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University of Wisconsin-Madison
Given a dataset containing Voronoi microstructure parameters and homogenized constitutive properties, a neural network is trained offline. The trained network is used as a surrogate during topology optimization to derive optimized Voronoi structures.
Abstract
Cellular structures found in nature exhibit remarkable properties such as high strength, high energy absorption, excellent thermal/acoustic insulation, and fluid transfusion. Many of these structures are Voronoi-like; therefore researchers have proposed Voronoi multi-scale designs for a wide variety of engineering applications. However, designing such structures can be computationally prohibitive due to the multi-scale nature of the underlying analysis and optimization. In this work, we propose the use of a neural network (NN) to carry out efficient topology optimization (TO) of multi-scale Voronoi structures. The NN is first trained using Voronoi parameters (cell site locations, thickness, orientation, and anisotropy) to predict the homogenized constitutive properties. This network is then integrated into a conventional TO framework to minimize structural compliance subject to a volume constraint. Special considerations are given for ensuring positive definiteness of the constitutive matrix and promoting macroscale connectivity. Several numerical examples are provided to showcase the proposed method.
Porosity vs Stiffness:
Porous structures are multifunctional, offering a combination of features such as strength, energy absorption, fluid circulation, and insulating properties. These capabilities are often achieved through the intricate arrangement of the pores. However, increasing porosity results in decreased stiffness. This tradeoff between porosity and stiffness is illustrated below.
Porosity vs Stiffness
VoroTO Framework
Offline Computation
The framework prevents concurrent homogenization during optimization by utilizing offline computation. This offline computation precomputes the mapping between Voronoi microstructures and their constitutive properties, which is then used during the optimization process. The steps of offline computation are as follows:
Dataset Generation:
In this step, a large and representative set of Voronoi microstructures associated with the macro elements (elements in the macroscale design domain) is created. To accurately represent a typical microstructure, the neighboring macro elements are also considered, as they influence the Voronoi microstructure. Therefore, we consider a macro element and its 8 neighboring macro elements and generate random cell sites in all 9 elements. Then, a Voronoi microstructure is generated in the central element using all the cell sites, and a random set of parameters thickness, anisotropy, and orientation, that are generated uniformly over a pre-defined range. The microstructure elements are discretized, and numerical homogenization is performed to obtain the homogenized elasticity matrix.
Data generation
Neural Network:
Following the data generation and Cholesky decomposition, we use neural networks (NNs) to establish a mapping between the Voronoi parameters and the Cholesky factors of the homogenized elasticity matrix. By mapping the parameters to the Cholesky factors rather than directly to the homogenized elasticity matrix, we ensure the positive definiteness of the homogenized elasticity matrix.
Neural network training
Multiscale TO
The cell sites and geometric parameters of macroscopic elements will be optimized to meet a desired objective. To avoid expensive homogenization, the trained neural network (NN) is utilized to predict the elasticity response of each element. Specifically, at each step of the optimization process, the locations of the cell sites of the macrostructure element and its eight neighbors, along with the current values of the three geometric parameters of the macrostructure element, are extracted to compute the homogenized elasticity matrix.
Multiscale topology optimization framework
Results
Dependency on parameter range
A central hypothesis of our current work is that better multiscale designs can be obtained with a broader range of parameters: thickness, anisotropy, and orientation. Additionally, the computational time is unaffected by the change in the upper bound. We illustrate the performance vs parameter range for thickness and anisotropy below:
Impact of anisotropy on objective.
Impact of thickness on objective.
Computational advantage
We consider a scenario where multiscale optimization is performed without relying on offline training, i.e., concurrent homogenization-based multiscale optimization. In this case, homogenization must be performed for every macroscopic element within the design domain across all optimization iterations. The plot below compares the time taken versus the number of macroscopic elements, demonstrating that the proposed offline NN-based multiscale optimization is computationally far superior.
Citation
@article{padhy2024voroto,
title={VoroTO: Multiscale Topology Optimization of Voronoi Structures using Surrogate Neural Networks},
author={Padhy, Rahul Kumar and Suresh, Krishnan and Chandrasekhar, Aaditya},
journal={arXiv preprint arXiv:2404.18300},
year={2024}
}
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