Structural and Multidisciplinary Optimization https://doi.org/10.1007/s00158-018-2009-0
RESEARCH PAPER
Concurrent topology optimization design of structures and non-uniform parameterized lattice microstructures Chuang Wang 1 & Ji Hong Zhu 1,2,3
&
Wei Hong Zhang 1 & Shao Ying Li 1 & Jie Kong 4
Received: 8 January 2018 / Revised: 9 May 2018 / Accepted: 13 May 2018 # Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract This paper presents a novel concurrent topology optimization approach for finding the optimum topologies of macrostructures and their corresponding parameterized lattice microstructures in an integrated manner. Considering the manufacturability of the structure designs and computational efficiency, additional parameters are introduced to define the microstructure unit cell patterns and their non-uniform distribution, which avoids expensive iterative numerical homogenization calculations during topology optimization and results in an easier modelling of structure designs as well. It is worth mentioning that the equivalent properties of material microstructures serve as a link between the macro and the micro scale with the help of homogenization theory and the Porous Anisotropic Material with Penalization (PAMP) model. Besides, sensitivities of global structure compliance with respect to the pseudo-density variables and the microstructure parameter variables are derived, respectively. Moreover, several numerical examples are presented and reasonable solutions have been obtained to demonstrate the efficiency of the proposed method. Finally, mechanical testing is conducted to investigate the better performance of the optimized structure which is fabricated by 3D printing. Keywords Topology optimization . Concurrent design . Parameterized microstructures . Multi-scale . Homogenization
1 Introduction Structural optimization is a classical engineering discipline of modifying the design of a structure in order to improve its performance with respect to some desirable behaviors, e.g. stiffness and strength (Alexandersen 2016). Topology optimization was employed for structural optimization by Bendsøe and Kikuchi (1988) via the homogenization approach and has * Ji Hong Zhu
[email protected] 1
State IJR Center of Aerospace Design and Additive Manufacturing, School of Mechanical Engineering, Northwestern Polytechnical University, Xian 710072, Shaanxi, China
2
MIIT Lab of Metal Additive Manufacturing and Innovative Design, NPU-QMUL Joint Research Institute, Northwestern Polytechnical University, Xian 710072, Shaanxi, China
3
Unmanned System Technologies, Northwestern Polytechnical University, Institute of Intelligence Material and Structure, Xian 710072, Shaanxi, China
4
Shaanxi Key Laboratory of Macromolecular Science and Technology, School of Science, Northwestern Polytechnical University, Xian 710072, Shaanxi, China
been developing rapidly due to the ability to find unintuitive and unanticipated designs over shape and size optimizations. Various topology optimization methods have been proposed over the last three decades, such as solid isotropic material with penalization (SIMP) (Bendsøe 1989; Zhou and Rozvany 1991; Rozvany et al. 1992), evolutionary structural optimization (ESO) (Xie and Steven 1993, 1996; Li et al. 2004; Huang and Xie 2007), level set method (LSM) (Wang et al. 2003; Yulin and Xiaoming 2004), moving iso-surface threshold method (MIST) (Tong and Lin 2011) and featuredriven method (Zhou et al. 2016). Comprehensive reviews on the development and applications of topology optimization can be seen in the literatures (Sigmund and Maute 2013; Deaton and Grandhi 2014; Zhu et al. 2016b). Topology optimization has also been extended to design material microstructures with prescribed or extreme properties, such as bulk modulus maximization, negative Poisson’s ratio or zero thermal expansion coefficients. The pioneering work of manipulating materials with specified properties was proposed by Sigmund (1994) via an inverse homogenization approach. This is followed by massive scholars to launch their works in materials design. Silva et al. (1997) designed periodic linear elastic microstructures for optimal properties subject
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to the constraint of material volume fraction. Nelli Silva et al. (1998) proposed an optimal design method for piezoelectric materials requiring an improvement in performance characteristics. Zhang et al. (2007) optimized the mechanical properties of material unit cells using topology optimization and strain energy-based method. Huang et al. (2011) extended the bidirectional evolutionary structural optimization (BESO) method to design microstructures of cellular materials for maximum bulk or shear modulus. Most of these studies are focused on how to obtain prescribed or the extreme properties of material microstructures without considering the specified macroscopic structures. As we all know, a structure under different loading and boundary conditions requires different properties of materials for best mechanical performance. Hence, the structure composed of these optimal material microstructures may be not as efficient as expected. Recent research progresses on multiscale structural design methods enable the consideration