Topology optimization of structures made of fiber-reinforced plates

Abstract

This work introduces a topology optimization method for the design of structures composed of fiber-reinforced, rectangular plates. Each of the plates has a predetermined, nondesignable reinforcement, and the proposed method determines an optimal layout of the plates within a prescribed three-dimensional region. A plate is modeled using a homogeneous, anisotropic material, whose properties are aligned relative to the plate’s orientation. This work builds upon existing geometry projection techniques with three notable additions. First, it introduces a novel parameterization of the plate orientation based on quaternions, which avoids numerical instabilities and $2\pi$-periodicity issues. Second, an overlap constraint for plates is formulated to prevent plate intersections that would make manufacturing of the structure impractical. Finally, the finite element assembly and sensitivity analysis are substantially accelerated by exploiting the structure of the material interpolation. This strategy is facilitated by the use of an adaptive mesh refinement technique. The efficacy of the proposed method is demonstrated with compliance minimization examples. The examples show the importance of considering material anisotropy in the design of composite structures. Moreover, it is demonstrated that naively replacing the material of the plates in an optimally stiff design made of an isotropic material with a composite can result in suboptimal performance.

Appendix: Smooth approximations

The smooth approximations of the Heaviside and argmax functions used in the formulation and their sensitivities are given by:

$$\begin{aligned} \widetilde{H}(x)&= {\left\{ \begin{array}{ll} 1 , &{}\text { if } x \ge 1 \\ (x + 1)^3(3x^2 - 9x + 8)/16 , &{}\text { if } |x| < 1\\ 0 , &{}\text { otherwise } \\ \end{array}\right. } \end{aligned}$$

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$$\begin{aligned} \frac{d\widetilde{H}}{dx}&= {\left\{ \begin{array}{ll} 15(x^2 - 1)^2/16 , &{}\text { if }|x|<1\\ 0, &{}\text { otherwise } \end{array}\right. } \end{aligned}$$

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$$\begin{aligned} w_i(\mathbf{x})&= \underset{j}{{\widetilde{\mathrm{argmax}}}_i} \ (x_j;\beta ) = \frac{e^{\beta x_i}}{\sum _je^{\beta x_j}} \end{aligned}$$

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$$\begin{aligned} \frac{\partial w_i}{\partial x_j}&= \beta w_i (\delta _{ij} - w_j). \end{aligned}$$

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