Topology optimization of 2D structures with nonlinearities using deep learning
Introduction
The pursuit of structures and materials with enhanced performance yet lightweight has been of high scientific and industrial interest [1], [2], [3]. Generally, such materials and structures can be obtained by selecting the constituents (a) materials, (b) volume fractions, and (c) architectures. The former two approaches have been studied extensively and are almost mature [4]. On the other hand, designing the architectures of materials is still an active area of research, as it allows for obtaining unique properties [5], [6], [7], [8]. The increased interest in architectured materials is related to their enhanced properties such as permeability, thermal and electrical conductivities, electromagnetic shielding effectiveness, stiffness-to-weight ratio, etc. [9], [10]. Recent advances in additive manufacturing have permitted the fabrication of such materials and structures with complex geometries [11], [12], [13], [14]. Attaining architectures resulting in structures and materials with enhanced performance is usually based on intuitions, experiments, and/or bioinspiration [15], [16].
Topology optimization offers a systematic platform for obtaining new designs of materials and structural systems with optimized responses [17], [18], [19], [20], [21], [22], [23]. Generally, solving the inverse problem is a difficult task to deal with, in which specific parameters need to be found to obtain an optimal response, and to do so, the forward problem has to be solved iteratively [24], regardless of using gradient-based or gradient-free optimization algorithms. In topology optimization problems, one aims at identifying the optimal material distribution yielding the desired properties such as maximization of energy absorption and minimization of compliance, while still, the design constraints are satisfied. James et al. [25] developed a framework for optimizing structures where they accounted for material damage. The failure is mitigated by enforcing a constraint on the maximum local damage intensity. Also, Russ et al. [26] used the phase-field method for the fracture to increase the structural fracture resistance and strength. Geometrically nonlinear structures have also been studied, as shown in [27], [28].
Another intriguing problem in the field of topology optimization is problems involving many load cases. Zhang et al. [29] proposed a computationally-efficient randomized approach for deterministic topology optimization with many load cases. Lately, manufacturing-oriented topology optimization has experienced an increasing interest by both industry and academia, especially with recent advances in the field of additive manufacturing [30]. Also, increasing attention is observed for developing topology optimization algorithms for multi-material structures. For example, Alberdi et al. [31] developed a bi-material topology optimization framework, where hyperelastic and viscoplastic phases are combined, for maximizing energy dissipation. Additionally, Conlan-Smith et al. [32] applied topology optimization to design compliant mechanisms using functionally graded materials.
Generally, topology optimization problems are very computationally expensive due to a large number of design variables and the need for many optimization iterations before obtaining the optimal one [33]. Also, gradient-based topology optimization algorithms may suffer from the dependency on the starting point, given that multiple local optima exist. In such a scenario, it is probable that the attained optimized solution is not the global optimum. These drawbacks urge many researchers to develop more efficient frameworks to determine the optimal solution. For instance, Lee et al. [34] proposed a new meta-heuristic optimization algorithm suitable for engineering applications.
Advances in high-performance computer (HPC) hardware and scalable solver algorithms have revolutionized various science and engineering fields in the last two decades allowing high fidelity nonlinear finite element (FE) simulations of highly heterogeneous materials [35] as well as multiphysics even on the petascale computing architecture [36], [37]. The field of machine learning (ML) is no exception, and particularly deep learning has benefited from these technological advances, especially on graphics processing units (GPU). ML has been successful and effective in spam detection, image and speech recognition, discoveries of diseases and drugs, remote sensing image analysis for traffic applications, and search engines [38], [39], [40].
Furthermore, ML has shown success in mechanics-related fields [41], [42], [43], [44], [45], [46], including and limited to predicting solidification defects [47] and effective thermal conductivities of composites [48], [49], solving multiphysics problems [50], and designing new materials [51], [52]. Bessa et al. [53] showed that obtaining material models using ML is possible, providing that the computational analyses of representative volume elements (RVEs) have high fidelity and enough efficiency required to generate sufficient data for supervised learning tasks. The use of ML algorithms has intriguingly been extended to the prediction and optimization of different materials and structural systems [54], [55], [56], [57], [58], [59], [60], [61], [62], [63]. Also, neural networks have been used to solve partial differential equations (forward boundary value problems), avoiding the conventional discretization involved in the finite element method, by using either energy approach [64] or collocation strategy [65].
Abueidda et al. [66] developed a convolutional neural network (CNN) model that is capable of quantitatively predicting the stiffness, strength, and toughness of a two-dimensional (2D) checkerboard composite. Also, they integrated the CNN model with a genetic algorithm to solve single- and multiple-objective optimization problems. The use of deep learning was taken one step further to precisely predict plasticity-constitutive laws as detailed in [67], in which the authors showed that sequence learning can obtain the evolution of stresses and plastic energy, given a deformation path.
