Boundary Element Method Technique Research Articles

Comparisons between direct interpolation and reciprocity techniques of the boundary element method for solving two-dimensional Helmholtz problems

PurposeThis work compares the performance of the three boundary element techniques for solving Helmholtz problems: dual reciprocity, multiple reciprocity and direct interpolation. All techniques transform domain integrals into boundary integrals, despite using different principles to reach this purpose.Design/methodology/approachComparisons here performed include the solution of eigenvalue and response by frequency scanning, analyzing many features that are not comprehensively discussed in the literature, as follows: the type of boundary conditions, suitable number of degrees of freedom, modal content, number of primitives in the multiple reciprocity method (MRM) and the requirement of internal interpolation points in techniques that use radial basis functions as dual reciprocity and direct interpolation.FindingsAmong the other aspects, this work can conclude that the solution of the eigenvalue and response problems confirmed the reasonable accuracy of the dual reciprocity boundary element method (DRBEM) only for the calculation of the first natural frequencies. Concerning the direct interpolation boundary element method (DIBEM), its interpolation characteristic allows more accessibility for solving more elaborate problems. Despite requiring a greater number of interpolating internal points, the DIBEM has presented higher-quality results for the eigenvalue and response problems. The MRM results were satisfactory in terms of accuracy just for the low range of frequencies; however, the neglected higher-order primitives impact the accuracy of the dynamic response as a whole.Originality/valueThere are safe alternatives for solving engineering stationary dynamic problems using the boundary element method (BEM), but there are no suitable comparisons between these different techniques. This paper presents the particularities and detailed comparisons approaching the accuracy of the three important BEM techniques, aiming at response and frequency evaluation, which are not found in the specialized literature.

Three-Temperature Boundary Element Modeling of Ultrasound Wave Propagation in Anisotropic Viscoelastic Porous Media

The main goal of this work is to develop a novel boundary element method (BEM) model for analyzing ultrasonic wave propagation in three-temperature anisotropic viscoelastic porous media. Due to the problems of the strong nonlinearity of ultrasonic wave propagation in three-temperature porous media, the analytical or numerical solutions to the problems under consideration are always challenging, necessitating the development of new computational techniques. As a result, we use a new BEM model to solve such problems. A time-stepping procedure based on the linear multistep method is obtained after solving the discretized boundary integral equation with the quadrature rule. The calculation of a double integral is required to obtain fundamental solutions, but this increases the total BEM computation time. Our proposed BEM technique is used to solve the current problem and improve the formulation efficiency. The numerical results are graphed to demonstrate the effects of viscosity and anisotropy on the nonlinear ultrasonic stress waves in three-temperature porous media. The validity, accuracy, and efficiency of the proposed methodology are demonstrated by comparing the obtained results to a corresponding solution obtained from the finite difference method (FDM).

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.

Semi-analytic integration for a parallel space-time boundary element method modelling the heat equation

The presented paper concentrates on the boundary element method (BEM) for the heat equation in three spatial dimensions. In particular, we deal with tensor product space-time meshes allowing for quadrature schemes analytic in time and numerical in space. The spatial integrals can be treated by standard BEM techniques known from three dimensional stationary problems. The contribution of the paper is twofold. First, we provide temporal antiderivatives of the heat kernel necessary for the assembly of BEM matrices and the evaluation of the representation formula. Secondly, the presented approach has been implemented in a publicly available library besthea allowing researchers to reuse the formulae and BEM routines straightaway. The results are validated by numerical experiments in an HPC environment.

