Boundary Element Method Formulation Research Articles

An axisymmetric model of ultrasound contrast agent collapse following shell rupture

Widely used spherically symmetric models of ultrasound contrast agents (UCA) are unable to capture complicated collapse behaviors when the UCA shell ruptures. Observations from acoustic passive cavitation detection analysis of collapsing UCAs as well as images from high speed videos of UCA destruction suggest that including spatial asymmetry in the shell interface conditions of models may be useful for studying collapse, onset of fragmentation, and initiation of postexcitation rebound. The concept of lipid shell rupture is incorporated into an axisymmetric boundary element method formulation as a circular hole developing during the growth phase of an initially spherical bubble. The different material interfaces are represented by a spatially varying surface tension due to different pressure discontinuity conditions from the inner gas region to the outer fluid region. Results from the simplified geometrical model simulations demonstrate that shell fragments may influence the evolution of collapsing UCA and indicate the possibility of microbubble jetting following lipid shell rupture. (NIH Grant R37EB002641.)

A dynamic formulation for the analysis of thick elastic plates by the boundary element method

A dynamic formulation for the analysis of thick elastic plates that is based on Reissner's theory is presented. The main objective is to include in the formulation the contribution of a new term of additional translational inertia that is commonly disregarded, and study its influence on the responses of thick elastic plate theory. Moreover, the boundary element method (BEM) formulation used to carry out time-domain analysis employs the static fundamental solution of the problem, while for the time marching the Houbolt's acceleration operator is considered. Thus, a system of equations is solved for the boundary and the domain simultaneously. To verify the importance of the new terms, the present methodology is compared with responses found in the literature.

Initial value problem formulation of time domain boundary element method for electromagnetic microwave simulations

To aim to obtain more stable solutions and wider area applications for the Time Domain Boundary Element Method (TDBEM), initial value problem formulation of the TDBEM is newly introduced for microwave simulations. The initial value problem formulation of the TDBEM allows us to solve transient microwave phenomena as interior region problems, which gives us well matrix property and interior resonance free solutions. This paper concentrates on applying the initial value problem formulation of the TDBEM to wake field phenomena in particle accelerator cavities.

Probabilistic crack growth analyses using a boundary element model: Applications in linear elastic fracture and fatigue problems

This paper addresses the numerical solution of random crack propagation problems using the coupling boundary element method (BEM) and reliability algorithms. Crack propagation phenomenon is efficiently modelled using BEM, due to its mesh reduction features. The BEM model is based on the dual BEM formulation, in which singular and hyper-singular integral equations are adopted to construct the system of algebraic equations. Two reliability algorithms are coupled with BEM model. The first is the well known response surface method, in which local, adaptive polynomial approximations of the mechanical response are constructed in search of the design point. Different experiment designs and adaptive schemes are considered. The alternative approach direct coupling, in which the limit state function remains implicit and its gradients are calculated directly from the numerical mechanical response, is also considered. The performance of both coupling methods is compared in application to some crack propagation problems. The investigation shows that direct coupling scheme converged for all problems studied, irrespective of the problem nonlinearity. The computational cost of direct coupling has shown to be a fraction of the cost of response surface solutions, regardless of experiment design or adaptive scheme considered.

Stochastic spline fictitious boundary element method in elastostatic problems with random fields

Mathematical formulation and computational implementation of the stochastic spline fictitious boundary element method (SFBEM) are presented for the analysis of plane elasticity problems with material parameters modeled with random fields. Two sets of governing differential equations with respect to the means and deviations of structural responses are derived by including the first order terms of deviations. These equations, being in similar forms to those of deterministic elastostatic problems, can be solved using deterministic fundamental solutions. The calculation is conducted with SFBEM, a modified indirect boundary element method (IBEM), resulting in the means and covariances of responses. The proposed method is validated by comparing the solutions obtained with Monte Carlo simulation for a number of example problems and a good agreement of results is observed.

