3 Results and Discussion
3.1 BESO Method for Fundamental Frequency Optimization
Vmac,q=Vmac,q−1ð1 ±ERmacÞ ð29Þ Vmic,q=Vmic,q−11 ±ERmic
ð30Þ
Vmac,qis the value of the volume in theq-th iteration, whileVmac,q−1is defined in the previous iteration (q− 1).Vmic,qand Vmic,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= 1ωqk−i+ 1−ωqk−N−i+ 1
∑Ni= 1ωqk−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.
Fig. 1 General scheme of the BESO method applied
Fig. 2 Problem 1, two-material beam with lumped mass and clamped at both ends
Optimization of the Fundamental Frequency… 149
The objective is to maximize the fundamental frequency, where each material occupies half of the domain. Table1gives the properties of both materials.
The input data for the BESO method are the evolutionary rate ER= 2%, the maximum addition ratio ARmax= 2%, τ= 0.1%, Vmac= 50%, Vmic,1 = 100%, Vmic,2= 100%, filtering radiusrmin= 0.0015 m and penalty factorP= 3.
The results of the optimization are shown in Fig.3, where black regions rep- resent Material 1 and yellow regions represent Material 2. Results of the BESO optimization implemented of this work were compared to those obtained by Huang et al. [7]. Table2 shows the fundamental frequencies.
Problem 2 Problem 2 considered the same beam shown in Fig.2, but this time taking the softer material density and stiffness close to zero to emulate voids. The domain is discretized with 280 × 40 square elements with unitary sides, similar to Problem 1. The beam is composed of a material whose properties are shown in Table3. The input data are: evolutionary rateER = 2%, addition ratioARmax= 2%, τ= 0.01%, Vmac= 50%, Vmic,1= 100%, Vmic,2 = 0%, filtering radius rmin=
0.0015 m and penalty factorP= 3.
The result obtained in this work is shown in Fig.4, where the material is colored black, while the simulated void is shown as white. Table4compares the obtained frequencies.
Table 1 Material properties for Problem 1
Properties Material 1 Material 2
Young modulus (E) (N/cm2) 100 20
Poisson ratio (ν) 0.3 0.3
Mass density (ρ) (kg/cm3) 10−6 10−7
Fig. 3 Problem 1, results obtained in this work
Table 2 Natural frequency
for Problem 1 (rad/s) Author ω1 ω2
Huang and Xie [2] 37.1 –
This work 38.2 115
Table 3 Material properties
for Problem 2 Properties Material
Young modulus (E) 10 N/cm2
Poisson ratio (ν) 0.3
Mass density (ρ) 1 kg/cm3
150 E. de Souza Lisboa et al.
Problem 3 Problem 3 aims to optimize the two material beam clamped at both ends, as illustrated in Fig.5. The domain was discretized with 80 × 40 square elements of unitary dimensions. The objective is to maximize the fundamental frequency where each material occupies half of the domain. Table5 shows prop- erties of both materials.
The input data are: evolutionary rate ER= 1%, addition ratio ARmax= 1%, τ= 0.01%, Vmac= 50%, Vmic,1= 100%, Vmic,2= 100%, filtering radius rmin= 0.0015 and penalty factorP= 3.
Thefinal structure found in this case is shown in Fig. 6, where the stiff material is shown in black while the softer material is shown in yellow. Table6compares the obtained frequencies.
Fig. 4 Problem 2,final optimized structure obtained in this work
Table 4 Natural frequency
for Problem 2 (rad/s) Author ω1 ω2 Iterations
Huang and Xie [2] 33.7 – 55
This work 34.7 104.3 47
Fig. 5 Problem 3,
two-material beam clamped at both ends
Table 5 Material properties
for Problem 3 Properties Material 1 Material 2
Young modulus (E) (N/cm2) 1 0.2
Poisson ratio (ν) 0.3 0.3
Mass density (ρ) (kg/cm3) 1 2
Optimization of the Fundamental Frequency… 151
4 Conclusions
The comparison shows that the BESO algorithm implemented in this work was capable of optimizing the fundamental natural frequency in different structures. The algorithm makes use of its own FEM module, thus being independent of com- mercial FEM solvers.
The averaged sensitivity used in this work enabled to optimize the fundamental natural frequency in agreement to most elaborated proposals of the literature, e.g.
the fundamental natural frequency obtained in this work was higher in 2.96, 2.97 and 8.21% in Problems 1, 2 and 3, when compared with values obtained by other authors.
Acknowledgements The research described in this paper was financially supported by the National Council of Scientific and Technological Development CNPq-Brazil (grants 140081/
2015-1, 148895/2016-6 and 148887/2016-3) from the Ministry of Science, Technology and Innovation.
References
1. Huang, X., Xie, Y.M.: Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Comput. Mech.43, 393–401 (2009)
2. Huang, X., Xie, Y.M.: Evolutionary Topology Optimization of Continuum Structures:
Methods and Applications. Wiley, Chichester (2010)
3. Xu, B., Xie, M.: Concurrent design of composite macrostructure and cellular microstructure under random excitations. Compos. Struct.123, 65–77 (2015)
4. Liu, Q., Chan, R., Huang, X.: Concurrent topology optimization of macrostructures and material microstructures for natural frequency. Mater. Des.106, 380–390 (2016)
Fig. 6 Problem 3, results obtained in this work
Table 6 Natural frequency
for Problem 3 (rad/s) Author ω1 ω2
Zuo et al. [6] 0.011317 –
This work 0.011246 0.019632
152 E. de Souza Lisboa et al.
5. Long, K., Han, D., Gu, X.: Concurrent topology optimization of composite macrostructure and microstructure constructed by constituent phases of distinct Poisson’s ratios for maximum frequency. Comput. Mater. Sci.129, 194–201 (2017)
6. Zuo, Z.H., Huang, X., Rong, J.H., Xie, Y.M.: Muti-scale design of composite materials and structures for maximum natural frequencies. Mater. Des.51, 1023–1034 (2013)
7. Huang, X., Zuo, Z.H., Xie, Y.M.: Evolutionary topological optimization of vibrating continuum structures for natural frequencies. Comput. Struct.88, 357–364 (2010)
8. Hassani, B., Hinton, E.: A review of homogenization and topology optimization I— homogenization theory for media with periodic structure. Comput. Struct.69, 707–717 (1998) 9. Hassani, B., Hinton, E.: A review of homogenization and topology optimization II— analytical and numerical solution of homogenization. Comput. Struct.69, 719–738 (1998) 10. Hassani, B., Hinton, E.: A review of homogenization and topology optimization III—
topology optimization using optimality criteria. Comput. Struct.69, 739–756 (1998) 11. Andreassen, E., Andreasen, C.S.: How to determine composite material properties using
numerical homogenization. Comput. Mater. Sci.83, 488–495 (2014)
Optimization of the Fundamental Frequency… 153