Boundary Element Method Approach Research Articles

Dipole‐based BEM formulation for three‐dimensional cohesive crack propagation modelling

AbstractThis study presents an alternative boundary element method (BEM) formulation for the cohesive crack propagation modelling in three‐dimensional structures. The proposed formulation utilises an initial stress field for representing the mechanical behaviour along the fracture process zone, which leads to a set of self‐equilibrated forces named as dipole. Cohesive laws govern the material nonlinear behaviour along the fracture process zone. The proposed dipole‐based formulation demonstrates some advantages in comparison to classical BEM approaches in this field. Among them, it is worth citing the requirement of solely three integral equations per collocation point positioned at the fracture process zone. The effectiveness of the dipole‐based formulation has been demonstrated by four applications. The results have been compared to numerical, experimental and analytical solutions available in the literature, in which excellent performance has been observed.

A boundary element analysis of quasi-potential inviscid incompressible flow in multiply connected airfoil wing

AbstractThe quasi-potential flow in a doubly linked airfoil wing with a sharp trailing edge can be solved using a high-order boundary-element method (BEM). A bicubic Hermite spline interpolation is used to express the unknown, with node derivatives given by finite difference method. The revolutionary traits apply to bleeding-edge settings; specifically, challenges relating to the three-dimensional problem of multiply connected domains are investigated. Numerical validation examples were utilized to assess the implemented BEM approach.

Propeller-Wing Interaction: A Simplified Method for Coupling BEM and CFD

This paper presents a numerical procedure for studying the interaction between the wake generated by a propeller and the wing, considering a tiltrotor model framed within the T-TECH Italian project, which has the scope to develop innovative technologies for a flight demonstrator. A tiltrotor is an aircraft that generates lift and propulsion by way of powered rotors mounted on rotating shafts or nacelles at the ends of a fixed wing; in which, in contrast with conventional aircraft, the dimension of the propeller is comparable with the vehicle one. The procedure proposes to couple a 3D low-order unsteady Boundary Element Method (BEM) approach, to evaluate the performance of the isolated propeller, and a Reynolds-Averaged Navier-Stokes (RANS) solution for the analysis of the flow field interaction between the propeller (simulated as actuator disk) and the wing.

Hydrodynamic forces in higher modes of a thin cantilever beam resonator

The hydrodynamic force in a thin uniform beam cantilever is essential quantity for designing a resonator. We obtained the hydrodynamic drag forces in a uniform cantilever vibrating at the first four transverse modes using a semi-analytical boundary element method (BEM) and finite element method (FEM) in ANSYS. The present three-dimensional (3D) BEM inertial and damping forces are compared with FEM over a frequency range with less than 9.5% deviation until frequency parameter of 100; thereafter, it deviates numerically due to neglecting the non-linear convective forces. Considering the hydrodynamic damping force, the quality factor is estimated for the first four transverse modes of a uniform cantilever beam. As a result, the present 3D BEM model behaves well at the first bending mode compared to FEM. We also study the effect of the aspect ratio (ratio of width to length) of the beam for all four modes. However, the first mode quality factor sufficiently correlates with the numerical value for all aspect ratios. For uniform beams operating with higher modes, the influence of nonlinear and resonance frequency by the numerical model is more pronounced at smaller lengths. Hence, the 3D BEM approach can estimate the hydrodynamic forces on cantilever-based resonators, for example, atomic force microscopy and biosensors.

The probabilistic mechanical-material modeling of complex non-homogeneous structures using the coupling of isogeometric boundary element method and reliability approach

Describing mechanical fields with precision in engineering analyses can be a challenging task. Analytical solutions fall short for real complex problems. The widespread use of numerical methods to overcome this obstacle has increased along with the advances in computational capacity. For instance, in geotechnical and composite materials problems, the use of a numerical approach becomes indispensable in several situations. For large volume-to-boundary ratio problems or for non-uniform fields evaluation, the boundary element method is a suitable tool. The method is based on the boundary discretization and on solutions to boundary integral equations. Thus, this article introduces the isogeometric boundary element method formulation and explores techniques for enhancing its physical problem descriptions, specially focusing in geotechnical issues and description of body forces. Two numerical examples are presented. Firstly, a multi-material dam, inspired by the real Itaipu dam section, is analyzed and the results are compared to the well-known ANSYS® FEM software. Furthermore, the article presents a tunnel case study in which a reliability assessment is conducted to account for the inherent uncertainties in the problem. This assessment is accomplished using the improved Hasofer, Lind, Rackwitz e Fiessler-first-order reliability method algorithm, which provides insight into the structural failure risk.

