TOPress: a MATLAB implementation for topology optimization of structures subjected to design-dependent pressure loads

Abstract

In a topology optimization (TO) setting, design-dependent fluidic pressure loads pose several challenges as their direction, magnitude, and location alter with topology evolution. This paper offers a compact 100-line MATLAB code, TOPress, for TO  of structures subjected to fluidic pressure loads using the method of moving asymptotes. The code is intended for pedagogical purposes and aims to ease the beginners’ and students’ learning toward the TO with design-dependent fluidic pressure loads. TOPress is developed per the approach first reported in Kumar et al. (Struct Multidisc Optim 61(4):1637–1655, 2020). The Darcy law, in conjunction with the drainage term, is used to model the applied pressure load. The consistent nodal loads are determined from the obtained pressure field. The employed approach facilitates inexpensive computation of the load sensitivities using the adjoint-variable method. Compliance minimization subject to volume constraint optimization problems is solved. The success and efficacy of the code are demonstrated by solving benchmark numerical examples involving pressure loads, wherein the importance of load sensitivities is also demonstrated. TOPress contains six main parts, is described in detail, and is extended to solve different problems. Steps to include a projection filter are provided to achieve loadbearing designs close to 0-1. The code is provided in Appendix 2 and can also be downloaded along with its extensions from https://github.com/PrabhatIn/TOPress.

Appendices

Appendix 1: Expressions for flow and transformation matrices: ${\textbf{K}}_\text{p}^i$, $\textbf{K}^i_\text{Dp}$ and $\textbf{T}_i$

The expressions for the flow matrices ${\textbf{K}}_\text{p}^i$ and $\textbf{K}^i_\text{Dp}$, and transformation matrix $\textbf{T}_i$ are provided for element i herein.

Figure 19 displays the local node numbers with nodal coordinates $(x_i,\,y_i)$, $(x_i +\text {D}x,\,y_i)$, $(x_i +\text {D}x,\,y_i+\text {D}y)$, and $(x_i,\,y_i+\text {D}y)$ for element i.

Fig. 19

Nodal coordinates of element i

As mentioned earlier, we employ the bi-linear shape functions for the finite element analysis. To determine, ${\textbf{K}}_\text{p}^i$ and $\textbf{K}^i_\text{Dp}$ as written in (5) and $\textbf{T}_i$ as provided in (9), numerical integration approach using the Gauss points is employed (Zienkiewicz et al. 2005). Mathematically,

$$\begin{aligned} {\textbf{K}}_\text{p}^i&= \int _{\Omega _{\rm e}} K\; \textbf{B}^\top _\text{p} \textbf{B}_\text{p} \text {d} V \\ &= t_i\int _{-1}^{1}\int _{-1}^{1} \left( K\; \textbf{B}^\top _\text{p} \textbf{B}_\text{p}\right) \det \textbf{J} \text{d}\xi \text{d}\zeta \\ \textbf{K}^i_\text{Dp}&= \int _{\Omega _{\rm e}}D \;\textbf{N}^\top _\text{p} \textbf{N}_\text{p} \text{d} V \\ &= t_i\int _{-1}^{1}\int _{-1}^{1} \left( D \;\textbf{N}^\top _\text{p} \textbf{N}_\text{p}\right) \det \textbf{J} \text{d}\xi \text{d}\zeta \\ \textbf{T}_i&=-\int _{{\Upomega}_{\rm e}} {\textbf{N}}^{\!\top }_\textbf{u} \textbf{B}_\text{p} \text {d} V =t_i\int _{-1}^{1}\int _{-1}^{1} {\textbf{N}}^{\!\top }_\textbf{u} \textbf{B}_\text{p} \det \textbf{J} \text{d}\xi \text{d}\zeta \end{aligned}$$

(22)

where $\textbf{J}$ is the Jacobian matrix given as

$$\textbf{J}= \frac{1}{4} \left[\begin{array}{ll} J11 &\quad J12\\ J21 & \quad J22, \end{array}\right]$$

