Element-boundary Integral Method Research Articles

Single-Layer Metasurface-Based Reflectarray Antenna with H-Shaped Slotted Patch for X-Band Communication.

In this study, a metasurface-based reflectarray is designed for X-band applications. The unit cells are equipped with an H-shaped slotted patch for additional resonance and phase range. Linear phase variation by altering the length of the patch is realized with a range exceeding 480∘. The reflectarray is designed and fabricated on a thin and high-quality Rogers 5880 substrate. The Finite Element Boundary Integral (FEBI) method is used to simulate a 23×23 element reflectarray and then fabricated to achieve the measured results using an anechoic chamber. The peak gain of the proposed reflectarray is 25.5 dBi recorded with an aperture efficiency of 63.7% at a center frequency of 10 GHz. The cross-polarization and side-lobe levels in the entire band are less than -33 dB and -21 dB, respectively. Moreover, the proposed reflectarray antenna achieves a 20% 1-dB gain bandwidth.

Numerical Computation of 2D Domain Integrals in Boundary Element Method by (α, β) Distance Transformation for Transient Heat Conduction Problems

When the time-dependent boundary element method, also termed the pseudo-initial condition method, is employed for solving transient heat conduction problems, the numerical evaluation of domain integrals is necessitated. Consequently, the accurate calculation of the domain integrals is of crucial importance for analyzing transient heat conduction. However, as the time step decreases progressively and approaches zero, the integrand of the domain integrals is close to singular, resulting in large errors when employing standard Gaussian quadrature directly. To solve the problem and further improve the calculation accuracy of the domain integrals, an (α, β) distance transformation is presented. Distance transformation is a simple and efficient method for eliminating near-singularity, typically applied to nearly singular integrals. Firstly, the (α, β) coordinate transformation is introduced. Then, a new distance transformation for the domain integrals is constructed by replacing the shortest distance with the time step. With the new method, the integrand of the domain integrals is substantially smoothed, and the singularity arising from small time steps in the domain integrals is effectively eliminated. Thus, more accurate results can be obtained by the (α, β) distance transformation. Different sizes of time steps, positions of source point, and shapes of integration elements are considered in numerical examples. Comparative studies of the numerical results for the domain integrals using various methods demonstrate that higher accuracy and efficiency are achieved by the proposed method.

Precise integration boundary element method for solving nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity

The precise integration boundary element method (PIBEM) is used to solve nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity in this article. At first, a boundary-domain integral equation can be obtained by using the fundamental solution for steady linear heat conduction problems. Secondly, the domain integrals are converted into boundary integrals by using the radial integration method to get a pure boundary integral equation, which can retain the advantage of dimension reduction of the boundary element method. Then, after discretization, a first-order nonlinear differential equation system in time is obtained, and the precise integration method with predictor-corrector technique is used to solve the system of nonlinear equations. In the end, several numerical examples are presented to verify the accuracy and stability of the presented method. The results obtained by the presented method are less affected by the time step size and satisfactory results can be obtained even for a large time step size.

POD method for solving elastodynamics problems of functionally gradient materials based on radial integration boundary element method

Functionally graded materials (FGMs) are widely utilized due to their excellent mechanical properties, prompting a need to explore the dynamic response of FGMs. To accurately and efficiently analyze elastodynamics problems, the radial integration boundary element method (RIBEM) is coupled with the proper orthogonal decomposition (POD) method in this paper. The POD method can transform a high-dimensional model into a low-dimensional system with high precision. Using the basic solution (Kelvin solution), the governing partial differential equation for the linear elastic dynamics problem of FGMs is transformed into a boundary-domain integral equation. The radial integration method is employed to convert the domain integral arising from material inhomogeneity and inertia terms into an equivalent boundary integral. This establishes a solution for the elastic dynamics problem of FGMs without the need for internal grid RIBEM. The Newmark time integration scheme is applied to solve the second-order ordinary differential equations. The displacement field obtained by RIBEM is utilized to construct the snapshots matrix, from which the POD reduced-order model is established. Numerical examples validate that the POD method significantly reduces the model's order, enhancing computational efficiency while also demonstrating the high accuracy of the POD method.

