Optimization method of thermo-elastic lattice structure based on surrogate models of microstructures
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摘要:
三维点阵材料是一种具有多尺度特性的新型轻质多功能材料，其有千变万化的微结构和高孔隙率，通过设计其细观尺度特征可以获得优良的宏观性能。为了发挥材料与结构的最大设计潜力，提出一种热弹性点阵结构优化方法。在材料细观研究尺度上，实现了三维点阵材料等效热弹性性能预测，利用周期性边界条件下的代表体元法进行数值求解，利用径向基函数代理模型构建细观结构和宏观材料性能的数学关系，并进行了预测误差验证，证明了所提方法具有良好的精确度。在结构宏观研究尺度上，建立了以等效材料填充的结构优化模型，考虑了热力载荷作用，以单胞等效性能代理模型作为材料插值模型，提出最小应变能热弹性点阵结构优化数学模型。在典型三维算例中得到了细观结构变密度分布的优化结果，结构热刚度在一定体积约束下显著提高，证明了所提方法的有效性。
Abstract:Lattice material is a new type of lightweight and multifunctional material, which has a variety of microstructures and high porosity. Excellent macroscopic properties can be obtained by designing its mesoscale features. To maximize the design potential of materials and structures, an optimization method for the thermo-elastic lattice structure is proposed. As for mesoscale material research, the effective thermo-elastic properties prediction of three-dimensional lattice materials is implemented. Relevant coefficients are solved using the idea of the representative volume method under periodic boundary conditions. Surrogate models are constructed to build the relationship between macroscopic responses and microstructures, and are proved to have good accuracy through error verification tests. As for macroscale material research, a structural optimization model filled with equivalent materials is established. Considering the thermal and mechanical loads, a mathematical model for structural optimization of thermo-elastic lattice structure with minimum strain energy is proposed using the surrogate models of effective properties as the material interpolation schemes. The result of an optimal spatially varying metamaterial is obtained in a typical three-dimensional structure example, and the thermal stiffness of the structure is improved under a certain volume constraint, demonstrating the effectiveness of the optimization method.
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图 2 不同约束下立方体单胞变形情况示意图
Figure 2. Schematic diagram of deformation of cubic cell under different constraints
图 7 热膨胀系数与细观结构的无关性验证
Figure 7. Independence verification of thermal expansion coefficient with microstructures
图 12 点阵填充工字悬臂梁优化迭代曲线
Figure 12. Iteration curves in optimization of lattice filled I-shaped cantilever beam
图 13 点阵填充工字悬臂梁最小应变能优化结果
Figure 13. Minimum strain energy optimization results of lattice filled I-shaped cantilever beam
图 14 最小应变能优化应力分布情况
Figure 14. Structure stress distribution under minimum strain energy optimization
图 15 最小应变能优化结构变形情况
Figure 15. Structure deformation under minimum strain energy optimization
表 1 TC4材料基本属性
Table 1. Properties of TC4 material
密度$\rho $ /
(kg·mm−3)泊松比$\mu $ 弹性模量E/
MPa热膨胀系数$\alpha $/
(10−6·K−1)4.43×10−6 0.30 115000 8.8 
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表 2 单胞等效热弹性系数计算方法
Table 2. Calculation method of effective thermal-elastic properties of lattice cells
