Topology optimization of electric vehicle chassis structure with distributed load-bearing batteries

Abstract

This paper presents a systematic design approach of conceptually forming a lightweight electric vehicle (EV) chassis topology integrated with distributed load-bearing batteries of different shapes and dimensions using a density-based topology optimization approach. A deformable feature description function tailored to commercial Li-ion batteries is proposed to describe cell features with desirable layouts, dimensions, and continuous shapes only from cylinder to cube by applying a handful of design variables. A Kreisselmeier–Steinhauser function Boolean operation and a gradient-norm method are sequentially leveraged to integrate multiple cells enclosed by reinforced shells into a unified battery set. Besides, a new non-overlap** constraint is developed to avoid the geometric overlaps between all cells and further restrict the minimum battery spacing through introducing an auxiliary density filter. By solving the optimization problem, an EV chassis with distributed various specification batteries can be obtained, which exhibits better comprehensive mechanical properties than that with centralized uniform specification batteries under the same battery capacity and structural weight. Numerical examples of different battery capacity requirements, battery shell thicknesses, and minimum battery spacing are given to demonstrate the applicability of the proposed approach.

Appendix: Sensitivities

The sensitivities of the compliance objective function based on adjoint analysis for a single working condition are written as follows:

$$\begin{aligned} \dfrac{\partial c}{\partial x_{e}}= & {} -{{\textbf {U}}}^{\text {T}} \dfrac{\partial {{\textbf {K}}}}{\partial x_{e}} {{\textbf {U}}} \\= & {} -\sum _{z} \dfrac{\partial E_{z}}{\partial x_{e}} {{\textbf {u}}}_{i}^{\text {T}} {{\textbf {k}}}_{0} {{\textbf {u}}}_{i}, x_{e} \in \{ \mu _{i}^{j},\upsilon _{e} \}, \end{aligned},$$

(26)

where ${{\textbf {k}}}_{0}$ is the element stiffness matrix for an element with unit Young’s modulus.

The element stiffness interpolation in Eq. 14 is rewritten as follows:

$$\begin{aligned} E(\varphi _{e}, \tau _{e}, \psi _{e})&=E_{m}+\underbrace{\psi _{e}^{P} (E_{1}-E_{m})}_{A} \\&\quad +\underbrace{\varphi _{e}^{P} [(E_{2}-E_{m})-\psi _{e}^{P} (E_{1}-E_{m})]}_{B} \\&\quad +\underbrace{\tau _{e}^{P} [(E_{3}-E_{m})-\psi _{e}^{P} (E_{1}-E_{m})]}_{C} \\&\quad +\underbrace{(\varphi _{e} \tau _{e})^{P}[\psi _{e}^{P} (E_{1}-E_{m})-(E_{2}-E_{m})]}_{D}. \end{aligned}.$$

(27)

Thus, $\partial E_{z} / \partial \mu _{i}^{j}$ and $\partial E_{z} / \partial \upsilon _{e}$ are calculated by applying the chain rule:

$$\begin{aligned} \dfrac{\partial E_{z}}{\partial \mu _{i}^{j}}= & {} \dfrac{\partial A}{\partial \varphi _{z}} \dfrac{\partial \varphi _{z}}{\partial \mu _{i}^{j}}+\dfrac{\partial B}{\partial \varphi _{z}} \dfrac{\partial \varphi _{z}}{\partial \mu _{i}^{j}} + \dfrac{\partial C}{\partial \tau _{z}} \dfrac{\partial \tau _{z}}{\partial \mu _{i}^{j}} \\{} & {} \quad + \dfrac{\partial D}{\partial (\varphi _{z} \tau _{z})} \left( \varphi _{z} \dfrac{\partial \tau _{z}}{\partial \mu _{i}^{j}} + \tau _{z} \dfrac{\partial \varphi _{z}}{\partial \mu _{i}^{j}}\right) , \end{aligned}$$

