Introducing Single Criterion Optimization Methods Into Mechanics Classes

Single criterion optimization problems are shown to be readily taught and understood at lower division course levels using algebra/calculus, exhaustive numerical searches, and solver-type tools in standard spreadsheet packages. Genetic algorithms provide another suitable numerical technique that is relatively easily understood by students in high-level form, directly applicable to the single criterion class of optimization problems and very helpful to the multiple criteria class of optimization problems. This paper describes the methods of lower-division mechanics classroom introduction of optimization methods including algebra/calculus, spreadsheet solver, exhaustive search, and genetic algorithms. A classical solid mechanics problem utilizing the simply supported beam with a central load is used as the baseline in this paper for presenting the optimization methods introduced. Several other more complex problems are described. Multiple criteria optimization problems, which can quickly exceed the capability of typical spreadsheet solver tools, require students to utilize the multiple criteria optimization capability of genetic algorithms software. A possible framework that would support both single criterion and multiple criteria optimization methods, based on using genetic algorithms software, is presented that could allow these powerful methods to be introduced and successfully utilized earlier in the student’s college experience. Introduction: Two major course learning objectives of a lower-division mechanics course, MET 211, Applied Strength of Materials, are: to understand the differences between analysis and design problems and to be able to properly address both types of problems. Analysis problems typically require a given input design with input design parameters. Design problems typically are much broader or open-ended in scope, often requiring the devising, analyzing, and testing of a series of design alternatives, before subsequent design analysis can begin. The analysis step, in either case, can be greatly enhanced if the student has a working knowledge of the utilization of optimization methods at their disposal. Single-criterion optimization methods have been shown to help analyze various design alternatives, helping selecting the best or optimal alternative. Single Criterion Optimization Methods Introduced: Four basic methods of optimization are introduced in this course to support the students analysis and design work, including algebra/calculus, exhaustive search, spreadsheet solvers, and genetic algorithms. All four methods are worked into this lower-division mechanics classroom in the sequence shown above. 1) Algebraic/Calculus Method: This method works well where there is a continuous function that is easily differentiable. Technology students typically take calculus as a corequisite making broad application of these methods difficult. Goldberg points out that P ge 749.1 Proceedings of the 2002 American Society for Engineering Education Annual Conference & Exposition Copyright ” 2002, American Society for Engineering Education calculus-based methods are based on the existence of “quadratic objective functions, ideal constraints and ever-present derivatives”. 2) Exhaustive Search Method: This method works very well in that the method is logical and all of the students have a good knowledge of spreadsheets. 3) Spreadsheet Solver Method: This method is typically unknown to all but one or two students coming into the course. After lecture and laboratory introduction, the speed with which optimal solutions can be obtained is welcomed compared with methods 1) and 2) above. 4) Genetic Algorithms Method: This method has both single-criterion and multiple-criteria capability, making it ultimately applicable to a much broader base of mechanics problems. Mechanics Problems Utilizing Optimization: There are many problems in the mechanics world that can utilize optimization methods. Mechanics and optimization textbooks carry a plethora of analysis and design problems that can also serve as good optimization problems and a sampling of these excellent texts is included in the Bibliography of this paper. A typical optimization will require a design to have a constant stress, minimum weight, and/or minimum cost, though there are many optimization fitness functions that can be envisioned. All four methods mentioned above have applicability to these classes of optimization problems with a single objective function, f(x), and one or more constraints, where x is a member the constraint set. Very few mechanical elements experience a constant stress state throughout the entire element, though this condition would better utilize the element’s material. Parts are typically loaded such that there is one point that has the maximum stress while the other material is stressed at a lower level. Thus, the material that is stressed less than the maximum stress level is not utilized optimally. This class of optimization problem requires a optimization fitness function that is looking to output a variation of stress across the element equal to or below some set threshold, usually set by establishing a suitable design factor or factor of safety with respect to the yield strength of the material, Syt. The stress states in various portions of the mechanical element are first parameterized and then the cross-section is varied locally to produce an almost-constant stress state. Initial mechanical element designs rarely exhibit the minimum possible weight. Often low weight can mean improved utility and/or reduced cost due to material savings. For beams of a given length, the cross-sectional area can be minimized, giving minimum weight for a given material. It is possible, of course, that changing the mechanical element’s material from steel (density ~0.3 lb/ft) to aluminum (density ~0.1 lb/ ft) can give lower weight even though the cross-section for the aluminum is larger than that of the steel. Assemblies of two-force mechanical linkages into pinned trusses made of a given material can utilize both minimization of cross-sectional area and producing constant stress as joint optimization goals. Cost minimization of individual parts and multiple part assemblies is often a major criterion in the marketability of a product or machine. Minimization of part count in an assembly, by eliminating redundant load paths and/or unloaded elements, is one approach to providing minimum cost. Minimizing weight sometimes can imply minimum cost, though extreme caution must be taken P ge 749.2 Proceedings of the 2002 American Society for Engineering Education Annual Conference & Exposition Copyright ” 2002, American Society for Engineering Education with the minimum weight approach with higher-cost, lighter-weight materials like titanium and beryllium. Optimization Problem Descriptions: An optimization learning progression from a simply supported beam with a central load to a simply supported beam with a distributed load to the more complex overhung beam with a distributed load is presented in this paper. These problems enable students to utilize analysis methods with given parameters, utilizing lower-division mechanics course lecture material and experimental procedures, to initially solve them. Subsequent discussion of optimization broadens both their interest and deepens their understanding of these problems. An overview sketch of these three mechanics problems involving beams is shown in Figure 1 below. A) Simply Supported Beam with Central Load B) Simply Supported Beam with Distributed Load C) Overhung Beam with Distributed Load Figure 1: Three mechanics problems involving beams utilized for optimization learning Detail Optimization Example Using a Simply Supported Beam with Central Load: This simple, straight-forward, and standard mechanics problem provides students an understanding of optimization in both analysis and design modes. The sketch of the loaded beam depicted in Figure 1A is analyzed using mechanics principles. This analysis, utilizing free-body, instantaneous load, distributed load, and moment diagrams is shown in Figure 2 below. P