Nonprobabilistic reliability oriented topological optimization for multi-material heat-transfer structures with interval uncertainties

Abstract

This study presents a nonprobabilistic reliability-based topology optimization (NRBTO) framework that combines a multi-material interpolation model and interval mathematics to achieve an optimal layout design for heat-transfer structures under unknown but bounded (UBB) uncertainties. In terms of the uncertainty quantification (UQ) issue, the interval dimension-wise method (IDWM) based on set collocation theory is first proposed to effectively determine the bounds of nodal temperature responses. For safety reasons, the interval reliability (IR) index corresponding to the thermal constraint is defined, and then a new design policy, i.e., the strategy of nonprobabilistic reliability oriented topological optimization is established. To circumvent problems of large-scale variable updating in a multi-material topology optimization procedure, theoretical deductions of the design sensitivity analysis are further given based on the adjoint-vector criterion and the chain principle. The validity and feasibility of the developed methodology are eventually demonstrated by several application examples.

Appendix 1

Since the temperature threshold t_cri is generally set to a large value in actual engineering, the detailed derivation of the last two cases shown in Eq. (29) is given above. Moreover, the derivation methods of the remaining four cases are all the same, and it is not necessary to again describe the detailed derivation process. For the case shown in Fig.6e, let ξ₁ = 1 in the limit-state function (namely Eq. (27)), yielding

$$ {\xi}_2=-\frac{t_{\mathrm{cri}}^c}{\varDelta {T}_{\mathrm{cri}}}+\frac{T_i^c+\varDelta {T}_i}{\varDelta {T}_{cri}}=-\frac{t_{\mathrm{cri}}^c}{\varDelta {T}_{\mathrm{cri}}}+\frac{\overline{T_i}\left(\mathbf{x}\right)}{\varDelta {T}_{cri}} $$

(42)

Since −1 < ξ₂ < 1, it can be obtained from Eq. (42) that

$$ -1<-\frac{t_{\mathrm{cri}}^c}{\varDelta {T}_{\mathrm{cri}}}+\frac{\overline{T_i}\left(\mathbf{x}\right)}{\varDelta {T}_{\mathrm{cri}}}<1\iff \underset{\_}{t_{cri}}<\overline{T_i}\left(\mathbf{x}\right)<\overline{t_{cri}} $$

(43)

Then, letting ξ₂ =  − 1 yields

$$ {\xi}_1=-\frac{T_i^c}{\varDelta {T}_i}+\frac{t_{\mathrm{cri}}^c-\varDelta {T}_{cri}}{\varDelta {T}_i}=-\frac{T_i^c}{\varDelta {T}_i}+\frac{\underset{\_}{t_{cri}}}{\varDelta {T}_i} $$

(44)

Since −1 < ξ₁ < 1, it can be obtained from Eq. (44) that

$$ -1<-\frac{T_i^c}{\varDelta {T}_i}+\frac{\underset{\_}{t_{cri}}}{\varDelta {T}_i}<1\iff \underset{\_}{T_i}\left(\mathbf{x}\right)<\underset{\_}{t_{cri}}<\overline{T_i}\left(\mathbf{x}\right) $$

(45)

From Eqs. (43) and (44), the fifth mathematical programming condition in the piecewise function Eq. (29) can be obtained, namely, with interference and $\underset{\_}{T_i}\left(\mathbf{x}\right)<\underset{\_}{t_{cri}}<\overline{T_i}\left(\mathbf{x}\right)<\overline{t_{cri}}$. For this case, according to the definition of R_{i, s} in Eq. (28), namely

$$ {R}_{i,s}=1-\frac{S_{shade}}{S_{total}} $$

(46)

S_shade and S_total can be calculated by geometric formulas, and then Eq. (46) can be converted to

$$ {R}_{i,s}=1-\frac{\frac{1}{2}\times \left(1+{\xi}_2\right)\times \left(1-{\xi}_1\right)}{2\times 2}=1-\frac{\left(1+{\xi}_2\right)\times \left(1-{\xi}_1\right)}{8} $$