of microscale material heterogeneities when pursuing high-performance macroscale structures. Compared with the design of mono-scale structures, an ideal multiscale design should be the structure which has the optimal topologies at macro-level and micro-level simultaneously (Chen et al. 2017). Regarded as a promising approach for concurrent design, multi-scale topology optimization has received great attentions and became a research hotspot in recent years. Rodrigues et al. (2002) proposed a hierarchical optimization model for optimizing material distribution at two scales. In this model, it underlined the designs of the microstructures and allowed the material variables to vary from point to point. This work was then extended to three-dimension structures by Coelho et al. (2008). Zhang and Sun (2006) studied the integrated optimization of lightweight cellular materials and structures considering scale-related effects. Furthermore, Xia and Breitkopf (2014) introduced a nonlinear multi-scale framework into concurrent material and structure design. In this work, the microstructures which were optimized to adapt the macroscopic structural physical response are defined at each element’s Gauss integration point. Although ideal optimal designs can be obtained by means of above mentioned methods, the high computational cost restricts theirs application range. Moreover, excessively flexible designs of microstructures bring expensive computation in optimization design and difficulties in the fabrication. On one hand, to circumvent the limitation of intensive computational cost, Xia and Breitkopf (2015a) constructed a reduced database model viewing the local material optimization process as a generalized constitutive behavior using separated representations. On the other hand, considering the manufacturability of the structure designs, Liu et al. (2008) proposed a concurrent topology optimization method which assumed that the macrostructure was composed of uniform microstructures to satisfy the requirements of manufacturing. Following this idea, Gao and Ma (2015) proposed a modified model for concurrent optimization by introducing microstructure orientation as a new type of design
variable. Furthermore, various topology optimization methods, such as the bidirectional evolutionary structural optimization method (Yan et al. 2014) and moving iso-surface threshold method (MIST) (Chen et al. 2017), etc., are applied for concurrent design of structures and materials using a uniform microstructure throughout the macro domain. These approaches are easily applied to wider range of problems by assuming the uniformity of material microstructures in the macro domain and result in easier manufacturing process of the designs. However, local tailoring of material properties is not possible and thus it limits potential benefits (Sivapuram et al. 2016). For more details, please refer to the recent research papers (Long et al. 2017; Wang et al. 2017; Li et al. 2018) and review (Xia and Breitkopf 2017), which summarize and highlight the developments and applications of multiscale topology design methods. To improve the performance of structures with lower computational costs and overcome the limitation of manufacturing difficulties in the existing multiscale structural designs, this paper presents a two-scale topology optimization approach for concurrently designing the macrostructures and their corresponding material microstructures. The approach combines two different viewpoints, i.e., “0–1” topology optimization on the macro scale and non-uniform lattice materials on the micro scale together. In order to achieve such a combination, two kinds of design variables are employed, including the topological design variables and the microstructure parameter variables. The two design variables are integrated into a common objective function for a minimum global compliance problem in two dimensions. Such a method will give more degrees of design, larger admissible design space, and will result in better design results with limited computational resources. Meanwhile, the structure designs will hold the advantage of being easy to model and manufacture. It is worth mentioning that, in the present work, the macrostructures are assumed to be consist of non-uniformly distributed lattice microstructures but with similar unit cell patterns. The distribution of the parameterized microstructures is controlled by new introduced additional parameters. In this way, the number of design variables sharply decreases and the challenges of high computational effort are alleviated immensely compared with other two-scale topology optimization approaches. The remainder of this paper is organized as follows. The homogenization method and the equivalent properties of parameterized lattice microstructures are first presented. The mathematical formulations of the concurrent optimization problemisthengivenindetail.Thereafter,asensitivityanalysis is conducted with respect to two kinds of design variables and the optimizationalgorithm isdescribed.Thenseveralnumerical examples are provided to demonstrate the efficiency of the proposedmethod.Mechanicaltestingisnextconductedtovalidate the better performance of the optimal structure. Conclusionsaregivenintheend.