Recently, deep learning has been implemented to perform optimization procedures directly without the need to involve an optimizer as in the work of Abueidda et al. [66] and Sasaki et al. [68]. This is accomplished by training the deep learning algorithms to produce images of the optimized designs given a set of boundary conditions and loads [69], [70]. For instance, Yu et al. [71] proposed a deep learning model that is capable of identifying optimal designs without using an iterative scheme. The model was trained on synthetic data generated by an open-source code for linear elastic optimization. Moreover, Rawad and Shen [72], [73] employed a generative adversarial network, which consists of a discriminator and a generator, to optimize two-dimensional (2D) and three-dimensional (3D) linear elastic structures. Also, Zhang et al. [74] developed a CNN model, composed of an encoder and decoder, that identifies the optimal designs in negligible time. The material they considered is a linear elastic one assuming infinitesimal strain theory. White et al. [75] developed a multiscale topology optimization framework for elastic structures using a neural network surrogate model.
So far, the implementation of machine learning algorithms in topology optimization has been limited to design spaces with linear elastic materials undergoing small deformation, with linear optimization constraints. Several studies have shown that geometric and material nonlinearities significantly influence the solution of the optimization, provided that the loads are large enough to onset system nonlinearities [76], [77], [78]. In this paper, we develop three CNN models to predict the material distribution possessing the optimized response, where the first model assumes linear elastic material and small deformations without stress constraint, while the second model accounts for large deformations. The CNN model accounting for large deformations is developed for materials obeying the hyperelastic neo-Hookean constitutive model. The third CNN model assumes a linear elastic material under a stress constraint [22], [79], [80]. The stress constraint is efficiently imposed using a smooth maximum function using global aggregation.
In this paper, we develop ML models that perform a real-time topology optimization of materials under large deformation and small deformation (with and without stress constraint). The remainder of the paper is organized as follows: Section 2 provides an overview of the general topology optimization problem we are interested in. Section 3 scrutinizes the sample space and associated training and testing datasets. Section 4 discusses the architectures of the CNN models and their corresponding hyperparameters and states the loss function and metrics employed in evaluating the performance of the CNN models. In Section 5, we present the results along with analysis and discussion. We conclude this study in Section 6 by summarizing the significant outcomes and discussing potential directions for future work.
Section snippets
Linear and nonlinear structures
Generally, topology optimization algorithms attempt to identify the optimal material distribution within a given design space that minimizes or maximizes single or multiple objective function(s) while a set of constraints are satisfied. Topology optimization problems are solved by directly optimizing the location of the material boundary inside a design space [81], or they are solved by determining elements to be contained within a material region [25]. In this study, the latter approach is
Elasticity and hyperelasticity
Here, two CNN models are developed, one for a linear elastic material and another for a neo-Hookean rubber-like material. A dataset is generated for each of these two material models. Each dataset is composed of many pairs of optimized designs and their corresponding boundary conditions, loads, and volume constraints. In this study, the proposed framework is illustrated using a single concentrated force (at a node on the right-hand side of the design space) while fixed displacements are imposed
ResUnet architecture
The primary objective of this paper is to develop deep CNN models to solve topology optimization problems. The adopted CNN model is based on the ResUnet proposed by Zhang et al. [40]. The ResUnet is a semantic segmentation convolutional neural network combining the privileges of the U-net and residual learning to improve the performance of U-net further. U-net was initially proposed by Ronneberger et al. [93]. U-net concatenates feature maps from different levels to improve segmentation
Results and discussion
Fig. 6 presents a flowchart showing the different stages of model development. The training of a CNN model is achieved by solving an optimization problem aiming at finding the parameters of the CNN model, so the loss function MSE is minimized. The CNN models developed for the linear elasticity (small deformation) with and without stress constraint and hyperelasticity are trained using 150 epochs. The data generated for each case are split into training (81%), validation (9%), and testing (10%)
Conclusions and future work
In this paper, we develop three CNN models to predict the optimized designs in the case of (a) linear elasticity with small deformation (without nonlinear constraints), (b) nonlinear hyperelasticity (neo-Hookean material) with geometric nonlinearity, and (c) linear elasticity with stress constraint, a nonlinear constraint. The developed machine learning models are robust, and they are in an excellent agreement with the results obtained from the mathematically rigorous nonlinear topology
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to thank the National Center for Supercomputing Applications (NCSA) Industry Program for software and hardware support.
References (102)
- et al.
Fabrication of 3D micro-architected/nano-architected materials. Three-Dimensional Microfabrication Using Two-photon Polymerization
(2016) - et al.
