Applications of Topology Optimization Techniques in Seismic Design of Structure - Structural Seismic Design Optimization and Earthquake Engineering - page 236

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level) stiffness matrix of the element e can be
expressed as

where λ , , , and N is the number

of elements. The second constraint restricts the
volume of the design to an upper limit denoted
by v . In this statement v e is the volume of the
element e .

= ω j = 1 N

j j d

( )

p

K x ) = K + x K K − (1.12)

(

e e e e e e

in which K e and K e are the stiffness matrices of
the element e when it is made of the base mate-
rial and void (in its solid and void states) respec-
tively. Similarly for the density and the mass
matrix of the element e we can write

4.1. Sensitivity Analysis

Solving Problem (1.15) requires finding the sen-
sitivities of λ 1 with respect to design variables x e .
Differentiating Kφ j = λ j Mφ j we can write

ρ x ) = x ρ (1.13) (

∂
∂

ϕ

= ∂
∂

λ ∂

∂

ϕ

e e e

∂
∂

K ∂

∂

M

j j j

K M M

ϕ + ϕ + λ ϕ + λ

j j j j j

x x x x x

e e e e e

(1.16)

and

Premultiplying by ϕ T and rearranging the
terms we obtain

M x ) = x M (1.14) (

e e e e

Where ρ is the density of the base material and
M e is the mass matrix of the element e in its
solid state.
Using these material interpolation schemes
one can change element e from solid to void and
backwards by changing the value of x e . Thus by
choosing x e -s as design variables, one can produce
different topologies without altering the finite
element mesh.
We can now formulate the optimization prob-
lem. The fundamental frequency optimization
problem can be stated as finding the best topol-
ogy of a structure to maximize its fundamental
frequency given a fixed amount of material. The
problem can thus be formulated as

= ∂
∂

λ

) ∂
∂

ϕ

∂
∂

K ∂

∂

M

(

T j T T j

ϕ − λ ϕ ϕ M ϕ − ϕ K − λ M

j j j j j j j

x x x x e

e e e

(1.17)

Using the symmetry of K and M , we can read-
ily conclude that

T

( ) = ( )

T

ϕ K − λ M K − λ M ϕ = 0 .

j j j j

We also use Equation (1.6) in Equation (1.17)
to finally express the sensitivities as

∂
∂

λ

∂
∂

K ∂

∂

M

j

T

= ϕ − λ ϕ (1.18)

j j j

x x x

e e e

{ }

max λ = min λ The stiffness and mass derivatives in Equa-

tion (1.18) can be calculated using Eqs. (1.12)
and (1.14).

1 j

x x , ,  , x j 1 2 , ,  , N

=

1 2 N d

such that K ϕ = λ M ϕ , j = 1 2 , ,  , N

j j j d

N

∑ ≤

x v v

4.2. Solution Method

e e

e = 1

0 ≤ ≤ x 1 , e = 1 2 , ,  , N

e

Having the sensitivities in Equation (1.18), the
optimization problem (1.15) can be solved using

(1.15)