Concurrent topology optimization design of structures and non-uniform parameterized lattice microstructures
x
Fig. 1 Lattice material and microstructure unit cell
x
2 Homogenization of lattice materials with parameterized microstructures Lattice materials are widely used in aerospace engineering, automotive engineering and biomedical engineering because of their outstanding strength-to-weight ratios and flexible designability. As shown in Fig. 1, the lattice microstructure mainly used in this paper is formed by a square frame with four branches, which possess superior in-plane mechanical properties (Wang and McDowell 2004). Assuming that the size of the microstructure unit cell is 1 × 1, the parameter x denotes the width of the horizontal and vertical branches. On the premise that the material volume fractions of different unit cells keep consistent, the microstructure unit cells with parameter x ranging 0 to 0.20 are depicted in Fig. 2. To design the distribution of non-uniform lattice microstructures, the parameter x is chosen as one kind of design variable and spatially varying during optimization. Therefore, it is cumbersome to perform numerical homogenization in each design iteration. Instead, we can calculate the effective elastic tensor DH of typical microstructure unit cells with gradient varying parameter x, and interpolate between them with explicit functions.
The constitutive behaviors of lattice materials are mainly governed by the microstructure unit cell pattern, which can be approximately replaced by an equivalent homogeneous medium. It satisfies the following conditions in the progress of homogenization: the stress and the strain tensors of the homogeneous medium are equivalent to the average stressσ and the average strainε of the microstructure unit cell (Zhang et al. 2007). σ¼
1 ∫Ω σdΩ jΩj
ð1Þ
ε¼
1 ∫Ω εdΩ jΩj
ð2Þ
where Ω denotes the volume of the microstructure unit cell. The average stress and strain of the microstructure unit cell follow the Hooke’s law σ¼DH ε
ð3Þ
where DH is the effective elastic tensor of the microstructure unit cell.
Fig. 2 Microstructure unit cells with gradient varying parameter x
x=0
x = 0.02
x = 0.04
x = 0.06
x = 0.08
x = 0.12
x = 0.14
x = 0.16
x = 0.18
x = 0.20
X2
C. Wang et al.
As illustrated in Fig. 2, the microstructure unit cells remain symmetric with the change of parameter x. Hence, the effective elastic matrix DH can be written as 2
D11 DH ¼ 4 D21 0
D12 D22 0
3 0 0 5 D33
ð4Þ
It is assumed that the property constants of the solid material which constructs the microstructure unit cells are Young’s modulus E = 1000 MPa and Poisson’s ratio μ = 0.3. The effective elastic tensor DH of the microstructure unit cell, which depends on the parameter x, can be obtained using the energy-based homogenization approach (Xia and Breitkopf 2015b). In this method, the microstructure unit cell is discretized into N finite elements and the effective elastic tensor DH is written in terms of element mutual energies Dij ¼
Table 1
1 N AðiÞ T ke uAe ð jÞ ∑ u jΩj e¼1 e
ð5Þ
where uAe ðiÞ are the elements displacement solutions corresponding to the unit test strain field ε0(i)(in 2D problem there are three: i.e., unit strain in the horizontal direction, unit strain in the vertical direction and unit shear strain) and ke is the element stiffness matrix. Table 1 shows the effective elastic tensor DH of the typical microstructure unit cells with parameter x ranging 0 to 0.20. The resulting curves for the 4 unique indices of the effective elastic tensor DH are shown in Fig. 3, where cubic polynomial interpolation is applied to obtain the values between the data-points.
3 Problem statement and optimization formulations 3.1 Concurrent topology optimization of structures and lattice materials In concurrent topology optimization, two kinds of design variables are employed, including the topological
The effective elastic tensor of typical microstructure unit cells
Parameter x
x=0
x=0.06
x=0.10
x=0.14
x=0.20
Microstructures
The effective elastic tensor DH (MPa)
Concurrent topology optimization design of structures and non-uniform parameterized lattice microstructures
105
300
90
280
75
D12
D11
320
260
60
240
45
220
0.00
0.05
0.10
0.15
30
0.20
0.00
0.05
0.10
0.15
0.20
0.15
0.20
Parameter x
320
100
300
80
280
D33
D22
Parameter x
260
60
40
240 20 220
0.00
0.05
0.10
0.15
0.20
0.00
0.05
0.10
Fig. 3 The indices of the effective elastic tensor DH for different x values
design variables describing spatial material layout within the macrostructure design domain, and the microstructure parameter variables defining patterns of underlying material microstructures at different locations, which can be defined as
X ¼ fη1 ; ⋯; ηn ; x1 ; ⋯; xn gT
ð6Þ
where ηi(i = 1, 2, ⋯, n) denotes the pseudo-density value for the i-th element andxi (i = 1, 2, ⋯, n)is the parameter x value of the underlying material microstructure for the i-th element. The configuration of the microstructure is controlled by only one parameter x and the effective elastic tensor DH of the microstructures with varying parameter x can be obtained from the polynomial interpolation curves shown in Fig. 3. Therefore, it is avoided to perform the numerical homogenization for calculating the effective properties of the material microstructures in each design iteration.
