Classical Boundary Element Method 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.

Geometric inverse estimation for the inner wall of a furnace under transient heat conduction based on dual reciprocity boundary element method

A boundary shape estimation problem for the furnace’s inner surface is solved using the dual reciprocity boundary element method (DRBEM) and the conjugate gradient method (CGM) under transient heat conduction. The DRBEM is utilized to eliminate the drawbacks of traditional numerical methods and classical boundary element method of discretizing the entire computational domain, with only boundary discretization. The inversion results are obtained by applying the CGM to minimize the objective function, in which the sensitivity coefficients are calculated with the complex variable derivation method (CVDM), making the calculation precise and independent of step size. To verify the accuracy of the DRBEM in solving the transient heat conduction problem, the influencing factors including radial-basis function, the number of internal collocation points, and time step size are investigated. The influences of measurement time interval, future time step, initial guess, measurement error, and the number and position of measurement points on the inversion results are analyzed. Meanwhile, the effectiveness of the proposed approach is tested by numerical examples, and the inversion results show that it is stable, accurate, and efficient for identifying different and complicated unknown boundary shapes of the furnace.

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.

Frequency‐domain fundamental solutions and boundary element method for consistent couple stress elastodynamic problems

AbstractIn this article, the elastodynamic problems are investigated within the framework of a consistent couple‐stress theory (CCST). The basic equations for the CCST elastodynamic problems are derived at first. These dynamic formulas are then analyzed in the frequency domain by the Fourier transform. The frequency‐domain fundamental solutions to the transformed elastodynamic problems are subsequently derived for the two‐dimensional (2D) plane‐strain state. Based on the fundamental solutions and the elastodynamic reciprocal theorem, the frequency‐domain boundary integral equations (BIEs) are established. Then, the BIEs are numerically solved by a collocation method in conjunction with analytical and numerical treatments of the arising singular integrals. To obtain the time‐domain dynamic responses, an exponential window method (EWM) is adopted to transform the frequency‐domain results to the time‐domain solutions. By using the developed boundary element method (BEM), several typical couple‐stress elastodynamic problems are numerically investigated and the size‐effects are studied by using different values of the length‐scale parameter in the CCST. Good agreements between the results obtained by the developed BEM for a tiny length‐scale parameter and the corresponding classical BEM results or analytical solutions are achieved, which validates the high accuracy of the present BEM.

On a time-discrete convolution—space collocation BEM for the numerical solution of two-dimensional wave propagation problems in unbounded domains

Abstract For the numerical solution of a classical two-dimensional Dirichlet exterior problem for the wave equation in the time domain, we consider a space-time boundary integral equation approach, based on its discretization by means of the coupling of a time discrete convolution quadrature of order 2, with the classical continuous piecewise linear boundary element method. The latter is either the Galerkin method or the collocation one, this being computationally much cheaper. For the efficient evaluation of most of the integrals generated by these two boundary element methods, but also for the subsequent analysis, we first define a new Gaussian quadrature. After recalling that stability and convergence have been proved for the Galerkin method, while for the collocation one we only have numerical evidences of these properties, to (partially) fill this gap, we first note that collocation can be obtained directly from the Galerkin system, after discretizing its inner product integral by using the composite trapezoidal rule, which in this case turns out to be the Gaussian rule mentioned above with $1$ node. Then, we show that for all (positive) time step-sizes $\varDelta _t$, when we let $h\rightarrow 0$, independently from $\varDelta _t$, the matrix of the linear system produced by the collocation method, properly normalized, converges, in the $\infty $-norm, to the corresponding Galerkin matrix. The behavior of the associated absolute and relative errors in the $\infty $-norm, as $h\rightarrow 0$, are $O\big (h^{\frac {5}{3}}+\varDelta _t^{-(1+\varepsilon )}h^{\frac {5+\varepsilon }{2}}[|\!\ln \varDelta _t|+|\!\ln h|]\big )$ and $O\big (h^{\frac {2}{3}}/|\!\ln \varDelta _t|+\varDelta _t^{-(1+\varepsilon )}h^{\frac {3+\varepsilon }{2}}[1+|\!\ln h|/|\!\ln \varDelta _t|]\big )$, respectively, with $\varepsilon>0$ as close as one likes to $0$ and the $O(\cdot )$ constant independent from $\varDelta _t, h$. Some possible extensions of the investigation we carried out are also mentioned.

