A new approach for flexoelectric material shape optimization is proposed in this study. In this work, a proxy model based on artificial neural network (ANN) is used to solve the parameter optimization and shape optimization problems. To improve the fitting ability of the neural network, we use the idea of pre-training to determine the structure of the neural network and combine different optimizers for training. The isogeometric analysis-finite element method (IGA-FEM) is used to discretize the flexural theoretical formulas and obtain samples, which helps ANN to build a proxy model from the model shape to the target value. The effectiveness of the proposed method is verified through two numerical examples of parameter optimization and one numerical example of shape optimization.

Optimization plays a vital role in the analysis of physical systems and in the science of decision making, aiming to maximize utility within resource constraints. Structural optimization problems can be broadly classified into three categories: size optimization [

Machine learning has gained significant attention in recent years as an alternative to classical optimization methods [

Kien et al. [

Flexoelectricity was first introduced by Mashkevich and Tolpygo in 1957 [

In the realm of flexoelectric effect analysis, numerous scholars have made remarkable contributions. El Dhaba et al. [

IGA represents substantial progress in computational mechanics [

Based on the aforementioned inspiration, we propose a method that combines ANN and IGA-FEM for solving shape optimization problems. In this method, the bending theoretical formulas are discretized using IGA-FEM to generate samples. These samples are then used to train the artificial neural network, which establishes a proxy model linking the model shape to the target value. Subsequently, this proxy model is utilized to solve shape optimization problems.

The content structure of this paper is set as follows: In the second section, the steps and improvements of ANN for optimization problems are introduced. In the third section, the theoretical formulas of flexographic problems and how to obtain initial samples by IGA-FEM method are expounded. Finally, in the fourth section, the effectiveness and accuracy of the method are verified by numerical examples.

In order to make it easier for readers to understand, we first give a brief introduction to the composition and working principle of ANN. Please see [

ANN typically consist of three layers: input, hidden, and output. Among them, the input layer receives data, while the output layer produces the final result. Multiple hidden layers can exist. Each layer contains multiple neurons, with the number depending on specific requirements. The number of hidden layers in ANN can be customized, often involving multiple layers. These layers are fully connected, meaning that there are connections between adjacent layers, but no connections within the same layer.

Neural networks utilize two main processes, known as forward propagation and back propagation, to achieve self-learning. Forward propagation involves inputting samples into the neural network, which then passes through hidden layers until the output layer generates the desired output. The quality of the model’s fit is evaluated by assessing the loss function. Typically, the mean square error between the output layer results and sample labels is commonly utilized as the loss function for ANN,

The input and output vectors are denoted as

In this study, we employed ANN to address the optimization problem. The methodology involved constructing a surrogate model of the optimization problem by training an artificial neural network, in order to find the optimal solution to the optimization problem. Notably, the ANN method is well-suited to address nonlinear optimization problems, thus enhancing its capability in handling complex systems. For instance,

Then, the mapping model of

Using the mapping capabilities of neural networks, the initial sample can be expanded to

The accuracy of models in artificial neural networks during training and prediction is influenced by the neural network structure, including hidden layers and neurons. However, the optimal neural network structures vary for different problems. To address this, we propose a data-driven approach, referred to as Algorithm 1. This approach efficiently discovers a neural network structure that is better suited to the specific model, reducing the required time. The implementation concept of the algorithm is provided in the appendix in

The DONN method first determines the optimal combination of a set of neural network layers and neurons through Algorithm 1 to establish a neural network structure, and then inputs the training data sets

The sigmoid function serves as the activation function for neuron nodes. Let us assume that the predicted value of the

Thus, as the value of the loss function approaches zero, the neural network’s predicted result and the actual result exhibit a smaller error. The role of a neural network optimizer is to continuously minimize the loss function through training on the data set. Different optimizers possess varying capabilities in reducing the loss function’s value. Furthermore, the choice of optimizer depends on the specific problem at hand, as different problems may require different optimizers. An appropriate optimizer not only enhances model accuracy but also minimizes the required training iterations and mitigates the risk of overfitting the neural network. In practice, it is possible to employ multiple optimizers for combined training within a neural network. Selecting the appropriate number of iterations further enhances training efficiency.

We divide the process of stochastic analysis with DONN into three stages:

(1) Preparation stage: Obtain a batch of training data sets through Latin-Hypercube sampling, leave samples that meet the constraints, obtain the initial data set

A small part of the normalized dataset is selected, and an artificial neural network is established through Algorithm 1.

(2) Model training and validation phase: the dataset is divided into training dataset and testing dataset. The former is utilized to construct the approximate function f that captures the relationship between the input and output. The latter is employed to assess the model’s generalization capability. The training process of the machine learning model involves searching for optimal parameters that minimize the loss function, while the testing process is conducted to validate the accuracy of the fitted function. Furthermore, it is necessary to post-process the predicted value to ensure its alignment with the input

(3) Application stage: The correctness of the mapping relationship ^{N}X_{ξ}^{N}X_{ξ}

For the input variable

In the upcoming section, we will present an overview of the flexoelectric problem theory and provide guidance on obtaining the initial sample.

In this section, we provide a comprehensive introduction to the theory of the flexoelectric problem and outline the process of obtaining initial samples using the IGA-FEM method.

