Image-Based Multiresolution Topology Optimization Using Deep Disjunctive Normal Shape Model

Highlights

• Generic framework for multiresolution topology optimization with varying parameters using deep neural networks.

• A systematic algorithm to choose important high resolution designs for training based on a quantitative image-based measure.

• Generation of near optimal high resolution designs using mainly inexpensive low resolution designs.

• Application to 3D topology optimization.

Abstract

We present a machine learning framework for predicting the optimized structural topology designs using multiresolution data. Our approach primarily uses optimized designs from inexpensive coarse mesh finite element simulations for model training and generates high resolution images associated with simulation parameters that are not previously used. Our cost-efficient approach enables the designers to effectively search through possible candidate designs in situations where the design requirements rapidly change. The underlying neural network framework is based on a deep disjunctive normal shape model (DDNSM) which learns the mapping between the simulation parameters and segments of multi resolution images. Using this image-based analysis we provide a practical algorithm which enhances the predictability of the learning machine by determining a limited number of important parametric samples (i.e. samples of the simulation parameters) on which the high resolution training data is generated. We demonstrate our approach on benchmark compliance minimization problems including the 3D topology optimization where we show that the high-fidelity designs from the learning machine are close to optimal designs and can be used as effective initial guesses for the large-scale optimization problem.

Introduction

Topology optimization has been extensively used as a popular computational design tool for generating optimal structural layouts. This powerful tool enables the creation of innovative structures with complex geometry that may not be realizable with traditional approaches such as shape or size optimization [1], [2]. This approach however suffers from significant computational cost especially when a detailed structural design is sought. In such cases, typically, many large scale finite element analyses need to be performed to reach to a satisfactory design which poses a significant challenge in terms of computational cost. Finding such high resolution designs in scenarios where the simulation parameters are variable is even more onerous.

In this paper, we adopt a scientific visualization/machine learning (ML) approach to overcome the computational challenges for large scale topology optimization, especially when optimal high-fidelity designs are required for variable simulation parameters. In particular, our approach utilizes multifidelity designs where mainly inexpensive low-fidelity designs in addition to judiciously chosen high-fidelity designs are used for training. Using the learning machine which encodes the relationship between the simulation parameters and the shape space we approximate optimized designs on parameters that are not previously considered in the simulations. This approach will enable the designers to efficiently traverse the simulation parameters and find the best solutions which satisfy the design objectives and criteria.

The result of this work contributes to the development of an efficient visualization tool for exploring optimized designs with variable parameters. Ideally the visualization tool is a platform that provides high resolution optimized designs interactively and this work is one stepping stone toward this objective. With such a tool, a designer could navigate through the design space interactively, assessing multiple designs qualitatively (i.e. visually) and quantitatively (through various evaluation metrics build into the tool).

In the context of engineering design, similar efforts have been devoted to embed the high-dimensional design space into semantic compact subspaces [3], [4] using a manifold clustering procedure and deep generative models. The idea of visual parameter space analysis has also been explored [5], [6] where visual interactive tools are introduced to navigate the data spaces or the space of quantities of interest as a function of reduced-dimensional parameter space. Similar to these efforts, several researchers have introduced visual analysis techniques for exploration of parameter-data space map in multi-criteria decision making [7], medical image analysis [8], [9], geoscientific simulation models [10], decision support in flood management [11], [12], inspection of fastening bolts on freight trains [13], etc. Our work in this paper can be viewed as a visual analysis technique which adopts a powerful function approximator, i.e. deep neural network to map the simulation parameters to the high-dimensional space of topology optimized designs.

Deep learning (DL) is known as a subset of ML. A large number of layers in the architecture of DL models enable them to estimate complex functions. However, there is a need for large datasets to train such complex models. In other words, DL fundamentally is neural networks that learn from large amounts of data with the use of various layers. An important class of DL, convolutional neural networks introduced in [14], are widely used in image processing. DL has also been employed in many different areas such as video processing, speech recognition, natural language processing, and computer vision [15], [16], [17], [18].

In this paper, we present a novel deep learning approach in conjunction with the disjunctive normal shape model (DNSM) for the generic topology optimization problem. We refer to this approach as Deep DNSM which is abbreviated as DDNSM. The DNSM is a shape model that first was introduced by Ramesh et al. [19] and reformulated by Mesadi et al. [20] to be used in a Bayesian framework for image segmentation tasks. The DNSM has also been used in conjunction with a convolutional neural network by Javanmardi et al. [21] for segmenting shapes with low quality and low signal-to-noise ratio. The main idea of DNSM is to model shapes as the disjunction of convex polytopes with each polytope as a conjunction of half-spaces. Inspired by [21] we have designed a new framework that uses the DNSM on top of a fully connected deep neural network as a multiresolution network with both low resolution and high resolution designs to be trained jointly. We provide more details on both DNSM and DDNSM in Section 3.

