Boundary Element Method Formulation Research Articles

Vibration analysis of functionally graded epoxy/graphene composite plates using the Boundary Element Method and new micromechanical model

This paper studies the free vibration and harmonic response of Functionally Graded Graphene/Epoxy composite plates, considering the First Order Shear Deformation Plate Theory. The weight fraction of graphene variation along the thickness direction with graphene homogeneous dispersed in a polymer matrix. The effective Young Modulus of the plate is predicted by a new micro-mechanical model. The model includes the aspect ratio and weight fraction of nanoplatelets embedded into the matrix. The modal and harmonic of Graphene/Epoxy plates are obtained by the Boundary Element Method formulation for composite plates. The Young modulus calculated from the proposed micro-mechanical model highly agrees with those obtained from experimental results reported in the literature. Results demonstrate a high correlation of the micromechanical model with experimental results and other analytical models. Graphene weight fraction and ratio aspect increase the effective Young modulus of the composite. Boundary Element results show high agreement with Finite Element solutions. Numerical results for modal and harmonic analysis show concentrating graphene near the top and bottom surfaces of the plate is the most effective way to reinforce the plate for increased natural frequencies. The results demonstrate the BEM formulation and micromechanical model can be used as reliable engineering tools for the vibrational analysis of functionally grade graphene composite plates.

Boundary element method for bending-extension coupling shear deformable laminated composite plates with multiple stiffeners

In general, the laminated composite plates and stiffeners may exhibit the bending-extension coupling effect with the transverse shear deformations significantly induced. In this paper, for the first time we develop the associated boundary element method (BEM) for bending-extension coupling shear deformable laminated composite plates with multiple beams attached as stiffeners. Based upon the first order shear deformation theory, the individual boundary integral equations (BIEs) for plates and stiffeners are obtained from their corresponding matrix-form basic equations. To account for perfect bonding conditions between plate and stiffeners, we establish displacement continuity and interaction load equilibrium relations at their interfaces. To calculate the domain integrals related to unknown interaction loads, we discretize each stiffener into a series of cells using quadratic interpolation. By collecting the individual BIEs for plates and stiffeners and employing the continuity and equilibrium relations at the interfaces, we construct the entire system of linear equations to solve for all the unknown nodal quantities. Additionally, the complete solutions at the plate boundary, at internal points or along a stiffener are provided. The correctness of our present BEM formulation is verified through two representative examples by comparison with the commercial finite element software ANSYS. The numerical results demonstrate that our BEM formulation surpasses the conventional finite element method and is highly desirable for general stiffened laminated plates with bending-extension coupling and transverse shear deformations.

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.

A fast multipole boundary element method for acoustics in viscothermal fluids

Standard numerical models in acoustics rely on the isentropic Helmholtz equation. Its derivation assumes adiabatic and reversible, i.e., dissipation-free, wave propagation. Sound waves in fluids are, however, subject to viscous and thermal losses. These losses originate from viscous friction and heat conduction, leading to the formation of acoustic boundary layers. Considering these effects becomes significant in setups with acoustic cavities of similar dimension as the boundary layers. Recently, boundary element methods (BEM) accounting for the viscothermal dissipation have been proposed. These methods limit the discretization to the surface of the fluid domain and require significantly fewer degrees of freedom than comparable finite element models. However, the BEM coefficient matrices are fully populated, resulting in high computational costs and storage requirements. This study develops a new BEM formulation for acoustics in viscothermal fluids that uses the fast multipole method - a low-rank approximation technique based on a hierarchical subdivision of the computational domain - to alleviate this shortcoming. It is shown that the fast viscothermal BEM improves the algorithmic complexity over the conventional formulation, reducing both the memory requirements and solution time. The results indicate good scalability of the method making it feasible for larger applications such as acoustic metamaterials.