The enthalpy density

Using the symbol

Consider the terms in

For purely piezoelectric dielectrics, we have

The normal electromechanical stresses

After substituting

The electrical enthalpy of a flexoelectric dielectric is given by

The work done by external forces, such as mechanical traction

The kinetic energy of a system is defined as

Upon substituting

By adding the variation operation within the integral operations, we obtain

Using the chain rule of variation and reordering the operations, we get

Now we can rewrite

To fulfill

The inertia element is ignored in the case of a static situation, resulting in the following:

We may derive the weak version of the governing equation for flexoelectricity by putting

In this section, we use B-spline basis functions to discretize the controlling

In

In

In this section, we use three models to analyze the optimization problem, the first two models to analyze the parameter optimization problem, and the last model to analyze the shape optimization problem using the flexoelectric effect as an example.

First example, we examine a truss structure consisting of bars with Young’s modulus E and density

The weight of the truss is given by

In addition, the following constraints were applied to the stress:

We use the Latin hypercube sampling method for initial sampling, and remove the sample points that do not meet the constraints to obtain S1, which is substituted into the neural network as a training set for model training. The number of hidden layers of the neural network and the number of nodes per layer are determined by Algorithm 1.

In the neural network, the size of the data set directly affects the time spent in model training and the training effect. If the data set is too large, more iterations are required in the neural network training stage, and the time consumption increases significantly. At the same time, if the dataset is too small, the neural network will not be able to learn all the features, resulting in poor prediction effect after model training. Therefore, choosing a dataset of suitable size can effectively solve the above two problems.

As can be seen from

After observing the decreasing law of the loss function and the relationship between the relative error and the number of iterations under different data and sizes, we make Adam optimizer perform 20,000 iterations, and GD optimizer perform 20,000 iterations. We restrict the range of

Using 1957 constrained samples as the training set, the solutions for DONN are given in

Design variable | Value | RE (%) | ||
---|---|---|---|---|

Exact | DLM | DONN | ||

0.2474 | 0.2480 | 0.2474 | ||

0.2474 | 0.2480 | 0.2474 | ||

min | 9.4300 | 9.4086 | 9.4250 | 0.05 |

Second example, we shift our focus to the cantilever beam illustrated in

In this case, we make an assumption that the thickness of the segment is significantly smaller in comparison to the side length, denoted as

Then, the weight of the beam can be calculated using the following equation:

The calculation of tip displacement can be expressed as [

The optimization problem of nested formulas can be expressed as follows:

In

Design variable | Value | RE (%) | ||
---|---|---|---|---|

Exact | DLM | DONN | ||

1.3797 | 1.3800 | 1.3821 | ||

2.2442 | 2.2400 | 2.2421 | ||

min | 3.6239 | 3.6200 | 3.6237 | 0.06 |

Similar to the previous example, we set the range for

The shape of flexoelectric materials that is most frequently researched is the truncated pyramid. In

Type | Symbol | Magnitude | Unit |
---|---|---|---|

Upper edge width | 750 | ||

Lower edge width | 2250 | ||

Thickness | 750 | ||

Distributed force | 6 |

0.37 | 100 GPa | −4.4 C/m^{2} |
1 |
11 nC/Vm | 12.48 nC/Vm | 1408 |

Note:

Firstly, we add control points to the truncated pyramid shape in

In terms of sample calculation, we select three control points on each waist, so that their x-direction coordinates fluctuate within a 20 percent range. 5000 sets of samples were calculated using IGA-FEM, and an ANN network was trained 50,000 times to obtain a mapping model of the maximum potential corresponding to the shape from the control point coordinates. The loss values of ANN are given in

It can be seen from the prediction error chart of loss value that the prediction error of neural network reaches a very low degree. Therefore, it can be used for the shape optimization of this problem. The shape change in the optimization process is shown in

With the increase of the number of iterations, the prediction accuracy and optimization effect of the neural network gradually increased, and became stable after the number of iterations reached 32,000.

In this paper, ANN is used to optimize the structural parameters and shapes, and some techniques to improve the fitting effect are proposed. ANN only needs the samples of the corresponding problem for optimization, and then a proxy model could be built to deal with the optimization problem. Since the reliability of the optimization in this method depends on the accuracy of ANN, the idea of pre-training is used to find a suitable network structure, and the method of combining multiple optimizers is used to improve the prediction accuracy of ANN. To ensure the accuracy of samples, IGA-FEM is used to obtain high-precision samples, and the mapping model from shape control points to maximum potentials is established to solve the shape optimization problem. In the future, we will introduce deeper neural networks and meta-heuristic optimization algorithms, and study the extension of this method to shape optimization problems of more complex models.

The authors wish to express sincere appreciation to the reviewers for their valuable comments, which significantly improved this paper.

The research in this article has been supported by a Major Research Project in Higher Education Institutions in Henan Province, with Project Number 23A560015.

The authors confirm contribution to the paper as follows: study conception and design: Yu Cheng, Yajun Huang, Xiaohui Yuan; data collection: Yu Cheng, Shuai Li, Zhongbin Zhou; analysis and interpretation of results: Yu Cheng, Yajun Huang, Yanming Xu; draft manuscript preparation: Yu Cheng, Xiaohui Yuan, Yanming Xu. All authors reviewed the results and approved the final version of the manuscript.

Data is available on request.

The authors declare that they have no conflicts of interest to report regarding the present study.