Multi-Material Topology Optimization Using Neural Networks

doi:10.1016/j.cad.2021.103017

Computer-Aided Design

Volume 136, July 2021, 103017

https://doi.org/10.1016/j.cad.2021.103017 Get rights and content

Highlights

•
A neural network (NN) based multi-material topology optimization (MMTO) method.
•
Spatial coordinates as NN inputs and material volumes as output.
•
The sensitivities are computed analytically using NN’s back-propagation.
•
Leads to a crisp and differentiable material interface.

Abstract

The focus of this paper is on multi-material topology optimization (MMTO), where the objective is to not only compute the optimal topology, but also the distribution of two or more materials within the topology. In the popular density-based MMTO, the underlying pseudo-density fields are typically represented using an underlying mesh. While mesh-based MMTO ties in well with mesh-based finite element analysis, there are inherent challenges, namely the extraction of thin features, and the computation of the gradients of the density fields.

The objective of this paper is to present a neural network (NN) based MMTO method where the density fields are represented in a mesh-independent manner, using the NN’s activation functions, with the weights and biases associated with the NN serving as the design variables. Then, by relying on the NN’s built-in optimization routines, and a conventional finite element solver, the MMTO problem is solved.

The salient features of the proposed method include: (1) thin features can be extracted through a simple post-processing step, (2) gradients and sensitivities can be computed accurately through back-propagation, (3) the NN construction implicitly guarantees the partition of unity between constituent materials, (4) the NN designs often exhibit better performance than mesh-based designs, and (5) the number of design variables is relatively small. Finally, the proposed framework is simple to implement, and is illustrated through several examples.

Introduction

Topology optimization (TO) is a rapidly evolving field, encompassing a rich set of methods including density-based methods that typically exploit material interpolation such as Solid IsotropicMaterial with Penalization (SIMP) [1], [2], level-set methods [3], [4], [5], evolutionary methods [6] and topological sensitivity methods [7], [8], [9], [10], [11]. The reader is referred to [12] for a critical review.

The main focus of this paper is on multi-material TO (MMTO), where the objective is to not only compute the optimal topology, but also the distribution of two or more materials within the topology. MMTO is partly motivated by the advent of additive manufacturing (AM) [13], [14], and in particular, multi-material AM [15], [16], [17]. Various single-material TO methods, most notably SIMP-based density methods, have been extended to multi-materials (see Section 2). In these methods, the unknown density fields are represented using an underlying mesh. While mesh-based MMTO ties in well with mesh-based finite element analysis, there are inherent challenges such as the extraction of thin topological features and boundary normals, elaborated later in the paper. To overcome these challenges we proposed here an MMTO method where the pseudo-density fields are represented using a mesh-independent neural-network (NN). The proposed framework is an extension of a single-material neural-network based TO framework recently proposed in [18] to multi-materials, and is discussed in Section 3. In Section 4, several numerical experiments are carried out to establish the validity, robustness and other characteristics of the method. Open research challenges, opportunities and conclusions are summarized in Section 5.

Section snippets

Literature review

Density-based TO methods, that typically exploit Solid Isotropic Material with Penalization (SIMP) material interpolation, are, by far, the most popular today. They were first extended to MMTO in 1992 [19]. This was followed by [20] where a single variable interpolation scheme for three-phase TO, with extreme thermal expansion, was proposed. The concept was extended in [21] to incorporate numerous candidate materials, and has been used to design multi-physics actuators [22], piezo-composites

Proposed method

Prior to describing the proposed method, the mathematical nomenclature used in the remainder of the paper is summarized below for convenience.

Symbol	Description
$Ω_{0}$	Design domain
$X$	Point (x,y) in $Ω_{0}$
$S$	Number of non-void materials
$E^{i}$	Young’s modulus of the $i$ th material $i = 0, 1, \dots, S$ ; $E^{0} \approx 0$
$\bar{E}$	Effective Young’s modulus (defined in Eq. (8))
$ρ^{i}$	Physical density (unit of kg/m $^{3}$ in SI) of the $i$ th material; $ρ^{0} = 0$
$v^{i} (X)$	Volume fraction (unit-less) of $i$ th material at location $X$ ; $v^{i} (X) \in (0, 1]$
$v (X)$	Volume fraction vector: ${v^{0} (X), v$

Numerical experiments

In this section, we conduct several experiments to illustrate the method and algorithm. The default parameters are as follows:

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All materials are assumed to exhibit a Poisson ratio of $ν = 0.3$ ; Young’s moduli and densities are specified for each example.
•
A mesh size of 60 × 30 is used for all experiments, unless otherwise stated; the force is assumed to be 1 unit.
•
The neural network is composed of 5 hidden layers with 25 nodes (neurons) per layer unless otherwise specified. This corresponds to 1774

Conclusion

The main contribution of this paper is a novel neural-network based multi-material topology optimization method. The method was validated by comparing it against published research. Some of the merits of the proposed method, including extraction of thin features, accurate gradient computations, and crisp material interfaces were demonstrated. However, only a simple total mass-constrained compliance minimization was considered in this paper; extensions to handle other objectives with multiple

Replication of results

The Python code used in generating the examples in this paper is available at www.ersl.wisc.edu/software/MM-TOuNN.zip.

