An efficient and easy-to-extend Matlab code of the Moving Morphable Component (MMC) method for three-dimensional topology optimization

Abstract

Explicit topology optimization methods have received ever-increasing interest in recent years. In particular, a 188-line Matlab code of the two-dimensional (2D) Moving Morphable Component (MMC)-based topology optimization method was released by Zhang et al. (Struct Multidiscip Optim 53(6):1243-1260, 2016). The present work aims to propose an efficient and easy-to-extend 256-line Matlab code of the MMC method for three-dimensional (3D) topology optimization implementing some new numerical techniques. To be specific, by virtue of the function aggregation technique, accurate sensitivity analysis, which is also easy-to-extend to other problems, is achieved. Besides, based on an efficient identification algorithm for load transmission path, the degrees of freedoms (DOFs) not belonging to the load transmission path are removed in finite element analysis (FEA), which significantly accelerates the optimization process. As a result, compared to the corresponding 188-line 2D code, the performance of the optimization results, the computational efficiency of FEA, and the convergence rate and the robustness of optimization process are greatly improved. For the sake of completeness, a refined 218-line Matlab code implementing the 2D-MMC method is also provided.

Appendices

Appendix 1: Exact expressions of $\frac{\partial \phi ^i}{\partial d_j^i}$ in the MMC3d code

For conciseness, the following notations are introduced

$$\begin{aligned} w \triangleq \left( \frac{x'}{L_1^i}\right) ^p + \left( \frac{y'}{L_2^i}\right) ^p + \left( \frac{z'}{L_3^i}\right) ^p \end{aligned}$$

(20)

$$\begin{aligned} {\tilde{x}}=x'/L_1^i, \quad {\tilde{y}}=y'/L_2^i, \quad {\tilde{z}}=z'/L_3^i \end{aligned}$$

(21)

$$\begin{aligned} R \triangleq \begin{bmatrix} \cos {\beta ^i}\cos {\gamma ^i} &{} \cos {\beta ^i}\sin {\gamma ^i} &{} -\sin {\beta ^i} \\ \sin {\alpha ^i}\sin {\beta ^i}\cos {\gamma ^i} - \cos {\alpha ^i}\sin {\gamma ^i} &{} \sin {\alpha ^i}\sin {\beta ^i}\sin {\gamma ^i} + \cos {\alpha ^i}\cos {\gamma ^i} &{} \sin {\alpha ^i}\cos {\beta ^i} \\ \cos {\alpha ^i}\sin {\beta ^i}\cos {\gamma ^i} + \sin {\alpha ^i}\sin {\gamma ^i} &{} \cos {\alpha ^i}\sin {\beta ^i}\sin {\gamma ^i} - \sin {\alpha ^i}\cos {\gamma ^i} &{} \cos {\alpha ^i}\cos {\beta ^i} \end{bmatrix} \end{aligned}$$

(22)

$$\begin{aligned} R^\alpha \triangleq \begin{bmatrix} 0 &{} 0 &{} 0 \\ \cos {\alpha ^i}\sin {\beta ^i}\cos {\gamma ^i} + \sin {\alpha ^i}\sin {\gamma ^i} &{} \cos {\alpha ^i}\sin {\beta ^i}\sin {\gamma ^i} - \sin {\alpha ^i}\cos {\gamma ^i} &{} \cos {\alpha ^i}\cos {\beta ^i} \\ -\sin {\alpha ^i}\sin {\beta ^i}\cos {\gamma ^i} + \cos {\alpha ^i}\sin {\gamma ^i} &{} -\sin {\alpha ^i}\sin {\beta ^i}\sin {\gamma ^i} - \cos {\alpha ^i}\cos {\gamma ^i} &{} -\sin {\alpha ^i}\cos {\beta ^i} \end{bmatrix} \end{aligned}$$