The fracture failure modelling of three-dimensional structures composed of quasi-brittle materials subjected to different loading velocities rates by the dipole-based BEM approach

This study presents a numerical formulation for representing the fracture-failure of three dimensional structures composed of quasi-brittle materials. In addition to the nonlinear nature of quasi-brittle fracture, the formulation accounts the mechanical effects of different loading velocities rates. Therefore, these developments enable the modelling of viscous-cohesive fracture processes. The Boundary Element Method (BEM) describes the mechanical fields by the dipole-based approach, which has been extended to the three dimensional problems herein. This BEM approach is effective in crack growth modelling because only one crack surface requires discretisation. The classical cohesive crack approach models the material resistance at the energy dissipation zone surrounding the crack front. Besides, a time dependent function updates the cohesive crack approach and incorporates the viscous behaviour by modifying the tensile material strength and the material fracture energy as a function of the loading velocity rate. Three applications demonstrate the adequate performance of the proposed formulation, in which the results obtained have been compared against analytical and experimental responses available in the literature.

Particle separation based on dielectrophoresis force using boundary element method and point-particle approach in a microfluidic channel

Particle separation based on dielectrophoresis force using boundary element method and point-particle approach in a microfluidic channel

Nanophotonic resonators in stratified media with the nanobem toolbox

nanobem is a recently developed matlab toolbox for the solution of Maxwell's equations using a boundary element method, which is particularly suited for nanophotonic resonators and the computation of resonance modes [Hohenester et al., CPC 276, 108337 (2022)]. In this paper we present an extension that allows simulating nanoparticles situated in stratified media, using the matrix-friendly approach of Chew. Possible simulation setups include nanoparticles on substrates or embedded in layer structures, holes in films, or void structures. We provide a detailed documentation of the various toolbox classes, and present selected showcase results and convergence studies. Program summaryProgram Title:nanobemCPC Library link to program files:https://doi.org/10.17632/f59yhhbjwx.1Licensing provisions: GNU General Public License 3Programming language: MatlabNature of problem: Solve Maxwell's equations for metallic or dielectric nanoparticles embedded in stratified media.Solution method: Galerkin implementation of boundary element method approach using the matrix-friendly approach of Chew for stratified media.Additional comments including restrictions and unusual features: Toolbox has been developed and tested for small to intermediate problems with a few thousand boundary elements.

ReLU Neural Network Galerkin BEM

We introduce Neural Network (NN for short) approximation architectures for the numerical solution of Boundary Integral Equations (BIEs for short). We exemplify the proposed NN approach for the boundary reduction of the potential problem in two spatial dimensions. We adopt a Galerkin formulation-based method, in polygonal domains with a finite number of straight sides. Trial spaces used in the Galerkin discretization of the BIEs are built by using NNs that, in turn, employ the so-called Rectified Linear Units (ReLU) as the underlying activation function. The ReLU-NNs used to approximate the solutions to the BIEs depend nonlinearly on the parameters characterizing the NNs themselves. Consequently, the computation of a numerical solution to a BIE by means of ReLU-NNs boils down to a fine tuning of these parameters, in network training. We argue that ReLU-NNs of fixed depth and with a variable width allow us to recover well-known approximation rate results for the standard Galerkin Boundary Element Method (BEM). This observation hinges on existing well-known properties concerning the regularity of the solution of the BIEs on Lipschitz, polygonal boundaries, i.e. accounting for the effect of corner singularities, and the expressive power of ReLU-NNs over different classes of functions. We prove that shallow ReLU-NNs, i.e. networks having a fixed, moderate depth but with increasing width, can achieve optimal order algebraic convergence rates. We propose novel loss functions for NN training which are obtained using computable, local residual a posteriori error estimators with ReLU-NNs for the numerical approximation of BIEs. We find that weighted residual estimators, which are reliable without further assumptions on the quasi-uniformity of the underlying mesh, can be employed for the construction of computationally efficient loss functions for ReLU-NN training. The proposed framework allows us to leverage on state-of-the-art computational deep learning technologies such as TENSORFLOW and TPUs for the numerical solution of BIEs using ReLU-NNs. Exploratory numerical experiments validate our theoretical findings and indicate the viability of the proposed ReLU-NN Galerkin BEM approach.

