Elsevier

Computers & Structures

Volume 247, 15 April 2021, 106480
Computers & Structures

Zooming method for FEA using a neural network

https://doi.org/10.1016/j.compstruc.2021.106480Get rights and content

Highlights

  • Zooming method for FEA using a neural network.

  • Local boundary displacement obtained accurately even when local nodes are outside of global meshes.

  • Simple zooming method that could be easily applied to FEA software.

  • Stress analysis of CFRP using a detailed model by parallel FEM.

  • A large-scale model with 70 million degree of freedom.

Abstract

In the analysis of carbon fiber reinforced composite materials (CFRP), zooming analysis is used to simplify a finite element model by dividing it into global coarse meshes and local fine meshes. The zooming method using a shape function has a problem that displacement of boundary nodes of a local model cannot be obtained accurately if the nodes are outside a global model. In addition, a large-scale finite element model is required to simulate their complex failure accurately by modeling fibers and resin matrices in a local model. Parallel finite element analysis (FEA) open-source software has been developed to analyze large-scale models, but to implement a zooming method into a finite element software is not easy. In this study, we have developed a zooming method using a neural network. The neural network learns the relationship between nodal coordinates and nodal displacements of a global model, and the displacements for boundary conditions of a local model are obtained using the trained neural network. We verified the proposed method using small-scale models. The analysis results from this method were in good agreement with analysis results when using fine meshes. It was found that the method had advantages even when a part of a local model was outside of a global model. In addition, it is a simple method that does not require rewriting software codes, and it can be applied to various pieces of software easily using frameworks for a neural network. We also evaluated if this method can be applied to the analysis of large-scale CFRP models with more than 70 million degrees of freedom. This zooming method and parallel FEM could evaluate the stress and strain of fibers and resin matrices in detail.

Introduction

Carbon fiber reinforced plastic composites (CFRP) are currently used for aircraft applications because of their excellent specific strength and specific elasticity. CFRPs are prepared by impregnating fibers of approximately 7 μm diameters with resin. CFRPs exhibit highly complex interactive failure mechanisms owing to their fiber–matrix interfaces (shown in Fig. 1), and hence, their experimental evaluation is essential. Approximately 10,000 tests are required for material specimens, before the safety certification of an airframe structure [1]. To reduce the number of experiments, simulation methods for failure analysis of CFRPs have been developed [2], [3], [4]. If the fibers and resin are modeled to evaluate all these damages, finite element modeling of even a single test piece would be difficult. Therefore, multiscale analysis is essential for evaluating the damage in composites. One of the multiscale analysis methods is a homogenization method using the finite element method. It is widely used in damage analysis of composite materials because it can evaluate the response of a microstructure that models fibers in detail and it can evaluate strength of overall structure composed of microstructure using the micro response [5], [6], [7]. In the method, a representative volume element (RVE) that models a microstructure is used. The RVE length scale is assumed to be much smaller than the characteristic length over which the macroscopic loading varies in space in the homogenization method [8]. It is assumed that the following relation is satisfied.lLlGlL is the representative length of RVE that models the microstructure, and lG is the representative length of a overall structure. RVE is a local model and a overall structure is a global model. In the homogenization method, the coordinate vector of RVE after the transformation x is calculated as follows [9], [10], [11], [12].x=F¯X+w~(X)F¯is a deformation gradient tensor at the element integration point of a global model.X is the coordinate vector of RVE before the transformation. w~(X) is a fluctuation field. The constraint for w~(X) is as follows.1VΩw~(X)dV=0Ω is the domain of RVE and V is the volume of the domain. The constraint (3) is satisfied by the alternative conditions.w~(X)=0inΩw~(X)=0onΩw~(X+)=w~(X-)onΩΩ is the boundary of RVE. The boundary is decomposed into two parts Ω=Ω+Ω- with outward normal n-=-n+ at associated points X+Ω+ and X-Ω-, respectively. Eq. (6) shows periodic boundary conditions. In the homogenization method, the displacement of a local model is limited. Moreover, it is difficult to evaluate the progress of the crack by taking a large area of a local model because the size of the local model is considerably smaller than that of the global model. The other of the multiscale methods is the zooming method [13], [14]. In this method, a local refined mesh model is analyzed using global model analysis results. An arbitrary boundary condition can be imposed to a local model without being limited by a periodic boundary condition. It is also possible to increase the size of a local model. The method has been applied to the strength analysis of CFRP [15], [16], [17], [18]. Large-scale finite elements for the local model of the fibers and resin matrices are required to more accurately predict the damage of CFRP laminates. Finite element analysis (FEA) software capable of large-scale structural analysis using parallel computing has been developed with improved computer performance lately. Using this, large-scale analysis cases with about a hundred of millions of degrees of freedom have been reported thus far [19], [20], [21]. Therefore, if the zooming method is applied to the large-scale parallel FEA, it can be used as one of the effective methods in progressive damage analysis of CFRPs.

Section snippets

Problem statement

In a zooming method based on displacement, boundary conditions of a local model are calculated from the displacement analysis results of the global model. First, a global analysis of a entire structure using coarse meshes is conducted. Second, a local analysis of small regions with refined meshes is conducted using the global displacement as the boundary condition. Nodal displacement for boundary conditions of the local model is calculated using the shape function of the finite element used in

A method combining FEM and neural networks

A zooming method based on displacement can be used if we can model a nonlinear function in which inputs are the coordinate values and outputs are nodal displacements. In recent years, A neural network [23], which is one of the machine learning methods, can model a nonlinear function [24], [25]. A neural network has garnered success especially in the field of image processing [26].Many frameworks such as Tensorflow [27] and Pytorch [28] help in the easy development of neural networks. A method

Methods

We verified the accuracy of the developed zooming method using small-scale models in static linear analysis. For this purpose, we compared our proposed method with a zooming method using a shape function, and the global model analysis using fine meshes (Fine mesh for global model). The global model used for this analysis is shown in Fig. 4. Here, one end face was fixed and a part of another face had a distributed load of 1 Mpa. Local models used are shown in Fig. 5, Fig. 6. The local models are

Methods

The method developed in this study was applied to the structural analysis of a large-scale CFRP model. FrontISTR is used for the large-scale parallel finite element analysis method. The OHT test specimen shown in Fig. 18 was used asthe global model. Fiber directions were as indicated in Fig. 18. The thickness of a unidirectional ply for the OHT test specimen was 0.02 mm. A total of 120 plies were stacked in the order of 45°, 0°, −45°, 90° from the two outermost layers. We analyzed a half

Conclusions

In this study, we developed a zooming method for FEA using a neural network. The neural network learnt nodal coordinates and nodal displacements of the global model, and the trained network predicted displacements for local model boundary conditions. We verified the method using small-scale models. The analysis results from the proposed method were in good agreement with the analysis results obtained using fine meshes. In a fillet part outside the global model, the proposed method obtained more

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.

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