Performance assessment of internal porous structures on liquid sloshing in various 3D tanks by multi-domain IGABEM

An isogeometric boundary element method (IGABEM) is extended in this work to study liquid sloshing characteristics in various 3D tanks with arbitrary internal bottom-mounted porous structures. Unlike the classical boundary element method (BEM), the present method uses the non-uniform rational B-splines(NURBS) instead of the piecewise Lagrange polynomials as the shape function to approximate both the domain boundary and field variables, it completely inherits the advantages of the traditional BEM and the isogeometric analysis (IGA), such as the properties of only boundary discretization needing, higher order continuity, self-adaptability and so on. All the information required in the IGABEM for the mesh generation can be obtained from the CAD software, which may evidently reduce the time and memory consumptions of preprocessing, meanwhile, since the same basis functions (NURBS) are used for both the IGABEM and CAD models, the proposed method can exactly reconstruct the geometry of the analysis domain without any error, and this substantively contributes to the high calculation accuracy of IGABEM. In this paper, by using a zoning method the entire domains of fluid are divided into several sub-domains with the consideration of compatibility boundary conditions besides the porous structures. Owning to the constant cross section of the container, impermeable boundary condition at the tank bottom and the linear boundary condition at the free surface, the 3D Laplace equation (which governs the 3D sloshing problem) is transformed into a couple of Helmholtz equation and modified Helmholtz equations (corresponding to the propagating and evanescent modes, respectively). Green's theorem and the divergence theorem of Gauss are used in this paper to derive the IGABEM system equations for the two types of equations. Moreover, by introducing a series of eigen-functions associated with the vertical coordinate, the potential and velocity boundary conditions are expanded into those associated with the propagating and evanescent modes, which can be applied upon the Helmholtz equation and modified Helmholtz equations. Finally, the present problem can be solved by solving these equations and combining the results linearly. Accuracy and convergence of the present IGABEM technique are verified through numerical tests by comparing the obtained results with analytical, experimental solutions or those calculated with other numerical method, thereafter, liquid sloshing problems in the cubic or circular cylindrical vessels with the coaxial or eccentric elliptical and circular cylindrical porous structures are analyzed. The main influences of the geometric parameter, porous-effect parameter, number and arrangement of the porous structures are investigated in detail, some conclusions are outlined in the end of this paper.

Helmholtz equation and non-singular boundary elements applied to multi-disciplinary physical problems

The famous scientist Hermann von Helmholtz was born 200 years ago. Many complex physical wave phenomena in engineering can effectively be described using one or a set of equations named after him: the Helmholtz equation. Although this has been known for a long time, from a theoretical point of view, the actual numerical implementation has often been hindered by divergence-free and/or curl-free constraints. There is further a need for a numerical method that is accurate, reliable and takes into account radiation conditions at infinity. The classical boundary element method satisfies the last condition, yet one has to deal with singularities in the implementation. We review here how a recently developed singularity-free three-dimensional boundary element framework with superior accuracy can be used to tackle such problems only using one or a few Helmholtz equations with higher order (quadratic) elements which can tackle complex curved shapes. Examples are given for acoustics (a Helmholtz resonator among others) and electromagnetic scattering.

Localized method of fundamental solutions for two- and three-dimensional transient convection-diffusion-reaction equations

This paper investigates the use of the localized method of fundamental solutions (LMFS) for the numerical solution of general transient convection-diffusion-reaction equation in both two-(2D) and three-dimensional (3D) materials. The method is developed as a generalization of the author's earlier work on Laplace's equation to transient convection-diffusion-reaction equation. The popular Crank-Nicolson (CN) time-stepping technology is adopted to perform the temporal simulations. The LMFS approach is then introduced for solving the resulting inhomogeneous boundary value problems, where a pseudo-spectral Chebyshev collocation scheme (CCS) is employed for the approximation of the corresponding particular solutions. As compared with the classical MFS and boundary element method (BEM), the present CN-CCS-LMFS approach produces sparse and banded stiffness matrix which makes the method possible to perform large-scale dynamic simulations. Several benchmark numerical examples are presented to demonstrate the efficiency and feasibility of the present method.

