smart technologies for safety engineering

VDM-Based Remodeling 247
6.4.3.3 Remodeling of Material Properties
Modification of the element cross-section influences both its stiffness and mass at the same
time. However, they are also related to the properties of the material, which is often modified
directly by exchanging structural elements while retaining the same element cross-sections.
In this case, the density and Young’s modulus modification parameters, μ ρ
i and μE i ,haveto
be used in Equations (41) instead of μA i . As before, the damping coefficients are assumed to
remain constant, ˆαN = αN and ˆλi = λi; the variables are the densities ˆρl and Young’s moduli
Êi of the elements of the modified structure. As in the previously considered case of material
redistribution, all four virtual distortions are nonvanishing and the general form of the system
of Equation (43) is retained, besides the same minor computational simplifications, which are
related to the fact that (αM)NM = αN MNM and (λS)ii = λiSii. The response of the
modified structure can be computed by solving the general system and by substituting the
resulting virtual distortions into Equations (42).
Differentiations of Equations (41) with respect to the densities and Young’s moduli of the
elements of the modified structure yield
∂MNM
∂ ˆρl
∂ (αM) NM
∂ ˆρl
∂ (λS) ii
∂ ˆρl
= Ml
NM
= αN
ˆρl
= ∂Sii
∂ ˆρl
,
M l
NM
ˆρl
= 0,
∂MNM
∂ Êl
∂ (λS) ii
∂ Êl
∂Sii
∂ Êl
= ∂ (αM) NM
∂ Êl
= δilλi Ai
= δil Ai
Differentiations of the original system of Equation (43) yield the two following linear systems:
F ω
⎡
∂ f
⎢
⎣
0 K
∂ ˆρl
∂d 0 K
∂ ˆρl
∂φ0 ⎤
⎥ ⎡
⎥
2 ∂MNM
⎥ ⎢
ω
⎥ ⎢ ∂ ˆρl
⎥ ⎢
⎥ ⎢
⎥ = ⎢−iω
⎥ ⎢
k ⎥ ⎢
∂
⎥ ⎣
ˆρl ⎥
⎦
∂ (αM) NM
∂ ˆρl
0
0
∂ε 0 k
∂ ˆρl
u M
u M
= 0
⎤
⎥ , F
⎥
⎦
ω
⎡
∂ f
⎢
⎣
0 K
∂ Êl
∂d 0 K
∂ Êl
∂φ0 ⎤
⎥ ⎡
⎥
0
⎥ ⎢
⎥ ⎢ 0
⎥ ⎢
⎥ ⎢
⎥ = ⎢−iω
⎥ ⎢
k ⎥ ⎢
⎥
∂
⎣
Êl ⎥
⎦
∂ (λS) ii
∂ Êl
− ∂Sii
εi
∂ Êl
which, after substitution of Equations (47), can be used to obtain the derivatives of the virtual
distortions and, by Equations (44), yield the derivatives of the response.
References
1. N. Kikuchi and M. P. Bendsøe, Generating optimal topologies in structural design using homogenization
method, Computer Methods in Applied Mechanics and Engineering, 71(2), November
1988, 197–224.
2. A. R. Diaz and M. P. Bendsøe, Shape optimization of multipurpose structures by homogenization
method, Structural Optimization, 4, 1992, 17–22.
∂ε 0 k
∂ Êl
εi
⎤
⎥
⎦
(47)