Boundary Element Method Formulation Research Articles

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).

Accelerated boundary element method for direct current interference of cathodic protections systems

The use of cathodic protection (CP) is a widely used practice to mitigate the corrosion of metals. In certain circumstances, stray-current induced corrosion can occur on a structure when it interacts with the CP system of another nearby structure. Due to the locality of the phenomenon, stray-current induced corrosion is one of the most severe types of corrosion. The boundary element method (BEM) is ideal for solving CP problems. In the present work, two accelerated BEM formulations are presented to treat direct current interference problems. The first formulation is more efficient in solving small and medium scale problems, while the second one is more suitable for large scale problems. Furthermore, the stray current induced corrosion in an offshore platform due to its interaction with a nearby moored oil tanker is studied utilizing the above methodologies. The obtained numerical results are extensively discussed, and interesting conclusions are outlined.

A novel high-performance quadrature rule for BEM formulations

This paper describes a general approach to compute the boundary integral equations that appear when the boundary element method is applied for solving common engineering problems. The proposed procedure consists of a new quadrature rule to accurately evaluate singular and weakly singular integrals in the sense of the Cauchy Principal Value by an exclusively numerical procedure. This procedure is based on a system of equations that results from the finite part of known integrals, that include the shape functions used to approximate the field variables. The solution of this undetermined system of equations in the minimum norm sense provides the weights of the quadrature rule. A MATLAB script to compute the quadrature rule is included as supplementary material of this work. This approach is implemented in a boundary element method formulation based on the Bézier–Bernstein space as an approximation basis to represent both geometry and field variables for verification purposes. Specifically, heat transfer, elastostatic and elastodynamic problems are considered.

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.

Acoustic radiation from a sphere pulsating near an impedance plane using a boundary integral equation method

In this study, a boundary integral equation method is proposed for investigating acoustic pressure radiation from a sphere pulsating near a free surface or an impedance plane. The half-space and free-space problems are investigated for the acoustic radiation of pulsating sphere. The effects of free surface and impedance boundaries are introduced into the mathematical model by employing three different half-space Green’s functions, respectively. These Green’s functions are derived, respectively, using the single image-source method, multiple equivalent-source method, and complex equivalent-source method. Green’s functions are implemented into the boundary element (BE) formulation. The surface of the pulsating sphere is discretized with linear and quadratic BEs, and the Combined Helmholtz Integral Equation Formulation (CHIEF) is employed to overcome the non-uniqueness problem. Four different case studies are considered for the sphere pulsating near a free surface or an impedance plane. The first case study involves the sphere pulsating near a free surface (perfectly reflective) and the single image-source method is used in the boundary element method (BEM) formulation. In the second case study, the sphere is assumed as pulsating near a perfectly reflecting and perfectly absorbing impedance planes, respectively. The multiple equivalent-source method is employed for the perfectly reflecting plane, but the multiple equivalent-source method and complex equivalent-source methods for the perfectly absorbing plane. The third case study involves a general impedance plane, and all the methods are employed, respectively, in the BE formulation. The final case study assumes a general impedance plane forming a perpendicular incidence and the complex equivalent-source method is used in this particular case. It is observed that there is a very good comparison between the results obtained from all these methods.

An isogeometric FE-BE method to investigate fluid–structure interaction effects for an elastic cylindrical shell vibrating near a free surface

In this study, an isogeometric finite element-boundary element (FE-BE) framework is proposed to investigate frequency-dependent hydrodynamic characteristics of elastic structures vibrating at free surface or in the vicinity of free surface. The overall numerical framework consists of two parts. In the first part, in vacuo dynamic characteristics are obtained by an isogeometric finite element method (IGAFEM) formulation. Then, in the second part, frequency-dependent generalized added mass and generalized hydrodynamic damping coefficients are obtained by an isogeometric boundary element method (IGABEM) formulation, assuming that the surrounding fluid is ideal, i.e., inviscid, incompressible and its motion is irrotational. To include free surface effects, the free surface boundary conditions are satisfied with an appropriate Green’s function in the mathematical formulation. To show the applicability of the presented framework, an elastic cylindrical shell which was studied previously in the literature is investigated considering that the structure is partially or fully submerged into fluid of infinite depth. It is shown that the results obtained in this study compare very well with the results available in literature.

