Topology optimization for 3D microstructures ofviscoelastic composite materialsJianglin Yang 

Jihua LaboratoryShiyang Zhang  ( shyyoung915@hotmail.com )

Hunan First Normal UniversityJian Li 

Jihua Laboratory

Original Article

Keywords: Topology optimization, 3D microstructure, viscoelastic composites, damping

Posted Date: October 28th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-1012567/v1

License: This work is licensed under a Creative Commons Attribution 4.0 International License.  Read Full License

Topology optimization for 3D microstructures of viscoelastic composite materials

Jianglin Yang1, ShiYang Zhang2*, Jian Li1

1Ji hua Laboratory, Foshan, 528200, China

2College of Physics and Chemistry, Hunan First Normal University, Changsha,

410205, China

*Corresponding author

E-mail: shyyoung915@hotmail.com

Abstract:Materials with high stiffness and good vibration damping properties are of great

industrial interest. In this paper, a topology optimization algorithm based on the

BESO method is applied to design viscoelastic composite material by adjusting its 3D

microstructures. The viscoelastic composite material is assumed to be composed of a

non-viscoelastic material with high stiffness and a viscoelastic material with good

vibration damping. The 3D microstructures of the composite are uniformly

represented by corresponding periodic unit cells (PUCs). The effective properties of

the 3D PUC are extracted by the homogenization theory. The optimized properties of

the composites and the optimal microscopic layout of the two materials phases under

the conditions of maximum stiffness and maximum damping are given by several

numerical examples.

Keywords: Topology optimization, 3D microstructure, viscoelastic composites,

damping.

1. Introduction

Viscoelastic materials combining high stiffness and vibration damping

can be widely used in construction and manufacturing industry, such as automotive,

aviation, machinery, etc [1-3]. However, it is difficult to find natural materials with

both high stiffness and high damping. Materials with favorable damping

characteristics usually do not have sufficient stiffness to construct engineering

products independently.

The practical difficulty inspired the idea of mixing two or more constituent

materials with different physical properties to produce composite material that

possesses desired mechanical properties[4]. It was found experimentally that the

metal-matrix composites with high stiffness and viscoelastic damping can be

synthesized by mixing lossy metal alloy and stiff silicon carbide[5-6]. Subsequently,

it was realized that not only the proportion and physical properties, but also the

geometric layout of the materials phases would affect the viscoelastic properties of the

composites[4, 7].

Topology optimization is a good tool for structural design. Bendsoe and Kikuchi

proposed the homogenization design method to conduct a topology optimization

model, which assumes that the material is periodically composed of microscopic unit

cells and the macroscopic properties of the material can be calculated from its

microstructure[8]. Topology optimization of continuum structure is transformed into

the optimization of the microstructure of a periodical unit cell. Based on the

homogenization theory, a series of topology optimization methods have been

proposed to design the microstructures of man-made materials to achieve the desired

properties[9-19]. Sigmund first adopted the Solid Isotropic Material with Penalization

(SMIP) method[9-10] to design the microstructures of elastic materials for prescribed

properties[20-21]. Yi et al. then applied this method to find the microstructures of

two-phase viscoelastic composites which exhibit improved stiffness/damping

characteristics[22-23]. In the SIMP method, the design variable is the relative material

densities of the elements of the unit cells, whose value varies from 0 to 1. Although it

is possible to make most element density values either close to 0 or close to 1 by

applying interpolation penalties, and then obtain a relatively clear optimal structure, a

few elements may escape penalty and take intermediate values between 0 and 1, thus

making the optimization results less refined in details and resulting in grey-scale

elements.

The Evolutionary Structural Optimization (ESO) method[16] and its extended,

the Bi-directional ESO method[17-19], can be used to avoid the above problems.

These two methods force the value of design variable to be binary, that is, 0 or 1.

There are no grey-scale elements in the process of calculation, so it is easy to get a

clear image of the optimized structure.

