multidisciplinary design problem that we introduce early in the semester and then use repetitively to teach and demonstrate the different aspects of the different optimization methods.
The use of the Mathematica TM program, which combines symbolic manipulation, programming, numerical calculations, and graphics features in a notebook environment that can be used to deliver electronic lectures, was crucial for the success of the course. A number of Mathematica TM notebooks were prepared to demonstrate the steps of various optimization algorithms. These notebooks were on linear programming, sequential linear programming, Powell's conjugate directions, steepest descent, Fletcher-Reeves conjugate gradients, Newton's, sequential simplex, penalty function approach, method of feasible directions, sequential quadratic programming, and global sensitivity derivatives. These notebooks were used to present live demonstrations during the lectures through a projection panel hooked up to a laptop computer. The notebooks prepared for the lectures were also made available to the students as tutorials that could be used outside the classroom environment to gain practical experience by performing the derivations and solving numerical examples. The advantage of the Mathematica TM notebooks over some of the multimedia programs which allow the students to go through only a fixed number of preestablished steps is the unlimited number of variations that the students and instructors can create during the tutorials.* Another advantage of the use of Mathematica TM was that it permitted the students to tackle nontrivial design problems as homework problems and projects.
*Indeed, several students taking the course used variations on the MDO class problems in their aircraft design projects in 1994- 95.
The results of our initial offerings are that the course works. It will play a key role in our program.
To find out more about our experience in MDO, look at our web page for the MAD center, and download a postscript file of a paper written for the ASEE meeting in June of 1995. The address is:
http://www.dept.aoe.vt.edu/research/collaborative/mad/postscripts/MDOExp.html