The objective of the project is to study and understand different methods in the optimization in mechanics and gain practical knowledge by implementing the methods in Matlab to optimize different problems. In this report, the optimization using dichotomous search for mono-variable equation and then extended to multi variable is demonstrated.
A program to find the optimized shortest trajectory between 2 targets having some points in between and some obstacles is also demonstrated. The best minimum path is found in such a way that the trajectory covers the least possible distance by avoiding collisions against obstacles. Obstacles here refers to 4 circles of varied radius defined by the user and two rectangular walls with dimensions being user defined. Thanks to the FMINCON keyword of MATLAB and the definition of Objective and Constraint functions, the trajectory is optimized in such a way that it travels from a start point to an end point so that it passes tangentially or away from the circles and in-between the walls. Future improvements can be done by implementing the technique of SIMULATED ANNEALING.
It’s not easy algebraically to solve for a maximum or a minimum using calculus. Various search methods are used to approximate solutions to non-linear optimizations problems with a single independent variable and it is extended to multi variable too. The idea is to divide the line to two parts of equal size. Then consider two points near the middle point (with the distance of epsilon) and calculate which one's function value is smaller. New end points will be chosen so that the smaller value is between those points, i.e. the other endpoint is one of the previous endpoints and the other end point is one of the middle points (The one with the greater value).
The following tasks are performed;
Optimization of shortest trajectory between 2 targets having a point and a circle.
Optimization of shortest trajectory between 2 targets having 2 points and a circle.
Optimization of shortest trajectory between 2 targets having multiple points and a circle.
Optimization of shortest trajectory between 2 targets having multiple points and 2 circles.
Optimization of shortest trajectory between 2 targets having multiple points and 2 circles, 2 rectangles as obstacle.
Optimization of shortest trajectory between 2 targets having multiple points and 4 circles, 2 rectangles as obstacle.
The optimization of mechanics theories is well implemented and demonstrated in MATLAB. Different cases are examined. All the results are achieved. As the optimization of mechanics is very hard to done by hand calculations, the tools like MATLAB plays very good role in making it easy. The in-build functions in MATLAB like FMINCON make this job very easier. As the importance of optimization is important in the modern technological world, these methods are crucial for the future engineering.