Topology optimization for microstructures of viscoelastic composite materials

The viscoelastic response of materials is often utilized for wide applications such as vibration reduction devices. This paper extends the bi-directional evolutionary structural optimization (BESO) method to the design of composite microstructure with optimal viscoelastic characteristics. Both storage and loss moduli of composite materials are calculated through the homogenization theory using complex variables. Then, the BESO method is established based on the sensitivity analysis. Through iteratively redistributing the base material phases within the unit cell, optimized microstructures of composites with the desirable viscoelastic properties will be achieved. Numerical examples demonstrate the effectiveness of the proposed optimization method for the design of viscoelastic composite materials. Various microstructures of optimized composites are presented and discussed. Meanwhile, the storage and loss moduli of the optimized viscoelastic composites are compared with available theoretical bounds. c ⃝ 2014 Elsevier B.V. All rights reserved.


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][2][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][10][11][12][13][14][15][16][17][18][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][18][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.

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] , . (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: , , 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.
where |Y| is the area of the unit cell. In Eq. (6), E H ijlk() 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: , where the form of isotropic material matrix E() and the stiffness matrix k ar e: , .

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: , .
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].

Statement of the optimization problem
The optimization problem of maximizing damping or stiffness at the operation frequency for viscoelastic materials can be defined as: .
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.

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: .
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).

Numerical examples and discussions
In    viscoelastic material 2 and 40% elastic material 1 is 92% of that of material 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 : 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.

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:  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.

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. 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.

-Availability of data and materials
All data generated or analysed during this study are included in this published article.

-Acknowledgments
This work is supported partially by the National Natural Science Foundation of China (Grant No. 11847090).

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