On the boundary element method for Stokes flow in domains with discontinuous boundaries

AbstractThe classical boundary element method (BEM) has difficulties capturing the solution accurately in the presence of boundary discontinuities. It exhibits higher error values in the proximity of corners than on the rest of the boundary. In an attempt to remedy this problem a two‐dimensional constant uniform flow across a square domain with an outflow condition on one side is considered and analyzed in depth. This leads to the conclusion that the free term from classical approaches needs to be reformulated for the degrees of freedoms lying at the corner points. This modification improves the accuracy of the BEM up to machine precision, which opens interesting directions for further research.

An adaptive model order reduction method for boundary element-based multi-frequency acoustic wave problems

The classical boundary element method (BEM) is widely used to obtain detailed information on the acoustic performance of large-scale dynamical systems due to the nature of semi-analytical characteristic. Its use, however, results in asymmetric and dense system matrix, which makes original full-order model evaluation very time consuming and memory demanding. Moreover, the frequency sweep analysis which is indispensable for the assessment of noise emission levels and the design of high-quality products requires the repetitive assembly and solution of system of equations, which further increases the computational complexity. In order to alleviate these problems, an adaptive structure-preserving model order reduction method is presented, which is based on an offline–online solution framework. In the offline phase, we first factor out the frequency term as a scalar function from the BE integral kernels of the Burton–Miller formulation, followed by integration to set up the system matrices. A global frequency-independent orthonormal basis is then constructed via the second-order Arnoldi (SOAR) method to span a projection subspace, onto which the frequency-decoupled system matrices are projected column by column to solve the memory problem arising from the frequency-related decomposition. In addition, the number of iterations required for convergence can be automatically determined by exploiting the condition number of an upper Hessenberg matrix in SOAR. In the online stage, a reliable reduced-order model can be quickly recovered by the sum of those offline stored reduced matrices multiplied by frequency-dependent coefficients, which is favored in many-query applications. Two academic benchmarks and a more realistic problem are investigated in order to demonstrate the potentials of the proposed approach.

Development and Application of a Pre-Corrected Fast Fourier Transform Accelerated Multi-Layer Boundary Element Method for the Simulation of Shallow Water Acoustic Propagation

Because of the complexities associated with the domain geometry and environments, accurate prediction of acoustics propagation and scattering in realistic shallow water environments by direct numerical simulation is challenging. Based on the pre-corrected Fast Fourier Transform (PFFT) method, we accelerated the classical boundary element method (BEM) to predict the acoustic propagation in a multi-layer shallow water environment. The classical boundary element method formulate the acoustics propagation problem as a linear equation system in the form of [A]{x}={b}, where [A] is an N×N dense matrix composed of influence coefficients. Solving such linear equation system requires O(N2/N3) computational cost for iterative/direct methods. The developed method, PFFT-BEM, can effectively reduce the computational efforts for direct numerical simulations from O(N2~3) to O(Nlog N), where N is the total number of boundary unknowns. To numerically simulate the sound propagation in a shallow water environment, we applied the first-order non-reflecting boundary condition in the truncated numerical domain boundary to eliminate the errors due to reflected waves. Multi-layer coupled formulation was used to include the environment inhomogeneity in PFFT-BEM. Through multiple convergence tests on the number of layers and elements, we validated and quantified the accuracy of PFFT-BEM. To demonstrate the usefulness and capability of the developed PFFT-BEM, we simulated three-dimensional (3D) underwater sound propagation through 3D geometries to check the efficacy of the established classical method: the 3D Parabolic equation model. Finally, PFFT-BEM was employed to simulate sound propagation through a complex multi-layer shallow water environment with internal waves. The “3D+T” results obtained by PFFT-BEM compared well with the physical test, thereby proving the capability and correctness of this method.