Noise field properties of marine propellers in open water

Aim of the paper is to gain a physical understanding of the main noise generating mechanisms related to vorticity and turbulence downstream the disk of a marine propeller in open water. Through the application of the Ffowcs Williams and Hawkings acoustic analogy (FWH) for permeable surfaces, the analysis of the noise field generated by the INSEAN E779A propeller model at different advancing ratios allows to detect correlations between acoustic signatures in the near/mid field and wake evolution. Turbulence noise-induced effects are captured by combining the FWH formulation with a Detached Eddy Simulation (DES) of the flow past the propeller. Besides, a phase-locked post-processing of the unsteady DES data on the acoustic surface is used to yield FWH signatures governed by vorticity effects, which are then compared with the pressure fluctuations predicted by the Bernoulli equation, within a potential hydrodynamics platform solved by a Boundary Integral Element Method (BIEM). It is found that turbulence is the most annoying source of propeller noise, whereas vorticity-driven phenomena are important only in a narrow region surrounding the propeller. Moreover, numerical results highlight that acoustic pressures are correlated to the vortex pairing hydrodynamic phenomenon associated to a blade-to-shaft harmonic energy transfer, detected in literature by time-resolved visualizations and velocimetry measurements.

A Fast Reduced-Order Model for Radial Integration Boundary Element Method Based on Proper Orthogonal Decomposition in the Non-Uniform Coupled Thermoelastic Problems

To efficiently address the challenge of thermoelastic coupling in functionally graded materials, we propose an approach that combines the radial integral boundary element method (RIBEM) with proper orthogonal decomposition (POD). This integration establishes a swift reduced-order model to transform the high-dimensional system of equations into a more manageable, low-dimensional counterpart. The implementation of this reduced-order model offers the potential for rapid numerical simulations of functionally graded materials (FGMs) under thermal shock loading. Initially, the RIBEM is utilized to resolve the thermal coupling issue within the FGMs. From these solutions, a snapshot matrix is constructed, capturing the solved temperature and displacement fields. Subsequently, the POD modes are established and a POD reduced-order model is constructed for the boundary element format of the thermally coupled problem. Finally, a system of low-order discrete differential equations is solved. Numerical experiments demonstrate that the results obtained from the reduced-order model closely align with those of the full-order model, even when considering variations in structural parameters or impact loads. Thus, the introduction of the reduced-order model not only guarantees solution accuracy but also significantly enhances computational efficiency.

Tidal Turbine Benchmarking Project: Stage I - Steady Flow Blind Predictions

This paper presents the first blind prediction stage of the Tidal Turbine Benchmarking Project being conducted and funded by the UK's EPSRC and Supergen ORE Hub. In this first stage, only steady flow conditions, at low and elevated turbulence (3.1%) levels, were considered. Prior to the blind prediction stage, a large laboratory scale experiment was conducted in which a highly instrumented 1.6m diameter tidal rotor was towed through a large towing tank in well-defined flow conditions with and without an upstream turbulence grid. Details of the test campaign and rotor design were released as part of this community blind prediction exercise. Participants were invited to use a range of engineering modelling approaches to simulate the performance and loads of the turbine. 26 submissions were received from 12 groups from across academia and industry using solution techniques ranging from blade resolved computational fluid dynamics through actuator line, boundary integral element methods, vortex methods to engineering Blade Element Momentum methods. The comparisons between experiments and blind predictions were extremely positive helping to provide validation and uncertainty estimates for the models, but also validating the experimental tests themselves. The exercise demonstrated that the experimental turbine data provides a robust data set against which researchers and design engineers can test their models and implementations to ensure robustness in their processes, helping to reduce uncertainty and provide increased confidence in engineering processes. Furthermore, the data set provides the basis by which modellers can evaluate and refine approaches.

