A new constraint on principal stress direction variance to improve load bearing capacity

Abstract

The present study addresses a simplified topology optimization method to improve load bearing capacity. Topology optimization taking account of structurally nonlinear and complex behavior is an important topic from the viewpoint of mechanics. However, most methodologies require complex analytical derivation of sensitivity analysis and high computational cost to obtain the optimal solution. In light of this, we propose a practical design method employing the principal direction of Cauchy stress and the directional statistics, to improve load bearing capacity indirectly with much lower computational cost than those based on complex nonlinear structural analysis. Finally, we discuss setting of optimization problems, and demonstrate the accuracy and performance of the proposed method with a series of numerical examples.

Appendix 1: Formulae for each derivative term

This section gives more details about how to derive the sensitivity of the constraint shown in Eqs. (34) to (38).

1.1 Sensitivity of $V^{\text{min}^{\prime }}$

First, it is expressed as follows:

$$\begin{aligned} \frac{\partial V^{\text{min}^{\prime }}}{\partial V_l}= & {} \frac{\partial V^{\text{min}^{\prime }}}{\partial V^{\text{PN}^{\prime }}} \frac{\partial V^{\text{PN}^{\prime }}}{\partial V_l} \end{aligned}$$

(39)

$$\begin{aligned} \frac{\partial V^{\text{min}^{\prime }}}{\partial \rho _j}= & {} \frac{\partial V^{\text{min}^{\prime }}}{\partial V^{\text{PN}^{\prime }}} \frac{\partial V^{\text{PN}^{\prime }}}{\partial \rho _j} \end{aligned}$$

(40)

Each derivative term is expressed by the following formulae:

$$\begin{aligned} \frac{\partial V^{\text{min}^{\prime }}}{\partial V^{\text{PN}^{\prime }}}&=-\frac{1}{\left( V^{\text{PN}^{\prime }}\right)^2} \end{aligned}$$

(41)

$$\begin{aligned} \frac{\partial V^{\text{PN}^{\prime }}}{\partial V_l}&= \left[ \frac{1}{n^{\text{loc}}}\sum _{m\in \Omega } \left( \frac{ \rho _m^{\text{loc}} }{V_m} \right)^p \right]^{\frac{1}{p}-1} \\&\quad \times \frac{1}{n^\text{loc}} \left( \frac{ \rho _l^{\text{loc}} }{V_l} \right)^{p-1} \times \left\{ - \frac{ \rho _l^{\text{loc}} }{\left( V_l \right)^2} \right\} \end{aligned}$$

(42)

$$\begin{aligned} \frac{\partial V^{\text{PN}^{\prime }}}{\partial \rho _j}&= \left[ \frac{1}{n^{\text{loc}}}\sum _{m\in \Omega } \left( \frac{ \rho _m^{\text{loc}} }{V_m} \right)^p \right]^{\frac{1}{p}-1} \\&\quad \times \frac{1}{n^\text{loc}} \left( \frac{ \rho _l^{\text{loc}} }{V_l} \right)^{p-1} \times \frac{ \left( \frac{\eta^{\prime} \rho _j^{\eta^{\prime}-1}}{ \sum _{k\in \mathbb{O}_l} 1} \right) }{V_l} \end{aligned}$$

(43)

1.2 Sensitivity of $V_l$

First, it is expressed as follows:

$$\begin{aligned} \frac{\partial V_l}{\partial \varvec{\sigma }_{r}}= -\frac{\partial \bar{R}_l}{\partial \varvec{\sigma }_{r}}. \end{aligned}$$

(44)

Next, term $\partial \bar{R}_l/\partial \varvec{\sigma }_{r}$ can be calculated according to the chain rule of differentiation as follows:

$$\begin{aligned} \frac{\partial \bar{R}_l}{\partial \varvec{\sigma }_{r}}&= \frac{\partial \bar{R}_l}{\partial \bar{C}_l} \frac{\partial \bar{C}_l}{\partial \varvec{\sigma }_{r}} + \frac{\partial \bar{R}_l}{\partial \bar{S}_l} \frac{\partial \bar{S}_l}{\partial \varvec{\sigma }_{r}} \\&= \frac{\partial \bar{R}_l}{\partial \bar{C}_l} \left( \frac{\partial \bar{C}_l}{\partial \sigma_{r}^{\text{vM}}} \frac{\partial \sigma_{r}^{\text{vM}}}{\partial \varvec{\sigma }_{r}} + \frac{\partial \bar{C}_l}{\partial \theta_{r}} \frac{\partial \theta_{r}}{\partial \varvec{\sigma }_{r}} \right) \\& \quad + \frac{\partial \bar{R}_l}{\partial \bar{S}_l} \left( \frac{\partial \bar{S}_l}{\partial \sigma_{r}^{\text{vM}}} \frac{\partial \sigma_{r}^{\text{vM}}}{\partial \varvec{\sigma }_{r}} + \frac{\partial \bar{S}_l}{\partial \theta_{r}} \frac{\partial \theta_{r}}{\partial \varvec{\sigma }_{r}} \right). \end{aligned}$$

(45)

