Multiphysics topology optimization of a multifunctional structural battery composite

Abstract

The structural battery composite (SBC) is a novel class of multifunctional materials with the ability to work as a lithium-ion battery that can withstand mechanical loads. The motivation of this study is to address one of the major challenges in the development of SBCs, which is a strong conflict in the structural and electrical demands for its electrolyte (i.e., high stiffness and high ionic conductivity). Furthermore, there is a design requirement that the electrochemical cycling should not result in overheating of the SBC. The novelty of this study is the development of an efficient multi-objective multiphysics density-based topology optimization framework that considers electrochemical/thermal/structural physics to identify the optimized design of a structural battery electrolyte (SBE). The optimization methodology is defined as solving a multi-objective problem of maximizing effective ionic conductivity and minimizing compliance of SBE. The problem is subjected to constraints on volume fraction and the maximum allowable temperature. The normalized-normal-constraint approach is utilized to generate a Pareto-front curve for this multi-objective problem. The proposed method is computationally efficient owing to utilizing a low-fidelity resistance network approach, for the electrochemical module and parallelizes the workload using portable, and extendable toolkit for scientific computing and message-passing interface. Several numerical examples are solved to demonstrate the applicability of the proposed methodology under different loading scenarios. The results reveal that the proposed methodology provides a better understanding of the required microstructural design of SBE for the performance improvement of structural battery composites.

Appendices

Appendix A: Finite element discretization of structural module

For the structural module explained in Sect. 2.2, the approximation of the displacement field in each element is given by

$$\begin{aligned} U^{h}_{e}({\textbf{X}})=\sum _{i=1}^{n_S^e}N_{i}({\textbf{X}})U_{i}={\textbf{N}}_S^{e}\left( {\textbf{X}}\right) {\textbf{U}}^{e} \end{aligned}$$

(A1)

where

$$\begin{aligned} {\textbf{N}}_{S}^{e}= & {} \left[ \begin{array}{cccccccccc} N_{1} &{} 0 &{} 0 &{} N_{2} &{} 0 &{} 0 &{} \ldots &{} N_{n_{S}^{e}} &{} 0 &{} 0\\ 0 &{} N_{1} &{} 0 &{} 0 &{} N_{2} &{} 0 &{} 0 &{} 0 &{} N_{n_{S}^{e}} &{} 0\\ 0 &{} 0 &{} N_{1} &{} 0 &{} 0 &{} N_{2} &{} 0 &{} 0 &{} 0 &{} N_{n_{S}^{e}} \end{array}\right] =[\begin{array}{cccc} N_{1}&N_{2}&\ldots&N_{n_{S}^{e}}\end{array}]\odot {\textbf{I}} \end{aligned}$$

(A2)

$$\begin{aligned} {\textbf{U}}^{e}= & {} [\begin{array}{cccccccccc} U_{1}^{x}&U_{1}^{y}&U_{1}^{z}&U_{2}^{x}&U_{2}^{y}&U_{2}^{z}&\ldots&U_{n_{S}^{e}}^{x}&U_{n_{S}^{e}}^{y}&U_{n_{S}^{e}}^{z}\end{array}]^{'} \end{aligned}$$

(A3)

$\odot$ is the Kronecker product, ${\textbf{I}}$ is a $3\times 3$ identity matrix, ${\textbf{X}}$ is the spatial coordinates, $n_S^e$ is the number of finite element shape functions, $N_i({\textbf{X}})$ is the finite element shape functions, and $U_{i}$ is the nodal displacement. The notation $(.)'$ denotes transpose.

As mentioned in Sect. 2.2, implementation of the FEM leads to Eq. (8), in which ${\textbf{K}}_S^e$ is a function of $\mathbf {{B}}_{S}^{e}$ and ${\mathbb {C}}$ defined by

$$\begin{aligned} \mathbf {{B}}_{S}^{e}=\left[ \begin{array}{ccccccc} \frac{\partial {N}_{1}}{\partial x} &{} 0 &{} 0 &{} \ldots &{} \frac{\partial {N}_{n_{S}^{e}}}{\partial x} &{} 0 &{} 0\\ 0 &{} \frac{\partial {N}_{1}}{\partial y} &{} 0 &{} \ldots &{} 0 &{} \frac{\partial {N}_{n_{S}^{e}}}{\partial y} &{} 0\\ 0 &{} 0 &{} \frac{\partial {N}_{1}}{\partial z} &{} \ldots &{} 0 &{} 0 &{} \frac{\partial {N}_{n_{S}^{e}}}{\partial z}\\ \frac{\partial {N}_{1}}{\partial x} &{} \frac{\partial {N}_{1}}{\partial y} &{} 0 &{} \ldots &{} \frac{\partial {N}_{n_{S}^{e}}}{\partial x} &{} \frac{\partial {N}_{n_{S}^{e}}}{\partial y} &{} 0\\ 0 &{} \frac{\partial {N}_{1}}{\partial y} &{} \frac{\partial {N}_{1}}{\partial z} &{} \ldots &{} 0 &{} \frac{\partial {N}_{n_{S}^{e}}}{\partial y} &{} \frac{\partial {N}_{n_{S}^{e}}}{\partial z}\\ \frac{\partial {N}_{1}}{\partial x} &{} 0 &{} \frac{\partial {N}_{1}}{\partial z} &{} \ldots &{} \frac{\partial {N}_{n_{S}^{e}}}{\partial x} &{} 0 &{} \frac{\partial {N}_{n_{S}^{e}}}{\partial z} \end{array}\right] \end{aligned}$$

(A4)

and

$$\begin{aligned} {\mathbb {C}} = 2\mu {\mathbb {P}}_{Sym}+\lambda {\textbf{I}}\otimes {\textbf{I}} \end{aligned}$$

(A5)

where $\otimes$ is dyadic product, $\lambda = \frac{\nu E}{(1+\nu )(1-2\nu )}$ is Lamé moduli, $\mu = \frac{E}{2(1+\nu )}$ is the shear modulus, E is the modulus of elasticity, $\nu$ is the Poisson ratio, and ${\mathbb {P}}_{Sym}$ is the 4-tensor symmetric projection, i.e., ${\mathbb {P}}_{Sym}[{\textbf{A}}]=({\textbf{A}}+{\textbf{A}}')/2$.

