Microstructural topology optimization for patch-based sandwich panel with desired in-plane thermal expansion and structural stiffness

Abstract

Apart from the lightweight and excellent mechanical properties, sandwich panels can be endowed with tailorable in-plane coefficient of thermal expansion (CTE) through an elaborate design of periodic face-sheets. However, albeit that the microstructural topology of their periodic face-sheets promises unique thermal expansion behaviors, it may also bring significant influences to the structural stiffness of sandwich panels. In this study, we apply the topology optimization method to design face-sheet microstructures to enable the sandwich panels to possess desired in-plane CTEs, lightweight and benign mechanical properties, simultaneously. By introducing the patch-based cell as initial configuration, the existing thermally bending adjustment mechanism for thermal deformation control is integrated to the process of topology optimization. The entire topology optimization process including the equivalent mechanical properties prediction and the sensitivity computation is performed within an in-house program coupled with commercial finite element analysis software. To this end, a matching numerical sensitivity analysis method to extract sensitivities straightforwardly from software’s output is also developed on the basis of asymptotic homogenization method. Three types of specific optimization problems in terms of different objective functions and constraint conditions are proposed, solved, and studied, namely, in-plane zero thermal expansion combining with maximum stiffness, the other for in-plane zero thermal expansion optimal specific stiffness, and minimizing in-plane isotropic thermal expansion. Some specific resulting topologies, microstructural features, and design details are subsequently obtained. In particular, the current strategy of integrating effective mechanism and topological technology can be extended to design more microstructures for simultaneously tailorable CTE and high mechanical performance by replacing present thermal deformation control mechanism with others.

Appendices

Appendix 1: Implementation steps of NSAM-CTE

Similar with the implementation mode of NIAH-CTE for predicting the effective CTEs of periodic microstructures, the present NSAM-CTE can be implemented using the simulation capacity of commercial FEA soft as a black box. The sensitivity information in the form of element strain energy can be extracted directly from the output of FEA software, which substantially reduces the computational cost compared with that of using traditional methods. The concrete implementation steps of NIAH-CTE for predicting the sensitivities of the effective CTE $\partial {\boldsymbol{\alpha}}_i^{\boldsymbol{H}}/\partial {\rho}_e$ are given as follows:

• Step 1: Build the finite element model for cell microstructure using standard modeling process.

• Step 2: Predict the effective elastic modulus E^H and thermos-elastic constant β^H of cell microstructure using numerical results given by NIAH-CTE.

• Step 3: Apply the generalized strain fields ${\overline{\boldsymbol{\varepsilon}}}^{\boldsymbol{i}+\alpha }$ to the finite element model for element strain energy ${WB}_e^{i+\alpha }$. Note that the element strain energy caused by the specific strain fields can be obtained through applying equivalent nodal displacement fields on each node due to the essence of the NIAH method. We can just construct the nodal displacement fields ${\chi}^{\boldsymbol{0}\left(\boldsymbol{i}\right)}-{\chi}^{\ast \left(\boldsymbol{i}\right)}-{\chi}^{\zeta }$ and thermal loads (−1 °C) for ${\overline{\boldsymbol{\varepsilon}}}^{\boldsymbol{i}+\alpha }$, and as input to the finite element model, the element strain energy ${WB}_e^{i+\alpha }$ can be extracted directly from the output of FEA software after one static analysis. The extractions for the element strain energies ${W}_e^{ii}$ and ${WB}_e^{\alpha }$ are subsequently performed using the same implementation procedure.

• Step 4: Calculate the sensitivities of the effective thermos-elastic constant ∂β^H/∂ρ_e from Eq.(17)–(23).

• Step 5: Extract the sensitivities of the effective elastic modulus $\partial {\boldsymbol{E}}_{ij}^{\boldsymbol{H}}/\partial {\rho}_e$ from the output of FEA software using original numerical sensitivity analysis method (Yi et al. 2016). For the purpose of the brevity, the concrete implementation steps for obtaining $\partial {\boldsymbol{E}}_{ij}^{\boldsymbol{H}}/\partial {\rho}_e$ are not presented herein.

• Step 6: Calculate the sensitivities of the effective CTE $\partial {\boldsymbol{\alpha}}_i^{\boldsymbol{H}}/\partial {\rho}_e$ from Eq. (16).

Appendix 2: Method verification of NSAM-CTE

In order to verify the effectiveness of present NSAM-CTE for computing the sensitivities, a simple verification example is performed through comparing the sensitivity results with that obtained by the finite difference method (FDM). After establishing the finite element model of original design configuration as shown in Fig. 3 (a), three arbitrary elements are taken as the testing cases for verifications. The comparison results of $\partial {\boldsymbol{E}}_{ij}^{\boldsymbol{H}}/\partial {\rho}_e$ and $\partial {\boldsymbol{\alpha}}_i^{\boldsymbol{H}}/\partial {\rho}_e$ given by NSAM-CTE and FDM are listed in Table 5 and Table 6, respectively. The finite-difference interval Δρfor FDM is taken as 1 × 10⁻⁴.

Table 5 The sensitivity comparison results of $\partial {\boldsymbol{E}}_{ij}^{\boldsymbol{H}}/\partial {\rho}_e$

Table 6 The sensitivity comparison results of $\partial {\boldsymbol{\alpha}}_i^{\boldsymbol{H}}/\partial {\rho}_e$

It can be seen from Table 5 and Table 6 that the sensitivity results of main in-plane coefficients of effective elastic modulus and CTEs match very well with the results computed using FDM. As such the effectiveness of the present numerical sensitivity analysis method and corresponding implementation steps are verified.