Direct Boundary Element Method Research Articles

음향방사문제에서 직접경계요소법의 비유일성 회피방법에 관한 고찰

직접경계요소법은 음향방사문제 해석에 널리 적용되는 수치해석기법중 하나이지만, 외부 음향방사 문제에 있어서는 방사면 내부공간의 고유주파수 근방에서 비유일성이 나타나게 된다. 이를 해결하기 위한 방법으로 CHIEF 방법이 일반적으로 적용되어 왔으며, 최근 들어서는 ICA-Ring 방법과 같은 새로운 기법이 제안되어 해의 정확도 향상에 기여하고 있다. 본 논문에서는 앞서 언급한 두 가지 비유일성 회피기법들의 특성을 살피고, 각각의 기법 구현을 포함한 직접경계요소법 기반의 수치해석 코드를 작성한다. 또한, 법선방향 균일속도로 진동하는 구의 음향방사문제에 대한 수치해석을 수행하고 그 결과를 해석 해와 비교하여 각각의 기법들에 대한 비유일성 회피 성능을 고찰한다. A direct boundary element method(DBEM) is widely applied for various acoustic wave problems. But this method has numerically non-unique solutions around the eigenfrequencies of the interior Dirichlet problem for the region enveloped with the acoustic boundary. A CHIEF method had been generally adopted to resolve the non-uniqueness problem and a new technique called ICA-Ring method has been suggested recently. In this paper, the characteristics of two techniques for avoiding the non-uniqueness of DBEM are examined and numerical codes embodying both techniques are developed. Numerical calculations are also carried out for an uniformly pulsating sphere, of which the results are investigated by including the comparisons with theoretical solutions.

Boundary integral equations for plane orthotropic bodies and exterior regions

Assuming linear displacements and constant strains and stresses at infinity, we re-formulate the equations of the direct boundary element method for plane problems of elasticity. We consider a body made of orthotropic material. The reformulated equations make it possible to attack plane problems on exterior regions without replacing the region by a bounded one.

Study on Numerical Methods for Acoustic Radiation of Open Structure

For acoustic radiation of open thin-walled structure, it was difficult to analyze directly by analytical method. The problem could be solved by several numerical methods. This paper had studied the basic theory of the numerical methods as FEM (Finite Element Method), BEM (Boundary Element Method) and IFEM (Infinite Element Method), and the numerical methods to solve open structure radiation problem. Under the premise of structure-acoustic coupling, this paper analyzed the theory and flow of the methods on acoustic radiation of open structure, including IBEM (Indirect Boundary Element Method), DBEM (Direct Boundary Element Method) coupling method of interior field and exterior field, FEM and BEM coupling method, FEM and IFEM coupling method. This paper took the open structure as practical example, and applied the several methods to analyze it, and analyzed and compared the several results to get relevant conclusions.

The evaluation of nearly singular integrals in the direct regularized boundary element method

The numerical analysis of boundary layer effect is one of the major concerned problems in boundary element method (BEM). The accuracy of this problem depends on the precision of the evaluation of t...

An enriched Dual Boundary Element Method for Fracture Mechanics

The direct collocation Boundary Element Method (BEM) has been enriched in the Partition of Unity sense by a set of functions known to describe the displacement field in the immediate vicinity of a crack tip. The Dual BEM approach is adopted to preclude the need for substructuring. The enrichment introduces a small number of additional degrees of freedom, and this therefore requires a corresponding number of auxiliary integral equations. The paper describes the enrichment strategies that may be employed and the results demonstrate that the enrichment provides for approximately one order of magnitude improvement in accuracy in comparison with the conventional, unenriched Dual BEM.

Parallelized fast multipole BEM based on the convolution quadrature method for 3-D wave propagation problems in time-domain

This paper presents a new time-domain boundary element method (BEM) using a convolution quadrature method (CQM) and a fast multipole method (FMM) in 3-D scalar wave propagation. In general, the use of direct time-domain BEM sometimes causes the numerical instability of time-stepping solutions and needs much computational time and memory. To overcome these difficulties, in this paper, the convolution quadrature method developed by Lubich is applied to establish the stability behavior of the time-stepping scheme. Moreover, the fast multipole method and parallelization techniques are adapted to improve the computational efficiency for large size problems. The formulation and numerical implementation of the new boundary element method, and the basic formulas for the fast multipole method such as the multipole expansion, the local expansion, and the translation relations of them in the fast multipole algorithm are presented. The accuracy, the computational efficiency and the applicability are checked by solving 3-D large scale wave scattering problems.

