Maximum stress minimization via data-driven multifidelity topology design

TL;DR

This work addresses the challenge of solving the maximum stress minimization problem without relaxation by introducing data-driven multifidelity topology design (MFTD), a gradient-free framework that uses low-fidelity gradient-based seeds and high-fidelity forward analysis on a body-fitted mesh, updated via a deep generative model. By combining NSGA-II-based selection with a variational autoencoder and latent crossover, the method progresses from initial seeds toward Pareto-optimal designs that exhibit reduced stress concentrations without sensitivity analysis. The main contribution is demonstrating that data-driven MFTD can yield significant volume reductions (e.g., 22.6%) at near-identical maximum stresses compared to gradient-based results, and can reveal novel topologies not typically found by traditional methods. This approach offers a practical route to high-performance, 0/1-design structures with manufacturable geometries, and extends to complex nonlinear problems in future work.

Abstract

The maximum stress minimization problem is among the most important topics for structural design. The conventional gradient-based topology optimization methods require transforming the original problem into a pseudo-problem by relaxation techniques. Since their parameters significantly influence optimization, accurately solving the maximum stress minimization problem without using relaxation techniques is expected to achieve extreme performance. This paper focuses on this challenge and investigates whether designs with more avoided stress concentrations can be obtained by solving the original maximum stress minimization problem without relaxation techniques, compared to the solutions obtained by gradient-based topology optimization. We employ data-driven multifidelity topology design (MFTD), a gradient-free topology optimization based on evolutionary algorithms. The basic framework involves generating candidate solutions by solving a low-fidelity optimization problem, evaluating these solutions through high-fidelity forward analysis, and iteratively updating them using a deep generative model without sensitivity analysis. In this study, data-driven MFTD incorporates the optimized designs obtained by solving a gradient-based topology optimization problem with the p-norm stress measure in the initial solutions and solves the original maximum stress minimization problem based on a high-fidelity analysis with a body-fitted mesh. We demonstrate the effectiveness of our proposed approach through the benchmark of L-bracket. As a result of solving the original maximum stress minimization problem with data-driven MFTD, a volume reduction of up to 22.6% was achieved under the same maximum stress value, compared to the initial solution.

Figures (5)

• Figure 1: Problem settings for L-bracket. (a) Design domain $D$ and the shape $\Omega$; (b) Discretization with structured mesh; (c) Discretization with body-fitted mesh; (d) Boundary conditions and dimensions for numerical examples in Section \ref{['sec5']}.
• Figure 2: Schematic illustration of data-driven multifidelity topology design
• Figure 3: Effect of the continuation method on the maximum von Mises stress. (a) Objective space of the relaxed maximum von Mises stress $\max(\hat{\sigma}_{\text{vm}})$ and volume fraction $V/V_{\max}$ with the stress distributions of the solutions; (b) Optimized designs with the filtered density $\tilde{\rho}$ and their stress distributions in the same volume constraint $\overline{V}/V_{\max} = 0.335$ under (i) the fixed parameter $P = 8$ and (ii) the continuation method $P = \textrm{8-16-32}$; (c) Convergence history of the objective function and the normalized volume constraint in the low-fidelity optimization using the continuation method $P = \textrm{8-16-32}$; (d) Initial solutions for data-driven MFTD generated by the low-fidelity optimization using the continuation method $P = \textrm{8-16-32}$.
• Figure 4: Effect of the fidelity on the maximum von Mises stress. (a) Objective space of the maximum von Mises stress $\sigma_{\max}$ and volume fraction $V/V_{\max}$, in which each stress value indicates the relaxed stress $\max(\hat{\sigma}_{\text{vm}})$ (low-fidelity model) and the true stress $\max(\sigma_{\text{vm}})$ (high-fidelity model), respectively; (b) Optimized design and its stress distribution in different analysis model: (i) the low-fidelity model with the filtered density $\tilde{\rho}$, (ii) the high-fidelity model; (c) Initial solutions analyzed with the high-fidelity model.
• Figure 5: Optimization results with data-driven MFTD. (a) Objective space with initial and final designs; (b)(i) An initial solution and (ii) A final solution under the similar maximum von Mises stress; (c) Convergence history of the hypervolume indicator; (d) Final solutions by deta-driven MFTD.