A new three-dimensional topology optimization method based on moving morphable components (MMCs)

Abstract

In the present paper, a new method for solving three-dimensional topology optimization problem is proposed. This method is constructed under the so-called moving morphable components based solution framework. The novel aspect of the proposed method is that a set of structural components is introduced to describe the topology of a three-dimensional structure and the optimal structural topology is found by optimizing the layout of the components explicitly. The standard finite element method with ersatz material is adopted for structural response analysis and the shape sensitivity analysis only need to be carried out along the structural boundary. Compared to the existing methods, the description of structural topology is totally independent of the finite element/finite difference resolution in the proposed solution framework and therefore the number of design variables can be reduced substantially. Some widely investigated benchmark examples, in the three-dimensional topology optimization designs, are presented to demonstrate the effectiveness of the proposed approach.

Appendix: The terms in the expression of shape sensitivity

The terms in the expression of Eqs. (11)–(13) can be calculated as follows:

$$\begin{aligned} A^{i}= & {} - \frac{{p\left( {\left( {x^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{1}^{i} } \right) ^{p} }}\frac{{\partial \left( {x^{\prime }} \right) ^{i} }}{{\partial x_{0}^{i} }} - \frac{{p\left( {\left( {y^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{2}^{i} } \right) ^{p} }}\frac{{\partial \left( {y^{\prime }} \right) ^{i} }}{{\partial x_{0}^{i} }}\nonumber \\&-\, \frac{{p\left( {\left( {z^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{3}^{i} } \right) ^{p} }}\frac{{\partial \left( {z^{\prime }} \right) ^{i} }}{{\partial x_{0}^{i} }}, \end{aligned}$$

(15)

$$\begin{aligned} B^{i}= & {} - \frac{{p\left( {\left( {x^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{1}^{i} } \right) ^{p} }}\frac{{\partial \left( {x^{\prime }} \right) ^{i} }}{{\partial y_{0}^{i} }} - \frac{{p\left( {\left( {y^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{2}^{i} } \right) ^{p} }}\frac{{\partial \left( {y^{\prime }} \right) ^{i} }}{{\partial y_{0}^{i} }}\nonumber \\&-\, \frac{{p\left( {\left( {z^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{3}^{i} } \right) ^{p} }}\frac{{\partial \left( {z^{\prime }} \right) ^{i} }}{{\partial y_{0}^{i} }}, \end{aligned}$$

(16)

$$\begin{aligned} C^{i}= & {} - \frac{{p\left( {\left( {x^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{1}^{i} } \right) ^{p} }}\frac{{\partial \left( {x^{\prime }} \right) ^{i} }}{{\partial z_{0}^{i} }} - \frac{{p\left( {\left( {y^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{2}^{i} } \right) ^{p} }}\frac{{\partial \left( {y^{\prime }} \right) ^{i} }}{{\partial z_{0}^{i} }}\nonumber \\&-\, \frac{{p\left( {\left( {z^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{3}^{i} } \right) ^{p} }}\frac{{\partial \left( {z^{\prime }} \right) ^{i} }}{{\partial z_{0}^{i} }}, \end{aligned}$$

(17)

$$\begin{aligned} D^{i}= & {} p\frac{{\left( {\left( {x^{\prime }} \right) ^{i} } \right) ^{p} }}{{(L_{1}^{i} )^{{p + 1}} }}, \end{aligned}$$

(18)

$$\begin{aligned} E^{i}= & {} p\frac{{\left( {\left( {y^{\prime }} \right) ^{i} } \right) ^{p} }}{{(L_{2}^{i} )^{{p + 1}} }}, \end{aligned}$$

(19)

$$\begin{aligned} F^{i}= & {} p\frac{{\left( {\left( {z^{\prime }} \right) ^{i} } \right) ^{p} }}{{(L_{3}^{i} )^{{p + 1}}}}, \end{aligned}$$

(20)

$$\begin{aligned} G^{i}= & {} - \frac{{p\left( {\left( {x^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{1}^{i} } \right) ^{p} }}\frac{{\partial \left( {x^{\prime }} \right) ^{i} }}{{\partial s_{a}^{i} }} - \frac{{p\left( {\left( {y^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{2}^{i} } \right) ^{p} }}\frac{{\partial \left( {y^{\prime }} \right) ^{i} }}{{\partial s_{a}^{i} }}\nonumber \\&-\, \frac{{p\left( {\left( {z^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{3}^{i} } \right) ^{p} }}\frac{{\partial \left( {z^{\prime }} \right) ^{i} }}{{\partial s_{a}^{i} }}, \end{aligned}$$

