5. Mass matrix

For elements MEMBRANE and GRILLE_MEMBRANE, the terms of the mass matrix are obtained after discretizing the following variational formulation:

(5.1)\[ \ begin {array} {c}\ mathrm {\ delta} {W} {W} _ {\ text {mass}} ^ {\ text {ac}} =\ underset {-h/2} {\ overset {+h/2} {\ h/2} {\ int}} {\ int}}\ underset {S} {\ mass}}} ^ {\ text {ac}}} =\ underset {-h/2} {\ overset {+h/2} {\ h/2} {\ int}} {\ int}} {\ int}}\ underset {S} {\ int}}\ mathrm {\ delta} u\ mathit {dz}\ mathit {dS} =\ underset {S} {\ s} {\ int} {\ int} {\ int} {\ int} {\ mathrm {\ delta} u+\ ddot {v}\ ddot {v}\ mathrm {v}\ mathrm {v}\ mathrm {\ delta} v+\ ddot {w}}\ mathrm {\ delta} w)\ mathit {dS}\ delta} u+\ ddot {v}\ v}\ mathrm {v}\ mathrm {v}}\ mathrm {v}\ mathrm {v}}\ mathrm {v}}\ mathrm {v}}\ mathrm {v}}\ mathrm {v}}\ mathrm {v}\]

With \({\mathrm{\rho }}_{m}=\underset{-h/2}{\overset{+h/2}{\int }}\mathrm{\rho }\text{dz}\). The displacement discretization (for \(N\) nodes) for this isoparametric element is:

(5.2)\[\begin{split} \ text {u} =\ sum _ {k=1} ^ {N} {N} {N} {N} _ {k}\ left (\ begin {array} {c} {u} _ {k}\\ {v} _ {k}\\ {w}\\ {w} _ {w} _ {k} _ {k}\ end {array}\ right)\} k=\ mathrm {1,}\ cdots, N\end{split}\]

The mass matrix, in the base where the degrees of freedom are grouped according to the directions of translation, then has the expression:

(5.3)\[\begin{split} \ text {M} =\ left (\ begin {array} {ccc} {ccc}} {\ text {M}} _ {m} & 0& 0\\ 0& {\ text {M}} _ {m} & 0\\ 0& 0\\ 0& 0& 0& 0& 0& {\ text {M}} _ {m}\ end {array}}} _ {m}\ end {array}}\ right)\end{split}\]

With: \({\text{M}}_{m}=\underset{S}{\int }{\mathrm{\rho }}_{m}{\text{N}}^{T}\text{N}\text{dS}\) and \(\text{N}=({N}_{1}\cdots {N}_{k})\).