# Introduction / examples

## Definition

A system is said to be nonlinear when one of the properties of the system is a function of the variable that is an unknown of the system. Two examples are presented in the following paragraph.

## Example 1 (thermal application)

The differential equation solved using finite element method in a Steady State Thermal application is the following:

where:

• [k] is the tensor of thermal conductivity
• q is the volume density of power of the heat source
• T is the temperature, respectively the state variable, i.e. the unknown of the system.

If the thermal conductivity k is a function of the temperature T, the system is a nonlinear system.

## Example 2 (magnetic application)

The differential equation solved using the finite element method in a Magneto Static application (with the scalar model) can be written:

where:

• [μ] is the tensor of magnetic permeability in the computation domain
• ϕ is a magnetic scalar potential, respectively the state variable
• is a term corresponding to sources (imposed field source or electric vector potential).

If the magnetic permeability μ is a function of the magnetic field H, respectively of the state variable ϕ, the system is a nonlinear system.

## Example 2 prime (magnetic application)

The differential equation solved using finite element method in a Magneto Static application (with vector model) can be written:

where:

• [ν] is the tensor of magnetic reluctivity of the computation domain
• is the density of current source
• is the vector potential, respectively the state variable, i.e. the unknown of the system

If the magnetic reluctivity ν is a function of the magnetic flux density B, respectively of the state variable A, the system is a nonlinear system.

## Different possibilities

A system is called nonlinear in Flux applications when:

• behavior laws of materials (constitutive equations) are nonlinear
• B(H) nonlinear law:

(magnetic permeability μ function of the magnetic field H)

• J(E) nonlinear law:

(electric conductivity σ function of the electric field E)

• D(E) nonlinear law:

(electric permitttivity ε function of the electric field E)

• a thermal property depends on temperature…
• thermal conductivity k function of T
• volume heat capacity function of T
• thermal exchange coefficients function of T

## ... in Flux

A brief description of the models proposed in Flux is presented in the two tables below:
• Models for the behavior laws (electromagnetic properties)
B(H) J(E) D(E)
Soft materials Hard materials
Linear Linear Linear Constant resistivity Linear
Linear complex

Linear complex

Linear with losses

Nonlinear

Saturation:

• analytic
• spline

Rayleigh

Demagnetization:

• analytic