2 Theoretical Background

2.1 BESO Method for Fundamental Frequency Optimization

The modal behavior of an undamped system can be analyzed by means of (1).

K−ω²M

u=0 ð1Þ

Thek-th natural frequencyωkand its corresponding modeu_kare related through Rayleigh’s quotient given by (2).

142 E. de Souza Lisboa et al.

ω²k= u^T_kKuk

u^T_kMu_k ð2Þ

Consider the objective of maximizing thek-th natural frequency of a structure described by a discrete mesh using a predetermined volume of material. To every element in the meshed domain is assigned a value of 1 or x_min indicating the presence or absence of material, respectively. In this context, the problem may be presented as in (3) and (4) according to Huang and Xie [2]:

Maximize:fðxÞ=ωk ð3Þ

Subject to: V*− ∑^N

i= 1

xiVi= 0 xi=xminor 1

ð Þ ð4Þ

V*is the prescribed volume fraction, meaning the ratio between the volume that the structure should occupy and the total volume of the domain.N is the number of elements in the domain, and V_i is the fraction of the total volume that the i-th element occupies. The binary design variablex_iindicates if the structure occupies thei-th element, and the small valuex_min(e.g. 10⁻⁶) corresponds to a region in the domain with no material.

2.2 Interpolation Scheme for the Material Properties

For the solid-void design, the material’s density and Young modulus are functions of the design variable. Given that with the SIMP (Solid Isotropic Material with Penalization) method the ratio between elementary mass and stiffness is extremely high for smallx_ivalues (e.g. after applying power law penalization to the stiffness and linear interpolation to the density), artificial localized modes appear in regions with low density. Huang et al. [7] proposed an alternative interpolation scheme where the mass/stiffness ratio is kept constant, as given by Eqs. (5) and (6):

ρðxminÞ=xminρ¹ ð5Þ E xð minÞ=xminE¹ ð6Þ A better interpolation scheme takes the explicit form shown in Eqs. (7) and (8), according to Huang and Xie [2]:

ρð Þx_i =x_iρ¹ ð7Þ

Optimization of the Fundamental Frequency… 143

E xð Þi = xmin−x^P_min

1−x^P_min 1−x^P_i +x^P_i

E¹ ð8Þ

Pis called the penalty exponent andx_iis 1 if thei-th element is composed of the corresponding material, 0 if otherwise. Further details may befind in [7].

2.3 Frequency Optimization in Hierarchical Structures

The objective function for dynamic problems in multiscale structures can be written as shown in Eqs. (9) through (14), according to Zuo et al. [6]:

Find: x=x^mac,x^{mic, 1},x^{mic, 2} x^mac_i ,x^{mic, 1}_i ,x^{mic, 2}_i =xminor 1

ð9Þ

Maximize:fðxÞ=ωk ð10Þ

Subject to: K−ω²kM

uk=0 ð11Þ

∑^M

i= 1

x^mac_i V_i^mac=V^mac ð12Þ

∑^N

j= 1

x^{mic, 1}_j V_j^{mic, 1}=V^{mic, 1} ð13Þ

∑^N

j= 1

x^{mic, 2}_j V_j^{mic, 2}=V^{mic, 2} ð14Þ

V is the volume, and the superscript mac represents the macromodel and mic represents the micromodel of the structure. The subscriptsiandjcorrespond to the i-th andj-th element of the macromodel and the micromodel, respectively.

In this problem the vectorx is composed byx^mac, which describes the macro- model layout, andx^mic,1andx^mic,2, which describe the micromodels corresponding to each of the phases present. This means that x_i^maccan assume a value of 0, in which case the properties for thei-th element are given by the homogenization of the microstructure defined by x^mic,1, or a value of 1, in which case the microstructure defined by x^mic,2is used instead.

For both macro and micromodels the design variable takes binary values cor- responding to the presence or absence of a certain phase. As the design variables in the two micromodels vary over the same domain, from now on they will be called onlyx^mic, requiring a distinction betweenx^mic,1or x^mic,2 when necessary.

144 E. de Souza Lisboa et al.

ωkrepresents thek-th natural frequency associated with the structure. Equation (12) describes the volume constraint on the macromodel, which controls the material distribution on the macroscale. V^mac correspond to the volume fraction of the predefined macro phase, whereV_i^macis the fraction of the total domain occupied by thei-th element in the macro domain. Similarly, Eqs. (13) and (14) describe volume constraints on the microscale of thefirst and second phases, respectively,i.e.,V_j^mic,1 is the volume of thej-th element in thefirst micromodel andV^mic,1is the prescribed fraction of volume of the first phase in the micromodel; V_j^mic,2 and V^mic,2 have analogous meanings but for the second micromodel.