An illustrative optimized structure is shown in Fig. 4. At the macro scale, the “blue domain” is material distribution area and the “white domain” is void area. The levels of the “blue” denote the variation of pseudodensity variables. At the micro scale, the “grey domain” is composed of parameterized lattice microstructures and the greyscale denotes the variation of microstructure parameter variables. Using the PAMP material model (Liu et al. 2008) at the of the i-th macro element macro scale, the elastic matrix DMA i can be expressed as ¼ ηpi DH ðxi Þ DMA i
ð7Þ
where DH (xi) is the parameterized effective elastic matrix of the microstructure when the parameter xi is applied. p is the penalty factor of pseudo-density variable and is set to be 4 in this study. The effective elastic matrix DH (xi) which serves as a link between the macro and the micro scale can be obtained from the interpolation curves shown in Fig. 3.
C. Wang et al. Fig. 4 An optimized structure composed of parameterized lattice materials
(a) Design topology and element representation by pseudo-density variables
(b) Design topology and element representation by microstructure parameter variables The global stiffness matrix K of the macrostructure assembled by the element stiffness matrix Ki can be written as n
n
i
i
K ¼ ∑ Ki ¼ ∑ ∫Ωi BT DMA i BdΩi
ð8Þ
where B is the strain matrix, DMA i is defined in (7) as a function H of D (xi), n is the total number of elements within the macro design domain, Ωi denotes the domain of the i-th element at the macro scale.
behavior of the macrostructure. V is the material volume fraction of the macro design domain with an upper limit of V. ηmin is a small positive value as an lower bound of pseudo-density, e.g., 0.001. xmin and xmax are the lower value and upper value of the parameter x of the proposed material microstructures. In this paper, xmin is set to 0 and xmax is set to 0.20, respectively.
4 Sensitivity analysis and solution algorithm 4.1 Sensitivity analysis
3.2 Formulation of optimization The concurrent optimization model can be mathematically expressed as find : X ¼ fη1 ; ⋯; ηn ; x1 ; ⋯; xn gT min : C ¼ FT U ¼ UT KU subject to : KU ¼ F V ≤V 0 < ηmin ≤ ηi ≤ 1; i ¼ 1; ⋯; n xmin ≤ xi ≤ xmax ; i ¼ 1; ⋯; n
Considering the governing equation for static behavior of the macrostructure in (9), the differentiation with respect to the pseudo-density variable ηi can be written as ∂K ∂U ∂ F UþK ¼ ∂ηi ∂ηi ∂ηi
ð10Þ
ð9Þ
where C denotes the global compliance of the macrostructure. U and F are the nodal displacement and force vectors of the macrostructure. KU = F is the governing equation for static
Assuming F = f + G, where f and G denote design independent external loads and design dependent internal loads, respectively, we can have ∂ F ∂G ¼ ∂ηi ∂ηi
ð11Þ
Concurrent topology optimization design of structures and non-uniform parameterized lattice microstructures
Then the derivative of the global structure compliance can be expressed as ∂C ∂U ∂K ¼ 2UT K þ UT U ∂ηi ∂ηi ∂ηi
ð12Þ
Substituting (10) and (11) into (12), we yield ∂C ∂G ∂K ¼ 2UT −UT U ∂ηi ∂ηi ∂ηi ∂G ∂ηi
ð13Þ
can be easily obtained from the design dependent inter-
nal loads. When the design independent external loads are applied separately, (13) can be simplified as ∂C ∂K ¼ −UT U ∂ηi ∂ηi
ð17Þ
f ¼1
The convolution operator (weight factor) H^f can be written as H f ¼ rmin −distðe; f Þ;
ð18Þ
f f ∈njdistðe; f Þ≤ rmin g; e ¼ 1; …; n