A new boundary element algorithm for modeling and simulation of nonlinear thermal stresses in micropolar FGA composites with temperature-dependent properties

The main aim of this article is to develop a new boundary element method (BEM) algorithm to model and simulate the nonlinear thermal stresses problems in micropolar functionally graded anisotropic (FGA) composites with temperature-dependent properties. Some inside points are chosen to treat the nonlinear terms and domain integrals. An integral formulation which is based on the use of Kirchhoff transformation is firstly used to simplify the transient heat conduction governing equation. Then, the residual nonlinear terms are carried out within the current formulation. The domain integrals can be effectively treated by applying the Cartesian transformation method (CTM). In the proposed BEM technique, the nonlinear temperature is computed on the boundary and some inside domain integral. Then, nonlinear displacement can be calculated at each time step. With the calculated temperature and displacement distributions, we can obtain the values of nonlinear thermal stresses. The efficiency of our proposed methodology has been improved by using the communication-avoiding versions of the Arnoldi (CA-Arnoldi) preconditioner for solving the resulting linear systems arising from the BEM to reduce the iterations number and computation time. The numerical outcomes establish the influence of temperature-dependent properties on the nonlinear temperature distribution, and investigate the effect of the functionally graded parameter on the nonlinear displacements and thermal stresses, through the micropolar FGA composites with temperature-dependent properties. These numerical outcomes also confirm the validity, precision and effectiveness of the proposed modeling and simulation methodology.

Three dimensional nonlinear BEM formulations for the mechanical analysis of nonhomogeneous reinforced structural systems

This study presents a coupling formulation for the mechanical modelling of nonhomogeneous reinforced 3D structural systems. The coupling of dissimilar reinforced materials and components provides efficient designs. Particularly, it enables high stiffness and low weight mechanical components, which are largely desired in several engineering applications. In the present study, the Boundary Element Method (BEM) mechanically represents 3D solids through elastic and viscoelastic approaches. The 1D truss BEM elements represent the reinforcements, which allow for elastoplastic behaviour. The sub-region BEM technique models the nonhomogeneous systems whereas connection 1DBEM elements allow for crossing reinforcements along materials interfaces. Bond-slip behaviour has been accounted in the formulation by a nonlinear scheme without special link elements. Different bond-slip laws can be accounted, which enables the mechanical modelling of different adherence behaviours. The proposed formulations are applied in the mechanical analysis of four complex 3D applications. The results achieved by the proposed formulations are compared against the responses of equivalent numerical methods and experimental results available in the literature. The proposed formulations lead to accurate and stable results. Besides, the applications complexity demonstrates it.

SLIM: A Well-Conditioned Single-Source Boundary Element Method for Modeling Lossy Conductors in Layered Media

The boundary element method (BEM) enables the efficient electromagnetic modelling of lossy conductors with a surface-based discretization. Existing BEM techniques for conductor modelling require either expensive dual basis functions or the use of both single- and double-layer potential operators to obtain a well-conditioned system matrix. The associated computational cost is particularly significant when conductors are embedded in stratified media, and the expensive multilayer Green's function (MGF) must be invoked. In this work, a novel single-source BEM formulation is proposed, which leads to a well-conditioned system matrix without the need for dual basis functions. The proposed single-layer impedance matrix (SLIM) formulation does not require the double-layer potential to model the background medium, which reduces the cost associated with the MGF. The accuracy and efficiency of the proposed method is demonstrated through realistic examples drawn from different applications.

A static boundary element solution for Bickford–Reddy beam

Bickford–Reddy beam theory is a refined model in which it is assumed that axial displacements vary cubically across the height of the cross section. Consequently, quadratic distribution for transverse shear stresses is automatically satisfied instead of Timoshenko beam model which predicts a constant distribution for those stresses, requiring the incorporation of a shear correction factor into the model. This article shows a new static solution which is based on direct boundary element method (BEM) for Bickford–Reddy beam theory. Mathematical steps required by this BEM technique are adequately addressed, for instance, a) integral equations are derived using Betti’s reciprocal theorem, b) fundamental solutions are obtained from the fundamental problem which has direct relationship to the real Bickford–Reddy beam problem; c) explicit influence matrices and load vectors are derived from source collocation at boundary points, and then at domain points, and d) BEM solutions are obtained for structures containing domain discontinuities such as stepped beams and continuous beams. Numerical results are presented for uniform beams having rectangular and circular cross sections and for problems having domain discontinuities.