Acoustic band gaps and elastic stiffness of PMMA cellular solids based on triply periodic minimal surfaces
Mater Des
(2018) - et al.
Architectured materials: Expanding materials space
Scr Mater
(2013) - et al.
3D printed spacers for organic fouling mitigation in membrane distillation
J Membr Sci
(2019) Mechanics of cellular and other low-density materials
Int J Solids Struct
(2000)- et al.
Topological shape optimization of 3D micro-structured materials using energy-based homogenization method
Adv Eng Softw
(2018) - et al.
Generating optimal topologies in structural design using a homogenization method
Comput Methods Appl Mech Eng
(1988) - et al.
The generation of hierarchic structures via robust 3D topology optimisation
Adv Eng Inf
(2017) - et al.
Failure mitigation in optimal topology design using a coupled nonlinear continuum damage model
Comput Methods Appl Mech Eng
(2014) - et al.
Topology optimization for brittle fracture resistance
Comput Methods Appl Mech Eng
(2019)
Stochastic sampling for deterministic structural topology optimization with many load cases: Density-based and ground structure approaches
Comput Methods Appl Mech Eng
Bi-material topology optimization for energy dissipation with inertia and material rate effects under finite deformations
Finite Elem Anal Des
A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice
Comput Methods Appl Mech Eng
Alya: Multiphysics engineering simulation toward exascale
J Comput Sci
Sparse matrix factorization in the implicit finite element method on petascale architecture
Comput Methods Appl Mech Eng
Prediction of aeroelastic response of bridge decks using artificial neural networks
Comput Struct
Linear and nonlinear model updating of reinforced concrete T-beam bridges using artificial neural networks
Comput Struct
A neural network surrogate model for the performance assessment of a vertical structure subjected to non-stationary, tornadic wind loads
Comput Struct
Deep long short-term memory networks for nonlinear structural seismic response prediction
Comput Struct
Data-driven approach for a one-dimensional thin-walled beam analysis
Comput Struct
Using deep neural network with small dataset to predict material defects
Mater Des
Predicting the effective thermal conductivities of composite materials and porous media by machine learning methods
Int J Heat Mass Transf
Predicting the effective thermal conductivity of composites from cross sections images using deep learning methods
Compos Sci Technol
A deep learning-based hybrid approach for the solution of multiphysics problems in electrosurgery
Comput Methods Appl Mech Eng
A framework for data-driven analysis of materials under uncertainty: Countering the curse of dimensionality
Comput Methods Appl Mech Eng
Post-buckling optimisation of composite stiffened panels using neural networks
Compos Struct
A novel deep learning based method for the computational material design of flexoelectric nanostructures with topology optimization
Finite Elem Anal Des
Structural optimization using evolution strategies and neural networks
Comput Methods Appl Mech Eng
Artificial neural network based hole image interpretation techniques for integrated topology and shape optimization
Comput Methods Appl Mech Eng
De novo composite design based on machine learning algorithm
Extreme Mech Lett
An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications
Comput Methods Appl Mech Eng
Prediction and optimization of mechanical properties of composites using convolutional neural networks
Compos Struct
Multiscale topology optimization using neural network surrogate models
Comput Methods Appl Mech Eng
Stress-based topology optimization using an isoparametric level set method
Finite Elem Anal Des
Structural optimization using sensitivity analysis and a level-set method
J Comput Phys
Stability of finite element models for distributed-parameter optimization and topology design
Comput Methods Appl Mech Eng
Topology optimization of non-linear elastic structures and compliant mechanisms
Comput Methods Appl Mech Eng
Statistical validation of image segmentation quality based on a spatial overlap index1: scientific reports
Acad Radiol
Mechanical response of 3D printed bending-dominated ligament-based triply periodic cellular polymeric solids
J Mater Eng Perform
Cellular Solids: Structure and Properties
Materials: Engineering, Science Processing and Design
Multidimensional phononic bandgaps in three-dimensional lattices for additive manufacturing
Materials
Viscoelastic properties of architected foams based on the Schoen IWP triply periodic minimal surface
Mech Adv Mater Struct
Shielding effectiveness and bandgaps of interpenetrating phase composites based on the Schwarz Primitive surface
J Appl Phys
An overview on additive manufacturing of polymers
JOM
Metal additive manufacturing: A review
J Mater Eng Perform
Architected cellular materials
Annu Rev Mater Res
Design and prototyping soft–rigid tendon-driven modular grippers using interpenetrating phase composites materials
Int J Robot Res
Topology Optimization: Theory Methods and Applications
Design of energy dissipating elastoplastic structures under cyclic loads using topology optimization
Struct Multidiscip Optim
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