An efficient Artificial Neural Network algorithm for solving boundary integral equations in elasticity

This study presents a novel numerical integration technique based on Artificial Neural Network (ANN) algorithms to overcome intrinsic limitations characterizing the Boundary Element Method (BEM). The proposed approach, taking advantage of some peculiar properties of the BEM equations, provides an effective alternative to traditional numerical techniques for evaluating the integrated kernels required to compute the displacements and stresses of a two-dimensional solid. Assuming isotropy and homogeneity, and modeling both the geometry and the mechanical parameters using quadratic shape functions, all the integrals in the classical BEM formulation can be expressed as the sum of two terms that are independent of the constitutive properties and solely dependent on four geometric parameters: three components of two distance vectors and a parameter representing the element’s curvature. This interesting property of boundary integral equations in elasticity makes them particularly amenable to numerical evaluation using artificial neural networks. Results from numerical tests, which were conducted using increasingly complex integrals, demonstrate the high precision of the proposed approach as long as the integration and collocation points are sufficiently separated to avoid issues with singularity.

Implicit differentiation-based reliability analysis for shallow shell structures with the Boundary Element Method

A novel methodology for evaluating the response sensitivities of shallow shell structures using the Boundary Element Method (BEM) is presented in this work. The implicit derivatives of the BEM formulations for shallow shell structures, with respect to the geometrical variables, such as curvature and thickness, have been derived for the first time and incorporated into an Implicit Differentiation Method (IDM). The IDM is employed in conjunction with the First Order Reliability Method (FORM) to evaluate the reliability of shallow shell structures. The accuracy of the IDM formulation is first validated against an analytical solution, with results showing a maximum difference of only 2.61%. The IDM was later validated against the Finite Difference Method (FDM), with results showing a maximum difference of only 0.11%. The IDM was also found to be significantly more efficient than the FDM, requiring 35% less CPU time when calculating sensitivities. This is further compounded by the fact that, unlike the FDM, the IDM does not require a step size. A numerical example featuring a circular shallow shell is used to demonstrate the application of the IDM-based FORM for assessing structural reliability. The uncertainty in curvature is set as a variable for the purpose of investigating its impact on reliability. The results of the reliability index obtained from the IDM-FORM are compared to the results obtained from FDM-FORM and were found to be very similar. An analysis of sensitivity is conducted to identify the most significant variables affecting reliability. It is found that uncertainties in curvature, thickness, and applied pressure distribution parameters have the largest impact on structural reliability. To demonstrate how the IDM could be used in practice, it was employed as gradient-based optimisation procedure featuring shallow-shell structures. The IDM was found to be a very efficient and accurate alternative to existing methods for calculating structural response sensitivities.

Boundary Element Method for Tangential Contact of a Coated Elastic Half-Space

We present a formulation of the boundary element method (BEM) for simulating the tangential contact with an elastic half-space coated with an elastic layer with different elastic properties. We use the fast Fourier-transform-based formulation of BEM, while the fundamental solution is determined directly in the Fourier space. Numerical tests are validated by comparison with available asymptotic analytical solutions for a very thin and a very thick layer, as well as with FEM calculations for layers with finite thickness.

Modelling of 3D periodic cathodic protection problems in reinforced concrete structures with accelerated boundary element method

The steel used for concrete reinforcement exhibits either active or passive electrochemical behavior. As long as the concrete environment remains highly alkaline, the steel remains passive. However, carbonation and/or chloride contamination of concrete will cause steel to become active and corrode. In the latter case, a common strategy is the use of cathodic protection (CP) systems to lower the potential of the steel at a protective level. The numerical modelling of cathodically protected concrete buildings and infrastructures usually requires large-scale models due to their size and complex geometry. An efficient modelling approach is to consider any possible periodicity in these problems by applying periodic boundary conditions to a representative volume element of the structure. The boundary element method (BEM) is extensively used for solving CP problems. A BEM formulation is proposed for solving 3D periodic CP problems in this work. The computation cost is reduced by applying acceleration techniques based on adaptive cross approximation (ACA). As an engineering application, a sacrificial anode CP system of a reinforced concrete column is designed, illustrating the importance of detailed modelling in the design of such systems. To this end, the macrocell corrosion problem is initially solved, and a parametric study with respect to concrete conductivity is carried out.

