Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2022.017708

ARTICLE

Multi-Material and Multiscale Topology Design Optimization of Thermoelastic Lattice Structures

1State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, International Research Center for Computational Mechanics, Dalian University of Technology, Dalian, 116024, China

2Ningbo Research Institute of Dalian University of Technology, Ningbo, 315016, China

3Institute of Structural Health Monitoring and Control, School of Mechanics, Civil Engineering & Architecture, Northwestern Polytechnical University, Xi'an, 710072, China

*Corresponding Author: Zunyi Duan. Email: duanzy@nwpu.edu.cn

Received: 05 June 2021; Accepted: 24 August 2021

Abstract: This study establishes a multiscale and multi-material topology optimization model for thermoelastic lattice structures (TLSs) considering mechanical and thermal loading based on the Extended Multiscale Finite Element Method (EMsFEM). The corresponding multi-material and multiscale mathematical formulation have been established with minimizing strain energy and structural mass as the objective function and constraint, respectively. The Solid Isotropic Material with Penalization (SIMP) interpolation scheme has been adopted to realize micro-scale multi-material selection of truss microstructure. The modified volume preserving Heaviside function (VPHF) is utilized to obtain a clear 0/1 material of truss microstructure. Compared with the classic topology optimization of single-material TLSs, multi-material topology optimization can get a better structural design of the TLS. The effects of temperatures, size factor, and mass fraction on optimization results have been presented and discussed in the numerical examples.

Keywords: Multi-material design optimization; thermoelastic lattice structure; multiscale topology optimization; mass constraint; strain energy

Analysis and design of thermoelastic structures are an important part of structural design in many mechanical and aerospace engineering applications. Topology optimization [1–4] design is an effective strategy to improve the performance of the thermoelastic structure. Many studies had been carried out with considering structure failure caused by thermal stresses [5]. Rodrigues et al. [6] established a topology optimization model to design the 2D linear thermoelastic structure. Deng et al. [7] investigated a multi-objective topology design optimization of the thermoelastic structure. Zhang et al. [8], and Deaton et al. [5] studied the minimum strain energy and minimum weight design of the thermoelastic structure, respectively. Takalloozadeh et al. [9] provided a level-set topology optimization model subjected to the thermal loading. Takezawa et al. [10] optimized the structure considering the structural strength and thermal conductivity constraints.

However, most researchers only focused on single-material topology optimization. For example, Zhang et al. [11,12] studied the single-material topology optimization based on Moving Morphable component method. But the single-material structure design cannot meet the special functional requirements, such as negative thermal expansion structure [13], zero expansion structure [14], etc. In order to seek optimal structural performance, the topology design optimization of multi-material lattice structures is proposed. The multi-material structure topology optimization was firstly studied by Thomsen [15]. Then, Sigmund et al. [16] studied the topology design optimization of three-phase material structures using the SIMP interpolation scheme. By improving the SIMP scheme, the Discrete Material Optimization scheme [17], Shape Functions with Penalization scheme [18], and Bi-value Coding Parameterization scheme [19] were proposed. Multi-material topology optimization considering thermal loads was conducted by Gao et al. [20] and Giraldo-Londoño et al. [21]. A multiscale topology design optimization with multiple micro materials [22] was presented. Multi-material topology optimization subjected to multiple volume constraints was investigated by Zhang et al. [23]. López et al. [24] designed multi-material structures considering structural cost and manufacturability. Ye et al. [25] proposed an effective method to optimise multi-material structures with the minimum weight as the objective function.

Most of the above research focus on isotropic material, and few types of research consider the multi-material lattice structure. However, the lattice structure is more and more widely used in aerospace and civil engineering because the lattice structure with a variety of porous microstructures [26,27] has the advantages of a high ratio of stiffness [28–30]. A large number of micro-members in the lattice structure will make the modeling and analysis of the lattice structure difficult. Therefore, multiscale analysis models are utilized to achieve the analysis and optimization of the lattice structure. Zhang et al. [31] used the EMsFEM to build the analysis model for the lattice structure based on the MsFEM [32,33]. The concurrent multiscale optimization of theTLS subjected to thermal and mechanical loads by using the EMsFEM was provided by Yan et al. [34,35]. Moreover, multiscale topology design optimization of TLSs based on clustering method was proposed by Yan et al. [36]. But there are few studies on multi-material TLSs. By using the same analysis method, the multiscale design optimization of the multi-material TLSs is achieved in the paper. The multi-material and multiscale design optimization of the TLS, where the microstructure is composed of multiple materials, is studied subjected to the thermal and mechanical loading, as shown in Fig. 1. At the macroscale, an artificial element density