(21)

$$\begin{aligned} H^{i}= & {} - \frac{{p\left( {\left( {x^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{1}^{i} } \right) ^{p} }}\frac{{\partial \left( {x^{\prime }} \right) ^{i} }}{{\partial s_{b}^{i} }} - \frac{{p\left( {\left( {y^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{2}^{i} } \right) ^{p} }}\frac{{\partial \left( {y^{\prime }} \right) ^{i} }}{{\partial s_{b}^{i} }}\nonumber \\&-\, \frac{{p\left( {\left( {z^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{3}^{i} } \right) ^{p} }}\frac{{\partial \left( {z^{\prime }} \right) ^{i} }}{{\partial s_{b}^{i} }}, \end{aligned}$$

(22)

$$\begin{aligned} I^{i}= & {} - \frac{{p\left( {\left( {x^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{1}^{i} } \right) ^{p} }}\frac{{\partial \left( {x^{\prime }} \right) ^{i} }}{{\partial s_{t}^{i} }} - \frac{{p\left( {\left( {y^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{2}^{i} } \right) ^{p} }}\frac{{\partial \left( {y^{\prime }} \right) ^{i} }}{{\partial s_{t}^{i} }}\nonumber \\&-\, \frac{{p\left( {\left( {z^{\prime }} \right) ^{i} } \right) ^{{p - 1}} }}{{\left( {L_{3}^{i} } \right) ^{p} }}\frac{{\partial \left( {z^{\prime }} \right) ^{i} }}{{\partial s_{t}^{i} }}, \end{aligned}$$

(23)

where the derivates of $x',y'$ and $z'$ with respective to the design variables can be calculated as

$$\begin{aligned} \frac{{\partial \left( {x^{\prime }} \right) ^{i} }}{{\partial x_{0}^{i} }}= & {} - R_{{11}}^{i} ,~~\frac{{\partial \left( {x^{\prime }} \right) ^{i} }}{{\partial y_{0}^{i} }} = - R_{{12}}^{i} ,~~\frac{{~\partial \left( {x^{\prime }} \right) ^{i} }}{{\partial z_{0}^{i} }} = - R_{{13}}^{i} , \end{aligned}$$

(24)

$$\begin{aligned} \frac{{\partial \left( {x^{\prime }} \right) ^{i} }}{{\partial s_{a}^{i} }}= & {} 0, \end{aligned}$$

(25)

$$\begin{aligned} \frac{{\partial \left( {x^{\prime }} \right) ^{i} }}{{\partial s_{b}^{i} }}= & {} - \frac{{s_{b}^{i} }}{{c_{b}^{i} }}c_{t} \left( {x - x_{0}^{i} } \right) - \frac{{s_{b}^{i} }}{{c_{b}^{i} }}s_{t}^{i} \left( {y - y_{0}^{i} } \right) + \left( {z - z_{0}^{i} } \right) , \end{aligned}$$

(26)

$$\begin{aligned} \frac{{\partial \left( {x^{\prime }} \right) ^{i} }}{{\partial s_{t}^{i} }}= & {} - c_{b}^{i} \frac{{s_{t}^{i} }}{{c_{t}^{i} }}\left( {x - x_{0}^{i} } \right) - c_{b}^{i} \left( {y - y_{0}^{i} } \right) , \end{aligned}$$

(27)

$$\begin{aligned} \frac{{\partial \left( {y^{\prime }} \right) ^{i} }}{{\partial x_{0}^{i} }}= & {} - R_{{21}}^{i} ,~~\frac{{\partial \left( {y^{\prime }} \right) ^{i} }}{{\partial y_{0}^{i} }} = - R_{{22}}^{i} ,~~\frac{{\partial \left( {y^{\prime }} \right) ^{i} }}{{\partial z_{0}^{i} }} = - R_{{23}}^{i} , \end{aligned}$$