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 support of National Science Foundation, United States of America through grant CMMI 1561899. Prof. Krishnan Suresh also serves as a consulting Chief Scientific Officer of SciArt, Corp.

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In this paper, a density-based topology optimization method using neural networks is used to design domains composed of multi-materials under combined thermo–mechanical loading. The neural network produces the topology density field by minimizing a loss function defined for the problem. Length scale control is conducted using a Fourier space projection. JAX, a high-performance automatic differentiation (AD) Python library is used to build an end-to-end differentiable neural network. The model can handle the sensitivity analysis automatically using the backpropagation process (automatic differentiation of loss function) and the need for solving adjoint equations manual sensitivity analysis is removed. The optimization problem is solved by minimizing the loss function of the network and other optimization algorithms such as the Method of Moving Asymptotes (MMA) are not required. The performance of the method is demonstrated through several examples and compared with those obtained from the popular Solid Isotropic Material with Penalization (SIMP) method. The network is able to handle high-resolution re-sampling, resulting in a more refined and smooth variation of optimal topologies.
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Physics-Informed Neural Networks (PINNs) have recently gained increasing attention in the field of topology optimization. The fusion of deep learning and topology optimization has emerged as a prominent area of insightful research, where minimization of the loss function in neural networks can be comparable to minimization of the objective function in topology optimization. Inspired by concepts of PINNs, this paper proposes a novel framework, ‘Complete Physics-Informed Neural Network-based Topology Optimization (CPINNTO)’, to address various challenges in topology optimization, particularly related to structural optimization. The key innovation of the proposed framework lies in introducing the first complete machine-learning-based topology optimization framework through integration of two distinct PINNs. Herein, the Deep Energy Method (DEM) PINN is implemented to determine the deformation state of corresponding structures numerically. In addition, derivation of the objective function with respect to design variables is replaced with automatic differentiation in sensitivity-analysis PINN (S-PINN). The feasibility and potential of the CPINNTO framework have been assessed through several case studies while highlighting strengths and limitations of utilizing PINNs in topology optimization. Subsequent findings indicate that CPINNTO can achieve optimal topologies without labeled data nor FEA. The numerical examples demonstrate that CPINNTO is capable of stably obtaining optimal structures for various topology optimization applications, including compliance minimization problems, multi-constrained problems, and three-dimensional problems. Resulting designs exhibit favorable compliance values comparable to the designs obtained via density-based topology optimization. In summary, the proposed CPINNTO framework opens up novel and interesting possibilities for structural topology optimization.
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^☆: This paper has been recommended for acceptance by Jean Claude Leon.

View full text

Multi-Material Topology Optimization Using Neural Networks☆

Highlights

Abstract

Introduction

Section snippets

Literature review

Proposed method

Numerical experiments

Conclusion

Replication of results

Declaration of Competing Interest

Acknowledgments

Comput Methods Appl Mech Engrg

J Comput Phys

J Comput Phys

Comput Struct

Addit Manuf

J Mech Phys Solids

Comput Methods Appl Mech Engrg

Comput Methods Appl Mech Engrg

Comput Methods Appl Mech Engrg

Comput Aided Des

Comput Methods Appl Mech Engrg

Eng Struct

Adv Eng Softw

Comput Methods Appl Mech Engrg

J Comput Phys

An analytical model to predict optimal material properties in the context of optimal structural design

J Appl Mech

A 99 line topology optimization code written in Matlab

Struct Multidiscip Optim

Efficient generation of large-scale pareto-optimal topologies

Struct Multidiscip Optim

Multi-constrained topology optimization via the topological sensitivity

Struct Multidiscip Optim

Topological derivatives in shape optimization

A pareto-optimal approach to multimaterial topology optimization

J Mech Des

Topology optimization approaches

Struct Multidiscip Optim

Current and future trends in topology optimization for additive manufacturing

Struct Multidiscip Optim

Additive manufacturing technologies: Rapid prototyping to direct digital manufacturing

Multiple-material topology optimization of compliant mechanisms created via PolyJet three-dimensional printing

J Manuf Sci Eng

OpenFab: a programmable pipeline for multi-material fabrication

ACM Trans Graph

Tounn: topology optimization using neural networks

Struct. Multidiscip. Optim.

Topology optimization of structures composed of one or two materials

Struct Optim

Discrete material optimization of general composite shell structures

Internat J Numer Methods Engrg

Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme

Struct Multidiscip Optim

Multi-material topology optimization using ordered SIMP interpolation

Struct Multidiscip Optim

Level-set method for design of multi-phase elastic and thermoelastic materials

Int J Mech Mater Design

Design of multimaterial compliant mechanisms using level-set methods

J Mech Des