ð15Þ
where Ci is the compliance of i-th element at the macro scale. It is similar to (10) ~ (15), the sensitivity of global structure compliance with respect to the parameter x variable xi can be expressed as n ∂C ∂Ki ¼ − ∑ Ui T Ui ∂xi ∂xi i¼1 n ∂ ¼ − ∑ Ui T ∫Ωi BT DMA i BdΩi Ui i¼1 ∂xi H p T T ∂D ðxi Þ ¼ −ηi Ui ∫Ωi B BdΩi Ui ∂xi
n ∂C 1 ∂C ¼ ∑ H f ηf n ∂ηe η ∑ ∂η f H f f ¼1 e
ð14Þ
Considering (7) and (8), the sensitivity of global structure compliance with respect to the pseudo-density variable ηi can be written as n ∂C ∂Ki ¼ − ∑ Ui T Ui ∂ηi ∂ηi i¼1 n ∂ ¼ − ∑ Ui T ∫Ωi BT DMA i BdΩi Ui ∂ηi i¼1 p T ¼ − Ui ∫Ωi BT DMA i BdΩi Ui ηi p p⋅C i ¼ − Ui T Ki Ui ¼ − ηi ηi
material distribution is used and all the pseudo-density variables are set to 0.4. Analogously, at the micro scale, the initial value of the microstructure parameter variable xi is set to 0.10. After finite iterations, the two kinds of design variables could converge to the optimal values. To suppress the checkerboard pattern and eliminate mesh dependency in the concurrent topology optimization problem, a filtering technique (Sigmund and Petersson 1998) is used for the pseudo-density variables at the macro scale. We can obtain the filtered sensitivities by modifying the element sensitivities as follows
ð16Þ
where the derivative of the effective elastic matrix DH(xi) can be easily obtained according to the polynomial interpolation curves shown in Fig. 3.
4.2 Optimization algorithm and filtering Based on the above sensitivity analysis with respect to two design variables at both scales, the concurrent optimization problem (9) can be solved by means of a gradient-based optimization algorithm. In the present work, the globally convergent version of moving asymptotes (GCMMA) optimization solver (Svanberg 1987) is used. An initial design is required to start the optimization procedure. At the macro scale, a uniform
where the operator dist(e, f) is defined as the distance between center of element e and element f. The convolution operator H^f is zero outside the filter area. The convolution operator decays linearly with the distance from element f. Now, we have established the mathematical formulations for the concurrent optimization model and completed the sensitivity analysis of the objective function with respect to two kinds of design variables at macro and micro length scales. Figure 5 shows the flow chart for the concurrent optimization method with each key step. It should be mentioned that, although the microstructures presented in Fig. 2 are used throughout the paper, the proposed optimization scheme is a common method and does not limit the application of different microstructures with different parameterizations.
5 Numerical examples In this section, we presented two examples for simultaneously designing the macrostructures and the underlying material microstructures to illustrate and validate the proposed method. It is assumed that the Young’s modulus for the solid material which constructs the lattice materials is 1000 MPa and the Poisson’s ratio is 0.3. The filter radius is set to two times of the average element size for the pseudo-density variables to suppress the checkerboard pattern and eliminate mesh dependency. Besides, the optimization problems are solved by means of the optimization algorithm GCMMA built within the Platform of Boss-Quattro (Radovcic and Remouchamps 2002).