In the present research, the minimum strain energy of the structure and structural mass are considered as the objective function and constraint, respectively. The paper mainly introduces the following five parts. Section 2 provides the basic formula of the computation of strain energy of the multi-material TLSs by using the EMsFEM. Section 3 elaborates the material interpolation model. The modified SIMP interpolation scheme of the multi-material microstructure is proposed. In Section 4, the problem formulation and sensitivity analysis of the topology optimization of the multi-material TLS are introduced. Section 5 provides numerical examples to achieve multiscale topology design optimization of multi-material TLSs. Conclusions are provided in Section 6.

2 Strain Energy of Multi-Material TLSs

According to the reference [8], it indicates that the strain energy of the thermoelastic structure is more useful than the compliance. Thus, the topology design optimization for the TLS with minimum strain energy is studied. This concurrent multiscale optimization for multi-material lattice structure combines the homogenization method with the EMsFEM using the framework of PAMP [37]. The EMsFEM proposed in Zhang et al. [31] extends the FEM method to multiscale analysis of the structure made of heterogeneous material. In the EMsFEM, the structure is discretized by a number of macro-element and material in macro-element can be heterogeneous. And the stiffness matrix of each macro-element is constructed in a similar way with the traditional FEM but based on numerical shape functions, which is obtained by applying microscale FEM analysis to each macro-element. On the micro-scale, the relationship between the macro-node displacement and micro-node displacement is obtained based on shape functions. On the macro-scale, the displacement of the macrostructure is obtained by traditional FEM analysis methods. As shown in Fig. 2, the multiscale shape function matrix N of a plane four-node element composed of a truss microstructure can be expressed as

where

where

Considering the discrete finite element format, the strain energy of the TLS is presented as

where

The microstructure consists of multiple materials, as shown in Fig. 1. Each rod is given a candidate material property, and there are M rods with up to M material properties. We can get the whole potential energy of the TLS by assembling the potential energy of all truss unit cells. The ESM and ETLV of the macro-element are determined as

where H is the number of macro-elements in the structure. M is the number of micro rods in the microstructure.

3 Material Interpolation Scheme

In order to get a better performance structure, the microstructure is comprised of multiple materials. For a microstructure with n solid materials, the material of each microrob is interpolated by n candidate base materials by using the material interpolation scheme. The SIMP interpolation scheme is modified by the VPHF [38] to make 0/1 discrete choices clearly. In the optimization problem subjected to mechanical and thermal loading, the elastic modulus and thermal expansion coefficient compute the ESM and ETLV of the macro-element, respectively. The interpolation for elastic modulus

and

where

where

The elastic modulus, the thermal expansion coefficient and density of the

where

4 Multi-Material and Multiscale Topology Optimization of TLSs

The topology optimization of the TLS with multi-material microstructure under mechanical and thermal loading is studied. The cross-sectional area of micro rods, macro-elements density, and the candidate base materials of microstructure are design variables. The minimum structural strain energy is chosen as the objective function subjected to the mass constraint. The optimization formulation of multi-material TLS is shown as

Find

Min

S.t.

where

To eliminate the checkerboard pattern at the macroscale, a linear density filter [39,40] is utilized. The filtered density

where

To reduce the number of gray elements at the macroscale, the VPHF projection is used. Substituting the Eq. (17) into the Eq. (12), the penalized element density

Sensitivity analysis is mainly divided into the adjoint sensitivity method and direct derivative method. Sensitivity analysis of multi-scale lattice structure can adopt the direct derivative method and the specific process can refer to Yan et al. [34]. The analytical sensitivity of the multi-material TLS can be directly given here. The finite element equilibrium equation of thermoelastic structure is presented as

where

The derivative of the strain energy with respect to the cross-sectional area of the microrod

where

Similarly, the sensitivity of micro-scale design variable

The derivative of the ESM, the ETLV, and the strain energy only due to the thermal loading with respect to the microscopic material design variable

where the derivative of the elastic modulus and thermal expansion coefficient with respect to the microscopic material variable are

where the derivative of filtered design variables

The derivative of the objective function with respect to the macro-density

4.4 Optimization Implementation

The topology optimization flow chart of multi-material TLS is shown in Fig. 5. The specific optimization process is given as follows

(1) Set the macro-element density, the cross-sectional area of the micro rods, and the number of candidate materials in the microstructure.