The BESO method was initially applied to the design of 2D microstructures of

viscoelastic composites by Huang in 2015[24]. In this paper, we follow the approach

proposed by Huang to design the 3D microstructure of viscoelastic composites. Our

optimization objective is to maximize the stiffness or damping properties of the

composites under volume fraction or/and stiffness constraints. In Section 2, the

homogenization theory is used to calculate the complex moduli (storage modulus and

loss modulus) of the viscoelastic composites with 3D microstructure. In Section3, we

formulate the material interpolation scheme. The optimization problem is solved

using the linear artificial two-phase material model, and the sensitivity analysis is

conducted. In Section 4, four numerical examples are presented to demonstrate the

design approach is feasible to optimize the 3D microstructure of the viscoelastic

composites. Clear optimal microstructures and the effective material properties of

composites under various constraints are given. This paper comes to conclusions in

Section 5.

2. Homogenization for viscoelastic composites

2.1. Properties of viscoelastic materials in the frequency domain

For the case of harmonic excitation the stress and strain of uniform viscoelastic

material can be represented by exponential functions: [24]

, (1)

. (2)

in which, is the phase angle of strain lag stress for viscoelastic materials.

The viscoelastic materials modulus of elasticity is now independent of time but

dependent on frequency ,which can be expressed as :

. (3)

According to Eq. (3), the elastic modulus of viscoelastic material can be written

as complex modulus:

, (4)

, (5)

where is the storage modulus and is the loss modulus. The loss tangent

tan is a damping evaluation indicator, which is equal to the ratio of loss modulus to

storage modulus and it is proportional to the energy loss of each cycle in the

framework of linear viscoelasticity.2.2. Effective complex modulus of viscoelastic composites

a b c

Fig. 1. (a) Macroscopic structure; (b) composite microstructure; (c) periodic unit cell.

The macroscopic structure in Fig. 1(a) is made up of a composite which is

formed by unit cells in a periodic pattern. The microstructure of the composite and the

unit cell of which it is composed are shown in Fig. 1(b) and Fig. 1(c), respectively. In

the unit cell, The red-phase based material represents the non-viscoelastic and the

black-phase based material represents the viscoelastic. The effective macro

constitutive matrix of the cellular microstructure can be obtained by using the

homogenization method [9, 24-26] (Hassani and Hinton, 1998), which can be

expressed by:

, (6)

where |Y| is the area of the unit cell. In Eq. (6), EHijlk() is the effective complex

modulus of the unit cell, I is a 6x6 unit matrix and B is the strain matrix on the

micro-scale. The displacement u matrix, which contains six columns corresponding to

the six test strains in 3D cases, is a periodic solution of the following problem:

, (7)

where the form of isotropic material matrix E() and the stiffness matrix k ar

e:

, (8)

. (9)

3. Topology optimization

3.1. Design variables and material interpolation scheme

The microstructure design problem of maximizing effective stiffness or damping

for viscoelastic materials based on two-phase materials is essentially the optimal

distribution of two materials within the unit cell.

:material 2

:material 1

Fig. 2. distribution of two materials within the unit cell.

As Fig. 2 shows, the unit cell is discrete into finite elements that each element is

assigned with either material 1(colored by red ) or material 2 (colored by black). The

modulus of the two materials can be expressed by:

, (10)

where and is the storage modulus and loss modulus of material m,

respectively (m = 1, 2). To assemble the finite element stiffness matrix and

differentiate the boundary of materials in unit cells more conveniently, we introduce

xe to characterize materials as a design variables. When xe = 1, the element is made of

material 1; when xe = 0, the element is made of material 2. With such a definition, Yi

et al.[10] employed a linear artificial two-phase material model as:

. (11)

A good rule of thumb is to set p1 = 3 and p2 = 1, which have been

proved the feasibility of the scheme in 2D viscoelastic composite materials work [23].

3.2. Statement of the optimization problem

The optimization problem of maximizing damping or stiffness at the operation

frequency for viscoelastic materials can be defined as:

. (12)

The f(xe) is the objective function. and is the storage effective

modulus and loss effective modulus of the unit cell, respectively. Vf is the prescribed

volume fraction of material 1 and Ve is the volume of the eth element, while the

volume fraction of material 2 is 1-Vf. N is the total number of elements in the unit cell.

is the operating frequency.

