Generalized Geometry Projection: A Unified Approach for Geometric Feature Based Topology Optimization

Abstract

Structural topology optimization has seen many methodological advances in the past few decades. In this work we focus on continuum-based structural topology optimization and more specifically on geometric feature based approaches, also known as explicit topology optimization, in which a design is described as the assembly of simple geometric components that can change position, size and orientation in the considered design space. We first review various recent developments in explicit topology optimization. We then describe in details three of the reviewed frameworks, which are the Geometry Projection method, the Moving Morphable Components with Esartz material method and Moving Node Approach. Our main contribution then resides in the proposal of a theoretical framework, called Generalized Geometry Projection, aimed at unifying into a single formulation these three existing approaches. While analyzing the features of the proposed framework we also provide a review of smooth approximations of the maximum operator for the assembly of geometric features. In this context we propose a saturation strategy in order to solve common difficulties encountered by all reviewed approaches. We also explore the limits of our proposed strategy in terms of both simulation accuracy and optimization performance on some numerical benchmark examples. This leads us to recommendations for our proposed approach in order to attenuate common discretization induced effects that can alter optimization convergence.

Appendices

Appendix 1: Characteristic Function and Local Volume Fraction Sensitivity Distribution

In this subsection we present the distribution of both characteristic function and local volume fraction sensitivity to the design variables in the example introduced in Sect. 2.6. The effect of both the sampling window size and the number of Gauss point is analyzed to compute $\delta$ from the same W. An important observation is that by increasing $N_{GP}$ one increases the ability of GGP to adequately capture the narrow distribution of characteristic function sensitivity.

Fig. 27

Cantilever beam parametric study using the AMMC approach. Effect of the sampling window size R and of the number of Gauss points $N_{GP}$ on the structural compliance and the volume fraction. In each graph we reported in green the true theoretical values based on the analytic beam model. (Color figure online)

Fig. 28

Cantilever beam parametric study using AGP method. Effect of the sampling window size R and of the number of Gauss points $N_{GP}$ on the structural compliance and the volume fraction. In each graph we reported in green the true theoretical values based on the analytic beam model. (Color figure online)

Fig. 29

Cantilever beam parametric study using the AMNA method. Effect of the sampling window size R and of the number of Gauss points $N_{GP}$ on the structural compliance and the volume fraction. In each graph we reported in green the true theoretical values based on the analytic beam model. (Color figure online)

Fig. 30

Initial configuration for the short cantilever topology optimization problem. Components are colored according to the value of m. Blue triangles represents clamped degrees of freedoms. The red arrow represents the applied load. 18 round ended bars are considered for the optimization, i.e. $6\times 18=108$ design variables for both AGP and AMNA and $5\times 18=90$ design variables for AMMC. (Color figure online)

Appendix 2: Parametric Study Results on the Cantilever Beam Case

In this section the full plot results from the parametric study on the cantilever beam presented in Sect. 3.1 are provided.

Fig. 31

Short Cantilever Beam Topology optimization using the AMMC method for variable number of Gauss points $N_{GP}$

Fig. 32

Short Cantilever Beam Topology optimization using the AGP method for variable number of Gauss points $N_{GP}$

Fig. 33

Short Cantilever Beam Topology optimization using the AMNA method for variable number of Gauss points $N_{GP}$

1.1 MMA Set-Up

In this subsection we provide details of implementations considered for the method of moving asymptotes (MMA, [44]). This approach makes local convex approximations at each iterations of both constraints and objective function. The convexity is adjusted by changing asymptotes’ positions during the optimization history. A move limit can also be chosen in order to control the optimization step and avoid divergence. A correct scaling of both design variables and compliance is recommended to avoid numerical issues. Here we propose to re-scale variables and gradients according to:

$$\begin{aligned}&{\hat{x}}_j= \frac{x_j-l_{j}}{u_j-l_j} \end{aligned}$$

(112)

$$\begin{aligned}&\frac{dC}{d{\hat{x}}_j}=\frac{1}{u_j-l_j}\frac{dC}{dx_j} \end{aligned}$$

(113)

$$\begin{aligned}&\frac{dV}{d{\hat{x}}_j}=\frac{1}{u_j-l_j}\frac{dV}{dx_j} \end{aligned}$$

(114)

where $l_j$ and $u_j$ are the jth—component respectively of the lower bound $\{l\}$ and of upper bound vector $\{u\}$. In order to avoid further MMA numerical issues one can either normalize the compliance dividing C and $\left\{ {\frac{dC}{d{\hat{x}}}}\right\}$ by a constant $C_0$ greater than 1 that ensures the compliance and its gradient are small enough. However this way of normalizing introduces the issue of a good choice of $C_0$, depending on the particular problem studied. To avoid this problem, here we considered the following normalization:

$$\begin{aligned}&{\hat{C}}=\log {\left( 1+C\right) } \end{aligned}$$

(115)

$$\begin{aligned}&\frac{d{\hat{C}}}{d{\hat{x}}_j}=\frac{1}{1+C}\frac{dC}{d{\hat{x}}_j} \end{aligned}$$

(116)

Note that since $C>0$, $\log {\left( 1+C\right) }$ is also greater than 0. This ensures the gradients to be smaller for higher values of C (that is the case of ill connected configurations). In order to avoid MMA divergence due to uncontrolled optimization step length, here we propose a strategy that is similar to the one taken by the globally convergent version of MMA (GCMMA) [45]. In the mmasub.m Matlab function called during the optimization loop we modified the updating of lowmin, lowmax, uppmin and uppmax formula, reducing the value of the coefficients that multiplies each variable range. Accordingly this ensures the control of the optimization step through the overestimation of the problem convexity. In this way MMA behaves more conservatively at each iteration and is less prone to oscillate or to skip local optima.⁷