(23)

$$\begin{aligned} R^\beta \triangleq \begin{bmatrix} -\sin {\beta ^i}\cos {\gamma ^i} &{} -\sin {\beta ^i}\sin {\gamma ^i} &{} -\cos {\beta ^i} \\ \sin {\alpha ^i}\cos {\beta ^i}\cos {\gamma ^i} &{} \sin {\alpha ^i}\cos {\beta ^i}\sin {\gamma ^i} &{} -\sin {\alpha ^i}\sin {\beta ^i} \\ \cos {\alpha ^i}\cos {\beta ^i}\cos {\gamma ^i} &{} \cos {\alpha ^i}\cos {\beta ^i}\sin {\gamma ^i} &{} -\cos {\alpha ^i}\sin {\beta ^i} \end{bmatrix} \end{aligned}$$

(24)

$$\begin{aligned} R^\gamma \triangleq \begin{bmatrix} -\cos {\beta ^i}\sin {\gamma ^i} &{} \cos {\beta ^i}\cos {\gamma ^i} &{} 0 \\ -\sin {\alpha ^i}\sin {\beta ^i}\sin {\gamma ^i} - \cos {\alpha ^i}\cos {\gamma ^i} &{} \sin {\alpha ^i}\sin {\beta ^i}\cos {\gamma ^i} - \cos {\alpha ^i}\sin {\gamma ^i} &{} 0 \\ -\cos {\alpha ^i}\sin {\beta ^i}\sin {\gamma ^i} + \sin {\alpha ^i}\cos {\gamma ^i} &{} \cos {\alpha ^i}\sin {\beta ^i}\cos {\gamma ^i} + \sin {\alpha ^i}\sin {\gamma ^i} &{} 0 \end{bmatrix} \end{aligned}$$

(25)

Then the exact formulations of $\partial \phi ^i/\partial d_j^i$ can be expressed as

$$\begin{aligned}&\frac{\partial \phi ^i}{\partial x_0^i}=w^{\frac{1-p}{p}}\left( \frac{{\tilde{x}}^{p-1}}{L_1^i}R_{11} + \frac{{\tilde{y}}^{p-1}}{L_2^i}R_{21} + \frac{{\tilde{z}}^{p-1}}{L_3^i}R_{31}\right) \end{aligned}$$

(26)

$$\begin{aligned} \frac{\partial \phi ^i}{\partial y_0^i}=w^{\frac{1-p}{p}}\left( \frac{{\tilde{x}}^{p-1}}{L_1^i}R_{12} + \frac{{\tilde{y}}^{p-1}}{L_2^i}R_{22} + \frac{{\tilde{z}}^{p-1}}{L_3^i}R_{32}\right) \end{aligned}$$

(27)

$$\begin{aligned} \frac{\partial \phi ^i}{\partial z_0^i}=w^{\frac{1-p}{p}}\left( \frac{{\tilde{x}}^{p-1}}{L_1^i}R_{13} + \frac{{\tilde{y}}^{p-1}}{L_2^i}R_{23} + \frac{{\tilde{z}}^{p-1}}{L_3^i}R_{33}\right) \end{aligned}$$

(28)

$$\begin{aligned} \frac{\partial \phi ^i}{\partial L_1^i}=\frac{1}{L_1^i}w^{\frac{1-p}{p}}{\tilde{x}}^p \end{aligned}$$

(29)

$$\begin{aligned} \frac{\partial \phi ^i}{\partial L_2^i}=\frac{1}{L_2^i}w^{\frac{1-p}{p}}{\tilde{y}}^p \end{aligned}$$

(30)

$$\begin{aligned} \frac{\partial \phi ^i}{\partial L_3^i}=\frac{1}{L_3^i}w^{\frac{1-p}{p}}{\tilde{z}}^p \end{aligned}$$

(31)

$$\begin{aligned} \frac{\partial \phi ^i}{\partial \alpha }=-w^{\frac{1-p}{p}}\left( \frac{{\tilde{x}}^{p-1}}{L_1^i} \frac{\partial x'}{\partial \alpha } + \frac{{\tilde{y}}^{p-1}}{L_2^i} \frac{\partial y'}{\partial \alpha } + \frac{{\tilde{z}}^{p-1}}{L_3^i} \frac{\partial z'}{\partial \alpha }\right) \end{aligned}$$