Direct BEM for high-resolution global gravity field modelling

The paper presents a high-resolution global gravity field modelling by the boundary element method (BEM). A direct BEM formulation for the Laplace equation is applied to get a numerical solution of the linearized fixed gravimetric boundary-value problem. The numerical scheme uses the collocation method with linear basis functions. It involves a discretization of the complicated Earth’s surface, which is considered as a fixed boundary. Here 3D positions of collocation points are simulated from the DNSC08 mean sea surface at oceans and from the SRTM30PLUS_V5.0 global topography model added to EGM96 on lands. High-performance computations together with an elimination of the far zones’ interactions allow a very refined integration over the all Earth’s surface with a resolution up to 0.1 deg. Inaccuracy of the approximate coarse solutions used for the elimination of the far zones’ interactions leads to a long-wavelength error surface included in the obtained numerical solution. This paper introduces an iterative procedure how to reduce such long-wavelength error surface. Surface gravity disturbances as oblique derivative boundary conditions are generated from the EGM2008 geopotential model. Numerical experiments demonstrate how the iterative procedure tends to the final numerical solutions that are converging to EGM2008. Finally the input surface gravity disturbances at oceans are replaced by real data obtained from the DNSC08 altimetryderived gravity data. The ITG-GRACE03S satellite geopotential model up to degree 180 is used to eliminate far zones’ interactions. The final high-resolution global gravity field model with the resolution 0.1 deg is compared with EGM2008.

Simulation of stage I-crack growth using a hybrid boundary element technique

Abstract Short fatigue crack propagation often determines the service life of cyclically loaded components and is highly influenced by microstructural features such as grain boundaries. A two-dimensional model to simulate the growth of these stage I-cracks is presented. Cracks are discretised by displacement discontinuity boundary elements and the direct boundary element method is used to mesh the grain boundaries. A superposition procedure couples these different boundary element methods to employ them in one model. Varying elastic properties of the grains are considered and their influence on short crack propagation is studied. A change in crack tip slide displacement determining short crack propagation is observed as well as an influence on the crack path.

Mathematical Approach to Investigate the Behaviour of the Principal Parameters in Axisymmetric Supercavitating Flows, Using Boundary Element Method

ABSTRACTIn this paper, a direct boundary element method (DBEM) is formulated numerically for the problems of the unbounded potential flows past supercavitating bodies of revolution (cones and also disks which are special case of cones with tip vertex angle of 180 degree) at zero degree angle of attack. In the analysis of potential flows past supercavitating cones and disks, a cavity closure model must be employed in order to make the mathematical formulation close and the solution unique. In the present study, we employ Riabouchinsky closure model. Since the location of the cavity surface is unknown at prior, an iterative scheme is used. Where, for the first stage, an arbitrary cavity surface is assumed. The flow field is then solved and by an iterative process, the location of the cavity surface is corrected. Upon convergence, the exact boundary conditions are satisfied on the body-cavity boundary. For this work, powerful software, based on CFD code, is developed in CAE center of IUST. The predictions of the software are compared with those generated by analytical solution and with the experimental data. The predictions of software for supercavitating cones and disks are seen to be excellent. Using the obtained data from software, we investigate the mathematical behavior of axisymmetric supercavitating flow parameters including drag coefficients of supercavitating cones and disks, cavitation number and maximum cavity width for a wide range of cone and disk diameters, cone tip angles and cavity lengths. The main objective of this study is to propose appropriate mathematical functions describing the behavior of these parameters. As a result, among all available functions such as linear, polynomial, logarithmic, power and exponential, only power functions can describe the behavior of mentioned parameters, very well.