Coupled VEM–BEM Approach for Isotropic Damage Modelling in Composite Materials

Numerical prediction of composite damage behaviour at the microscopic level is still a challenging engineering issue for the analysis and design of modern materials. In this work, we document the application of a recently developed numerical technique based on the coupling between the virtual element method (VEM) and the boundary element method (BEM) within the framework of continuum damage mechanics (CDM) to model the in-plane damage evolution characteristics of composite materials. BEM is a widely adopted and efficient numerical technique that reduces the problem dimensionality due to its underlying formulation. It substantially simplifies the pre-processing stage and decreases the computational effort without affecting the solution’s accuracy. VEM is a recent generalization to general polygonal mesh elements of the finite element method that ensures noticeable simplification in the data preparation stage of the analysis, notably for computational micro-mechanics problems, whose analysis domain often features complex geometries. The numerical technique has been applied to artificial microstructures, representing the transverse section of composite material with stiffer circular-shaped inclusions embedded in a softer matrix. BEM is used to model the inclusions that are supposed to behave within the linear elastic range, while VEM is used to model the surrounding matrix material, developing nonlinear behaviours. Numerical results are reported and discussed to validate the proposed method.

Power absorption of combined wind turbine and wave energy converter mounted on braceless floating platform

This work investigates the dynamic response and power absorption of a combined wind turbine and wave energy converter (WEC) mounted on a braceless semisubmersible floating platform. The floating platform consists of a Semi-submersible Wind turbine, a Wave energy converter Combination (SWWC) and a NREL 5-MW Current Source Converter (CSC) wind turbine. In this study, ANSYS/AQWA software is used in cooperation with a potential-based boundary element method (BEM) approach. The main targeted results include average power output and response amplitude of surge, heave and pitch motions of the combined concept at various wave and wind directions plus operational conditions. The numerical results are compared with experimental data, suggesting that the current numerical model predicts heave amplitude, average power, power take-off (PTO) damping force and response amplitude operator (RAO) of the platform motions with an acceptable agreement. Meanwhile, six environmental conditions (ECs) and five PTO damping coefficients are considered to optimize energy production. An optimal damping coefficient of 4000 kN s/m was found for the maximum average power generation for all the ECs. In addition, the effect of the wave/wind directions on the response motions and power absorption are obtained and presented.

A Study on Plate Bending Analysis Using Boundary Element Method

This study presents a method for level ice-structure interaction analysis to estimate the fatigue damage of arctic structures by applying plate theory to the behavior of level ice. The boundary element method (BEM), which incurs a lower computational cost than the finite element method (FEM), was introduced to solve the plate bending problem. The BEM formulation was performed by applying the BEM to plate theory. Finally, to check the validity of the proposed method, the BEM results and FEM results obtained using the ABAQUS commercial software were compared. The response results of the BEM analysis agreed well with those of the FEM analysis. Based on the results of the analysis, the BEM approach is considered to be very powerful in level ice-structure interaction analysis for estimating level ice-induced fatigue damage. Further work is being conducted to perform level ice fracture analysis based on the stress field calculated using the boundary element method.

Fully nonlinear dynamics of floating solar platform with twin hull by tubular floaters in ocean waves

Sustainability in harvesting solar power offshore depends on the survival of their platforms under environmental loads such as ocean waves. Therefore, a framework is established in this study to estimate the dynamics of floating solar platforms under ocean waves experimentally and numerically. Hence the platform is taken twin hull with double circular cylinders which exposed to a broad range of waves in the calibrated physical wave tank. On the other hand, the high order boundary element method and mixed Eulerian-Lagrangian approach are used in the numerical model to solve the fully nonlinear fluid field and the acceleration potential method is employed to approximate the response of the platform implicitly. To validate the framework, numerical and experimental results are compared for a single cylinder under the incident waves. Then, the comparison is extended for a twin hull platform. The effect of nonlinear free surface flow amongst the twin cylinders on the dynamics of the platform is studied parametrically. Afterwards, the performance of the present framework is assessed for a near breaking incident wave and an irregular wave. The stream function theory and flux of superposed wave components are used to emulate the high steep wave and the irregular wave respectively.