Boundary element method applied to three-dimensional crack analysis in exponentially graded viscoelastic materials

In the present work, three-dimensional crack problems in viscoelastic isotropic exponentially graded solids are investigated using the boundary element method (BEM). To reproduce the viscoelastic behaviour of the material, the BEM formulation is incorporated with an approach based on the differential constitutive relations for linear viscoelasticity employing Kelvin–Voigt and Boltzmann models. Moreover, the special case of material exponential gradation which is governed by a fundamental solution available in the literature is included into the boundary integral kernels. Because this methodology allows the material response to vary in time and space, it can be used to obtain the effects on the stress intensity factors, crack opening displacements, and energy release rates in practical fracture problems. Numerical examples are presented to demonstrate the applicability of the used methodology.

Electromagnetic Modeling of Lossy Materials With a Potential-Based Boundary Element Method

The boundary element method (BEM) enables solving three-dimensional electromagnetic problems using a two-dimensional surface mesh, making it appealing for applications ranging from electrical interconnect analysis to the design of metasurfaces. The BEM typically involves the electric and magnetic fields as unknown quantities. Formulations based on electromagnetic potentials rather than fields have garnered interest recently, for two main reasons: (a) they are inherently stable at low frequencies, unlike many field-based approaches, and (b) potentials provide a more direct interface to quantum physical phenomena. Existing potential-based formulations for electromagnetic scattering have been proposed primarily for perfect conductors. We develop a potential-based BEM formulation which can capture both dielectric and conductive losses, and accurately models the skin effect over broad ranges of frequency. The accuracy of the proposed formulation is validated through canonical and realistic numerical examples.

A Direct Method for Solving Singular Integrals in Three-Dimensional Time-Domain Boundary Element Method for Elastodynamics

The analytically time integrable time-space domain (ATI-TSD) is discovered based on which the minimum time-space domain is identified for treatment on singularities in the three-dimensional time-domain boundary element method (3D TD-BEM) formulation. A direct method to solve singular integrals in the 3D TD-BEM formulation for elastodynamic problems is proposed. The wavefront singularity can be analytically eliminated in ATI-TSD, while the dual singularity can be treated by the direct method using Kutt’s quadrature in the identified minimum time-space domain. Three benchmark examples are presented to verify the correctness and the applicability of the direct method for solving the singular integrals in 3D TD-BEM.

Two accelerated isogeometric boundary element method formulations: fast multipole method and hierarchical matrices method

This work presents two fast isogeometric formulations of the Boundary Element Method (BEM) applied to heat conduction problems, one accelerated by Fast Multipole Method (FMM) and other by Hierarchical Matrices. The Fast Multipole Method uses complex variables and expansion of fundamental solutions into Laurant series, while the Hierarchical Matrices are created by low rank CUR approximations from the k−Means clustering technique for geometric sampling. Both use Non-Uniform Rational B-Splines (NURBS) as shape functions. To reduce computational cost and facilitate implementation, NURBS are decomposed into Bézier curves, making the isogeometric formulation very similar to the conventional BEM. A description of the hierarchical structure of the data and the implemented algorithms are presented. Validation is performed by comparing the results of the proposed formulations with those of the conventional BEM formulation. The computational cost of both formulations is analyzed showing the advantages of the proposed formulations for large scale problems.

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.

A NURBS-discontinuous and enriched isogeometric boundary element formulation for two-dimensional fatigue crack growth

A new extended isogeometric boundary element method (XIGABEM) formulation is proposed for simulating multiple fatigue crack propagation in two-dimensional domains. The classical use of NURBS in isogeometric formulations is further extended by repeated knot insertion to introduce a C−1 continuity within the approximation space as an elegant approach to representing geometrical discontinuities at crack intersections. This strategy is also used to restrict the enrichment term to the portion of the NURBS defining the tip, where it is necessary. At this near-tip zone, the linear elastic fracture mechanics solutions are embedded into the displacement approximation to represent the theoretical square root behaviour. The enrichment procedure introduces just two degrees of freedom per crack tip, and a tying constraint is used to yield a square linear system. In this direct approach, the stress intensity factors (SIFs) are found as terms in the solution vector without requiring post-processing techniques.Several examples are presented to illustrate the application of the XIGABEM. The accuracy of the results compares favourably against those from the literature, and also against solutions obtained from unenriched and enriched indirect methods that employ the J-integral for SIF extraction. Furthermore, the proposed direct approach is capable of significantly reducing the execution time.