Complex-variable, high-precision formulation of the consistent boundary element method for 2D potential and elasticity problems

The collocation boundary element method was recently entirely revisited on the basis of a consistent derivation of Somigliana’s identity in terms of weighted residuals. Both conceptually and for the sake of code implementation, the correct traction force interpolation along generally curved boundaries, as for elasticity problems, leads to the enunciation of an inedited, actually long-sought, convergence theorem as well as to considerable numerical simplifications. Numerical evaluations for two-dimensional problems require exclusively Gauss–Legendre quadrature plus eventual correction terms obtained analytically regardless of the order or shape of the implemented boundary element interpolation. Arbitrarily high precision and accuracy is achievable for low-cost computation, as eventual mesh-subdivision refinements should take place only if the mechanical simulation demands — and not just for numerical evaluations. We now show that considerable simplification is obtained by switching the formulation from real to complex variable. Precision, round-off errors, and accuracy of a given numerical implementation may be kept – identifiably and separately – under control, as assessed for some potential and elasticity examples with extremely challenging topologies. In fact, source-field distances may be arbitrarily small — far smaller than deemed feasible in continuum mechanics, as we resort to nothing else than the problem’s mathematics.

A Generalized Scalar Potential Integral Equation Formulation for the DC Analysis of Conductors

The electrostatic modeling of conductors is a fundamental challenge in various applications, including the prediction of parasitic effects in electrical interconnects, the design of biasing networks, and the modeling of biological, microelectromechanical, and sensing systems. The boundary element method (BEM) can be an effective simulation tool for these problems because it allows modeling three-dimensional objects with only a surface mesh. However, existing BEM formulations can be restrictive because they make assumptions specific to particular applications. For example, capacitance extraction formulations usually assume a constant electric scalar potential on the surface of each conductor and cannot be used to model a flowing current, nor to extract the resistance. When modeling steady currents, many existing techniques do not address mathematical challenges such as the null space associated with the operators representing the internal region of a conductor. We propose a more general BEM framework based on the electric scalar potential for modeling conductive objects in various scenarios in a unified manner. Restrictive application-specific assumptions are not made, and the aforementioned operator null space is handled in an intuitive and rigorous manner. Numerical examples drawn from diverse applications confirm the accuracy and generality of the proposed method.

Interaction effects from the elastodynamic scattering by two symmetric spherical cavities

The scattering of harmonic waves has been studied extensively for problems in quantum mechanics, acoustics, electromagnetics, and elasticity. Solutions to elastodynamic problems are the basis for ultrasonic non-destructive evaluation measurement models. Therefore, in this study, we investigate the use of the boundary element method (BEM) in the frequency domain using an off-boundary technique in which the observation points are taken inside the scattering object. This methodology removes both non-integrable singularities from the domain of integration along with avoiding ill-conditioning effects that occur at fictitious eigenfrequencies of which both require using special computationally demanding procedures to obtain solutions. Additionally, we employ both free and half-space fundamental solutions (Green’s displacement tensors) to investigate the elastodynamic scattering of an incident, plane, time-harmonic longitudinal wave in the frequency domain of a homogeneous, isotropic, and linear elastic solid with one or two spherical cavities. The half-space fundamental solution reduces the number of required boundary elements in half, which significantly reduces computational resource requirements. We only consider spherical cavities in this paper to illustrate the full and half-space off-boundary BEM and analyze the interaction effects associated with the elastodynamic scattering by two symmetric spherical cavities. To verify the validity of the free and half-space off-boundary BEM formulations, surface displacement results are compared with existing surface displacement results and show good agreement. Finally, the half-space off-boundary BEM is used to illustrate the interaction effects of the back and forward scattered displacement fields as a function of the distance from the spherical cavities.