宏观
应力载荷
施加等效热弹性性能 $\sigma _x^0$ ${F_x}( B)$ ${E_x} = \dfrac{ {\sigma _x^0} }{ {\varepsilon _x^0} } = \dfrac{ { {F_x}{a^2} } }{ { {V_{\text{c} } }{u_B} } },\;{\mu _{xy} } = - \dfrac{ {\varepsilon _y^0} }{ {\varepsilon _x^0} } = - \dfrac{ { { { {v_D} } / b} } }{ { { { {u_B} } /a} } },\;{\mu _{x{\textit{z} } } } = - \dfrac{ {\varepsilon _{\textit{z} }^0} }{ {\varepsilon _x^0} } = - \dfrac{ { { { {w_E} }/ c} } }{ { { { {u_B} } / a} } }$ $\sigma _y^0$ ${F_y}( D )$ ${E_y} = \dfrac{ {\sigma _y^0} }{ {\varepsilon _y^0} } = \dfrac{ { {F_y}{b^2} } }{ { {V_{\text{c} } }{v_D} } },\;{\mu _{yx} } = {\mu _{xy} },\;{\mu _{y{\textit{z} } } } = - \dfrac{ {\varepsilon _{\textit{z} }^0} }{ {\varepsilon _y^0} } = - \dfrac{ { { { {w_E} } /c} } }{ { { { {v_D} } / b} } }$ $\sigma _{\textit{z}}^0$ ${F_{\textit{z} } }( E )$ ${E_{\textit{z} } } = \dfrac{ {\sigma _{\textit{z} }^0} }{ {\varepsilon _{\textit{z} }^0} } = \dfrac{ { {F_{\textit{z} } }{c^2} } }{ { {V_{\text{c} } }{w_E} } },\;{\mu _{ {\textit{z} }x} } = {\mu _{x{\textit{z} } } },\;{\mu _{ {\textit{z} }y} } = {\mu _{y{\textit{z} } } }$ $\tau _{xy}^0$ ${F_y}( B )$ ${G_{xy}} = \dfrac{{\tau _{xy}^0}}{{\gamma _{xy}^0}} = \dfrac{{{F_y}{a^2}}}{{{V_{\text{c}}}{v_B}}}$ $\tau _{y{\textit{z}}}^0$ ${F_{\textit{z} } }( D )$ ${G_{y{\textit{z}}}} = \dfrac{{\tau _{y{\textit{z}}}^0}}{{\gamma _{y{\textit{z}}}^0}} = \dfrac{{{F_{\textit{z}}}{b^2}}}{{{V_{\text{c}}}{w_D}}}$ $\tau _{{\textit{z}}x}^0$ ${F_x}( E )$ ${G_{{\textit{z}}x}} = \dfrac{{\tau _{{\textit{z}}x}^0}}{{\gamma _{{\textit{z}}x}^0}} = \dfrac{{{F_x}{c^2}}}{{{V_{\text{c}}}{u_E}}}$ $\sigma _{{\text{th}}}^0$ $\Delta T$ ${\alpha _x} = \dfrac{ {\varepsilon _x^0} }{ {\Delta T} } = \dfrac{ { {u_B} } }{ {a\Delta T} },\;{\alpha _y} = \dfrac{ {\varepsilon _y^0} }{ {\Delta T} } = \dfrac{ { {v_D} } }{ {b\Delta T} },\;{\alpha _{\textit{z} } } = \dfrac{ {\varepsilon _{\textit{z} }^0} }{ {\Delta T} } = \dfrac{ { {w_E} } }{ {c\Delta T} }$
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表 3 单胞等效性能代理模型测试误差
Table 3. Test error of surrogate model of effective properties for lattice cells
输出变量 相对误差${\varepsilon '_{{\text{RMSE}}}}$/% ${E_1}{\text{, }}{E_2}$ 0.86 ${E_3}{\text{ }}$ 1.84 ${\mu _{12}}{\text{ }}$ 3.88 ${\mu _{13}}{\text{, }}{\mu _{23}}$ 3.29 ${G_{13}}{\text{, }}{G_{23}}$ 0.85 ${G_{12}}{\text{ }}$ 1.40 ${V_{\text{f}}}$ 0.23
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表 4 算例参数设置
Table 4. Parameter settings in the example
上翼缘温升/
K下翼缘温升/
K机械载荷/
MPa初始半径/
mm体积分数
约束/%$350$ ${\text{10}}$ $0.5$ $1.8$ $26.3$
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表 5 点阵填充工字悬臂梁优化前后对比分析
Table 5. Comparative analysis before and after structural optimization of lattice filled I-shaped cantilever beam
优化时间 ${S_{\max }}{\text{/MPa}}$ ${U_{\max }}{\text{/mm}}$ ${S_{{\text{web}}}}{\text{/MPa}}$ ${U_{{\text{web}}}}{\text{/mm}}$ 优化前 $19.34$ $19.8$ $1.92$ $5.8$ 优化后 $15.75$ $15.4$ $1.55$ $3.9$
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