In this paper, an adaptive polynomial chaos expansion method (PCE) based on the method of moments (MoM) is proposed to construct surrogate models for electromagnetic scattering and further sensitivity analysis. The MoM is applied to accurately solve the electric field integral equation (EFIE) of electromagnetic scattering from homogeneous dielectric targets. Within the bistatic radar cross section (RCS) as the research object, the adaptive PCE algorithm is devoted to selecting the appropriate order to construct the multivariate surrogate model. The corresponding sensitivity results are given by the further derivative operation, which is compared with those of the finite difference method (FDM). Several examples are provided to demonstrate the effectiveness of the proposed algorithm for sensitivity analysis of electromagnetic scattering from homogeneous dielectric targets.

Electromagnetic scattering simulation of dielectric conductors has become an essential tool in remote sensing, radar detection, and target stealth design. Currently, the primary numerical methods for electromagnetic scattering include the finite-difference time-domain (FDTD) method, finite element method (FEM), and the method of moments (MoM). The FDTD analysis method [

The MoM distinguishes itself from FDTD and FEM in that it specifically focuses on discretizing the surface or volume of the antenna or scatterer, as the electromagnetic source, whether it be surface or volume current, is the primary quantity of interest. As a result, MoM has gained wide usage in solving radiation and scattering problems, spanning across various fields such as dynamics [

It is worth noting that although the MoM can achieve the desired accuracy, it entails solving a large matrix equation consisting of full elements [

In recent years, extensive research has been conducted by both domestic and international scholars to address the challenges of using surrogate models to reduce computational costs, improve optimization efficiency, and improve approximation accuracy. Commonly employed surrogate models encompass Polynomial Regression, Artificial Neural Networks (ANN) [

The PCE was initially introduced by Wiener [

It is worth noting that in the process of determining coefficients, selecting an improper order can lead to overfitting or underfitting phenomena, which significantly impact the accuracy of the PCE method. Consequently, this paper aims to bridge the gap between the surrogate model and sensitivity analysis and explicitly express the sensitivity of the EM scattering problem by employing an adaptive PCE algorithm to construct a surrogate model of EM scattering with appropriate orders. In summary, the key novelties of this paper are as follows:

The adaptive PCE method based on the method of moments is proposed for electromagnetic scattering analysis of homogeneous dielectric (DIE) targets.

This work presents the first sensitivity analysis method for electromagnetics with multivariate surrogate model.

The remainder of this paper is structured as follows.

In this section, the MoM is applied to discretize the dielectric EFIE and obtain the surface current. The electric field radiation equations are subsequently coming into service for the scattering field, and the two-dimensional bistatic radar cross section (RCS) is eventually presented.

Considering a homogeneous dielectric body in the infinite domain, the simplified 2D schematic of electromagnetic scattering analysis is depicted in

This bounded-connected DIE region

The above model can be simplified for computational convenience. Assuming the source point is the origin and the direction towards the field point is the positive x-axis, as shown in

It is obvious that the above integral equation is a variable coefficient, which is not easy to solve directly. To ensure the accuracy of the solution and the stability of the calculation process, the discretization method is used to transform it into a system of equations.

The MoM is used to convert

The matrices

There are two unknowns in each EFIE in the two regions, which cannot be solved independently. Additionally, the tangential components of the electric and magnetic fields across the dielectric interface are by an amount equal to the magnetic and electric surface currents on that interface, respectively. The surface currents on dielectric interfaces are fictitious in

Combining the equations above of the two regions

The coefficient

Here is the 2-D near-field radiation equations. The magnetic field radiated by an electric current is expressed as

It can be rewritten by expanding into Cartesian components as

where

Sometimes people are interested in RCS, for which electromagnetics plays a critical important role. It is generally regarded as the equivalent scattering area of the object under a specific incident wave. It can be expressed as

Generally speaking, the computational procedure for RCS is intricate, particularly when it involves computing a substantial full-element matrix. This complexity is further compounded by the extensive data computations required for uncertainty analysis, sensitivity analysis, and optimization. In light of these challenges, we propose the development of a surrogate model that can swiftly generate large-scale sample data.