Nonlinear IGABEM formulations for the mechanical modelling of 3D reinforced structures

The adequate coupling of dissimilar materials and structural element types provides effective engineering designs. Particularly, it enables high stiffness and low weight mechanical components, which are largely desired in numerous applications. Besides, Isogeometric Analysis (IGA) has recently stood out as a robust and accurate approach for the mechanical modelling of complex 3D engineering designs directly from Computer-Aided Design (CAD) tools. In this context, this study presents nonlinear Isogeometric Boundary Element Method (IGABEM) formulations for the effective mechanical modelling of nonhomogeneous 3D structural systems reinforced by embedded fibres. The IGABEM based on NURBS surfaces represents the solid whereas the 1D approach of BEM models the reinforcements, which characterises a 1DBEM/IGABEM coupling technique. The sub-region technique enables the nonhomogeneous system modelling whereas connection 1DBEM elements allow for crossing reinforcements along boundary interfaces without the need for surfaces remeshing. Thus, the CAD model information can be straightforwardly used in the mechanical model. Nonlinear time independent formulations are presented accounting for both elastoplastic and bond-slip behaviours. Besides, linear and nonlinear time dependent formulations are presented herein applying the Kelvin-Voigt and Boltzmann viscoelastic rheological models. Five applications illustrate the accuracy and robustness of the proposed formulations. The IGABEM formulations lead to accurate and stable results, in addition to efficient modelling. Besides, it demonstrates excellent performance considering the computational time consuming and the total number of degrees of freedom in each application.

Numerical modelling of convection-diffusion problems with first-order chemical reaction using the dual reciprocity boundary element method

PurposeThe purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed for one- and two-dimensional steady-state problems, to analyse transient convection–diffusion problems associated with first-order chemical reaction.Design/methodology/approachThe mathematical modelling has used a dual reciprocity approximation to transform the domain integrals arising in the transient equation into equivalent boundary integrals. The integral representation formula for the corresponding problem is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. The finite difference method is used to simulate the time evolution procedure for solving the resulting system of equations. Three different radial basis functions have been successfully implemented to increase the accuracy of the solution and improving the rate of convergence.FindingsThe numerical results obtained demonstrate the excellent agreement with the analytical solutions to establish the validity of the proposed approach and to confirm its efficiency.Originality/valueFinally, the proposed BEM and DRBEM numerical solutions have not displayed any artificial diffusion, oscillatory behaviour or damping of the wave front, as appears in other different numerical methods.

Boundary Element Formulation Enforcing High-Order Surface Impedance Boundary Conditions for Axisymmetric Eddy Current Problems

This article presents a boundary element method (BEM) formulation coupled with high-order surface impedance boundary conditions (SIBCs) for axisymmetric eddy current problems. The formulation is based on the perturbation theory (PT) and the reduced magnetic scalar potential. A big advantage of the proposed approach is that if the solution is sought at different frequencies, the boundary integral equations can be solved only once and, then, the fields are easily computed at any frequency using a series expansion according to PT. Three test problems are solved, for which the reference solution is given by the standard finite-element method (FEM). The convergence of BEM-SIBC results toward FEM solution with respect to frequency is verified. The proposed approach requires reduced memory and computational times compared to the standard FEM and can be applied with linear, isotropic, and homogeneous materials.

A boundary element method formulation based on the Caputo derivative for the solution of the diffusion-wave equation

A boundary element method formulation based on the Caputo derivative for the solution of the diffusion-wave equation

A new boundary element formulation for modeling and simulation of three‐temperature distributions in carbon nanotube fiber reinforced composites with inclusions

A new boundary element method (BEM) formulation is developed in this paper for modeling and simulation of nonlinear three‐temperature (3T) distributions in carbon nanotube (CNT) fiber‐reinforced composites embedded with many rigid‐line inclusions that can be treated as having constant temperature distributions. The fast multipole method (FMM) has been applied to accelerate the proposed BEM solution of the considered problem of a large number of nano‐inclusions, and therefore, the computational complexity and computational time are reduced. The CPU time and memory requirement are also reduced by implementing generalized modified shift‐splitting (GMSS) iteration method which does not require the entire matrix to be stored in the memory. Then, we introduced a new formula which can play an important role in solving thermal problems arising in CNT nanotechnology. Effective thermal conductivity of fiber‐reinforced composites can be evaluated using the computed temperature and heat flux fields. The fast multipole BEM results are depicted graphically to show the effects of CNT manufacturing methods and thermal conductivities of different CNT composites on the temperature and heat flux distributions. The computational performance of the proposed methodology has been demonstrated. The fast multipole BEM results are compared graphically with the finite element method (FEM) results and with the experimental results reported in the literature. Excellent agreements are found in both cases, which clearly demonstrate the validity, accuracy, and efficiency of our proposed algorithm and modeling and simulation technique. The findings and solutions of this study contribute to the further development of CNT nanotechnology applications and devices.