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We present a virtual element method (VEM)-based topology optimization framework using polyhedral elements, which allows for convenient handling of non-Cartesian design domains in three dimensions. We take full advantage of the VEM properties by creating a unified approach in which the VEM is employed in both the structural and the optimization phases. In the structural problem, the VEM is adopted to solve the three-dimensional elasticity equation. Compared to the finite element method, the VEM does not require numerical integration (when linear elements are used) and is less sensitive to degenerated elements (e.g., ones with skinny faces or small edges). In the optimization problem, we introduce a continuous approximation of material densities using the VEM basis functions. When compared to the standard element-wise constant approximation, the continuous approximation enriches the geometrical representation of structural topologies. Through two numerical examples with exact solutions, we verify the convergence and accuracy of both the VEM approximations of the displacement and material density fields. We also present several design examples involving non-Cartesian domains, demonstrating the main features of the proposed VEM-based topology optimization framework. The source code for a MATLAB implementation of the proposed work, named PolyTop3D, is available in the (electronic) Supplementary Material accompanying this publication.
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This content will become publicly available on September 1, 2025
Topology optimization with graph neural network enabled regularized thresholding
Topology optimization algorithms often employ a smooth density function to implicitly represent geometries in a discretized domain. While this implicit representation offers great flexibility to parametrize the optimized geometry, it also leads to a transition region. Previous approaches, such as the Solid Isotropic Material Penalty (SIMP) method, have been proposed to modify the objective function aiming to converge toward integer density values and eliminate this non-physical transition region. However, the iterative nature of topology optimization renders this process computationally demanding, emphasizing the importance of achieving fast convergence. Accelerating convergence without significantly compromising the final solution can be challenging. In this work, we introduce a machine learning approach that leverages the message-passing Graph Neural Network (GNN) to eliminate the non-physical transition zone for the topology optimization problems. By representing the optimized structures as weighted graphs, we introduce a generalized filtering algorithm based on the topology of the spatial discretization. As such, the resultant algorithm can be applied to two- and three-dimensional space for both Cartesian (structured grid) and non-Cartesian discretizations (e.g. polygon finite element). The numerical experiments indicate that applying this filter throughout the optimization process may avoid excessive iterations and enable a more efficient optimization procedure.
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- Award ID(s):
- 1846875
- PAR ID:
- 10574839
- Editor(s):
- Hsia, KJ; Rogers, JA; Suo, Z; Zhao, X
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Extreme Mechanics Letters
- Volume:
- 71
- Issue:
- C
- ISSN:
- 2352-4316
- Page Range / eLocation ID:
- 102215
- Subject(s) / Keyword(s):
- Topology optimization Graph neural networks SIMP
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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