Simulation of short crack propagation using a hybrid boundary element technique

AbstractService life of cyclically loaded components is often determined by the propagation of short fatigue cracks, which is highly influenced by microstructural features such as grain boundaries. A two‐dimensional model to simulate the growth of such stage I‐cracks is presented. The crack is discretised by dislocation discontinuity boundary elements and the direct boundary element method is used to mesh the grain boundaries. A superposition procedure couples these different boundary element methods to employ them in one model. Varying elastic properties of the grains are considered and their influence on short crack propagation is studied. A change in crack tip slide displacement determining short crack propagation is observed. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Simulation of Wave Propagation in a Fluid-Filled Borehole Embedded in a Cracked Medium Using a Coupled BEM/TBEM Formulation

Abstract This article simulates the 3D wave propagation in a fluid-filled borehole, embedded in an elastic medium containing an empty crack, when excited by a sonic tool placed inside the borehole. The crack is assumed to be empty with thickness tending toward zero. The geometry of the cross section of the borehole and the crack are assumed to be constant along the borehole axis, allowing the 3D problem to be solved as a summation of 2D responses for different wavenumbers along the z direction (2.5D formulation). The problem is formulated in the frequency domain using a coupled formulation incorporating the traction boundary element method, capable of modeling thin-body geometries, and the conventional direct boundary element method. In this formulation all singular and hypersingular integrals are computed analytically. This article assesses the influence of the length of the crack, its orientation, and its position in relation to the acoustic well in the wave field recorded inside the borehole.

Finite separation method: an efficient boundary element crack modeling technique

In computational fracture mechanics, great benefits are obtained from the reduced modeling dimension order and the accurate integral formulation of the boundary element method (BEM). However, the direct representation of co-planar surfaces (i.e., cracks) causes a degeneration of the standard displacement BEM formulation which can only be circumvented with special modeling techniques. Aiming to simplify the generalized application of the BEM to fracture mechanics problems, this paper presents a two-dimensional crack modeling approach. The method uses the direct BEM displacement formulation within a single-domain model to efficiently and precisely calculate any mixed mode crack tip stress intensity factor. Details of the application of the method are presented, while its accuracy and reliability are demonstrated through numerous comparisons with benchmark results.

Solution of potential flow problems by the modified method of fundamental solutions: Formulations with the single layer and the double layer fundamental solutions

This paper describes an application of the recently proposed modified method of fundamental solutions (MMFS) to potential flow problems. The solution in two-dimensional Cartesian coordinates is represented in terms of the single layer and the double layer fundamental solutions. Collocation is used for the determination of the expansion coefficients. This novel method does not require a fictitious boundary as the conventional method of fundamental solutions (MFS). The source and the collocation points thus coincide on the physical boundary of the system. The desingularised values, consistent with the fundamental solutions used, are deduced from the direct boundary element method (BEM) integral equations by assuming a linear shape of the boundary between the collocation points. The respective values of the derivatives of the fundamental solution in the coordinate directions, as required in potential flow calculations, are calculated indirectly from the considerations of the constant potential field. The normal on the boundary is calculated by parametrisation of its length and the use of the cubic radial basis functions with the second-order polynomial augmentation. The components of the normal are calculated in an analytical way. A numerical example of potential flow around a two-dimensional circular region is presented. The results with the new MMFS are compared with the results of the classical MFS and the analytical solution. It is shown that the MMFS gives better accuracy for the potential, velocity components (partial derivatives of the potential), and absolute value of the velocity as compared with the classical MFS. The results with the single layer fundamental solution are more accurate than the results with the double layer fundamental solution.

A practical examination of the errors arising in the direct collocation boundary element method for acoustic scattering

For many engineers and acousticians, the boundary element method (BEM) provides an invaluable tool in the analysis of complex problems. It is particularly well suited for the examination of acoustical problems within large domains. Unsurprisingly, the widespread application of the BEM continues to produce an academic interest in the methodology. New algorithms and techniques are still being proposed, to extend the functionality of the BEM, and to compute the required numerical tasks with greater accuracy and efficiency. However, for a given global error constraint, the actual computational accuracy that is required from the various numerical procedures is not often discussed. Within this context, this paper presents an investigation into the discretisation and computational errors that arise in the BEM for acoustic scattering. First, accurate routines to compute regular, weakly singular, and nearly weakly singular integral kernels are examined. These are then used to illustrate the effect of the requisite boundary discretisation on the global error. The effects of geometric and impedance singularities are also considered. Subsequently, the actual integration accuracy required to maintain a given global error constraint is established. Several regular and irregular scattering examples are investigated, and empirical parameter guidelines are provided.