Effects of contact behaviors on tolerance analysis of mechanism based on skin model shapes and a boundary element method

Abstract Tolerance analysis plays an important role in predicting the quality of mechanical products in early design stage. However, previous researches on tolerance analysis have mostly ignored the effects of contact behaviors between mating surfaces. This may lead to unrealistic results, especially when analyzing products with kinematic pairs such as rotating mechanisms. In order to obtain more accurate and realistic analysis results, this paper proposes a tolerance analysis approach that considers the effects of contact behaviors between mating surfaces. Geometric features with machining and assembly errors are modeled based on the concept of Skin Model Shapes. Contact behaviors between mating surfaces are calculated by the Conjugate Gradient-Fast Fourier Transform (CG-FFT) method, a classic Boundary Element Method (BEM) for solving contact problems. On these bases, a cam mechanism is adopted as an example to illustrate the effects of contact behaviors on transmission performance. Through the tolerance analysis, not only the transmission error of cam mechanism is analyzed, but also the contact status and pressure distribution between kinematic pairs can be obtained, which provide references for other aspects such as reliability analysis.

Effect of Torso Non-Homogeneities in the quasi-static inverse problems arising in electrocardiology

Abstract In the present paper, an homogeneous and non-homogeneous inverse problem constrained by the stationary problem in electrocardiology representing the heart, lungs surfaces, and torso model is investigated. Our goal is to reconstruct the electrical potentials on the surface of the heart from the information obtained non invasively on the torso surface. The existence and uniqueness of solution for the heart-torso problem and the related inverse problem is assessed, and the primal and dual problems are discretized using a finite element method. We present some preliminary numerical experiments using an efficient implementation of the proposed scheme in homogeneous and non-homogeneous cases. Finally, we demonstrate the effect of the non-homogeneity on the reconstructed epicardial potential and show that the inverse ECG problem cannot be solved by the classical BEM (boundary element method).

A reduced‐order extrapolated natural boundary element method based on POD for the 2D hyperbolic equation in unbounded domain

In this article, we primarily focuses to study the order‐reduction for the classical natural boundary element (NBE) method for the two‐dimensional (2D) hyperbolic equation in unbounded domain. To this end, we first build a semi‐discretized format about time for the hyperbolic equation and discuss the existence, stability, and convergence of the time semi‐discretized solutions. We then establish the classical fully discretized NBE format from the time semi‐discretized one and analyze the existence, stability, and convergence of the classical NBE solutions. Next, using proper orthogonal decomposition method, we build a reduced‐order extrapolated NBE (ROENBE) format containing very few unknowns but having adequately high accuracy, and we also discuss the existence, stability, and convergence of the ROENBE solutions. Finally, we use some numerical examples to show that the ROENBE method is far superior to the classical NBE one. It shows that the ROENBE method is reliable and effective for solving the 2D hyperbolic equation with the unbounded domain.

A tailored fast multipole boundary element method for viscous layer mesh generation

In this study, the fast multipole boundary element method (BEM) is tailored to solve Laplacian equations, aimed at providing smoothed marching directions for the generation of boundary layer elements and overcoming the performance bottleneck in applying classic BEM in this process. A set of speeding-up techniques are thus developed to enable the development of a fast enough viscous mesh generator. The first technique exploits different levels of parallelism within the FMM to fuel the efficient execution of the method on vastly available multi-core computers. The second technique reschedules the marching-direction computing procedure such that overlapped computations are treated only once. The third technique is built on the detailed experimental studies on two parameters of the FMM such that a good trade-off could be achieved between computing efficiency and accuracy. Applying the above techniques as a whole, the developed mesh generator could now treat real configurations at a comparable speed as that of state-of-the-art commercial tools but provide elements with much higher quality. This result convinces us, if conquering the bottleneck in terms of computing time, the BEM is extremely useful for the development of a wide range of meshing techniques.