3.3. Sensitivity analysis

Sensitivity is necessary for guiding the search direction during the iteration

process in the BESO topology optimization. In this section, the sensitivity of the

viscoelastic material on the micro-level will be discussed. The sensitivity (dc) is the

differential of objective function to design variables xe, which can be expressed by:

. (13)

According to Eqs. (12) and (13), the sensitivity with respect to design variables

xe can be further expressed as:

, (14)

where dc1 and dc2 are the sensitivity of maximizing damping and stiffness objective

functions. From the material interpolation scheme in Eq. (11) and the effective macro

constitutive matrix of cellular microstructure given by homogenization calculation

method in Eq. (6), the differential of the complex modulus to the design variables xe

in Eq. (14)can be obtained[10,12]:

. (15)

With the operation frequency condition, final expression of sensitivity with the

operation frequency constrain can be obtained by combining Eq. (14) and Eq. (15).

4. Numerical examples and discussions

In this section, numerical examples are carried out for demonstrating the

effectiveness of the proposed optimization approach in microstructural of 3D

viscoelastic composites design work. The viscoelastic composite is composed of pure

elastic material(material 1) and viscoelastic material(material 2). Both of the two

materials are isotropic. The Young’s modulus of material 1 and material 2 is 70 GPa

and 1.5+1i GPa respectively, setting excitation frequency as 0.5 rad/s. The Poisson’s

ratio of material 1 and material 2 is 0.22 and 0.35 respectively. The matrices of the

two materials are given in Table1 and Eq. (16).Table 1The properties of materials1 and2 at the operation frequencies.

Operation frequencyrad/s

Storage modulusE’1111 (GPa)

Loss modulusE”1111 (GPa)

Loss tangent Bulk modulusE (GPa)

material 1 all 79.91 0.00 0.00 41.66

material 2 0 2.40 0.00 0.00 1.66

0.5 2.40 1.60i 0.67 1.66+1.11i

(16)

Without loss of accuracy，the model with 40×40×40 finite elements is selected

for the optimization work. In the following figures of the unit cells, red elements

represent material 1(stiff and elastic) and black elements represent material 2(soft and

viscoelastic). The damping( f ) and stiffness( E* ) of composites can then be

calculated respectively as follows:

. (17)

4.1. Examples for maximizing damping of composites

In this part, the optimization objective is to find the 3D microstructure of

composites with high damping under various volume fractions of material 1. The

optimization function of maximizing damping at 0.5rad/s operation frequency for

viscoelastic materials can be defined as:

(18)

Table 2Optimized results for maximizing damping of composites

Vf1 Unit cell 3x3 Unit cells Elasticity matrix (GPa) f

80% 0.41

70% 0.51

60% 0.56

50% 0.59

40% 0.62

The optimized microstructures of the composites and calculation results are

shown in table 2. It is noticeable that the material 1(stiff and elastic) appears not only

in the center but also at the eight vertices of the unit cell surrounding with material

2(soft and viscoelastic). As the volume of material 1 decreases, so does the storage

modulus of the composite, and thus the ratio of the loss modulus to the storage

modulus increases. The structural optimization proves the utilization efficiency of the

materials. The damping coefficient f of the viscoelastic composite made of 60%

viscoelastic material 2 and 40% elastic material 1 is 92% of that of material 2.

4.2. Examples for maximizing stiffness of composites

In this part, the optimization objective is to maximizing the stiffness of

viscoelastic composite. Still setting operation frequency = 0.5 rad/s, then the

optimization function of maximizing stiffness can be expressed by :

(19)

For maximizing stiffness, the optimized results of the microstructures of

composites under various volume fractions of material 1 are shown in table3.