PCE is a technique utilized to characterize the uncertainty inherent in random variables by formulating random spaces underpinned by polynomial bases. This method transforms the original nonlinear issue into a weighted sum of polynomials, essentially embodying a process of fitting coefficients. For any discrete data

In practice, the above expansion should be truncated to contain a subset of polynomials, whose total degree does not exceed order

It is obvious that two primary factors must be taken into account in constructing the surrogate model with

The dimensions of the independent variables differ greatly in the electromagnetic scattering system. For example, the bistatic angle

The choice of one-dimensional orthogonal polynomials is often influenced by the distribution of the independent variables. For instance, Legendre polynomials are typically employed when the inputs exhibit a uniform distribution, whereas Hermite polynomials are favored for inputs that follow a Gaussian distribution. The standard recurrence relation for the Legendre polynomial system denoted as

The formula for the general term of the Hermite polynomial system

Assuming

It can be easily obtained that the one-dimensional orthogonal polynomials

When

The truncated polynomial chaos expansion, as depicted in

Thus, the PCE coefficient

The response from the surrogate model can be obtained with the solved coefficient

Based on the upper limit of CV that represents the accuracy of the surrogate model, the order can be iterated until CV

The explicit expression of the surrogate model for the electromagnetic scattering problem can be derived using the MoM and PCE methods according to the above description. The model’s sensitivity concerning a specific variable can be obtained through differentiation, the calculation equations and precision of this method are described in detail in

This section begins by using an example of a dielectric circular cylinder to validate the accuracy of the solutions obtained through solving the EFIE based on the MoM. Subsequently, the MoM solutions are utilized as input samples for the PCE method to construct univariate and bivariate surrogate models for electromagnetic scattering problems. The sensitivity results of each surrogate model are compared with those obtained from the finite difference method (FDM), followed by the verification of the PCE method for sensitivity analysis.

Both the dielectric cylinder and 5-bladed gear in this section are modeled by Non-Uniform Rational B-Splines (NURBS) in CAD. Both the basis function and the testing function in MoM are triangle functions, and the number of them are both 100 for interpolation calculation in CAE. The framework of the MoM is written in Fortran 90 language, and the adaptive PCE method is written using Matlab, which is run on a PC with an Intel (R) Core(TM) i7-7700 CPU and 128 GB RAM.

In this section, a dielectric cylinder model constructed using the control points (see

The following polar plots

The dielectric 5-bladed gear model (see

The 5-bladed gear has 5-fold symmetry.

The bistatic RCS with a fixed observation point of the 5-bladed gear is not as symmetric as the cylinder model. The observation point is fixed at

In addition, it can be seen that the bistatic RCS is related to the permittivity

In this section, the sensitivity analysis is performed with PCE and the global FDM. The objective function

The sensitivity sequence of bistatic RCS concerning the variable

The sensitivity analysis of the homogeneous dielectric 5-bladed gear under TM-polarized incident plane wave is carried out here. The frequency

The sensitivity is obtained by taking the derivative of the explicit expression of the surrogate model, which is compared with the results of the FDM method. It can be seen from

In addition, the sensitivity D is the largest while

As far as this surrogate model is concerned, the time needed to obtain a sample is about 0.455 s with MoM. The time required by combining the PCE method is mainly divided into three parts:

Calculate 38 samples by MoM. This is a preparation for the next step and takes about 17.3 s.

Construct surrogate model by PCE. The time required for the PCE method is little related to the number of sample points but mainly depends on the order of the surrogate model. In this example, a 15th-order surrogate model takes approximately 1.357 s.

Solve the response. This process is based on the constructed surrogate model and requires about 2.04 s for 10,000 independent variables.

The “FDM” in

It is clear that higher-order surrogate models provide better fitting results, but this also leads to an increase in computational cost.