(32)

$$\begin{aligned} \frac{\partial \phi ^i}{\partial \beta }=-w^{\frac{1-p}{p}}\left( \frac{{\tilde{x}}^{p-1}}{L_1^i} \frac{\partial x'}{\partial \beta } + \frac{{\tilde{y}}^{p-1}}{L_2^i} \frac{\partial y'}{\partial \alpha } + \frac{{\tilde{z}}^{p-1}}{L_3^i} \frac{\partial z'}{\partial \beta }\right) \end{aligned}$$

(33)

$$\begin{aligned} \frac{\partial \phi ^i}{\partial \gamma }=-w^{\frac{1-p}{p}}\left( \frac{{\tilde{x}}^{p-1}}{L_1^i} \frac{\partial x'}{\partial \gamma } + \frac{{\tilde{y}}^{p-1}}{L_2^i} \frac{\partial y'}{\partial \gamma } + \frac{{\tilde{z}}^{p-1}}{L_3^i} \frac{\partial z'}{\partial \gamma }\right) \end{aligned}$$

(34)

with the following identities

$$\begin{aligned} \frac{\partial x'}{\partial \zeta }= R_{11}^\zeta \left( x-x_0\right) + R_{12}^\zeta \left( y-y_0\right) + R_{13}^\zeta \left( z-z_0\right) \end{aligned}$$

(35)

$$\begin{aligned} \frac{\partial y'}{\partial \zeta }= R_{21}^\zeta \left( x-x_0\right) + R_{22}^\zeta \left( y-y_0\right) + R_{23}^\zeta \left( z-z_0\right) \end{aligned}$$

(36)

$$\begin{aligned} \frac{\partial z'}{\partial \zeta }= R_{31}^\zeta \left( x-x_0\right) + R_{32}^\zeta \left( y-y_0\right) + R_{33}^\zeta \left( z-z_0\right) \end{aligned}$$

(37)

where $\zeta =\alpha ,\beta ,\gamma$ respectively.

Appendix 2: Modifications of the MMC3d code for the initial design 2 of compliant mechanism design example

The main code is updated as

vInt = [1.2 0.2 0.2 atan(2.5) atan(0.75) atan(0.3)];

MMC3d(10, 1, 5, 200, 20, 100, 5/6, 1/4, 5/8, vInt, 0.2);

In the MMC3d.m, the modifications in SEC 2, SEC 3, the finite element analysis and sensitivity analysis subsections are the same as the modifications for initial design 1. Nevertheless, lines 44-53 in SEC 4 for the generation of initial design variables are modified as:

x0 = xInt:2*xInt:DL; y0 = yInt:2*yInt:DW; z0 = zInt:2*zInt:DH;

xn = length(x0); yn = length(y0); zn = length(z0);

x0 = kron(x0,ones(1,1*yn*zn));

y0 = repmat(repmat(y0,1,1*zn),1,1*xn);

z0 = repmat(reshape(repmat(z0,2,1),1,8),1,xn);

N = length(x0);

l1 = repmat(vInt(1),1,N);

l2 = repmat(vInt(2),1,N);

l3 = repmat(vInt(3),1,N);

alp = repmat(vInt(4),1,N);

bet = repmat([repmat([-1 -1 1 1],1,2) repmat([1 1 -1 -1],1,2)]*vInt(5),1,3);

gam = repmat([repmat([1 -1],1,4) repmat([-1 1],1,4)]*vInt(6),1,3);

Appendix 3: Exact expressions of $\frac{\partial \phi ^i}{\partial d_j^i}$ in the MMC2d code

For conciseness, the following notations are introduced

$$\begin{aligned} w \triangleq \left( \frac{x'}{L_1^i}\right) ^p + \left( \frac{y'}{L_2^i}\right) ^p \end{aligned}$$

(38)

$$\begin{aligned} {\tilde{x}}=x'/L^i, \quad {\tilde{y}}=y'/l^i \end{aligned}$$

(39)

And the derivatives of intermediate variables $x',y',l^i$ with respect to design variables are first detailed as