The fast multipole method–accelerated line integration boundary element method for 3D heat conduction analysis with heat source

Purpose3D steady heat conduction analysis considering heat source is conducted on the fundamental of the fast multipole method (FMM)-accelerated line integration boundary element method (LIBEM).Design/methodology/approachDue to considering the heat source, domain integral is generated in the traditional heat conduction boundary integral equation (BIE), which will counteract the well-known merit of the BEM, namely, boundary-only discretization. To avoid volume discretization, the enhanced BEM, the LIBEM with dimension reduction property is introduced to transfer the domain integral into line integrals. Besides, owing to the unsatisfactory performance of the LIBEM when it comes to large-scale structures requiring massive computation, the FMM-accelerated LIBEM (FM-LIBEM) is proposed to improve the computation efficiency further.FindingsAssuming N and M are the numbers of nodes and integral lines, respectively, the FM-LIBEM can reduce the time complexity from O(NM) to about O(N+ M), and a full discussion and verification of the advantage are done based on numerical examples under heat conduction.Originality/value(1) The LIBEM is applied to 3D heat conduction analysis with heat source. (2) The domain integrals can be transformed into boundary integrals with straight line integrals by the LIM. (3) A FM-LIBEM is proposed and can reduce the time complexity from O(NM) to O(N+ M). (4) The FM-LIBEM with high computational efficiency is exerted to solve 3D heat conduction analysis with heat source in massive computation successfully.

A reduced-order modeling for thermo-mechanical coupling analyses by using radial integration boundary element method

In order to efficiently solve thermo-mechanical coupling problems, this work combines the radial integration boundary element method (RIBEM) and the proper orthogonal decomposition (POD) to construct a fast reduced-order model. This model transforms high-dimensional systems into low-dimensional ones, enabling faster computation of thermo-mechanical coupling problems under thermal loads. Firstly, RIBEM is employed to solve coupled thermoelastic dynamic problems under thermal shock loads. Domain integrals are converted to boundary integrals using the radial integration method, establishing a purely boundary element algorithm without internal cells. Subsequently, finite difference techniques are applied to calculate the discretized algebraic equations, and the obtained field variables are used as a snapshot matrix to establish POD modes and construct the POD reduced-order model. Numerical examples demonstrate that the results of the reduced-order model are essentially consistent with the full-order model across various physical parameters and boundary conditions. Therefore, this reduced-order model significantly improves computational efficiency while maintaining high accuracy, especially when solving large-scale problems.

Buckling analysis of thin plates using direct interpolation boundary element method

The present study employs the direct integration boundary element method (DIBEM) to analyze the buckling behavior of thin plates. Unlike conventional approaches, this method eliminates the need for domain discretization or specialized solutions. Furthermore, it constitutes a continuation from the previous research in Doval et al. (2013), which employed the radial integration method for the same investigation.To showcase the efficacy of the proposed approach, it is utilized in the stability analysis of both perforated and non-perforated plates. The findings are then compared against analytical responses, in the case of non-perforated plates, and finite element method results, for perforated plates.Moreover, the use of boundary elements reduces the size of the eigenvalue problem involved in stability analysis, as compared to finite elements, provided that only the domain and a few internal points are needed for discretization.Overall, this paper presents a novel and effective approach to the analysis of buckling plates.

Prediction of Added Resistance of Container Ships in Regular Head Waves Using an Artificial Neural Network

In this paper, an artificial neural network was used to predict the added resistance coefficient for container ships in regular head waves for various speeds. The data used for training the neural network were gathered based on performed numerical calculations using the Boundary Integral Element Method for various hull forms of container ships. The numerically obtained results were validated against the available experimental data for three benchmark container ships. The data were divided into three classes based on the ship length, and the expressions for the prediction of the added resistance coefficient for each container ship class were provided. The performance and generalization properties of the neural network were evaluated based on the normalized value of the root mean square error. The model enables reliable prediction of the added resistance coefficient within the preliminary design stage of a ship based on the ship characteristics and speed.

A Kernel-Free Boundary Integral Method for 2-D Magnetostatics Analysis

Performing magnetostatic analysis accurately and efficiently is very important for multi-objective optimization of electromagnetic device designs. In this research, a kernel-free boundary integral method (KFBIM) has been introduced for solving magnetic computations in a toroidal core geometry in 2-D. This study is very relevant in the design and optimization of toroidal inductors or transformers used in electrical systems, where lighter weight, higher inductance, higher efficiency, and lower leakage flux are required. The governing partial differential equations (PDEs) have been formulated as a system of the Fredholm integral equations of the second kind. Unlike traditional boundary integral methods or boundary element methods, KFBIM does not require an analytical form of Green’s function for evaluating integrals via numerical quadrature. Instead, KFBIM computes integrals by solving an equivalent interface problem on a Cartesian mesh. Compared with traditional finite difference methods for solving the governing PDEs directly, KFBIM produces a well-conditioned linear system. Therefore, the KFBIM requires only a fixed number of iterations when an iterative method [e.g., generalized minimal residual method (GMRES)] is applied for solving the linear system, and the numerical solution is not sensitive to computer round-off errors. The obtained results are then compared with a commercial finite element solver (ANSYS), which shows excellent agreement. It should be noted that, compared with FEM, the KFBIM does not require a body-fit mesh and can achieve high accuracy with a coarse mesh. In particular, the calculations of the magnetic potential and the tangential field intensity on the boundaries are more stable and exhibit almost no oscillations.

