Optimization in Practice with MATLAB® For Engineering Students and Professionals

Overview

This chapter covers the practice of optimization in diverse technical areas. What has been learned thus far is the practical knowledge that is needed to apply optimization to different types of designs or systems. We will explore optimizing systems/designs in the following disciplines: chemistry, mechanics, aerospace, automotive, mathematics (data fitting), nuclear, electrical, portfolio management, and business (Ref. [1]). Our ability to optimize these various designs provides us with the skills to apply optimization beyond the confines of this book or of the class you might be taking, and successfully venture into the real world!

Mechanical Engineering Example

Structural Example

Figure 10.1 represents a ten-bar truss. The members are connected to each other at six nodes, numbered 1 to 6. The truss is assumed to be planar, and every node has two degrees of freedom (i.e., it is free to move both in the x and y directions). The left edge of the truss is fixed to the wall. The displacements u1 to u8 are specified at the specific non-fixed nodes. F1 to F8 represent loads applied to these nodes. The variables by x1 to x10 denote the cross sectional areas of the truss members. The Young's modulus of the material is E = 1 × 106N/mm2. The maximum allowable stress in each bar is σult = ±100N/mm2 (i.e., tension/compression), and the maximum allowable deflection is δmax = ±2 mm. We note that the unit of length in the figure and in this problem is mm, and that of force is N. While the current example is commonly used in the literature, the reader is also encouraged to review the topology optimization survey by Rozvany in Ref. [2].

Imagine you are hired as the optimization expert for the ten-bar truss project. The structural engineer working on the project provides you with a program blackbox−10bar.m, which allows you to enter the loads at the nodes and the area of each member, and obtain the corresponding nodal displacements and stresses in the members. Figure 10.2 illustrates this task. The ten-bar truss is subjected to 10 different loading conditions given in Table 10.1.