SDT-visc         Contents     Functions         Previous Next     PDF Index

3.4  Topology optimization tools

xxx rewrite formula (9) of [48] .

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

K(pg,DD) = K0 + 
 
∑
e
 ⎛
⎝		(єκ+ (1−єκ) gke(pg)  ⎞
⎠		Ke 
M(p) = M0 + 
 
∑
e
 ⎛
⎝		(єκ+ (1−єκ) gme(pg)  ⎞
⎠		Me
    (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

 
  
∂ K
∂ p
 = 
 
∑
e
 (1−κ) 
∂ gke
∂ p
 Ke 
  
∂ M
∂ p
 = 
 
∑
e
 (1−κ) 
∂ gme
∂ p
 Me
      (3.21)

The MSE approximation of loss is given by

  ηj = 
φjT Kv φj
ωj2
     (3.22)

but one minizes the objective function combining multiple modes with positive weights µj

J  = 
 
∑
j
 
µj
ηj
     (3.23)

whose derivative is given by

∂ J
∂ p
= 
 
∑
j
 µj 
∂ ηj−1
∂ p
 = 
 
∑
j
 µj  
−1
ηj2
 
∂ ηj
∂ p
      (3.24)

with the detailed expression of the contribution of each mode (a negative sensitivity corresponds to more added damping)

∂ ηj−1
∂ p
 = 
−1
ηj2
 ⎡
⎢
⎢
⎣		−
∂ ωj2
∂ p
 
1
ωj4
 φjT Kv φj + 
1
ωj2
φjT 
∂ Kv
∂ p
 φj + 
2
ωj2
 φjT Kv 
∂ φj
∂ p
 ⎤
⎥
⎥
⎦		     (3.25)

where the squared frequency sensitivity is

∂ ωj2
∂ p
 = φjT ⎛
⎜
⎜
⎝
∂ K
∂ p
 − ωj2 
∂ M
∂ p
  ⎞
⎟
⎟
⎠		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

∂ φj
∂ p
 = ⎡
⎣		K −ωj2 M ⎤
⎦		−1 ⎡
⎢
⎢
⎣
∂ K
∂ p
 − ωj2
∂ M
∂ p
 −
∂ ωj2
∂ p
 M ⎤
⎥
⎥
⎦		⎧
⎨
⎩		φj⎫
⎬
⎭		    (3.27)

©1991-2025 by SDTools
Previous Up Next