The solution of elastodynamic problems using modified radial integration boundary element method (MRIBEM)

In this paper, the modified radial integration boundary elements method (MRIBEM) is applied to analyze two-dimensional (2D) elastodynamic problems. The boundary and domain variables are approximated by Lagrangian shape functions and global radial basis functions (RBFs), respectively. As a key idea of this study, the domain integrals that result from implying the inertia force effect are computed by means of a modified radial integration method (MRIM). With the MRIBEM, we are able to deal with problems with concave shapes when the domain variables are approximated by global RBFs by defining an auxiliary point or, in case of complex domain shapes, two or more auxiliary points as the origin for radial integration. Two-time marching schemes, i.e. Newmark method and the Houbolt method, are implemented for the solution of governing differential equations along time. The performance of the proposed method is examined in the solution of four numerical examples.

Surface Integral Equation Solutions for Electromagnetic Scattering From a Special Class of Anisotropic Media

We present a surface integral equation (SIE) solution for a special class of anisotropic media, whose permittivity and permeability are both symmetric full tensors with the same or multiple values. This kind of media is generated from the invisible cloaking or absorbing material of a perfectly matched layer (PML) design. A new derivation based on the potential functions is presented to obtain the closed-form Green’s function. Detailed and rigorous proofs of some vector identities are given for the first time. Special treatments are detailed to address the singularity of the anisotropic Green’s function for discretizing the SIE. Numerical experiments validate the accuracy, stability, and high efficiency of the proposed solution compared with the analytical solution or the finite element-boundary integral (FE-BI) method.

Numerical solution of water wave propagation problems over variable bathymetries using the modified radial integration boundary element method

In this paper, a modified radial integration boundary element method (MRIBEM) is applied to the solution of water wave propagation problem. The mild slope equation which has proved to be a very efficient equation for a large number of water wave problems is solved using the MRIBEM over arbitrary seabed geometries. The infinite region is subdivided into a central finite area with irregular seabed and a semi-finite one with parallel depth contour lines. The interior part is dealt with using MRIBEM and the exterior one is solved using the boundary element method. This method benefits from the meshfree characteristic of MRIBEM for the inner domain and the fulfillment of Sommerfeld's radiation condition for the exterior domain at the same time. Numerical examples are compared with experimental results and other researches which are indicative of the validity and efficiency of the present method.

Shape reconstruction in transient heat conduction problems based on radial integration boundary element method

In order to accurately identify the geometric boundary, the radial integration boundary element method (RIBEM) combined with the modified Levenberg-Marquardt (LM) algorithm is proposed for shape reconstruction in transient heat conduction problems. Compared with the finite element method (FEM), the boundary element method (BEM) only discretizes the boundary rather than the whole domain, so it has incomparable advantages in the shape reconstruction. Especially, the radial integration method still maintains the superiority of boundary discretization in transient heat conduction problems. For the iterative scheme of LM algorithm, the shape reconstruction greatly depends on whether the sensitivity matrix can be obtained accurately. Therefore, complex variable derivation method (CVDM) is firstly introduced to carry out shape reconstruction in the transient heat conduction problems, which accurately solves the sensitivity coefficients of node temperature to the reconstructed boundary, and the RIBEM is transformed from the real domain to the complex domain to solve the transient heat conduction problems. The modified LM algorithm effectively overcomes the shortcoming of the excessive dependence on the step size of finite difference scheme. It should be noted that compared with the finite difference scheme, the proposed method can greatly reduce the computational cost of the direct problem in the multi-variate optimization process, thereby greatly improving the efficiency of identifying the boundary shape. Finally, different reconstruction strategies are applied in numerical models with three types of variable boundary, and reconstruction results demonstrate the great accuracy and efficiency of the proposed method in 2D and 3D shape reconstruction, and it also shows good robustness in the presence of measurement errors.