Stabilized solution of the 3-D MHD flow problem with FEM–BEM coupling approach

In this study, we obtain the numerical solution of the different problem configurations of the 3-D Magnetohydrodynamic (MHD) flow. We solve the MHD flow inside an unbounded conducting domain or around a conducting solid problems which require to consider both MHD equations and the Laplace equation on the connected domains with the coupled boundary conditions. As a numerical procedure we will propose the BEM (Boundary Element Method) and FEM (Finite Element Method) coupling approach with stabilization in order to eliminate the instabilities due to the numeric formulation. We consider the formulations of the MHD equations on the spherical or cubic domains inside an unbounded domain as a first problem and also on the annular spherical or annular cubic region around a conducting solid for the second problem with stabilized FEM. Then obtained discretized equations are combined with BEM formulation of the Laplace equations just only on the boundary. Finally, the BEM is used in order to obtain the rest of the unknown values on the Laplace domain using the boundary values obtained from the coupled formulation. Different values of the problem parameters and configurations are tested, and obtained solutions are displayed in terms of the figures as the 2-D slices of the 3-D solutions.

Nanophotonic resonance modes with the nanobem toolbox

nanobem is a matlab toolbox for the solution of Maxwell's equations for nanophotonic systems and the computation of resonance modes, sometimes also referred to as quasinormal modes or resonance states. It is based on a Galerkin scheme for the boundary element method, using Raviart-Thomas shape elements for the representation of the tangential electromagnetic fields at the particle boundary. The toolbox is written in an object-oriented manner with the focus on clarity rather than speed, and has been developed and tested for small to intermediate problems with a few thousand boundary elements. The computation of the resonance modes uses the contour integral method of Beyn. Program summaryProgram Title:nanobemCPC Library link to program files:https://doi.org/10.17632/63cwtv93ry.1Licensing provisions: GNU General Public License v3Programming language: MatlabNature of problem: Solve Maxwell's equations and compute resonance modes for optical resonators and nanophotonic systems with linear, homogeneous, and local materials separated by abrupt interfaces.Solution method: Galerkin implementation of boundary element method approach using Raviart-Thomas shape elements, and contour integral method for the computation of nanophotonic resonance modes.Additional comments including restrictions and unusual features: Toolbox has been developed and tested for small to intermediate problems with a few thousand boundary elements.

Nanoscale electromagnetism with the boundary element method

In Yang et al. [Nature 576, 248 (2019)], the authors introduced a general theoretical framework for nanoscale electromagnetism based on Feibelman parameters. Here quantum effects of the optically excited electrons at the interface between two materials are lumped into two complex-valued and frequency-dependent parameters, which can be incorporated into modified boundary conditions for Maxwell's equations, the so-called mesoscopic boundary conditions. These modifications can in principle be implemeted in any Maxwell solver, although the technicalities can be subtle and depend on the chosen computational approach. In this paper we show how to implement the mesoscopic boundary conditions in a boundary element method approach, based on a Galerkin scheme with Raviart-Thomas shape elements for the representation of the tangential electromagnetic fields at the boundary. We demonstrate that the results of our simulations are in perfect agreement with Mie theory including Feibelman parameters, and that for typical simulation scenarios the computational overhead is usually small.

Probabilistic chloride diffusion modelling in cracked concrete structures by transient BEM formulation

Abstract Reinforcement corrosion is a concern in the structural engineering domain, since it triggers several pathological manifestations, reducing the structural service life. Chloride diffusion has been considered one of main causes of reinforcements' corrosion in reinforced concrete. Corrosion starts when the chloride concentration at the reinforcements interface reaches the threshold content, leading to depassivation, whose assessment of its time of starts is a major challenge. This study applied the transient Boundary Element Method (BEM) approach for modelling chloride diffusion in concrete pores. The subregion BEM technique effectively represented the cracks inherent to the material domain, and environmental effects were also considered. Because of the inherent randomness of the problem, the service life was evaluated within the probabilistic context; therefore, Monte Carlo Simulation (MCS) assessed the probabilistic corrosion time initiation. Three applications demonstrated the accuracy and robustness of the model, in which the numerical results achieved by BEM were compared against numerical, analytical, and experimental responses from the literature. The probabilistic modelling substantially reduced the structural service life when the cracks length was longer than half of concrete cover thickness in highly aggressive environments.