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Fig. 5 Flow chart for concurrent topology optimization
5.1 MBB beam We consider an MBB beam problem as shown in Fig. 6 as the first example. The geometric parameters of the beam are L = 0.96 m and H = 0.25 m. A uniform pressure load P = 0.2 MPa is applied at the center of the top edge and the length of the loading domain is d = 0.3 m. Due to the symmetry of the problem, only the right half part of the beam is considered as the macro design domain which is discretized into a mesh of 48 × 25 elements. It is assumed that the MBB beam is composed of lattice materials with parameterized microstructures as shown in Fig. 2. In this example, we will design the macrostructure and the underlying material microstructures simultaneously. The objective is to minimize the global
compliance of the macro design domain with a volume constraint of 40%. Fig. 7 plots the iteration histories of the global compliance and volume fraction of the MBB beam. The value of the global compliance decreases quickly before the 25th iteration and then keeps decreasing until it converges to the final value C = 261.54 J after 80 iterations. Obviously, the value of volume fraction keeps quite stable during the iterations. The optimized topologies of macrostructure and material microstructures are given in Fig. 8. The non-uniform microstructures are used at different positions in the design domain. For example, the microstructure with the parameter x = 0.2 is applied for the typical point A, which possesses superior mechanical properties in horizontal direction. The shear resistant microstructure with the parameter x = 0 is used for the typical point B. Besides, the microstructures with different parameter x are applied for the transitional regions, such as typical points C ~ E. In order to show the advantages of the method proposed in this study, we consider the following five solutions as comparisons. It is assumed that the MBB beam is composed of uniform lattice microstructures. We will solve the problem shown in Fig. 6 at single macro-scale using the microstructure with constant value of the parameter x in every solution. Table 2 lists two-scale structure topologies and the objective C values for the five solutions. The material microstructures greatly affect the macro structure design and the objective C values in turn. Obviously, Solution C has the smallest objective value, 39.3% lower than Solution E which has the largest objective value. It generally proves that the microstructure in Solution C have balanced mechanical properties. But compared with the concurrent design, the global compliance of the structures obtained by solution A ~ E turn out to be less optimal, as shown in Fig. 9. The optimized design shown in Fig. 8 possesses better stiffness with the objective value C = 261.54 J, 16.2% lower than Solution C and 49.1% lower than Solution E. Furthermore, we also consider the traditional optimal design using solid material only by assuming with the same weight as comparison. Fig. 10 shows the optimized structural topology of the solid material design and the objective C d
Fig. 6 MBB beam with loading and boundary conditions
P
Macro design domain
L
H
Concurrent topology optimization design of structures and non-uniform parameterized lattice microstructures 1.0 Global structure compliance Volume fraction
6000
features, e.g., good energy absorption characteristics and good heat dissipation performance, by using lattice microstructures.
0.8
5000 0.6
4000 3000
0.4
2000
261.54
1000
0.2
0 0
20
40
60
Volume fraction
Global structure compliance (J)
7000
80
0.0
Iteration
Fig. 7 Iteration histories of global compliance and volume fraction of MBB beam
converges to 225.09 J. It can be concluded that the structural stiffness is slightly better than the optimized design shown in Fig. 8. This is mainly due to the limitations of configurations and fixed volume fraction of the microstructure unit cells used in this work. Even though the stiffness of the concurrent design is slightly lower than the solid material design, it can benefit from the improvement of other high performance
5.2 L-shaped beam In this example, the two-scale optimized design for the Lshaped beam as shown in Fig. 11 is considered. The geometric parameters of the L-shaped beam are L1 = 1.0 m, L2 = 0.7 m, H1 = 1.0 m and H2 = 0.7 m. The upper edge is fixed and a concentrated vertical load F = 10 KN is applied at the middle of the right edge. For simplicity, a uniform mesh with element size 0.01 m × 0.01 m is assigned to the macro design domain. Fig. 12 plots the iteration histories of the global compliance with the volume fraction of 40%. The global compliance has a significant drop before the 20th iteration and then keeps decreasing until it converges to a final value C = 193.04 J after 88 iterations. It is obvious that the volume fraction keeps quite stable throughout the whole iterations. Fig. 13 shows the optimized design of the macrostructure and its material microstructures’ distribution. It can be seen that point A is composed of the material with the high stiffness in horizontal and vertical directions so as to improve the bending stiffness of the structure. The shear resistant material microstructure with the parameter x = 0 is selected for the typical
Fig. 8 Optimized macrostructure and its material microstructures for the MBB beam
(a) Structural topology on the macro scale
(b) Material distribution on the micro scale
C. Wang et al. Table 2
Topologies of micro and macro structures and objective values with various parameter x
Solution
Parameter x
A
0
344.82J
B
0.05
316.09J
C
0.10
312.22J
D
0.15
328.19J
E
0.20
513.98J
Microstructures
Global structure compliance (J)
B. As for point C ~ G, different material microstructures which have a balance among tensile strength, compressive strength and shear strength are used. These results bring the improvement of stiffness to the structure.