Structural topology optimization has evolved significantlysince the time it first introduced. A comprehensive review of current and future trends in this field has recently been published [22]. Among these works many researchers have addressed the problem of parametric topology optimization and topology optimization under uncertainty where the main focus is to generate robust and reliable designs [23], [24], [25], [26], [27], [28]. Such problems pose a similar challenge to the one in this paper where the simulation parameters are variable. In the same vein, several research works have also attempted to tackle the problem of data-driven topology optimization and proposed novel approaches for prediction improvement in such data-driven approximations [29], [30]. A number of researchers have proposed to use computational capabilities of a Graphics Processing Units (GPU) for large scale topology optimization where the main goal is to achieve optimal high resolution structures with minimal process time [31], [32], [33]. These efforts are in line with approaches which use machine learning techniques for large scale problems where powerful computational resources are required.

Machine learning techniques have achieved compelling results in data-based approximations such as image processing, pattern recognition, finance, etc. [34], [35], [36]. Complementing these efforts, novel techniques have been proposed for prediction/approximation with physics-based or simulation-based learning machines. The main goal in these works is to predict physics-based simulations which are typically governed by partial differential equations (PDE) [37], [38]. This type of approaches has close connection to the topology optimization problem where the solution of the optimization problem is sought for a physical systems described with PDEs.

Topology optimization using machine learning is a relatively new research direction. As examples, authors in [39], [40], [41] use generative modeling techniques for topology optimization where they use variational auto encoders (VAE) and generative adversarial networks (GANs) to predict the optimized topology designs. A similar work [30] proposes the use of Principal Component Analysis (PCA) and a fully connected neural network to learn the mapping between loading configurations and optimal topologies. Authors in [42] use convolutional encoder–decoder architecture to solve the topology optimization problem by posing it as an image segmentation task. Finally authors in [43] introduce a theory-driven learning mechanism based on the optimality criteria in topology optimization for generation of near-optimal topology designs.

It is also noteworthy to mention approaches based on multiresolution analysis in the context of shape and topology optimization [44], [45], [46], [47], [48], [49], [50], [51], [52]. The main theme among these works is to leverage the computational efficiency of low resolution models to obtain more expensive high resolution designs. Our multiresolution approach in this paper is indeed similarly motivated by reducing the computational cost; however unlike the aforementioned references which consider a deterministic setting, our strategy is devised to incorporate parametric variations in design such as variations in load and boundary conditions [28], [53]. Another important line of research which is relevant to this paper is the incorporation/representation of geometric features in topology optimization which is done via combining free-form topology optimization with embedded components/holes or representing the structure via union of geometric primitives, e.g. rectangles with straight or semicircular ends [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69]. We however remark that our approach in this paper, DDNSM which is an effective way to represent binary images is solely used to learn from and subsequently approximate the topology optimized designs as binary images. We note that we do not use this approach for the task of topology optimization (or representing the geometry) itself; we instead use a well-known density-based approach to generate the optimized designs [70].

In this paper, we mainly focus on the computational challenges associated with the high resolution designs with variable parameters. To this end we propose an algorithm which utilizes a limited number of high resolution images corresponding to previously found optimized designs in conjunction with a large number of inexpensive low resolution optimized topology images for network training and generate a learning machine which predicts the near-optimal high resolution images for previously unseen simulation parameters. Our work contributes to the existing literature on topology optimization using machine learning by providing a novel approach which connects topology optimization with scientific visualization. In particular our work can be considered among the few recent studies which leverage deep learning for predicting near optimal designs [41], [43]. These approaches are particularly useful when there is variation in design parameters and could be effective for fast exploration of optimal design manifold. We remark that in this paper we provide a systematic computational cost analysis which serves as a practical guideline to assess the learning cost in comparison with the cost of direct high resolution designs. We also remark that our deep learning approach is, in essence, different from the zero-order optimization approaches (or population-based approaches) such as genetic programming. Our approach similarly to any deep learning approach is based on the minimization of a loss function which is performed via a gradient-descent approach. As mentioned earlier we also use a gradient-based approach to generate the training data which renders our approach entirely gradient-based [71].

The organization of this paper is as follows. In Section 2 we briefly discuss the topology optimization problem, in particular the compliance minimization. We also discuss the details of data generation for network training in this section. Section 3 discusses the details of DNSM and DDNSM approach in conjunction with an algorithm for selection of important high resolution samples. Section 4 presents numerical results on topology design of an L-bracket elastic structure, a heat sink structure, and a 3D linear elastic structure. Finally Section 5 discusses the concluding remarks.