In addition, the normalization is first performed in PCE box to reduce the influence of the range of the independent variables on the fitting effect and improve the performance of the polynomial chaos expansion algorithm. In this example, CV = 8.07

Another three sensitivity analysis of dielectric 5-bladed gear under TM-polarized incident plane wave is carried out here (see

Range | Order | ||||||
---|---|---|---|---|---|---|---|

Case 1 | 5 | 100 MHz | 180 | 37 | 143 | 4 | |

Case 2 | 0 | [1, 30] | 100 MHz | 290 | 73 | 217 | 18 |

Case 3-1 | 0 | 5 | [50, 76] MHz | 261 | 54 | 207 | 8 |

Case 3-2 | 0 | 5 | [76, 90] MHz | 141 | 37 | 104 | 19 |

Case 3-3 | 0 | 5 | [90, 150] MHz | 601 | 122 | 479 | 5 |

It can be seen from

For Case 1 and Case 2, the overall change is relatively gentle and the curvature is small, a satisfied surrogate model can be obtained with a low order, which is 4 for Case 1 and 18 for Case 2. The initial value of the training set, the sensitivity value solved by FDM with the training set, the response at the test set with the surrogate model which is constructed using the training set, and the derivatives of the surrogate model at the test set are shown in

The middle segment of Case 3 fluctuates more violently than the other two segments, it is hard to fit the whole range well using a lower order of surrogate model. It is divided into three parts and the PCE method is used to expand the samples. The result is shown in

The statistical characteristics of the sensitivity results by the FDM and 20th-order PCE methods are shown in

Mean (FDM) | Mean (PCE) | Relative error | Var (FDM) | Var (PCE) | Relative error | |
---|---|---|---|---|---|---|

Case 1 | −0.0001 | −0.0001 | 0.2369 |
0.0019 | 0.0019 | 0.0194 |

Case 2 | −0.1064 | −0.1070 | 0.5458 |
2.7498 | 2.8047 | 1.9967 |

Case 3 | −0.1317 | −0.1348 | 2.3153 |
5.1465 | 5.2940 | 2.8656 |

Case 3-1 | −0.6790 | −0.6788 | 0.0345 |
1.7670 | 1.7636 | 0.1921 |

Case 3-2 | 0.2791 | 0.2596 | 6.9907 |
32.5062 | 33.0439 | 1.6541 |

Case 3-3 | 0.0096 | 0.0096 | 0.0214 |
0.0866 | 0.0866 | 0.0005 |

Based on the above conclusions, taking frequency

Using the method of moments, the time needed to calculate 10000 test samples is 48084.78 s, and the time to calculate 1681 groups of training samples is 8091.05 s, while the time used to build the two-dimensional 15th-order surrogate model by PCE is 215 s, which greatly reduces the calculation time of data expansion.

This paper presents a sensitivity analysis method for homogeneous dielectric targets. The triangle functions are utilized to discretize and test the EFIE, and the fictitious current on the dielectric surface is obtained by solving the coefficients. The RCS samples obtained from MoM considering several different variables are subsequently divided into the training set and the testing set and inputted into the adaptive PCE box. CV

Numerical examples show that the Bi-RCS is sensitive to frequency and permittivity. The correctness and efficiency of the proposed sensitivity analysis method for electromagnetic scattering are verified by comparing with the results obtained by the MoM, analytical solutions, and FDM. The accuracy of the surrogate model is affected by the number of training samples, the order of the model, and the complexity of the model. The calculation time is mainly related to the order, so the calculation cost and accuracy can be balanced by improving the sample quality or fitting the complex interval piecewise. This method can efficiently acquire large-scale samples in the bounded range. We believe that this framework is user-friendly and conducive to future development.

In future research, the proposed technique will be optimized and extended to address three-dimensional electromagnetic problems, further enhancing its versatility and applicability across various engineering fields.

The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

This work was supported by the Young Scientists Fund of the National Natural Science Foundation of China (No. 62102444) and a Major Research Project in Higher Education Institutions in Henan Province (No. 23A560015).

The authors confirm contribution to the paper as follows: study conception and design: Yujing Ma, Xiaohui Yuan; data collection: Zhongwang Wang, Ruijin Huo; analysis and interpretation of results: Zhongwang Wang, Jieyuan Zhang, Yujing Ma; draft manuscript preparation: Yujing Ma, Xiaohui Yuan, Jieyuan Zhang. 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.

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_{3}nanofluid flow through expanding/contracting rectangular channel

The dielectric infinite cylinder, impinged by electromagnetic plane waves with different polarization directions, is considered and its analytical solutions are obtained in this section.

The scattered electric field under TE-polarized incident waves can be expressed as

The expression of the scattered magnetic field under TM polarization is obtained by

The induced azimuthal electric current is

The recurrence relationship