# Topology Optimisation under Uncertainties with Neural Networks
## 3 Neural Network Architectures
### Topology Convolutional Neural Networks (TCNN)
#### 3.1.2 Data Preparation
As can be seen by the finer edges in the images, in the course of optimising the topology (compare, e.g., \(\varphi_{200}\) and \(\varphi_{538}\)), an adapted mesh is used which is refined depending on \(\varphi_{n}\) in order resolve fine details of the topology and to save computational costs (see Appendix B). Algorithm A1 took 538 iterations to converge. Figure A2 illustrates other metrics in the optimisation. The convergence of \(J^{\varepsilon}(\varphi)\) is clearly visible. Through the adaptation method of the step size \(\tau_{n}\) (see [2]), it becomes increasingly larger when \(\varphi_{n}\) begins to converge towards the optimal mesh \(\mathcal{T}_{m}\). The small spikes in all time series are due to the refinement of the mesh \(\mathcal{T}_{m}\). In the lower-right corner, it can be seen that the calculation time increases with the fineness of the mesh \(\mathcal{T}_{m}\). The lower-left part of Figure A2 shows \(\gamma_{n}\), which stabilises the form of \(\varphi_{n}\), but plays no further role in our investigations. ### Stochastic Bridge Problem

This modification of the experiment described in Appendix A.1 introduces uncertainties in the data, which render the problem much more involved. The Lame-coefficients \(\mu_{\text{mat}}=\lambda_{\text{mat}}\) are modelled as a truncated lognormal Karhunen-Loeve expansion with 10 modes, a mean value of 150 and a covariance length of 0.1 which is scaled by a factor of 100. The load \(g\) is assumed as a vector with mean \((0,-5000)\) and a random rotation angle simulated through a truncated normal distribution with bounds \([\frac{-\pi}{2},\frac{\pi}{2}]\), standard deviation 0.3 and mean 0. In each iteration we use \(N=224\) samples for the evaluation of the risk functional. Some of the resulting iterations of the optimisation process are depicted in Figure A.3. By calculating the expected value of the functional in Equation (7) (with parameter \(\beta=0\)), one can see a loss of symmetry in the resulting topology compared to the deterministic setting from Example 2 since the load is almost always not perpendicular to the load-bearing boundaries. The main difference is the strain on the Dirichlet boundary, which is introduced by the moved left-most spoke. In contrast to this, the right-hand side closely resembles the deterministic case since the slip boundary cannot absorb energy in the tangential direction. In this particular stochastic setting, an additional spoke is formed. ## Appendix B Finite Element Discretization

The physical space \(D\subset\mathbb{R}^{2}\) is discretised with first-order conforming finite elements with the FEniCS framework [24]. The reason for the favourable performance of the optimiser from [1,2] is the adaptive mesh refinement based on gradient information of the phase field. The idea is that the optimisation is started on a coarse mesh and refined over the course of the optimisation depending on the topology (more precisely on the phase transitions of \(\varphi\)). For this purpose it is assumed that the domain \(D\) is a convex polygon and is described with first-order conforming elements by the mesh \(\mathcal{T}_{m}=(V(\mathcal{T}_{m}),E(\mathcal{T}_{m}))\) at iteration step \(m\in\mathbb{N}\), which also can be understood as a graph. The mesh consists of triangles \(T\in\mathcal{T}_{m}\) and an associated set of edges \(E(T)\subset E(\mathcal{T}_{m})\subset D\times D\) and vertices \(V(T)\subset V(\mathcal{T}_{m})\subset D\). Based on this mesh, the next mesh \(\mathcal{T}_{m+1}\) is generated with a simple error indicator using the bulk criterion for the Dorfler marking [26] to determine which triangles \(T\) should be refined. Since \(\varphi\) moves within the domain \(D\), the mesh refinement necessarily reflects this.