Modeling and Optimization of Anisotropic Viscoelastic Porous Structures Using CQBEM and Moving Asymptotes Algorithm

The aim of this paper is to develop a novel numerical modeling optimization technique and implement it with the boundary element method (BEM) based on convolution quadrature method for studying optimization of anisotropic viscoelastic porous permeable structures. After applying the quadrature rule to the discretized boundary integral equation, a time stepping procedure based on the use of the linear multistep method is obtained. To obtain anisotropic fundamental solutions, the calculation of a double integral should be performed, but it increases the total computation time in the BEM. In order to overcome this problem and improve the computational efficiency of the formulation, our new proposed BEM technique is implemented. The method of implicit differentiation with respect to design variables has been implemented to calculate the displacements and pore pressure design sensitivities, and the effects of viscosity on these sensitivities are discussed. The resulting topology optimization problem has been solved using the method of moving asymptotes algorithm with adjoint variable method to optimize material distribution and find the influence of viscosity on the optimal design. The validity, efficiency and accuracy of the proposed BEM technique were confirmed by comparing obtained results for elliptical sandwich structure and multi-well reservoir with the corresponding results of finite difference method, lattice Boltzmann method and finite element method, which are special cases of our general and complex problem.

Acoustic Scattering for Rotational and Translational Symmetric Structures in Nonuniform Potential Flow

An efficient approach is proposed to predict acoustic scattering with nonuniform potential flow effects for structures with rotational and translational symmetries. The convected wave equation is transformed to Helmholtz and Laplace equations using a time transformation. The boundary-element method is used to formulate scattering by rotationally symmetric structures as two separate block circulant matrix equations and, similarly, as two separate block Toeplitz matrix equations for structures with translational symmetry. Discrete Fourier transform is employed to solve the block circulant systems. The block Toeplitz systems are solved using the generalized minimal residual method along with the discrete Fourier transform. Solving the convected wave equation using structured matrices significantly reduces computational time and storage requirements. To demonstrate the application of the formulation, two exterior acoustic case studies are considered. The first case study examines acoustic scattering from a sphere submerged in potential flow under monopole source excitation. Directivity plots obtained using the proposed technique are compared with analytical results. The second case study examines flow-induced noise generated by a rigid cylinder immersed in low-Mach-number flow, with the effect of mean flow on the scattered acoustic field taken into account using nonuniform potential flow. The fluctuating flowfield is obtained using an incompressible computational fluid dynamics solver. Acoustic sources based on Lighthill’s analogy are extracted from the flowfield data using a high-order reconstruction scheme. Results from the hybrid computational fluid dynamics– boundary-element method technique are presented for turbulent flow past the cylinder, with Reynolds number based on cylinder diameter of and Mach number . The aeroacoustic results are compared with data from literature.

COMPARISON BETWEEN THE CLASSICAL SUB REGIONS TECHNIQUE AND A NEW APPROACH WITH DOMAIN SUPERPOSITION TO SOLVE SECTORIAL INHOMOGENEOUS LAPLACE’S PROBLEMS

Abstract. The Boundary Element Method (BEM) has excellent performance in applications where the variable field is scalar and stationary. However, there is a wide range of issues in science and engineering that are difficult to solve by the BEM. Among these issues, there are the constitutive non-homogeneous problems, where the physical properties vary sectorally. In these kind of problems, the domain techniques, such as Finite Element Method (FEM), Finite Volume Method (FVM) or Finite Difference Method (FDM), present considerable advantages. However, even for these cases, there is a consistent BEM formulation, the classic sub-region technique. This work presents numerical comparisons between the classic subregion technique and an alternative BEM technique that is not based on a partition of the domain. Results are compared with analytical results and other achieved by domain methods using finer meshes. Keywords: Boundary Element Method, Inhomogeneous Laplace’s Problems, Sub-regions Technique

Aerodynamic Boundary Element Method for Bluff Bodies with Wind Tunnel Data Correlation

