Hollow structural design in topology optimization via moving morphable component method

Abstract

Hollow structures are widely used in industry because of the high stiffness-to-mass ratio and mature joining technology. However, in topology optimization (TO), the TO results are almost solid structures because the geometries of hollow structures cannot be represented in an explicit way. Therefore, this paper innovatively presents an explicit three-dimensional TO method to obtain the hollow structures using moving morphable components (MMCs), which is achieved by combining two topology description functions, namely, internal and external topology description functions. Especially, the combining topology description functions of the hollow component are derived for the first time. Compared to the solid TO results, the structural stiffness of the hollow TO results can be further improved. Finally, the TO problems of the three-dimensional square Mindlin cantilever plate, cantilever beam and miniature bending beam demonstrate the effectiveness of the proposed method.

Sensitivities of internal and external TDFs

The sensitivities of TDFs with respect to L can be calculated as follows:

$$ \Big\{{\displaystyle \begin{array}{l}\frac{\partial {\phi}_i^1\left({\mathbf{x}}_i^e\right)}{\partial L}=p{\left(x\hbox{'}\right)}^p{(L)}^{-p-1}\kern0.1em \\ {}\frac{\partial {\phi}_i^2\left({\mathbf{x}}_i^e\right)}{\partial L}=p{\left(x\hbox{'}\right)}^p{(L)}^{-p-1}\end{array}} $$

(29)

The sensitivities of TDFs with respect to α can be calculated as follows:

$$ \Big\{{\displaystyle \begin{array}{l}\frac{\partial {\phi}_i^1\left({\mathbf{x}}_i^e\right)}{\partial \alpha }=-\frac{p}{L}{\left(\frac{x^{\prime }}{L}\right)}^{p-1}\frac{\partial {x}^{\prime }}{\partial \alpha }-p{\left(\frac{y^{\prime }}{f_1\left({x}^{\prime}\right)}\right)}^{p-1}\left(\frac{\frac{\partial {y}^{\prime }}{\partial \alpha }{f}_1\left({x}^{\prime}\right)-\frac{\partial {f}_1\left({x}^{\prime}\right)}{\partial {x}^{\prime }}\frac{\partial {x}^{\prime }}{\partial \alpha }{y}^{\prime }}{f_1\left({x}^{\prime}\right){f}_1\left({x}^{\prime}\right)}\right)\\ {}\kern4.399998em -p{\left(\frac{z^{\prime }}{g_1\left({x}^{\prime}\right)}\right)}^{p-1}\left(\frac{\frac{\partial {z}^{\prime }}{\partial \alpha }{g}_1\left({x}^{\prime}\right)-\frac{\partial {g}_1\left({x}^{\prime}\right)}{\partial {x}^{\prime }}\frac{\partial {x}^{\prime }}{\partial \alpha }{z}^{\prime }}{g_1\left({x}^{\prime}\right){g}_1\left({x}^{\prime}\right)}\right)\kern0.1em \\ {}\frac{\partial {\phi}_i^2\left({\mathbf{x}}_i^e\right)}{\partial \alpha }=-\frac{p}{L}{\left(\frac{x^{\prime }}{L}\right)}^{p-1}\frac{\partial {x}^{\prime }}{\partial \alpha }-p{\left(\frac{y^{\prime }}{f_2\left({x}^{\prime}\right)}\right)}^{p-1}\left(\frac{\frac{\partial {y}^{\prime }}{\partial \alpha }{f}_2\left({x}^{\prime}\right)-\frac{\partial {f}_2\left({x}^{\prime}\right)}{\partial {x}^{\prime }}\frac{\partial {x}^{\prime }}{\partial \alpha }{y}^{\prime }}{f_2\left({x}^{\prime}\right){f}_2\left({x}^{\prime}\right)}\right)\\ {}\kern4.299998em -p{\left(\frac{z^{\prime }}{g_2\left({x}^{\prime}\right)}\right)}^{p-1}\left(\frac{\frac{\partial {z}^{\prime }}{\partial \alpha }{g}_2\left({x}^{\prime}\right)-\frac{\partial {g}_2\left({x}^{\prime}\right)}{\partial {x}^{\prime }}\frac{\partial {x}^{\prime }}{\partial \alpha }{z}^{\prime }}{g_2\left({x}^{\prime}\right){g}_2\left({x}^{\prime}\right)}\right)\end{array}} $$

(30)

with

$$ \left\{\begin{array}{c}\partial {x}^{\prime }/\partial \alpha \\ {}\partial {y}^{\prime }/\partial \alpha \\ {}\partial {z}^{\prime }/\partial \alpha \end{array}\right\}=\frac{\partial \mathbf{T}}{\partial \alpha}\left\{\begin{array}{c}x-{x}_0\\ {}y-{y}_0\\ {}z-{z}_0\end{array}\right\} $$

