Clustering-based multiscale topology optimization of thermo-elastic lattice structures

Abstract

A multiscale clustering-based topology optimization for thermo-elastic lattice structures is studied based on the extended multiscale finite element method (EMsFEM). The strain energy of thermo-elastic lattice structures is chosen as the objective function. The microstructural configuration and the macrostructural distribution of the thermo-elastic lattice material are designed through topology optimization concurrently. The K-means clustering-based method is proposed to group the microstructures of the lattice materials. The effects of the number of clusters (groups), magnitude of the thermal loads, size factor of the microstructure, and material volume fraction on the optimization results are discussed. The results show that the clustering-based multiscale design optimization is superior to the classical multiscale design optimization of lattice structures.

Appendices

Appendix A: Equivalent stiffness matrix of the truss microstructure

The rod m–n within the truss unit cell in the local and global coordinate systems is shown in Fig. 20. According to Eqs. (3–6), the deformation $\Delta L$ of the rod m–n is given as

$$ \Delta L = {\varvec{\uptheta}}^{\text{e}} {\mathbf{R}}^{\text{e}} {\mathbf{u}}_{\text{e}}^{'} $$

(A1)

where

$$ {\varvec{\uptheta}}^{\text{e}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - \cos \theta } & { - \sin \theta } \\ \end{array} } & {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ \end{array} } \\ \end{array} } \right] $$

(A2)

$$ {\mathbf{R}}^{\text{e}} = \left[ {\begin{array}{*{20}c} {R_{x} \left( p \right)^{\text{T}} } & {R_{y} \left( p \right)^{\text{T}} } & {R_{x} \left( q \right)^{\text{T}} } & {R_{y} \left( q \right)^{\text{T}} } \\ \end{array} } \right] $$

(A3)

where $\theta^{\text{e}}$ is the trigonometric function between the microrod and X axis.

Fig. 20

Microscopic rod and coordinate systems of the lattice structure

Only consider the body load, the potential energy of the thermo-elastic lattice structure can be written as follows,

$$ \varPi_{\text{p}}^{\text{e}} \left( {\mathbf{u}} \right) = \mathop \int \limits_{{\varOmega^{e} }}^{{}} \left( {\frac{1}{2}{\varvec{\upvarepsilon}}^{\text{T}} E{\varvec{\upvarepsilon}} - {\varvec{\upvarepsilon}}^{\text{T}} E{\varvec{\upvarepsilon}}_{0} - {\mathbf{u}}_{\text{e}}^{\text{T}} {\mathbf{f}}_{\text{e}} } \right) $$

(A4)

The elemental strain–displacement can be expressed as

$$ \varepsilon = \frac{\Delta L}{L} $$

(A5)

Substitute Eq. (A1–A3) into Eq. (A4), then the potential energy can be expressed as

$$ \Pi_{\text{p}}^{\text{e}} \left( {\mathbf{u}} \right) = \frac{1}{2}\left( {{\mathbf{u}}_{\text{e}}^{ '} } \right)^{\text{T}} \left[ {\left( {{\varvec{\uptheta}}^{\text{e}} {\mathbf{R}}^{\text{e}} } \right)^{\text{T}} k_{\text{e}} \left( {{\varvec{\uptheta}}^{\text{e}} {\mathbf{R}}^{\text{e}} } \right)} \right]{\mathbf{u}}_{\text{e}}^{ '} - \left( {{\mathbf{u}}_{\text{e}}^{ '} } \right)^{\text{T}} \left[ {\left( {{\varvec{\uptheta}}^{\text{e}} {\mathbf{R}}^{\text{e}} } \right)^{\text{T}} EA\alpha \Delta T} \right] - {\mathbf{u}}_{\text{e}}^{\text{T}} {\mathbf{f}}_{\text{e}} $$

(A6)

Then the equivalent stiffness matrix and thermal load by assembling the corresponding components of each rod in the microstructure are given as

$$ {\mathbf{K}}_{\text{e}} = \mathop \sum \limits_{i = 1}^{\text{M}} \left( {{\varvec{\uptheta}}^{{{\text{e}}i}} {\mathbf{R}}^{{{\text{e}}i}} } \right)^{\text{T}} k_{\text{e}}^{i} {\varvec{\uptheta}}^{{{\text{e}}i}} {\mathbf{R}}^{{{\text{e}}i}} $$

(A7)

$$ {\mathbf{F}}_{\text{e}}^{\text{th}} = \mathop \sum \limits_{i = 1}^{\text{M}} \left( {{\varvec{\uptheta}}^{{{\text{e}}i}} {\mathbf{R}}^{{{\text{e}}i}} } \right)^{\text{T}} \left( {EA\alpha \Delta T} \right) $$

(A8)

where $k_{\text{e}}^{i}$ denotes the elastic stiffness matrix of the microrod in the local coordinate.

Appendix B: L-shaped beam example

An L-shaped beam, shown in Fig. 21, is considered as an example for optimization with the proposed clustering-based multiscale optimization method. The upper and right edges are fixed. A mechanical load is applied at the node in the lower left. Horizontal distributed loads are applied to the bottom left corner as $F_{x} = - 200 \times 8 = 1600$. Similarly, vertical distributed loads are applied to the bottom left corner as $F_{y} = - 200 \times 8 = 1600$. A uniform temperature rise of $\Delta T$ = 50 °C is applied to the whole structure. The base material available for the micro unit cell is ${\text{V}}^{\text{MI}} = 5.8284.$ For the geometry, a = 15, and the area of each macro-element is unity (the length of each side of the macro-element was unity). The mesh is composed of 675 macro-elements. Therefore, the area of macro design domain is $V^{MA} = 675$. The amount of base material is retained during the optimization process. The upper limit of base material volume fraction was $\overline{\varphi }$ = 0.1, which is kept as an active constraint. The other parameters are the same as in the example presented in Sect. 5.

Fig. 21

Macro model of the L-shaped beam and its loading/boundary conditions

The strain energy with the classical multiscale topology optimization is 60.07. The configuration of the lattice structure of the classical multiscale optimization with uniform microstructures is shown in Fig. 22. The strain energies with the clustering-based multiscale topology optimization by the average clustering-based method and the K-means clustering-based method with four clusters were 55.78 and 54.90, respectively. The K-means clustering-based multiscale optimization is better than the average clustering-based multiscale optimization. The results of the clustering-based multiscale designed by the average clustering-based method and the K-means clustering-based method are shown in Figs. 23, 24, respectively. They are both assembled with three types of microstructures as shown in Figs. 23b, 24b, respectively. The clustering-based multiscale optimization design of the lattice structure is clearly superior to that by the classical multiscale optimization. Furthermore, the clustering-based multiscale design optimization can achieve a better distribution of the base material. The macro-configuration from the three multiscale optimizations is the same, namely the oblique V-shaped configuration.

Fig. 22

Minimum strain energy with the classical multiscale optimization of the L-shaped beam with uniform microstructures (60.07)

Fig. 23

Minimum strain energy of the clustering-based multiscale topology optimization by the average clustering-based method with four clusters is 55.78

Fig. 24

Minimum strain energy of the clustering-based multiscale topology optimization by the K-means clustering-based method with four clusters is 54.90