(28)

$$\begin{aligned} \dfrac{\partial E_{z}}{\partial \upsilon _{e}}= & {} \dfrac{\partial E_{z}}{\partial \psi _{z}} \dfrac{\partial \psi _{z}}{\partial \upsilon _{e}}, \end{aligned}$$

(29)

where $\partial A / \partial \varphi _{z}=0$, $\partial B / \partial \varphi _{z}=P\varphi _{z}^{P-1}[(E_{2}-E_{m})-\psi _{z}^{P} (E_{1}-E_{m})]$, $\partial C / \partial \tau _{z}=P\tau _{z}^{P-1}[(E_{3}-E_{m})-\psi _{z}^{P} (E_{1}-E_{m})]$, $\partial D / \partial (\varphi _{z} \tau _{z})=P(\varphi _{z} \tau _{z})^{P-1}[\psi _{z}^{P} (E_{1}-E_{m})-(E_{2}-E_{m})]$, and $\partial E_{z} / \partial \psi _{z} = P\psi _{z}^{P-1}(1-\varphi _{z}^{P})(1-\tau _{z}^{P})(E_{1}-E_{m})$.

Considering ${\varvec{\tau }}$ is a function of ${\varvec{\varphi }}$ correlated to the first design variable ${\varvec{\mu }}$, the sensitivity $\partial \varphi _{z} / \partial \mu _{i}^{j}$ is a vital item for the TO, derived as follows:

$$\dfrac{\partial \varphi _{z}}{\partial \mu _{i}^{j}} = \dfrac{\partial \varphi _{z}}{\partial ({\bar{\xi }}_{i})_{l}} \dfrac{\partial ({\bar{\xi }}_{i})_{l}}{\partial (\xi _{i})_{k}} \dfrac{\partial (\xi _{i})_{k}}{\partial (T_{i})_{b}} \dfrac{\partial (T_{i})_{b}}{\partial \mu _{i}^{j}},$$

(30)

with two critical items:

$$\begin{aligned} \dfrac{\partial \varphi _{z}}{\partial ({\bar{\xi }}_{i})_{l}}= & {} \dfrac{{\text {e}}^{K_{{\text {S}}1}}({\bar{\xi }}_{i})_{l}}{\sum _{i=1}^{N_{\text {b}}}{{\text {e}}^{K_{{\text {S}}1} ({\bar{\xi }}_{i})_{l}}}}, \end{aligned},$$

(31)