Boundary Element Method (BEM) is a widely used technique for the aerodynamic analysis of lifting bodies, due to the good compromise between fidelity and computational costs. This is especially true in rotorcraft applications, where the effects of the rotor wake play a crucial role in the evaluation of aerodynamic loads, and CFD solvers suffer for dissipation problems. However, potential formulations are unable to simulate many phenomena, like stall or separation around non-aerodynamic (bluff) bodies. This limits they application to the aerodynamic analysis of complete aircraft/rotorcraft configurations, where massive separations might arise downstream of the fuselage. In this paper, several potential-flow, BEM methodologies are proposed, with the objective of overcoming these limitations, at the cost of receiving information concerning the position of separation point and/or the value of pressure in the separated-flow region. Numerical results concerning the assessment of the proposed BEM techniques are presented: these consist in comparisons with data available in the literature for simple-geometry bodies, and with measurements obtained by experimental tests performed in a semi-open wind tunnel facility on a light-weight helicopter fuselage.

A Study on Fourier Spectral and Fourier-Hankel Spectral Field Variable Approximations in BEM for 2D Interior Helmholtz Problems

In this work the numerical modeling of the time harmonic problems in acoustics using boundary element method (BEM) is studied. The different computational techniques and methods for the solution of acoustic problems are reviewed. The formulation of some of the conventional BEM techniques is presented. The mathematical formulation and numerical implementation of Fourier Spectral BEM (FSBEM) and Fourier-Hankel Spectral BEM (FHSBEM) is presented for 2D problems. The numerical implementation of the conventional C-1 and linear C0 BEM formulations and FSBEM/FHSBEM is carried out by developing MATLAB codes. The verification of the numerical implementation of C-1 and linear C0 BEM formulations is carried out by comparing with analytical solutions and FEM. The comparison of FSBEM and FHSBEM is carried out with C-1 and linear C0 BEM formulations.

BEM and Tangent Operator Technique Applied to Analysis of Contact Problems

The boundary element method (BEM) is a robust and accurate numerical technique to deal contact problems, because the contact among solids occurs along its boundaries. In this regard, this work presents a nonlinear BEM formulation applied to contact problems simulation. The formulation is based on the use of singular or hyper-singular integral equations of BEM, for multi-region contact, and the dual version of BEM to simulate the contact between crack surfaces. The mechanical nonlinear behaviour introduced by the contact is represented by the Coulomb’s friction law. The nonlinear formulation uses the tangent operator technique, in which one uses the derivate set of algebraic equations to construct the corrections field for the nonlinear process. This implicit formulation has shown accurate as the classical approach. However it is faster, in terms of computational time consuming, than the classical nonlinear approach. Examples of simple and multi-region contact problems are presented in order to illustrate the applicability of the proposed nonlinear numerical scheme.

Innovative image processing techniques applied to the thermographic inspection of PFC with SATIR facility

The components used in fusion devices, especially high heat flux Plasma Facing Components (PFC), have to withstand heat fluxes in the range of 10–20 MW/m 2. So, they require high reliability which can be only guaranteed by accurate Non Destructive Examinations (NDE). The SATIR test bed operating at Commissariat à l’Energie Atomique (CEA) Cadarache performs NDE using transient infrared thermography sequence which compares the thermal response of a tested element to a Reference element assumed to be defect free. The control parameter is called DT ref max. In this paper, we present two innovative image processing techniques of the SATIR signal allowing the qualification of a component without any Reference element. The first method is based on a spatial image autocorrelation and the second on the resolution of an Inverse Heat Conduction Problem (IHCP) using a BEM (Boundary Element Method) technique. After a validation step performed on numerical data, these two methods have been applied to SATIR experimental data. The results show that these two techniques allow accurate defect detection, without using a Reference tile. They can be used in addition to the DT ref max, for the qualification of plasma facing components.