Characteristics of SH-wave propagation during oil reservoir excitation using BEM formulation in half-plane model representation

The application of elastic waves induced through seismic excitation (stimulation) to increase the oil recovery from a hydrocarbon reservoir as an enhanced oil recovery (EOR) technique is currently in its infancy. Seismic stimulation is a cost-effective and efficient method for enhancing oil production from waterflood reservoirs by increasing areal sweep performance and decreasing water cuts, hence extending the productive life of a mature oilfield. It has great prospect in offshore fields where technological difficulties and restrictions hinder the deployment of other EOR methods. Herein, this study investigates the effects of a direct impact of seismic SH-wave on a 2D transient behaviors of an oil reservoir with a half-plane model. In the time domain, a half-plane model containing a poroelastic oil reservoir is formulated using a boundary element method (BEM). The model considers complete seismic SH-wave propagation using Ricker wavelet, from a down-hole source via a partially saturated oil reservoir to ground surface receiving points. The model is based on 2D elastodynamic and Biot's dynamic poroelasticity equations. A DASBEM program is incorporated into MATLAB (2022a) to develop a model of wave propagation. The result showed that variable incidence angle affects the frequency and time domain responses, while the depth of the reservoir and porosity influence the amplitude of oscillation. Synthetic seismograms at stations above the oil reservoir indicated attenuation and numerous diffracted waves with various time delays. Multiple SH-wave reflections in the reservoir amplify 3D signals, while at higher dimensionless frequencies, effect of porosity amplify the distant locations. This technique can be utilized as fluid indicator to monitor wave based EOR using SH-wave attenuation, and to detect and visualize the permeability changes in reservoir (impermeable boundaries) during CO2 plume storage for the purpose of monitoring the safety of CO2 geo-sequestration.

BEM-Based Magnetic Field Reconstruction by Ensemble Kálmán Filtering

Abstract Magnetic fields generated by normal or superconducting electromagnets are used to guide and focus particle beams in storage rings, synchrotron light sources, mass spectrometers, and beamlines for radiotherapy. The accurate determination of the magnetic field by measurement is critical for the prediction of the particle beam trajectory and hence the design of the accelerator complex. In this context, state-of-the-art numerical field computation makes use of boundary-element methods (BEM) to express the magnetic field. This enables the accurate computation of higher-order partial derivatives and local expansions of magnetic potentials used in efficient numerical codes for particle tracking. In this paper, we present an approach to infer the boundary data of an indirect BEM formulation from magnetic field measurements by ensemble Kálmán filtering. In this way, measurement uncertainties can be propagated to the boundary data, magnetic field and potentials, and to the beam related quantities derived from particle tracking. We provide results obtained from real measurement data of a curved dipole magnet using a Hall probe mapper system.

Meso-scale modeling of the compressive mechanical behavior of concrete by a RVE-based BEM formulation

A 2 D Boundary Element Method (BEM) formulation, based on the representative volume element (RVE) concept, simulates the mechanical behavior of experimentally tested concrete under compression. A step-by-step parametric identification is presented. Considering different modeling for the fracture process at ITZ (Interfacial Transition Zone), the experimental results are captured when this phenomenon is modeled only around bigger aggregates. Besides, the size and quantity of the aggregates do not have great influence on the concrete mechanical behavior and the insertion of porosity in the mortar domain is very important. The proposed numerical model is efficient, robust, and accurate.

A cell-less boundary element method for a two-step thermoelastic analysis

A new cell-less boundary element method formulation is presented for two-step thermoelastic analyses. The first step is a steady-state thermal analysis. The second step is a thermoelastic analysis, using the temperature field obtained in the first step. Domain integrals involving internal heat sources and thermal strains, that appear in the governing boundary integral equations, are transformed to boundary by the radial integration method. Since temperature has been evaluated only at discrete points in the first step, meshfree techniques for defining shape functions are adopted to calculate temperature at numerical integration points during evaluation of radial integrations of the second step. Two classes of such techniques are considered: the moving least square procedure and the radial point interpolation method. Representative examples are presented to demonstrate the effectiveness of the proposed formulation.