(31)

and

$$ \frac{\partial \mathbf{T}}{\partial \alpha }=\left[\begin{array}{ccc}0& 0& 0\\ {}{c}_a\cdot {s}_b\cdot {c}_t-{s}_a\cdot {s}_t& -{c}_a\cdot {s}_b\cdot {s}_t-{s}_a\cdot {c}_t& -{c}_a\cdot {c}_b\\ {}{s}_a\cdot {s}_b\cdot {c}_t+{s}_a\cdot {s}_t& -{s}_a\cdot {s}_b\cdot {s}_t+{s}_a\cdot {c}_t& -{s}_a\cdot {c}_b\end{array}\right] $$

(32)

The sensitivities of TDFs with respect to β or θ can be obtained by replacing ∂/∂α with ∂/∂β or ∂/∂θ. ∂T/∂β and ∂T/∂θ are derived as follows:

$$ \frac{\partial \mathbf{T}}{\partial \alpha }=\left[\begin{array}{ccc}-{s}_b\cdot {c}_t& {s}_b\cdot {s}_t& {c}_b\\ {}{s}_a\cdot {c}_b\cdot {c}_t& -{s}_a\cdot {c}_b\cdot {s}_t& {s}_a\cdot {s}_b\\ {}-{c}_a\cdot {c}_b\cdot {c}_t& {c}_a\cdot {c}_b\cdot {s}_t& -{c}_a\cdot {s}_b\end{array}\right] $$

(33)

$$ \frac{\partial \mathbf{T}}{\partial \alpha }=\left[\begin{array}{ccc}-{c}_b\cdot {s}_t& -{c}_b\cdot {c}_t& 0\\ {}-{s}_a\cdot {s}_b\cdot {s}_t+{c}_a\cdot {c}_t& -{s}_a\cdot {s}_b\cdot {c}_t-{c}_a\cdot {s}_t& 0\\ {}{c}_a\cdot {s}_b\cdot {s}_t+{s}_a\cdot {c}_t& {c}_a\cdot {s}_b\cdot {c}_t-{s}_a\cdot {s}_t& 0\end{array}\right] $$

(34)

The sensitivities of TDFs with respect to x₀ can be expressed as follows:

$$ \Big\{{\displaystyle \begin{array}{l}\frac{\partial {\phi}_i^1\left({\mathbf{x}}_i^e\right)}{\partial {x}_0}=-\frac{p}{L}{\left(\frac{x^{\prime }}{L}\right)}^{p-1}\frac{\partial {x}^{\prime }}{\partial {x}_0}-p{\left(\frac{y^{\prime }}{f_1\left({x}^{\prime}\right)}\right)}^{p-1}\left(\frac{\frac{\partial {y}^{\prime }}{\partial {x}_0}{f}_1\left({x}^{\prime}\right)-\frac{\partial {f}_1\left({x}^{\prime}\right)}{\partial {x}^{\prime }}\frac{\partial {x}^{\prime }}{\partial {x}_0}{y}^{\prime }}{f_1\left({x}^{\prime}\right){f}_1\left({x}^{\prime}\right)}\right)\\ {}\kern4.399998em -p{\left(\frac{z^{\prime }}{g_1\left({x}^{\prime}\right)}\right)}^{p-1}\left(\frac{\frac{\partial {z}^{\prime }}{\partial {x}_0}{g}_1\left({x}^{\prime}\right)-\frac{\partial {g}_1\left({x}^{\prime}\right)}{\partial {x}^{\prime }}\frac{\partial {x}^{\prime }}{\partial {x}_0}{z}^{\prime }}{g_1\left({x}^{\prime}\right){g}_1\left({x}^{\prime}\right)}\right)\kern0.1em \\ {}\frac{\partial {\phi}_i^2\left({\mathbf{x}}_i^e\right)}{\partial {x}_0}=-\frac{p}{L}{\left(\frac{x^{\prime }}{L}\right)}^{p-1}\frac{\partial {x}^{\prime }}{\partial {x}_0}-p{\left(\frac{y^{\prime }}{f_2\left({x}^{\prime}\right)}\right)}^{p-1}\left(\frac{\frac{\partial {y}^{\prime }}{\partial {x}_0}{f}_2\left({x}^{\prime}\right)-\frac{\partial {f}_2\left({x}^{\prime}\right)}{\partial {x}^{\prime }}\frac{\partial {x}^{\prime }}{\partial {x}_0}{y}^{\prime }}{f_2\left({x}^{\prime}\right){f}_2\left({x}^{\prime}\right)}\right)\\ {}\kern4.299998em -p{\left(\frac{z^{\prime }}{g_2\left({x}^{\prime}\right)}\right)}^{p-1}\left(\frac{\frac{\partial {z}^{\prime }}{\partial {x}_0}{g}_2\left({x}^{\prime}\right)-\frac{\partial {g}_2\left({x}^{\prime}\right)}{\partial {x}^{\prime }}\frac{\partial {x}^{\prime }}{\partial {x}_0}{z}^{\prime }}{g_2\left({x}^{\prime}\right){g}_2\left({x}^{\prime}\right)}\right)\end{array}} $$