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial x'}{\partial x_0^i} = - \cos {\theta ^i} \\ \frac{\partial x'}{\partial y_0^i} = - \sin {\theta ^i} \\ \frac{\partial x'}{\partial \theta ^i} = y' \end{array}\right. } \qquad {\left\{ \begin{array}{ll} \frac{\partial y'}{\partial x_0^i} = \sin {\theta ^i} \\ \frac{\partial y'}{\partial y_0^i} = -\cos {\theta ^i} \\ \frac{\partial y'}{\partial \theta ^i} = -x' \end{array}\right. } \end{aligned}$$

(40)

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial l^i}{\partial x_0^i} = \frac{\partial l^i}{\partial x'}\frac{\partial x'}{\partial x_0^i} = -\frac{t_2^i-t_1^i}{2L^i}\cos {\theta ^i}\\ \frac{\partial l^i}{\partial y_0^i} = -\frac{t_2^i-t_1^i}{2L^i} \sin {\theta ^i} \\ \frac{\partial l^i}{\partial L^i} = -\frac{t_2^i-t_1^i}{2(L^i)^2}x' \\ \frac{\partial l^i}{\partial t^i_1} = \frac{1}{2}-\frac{x'}{2L^i} \\ \frac{\partial l^i}{\partial t^i_2} = \frac{1}{2}+\frac{x'}{2L^i} \\ \frac{\partial l^i}{\partial \theta ^i} = \frac{t_2^i-t_1^i}{2L^i}y' \end{array}\right. } \end{aligned}$$

(41)

Then the exact expressions of $\frac{\partial \phi ^i}{\partial d_j^i}$ in the MMC2d code can be formulated as

$$\begin{aligned}&\frac{\partial \phi ^i}{\partial x_0^i}=w^{\frac{1-p}{p}}\left( \frac{{\tilde{x}}^{p-1}}{L^i}\cos {\theta ^i} - \frac{{\tilde{y}}^{p-1}}{l^i}\sin {\theta ^i} + \frac{{\tilde{y}}^p}{l^i} \frac{\partial l^i}{\partial x_0^i}\right) \end{aligned}$$

(42)

$$\begin{aligned} \frac{\partial \phi ^i}{\partial y_0^i}=w^{\frac{1-p}{p}}\left( \frac{{\tilde{x}}^{p-1}}{L^i}\sin {\theta ^i} + \frac{{\tilde{y}}^{p-1}}{l^i}\cos {\theta ^i} + \frac{{\tilde{y}}^p}{l^i} \frac{\partial l^i}{\partial y_0^i}\right) \end{aligned}$$

(43)

$$\begin{aligned} \frac{\partial \phi ^i}{\partial L^i}=w^{\frac{1-p}{p}}\left( \frac{{\tilde{x}}^{p}}{L^i} + \frac{{\tilde{y}}^p}{l^i} \frac{\partial l^i}{\partial L^i}\right) \end{aligned}$$

(44)

$$\begin{aligned} \frac{\partial \phi ^i}{\partial t^i_1}=w^{\frac{1-p}{p}} \frac{{\tilde{y}}^p}{l^i} \frac{\partial l^i}{\partial t^i_1} \end{aligned}$$

(45)

$$\begin{aligned} \frac{\partial \phi ^i}{\partial t^i_2}=w^{\frac{1-p}{p}} \frac{{\tilde{y}}^p}{l^i} \frac{\partial l^i}{\partial t^i_2} \end{aligned}$$

(46)

$$\begin{aligned} \frac{\partial \phi ^i}{\partial \theta ^i}=w^{\frac{1-p}{p}}\left( -\frac{{\tilde{x}}^{p-1}}{L^i}y' + \frac{{\tilde{y}}^{p-1}}{l^i}x' + \frac{{\tilde{y}}^p}{l^i} \frac{\partial l^i}{\partial \theta ^i}\right) \end{aligned}$$

(47)

Appendix 4: A 256-line MATLAB code for the 3D-MMC method

Appendix 5: A 218-line MATLAB code for the 2D-MMC method