2.4 Sensitivity Analysis

The element mass matrix m and the element stiffness matrix k, both of the macromodel, and the material constitutive matrixDof the micromodel, are defined respectively by Eqs. (15), (16) and (17), as shown by Zuo et al. [6], being necessary components for the sensitivity analysis. The penalized relations for the stiffness and constitutive matrix come from the SIMP model and are valid for both macro and micro levels of the structure.

mx^mac_i

=x^mac_i m¹_i + 1 −x^mac_i

m²_i ð15Þ

kx^mac_i

=x^mac_i P

k¹_i + 1h −x^mac_i Pi

k²_i ð16Þ

D x^mic_j

= x^mic_j P

D¹_j+ 1− x^mic_j P

D²_j ð17Þ

The derivatives for the global mass matrixMand stiffness matrixK, and for the material constitutive matrixDof the micromodel are obtained through Eqs. (18), (19) and (20), respectively. According to Zuo et al. [6]:

∂M

∂x^mac_i =m¹_i−m²_i ð18Þ

∂K

∂x^mac_i =P x ^mac_i P−1

k¹_i−k²_i

ð19Þ

∂D

∂x^mic_j =P x^mic_j P−1

D¹_j −D²_j

ð20Þ

Optimization of the Fundamental Frequency… 145

2.5 Macroscale Sensitivity Analysis

Deriving thek-th natural frequency from Rayleigh’s quotient (2) with respect to the i-th design variable in the macrolevel, normalizing the eigenvectors with respect to the mass matrix, and using the definition of the derivatives given by Eqs. (18) and (19), the sensitivity of thek-th natural frequency for the macromodel is obtained in Eq. (21), as shown by Zuo et al. [6]:

α^maci = ∂ωk

∂x^mac_i = 1 2ωk

u^T_k P x ^mac_i P−1

k¹_i −k²_i

−ω²km¹_i −m²_i

h i

uk ð21Þ

2.6 Microscale Sensitivity Analysis

The micromodel describes the microstructure of the phases present in the macro- model. This microstrucuture is then homogenized to get the effective material properties used in the macrostructure. The homogenized matrix D^H is calculated according to Eq. (22), over the domainYof the base cell described by the variable x^mic. The procedure used to calculate D^H is explained in a series of papers by Hassani and Hinton [8–10], and a detailed computational implementation may be found, for instance, in Andreassen and Andreasen [11].

D^H= 1 j jY

Z

Y

D Ið −buÞdY ð22Þ

D represents the constitutive matrix of the material in the microstructure, I is identity matrix,bis strain matrix in the micromodel anduthe displacementfield.

The stiffness matrixkimay be calculated according to Eq. (23), assuming a 2D domain, by imposition of a periodic boundary condition, where the displacement fieldsuare chosen so that the strain bis uniform [1,0,0]^T, [0,1,0]^Tand [0,0,1]^T:

ki= Z

V_i

B^TD^HBdVi ð23Þ

Brepresents the strain matrix andVithei-th element volume.

Differentiating the k-th natural frequency from Eq. (2) with respect to the j-th variable of the micromodel, Eq. (24) is obtained:

∂ωk

∂x^mic_j = 1 2ωk

u^T_k ∂K

∂x^mic_j −ω²k

∂M

∂x^mic_j

! uk

" #

ð24Þ

146 E. de Souza Lisboa et al.

In afinite element analysis the global stiffness matrixKis composed from the element stiffness matrices, and the global mass matrix M is composed from the element mass matrices. The sensitivity of the frequency with respect to the microstructure design variable was developed by Zuo et al. [6], as given by Eq. (25):

α^micj = ∂ωk

∂x^mic_j = 1 2ωk ∑^M

i= 1

u^T_k,i Z

Vi

B^T∂D^H

∂x^mic_j BdViuk,i ð25Þ

The sensitivity of the homogenized matrix can be obtained from Eqs. (20) and (22), obtaining Eq. (26):

∂D^H

∂x^mic_j = 1 j jY

Z

Y

I−bu ð Þ^T ∂D

∂x^mic_j ðI−buÞdY

= P x^mic_j

P−1

j jY Z

Y

I−bu

ð Þ^T D¹_j−D²_j

I−bu

ð ÞdY

ð26Þ

Thus, the sensitivity of the fundamental frequency with respect to the microstructure design variables is found by substituting Eq. (25) into Eq. (26), producing Eq. (27):

α^micj = P x^mic_j

P−1

2ωkj jY ∑^M

i= 1

u^T_k,i Z

V_i

B^T Z

Y

I−bu

ð Þ^T D¹_j−D²_j

I−bu

ð ÞdY

2 4

3 5BdV_i 8<

:

9=

;uk,i

ð27Þ To minimize oscillatory effects that occur during the optimization process, where the same elements are repeatedly added and removed, a filtering technique is applied. The averaged sensitivity with respect to time is given in Eq. (28), where qis the iteration numerical index:

α̃=1

2 α^qi+α^qi⁻¹

ð28Þ

2.7 Convergence Criteria

The evolution ratio ER is an algorithm parameter, and its importance resides in controlling the volume variation between iterations, given in Eq. (29) for the macromodel and in Eq. (30) for the micromodel:

Optimization of the Fundamental Frequency… 147

V^mac,q=V^mac,q−1ð1 ±ER^macÞ ð29Þ V^mic,q=V^mic,q−11 ±ER^mic

ð30Þ

V^mac,qis the value of the volume in theq-th iteration, whileV^mac,q−1is defined in the previous iteration (q− 1).V^mic,qand V^mic,q−1are defined in the same way.

After the final volume fractions are reached the element addition/removal pro- cess continues until the variation in the objective function for consecutive iterations is lower than a certain threshold value. The considered variation ofωktakes into account the lastNiterations, as shown in Eq. (31):

τ=∑^Ni= 1ω^qk^{−i+ 1}−ω^qk^{−N−i+ 1}

∑^Ni= 1ω^qk^{−i+ 1}

≤τ* ð31Þ

τ represents the relative variation of the objective function,τ* is the tolerance or threshold andN is a predefined number of iterations. When this condition is sat- isfied the system is considered to have converged and the optimization process is stopped.

2.8 Flowchart

Theflowchart in Fig.1shows the implementation of the BESO algorithm.