Drop deformation in Stokes flow through converging channels

This work presents an application of a direct boundary element method (BEM) formulation for drop deformation and interaction in Stokes flows through converging channels. Parametric studies are conducted to investigate the effect, on drop deformation, of the channel's convergence ratio, the drop-fluid viscosity ratio, the interfacial tension and the initial relative position of the drops.

Comparison of boundary collocation methods for singular and non-singular axisymmetric heat transfer problems

This paper presents a computational study of some boundary collocation solution methods for the Laplace equation in cylindrical coordinates with axisymmetry. The methods compared are (i) the direct boundary element method (BEM), (ii) the method of fundamental solutions (MFS) with fixed sources and (iii) the Trefftz method. Relative accuracy of these methods are compared for two test problems. The first problem is a simple problem of heat transfer through a cylindrical rod which is a standard benchmark problem in this field. The second problem deals with heat transfer in silicon melt for Czochralski (CZ) process which involves a singularity in the boundary conditions at the corner of the crystal–melt interface. All the three methods indicated above are highly successful for the simple (first) problem with MFS and Trefftz being simpler to implement than the BEM. However, the Trefftz method was not effective for the second problem due to the boundary singularity and the MFS showed oscillations near the singularity point. Hence the use of higher order non-conforming elements with accurate Gauss–Kronrod integration schemes in the direct BEM method was investigated. It was found that the boundary singularity does not deteriorate the accuracy of the results if this improved numerical integration procedure is used in the direct BEM. Hence higher order elements with Gauss–Kronrod integration schemes can be used for the solution of many free interface problems encountered in crystal growth.

Analysis of Cracks with Bridged Zone of Large Scale

AbstractABSTRACT. The direct boundary element method is used to analyze of the stress–strain state and fracture parameters of fiber–reinforced and layered composites with bridged interface cracks under transient thermal and mechanical loadings. It is supposed that unbroken fibers make up a bridged zone along of whole or part of a delamination region between layers of composite materials. For modelling purposes a bridged zone is considered as a part of a crack between joined materials and supposed that distributed spring–like bonds with given nonlinear deformation law link and constrain crack surfaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Integrated virtual-design methods for forecasting radiated noise of single cylinder diesel block

The two cycle dynamical results, such as the bearing load, the piston thrust, and the load spatial distribution etc., were obtained by hybrid dynamical simulation of the flexible assembly of the block and cranktrain. The finite element model of the block was validated by modal test. The frequency response of the block was calculated using the finite element method (FEM). Finally, the radiated noise such as sound power level and efficiency in the out sound field were obtained using the direct boundary element method (BEM).

BEM solution of thin film flows on an inclined plane with a bottom outlet

This article considers the numerical analysis of thin films flowing on an inclined plane with an opening (a gap) at the bottom of the plane, representing an outlet. The film is subject to gravitational force and a constant surface shear stress at the film surface. The present problem can be regarded as a special draining-type problem, in which there is a flow bifurcation from one side of the domain (inlet flux) to the outlet and the other side of the domain. The thin film flow is modelled as governed by the two-dimensional Stokes system of equations and an investigation into the effects of surface tension, surface shear stress and geometry variation on the flow of the film into an outlet is attempted. Numerical simulations are obtained in terms of a direct boundary element method (BEM) with primitive variables of velocity and surface traction.

Efficient Extraction of Frequency-Dependent Substrate Parasitics Using Direct Boundary Element Method

An efficient method based on a direct boundary element method is proposed for extracting frequency-dependent substrate coupling parameters. A frequency-independent real-valued linear equation system is first solved. Then, the solution is transformed into frequency-dependent parameters at a specified frequency with the Sherman-Morrison-Woodbury formula. The first step is performed only once for a given structure, and the method is very efficient for extraction with multiple frequencies. The proposed method is compared with the Green's function-based method and the approach in our earlier paper for typical substrate structures. Numerical results demonstrate its accuracy, efficiency, and versatility.