(28)

$$\begin{aligned} \frac{{\partial \left( {y^{\prime }} \right) ^{i} }}{{\partial s_{a}^{i} }}= & {} \left( {s_{b}^{i} c_{t}^{i} - \frac{{s_{a}^{i} }}{{c_{a}^{i} }}s_{t}^{i} } \right) \left( {x - x_{0}^{i} } \right) \nonumber \\&+\, \left( { - s_{b}^{i} s_{t}^{i} - \frac{{s_{a}^{i} }}{{c_{a}^{i} }}c_{t}^{i} } \right) \left( {y - y_{0}^{i} } \right) - c_{b}^{i} \left( {z - z_{0}^{i} } \right) , \end{aligned}$$

(29)

$$\begin{aligned} \frac{{\partial \left( {y^{\prime }} \right) ^{i} }}{{\partial s_{b}^{i} }}= & {} s_{a}^{i} c_{t}^{i} \left( {x - x_{0}^{i} } \right) - s_{a}^{i} s_{t}^{i} \left( {y - y_{0}^{i} } \right) + s_{a}^{i} \frac{{s_{b}^{i} }}{{c_{b}^{i} }}\left( {z - z_{0}^{i} } \right) , \end{aligned}$$

(30)

$$\begin{aligned} \frac{{\partial \left( {y^{\prime }} \right) ^{i} }}{{\partial s_{t}^{i} }}= & {} \left( { - s_{a}^{i} s_{b}^{i} \frac{{s_{t}^{i} }}{{c_{t}^{i} }} + c_{a}^{i} } \right) \left( {x - x_{0}^{i} } \right) \nonumber \\&+\, \left( { - s_{a}^{i} s_{b}^{i} - c_{a}^{i} \frac{{s_{t}^{i} }}{{c_{t}^{i} }}} \right) \left( {y - y_{0}^{i} }\right) \end{aligned}$$

(31)

and

$$\begin{aligned} \frac{{\partial \left( {z^{\prime }} \right) ^{i} }}{{\partial x_{0}^{i} }}= & {} - R_{{31}}^{i} ,~~\frac{{\partial \left( {z^{\prime }} \right) ^{i} }}{{\partial y_{0}^{i} }} = - R_{{32}}^{i} ,~~\frac{{\partial \left( {z^{\prime }} \right) ^{i} }}{{\partial z_{0}^{i} }} = - R_{{33}}^{i} , \end{aligned}$$

(32)

$$\begin{aligned} \frac{{\partial \left( {z^{\prime }} \right) ^{i} }}{{\partial s_{a}^{i} }}= & {} \left( {\frac{{s_{a}^{i} }}{{c_{a}^{i} }}s_{b}^{i} c_{t}^{i} + s_{t}^{i} } \right) \left( {x - x_{0}^{i} } \right) + \left( { - \frac{{s_{a}^{i} }}{{c_{a}^{i} }}s_{b}^{i} s_{t}^{i} + c_{t}^{i} } \right) \nonumber \\&\times \left( {y - y_{0}^{i} } \right) ~ - \frac{{s_{a}^{i} }}{{c_{a}^{i} }}c_{b}^{i} \left( {z - z_{0}^{i} } \right) , \end{aligned}$$

(33)

$$\begin{aligned} \frac{{\partial \left( {z^{\prime }} \right) ^{i} }}{{\partial s_{b}^{i} }}= & {} - c_{a}^{i} c_{t}^{i} \left( {x - x_{0}^{i} } \right) + c_{a}^{i} s_{t}^{i} \left( {y - y_{0}^{i} } \right) ~~ - c_{a}^{i} \frac{{s_{b}^{i} }}{{c_{b}^{i} }}\left( {z - z_{0}^{i} } \right) ,\nonumber \\ \end{aligned}$$

(34)

$$\begin{aligned} \frac{{\partial \left( {z^{\prime }} \right) ^{i} }}{{\partial s_{t}^{i} }}= & {} \left( {c_{a}^{i} s_{b}^{i} \frac{{s_{t}^{i} }}{{c_{t}^{i} }} + s_{a}^{i} } \right) \left( {x - x_{0}^{i} } \right) + \left( {c_{a}^{i} s_{b}^{i} - s_{a}^{i} \frac{{s_{t}^{i} }}{{c_{t}^{i} }}} \right) \left( {y - y_{0}^{i} } \right) .\nonumber \\ \end{aligned}$$

(35)