Each derivative term is expressed by the following formulae:

$$\begin{aligned}&\frac{\partial \bar{R}_l}{\partial \bar{C}_l} =\frac{\bar{C}_l}{\sqrt{\bar{C}_l^2+\bar{S}_l^2}} \end{aligned}$$

(46)

$$\begin{aligned}&\frac{\partial \bar{R}_l}{\partial \bar{S}_l} =\frac{\bar{S}_l}{\sqrt{\bar{C}_l^2+\bar{S}_l^2}} \end{aligned}$$

(47)

$$\begin{aligned}&\frac{\partial \bar{C}_l}{\partial \sigma_{r}^{\text{vM}}} =\frac{ \zeta^{\prime}(\sigma_{r}^{\text{vM}})\cos 2\theta_{r} \left\{ \sum _{r\in \mathbb{O}_l}\zeta (\sigma_{r}^{\text{vM}}) \right\} }{ \left\{ \sum _{r\in \mathbb{O}_l}\zeta (\sigma_{r}^{\text{vM}}) \right\}^2 } \\& -\frac{ \left\{ \sum _{r\in \mathbb{O}_l} \zeta (\sigma_{r}^{\text{vM}}) \cos 2\theta_{r} \right\} \zeta^{\prime}(\sigma_{r}^{\text{vM}}) }{ \left\{ \sum _{r\in \mathbb{O}_l}\zeta (\sigma_{r}^{\text{vM}}) \right\}^2} \end{aligned}$$

(48)

$$\begin{aligned}&\frac{\partial \sigma_{r}^{\text{vM}}}{\partial \varvec{\sigma }_{r}^{\text{vec}}} =\frac{\varvec{P}\varvec{\sigma }_{r}^{\text{vec}}}{\sigma_{r}^{\text{vM}}} \end{aligned}$$

(49)

$$\begin{aligned}&\frac{\partial \bar{C}_l}{\partial \theta_{r}} =\frac{-2}{\sum _{r\in \mathbb{O}_l}\zeta (\sigma_{r}^{\text{vM}})}\zeta (\sigma_{r}^{\text{vM}})\sin 2\theta_{r} \end{aligned}$$

(50)

$$\begin{aligned}&\frac{\partial \bar{S}_l}{\partial \sigma_{r}^{\text{vM}}} =\frac{ \zeta^{\prime}(\sigma_{r}^{\text{vM}})\sin 2\theta_{r} \left\{ \sum _{r\in \mathbb{O}_l}\zeta (\sigma_{r}^{\text{vM}}) \right\} }{ \left\{ \sum _{r\in \mathbb{O}_l}\zeta (\sigma_{r}^{\text{vM}}) \right\}^2 } \\& -\frac{ \left\{ \sum _{r\in \mathbb{O}_l} \zeta (\sigma_{r}^{\text{vM}}) \sin 2\theta_{r} \right\} \zeta^{\prime}(\sigma_{r}^{\text{vM}}) }{ \left\{ \sum _{r\in \mathbb{O}_l}\zeta (\sigma_{r}^{\text{vM}}) \right\}^2} \end{aligned}$$

(51)

$$\begin{aligned}&\frac{\partial \bar{S}_l}{\partial \theta_{r}} =\frac{2}{\sum _{r\in \mathbb{O}_l}\zeta (\sigma_{r}^{\text{vM}})}\zeta (\sigma_{r}^{\text{vM}})\cos 2\theta_{r} \end{aligned}$$

(52)

$$\begin{aligned}&\frac{\partial \zeta (\sigma_{r}^{\text{vM}})}{\partial \sigma_{r}^{\text{vM}}} =\frac{a}{4\cosh^2 \left\{ \frac{1}{2}a \left( \sigma_{r}^{\text{vM}}-\bar{\sigma } \right) \right\} } \end{aligned}$$

(53)

1.3 Sensitivity of $\theta _{r}$

The formulae are expressed as follows:

$$\begin{aligned}&\frac{\partial \theta _{r}}{\partial \varvec{\sigma }_{r}} = \frac{1}{1+\left( \frac{N_{{r}_y}}{N_{{r}_x}} \right)^2} \times \left[ \frac{\frac{\partial N_{{r}_y}}{\partial \varvec{\sigma }}N_{{r}_x} -N_{j_y}\frac{\partial N_{{r}_x}}{\partial \varvec{\sigma }}}{\left( N_{{r}_x}\right)^2} \right] \end{aligned}$$

(54)

$$\begin{aligned}& \left( \varvec{\sigma }_{r} - \lambda _{r} \varvec{I} \right) \frac{\partial \varvec{N}_{r}}{\partial \varvec{\sigma }_{r}} = \varvec{N}_{r} \otimes \frac{\partial \lambda _{r}}{\partial \varvec{\sigma }_{r}} - \varvec{I} \otimes \varvec{N}_{r}, \end{aligned}$$

(55)

where $\varvec{I}$ represents an identity matrix. It should also be noted that, with respect to Eq. (55), $\left( \varvec{\sigma }_{r} - \lambda _{r} \varvec{I} \right)$ has no invert matrix and therefore $\partial {\varvec{N}_{r}}/\partial {\varvec{\sigma}_{r}}$ gives indefinite solutions.