HoneyTop90: A 90-line MATLAB code for topology optimization using honeycomb tessellation

Abstract

This paper provides a simple, compact and efficient 90-line pedagogical MATLAB code for topology optimization using hexagonal elements (honeycomb tessellation). Hexagonal elements provide nonsingular connectivity between two juxtaposed elements and, thus, subdue checkerboard patterns and point connections inherently from the optimized designs. A novel approach to generate honeycomb tessellation is proposed. The element connectivity matrix and corresponding nodal coordinates array are determined in 5 (7) and 4 (6) lines, respectively. Two additional lines for the meshgrid generation are required for an even number of elements in the vertical direction. The code takes a fraction of a second to generate meshgrid information for the millions of hexagonal elements. Wachspress shape functions are employed for the finite element analysis, and compliance minimization is performed using the optimality criteria method. The provided MATLAB code and its extensions are explained in detail. Options to run the optimization with and without filtering techniques are provided. Steps to include different boundary conditions, multiple load cases, active and passive regions, and a Heaviside projection filter are also discussed. The code is provided in Appendix A, and it can also be downloaded along with supplementary materials from https://github.com/PrabhatIn/HoneyTop90.

Appendices

Appendix A: HoneyTop90 MATLAB code

Appendix B: Sensitivity analysis

In this paper, the optimality criteria approach is employed for the optimization. Therefore, derivatives of the objective and constraint with respect to the design variables are required. Herein, the adjoint-variable method is used to determine the sensitivity of the objective, $C({{\varvec{\rho }}}) = {\mathbf {u}}^\top {\mathbf {K}}({\varvec{\rho }}){\mathbf {u}}$. The overall performance function ${\mathcal {L}}$ in conjunction with the state equation, ${\mathbf {K}} {\mathbf {u}} - {\mathbf {F}} = {\mathbf {0}}$ is written as

$$\begin{aligned} {\mathcal {L}} = C + {\varvec{\lambda }}^\top ({\mathbf {K}} {\mathbf {u}} - {\mathbf {F}}), \end{aligned}$$

(B.1)

where ${\varvec{\lambda }}$ is the Lagrange multiplier vector. In view of (B.1), one finds derivative of ${\mathcal {L}}$ with respect to ${\varvec{\rho }}$ as

$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial {\varvec{\rho }}}= & {} \frac{\partial C}{\partial {\varvec{\rho }}} + \frac{\partial C}{\partial {\mathbf {u}}}\frac{\partial {\mathbf {u}}}{\partial {\varvec{\rho }}} + {\varvec{\lambda }}^\top \left( \frac{\partial {\mathbf {K}}}{\partial {\varvec{\rho }}} {\mathbf {u}}+ {\mathbf {K}} \frac{\partial {\mathbf {u}}}{\partial {\varvec{\rho }}}\right) \nonumber \\= & {} {\mathbf {u}}^\top \frac{\partial {\mathbf {K}}}{\partial {\varvec{\rho }}}{\mathbf {u}} + 2{\mathbf {u}}^\top {\mathbf {K}}\frac{\partial {\mathbf {u}}}{\partial {\varvec{\rho }}} + {\varvec{\lambda }}^\top \frac{\partial {\mathbf {K}}}{\partial {\varvec{\rho }}}{\mathbf {u}} + {\varvec{\lambda }}^\top {\mathbf {K}} \frac{\partial {\mathbf {u}}}{\partial {\varvec{\rho }}} \qquad \left( \text {using},\, C({{\varvec{\rho }}}) = {\mathbf {u}}^\top {\mathbf {K}} ({\varvec{\rho }}){\mathbf {u}}\right) \nonumber \\= & {} {\mathbf {u}}^\top \frac{\partial {\mathbf {K}}}{\partial {\varvec{\rho }}}{\mathbf {u}} + {\varvec{\lambda }}^\top \frac{\partial {\mathbf {K}}}{\partial {\varvec{\rho }}}{\mathbf {u}} + \underbrace{\left( 2{\mathbf {u}}^\top {\mathbf {K}} + {\varvec{\lambda }}^\top {\mathbf {K}}\right) }_{\Theta }\frac{\partial {\mathbf {u}}}{\partial {\varvec{\rho }}} \end{aligned}$$

(B.2)

In (B.2), $\Theta = 0$, the adjoint equation, yields ${\varvec{\lambda }} = -2{\mathbf {u}}$ and thus, one writes (B.1) as

$$\begin{aligned} \begin{aligned} \frac{\partial {\mathcal {L}}}{\partial {\varvec{\rho }}}&= {\mathbf {u}}^\top \frac{\partial {\mathbf {K}}}{\partial {\varvec{\rho }}}{\mathbf {u}} -2{\mathbf {u}}^\top \frac{\partial {\mathbf {K}}}{\partial {\varvec{\rho }}}{\mathbf {u}} \\&= - {\mathbf {u}}^\top \frac{\partial {\mathbf {K}}}{\partial {\varvec{\rho }}}{\mathbf {u}} \end{aligned} \end{aligned}$$

(B.3)

Therefore, in view of (B.3), the derivative of objective C with respect to design variable $\rho _j$ can be written as

$$\begin{aligned} \frac{\partial C}{\partial \rho _j} = -{\mathbf {u}}_j^\top \frac{\partial {\mathbf {k}}_j}{\partial \rho _j} {\mathbf {u}}_j \end{aligned}$$

(B.4)

where ${\mathbf {u}}_j$ and ${\mathbf {k}}_j$ are the displacement vector and the stiffness matrix of element j, respectively.