513.98
500
Macrostructures
Objective C
Fig. 14 shows the optimized designs with different volume fractions. It is obvious that as the value of volume fraction increases, the global compliance decreases. Nevertheless, the topologies of macrostructures and spatial distribution of microstructures have no significant difference among various volume fractions. Similar to the previous MBB beam problem, the following five solutions are used to solve the L-shaped beam problem
261.54 400 344.82 300
316.09
312.22
328.19
Solid material 200
100
0
A
B
C
D
E
Solution Fig. 9 Comparison of global compliances of different optimized structures of the MBB beam
Fig. 10 Optimized structural topology using solid material for the MBB beam
Concurrent topology optimization design of structures and non-uniform parameterized lattice microstructures
H2 H1 L2
Macro design domain F
(a) Structural topology on the macro scale
L1 Fig. 11 L-shaped beam with loading and boundary conditions
shown in Fig. 11 with the volume fraction of 40%. Table 3 lists two-scale structure topologies and the objective C values for five solutions with various parameter x values. From the results in Table 3, it can be seen that optimized topologies of macrostructures have significant differences among the solution A ~ E. Especially, the macrostructure obtained by solution E is drastically different from others. This is due to the weakness of shear strength of the material microstructure used in solution E. With the arguments above, we can safely come to the conclusion that the material microstructures greatly affect the macro structure design. Furthermore, the performance of the designs depends on the used material microstructures. It can be seen that Solution D has the smallest objective value, which is 21.6% lower than Solution E with the largest 1.0 Global structure compliance Volume fraction
5000
0.6 3000 0.4
1000
0.2
193.04
0 0
20
40
60
Fig. 13 Optimized macrostructure and its material microstructures for the L-shaped beam
0.8
4000
2000
(b) Material distribution on the micro scale
Volume fraction
Global structure compliance (J)
6000
80
0.0
Iteration
Fig. 12 Iteration histories of global compliance and volume fraction of Lshaped beam
objective value. It generally proves that the microstructure in Solution D have balanced mechanical properties. Fig. 15 shows the global compliance of the structures obtained by solution A ~ E. And the dash line represents the global compliance of the structure shown in Fig. 13, which possesses better stiffness with the objective value C = 193.04 J. It is 15.1% lower than Solution D and 33.5% lower than Solution E. Moreover, we also consider the traditional optimal design using solid material only by assuming with the same weight as comparison. Fig. 16 shows the optimized structural topology
C. Wang et al.
Global structure compliance (J)
400
193.04 300
240.17
Table 3
228.37
227.44
C
D
200
100
0
Fig. 14 Optimized designs versus different volume fractions and their global structure compliance
290.26
274.03
A
B
E
Solution Fig. 15 Comparison of global compliances of different optimized structures of the L-shaped beam
Topologies of micro and macro structures and objective values with various parameter x
Solution
Parameter x
A
0
274.03J
B
0.05
240.17J
C
0.10
228.37J
D
0.15
227.44J
E
0.20
290.26J
Microstructures
Macrostructures
Objective C
Concurrent topology optimization design of structures and non-uniform parameterized lattice microstructures
6 Experimental validation
Solid material
Fig. 16 Optimized structural topology using solid material for the Lshaped beam
of the solid material design and the objective C converges to 170.05 J. Similar to the previous MBB beam problem, the solid material design can achieve slightly better stiffness compared with the concurrent design using lattice microstructures.
(a)
(b)
In this section, we intend to validate the performance improvement of the structure designs obtained by proposed concurrent topologyoptimizationmethod.Here,wechoosestereolithography based on 3D printing technology to fabricate the structures designs due to its high processing efficiency and low cost. The unique capability of 3D printing provides an opportunity to manufacture highly complex structures that are obtained by topology optimization (Gu et al. 2012; Zhu et al. 2016a). We take the optimized MBB beam design shown in Fig. 8 as experimental subject. In addition to the optimal design, other two designs with uniform lattice material (taken from Table 2) and a traditional optimal design using solid material are selected for comparison. The reconstructed models make some appropriate simplification compared with the optimized designs. Fig. 17 shows the geometric models of the specimens. As we all know, the microstructure unit cells should be infinitesimally small upon homogenization theory compared with the macroscopic structures. However, due to the limits of the 3D printer’s resolution, the size of the microstructure unit cells is set as 6.4 mm × 6.4 mm. The length, height and thickness of the models are 307.2 mm, 80 mm and 16 mm, respectively. The four MBB beam demonstrators were fabricated using a SPS350B 3D printer with SPR6000B epoxy resin material in same condition and same print direction. The material is a low viscosity about 355 cps at 28 °C liquid photopolymer that produces strong, tough, water-resistant parts. Table 4 lists the specific information about SPR6000B epoxy resin material. Fig. 18 shows the physical models of the specimens fabricated by stereolithography. The static structure tests were performed with a 5kN TestResources™ Testing machine under quasi-static conditions (see Fig. 19). A digital camera was used to record the deformation of the specimen structures. The loading rate was set as 3.6 mm/min. All the specimen structures were tested under the same condition.