Integrating form errors and local surface deformations into tolerance analysis based on skin model shapes and a boundary element method

Tolerance analysis aims to ensure the performance of mechanical products. However, traditional analysis methods largely ignore the effects of form errors and local surface deformations, which are inevitably introduced during manufacturing and assembly phases. Therefore, this paper proposes a novel tolerance analysis framework integrating form errors and local surface deformations based on skin model shapes and a Boundary Element Method (BEM) for rectangular surfaces. First, it introduces a method for modeling skin model shapes considering form errors for rectangular surfaces. Next, it presents the calculation of local surface deformations based on the Conjugate Gradient-Fast Fourier Transform (CG-FFT) method, a classic BEM method for solving contact problems. Together, these can be used to obtain the relative positioning of mating surfaces considering the effects of form errors and local surface deformations. Case studies revealed the considerable effects of form errors and local surface deformations on assembly deviations, and demonstrated the efficiency of the proposed tolerance analysis process. Because the proposed tolerance analysis framework couples geometric deviations introduced during the manufacturing phase with mechanical deformations introduced during the assembly phase, it is expected to help control the geometric deviations of mechanical assemblies more accurately and also ensure product performance.

Tolerance analysis of annular surfaces considering form errors and local surface deformations

This paper proposes a tolerance analysis method of annular surfaces considering form errors and local surface deformations. Specifically, form errors are properly considered based on skin model shapes. Local surface deformations are calculated based on a Conjugate Gradient-Fast Fourier Transform (CG-FFT) method, which is a classic Boundary Element Method (BEM). Based thereon, tolerance analysis of annular surfaces is conducted based on Monte Carlo simulation, and efficiency of the process is also demonstrated.

РОЗРОБКА АЛГОРИТМІВ ТА ПРОГРАМНИХ КОМПОНЕНТІВ МОДЕЛЮВАННЯ КВАЗІСТАЦІОНАРНИХ МАГНІТНИХ ПОЛІВ

The magnetometric section of the marine areas geophysics is characterized by the transition to a qualitatively new level of research, pipeline tracing, search for sunken ships, aerial bombs and unexploded ordnance, study of sunken objects of cultural heritage, identification of such sunken objects (inverse magnetometry problem), etc. The identification of such objects is called the inverse problem of marine magnetometry, and it belongs to the class of incorrectly posed task. The first step in solving this task is to find solution of the direct problem. A direct problem is to determine the anomalous field created by objects of a certain shape, density under certain conditions of immersion and accumulation of information about these objects. Problems of determining the anomalous field of a thin-walled objects are of particular interest. The solution to this problem is largely determined by the existence of an effective system of programs for solving three-dimensional direct magnetometry problems – finding the fields of typical objects (by using sets of such objects you can approximate real objects). The article describes the operation of the main algorithm and subroutines that are part of the main algorithm of the developed MBEM software for modeling the magnetic fields of thin-walled structures. MBEM is based on the developed modified boundary element method, based on the classical boundary element method and the secondary source method. Numerical verification has been carried out and the efficiency of the developed algorithms and software components has been established on the problems of calculating the magnetic field of thin-walled structures for which solutions are known. The example of the software complex work is given – the calculation of the resulting field of a thin-walled pipe, namely the simulation of the tension vector in the plane that is under the object at a distance of one half meter.

Application of a Precoated Permeable Layer to a Pipeline Partially Buried in a Porous Seabed

AbstractThe flow characteristics around a pipeline partially buried in a rigid porous seabed of infinite depth are numerically examined using the classical boundary-element method. A simplified mod...

An improved axisymmetric convected boundary element method formulation with uniform flow

This paper presents an improved form of the convected Boundary Element Method (BEM) for axisymmetric problems in a subsonic uniform flow. The proposed formulation based on the axisymmetric Convected Helmholtz Equation (CHE) and its fundamental solution that describes the sound radiation from a monopole source. The variables in the new axisymmetric boundary integral formulation can be expressed explicitly in terms of the acoustic pressure and its particular normal derivative. Also, the constant coefficients are expressed only in terms of the axisymmetric convected Green's function and its convected normal derivative. The particular and convected derivatives reduce the flow effects of the normal and the flow direction derivatives incorporated in the conventional convected boundary integral formulas. The advanced form of the axisymmetric boundary integral representation with flow is a similar form of the axisymmetric boundary element method without flow. Precisely, the two new operators significantly reduce the computational burden of the classical BEM and then becomes the CPU time of BEM without flow. The formula is verified comparing to both analytical and Finite Element Methods (FEM) of an axisymmetric infinite rigid duct in a subsonic uniform flow.