(23)

where $J11 = -(1-\zeta )x_1 + (1-\zeta ) x_2 + (1+\zeta )x_3 - (1+\zeta )x_4$, $J12 = \quad -(1-\zeta )y_1 + (1-\zeta ) y_2 + (1+\zeta )y_3 - (1+\zeta )y_4$, $J21 = -(1-\xi )x_1 + (1-\xi ) x_2 + (1+\xi )x_3 - (1+\xi )x_4,$ and $J22 = -(1-\xi )y_1 + (1-\xi ) y_2 + (1+\xi )y_3 - (1+\xi )y_4$, with $N_k = \frac{1}{4}(1+\xi \xi _k)(1+ \zeta \zeta _k)$, where $(\xi _k,\,\zeta _k)$ are natural coordinates with (− 1, − 1), (1, − 1), (1, 1) and (− 1, 1) for node k. (22) yields the following with $2\times 2$ Gauss-quadrature rule:

$$\begin{aligned} {\textbf{K}}_\text{p}^i=\frac{t_i}{6 Dx Dy} \begin{bmatrix} K_{p1} &{}\quad K_{p2} &{}\quad -0.5K_{p1}&{}\quad K_{p3} \\ &{}\quad K_{p1} &{}\quad K_{p3}&{}\quad -0.5K_{p1}\\ &{}\text {Sym.} &{}K_{p1} &{}\quad K_{p2}\\ {} &{} &{} &{}\quad K_{p1} \end{bmatrix} \end{aligned}$$

(24)

where $K_{p1} = 2(Dx^2+ Dy^2)$,  $K_{p2} = Dx^2 -2Dy^2$, $K_{p3} = Dy^2 - 2Dx^2$.

$$\begin{aligned} {\textbf{K}}_\text{Dp}^i=\frac{t_i Dx Dy}{36} \begin{bmatrix} 4 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ &{}\quad 4 &{}\quad 2 &{}\quad 1\\ &{} \text {Sym.} &{}\quad 4 &{}\quad 2\\ &{} &{} &{}\quad 4 \end{bmatrix} \end{aligned}$$

(25)

and

$$\begin{aligned} \textbf{T}_i=\frac{t_i}{12} \begin{bmatrix} -2Dy &{}\quad 2Dy &{}\quad Dy &{}\quad -Dy \\ -2Dx &{}\quad -Dx &{}\quad Dx &{}\quad 2Dx\\ -2Dy &{}\quad 2Dy &{}\quad Dy &{}\quad -Dy\\ -Dx&{}\quad -2Dx &{}\quad 2Dx &{}\quad Dx\\ -Dy &{}\quad Dy &{}\quad 2Dy &{}\quad -2Dy\\ -Dx&{}\quad -2Dx &{}\quad 2Dx &{}\quad Dx\\ -Dy &{}\quad Dy &{}\quad 2Dy &{}\quad -2Dy\\ -2Dx &{}\quad -Dx &{}\quad Dx &{}\quad 2Dx \end{bmatrix} \end{aligned}$$

(26)

For the square elements which we consider to parameterize the design domain with $t_i =1$ (plane-stress cases). The expressions for ${\textbf{K}}_\text{p}^i$, ${\textbf{K}}_\text{Dp}^i$ and $\textbf{T}_i$ transpire to

$$\begin{aligned}{} & {} {\textbf{K}}_\text{p}^i=\frac{1}{6} \begin{bmatrix} 4 &{}\quad -1 &{}\quad -2&{}\quad -1 \\ &{}4 &{}\quad -1&{}\quad -2\\ &{}\text {Sym.} &{}4 &{}\quad -1\\ {} &{} &{} &{}4 \end{bmatrix} \end{aligned}$$

(27)

$$\begin{aligned}{} & {} {\textbf{K}}_\text{Dp}^i=\frac{1}{36} \begin{bmatrix} 4 &{}\quad 2 &{}\quad 1 &{}\quad 2 \\ &{}\quad 4 &{}\quad 2 &{}\quad 1\\ &{} \text {Sym.} &{}\quad 4 &{}\quad 2\\ &{} &{} &{}\quad 4 \end{bmatrix} \end{aligned}$$

(28)

and

$$\begin{aligned} \textbf{T}_i=\frac{1}{12} \begin{bmatrix} -2 &{}\quad 2 &{}\quad 1 &{}\quad -1 \\ -2 &{}\quad -1 &{}\quad 1 &{}\quad 2\\ -2 &{}\quad 2 &{}\quad 1 &{}\quad -1\\ -1&{}\quad -2 &{}\quad 2 &{}\quad 1\\ -1 &{}\quad 1 &{}\quad 2 &{}\quad -2\\ -1&{}\quad -2 &{}\quad 2 &{}\quad 1\\ -1 &{}\quad 1 &{}\quad 2 &{}\quad -2\\ -2 &{}\quad -1 &{}\quad 1 &{}\quad 2 \end{bmatrix} \end{aligned}$$

(29)

Appendix 2: The MATLAB code: TOPress