A stochastic method of moving asymptotes for topology optimization under uncertainty

Abstract

Topology optimization under uncertainty or reliability-based topology optimization is usually numerically very expensive. This is mainly due to the fact that an accurate evaluation of the probabilistic model requires the system to be simulated for a large number of varying parameters. Traditional gradient-based optimization schemes thus face the difficulty that reasonable accuracy and numerical efficiency often seem mutually exclusive. In this work, we propose a stochastic optimization technique to tackle this problem. To be precise, we combine the well-known method of moving asymptotes (MMA) with a stochastic sample-based integration strategy. By adaptively recombining gradient information from previous steps, we obtain a noisy gradient estimator that is asymptotically correct, i.e., the approximation error vanishes over the course of iterations. As a consequence, the resulting stochastic method of moving asymptotes (sMMA) allows us to solve chance constraint topology optimization problems for a fraction of the cost compared to traditional approaches from literature. To demonstrate the efficiency of sMMA, we analyze structural optimization problems in two and three dimensions.

Figures (19)

• Figure 1: Indicator function $\chi_{(0,\infty)}$ as well as the smooth approximation $h_{a_1}$ and steepened smooth approximation $h_{a_1,a_2,a_3}$. In this illustration, we chose ${a_1=35}$, ${a_2=\tfrac{1}{20}}$ and ${a_3=5}$.
• Figure 2: Nearest neighbor approximation for $K=10$, ${\mathop{\mathrm{dim}}\nolimits(\mathcal{U})=\mathop{\mathrm{dim}}\nolimits(\mathcal{X})=1}$ and ${\Vert\cdot\Vert_{\mathcal{U}\times\mathcal{X}}=\Vert\cdot\Vert_2}$. Piecewise constant regions of $\widehat{j}_{10}$ are indicated by the cells in the background. The approximation $\widehat{J}_{10}(u_{10})$ of $J(u_{10})$ is obtained by integrating $\widehat{j}_{10}$ along the solid line. The sets $M_k^{10}$ are given by the colored line segments. Grey cells correspond to cases where $M_k^{10}$ is empty, resulting in an integration weight $\alpha_{k,10}=0$.
• Figure 3: For pseudoexact integration weights, the domain $\mathcal{X}$ is first discretized by the points $(\mathbf{x}_t)_{t=1,\ldots,T}$ (colored diamonds). For each $\mathbf{x}_t$, it is easy to check the closest sample point (indicated by the diamond's color). The measure of a line segment $\mu(M_k^K)$ is then approximated by adding the weights $w_t$ of all discretization points of the same color.
• Figure 4: Design domain $\mathscr{D}$ (light grey) with fixed inner Dirichlet boundary (green) and material at the outer rim (dark grey). Depending on $\omega$, the structure is loaded by the force $F(\omega)$, acting in normal direction of the Neumann boundary (red). The force intensity $f_\omega$ associated to $F(\omega)$, see \ref{['eq:force_intensity_wheel']}, is indicated by the blue curve.
• Figure 5: Final relative physical volumes and chance constraint values (evaluated using a trapezoidal rule with 1080 load cases) for sMMA (blue circles) and MMA (red triangles). In the right diagram, only feasible designs (all sMMA results and MMA results for ${\mathcal{B}\in\{32,64\}}$) are shown. Of all feasible designs, the lowest relative physical volume corresponds to sMMA with ${\mathcal{B}=8}$ and ${\tau=1}$, while the highest value of $\mathop{\mathrm{phyvol}}\nolimits$ is associated to sMMA with ${\mathcal{B}=8}$ and ${\tau=\tfrac{1}{4}}$. Values of $\mathop{\mathrm{relvol}}\nolimits$ and $\mathop{\mathrm{phyvol}}\nolimits$, sorted by batch size, can be found in \ref{['fig:wheel_objphyscatter']}.

Theorems & Definitions (2)

• Proposition 3.1
• proof