A novel data-driven nonlinear solver for solid mechanics using time series forecasting

https://doi.org/10.1016/j.finel.2019.103377Get rights and content

Highlights

  • A novel data-driven nonlinear solver (DDNS) for solid mechanics using time series forecasting is first proposed.

  • The key concept behind this work is to modify the starting point of iterations of the modified Riks method.

  • The modified Riks method starts iterations at the previously converged solution point.

  • Using DDNS, the new starting point is very close to the converged solution of the current step.

  • The proposed method reduces significantly number of iterations compared with the conventional modified Riks method.

Abstract

In this paper, a novel data-driven nonlinear solver (DDNS) for solid mechanics using time series forecasting is first proposed. The key concept behind this work is to modify the starting point of iterations of the modified Riks method (M-R). The modified Riks method starts iterations at the previously converged solution point while the proposed method starts at a predicted point which is very close to the converged solution of the current step. In the prediction phase, the predicted starting point of the current step is simply determined only based on the previously converged solutions and the predictive networks built via group method of data handling (GMDH) known as a self-organizing deep learning method for time series forecasting problems. Then, the correction phase of the modified Riks method is used to obtain the converged solution via an iterative procedure starting at the predicted point. In this work, the training and applying processes of networks are continuously performed during the analysis to predict the starting point of each increment. It is interesting that the present deep learning networks are built with small data in very short time. Especially, the proposed method is not only simple in implementation but also reduces significantly number of iterations and computational cost compared with the conventional modified Riks method. Some benchmark problems on geometrically nonlinear analysis of shells are provided and solved by using isogeometric analysis (IGA) in conjunction with the first-order shear deformation shell theory (FSDT). The high accuracy, efficiency and stability of the proposed method are confirmed.

Introduction

In structural engineering, several structures are sensitive to change in geometry or change their shapes significantly during loading. For these structures, geometrically nonlinear analysis should be considered to assess accurately and comprehensively the structural real behavior. The geometric nonlinearity is described via the relationship between strain and displacement. As a result, equilibrium paths are obtained via analytical or numerical approaches. For geometrically nonlinear analysis using numerical method, an appropriate iterative algorithm should be chosen to trace equilibrium paths completely. It is well-known that the Newton-Raphson method can be used to trace successfully the monotonous equilibrium path [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]]. As an disadvantage, this method fails to trace the equilibrium path through the limit point. This is due to the tangent stiffness matrix becomes singular at this point. To overcome this difficulty, the modified Newton-Raphson [12] and Quasi-Newton methods [13] were proposed. However, these two methods still fail to trace the equilibrium path of nonlinear problem in which “snap-back type of instability” occurs. Thereafter, the Riks method [[14], [15], [16]] was proposed as a good method for geometrically nonlinear problems in which various types of instabilities occur such as softening-hardening, snap-through, snap-back, etc. It was found that the Rik's technique was still not suitable for use with finite element method (FEM). Then, the modified Riks method considered as one of the best methods was proposed [17] to overcome the disadvantage of the Riks method. As seen in Ref. [17], using the modified Riks method requires a high computational cost due to numerous iterations performed. This obstructs the convenient application of the modified Riks method to geometrically nonlinear problems. Today, many iterative algorithms have been proposed to reduce computational cost as well as number of iterations for nonlinear problems such as iterative methods based on optimization technique [18] and residual areas [19], an iterative method without predictor step [20], dynamic relaxation methods [[21], [22], [23]], multi-point methods [24] and a new method to transform the discretized governing equations [25], etc.

According to the above literature review, it is observed that using time series forecasting to improve iterative methods is very limited. This paper proposes a novel data-driven nonlinear solver for solid mechanics using time series forecasting. As the advantages of the proposed method, number of iterations is significantly saved while the high accuracy and stability of solutions are always ensured compared with the conventional modified Riks method. The key concept behind this work is to modify the starting point of iterations of the modified Riks method. The modified Riks method starts iterations at the previously converged solution point while the proposed method starts at a predicted point which is very close to the converged solution of the current step. In the prediction phase, the predicted starting point of the current step is determined based on the previously converged solutions and the predictive networks built via group method of data handling (GMDH) known as a self-organizing deep learning method for time series forecasting problems. Then, the correction phase of the modified Riks method is used to obtain the converged solution via an iterative procedure starting at the predicted point. It should be emphasized that the predicted starting point in this paper is determined straightforwardly and simply based on the converged displacement and force vectors without requirement of any additional variables. The proposed method is investigated for geometrically nonlinear problems which have some characteristics similar to time series forecasting problems in deep learning area. Deep learning is drawing much attention of many researchers and known as one of the most popular fields. It is not only used in academia but also in industry such as machine health monitoring [26], natural language processing [27], fault-tolerant control [28,29], bankruptcy prediction [[30], [31], [32]], computer vision and pattern recognition [33], material design [34,35], structural engineering [36], etc. As an alternate approach to model selection, bayesian analysis was successfully investigated for computational mechanics [[37], [38], [39]]. In addition, deep neural network (DNN) was applied to solve directly for the partial differential equation without using a discretization method such as FEM [40]. Further developments of deep learning for computational mechanics are found in Refs. [41,42]. Deep learning is a branch of machine learning based on a set of algorithms [43]. Machine learning is the field which gives computers the ability to learn without being explicitly programmed [44]. As known, the application of deep learning to time series forecasting problems is very promising. This application was investigated for music recognition, speech recognition, stock market prediction, etc. A good review of deep learning for time series modeling is presented in Ref. [45]. In attempts to apply deep learning to time series problems, many types of networks have been proposed and developed. In which, long short-term memory (LSTM) network [46] is well-suited to classifying, processing and making predictions based on time series data. In addition, convolutional neural network (CNN) is considered as the most commonly adapted deep learning model [[47], [48], [49], [50], [51], [52], [53], [54]]. As an efficient network for time series forecasting problems, group method of data handling was proposed and known as a self-organizing deep learning method [55]. It is used widely in fields: optimization, data mining, forecasting and pattern recognition, etc. GMDH-based neural network is considered as a polynomial neural network. Several advantages of GMDH network can be listed as: good identification for high-order nonlinear systems, high accuracy in forecasting, self organization in the training process, etc. In addition, the GMDH network can be built and predict with a small amount of data thanks to the principle of self-organization [56]. In solid mechanics, a novel analysis-prediction approach for geometrically nonlinear problems using group method of data handling is found in Ref. [57]. The analysis-prediction approach (ANP) was proposed in Ref. [57] to trace equilibrium paths of geometrically nonlinear problems. Using ANP approach, a part of equilibrium path is traced by numerical analysis (IGA, finite element method or meshless, etc.) and the rest of equilibrium path is purely predicted by GMDH without using any analysis.

