[Paper Review] A Parametric Level Set Method for Topology Optimization based on Deep Neural Network (DNN)

Hao Deng, Albert C. To|arXiv (Cornell University)|Jan 9, 2021

Topology Optimization in Engineering61 references4 citations

TL;DR

This paper proposes a DNN-based parametric level set method for structural topology optimization, where deep neural networks parameterize the level set function and evolve it via learned weights and biases. By transforming Hamilton-Jacobi PDEs into parametrized ODEs and applying periodic reinitialization, the method enables diverse, high-quality designs with improved convergence and flexibility over traditional level set approaches.

ABSTRACT

This paper proposes a new parametric level set method for topology optimization based on Deep Neural Network (DNN). In this method, the fully connected deep neural network is incorporated into the conventional level set methods to construct an effective approach for structural topology optimization. The implicit function of level set is described by fully connected deep neural networks. A DNN-based level set optimization method is proposed, where the Hamilton-Jacobi partial differential equations (PDEs) are transformed into parametrized ordinary differential equations (ODEs). The zero-level set of implicit function is updated through updating the weights and biases of networks. The parametrized reinitialization is applied periodically to prevent the implicit function from being too steep or too flat in the vicinity of its zero-level set. The proposed method is implemented in the framework of minimum compliance, which is a well-known benchmark for topology optimization. In practice, designers desire to have multiple design options, where they can choose a better conceptual design base on their design experience. One of the major advantages of DNN-based level set method is its ability to generate diverse and competitive designs with different network architectures. Several numerical examples are presented to verify the effectiveness of the proposed DNN-based level set method.

Motivation & Objective

• To develop a novel parametric level set method that integrates deep neural networks (DNNs) into topology optimization for enhanced design flexibility.
• To address the limitations of conventional level set methods in generating diverse and competitive design concepts.
• To enable efficient evolution of the level set function through trainable DNN parameters (weights and biases).
• To maintain numerical stability by applying parametrized reinitialization to control the steepness or flatness of the level set function.
• To demonstrate the method’s effectiveness in generating multiple competitive designs under the minimum compliance benchmark.

Proposed method

• The level set function is implicitly represented using a fully connected deep neural network, where the zero-level set defines the structural boundary.
• The evolution of the level set is governed by transforming Hamilton-Jacobi PDEs into parametrized ODEs, with time derivatives computed via backpropagation through the DNN.
• The weights and biases of the DNN are updated iteratively using gradient-based optimization to minimize the compliance objective.
• A parametrized reinitialization procedure is applied periodically to maintain the level set function’s regularity near the zero-level set.
• The method is embedded within a minimum compliance topology optimization framework to evaluate structural performance.
• Different network architectures are used to generate diverse design options, leveraging the DNN’s capacity for structural variation.

Experimental results

Research questions

• RQ1 Can a deep neural network effectively parameterize the level set function to enable efficient and stable topology optimization?
• RQ2 How does the DNN-based level set method compare to traditional level set methods in terms of design diversity and convergence?
• RQ3 To what extent can different DNN architectures generate competitive and distinct structural designs?
• RQ4 Can parametrized reinitialization maintain numerical stability during the optimization process?
• RQ5 Does the proposed method produce high-performance designs under the minimum compliance benchmark?

Key findings

• The DNN-based level set method successfully generates multiple diverse and competitive structural designs by varying network architectures.
• The method achieves stable convergence through the use of parametrized reinitialization, preventing ill-conditioning of the level set function.
• The optimization process is accelerated by leveraging backpropagation to compute gradients of the compliance with respect to DNN parameters.
• The approach demonstrates superior flexibility in design exploration compared to conventional level set methods.
• Numerical examples confirm the method’s effectiveness in producing high-performance topologies under the minimum compliance criterion.
• The integration of DNNs enables a seamless transition from implicit geometry representation to optimization, enhancing design diversity.