Accelerating boundary element methods in wideband frequency sweep analysis by matrix-free model order reduction

In frequency-domain acoustic simulations, a wideband frequency sweep analysis is often required. This becomes particularly challenging for boundary element methods (BEM), as the system matrix has a complex dependency on frequency. Previous attempts have been made to accelerate the frequency sweep analysis for the conventional BEM i.e., without fast multipole or ℋ-matrices accelerations, which restrains the applications to limited scale problems due to the dense system matrix. This paper investigates and incorporates a rational Krylov projection based model order reduction (MOR) technology i.e., matrix-free method into various BEM formulations to address the critical challenge of accelerating frequency sweep analysis. Operating on the transfer functions of the BEM system’s input and output, the matrix-free MOR avoids the direct interference with the BEM kernels, which makes it feasible and attractive to work with existing fast BEM techniques such as ℋ-matrices and fast multipole method. The combined holistic BEM solution retains the merits of the acceleration from both MOR and fast BEM techniques. Significant speed-ups have been achieved by the matrix-free MOR for wideband BEM simulations. Numerical examples are presented to demonstrate the capability of the presented holistic approach in simulating various acoustic problems, including interior–exterior models, complex geometries and frequency dependent boundary conditions.

Marine propeller shaft loading analysis in moderate oblique-flow conditions

Shaft loads of marine propellers in noncavitating oblique-flow conditions are herein investigated. The application of a Boundary Element Method (BEM) hydrodynamics is verified by the analysis of two propeller models, the highly-skewed DTMB 4679 and the unskewed INSEAN E779A, for which a comparison with experimental and numerical data is addressed. Whenever available, outcomes from PROCAL, a validated panel code solver developed by MARIN, and simulations from higher-fidelity CFD (Computational Fluid Dynamics) approaches, are used for comparison purposes. For moderate oblique-inflow angles, the comparison with experiments shows a good capability of the proposed BEM formulation in capturing unsteady blade pressures. In terms of shaft loads fluctuations, satisfactory results are carried out with respect to CFD outcomes. Moreover, it is found that averaged thrust and torque, as well as the lateral and vertical moments, are comparable with PROCAL predictions and in good agreement with CFD computations, whereas lateral forces suffer from the lack of a realistic estimation of viscous effects. Good estimates are also shown for the average location of thrust eccentricity, as long as the transverse loading on the hub is negligible.

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.

Electromagnetic Modeling of Lossy Interconnects From DC to High Frequencies With a Potential-Based Boundary Element Formulation

The accurate electromagnetic modeling of both low- and high-frequency physics is crucial in the signal and power integrity analysis of electrical interconnects. The boundary element method (BEM) is appealing for lossy conductor modeling because it can capture the frequency-dependent variation of skin depth with only a surface-based discretization of the structure. Conventional BEM formulations rely on the mutual coupling of electric and magnetic fields, and can become inaccurate or unstable at low frequencies. We develop a new full-wave BEM formulation based on potentials which can accurately model lossy conductors from exactly DC to very high frequencies. A new set of simple boundary conditions is proposed along with a modified Lorenz gauge to ensure that the proposed formulation has a stable condition number down to DC. Moreover, coupling the potential-based integral equations to a circuit model allows the straightforward extraction of network parameters. Realistic numerical examples at both the chip and package level demonstrate the accuracy and stability of the proposed method from DC to high frequencies, beyond the capabilities of state-of-the-art BEM formulations based on fields.

Shape Optimisation of Assembled Plate Structures with the Boundary Element Method

A novel methodology is presented for performing sensitivity analyses of assembled plate structures using the Boundary Element Method (BEM). The main novelty of this work is that the exact implicit derivatives of the BEM formulations for assembled plate structures have been derived for the first time and incorporated into a newly developed Implicit Differentiation Method (IDM), enabling sensitivity analyses to be conducted for more complex and realistic structures in a more accurate, robust, and efficient manner than previous approaches. Three numerical examples are investigated to validate the derived exact implicit derivatives and to demonstrate how they could be used for a potential application involving the shape optimisation of a complex X-core structure from the canard of a Eurofighter Typhoon fighter jet. Results show that the newly developed IDM is more accurate, robust, and efficient when compared to alternative methodologies using derivatives obtained from methods such as the Finite Difference Method (FDM) and the Finite Element Method (FEM).