PDF Index| SDT-visc
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Functions
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CaseParVeCut : layers for visco and constraining layer
xxx rewrite formula (9) of [49] .
The parametrization of mass and stiffness is of the form DD(comp,orientMap,orientIncrement) : to represent viscoelastic deviatoric kvd, viscous isochore kvi, contraining layer kcl, base structure ke
| (3.20) |
where parameters p may link multiple elements while gke(p) designates the coefficient associate with a given integration point. The matrix derivatives are given by
| (3.21) |
The MSE approximation of loss is given by
| ηj = |
| (3.22) |
but one minizes the objective function combining multiple modes with positive weights µj
| J = |
|
| (3.23) |
whose derivative is given by
| = |
| µj |
| = |
| µj |
|
| (3.24) |
with the detailed expression of the contribution of each mode (a negative sensitivity corresponds to more added damping)
| = |
| ⎡ ⎢ ⎢ ⎣ | − |
|
| φjT Kv φj + |
| φjT |
| φj + |
| φjT Kv |
| ⎤ ⎥ ⎥ ⎦ | (3.25) |
where the squared frequency sensitivity is
| = φjT | ⎛ ⎜ ⎜ ⎝ |
| − ωj2 |
| ⎞ ⎟ ⎟ ⎠ | φj (3.26) |
and a negative frequency sensitivity corresponds to an added damping (because in (3.22) decreasing ωj will increase damping). and the shape derivative is found as
| = | ⎡ ⎣ | K −ωj2 M | ⎤ ⎦ | −1 | ⎡ ⎢ ⎢ ⎣ |
| − ωj2 |
| − |
| M | ⎤ ⎥ ⎥ ⎦ | ⎧ ⎨ ⎩ | φj | ⎫ ⎬ ⎭ | (3.27) |
This is used to define implicit matrices allowing fast variations of constitutive laws.