(35)

where

$$ \left[\begin{array}{ccc}\partial {x}^{\prime }/\partial {x}_0& \partial {x}^{\prime }/\partial {y}_0& \partial {x}^{\prime }/\partial {z}_0\\ {}\partial {y}^{\prime }/\partial {x}_0& \partial {y}^{\prime }/\partial {y}_0& \partial {y}^{\prime }/\partial {z}_0\\ {}\partial {z}^{\prime }/\partial {x}_0& \partial {z}^{\prime }/\partial {y}_0& \partial {z}^{\prime }/\partial {z}_0\end{array}\right]=-\mathbf{T} $$

(36)

and

$$ \frac{\partial {f}_1\left({x}^{\prime}\right)}{\partial {x}^{\prime }}=\frac{\partial {g}_1\left({x}^{\prime}\right)}{\partial {x}^{\prime }}=\frac{\partial {f}_2\left({x}^{\prime}\right)}{\partial {x}^{\prime }}=\frac{\partial {g}_2\left({x}^{\prime}\right)}{\partial {x}^{\prime }}=0 $$

(37)

The sensitivities of TDFs with respect to y₀ or z₀ can be obtained by replacing ∂/∂x₀ with ∂/∂y₀ or ∂/∂z₀.

The sensitivities of TDFs with respect to a₁, c_b1, and c_t1 can be expressed as follows:

$$ \Big\{{\displaystyle \begin{array}{l}\frac{\partial {\phi}_i^1\left({\mathbf{x}}_i^e\right)}{\partial {a}_1}=p{\left(y\hbox{'}\right)}^p{\left({f}_1\left({x}^{\prime}\right)\right)}^{-p-1}+p{c}_{b_1}{\left({z}^{\prime}\right)}^p{\left({g}_1\left({x}^{\prime}\right)\right)}^{-p-1}\kern0.1em \\ {}\frac{\partial {\phi}_i^2\left({\mathbf{x}}_i^e\right)}{\partial {a}_1}=p\left(1-{c}_{t_1}{c}_{b_1}\right){\left(y\hbox{'}\right)}^p{\left({f}_2\left({x}^{\prime}\right)\right)}^{-p-1}+p\left({c}_{b_1}-{c}_{t_1}{c}_{b_1}\right){\left({z}^{\prime}\right)}^p{\left({g}_2\left({x}^{\prime}\right)\right)}^{-p-1}\end{array}} $$

(38)

$$ \Big\{{\displaystyle \begin{array}{l}\frac{\partial {\phi}_i^1\left({\mathbf{x}}_i^e\right)\pi }{\partial {c}_{b_1}}={a}_1p{\left(z\hbox{'}\right)}^p{\left({g}_1\left(x\hbox{'}\right)\right)}^{-p-1}\kern0.2em \\ {}\frac{\partial {\phi}_i^2\left({\mathbf{x}}_i^e\right)}{\partial {c}_{b_1}}=-p{c}_{t_1}{a}_1{\left(y\hbox{'}\right)}^p{\left({f}_2\left(x\hbox{'}\right)\right)}^{-p-1}+p\left({a}_1-{c}_{t_1}{a}_1\right){\left(z\hbox{'}\right)}^p{\left({g}_2\left(x\hbox{'}\right)\right)}^{-p-1}\end{array}} $$

(39)

$$ \Big\{{\displaystyle \begin{array}{l}\frac{\partial {\phi}_i^1\left({\mathbf{x}}_i^e\right)}{\partial {c}_{t_1}}=0\kern0.1em \\ {}\frac{\partial {\phi}_i^2\left({\mathbf{x}}_i^e\right)}{\partial {c}_{t_1}}=-p{a}_1{c}_{b_1}\left[{\left({y}^{\prime}\right)}^p{\left({f}_2\left({x}^{\prime}\right)\right)}^{-p-1}+{\left({z}^{\prime}\right)}^p{\left({g}_2\left({x}^{\prime}\right)\right)}^{-p-1}\right]\kern0.1em \end{array}} $$

(40)