Neural Level Set Topology Optimization Using Unfitted Finite Elements

TL;DR

This work presents a differentiable neural level-set topology optimization framework that couples a U-Net-based neural parameterization with an unfitted finite-element method for multiphysics problems. The design map from parameters to the level-set is differentiable, enabling gradient-based optimization with a backward pass that uses adjoint PDEs and automatic differentiation; gradient cost remains comparable to a forward solve. The method yields more regular, multi-scale geometries and demonstrates generality by solving interface-coupled multiphysics problems, validated against standard benchmarks and a fluid-structure interaction case. While convergence may be slower than traditional SIMP on simple problems, the approach naturally handles complex interfaces and boundary conditions without remeshing, and is available as open-source for replication.

Abstract

To facilitate widespread adoption of automated engineering design techniques, existing methods must become more efficient and generalizable. In the field of topology optimization, this requires the coupling of modern optimization methods with solvers capable of handling arbitrary problems. In this work, a topology optimization method for general multiphysics problems is presented. We leverage a convolutional neural parameterization of a level set for a description of the geometry and use this in an unfitted finite element method that is differentiable with respect to the level set everywhere in the domain. We construct the parameter to objective map in such a way that the gradient can be computed entirely by automatic differentiation at roughly the cost of an objective function evaluation. The method produces optimized topologies that are similar in performance yet exhibit greater regularity than baseline approaches on standard benchmarks whilst having the ability to solve a more general class of problems, e.g., interface-coupled multiphysics.

Figures (10)

• Figure 1: Computational graph of the optimization loop. We start with an input to the system $\mathbf{p}$, and perform the forward pass by descending through the blue boxes on the right side to obtain a performance measure $J$ as explained in Section \ref{['forward-pass']}. The convergence criteria are used to decide whether this $J$ represents an acceptable minimum. If not, the backward pass is performed to compute an update for the parameters as explained in Section \ref{['backwards-pass']} and the loop is continued.
• Figure 2: Architecture of the nn. A trainable input vector $\boldsymbol{\Theta}$ is fed into the network. The light blue arrow involves a set of operations that include a fully connected layer and the dark blue arrow involves a set of operations that include a convolutional layer. The intermediate data structures are of size $(c_l,w_l,h_l)$ and the final output, after $n$ layers, gives $\mathbf{\varphi}$.
• Figure 3: Illustration of the problem domains. The background domain is segmented into the domains $\Omega_{\rm in}$ and $\Omega_{\rm out}$. In $\Omega_{\rm in}$, the physical terms $\mathcal{R}(\boldsymbol{u},\boldsymbol{v})$ are integrated. In $\Omega_{\rm out}$, the stabilization terms $\mathcal{R}^{\mathrm{stb}}(\boldsymbol{u},\boldsymbol{v})$ are integrated.
• Figure 4: A small perturbation with size $\epsilon$ to the ls to form a new domain $\Omega_{\rm in}^\epsilon$.
• Figure 5: The change in a portion of the ghost skeleton triangulation $\Gamma_G$, depicted by the red faces, before and after a perturbation to the boundary.