Day 4: ODE and Optimization

ODE

Differential Equations is one of the more important mathematical topics in Engineering. It is fundamental in Design and Simulation. It is incorporated in many of the software the engineers use for Design and Analysis.

ODE and PDE

Must be accompanied by boundary conditions and/or initial conditions to make the problem unique

• The differential equation can be single or coupled
• Can be linear or nonlinear
• Can respond naturally or forced
• Can have constant coefficients or varying coefficients
• Is categorized by the highest derivative in the expression

Symbolic approach to obtain the solution is limited

OPTIMIZATION

Optimization is another one of important mathematical topics in Engineering. It is fundamental in Design and Data Reduction and Modeling. Every mechanical device must be designed to be optimal if it is to exist today. Optimization is usually software driven and is performed numerically.

Optimization is fundamentally a search for optimal parameters that define the design. It may be material properties, dimensions, connections, constraints, costs, and manufacturing process etc. These are usually considered design variables. Traditionally it assumes that a mathematical expression exists that relates all or some of the above elements and the search exploits this relations to identify the best value of the parameters.

Optimization Methods:

1. Usually deal with nonlinear equations
2. Can include differential equations
3. Usually involves an iterative approach to the solution
4. Can involve continuous, discrete, or both kinds of variables
5. Requires computer resources and takes a long time
6. Determines mostly locally optimal values for the design
7. Global optimal values takes a long time to identify
8. Solution is based on numerical techniques

The numerical methods are implemented in the Scipy Optimization Module. The module is available as part of the standard Anaconda implementation. There are several types of optimization problems that can be solved and there are many algorithms available to solve them. Here we explore the following exercises.