Local heat transfer estimation in microchannels during convective boiling under microgravity conditions: 3D inverse heat conduction problem using BEM techniques

Two-phase and boiling flow instabilities are complex, due to phase change and the existence of several interfaces. To fully understand the high heat transfer potential of boiling flows in microscale's geometry, it is vital to quantify these transfers. To perform this task, an experimental device has been designed to observe flow patterns. Analysis is made up by using an inverse method which allows us to estimate the local heat transfers while boiling occurs inside a microchannel. In our configuration, the direct measurement would impair the accuracy of the searched heat transfer coefficient because thermocouples implanted on the surface minichannels would disturb the established flow. In this communication, we are solving a 3D IHCP which consists in estimating using experimental data measurements the surface temperature and the surface heat flux in a minichannel during convective boiling under several gravity levels (g, 1g, 1.8g). The considered IHCP is formulated as a mathematical optimization problem and solved using the boundary element method (BEM).

An alternative multi-region BEM technique for three-dimensional elastic problems

The main objective of this work is to present an alternative boundary element method (BEM) formulation for the static analysis of three-dimensional non-homogeneous isotropic solids. These problems can be solved using the classical boundary element formulation, analyzing each subregion separately and then joining them together by introducing equilibrium and displacements compatibility. Establishing relations between the displacement fundamental solutions of the different domains, the alternative technique proposed in this paper allows analyzing all the domains as one unique solid, not requiring equilibrium or compatibility equations. This formulation also leads to a smaller system of equations when compared to the usual subregion technique, and the results obtained are even more accurate.

Boundary Element Computations in the Forward and Inverse Problems of Electrocardiography: Comparison of Collocation and Galerkin Weightings

In electrocardiographic imaging, epicardial potentials are reconstructed computationally from electrocardiographic measurements. The reconstruction is typically done with the help of the boundary element method (BEM), using the point collocation weighting and constant or linear basis functions. In this paper, we evaluated the performance of constant and linear point collocation and Galerkin BEMs in the epicardial potential problem. The integral equations and discretizations were formulated in terms of the single- and double-layer operators. All inner element integrals were calculated analytically. The computational methods were validated against analytical solutions in a simplified geometry. On the basis of the validation, no method was optimal in all testing scenarios. In the forward computation of the epicardial potential, the linear Galerkin (LG) method produced the smallest errors. The LG method also produced the smallest discretization error on the epicardial surface. In the inverse computation of epicardial potential, the electrode-specific transfer matrix performed better than the full transfer matrix. The Tikhonov 2 regularization outperformed the Tikhonov 0. In the optimal modeling conditions, the best BEM technique depended on electrode positions and chosen error measure. When large modeling errors such as omission of the lungs were present, the choice of the basis and weighting functions was not significant.

Cavity detection in biomechanics by an inverse evolutionary point load BEM technique

An efficient solution of the inverse geometric problem for cavity detection using a point load superposition technique in the elastostatics boundary element method (BEM) is presented in this article. A superposition of point load clusters technique is used to simulate the presence of cavities. This technique offers tremendous advantages in reducing the computational time for the elastostatics field solution as no boundary re-discretization is necessary throughout the inverse problem solution process. The inverse solution is achieved in two steps: (1) fixing the location and strengths of the point loads, (2) locating the cavity geometry. For a current estimated point load distribution, a first objective function measures the difference between BEM-computed and measured deformations at selected points. A genetic algorithm is employed to automatically alter the locations and strengths of the point loads to minimize the objective function. Upon convergence, a second objective function is defined to locate the cavity geometry modelled as traction-free surface. Results of cavity detection simulations using numerical experiments and simulated random measurement errors validate the approach in regular and irregular geometrical configurations.

Application of BEM and optimization technique to wear problems

In this paper a new method for solving wear problems, using the Boundary Element Method (BEM) is proposed. The proposed method uses a shape optimization technique for rapid calculation of the maximum wear depth and the final geometry of the bodies in contact after a specified service period. The optimization method solves the wear problem directly, without using increments of the sliding distance as in the classical methods. The BEM is used for modelling both bodies in contact, treating the problem as a multi-region problem. The material loss is presented in terms of the applied load and the sliding distance, and is modelled using a linear wear model. The proposed method is shown to be robust as it requires only few iterations to achieve a converged solution. The reduction in computational effort, as compared to the classical incremental method, becomes more significant as the sliding distance increases.