Fig. 13

A regular hexagonal element with vertices $\hbox {V}_i|_{i = 1,\,2,\,\cdots ,\,6}$ and circumscribing circle ${C}_\text {c}$ with radius 1 unit. Coordinates of vertex $\hbox {V}_i$ are $\left( (\eta _1^i,\,\eta _2^i) \equiv \left( \cos (\frac{(2i-1)\pi }{6}),\,\sin (\frac{(2i-1)\pi }{6})\right) \right)$. Straight lines $l_i$ pass through vertices $\hbox {V}_i$ and $\hbox {V}_{i-1}$. $\hbox {P}_{i\,{i+2}}$ are the intersection points of straight lines $l_i$ and $l_{i+2}$. Circle ${C}_\text {s}$ with radius $\sqrt{3}$ unit is drawn that passes through points $\hbox {P}_{i\,{i+2}}$

Appendix C: Wachspress shape functions

Figure 13 depicts a hexagonal element with vertices $\hbox {V}_i|_{i = 1,\,2,\,3,\,\cdots ,\,6}$ in ${\varvec{\eta }}$ co-ordinates system. Coordinates of vertex $\hbox {V}_i$ are $\left( (\eta _1^i,\,\eta _2^i) \equiv \left( \cos (\frac{(2i-1)\pi }{6}),\,\sin (\frac{(2i-1)\pi }{6})\right) \right)$. The circumscribing circle with radius 1 unit is represented via ${C}_\text {c}$. Let Wachspress shape function for vertex $\hbox {V}_i$ (Fig. 13) be $N_i$. Using the fundamentals of coordinate geometry and in view of coordinates of $\hbox {V}_i$, the equations of straight lines $l_i$ (Fig. 13) can be written as

(C.1)

and likewise, the equation of circle ${C}_\text {s}$ (cf. Fig. 13, passing through $\hbox {P}_{i i+2}$) can be written as

$$\begin{aligned} {C}_\text {s}({\varvec{\eta }}) \equiv \eta _1^2 + \eta _2^2 - 3 = 0. \end{aligned}$$

(C.2)

Straight lines $l_i$ and $l_{i+2}$ intersect at points $\hbox {P}_{i\,{i+2}}$ (Fig. 13). The shape function of node 1, i.e., $N_1$ is determined as (Wachspress 1975)

$$\begin{aligned} N_1 = s_1\frac{l_2({\varvec{\eta }}) l_3({\varvec{\eta }}) l_4({\varvec{\eta }}) l_5({\varvec{\eta }})}{{C}_\text {s}({\varvec{\eta }})}, \end{aligned}$$

(C.3)

where $s_1$, a constant, is calculated using the Kronecker-delta property of a shape function, which is defined as

$$\begin{aligned} N_i({\varvec{\eta }}_j) = \delta _{ij} = {\left\{ \begin{aligned} 1, \quad \text {if}\,\, i = j\\ 0, \quad \text {if}\,\,i\ne j\\ \end{aligned}\right. }. \end{aligned}$$

(C.4)

Now, in view of coordinates³ of node 1, i.e., $\left( \eta _1^1,\,\eta _2^1\right)$ and (C.4), (C.3) yields

$$\begin{aligned} s_1 = \frac{{C}_\text {s}(\eta _1^1,\,\eta _2^1)}{l_2(\eta _1^1,\,\eta _2^1) l_3(\eta _1^1,\,\eta _2^1) l_4(\eta _1^1,\,\eta _2^1) l_5(\eta _1^1,\,\eta _2^1)} = \frac{1}{18}. \end{aligned}$$

(C.5)

Likewise, one can determine Wachspress shape functions for all other nodes with their respective constants. The final expressions of the shape functions are:

(C.6)

Appendix D: Numerical Integration

To evaluate stiffness matrix of an element, numerical integration approach using the quadrature points is employed. As per (Lyness and Monegato 1977), quadrature points for a hexagonal element are given in Table 4, and a function $f\left( \eta _1,\,\eta _2\right)$ can be integrated as

$$\begin{aligned} \int _{A_6}f\left( \eta _1,\,\eta _2\right) \text {d}\Omega \approx A_6\left( w_o f(0,\,0) + \sum _{k=2}^{k_\text {max}}\sum _{i=1}^{6} w_k f\left( r_k,\,\alpha _k + \frac{\pi i}{3}\right) \right) . \end{aligned}$$

(D.1)

where $A_6$ is the area of a regular hexagonal element. Table 4 notes the quadrature points N with coordinates $\left( \eta _1^a,\,\eta _2^a\right)$ $= \left( r_k\cos (\alpha _k + \frac{i\pi }{3}),r_k\sin (\alpha _k + \frac{i\pi }{3})\right)$. $w_k$ indicate the weights for these points. $i =1,\,2,\,3,\,\cdots ,\,6$, if $k>1$, and $i =1,\,\text {if}\, k=1$, corresponds to single integration point at center $(0,\,0)$. For $k>1$, six integration points lie on the a circle with center at (0, 0) and radius $r_k$. Note that the quadrature rule is invariant under a rotation of $60^o$ for a hexagonal element.

Table 4 Quadrature points for a hexagonal element