Different from the results in section 4.1, material 2 is separated by material 1 and

material 1 is connected with each other in the optimized 3x3 unit cells. The storage

modulus of composites with maximized stiffness is much higher than which with

maximized damping under the same material 1 volume fraction constraint, while the

loss modulus has the opposite.Table 3Optimized results for maximizing stiffness of composites

Vf1 Unit cell 3x3 Unit cells Elasticity matrix (GPa) f

60% 0.03

50% 0.04

40% 0.06

20% 0.13

4.3. Examples for maximizing damping with stiffness constraint

Both the damping and stiffness of viscoelastic will affect its vibration reduction.

Instantly, the material stiffness is reflected by the material storage modulus. In the

manufacturing industries, the value of the storage modulus is in a certain range.

Therefore, the design of viscoelastic composites is a multi-objective optimization

problem including damping and stiffness considerations. With volume constraint and

stiffness constraints setting in the maximizing composite damping optimization object,

the optimization problem can be stated as:

(20)

Table4Optimized results for maximizing damping of composites with stiffness constraint

E* Unit cell 3x3 Unit cells Elasticity matrix (GPa) f

20 0.43

30 0.26

40 0.11

50 0.03

For maximizing the damping, the optimized results of the microstructures of

composites under different E* are shown in table 4. material 1(stiff and elastic,

colored by red) is distributed at the center and eight corners of the unit cell, while

material 2(soft and viscoelastic, colored by black) is embedded in it. With the E*

increasing, the volume fraction of material 2 is decreased and the region of isolation

material 1 is getting smaller. The resulting storage modulus showed in table 4 is very

close to the corresponding constraint values.

4.4. Examples for maximizing damping with stiffness and volume constraints

Taking into account the cost of materials, volume constraint is added to the

optimization problem, which can be stated as:

(21)

Table 5Optimized results for maximizing damping of composites with stiffness and volume constraint

E* Unit cell 3x3 Unit cells Elasticity matrix(GPa) f

10 0.53

15 0.19

20 0.12

25 0.07

We set the volume constraint for material 1 at 50%. As the E* increases, the

connection between the different parts of material 1 becomes stronger. When the E*

=10Gpa, material 1 is wrapped in soft and viscoelastic material 2. material 1

beginning grows on the boundary of the unit cell when the E* =15Gpa. As E*

continues to increase, material 1 keeps staying on the boundary of the unit cell and

becomes more and more crowded and strongly connected. Finally, the resulting

storage modulus is also very close to the corresponding constraint values.

5. Conclusions

In this paper, we have extended the topology optimization algorithm to designing

3D microstructures of composites. The objective function is defined and the

optimization problem becomes to find the microstructure of the viscoelastic

composites with maximum damping/stiffness at the specified operation frequency.

Several numerical design examples are presented. Design constraints include

volume fraction, operation frequency, and stiffness (the real part of the effective

complex moduli). The examples fully show that this topology optimization method is

effective for obtaining the clear 3D microstructures of viscoelastic composites with

our desired properties. Through microstructural optimization, only a small amount of

viscoelastic/elastic material can be mixed to increase the damping/stiffness of the

composites significantly.

From the results of the four numerical examples in Sec. 4, we come to a general

rule to constructing ideal viscoelastic composites, that is, to get composites with high

damping, we should make the viscoelastic material cut off the stiff material in the unit

cell, and to get composites with high stiffness, the stiff material should be joined

together and run through the composite to form a supporting skeleton.

Declarations- Availability of data and materials

All data generated or analysed during this study are included in this published

article.

- AcknowledgmentsThis work is supported partially by the National Natural Science Foundation of

China (Grant No. 11847090).

- Conflict of interest statementOn behalf of all authors, the corresponding author states that there is no conflict of

interest.

References

[1] Rao M. D. (2003) Recent applications of viscoelastic damping for noise control in

automobiles and commercial airplanes. Journal of Sound and Vibration, 262(3),

457-474.

[2] Fan R., Meng G., Yang J. & He C. (2009) Experimental study of the effect of

viscoelastic damping materials on noise and vibration reduction within railway

vehicles. Journal of Sound and Vibration, 319(1-2), 58-76.