Appendix B: Finite element discretization of thermal module

The temperature field approximation in each element can be written as

$$\begin{aligned} T^{h}_{e}({\textbf{X}})=\sum _{i=1}^{n_T^e}N_{i}({\textbf{X}})T_{i}={\textbf{N}}_T^{e}\left( {\textbf{X}}\right) {\textbf{T}}^{e} \end{aligned}$$

(B6)

where

$$\begin{aligned} {\textbf{N}}_{T}^{e}= & {} [\begin{array}{cccc} N_{1}&N_{2}&\ldots&N_{n_{T}^{e}}\end{array}] \end{aligned}$$

(B7)

$$\begin{aligned} {\textbf{T}}^{e}= & {} [\begin{array}{cccc} T_{1}&T_{2}&\ldots&T_{n_{T}^{e}} \end{array}]^{'} \end{aligned}$$

(B8)

$n_T^e$ is the number of degrees of freedoms (dof) in each element and $T_{i}$ is the nodal temperature.

As mentioned in Sect. 2.3, implementing FEM results leads to Eq. (12), in which ${\textbf{K}}^e_T$ is a function of $\mathbf {{B}}_{T}^{e}$ defined by

$$\begin{aligned} \mathbf {{B}}_{T}^{e}=\left[ \begin{array}{cccc} \frac{\partial {N}_{1}}{\partial x} &{} \frac{\partial {N}_{2}}{\partial x} &{} \ldots &{} \frac{\partial {\hat{N}}_{n_{T}^{e}}}{\partial x} \\ \frac{\partial {N}_{1}}{\partial y} &{} \frac{\partial {N}_{2}}{\partial y} &{} \ldots &{} \frac{\partial {\hat{N}}_{n_{T}^{e}}}{\partial y} \\ \frac{\partial {N}_{1}}{\partial z} &{} \frac{\partial {N}_{2}}{\partial z} &{} \ldots &{} \frac{\partial {\hat{N}}_{n_{T}^{e}}}{\partial z} \\ \end{array}\right] \end{aligned}$$

(B9)

Appendix C: Verification of analysis modules

The structural and thermal modules developed in this study are verified against ANSYS structural and ANSYS FLUENT, respectively. The problem setup for this verification study is presented in Fig. 11. Thermal conductivity and modulus of elasticity of the sample are assumed to be $1\;Wm^{-1}K^{-1}$ and $0.5\;GPa$, respectively. The boundary conditions are presented in Fig. 12. For the structural problem, the sample is assumed to be under a tensile load in the X-direction. And in the thermal problem, the sample is assumed to have convection heat transfer on front and back surfaces, and the rest of the surfaces are considered to be insulated. Constant heat generation of $24\;W/m^3$ is considered for this problem.

Fig. 11

Problem setup for verification study for a structural and b thermal modules

Figure 12 compares temperature and displacement distributions obtained from ANSYS and the analysis code in this study. The results are quite close to each other as the L2-norm difference between the in-house code and ANSYS for T, $U_x$, $U_y$, and $U_z$ are 0.02$^\circ C$, 0.001 $\mu m$, 0.0004 $\mu m$, and 0.0004 $\mu m$, respectively. Hence, we can conclude that the structural and thermal analysis modules developed in this study have acceptable accuracy. Note that the accuracy of the electrochemical module is checked and confirmed against the test cases provided in Rhazaoui et al. (2013).

Fig. 12

Comparison between ANSYS and the multiphysics code for a temperature, b x-component of displacement, c y-component of displacement, and d z-component of displacement distributions

Appendix D: Verification of sensitivity analysis

We perform a verification study of the analytic adjoint sensitivity analysis developed in this study by comparing it with the central finite difference method. The error between the adjoint and finite difference sensitivity analysis is given by

$$\begin{aligned} \epsilon = \mid \frac{(\frac{d\theta }{d{d_i}})^{Adj}-(\frac{d\theta }{d{d_i}})^{FD}}{(\frac{d\theta }{d{d_i}})^{Adj}} \mid \end{aligned}$$

(D10)

where Adj and FD indicate the adjoint method and finite difference approach, respectively. The error is plotted in Fig. 13 for a sequence of perturbations from $\Delta {d}_{i}=10^{-2}-10^{-7}$ for all three physics of electrochemical, thermal, and structural. Note that $\theta$ in Eq. (D10) indicates ionic conductivity, compliance, and p-mean temperature for electrochemical, structural, and thermal physics, respectively. The relative error has a small value, which indicates that the sensitivity analysis is correctly derived and implemented.

Fig. 13

The relative error, $\epsilon$, between the adjoint sensitivity analysis and approximated central finite difference sensitivity versus the magnitude of the perturbation in the design parameter, $\Delta {d}_i$. The amount of error is quite small and we can conclude that the sensitivity analysis is correctly derived and implemented in this study