4. Clean modes and disposal

This section echoes section §7 of the R3.03.01 documentation, and justifies the elimination method for the calculation of natural modes. Here we are interested in the following general problem:

(4.1)\[ \ textrm {Find couples} \ left ({\ omega} _ {0} ^ {2},\ {\ varphi\}\ right)\]

The matrices \({M}_{c}\) and \({K}_{c}\) must take into account the boundary conditions, and are here assumed to be obtained from functions of the forms \({u}_{N}\) satisfying \({\mathit{Bu}}_{N}=0\). However, in the present case, the assembled problem is not the problem (), but a problem of greater size, for which the shape functions do not verify the kinematic constraints. The matrices \({M}_{b}\) and \({K}_{b}\) have dimensions \((N-\mathit{Nb})\times (N-\mathit{Nb})\).

Starting from the assembled matrices \(M\) and \(K\), of size \(N\times N\), built on the basis of functions of shapes \(u\) that do not verify step \(\mathit{Bu}=0\), there are two alternatives to solve the problem (). The first consists in increasing the \(K\) matrix to reveal the kinematic constraints, by adding unknowns in the form of Lagrange multipliers. This has been the approach taken so far in the code. The problem to be solved is written, using the double dualization technique:

(4.2)\[\begin{split} \ left (\ left [\ begin {array} {ccc} {ccc} K& {B} ^ {T} & {B} ^ {T}\\ B& -\ mathit {Id} &\ mathit {Id}\\ B&\ mathit {Id}} & -\ mathit {Id} ^ {Id} & -\ mathit {Id}} & -\ mathit {Id}} & -\ mathit {Id}} & -\ mathit {Id}} & -\ mathit {Id}} & -\ mathit {Id}} & -\ mathit {Id}} & -\ mathit {Id} & -\ mathit {Id}} & -\ mathit {Id} & -\ mathit {Id}} &\ left [\ begin {array} {ccc} M& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0\ end {array}\ right]\ right)\ left\ {\ begin {array} {c}\ mathrm {\ varphi}\\ mathrm {\ varphi}}\\ {\ mathrm {\ varphi}}\\ {\ mathrm {\ varphi}}\\ {\ mathrm {\ varphi}}\\ {\ mathrm {\ varphi}}\\ {\ mathrm {\ varphi}}\\ {varphi}}\\ {\ mathrm {\ varphi}}\\ {\ mathrm {\ varphi}}\\ {\ mathrm {\ varphi}}\\ {\ 2}\ end {array}\ right\} =\ left\ {\ begin {array} {c} 0\\ 0\\ 0\\ 0\ end {array}\ right\}\end{split}\]

The approach by adding Lagrange multipliers, in the case of the search for natural modes, is not satisfactory, since in addition to the loss of the positivity properties of the operator, a large number of degrees of freedom are added, and the spectrum of the problem is therefore broadened. This broadening of spectrum poses numerical problems, which require significant work in this case.

The second approach, the one proposed here, is more natural, and is closer to the initial problem. The mass and stiffness matrices are certainly assembled on the basis of shape functions that do not satisfy the limit conditions, but the search for eigenmodes and eigenvalues can be done in a suitable subspace. All you have to do is build a subspace base for vectors \(v\) verifying \(\mathit{Bv}=0\). This subspace is naturally the core of \(B\). Then all you have to do is look for the modes of the unconstrained problem projected into the core of \(B\). Let \(Z\) be a core base, we then look for \(\left({\mathrm{\omega }}_{0}^{2},\{\mathrm{\psi }\}\right)\) couples that check

\[\]

: label: eq-20

left [{Z} ^ {T}left (K- {mathrm {omega}}} _ {0} ^ {2} Mright) Zright]left{begin {array} {c}mathrm {psi}mathrm {psi}mathrm {psi}}end {psi}}end {psi}end {psi}end {psi}end {psi}end {psi}end {array}right} =left{begin {array} {c} 0end {array} {c} {c}}end {array} {c} {c}}mathrm {psi}mathrm {psi}}end {psi}

We then identify \({M}_{b}={Z}^{T}MZ\) and \({K}_{b}={Z}^{T}KZ\).

After solving the reduced modal problem, we can calculate the value of the Lagrange multipliers \(\mathrm{\lambda }\) by the relationship:

\[\]

: label: eq-21

left{mathrm {lambda}right} = -left (BB^Tright) ^ {-1} Bleft (K- {mathrm {omega}} _ {0}} ^ {2} ^ {2} Mright) Zleft{mathrm {psi}omega}} _ {0} ^ {2} ^ {2} Mright) Zleft{mathrm {omega}}