(2) According to the Eq. (17), filter the macro-element density. And penalize the macro-element density and microscopic material design variables by the VPHF [38] projection.

(3) According to the multi-material interpolation scheme Eqs. (13)–(15) to calculate the elastic modulus matrix

(4) According to the Eqs. (4)–(6), calculate the structural ESM

(5) Solve the equilibrium equation of the multi-material TLS to get the macro-node displacement and get the micro-node displacement through the shape function.

(6) According to Eqs. (21)–(29), calculate structural sensitivity analysis to obtain

(7) Optimize the multi-material TLS with the sequential quadratic programming method [41] and judge the iterative convergence condition. If the program is not convergent, update the design variables and go to Step 2; otherwise, the optimization is finished.

We take the microstructure with two materials as an example. The elastic modulus, thermal expansion coefficients and densities of the two candidate base materials are given in Table 1. Comparing the ratio of elastic modulus to the density of the two materials, it is found that Material 1 is higher than Material 2. The product of the thermal expansion coefficient and elastic modulus is higher than Material 2 compared to Material 1. Thus, Material 2 has the advantage in mechanical properties, and Material 1 has better resistance to thermal deformation. Elastic modulus and thermal expansion coefficients of the microstructure can be obtained by the material interpolation scheme in Section 3, as shown in Fig. 6.

In this section, a two-end clamped beam subjected to both mechanical and thermal loading is considered. The length and width of the two-end clamped beam are L = 40 and W = 20, respectively. The mesh is composed of

5.1 Comparison of Single-Material Optimization and Multi-Material Optimization

In this example, a mechanical load

5.2 Multi-Material Optimization with Different Ground Structures Considering Size Factors

In this example, a mechanical load

5.3 Multi-Material Optimization Results with Different Temperatures

In this example, a mechanical load

5.4 Multiscale Optimization of the Multi-Material Structure with Different Mass Fractions

In this example, a mechanical load

In Fig. 13, the curves of the microstructure acturl mass fraction and the macroscopic actural mass fraction change with the constrained mass fraction. The microstructure acturl mass fraction is the ratio of actul microstructure mass to initial microstructure mass. With the increase of the total mass constraint, the microstructure mass fraction gradually increase. Due to the more material distribution into the microstructure, the cross-sectional areas of the micro rods will increase. Comparing the actual total mass fraction with the constrained total mass fraction, we found that the total mass constraint is not active when the constrained mass fraction is larger than 0.50 in this example.

A multiscale and multi-material topology design optimization model of TLSs subjected to mechanical and thermal loading based on the EMsFEM is established in this research. Different from the traditional minimum compliance optimization, the corresponding multi-material and multiscale mathematical formulation has been established with minimizing strain energy as the objective function. And the mass constraint is considered. The following conclusions are obtained.

The SIMP interpolation scheme has been adopted to realize micro-scale multi-material selection of truss microstructure. The modified VPHF is utilized to obtain a clear 0/1 material of truss microstructure. Compared with the traditional single-material topology optimization, multiscale topology optimization of multi-material TLSs allow increase design freedom for potentially better solutions to obtain the structure for better performance. The influence of temperatures, size factors, and structural mass fraction on the optimization results are introduced. As the temperature and mass fraction increase, the strain energy gradually increases. Temperature and mass fraction have little effect on the macro topological configuration, but have a more significant effect on the distribution of microscopic materials. The macro topological configurations and micro material distributions of multi-material TLSs are presented. The topology optimization of TLS with the microstructure composed of multiple materials realizes the improvement of structural performance.

Funding Statement: This research was financially supported by the National Natural Science Foundation of China (Nos. U1906233, 11732004, Jun Yan; No. 12002278, Zunyi Duan), the Key R&D Program of Shandong Province (2019JZZY010801, Jun Yan), the Fundamental Research Funds for the Central Universities (DUT20ZD213, DUT20LAB308, DUT21ZD209, Jun Yan; G2020KY05308, Zunyi Duan). These supports are gratefully acknowledged. Prof. Jun Yan and Associate Prof. Zunyi Duan received the grants and the URLs to sponsors’ websites are https://www.dlut.edu.cn/, and https://www.nwpu.edu.cn/, respectively.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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