Different from ANP approach, the aim of this paper is using the GMDH network to improve the modified Riks method of nonlinear computational mechanics. It should be emphasized that the data-driven nonlinear solver (DDNS) can solve any nonlinear problems. The disadvantage of the analysis-prediction approach (ANP) proposed in Ref. [57] is analyzer should recognize type of nonlinear problem before applying ANP. However, in some cases we can not recognize type of nonlinear problem as well as instable possibility of a given structure due to complexity of nonlinear problems. Therefore, DDNS overcomes the difficulty of the analysis-prediction approach in Ref. [57]. To the best of author's knowledge, this is the first study that proposes a novel data-driven nonlinear solver for solid mechanics based on the modified Riks method and the GMDH network built with a small amount of data in a very short time. Therefore, the proposed method can be considered as a bridge of the existing gap between deep learning and iterative methods of nonlinear computational mechanics. The idea of this paper can be used straightforwardly for improvements of other iterative methods. The outline of this paper is organized as follows. Geometrically nonlinear analysis of shells using FSDT is mentioned in the next section. Section 3 presents the shell formulation using FSDT and IGA. A novel data-driven nonlinear solver for solid mechanics using group method of data handling is proposed in Section 4. Some benchmark problems on geometrically nonlinear analysis of shells are solved and discussed in Section 5 by using isogeometric analysis (IGA) [58] and the first-order shear deformation shell theory (FSDT) [[59], [60], [61]]. The high accuracy, efficiency and stability of the proposed method are verified. The paper is closed with several remarkable conclusions in the last section.

Section snippets

Geometrically nonlinear shells analysis using FSDT

In this paper, the shell formulation is based on FSDT. In case of thick plate/shell, the comprehensive shear stress/strain through the thickness can be obtained via higher-order shear deformation theories [[62], [63], [64], [65], [66], [67], [68], [69], [70]]. We first consider a singly curved shell described as in Fig. 1. Based on the FSDT, the strain components of shell using von Karman assumption are written briefly as follows [61].ɛ=ɛxxɛyyγxyT=ɛ0+zκbγ=γxzγyzT=ɛswhereɛ0=ɛL+ɛN;ɛL=u0,x+w0Rv0,yu

The isogeometric shell formulation based on FSDT

In this section, geometrically nonlinear analysis of isotropic shells using IGA and the FSDT is briefly presented. The detailed formulation is found in Ref. [61] for geometrically nonlinear and postbuckling analyses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells. Because of isotropic shell is a special case of FG-CNTRC shell, the present formulation for isotropic shell is as same as that for FG-CNTRC shell in Ref. [61] except for a small difference in the material

A brief introduction

Group method of data handling [55,71] is known as a self-organizing deep learning method for time series forecasting problems. As a difference with other networks, the GMDH network changes continually during the training process. We first consider a nonlinear system as followsφ=f(x1,x2,,xn)where x1, x2, …, xn and φ are the input and output variables of system, respectively. These variables are connected via a function f. Eq. (21) can be re-expressed based on the Kolmogorov-Gabor form as

Results and discussions

The aim of this section is to demonstrate the high accuracy and efficiency of DDNS via some benchmark geometrically nonlinear problems with various types of equilibrium paths. All the numerical examples in this paper are performed with a mesh of 14 × 14 cubic NURBS elements and 4 × 4 Gauss points per element. In addition, the material characteristics of shells are Young's modulus E = 3.103 kN/mm2 and Poisson's ratio v = 0.3. Boundary conditions are simply imposed as in the standard finite

Conclusions

A novel data-driven nonlinear solver (DDNS) for solid mechanics using time series forecasting has been first proposed in this paper. The core idea behind this work is to modify the starting point of iterations of the modified Riks method (M-R). The modified Riks method starts iterations at the previously converged solution point. In this study, the DDNS starts iterations at a predicted point which is simply determined only based on the previously converged solutions and the predictive networks

CRediT authorship contribution statement

Tan N. Nguyen: Investigation, Writing - original draft. H. Nguyen-Xuan: Writing - review & editing. Jaehong Lee: Writing - original draft, Supervision.

Acknowledgments

This research was supported by a Grant (NRF-2018R1A2A1A05018287) from NRF (National Research Foundation of Korea) funded by MEST (Ministry of Education and Science Technology) of Korean government. The support is gratefully acknowledged.

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