Advances in nonlinear ultrasonic detection of microcracks

<p indent="0mm">A large number of key components are widely operated under extreme conditions of high temperature and high pressure in power, nuclear energy, aerospace and other important industrial fields. Microdefects are inevitably generated in these key components during the process of manufacturing. Due to the periodic loading and thermal stress, microcracks are generally initiated from these microdefects at the location with stress concentration. It has been found that the initiation and propagation of fatigue microcracks usually occur prior to 70%–90% of fatigue life. It is difficult to observe the evidence before the formation of macrocracks, which may result in catastrophic accidents. Thus, the risk of sudden failure of key components increases greatly. Therefore, the accurate detection of microcracks is crucial for guaranteeing the safe operation of key components. Based on the diffraction, reflection and transmission, traditional ultrasonic testing method is merely validated to effectively detect macroscopic defects within half a wavelength. The nonlinear ultrasonic testing method is found to be sensitive to microcracks, as high order harmonics, sub-harmonics, DC components and frequency mixing waves can be generated due to the strong nonlinear interaction microcracks and ultrasonic waves. With the outstanding effectiveness to detect microcracks, the nonlinear ultrasonic testing method has attracted much attention in recent years. In this work, the research on detection of microcracks using nonlinear ultrasonic testing method is systematically reviewed in terms of theoretical models, numerical simulations and experimental measurements. Firstly, according to the breathing effect and sliding mechanism of microcrack surfaces, we present several theoretical models of nonlinear interaction between ultrasonic waves and microcracks, namely contact acoustic nonlinearity (CAN), bi-linear stiffness model, nonlinear spring model, Hertz contact theory, and rough contact surface model. The advantages and disadvantages of these theoretical models are discussed comparatively. Two mechanisms are mainly considered for the nonlinear interaction of microcracks and ultrasonic waves, namely the macroscopic opening and closing of crack surfaces and the microscopic contact of rough microcracks surfaces. Secondly, we introduce numerical simulation studies on the nonlinear ultrasonic generation induced by microcracks. The applicability is analyzed for several simulation methods, such as finite element method, spectral finite element method, finite difference method, boundary element method and local interaction simulation approach, and the generation and propagation of nonlinear ultrasonic waves are analyzed as well. The effect of microcrack geometries on the nonlinear ultrasonic waves is then summarized. The microcrack geometries are considered mainly clouding the microcrack length, width, depth, density and direction. Furthermore, we present the experimental measurements on the generation of nonlinear ultrasonic waves induced by microcracks and on the localization and imaging of microcracks based on the nonlinear ultrasonic waves. The generation of high order harmonics, sub-harmonics, DC components and frequency mixing waves was observed in the specimens with microcracks. The nonlinear ultrasonic detection is presented to evaluate fatigue microcracks with various microcrack length, width, depth, density and direction. Based on this phenomenon, nonlinear ultrasonic mixing method, transducer array imaging method and nonlinear ultrasonic phased array method are introduced to localize the microcracks. Finally, further studies on evaluation of microcracks using nonlinear ultrasonic waves are proposed, including the comprehensive physical model for actual microcracks in engineering components, robust detection method, and quantitative evaluation of microcracks and service life.

A new hybrid method between equivalent source method and boundary element method for modeling diffraction

The Equivalent Source Method (ESM) allows to model the acoustical radiation of an object with a set of monopole sources inside the object. This method efficiently describes acoustical radiation of an object with smooth surface, but is not performant for addressing diffraction induced by sharp edge. In this study, we use the simple source formulation to establish ESM as an integral formulation. In this form, ESM belong to the Boundary Element Method (BEM) family, with an implicit mesh. Therefore, we propose an alternative method similar to ESM with sources on the surface object, as the singularities can now be handled with the integral formulation. This new method is more performant than ESM for modeling diffraction, and is easier to implement than standard BEM based on Gauss Quadrature while it still takes advantage of the ESM implicit mesh. Comparisons of this hybrid method with ESM and standard BEM approaches show that the developed method is a good compromise between the two other approaches: the new method is more accurate than BEM, but less than ESM for a smooth object; while it is equivalent to BEM and more accurate than ESM for an edgy object. Work Funded by AMIDEX foundation.

High-order Chebyshev-based Nyström Methods for Electromagnetics

Boundary element methods (BEM) have been successfully applied towards solving a broad array of complicated electromagnetic problems. Most BEM approaches rely on flat triangular discretizations and discretization via the Method of Moments (MoM) and low-order basis functions. Although more complicated from an implementation standpoint, it has been shown that high-order methods based on curvilinear patch mesh discretizations can significantly outperform low-order MoM in both accuracy and computational efficiency. In this work, we review a new high-order Nystrom method based on using Chebyshev basis functions with curvilinear elements that we have recently developed, present a few scattering examples, and discuss related on-going and future work.