topology optimization for 3d microstructures of
TRANSCRIPT
Topology optimization for 3D microstructures ofviscoelastic composite materialsJianglin Yang
Jihua LaboratoryShiyang Zhang ( [email protected] )
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: [email protected]
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.