[3] Nakra B. C. & Dig S. V. (1984) Vibration control with viscoelastic

materials-III. The Shock and Vibration Digest: A Publication of the Shock and

Vibration Information Center, Naval Research Laboratory.

[4]Lakes R. S. (2002) High damping composite materials: effect of structural

hierarchy. Journal of Composite Materials, 36(3), 287-297.

[5]Ludwigson M. N., Lakes R. S. , & CC Swan. (2002) Damping and stiffness of

particulate sic-insn composite. Journal of Composite Materials, 36(19), 2245-2254.

[6] Jaglinski T. , & Lakes, R. . (2012). Zn-al-based metal-matrix composites with high

stiffness and high viscoelastic damping. Journal of Composite Materials, 46(7),

755-763.

[7]Chung, D. (2001) Review: materials for vibration damping. Journal of Materials

Science, 36(24), 5733-5737.

[8]Bendsøe, M. P. & Kikuchi N. (1988) Generating optimal topology in structural

design using a homogenization method. Computer Methods in Applied Mechanics

and Engineering, 71(2): 197-224.

[9] Bendsøe M. P. (1989) Optimal shape design as a material distribution

problem. Structural optimization, 1(4), 193-202.

[10]Zhou M. & Rozvany G.I.N. (1991) The COC algorithm, part II: Topological,

geometrical and generalized shape optimization. Computer Methods in Applied

Mechanics and Engineering, 89(103): 309–336.

[11]Díaz A R & Bendsøe M. P. (1992) Shape optimization of structures for multiple

loading conditions using a homogenization method. Structural & Multidisciplinary

Optimization, 4(1): 17-22.

[12]Bendsøe, M. & Sigmund O. (2004) Topology Optimization: Theory, Methods

and Applications, Springer-Verlag, Berlin.

[13]Rietz A. (2001) Sufficiency of a finite exponent in SIMP (power law) methods.

Structural & Multidisciplinary Optimization, 21(2): 159-163.

[14]Sethian J. A. &Wiegmann A. (2000) Structural boundary design via level set

and immersed interface methods. Journal of Computational Physics, 163(2): 489-528.

[15]Wang M. Y., Wang X, Guo D. (2003) A level set method for structural topology

optimization. Computer Methods in Applied Mechanics and Engineering,

192:227–246.

[16]Xie Y M, Steven G. P. (1993) A simple evolutionary procedure for structural

optimization. Computers & Structures, 49(5): 885-896.

[17]Querin O. M., Young V., Steven G P, et al. (2001) Computational efficiency and

validation of bi-directional evolutionary structural optimisation. Computer Methods in

Applied Mechanics and Engineering, 189(2): 559-573.

[18]Huang X. & Xie Y.M. (2010) Evolutionary Topology Optimization of Continuum

Structures: Methods and Applications, John Wiley & Sons, Chichester.

[19]Huang X. & Xie Y. M. (2007) Convergent and mesh-independent solutions for

the bi-directional evolutionary structural optimization method, Finite Elements in

Analysis and Design, 43(14): 1039-1049.

[20]Sigmund O. (1994) Materials with prescribed constitutive parameters: an inverse

homogenization problem. International Journal of Solids and Structure, 31(17),

2313–2329.

[21]Sigmund O. (1995) Tailoring materials with prescribed elastic proper ties.

Mechanics of Materials, 20(4):351–368.

[22]Yi Y. M., Park, S. H., & Youn, S. K. (1998). Asymptotic homogenization of

viscoelastic composites with periodic microstructures. International Journal of Solids

and Structures, 35(17), 2039-2055.

[23]Yi, Y. M., Park, S. H., & Youn, S. K. (2000). Design of microstructures of

viscoelastic composites for optimal damping characteristics. International Journal of

Solids and Structures, 37(35), 4791-4810.

[24]Huang X. D., Zhou S. W., Sun G. Y., et al. (2015) Topology optimization for

microstructures of viscoelastic composite materials. Computer Methods in Applied

Mechanics and Engineering, 283: 503–516.