2. What is a linear relationship?

This expression refers to a linear constraint on the degrees of freedom of the system under study:

• the degrees of freedom of the quantity TEMP_Rpour the thermal phenomenon,

• the degrees of freedom of quantities DEPL_Rou DEPL_Cpour the mechanical phenomenon,

• the degrees of freedom of the quantity PRES_Cpour the acoustic phenomenon.

The coefficients of this linear relationship are real (or complex) constants, the second member can be real, complex, or of the « function » type (K8).

A linear relationship can be written as:

\({\alpha }_{1}{\mathit{ddl}}_{1}+{\alpha }_{2}{\mathit{ddl}}_{2}+\mathrm{...}+{\alpha }_{n}{\mathit{ddl}}_{n}\mathrm{=}{\alpha }_{0}\)

where

\({\alpha }_{i}\mathrm{\in }\mathrm{ℝ}\) (or \(\mathrm{ℂ}\)) (\(i\mathrm{=}\mathrm{1,}n\))

\({\alpha }_{0}\mathrm{\in }\mathrm{ℝ}\) (or \(\mathrm{ℂ}\)) (or function)

Degrees of freedom \({\mathit{ddl}}_{i}\) are degrees of freedom carried by one or more different nodes.

Examples of linear relationships:

\(\mathit{DX}(\mathit{N1})\mathrm{=}0.\)	blocking the \(\mathit{DX}\) component of node \(\mathit{N1}\)
\(\mathit{TEMP}(\mathit{N3})\mathrm{=}100.\)	temperature set to \(100\mathrm{.}\) for node \(\mathit{N3}\)
\(\mathit{DY}(\mathit{N1})\mathrm{-}\mathit{DY}(\mathit{N2})\mathrm{=}0.\)	the nodes \(\mathit{N1}\) and \(\mathit{N2}\) have the same displacement \(\mathit{DY}\)
\(\mathrm{cos}\alpha \mathit{DX}(\mathit{N1})+\mathrm{sin}\alpha \mathit{DY}(\mathit{N1})\mathrm{=}0.\)	the node \(\mathit{N1}\) is forced to move on the line perpendicular to the vector (\(\mathrm{cos}\alpha\), \(\mathrm{sin}\alpha\)) (in 2D).