Nested kernel degeneration-based boundary element method solver for rapid computation of eddy current signals

In this article, the nested kernel degeneration (NKD) based integral equation fast solver is proposed to compress the dense system matrix generated by boundary element method (BEM) for calculating eddy current signals in 3D nondestructive evaluation (NDE) problems. The near and diagonal block interactions are computed and stored by full matrices. However, for the far block interactions, not like in the single level kernel degeneration-based method that all the interaction matrices are compressed by the interpolations, the nested approximation framework makes the expression of the higher-level matrices by the ones at leaf levels and the transfer matrices between neighboring levels. Different integral kernels are degenerated and nested to ensure the performance of NKD based method. The robustness and efficiency of the proposed solver are validated and verified by making numerical predictions and comparing them with the ones achieved by analytical method, numerical methods and experiments. Our numerical experiments show that the NKD solver is more efficient than the butterfly solver (multilevel adaptive cross approximation algorithm (MLACA)) for electrically small eddy current NDE problem. With the NKD based method, O(N) computational complexity is achieved for multiscale low frequency eddy current simulations without sacrificing accuracy, where N is the number of unknowns. So far, to our best knowledge, the proposed method with linear complexity is the most efficient integral equation-based BEM forward solver to predict the eddy current signals from cracks and flaws.

Ray-Optical Evaluation of Scattering From Electrically Large Metasurfaces Characterized by Locally Periodic Surface Susceptibilities

This work continues the development of the ray-tracing method of de Jong (2021) for computing the scattered fields from metasurfaces characterized by locally periodic reflection and transmission coefficients. In this work, instead of describing the metasurface in terms of scattering coefficients that depend on the incidence direction, its scattering behavior is characterized by the surface susceptibility tensors that appear in the generalized sheet transition conditions (GSTCs). As the latter quantities are constitutive parameters, they do not depend on the incident field and, thus, enable a more compact and physically motivated description of the surface. The locally periodic susceptibility profile is expanded into a Fourier series subject to a spatially varying phase parameter, and the GSTCs are rewritten in a form that enables them to be numerically solved for the reflected and transmitted surface fields. The phase parameter can either be determined from a prescribed surface transformation or extracted from known surface susceptibilities. A method for the extraction of this phase parameter from a susceptibility profile using a spatial variant of the short-time Fourier transform (STFT) is proposed. The scattered field at arbitrary detector locations is constructed by evaluating critical-point contributions of the first and second kinds using a forward ray-tracing (FRT) scheme. The accuracy of the resulting framework has been verified with an integral equation-based boundary element method (BEM)-GSTC full-wave solver for a variety of examples, such as a periodically modulated metasurface, a metasurface diffuser, and a beam collimator.

Shape Reconstruction in Transient Heat Conduction Problems Based on Radial Integration Boundary Element Method

Shape Reconstruction in Transient Heat Conduction Problems Based on Radial Integration Boundary Element Method

Fully Dynamic Earthquake Cycle Simulations on a Nonplanar Fault Using the Spectral Boundary Integral Element Method (sBIEM)

ABSTRACT One of the most suitable methods for modeling fully dynamic earthquake cycle simulations is the spectral boundary integral element method (sBIEM), which takes advantage of the fast Fourier transform (FFT) to make a complex numerical dynamic rupture tractable. However, this method has the serious drawback of requiring a flat fault geometry due to the FFT approach. Here, we present an analytical formulation that extends the sBIEM to a mildly nonplanar fault. We start from a regularized boundary element method and apply a small-slope approximation of the fault geometry. Making this assumption, it is possible to show that the main effect of nonplanar fault geometry is to change the normal traction along the fault, which is controlled by the local curvature along the fault. We then convert this space–time boundary integral equation of the normal traction into a spectral-time formulation and incorporate this change in normal traction into the existing sBIEM methodology. This approach allows us to model fully dynamic seismic cycle simulations on nonplanar faults in a particularly efficient way. We then test this method against a regular BIEM for both rough-fault and seamount-fault geometries and demonstrate that this sBIEM maintains the scaling between the fault geometry and slip distribution.