Table 4
(c)
(d) Fig. 17 Specimen designs: (a) uniform microstructure with parameter x = 0.10, (b) uniform microstructure with parameter x = 0.20, (c) optimized design with mixed parameterized microstructures, (d) optimized design with solid material
The physical characteristics of SPR6000B epoxy resin
Property
Test standard
Magnitude
Tensile modulus Tensile strength Elongation at break Flexural modulus Flexural strength Glass transition (Tg) coefficient of thermal expansion Poisson’s ratio Density
ASTM D 638 ASTM D 638 ASTM D 638 ASTM D 790 ASTM D 790 DMA, E”peak TMA (T < Tg) – –
2189–2395 MPa 27–31 MPa 12–20% 2692–2775 MPa 69–74 MPa 62 °C 97 × 10−6/°C 0.39 1.16 g/cm3
C. Wang et al. structure (a) structure (b) structure (c) structure (d)
2500
Force (N)
2000
1500
1000
500
0
Fig. 18 The physical models of the specimens fabricated by stereolithography
The experimental force-displacement characteristics of all four MBB beam structures are shown in Fig. 20. Among the designs (structure a ~ c) using lattice microstructures, it can be seen that the force-displacement curve of the structure (c) has much bigger slope than the other two curves. It indicates that the structure obtained by the concurrent design has the better stiffness performance, followed by the structure (a) and then structure (b). Hence, the experimental results are in good agreement with the results of numerical simulation shown in Fig. 9. Moreover, the strength of structure (c) is even better
0
2
4
6
8
10
12
Displacement (mm) Fig. 20 Load-displacement curves of the specimen structures
with the bearable load increasing by 21.1 and 65.7%, compared with structure (a) and structure (b). Even though the structure (d) can achieve slightly better stiffness and strength, the fracture failure occurs earlier than the designs with lattice microstructures. Besides, it can be seen that there is a long plastic plateau after the bearable load in the forcedisplacement curves of the structure (a) ~ (c). It is the long plastic plateau that makes the designs with lattice microstructures particularly attractive for the purpose of impact protection, as it contributes the majority of the energy absorption.
7 Conclusions
Fig. 19 The static loading test of optimized structures
In this work, we developed a two-scale concurrent optimization approach for the macrostructures and the underlying lattice microstructures. Additional parameters are introduced to define the non-uniform distribution and the microstructure unit cell patterns in a parameterized way. The effective properties of material microstructures are obtained by using homogenization theory before the optimization. As a result, it avoids performing the numerical homogenization in each design iteration and the computational efficiency is remarkably improved. In this way, two kinds of design variables, i.e., topological design variables in the macro scale and microstructure parameter variables, are employed and the mathematical model of the concurrent topology optimization problem is thus established. Two numerical examples were presented to demonstrate the validity and advantage of the proposed method. Compared with the designs with uniform lattice microstructure, the performance of the concurrent designs is improved obviously. In addition, four MBB beam specimens were fabricated by 3D printing technology and then the mechanical tests were performed under quasi-static conditions. From the
Concurrent topology optimization design of structures and non-uniform parameterized lattice microstructures
experimental results, we can see that the optimized design obtained by concurrent topology optimization method has the better stiffness and strength compared with the structure designs using uniform lattice microstructure. Even though the sold material design can achieve slightly better stiffness and strength, the fracture failure occurs earlier than the designs with lattice microstructures. Besides, the long plastic plateau makes the designs with lattice microstructures particularly attractive for the purpose of impact protection. It is a valuable research field and it will guide our future research work. Acknowledgements This work is supported by National Key Research and Development Program (2017YFB1102800), NSFC for Excellent Young Scholars (11722219), Key Project of NSFC (51790171, 5171101743, 51735005, 11620101002). Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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