(47)

Substituting Eq. (42) and Eq. (43) into Eq. (47) yields

$$ {R}_{i,s}=1-\frac{{\left({t}_{\mathrm{cri}}^c-{t}_i^c-\varDelta {T}_{cri}-\varDelta {T}_i\right)}^2}{8\varDelta {T}_{\mathrm{cri}}\varDelta {T}_i},\mathrm{with}\ \mathrm{interference}\ \mathrm{and}\ \underset{\_}{T_i}\left(\mathbf{x}\right)<\underset{\_}{t_{cri}}<\overline{T_i}\left(\mathbf{x}\right)<\overline{t_{cri}} $$

(48)

For the case shown in Fig. 6f, let ξ₁ = 1 in the limit-state function (namely Eq. (27)), which yields

$$ {\xi}_2=-\frac{t_{\mathrm{cri}}^c}{\varDelta {T}_{cri}}+\frac{T_i^c+\varDelta {T}_i}{\varDelta {T}_{cri}}=-\frac{t_{\mathrm{cri}}^c}{\varDelta {T}_{cri}}+\frac{\overline{T_i}\left(\mathbf{x}\right)}{\varDelta {T}_{cri}} $$

(49)

Since ξ₂ ≤  − 1, it can be obtained from Eq. (48) that

$$ \overline{T_i}\left(\mathbf{x}\right)\le \left(\frac{t_{\mathrm{cri}}^c}{\varDelta {T}_{cri}}-1\right)\times \varDelta {T}_{cri}={t}_{\mathrm{cri}}^c-\varDelta {T}_{cri}=\underset{\_}{t_{\mathrm{cri}}} $$

(50)

Then, letting ξ₂ =  − 1 yields

$$ {\xi}_1=-\frac{T_i^c}{\varDelta {T}_i}+\frac{t_{\mathrm{cri}}^c-\varDelta {T}_{\mathrm{cri}}}{\varDelta {T}_i}=-\frac{T_i^c}{\varDelta {T}_i}+\frac{\underset{\_}{t_{\mathrm{cri}}}}{\varDelta {T}_i} $$

(51)

Since ξ₁ ≥ 1, it can be obtained from Eq. (50) that

$$ \underset{\_}{t_{cri}}\ge \left(\frac{T_i^c}{\varDelta {T}_i}+1\right)\times \varDelta T=\frac{\overline{T_i}\left(\mathbf{x}\right)}{\varDelta {T}_i}\times \varDelta {T}_i=\overline{T_i}\left(\mathbf{x}\right) $$

(52)

From Eqs. (49) and (51), the sixth mathematical programming condition in the piecewise function Eq. (29) can be obtained, namely, non-interference and $\overline{T_i}\left(\mathbf{x}\right)\le \underset{\_}{t_{cri}}$. For this case, according to the definition of R_{i, s} in Eq. (28), namely

$$ {R}_{i,s}=1+{D}_{safe} $$

(53)

According to the formula of the distance from the point (namely, (1, −1)) to the line (namely ${\widehat{G}}_i\left(\boldsymbol{\upxi} \right)={t}_{cri}^c-{T}_i^c+\varDelta {T}_{cri}{\xi}_2-\varDelta {T}_i{\xi}_1=0$) in the Cartesian coordinate system, the following expression of R_{i, s} can be obtained

$$ {R}_{i,s}=1+\frac{t_{\mathrm{cri}}^c-{t}_i^c-\varDelta {T}_{cri}-\varDelta {T}_i}{\sqrt{{\left(\varDelta {T}_{\mathrm{cri}}\right)}^2+{\left(\varDelta {T}_i\right)}^2}},\kern1em \mathrm{non}-\mathrm{interference}\ \mathrm{and}\ \overline{T_i}\left(\mathbf{x}\right)\le \underset{\_}{t_{cri}} $$

(54)