$$\begin{aligned} \dfrac{\partial (T_{i})_{b}}{\partial \mu _{i}^{j}}= & {} {\left\{ \begin{array}{ll} \dfrac{\partial (T_{i})_{b}}{\partial p_{i}} \\ \dfrac{\partial (T_{i})_{b}}{\partial q_{i}} \\ \dfrac{\partial (T_{i})_{b}}{\partial L_{i}} \\ \dfrac{\partial (T_{i})_{b}}{\partial t_{i}} \\ \dfrac{\partial (T_{i})_{b}}{\partial \theta _{i}} \\ \dfrac{\partial (T_{i})_{b}}{\partial m_{i}} \end{array}\right. } \\= & {} {\left\{ \begin{array}{ll} C_{T} \left[ \dfrac{1}{2L_{a}} \dfrac{2X_{i}}{L_{i}/2} \left( -\dfrac{\cos \theta _{i}}{L_{i}/2}\right) + \dfrac{1}{2t_{a}} \dfrac{2Y_{i}}{f_{i}/2} \left( \dfrac{\sin \theta _{i}}{f_{i}/2}\right) \right] \\ C_{T} \left[ \dfrac{1}{2L_{a}} \dfrac{2X_{i}}{L_{i}/2} \left( -\dfrac{\sin \theta _{i}}{L_{i}/2}\right) + \dfrac{1}{2t_{a}} \dfrac{2Y_{i}}{f_{i}/2} \left( -\dfrac{\cos \theta _{i}}{f_{i}/2}\right) \right] \\ C_{T} \left[ \dfrac{1}{2L_{a}} \dfrac{2X_{i}}{L_{i}/2} \left( -\dfrac{X_{i}}{L_{i}^{2}/2}\right) + \dfrac{1}{2t_{a}} \dfrac{2Y_{i}}{f_{i}/2} \left( -\dfrac{Y_{i}}{f_{i}^{2}/2}\right) \left( \dfrac{m_{2}-m_{i}}{m_{2}-m_{1}}\right) \right] \\ C_{T} \left[ \dfrac{1}{2t_{a}} \dfrac{2Y_{i}}{f_{i}/2} \left( -\dfrac{Y_{i}}{f_{i}^{2}/2}\right) \left( \dfrac{m_{i}-m_{1}}{m_{2}-m_{1}}\right) \right] \\ C_{T} \left[ \dfrac{1}{2L_{a}} \dfrac{2X_{i}}{L_{i}/2} \left( \dfrac{-\sin \theta _{i} (x_{l}-p_{i})+\cos \theta _{i} (y_{l}-q_{i})}{L_{i}/2}\right) + \dfrac{1}{2t_{a}} \dfrac{2Y_{i}}{f_{i}/2} \left( \dfrac{-\cos \theta _{i} (x_{l}-p_{i})+\sin \theta _{i} (y_{l}-q_{i})}{f_{i}/2}\right) \right] \\ -\left[ \ln {(L_{a})} L_{a}^{m_{i}} + \ln {(t_{a})} t_{a}^{m_{i}} + t_{a}^{m_{i}-1} \dfrac{1}{2t_{a}} \dfrac{2Y_{i}}{f_{i}/2} \left( -\dfrac{Y_{i}}{f_{i}^{2}/2}\right) \left( \dfrac{m_{2}-m_{i}}{m_{2}-m_{1}}\right) \right] \\ \end{array}\right. }, \end{aligned},$$

(32)

where $C_{T}=-(T_{i})_{b} \left[ m_{i} (L_{a}^{m_{i}-1} + t_{a}^{m_{i}-1}) \right]$, $L_{a}=\left| \dfrac{X_{i}}{L_{i}/2} \right|$, and $t_{a}=\left| \dfrac{Y_{i}}{f_{i}/2} \right|$.

Besides, the sensitivities of the proposed non-overlap** constraint in Eq. 19 associated with the first design variable are expressed as follows:

$$\dfrac{\partial V_{\text {ov}}}{\partial \mu _{i}^{j}} = \sum _{k} \sum _{l} \sum _{r} \dfrac{\partial V_{\text {ov}}}{\partial \chi _{k}} \dfrac{\partial \chi _{k}}{\partial (\tilde{{\bar{\xi }}}_{i})_{l}} \dfrac{\partial (\tilde{{\bar{\xi }}}_{i})_{l}}{\partial ({\bar{\xi }}_{i})_{r}} \dfrac{\partial ({\bar{\xi }}_{i})_{r}}{\partial \mu _{i}^{j}},$$

(33)

with another critical item:

$$\dfrac{\partial \chi _{k}}{\partial (\tilde{{\bar{\xi }}}_{i})_{l}} = \dfrac{\sum _{r=1}^{N_{\text {b}}-1} {\text {e}}^{K_{{\text {S}}2} (\gamma _{z})_{l}} (\tilde{{\bar{\xi }}}_{r})_{l}}{\sum _{z=1}^{N_{\text {b}}\atopwithdelims ()2} {\text {e}}^{K_{{\text {S}}2} (\gamma _{z})_{l}}},\quad (\gamma _{z})_{l}=(\tilde{{\bar{\xi }}}_{i})_{l} (\tilde{{\bar{\xi }}}_{r})_{l},\quad i \ne r,$$

(34)

where the $\varvec{\gamma }_{z}$ in the numerator item represent all the overlap** sub-regions between the ith cell and other cell (marked as rth cell).