5. Theoretical elements#

In this section, the theoretical foundations of the approach are presented.

5.1. Introductory#

5.1.1. General expression of the problem#

We start by recalling the formulation of a linear dynamics problem in the form of minimizing the Hamiltonian:

(5.1)#\[ \ underset {\ mathit {0}}} {\ mathit {min}}}\ underset {{\ mathit {enc}} _ {1}} {\ overset {{\ mathit {enc}} _ {2}} {2}}} {\ int}}} {\ int}}} {\ int}}}\ left ({\ mathit {0}}}\ left ({\ mathit {0}}} ^ {\ mathit {halt}}} {\ mathit {1}}} {\ mathit}} {\ mathit}} {\ mathit}} {\ mathit}} {\ mathit}}\ mathit {}} - {\ dot {\ mathit}} - {\ dot {\ mathit}} √}}} ^ {\ mathit {halt}}}\ mathit {residue}}\ dot {\ mathit {0}}\ right)\ mathit {}}\ right)\ mathit {}}\]

With a few algebraic manipulations, we can obtain the modal calculation expression that is important to us:

(5.2)#\[ \ mathit {thanx} - {\ mathit {8,5}}} ^ {2}\ mathit {forx} =0\]

5.1.2. Generalized modes and unknowns#

We also recall that we denote the rectangular matrix of the \({n}_{m}\) modes as follows:

(5.3)#\[ \ mathrm {\ Phi} =\ left [\ begin {array} {ccc} {\ mathrm {\ Phi}} _ {1} &\ text {...} & {\ mathrm {\ Phi}}} _ {{n}} _ {m}}\ end {array}\ right]\]

Since each vector \({\mathrm{\Phi }}_{i}\) is of length \({n}_{\text{eq}}\), the matrix \(\mathrm{\Phi }\) has dimensions \({n}_{\text{eq}}\times {n}_{m}\). Finally, we recall the relationship between a vector of unknowns in physical space \(x\) and a vector of generalized unknowns \(\mathrm{\eta }\):

\[\]

: label: eq-4

x=mathrm {Phi}mathrm {eta} = {mathrm {eta}} _ {1} {mathrm {Phi}} _ {1} + {mathrm {eta}}} + {mathrm {eta}}} _ {2}} +mathrm {Phi}} _ {1} +mathrm {eta}} + {mathrm {eta}} + {mathrm {eta}} + {mathrm {eta}} + {mathrm {eta}} + {mathrm {eta}} + {mathrm {eta}} _ {{n}} _ {n} _ {m}} {mathrm {Phi}} _ {{n} _ {m}}

5.1.3. Problem under constraints#

The last point presented concerns the resolution of a dynamic problem under constraints. The latter is expressed in the general form:

\[\]

: label: eq-5

mathit {Bx} =0

In the present case, the matrix \(B\) is a rectangular matrix whose number of rows is equal to the number of constraints applied to the unknowns. The problem under constraints is expressed as follows:

(5.4)#\[ \ underset {\ mathit {740}} =0} {\ mathit {min}}}\ underset {{\ mathit {enc}} _ {1}} {\ overset {{\ mathit {enc}}} _ {2}}} {2}}} {\ int}}} {\ int}}}\ left ({\ mathit {8}}} ^ {\ mathit {10}}} {\ mathit {10}}} {\ mathit {1}}} {\ mathit {10}}} {\ mathit {2}}} _ {\ mathit {mathit {0}}}} ^ {\ mathit {‡ {‡}}\ mathit {10}}\ dot {\ mathit {0}}\ right)\ mathit {]}\]

We assume that we know how to build a \({n}_{\mathit{ker}}\) dimensional base of the core of the constraint matrix. It is noted:

(5.5)#\[ \ mathrm {\ Psi} =\ left [{\ mathrm {\ Psi}} _ {1}\ mathrm {...} {\ mathrm {\ Psi}} _ {{n} _ {\ mathit {ker}}}\ right]\]

As the solutions sought verify the \(\mathit{Bx}=0\) constraints, they belong to the core of \(B\). So we can write them on this basis:

(5.6)#\[ x=\ mathrm {\ Psi}\ mathrm {\ eta}\]

By injecting this expression into the minimization problem, we see that we can then relax the constraint and write the problem in the form:

(5.7)#\[ \ underset {\ mathrm {\ eta}}} {\ mathit {min}}} {\ mathit {min}}}\ underset {{\ mathit {enc}}} {\ mathit {enc}} _ {2}} {\ int}}} {\ int}}}\ left ({\ mathrm {\ int}}}\ left [{\ mathrm {\ Psi}}\ left [{\ mathrm {\ psi}}} ^ {\ mathit {REGULO}}\ mathit {332}\ mathrm {\ Psi}\ right]\ mathrm {\ eta} - {\ dot {\ mathrm {\ eta}}}}}} ^ {\ mathit {scrap}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}}} ^ {\ mathit {\ eta}}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}} ^ {\ mathit {\ eta}}}} ^ {\ right]\ dot {\ mathrm {\ eta}}}\ right)\ mathit {\]

The problem is therefore solvable using conventional techniques (minimization without constraints).

5.2. Implementation#

We consider a structure divided into 2 substructures. For each, the following were determined:

  • the matrices of mass \({M}_{1}\) \({M}_{2}\) and stiffness \({K}_{1}\) \({K}_{2}\), of respective dimensions \({n}_{{\text{eq}}_{1}}\times {n}_{{\text{eq}}_{1}}\) and \({n}_{{\text{eq}}_{2}}\times {n}_{{\text{eq}}_{2}}\)

  • the matrices of the modes with blocked interface \({\mathrm{\Phi }}_{1}\) and \({\mathrm{\Phi }}_{2}\) of respective dimensions \({n}_{{\mathit{eq}}_{1}}\times {n}_{{m}_{1}}\) and \({n}_{{\mathit{eq}}_{2}}\times {n}_{{m}_{2}}\). They are related to generalized unknowns \({\mathrm{\eta }}_{1}\) and \({\mathrm{\eta }}_{2}\) respectively.

  • the matrices of the interface modes \(\overline{{\mathrm{\Phi }}_{1}}\) and \(\overline{{\mathrm{\Phi }}_{2}}\) of respective dimensions \({n}_{{\mathit{eq}}_{1}}\times {n}_{i}\) and \({n}_{{\mathit{eq}}_{2}}\times {n}_{i}\). They are related to generalized unknowns \(\overline{{\mathrm{\eta }}_{1}}\) and \(\overline{{\mathrm{\eta }}_{2}}\) respectively.

Based on these definitions, we can write the link between the physical and generalized degrees of freedom of each of the substructures:

(5.8)#\[ {\ mathit {x}} _ {1} =\ left [{\ mathrm {\ Phi}}} _ {1} {\ overline {\ mathrm {\ Phi}}}} _ {1}\ right]\ right]\ left\ left\ {\ begin {array} {\ left\ {\ begin {array} {c}} {c}} {\ mathrm {\ eta}}} _ {1}\ {\ overline {\ mathrm {\ Phi}}} _ {1}\ {\ overline {\ mathrm {\ eta}}} _ {1}\ end {array}\ right\}\]

By noting \(\stackrel{~}{{\mathrm{\Phi }}_{i}}=\left[{\mathrm{\Phi }}_{i}{\overline{\mathrm{\Phi }}}_{i}\right]\) and \(\stackrel{~}{{\mathrm{\eta }}_{i}}=\left\{\begin{array}{c}{\mathrm{\eta }}_{i}\\ {\overline{\mathrm{\eta }}}_{i}\end{array}\right\}\), we can write the previous relationship as:

(5.9)#\[ {\ mathit {x}} _ {1} = {\ stackrel {~} {\ stackrel {~}} {\ mathrm {\ Phi}}} _ {1} {\ stackrel {~}} {\ stackrel {~}} {\ mathrm {\ eta}}}} _ {1}\]

For each substructure, it is then possible to calculate the mass and stiffness matrices projected onto the blocked interface modes and the interface modes:

(5.10)#\[ {\ stackrel {~} {\ mathit {492}}}} _ {1}} _ {1}} = {\ stackrel {~} {\ mathrm {\ Phi}}} _ {1} ^ {T} {\ mathit {390}} {\ mathit {390}}} {\ mathit {}}} {\ mathit {}}} {\ mathit {}}} {\ mathit {}}}} {\ mathit {390}}} _ {1}}\]

We then build the global matrices that relate to the 2 substructures:

(5.11)#\[\begin{split} {\ mathit {492}} _ {\ mathit {zern}} =\ left [\ begin {array} {cc} {\ stackrel {~} {\ mathit {390}}}} _ {1}} & 0\\ 0& 0\\ 0& {\ stackrel {\ 0& {\ stackrel {~}} {\ mathit {902}}} _ {2}\ end {array}}} _ {1}} & 0\\ 0& 0\\ 0& {\ stackrel {~}} {\ mathit {902}}} _ {2}\ end {array}\ right]\end{split}\]

The generalized degrees of freedom are constructed in the same way:

(5.12)#\[\begin{split} {\ mathrm {\ eta}} _ {g} =\ left\ {\ begin {array} {c} {\ stackrel {~} {\ mathrm {\ eta}}}} _ {1}\\ {\ stackrel {\ stackrel {~} =\\ {\ stackrel {~}} =\\ {\ {\ stackrel {~}} {\ stackrel {~}}} _ {1}\\ {\ {\ stackrel {~}}} _ {1}\\ {\\ stackrel {~}} =\\ {\ {\ stackrel {~}} {\ stackrel {~}}} _ {1}\\ {\ {\ stackrel {~}} =\ left\ {\\ stackrel {~}} c}\ left\ {\ begin {array} {c} {c} {\ mathrm {\ eta}}} _ {1}\\ {\ overline {\ eta}}}} _ {1}\ end {array}\\ array}\ right\\ array}\ right\ right\}\ right\}\ right\}\ right\}}\\ {\ overline}}\\ {\ overline {\ mathrm {\ eta}}} _ {2}\\ {\ array}\ right\ right\}\ right\}}\ right\}\ right\}}\ right\}\ right\}}\ right\}}\ right\}\ right\}}\ right\}\ right\}}\ right\}\ right\}}\ right\}\ right\}}\ right\ thrm {\ eta}}} _ {2}\ end {array}\ right\}\ end {array}\ right\}\end{split}\]

We recall that \({\mathrm{\eta }}_{i}\) are the generalized unknowns associated with blocked interface modes and \(\overline{{\mathrm{\eta }}_{i}}\) those associated with interface modes. So to ensure the continuity of movements in physical space when crossing the interface, we must impose:

(5.13)#\[ \ overline {{\ mathrm {\ eta}}} _ {1}} =\ overline {{\ mathrm {\ eta}}} _ {2}}\]

This can be expressed simply using a constraint matrix in the form:

(5.14)#\[ B {\ mathrm {\ eta}} _ {g} =0\]

with:

+—————————————————————————————————————————-++ | || + .. image:: images/100002000000015C0000009BADCF883F8884610B.png ++ | :width: 1.622in || + :height: 0.7217in ++ | || + ++ | || +—————————————————————————————————————————-++

The global problem can then be expressed in the form of:

(5.15)#\[ \ underset {\ mathit {bola} {\ mathrm {\ eta}}} _ {g} =0} {\ mathit {min}}\ underset {{t} _ {1}} {\ overset {{t} _ {t} _ {2} _ {2}}} {2}}} {\ int}}} {\ int}} {\ int}}\ left ({\ mathrm {\ eta}}} _ {g} ^ {\ mathit {‡}} {\ mathit {‡}} {332}} _ {g} {\ mathrm {\ eta}}} _ {eta}} _ {\ eta}} - {\ dot {\ mathrm {\ eta}}} {\ mathit {halt}}} {\ mathit {exactly}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {147}}}} {\ mathit {146}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}} {\ mathit {exactly}}}\]

By calculating the core of \(B\), the problem above is solvable as presented in section 5.1.3.

Implementation examples can be found in the sdlv102 and sdlv103 tests.