# Structural optimization in Flux

## Overview

An optimization can be defined as a mathematical problem allowing to minimize an objective function while respecting or not some constraints. The objective function and the constraints may be defined with physical or structural quantities. The aim of this mathematical problem description is to find the best element which satisfies all the defined constraints. Such a problem mathematical problem may be written as:

With F(x), the objective function, G(x) and H(x) respectively the equality and inequality constrains.

The mathematical equations mentioned above are the basis of several appoaches of optimization that could be classified in one of the following categories:
• Parametric Optimization: the shape of the design is parametrized with geometrical parameters, accross all the given solutions the best which respects contraints is keepen
• Shape Optimization: is an approach that optimizes material boudary from a given design, for a given set of constraints with the goal of maximizing the efficiency of the system.
• Topological Optimisation: is a mathematical approach that optimizes material layout within a given design space, for a given set of loads, boundary conditions and constraints with the goal of maximizing the performance of the system.
In electrical engineering, these three approaches are used for several applications:
• Electromechanical energy conversion: equipment such as rotating electrical machinery and actuators rely upon a magnetic field to convert electrical power into mechanical work, and vice-versa. In these devices, coils establish a magnetic flux density field in their ferromagnetic parts and in their air gaps. Variations in the energy stored in the magnetic field result in the manifestation of forces and torques in the moving parts, and the system is generally designed to maximize the efficiency of the electromechanical conversion while decreasing the weight of the system.
• Power conversion and conditioning: this category includes coils and inductors belonging to solid-state converters. In this context, coils are designed to exhibit a given current-carrying capacity and a required inductance, providing smooth and harmonic-free current wave-forms. In other applications, such as power transformers, induction heating, and welding, the coils are designed to exhibit optimal magnetic coupling factors and reduced flux leakage.