Two‐Dimensional Fracture Mechanics Analysis Using a Single‐Domain Boundary Element Method

This work calculates the stress intensity factors (SIFs) at the crack tips, predicts the crack initiation angles, and simulates the crack propagation path in the two‐dimensional cracked anisotropic materials using the single‐domain boundary element method (BEM) combined with maximum circumferential stress criterion. The BEM formulation, based on the relative displacements of the crack tip, is used to determine the mixed‐mode SIFs and simulate the crack propagation behavior. Numerical examples of the application of the formulation for different crack inclination angles, crack lengths, degree of material anisotropy, and crack types are presented. Furthermore, the propagation path in Cracked Straight Through Brazilian Disc (CSTBD) specimen is numerically predicted and the results of numerical and experimental data compared with the actual laboratory observations. Good agreement is found between the two approaches. The proposed BEM formulation is therefore suitable to simulate the process of crack propagation. Additionally, the anisotropic rock slope failure initiated by the tensile crack can also be analyzed by the proposed crack propagation simulation technique.

OS0715 空気超音波法に対する3次元数値シミュレーション

This paper presents a 3D numerical simulation for air-coupled ultrasonic testing. Air-coupled ultrasonic testing has some advantages because this method is one of non-contact ultrasonic NDE methods. However, the positioning of air-coupled transducers is very sensitive, because the difference of acoustic impedance between air and solid is very large. In order to use this method for evaluation of various materials, it needs to simulate wave propagation in a solid medium. A BEM (boundary element method) is an effective method for wave propagation problems. Therefore, the BEM formulation using the CQM (convolution quadrature method) is developed for air-coupled ultrasonic testing and the numerical results for 3D wave simulation are shown.

Application of the BEM approach for a determination of the regional marine geoid model and the mean dynamic topography in the Southwest Pacific Ocean and Tasman Sea

Application of the BEM approach for a determination of the regional marine geoid model and the mean dynamic topography in the Southwest Pacific Ocean and Tasman SeaWe apply a novel approach for the gravimetric marine geoid modelling which utilise the boundary element method (BEM). The direct BEM formulation for the Laplace equation is applied to obtain a numerical solution to the linearised fixed gravimetric boundary-value problem in points at the Earth's surface. The numerical scheme uses the collocation method with linear basis functions. It involves a discretisation of the Earth's surface which is considered as a fixed boundary. The surface gravity disturbances represent the oblique derivative boundary condition. The BEM approach is applied to determine the marine geoid model over the study area of the Southwest Pacific Ocean and Tasman Sea using DNSC08 marine gravity data. The comparison of the BEM-derived and EGM2008 geoid models reveals that the geoid height differences vary within -25 and 18 cm with the standard deviation of 6 cm. The DNSC08 sea surface topography data and the new marine geoid are then used for modelling of the mean dynamic topography (MDT) over the study area. The local vertical datum (LVD) offsets estimated at 15 tide-gauge stations in New Zealand are finally used for testing the coastal MDT. The average value of differences between the MDT and LVD offsets is 1 cm.

Transient heat conduction under nonzero initial conditions: A solution using the boundary element method in the frequency domain

A boundary element method formulation is proposed to solve the diffusion equation under nonzero initial conditions. The problem is solved in the frequency domain, considering only the conduction phenomenon. Complex frequencies are used to avoid aliasing and to allow the computation of the static response. Two numerical examples are given to illustrate the effectiveness of this approach for solving 2-D diffusion equations.

Analysis of anisotropic Kirchhoff plates using a novel hypersingular BEM

In this article a hypersingular boundary element method (BEM) for bending of thin anisotropic plates is presented. A new complex variable fundamental solution is implemented in the algorithm. For spatial discretization a collocation method with discontinuous quadratic elements is adopted. The domain integrals arising from the transversely applied load are transformed analytically into boundary integrals by means of the radial integration technique. The considered numerical examples prove that the novel BEM formulation presented in this study is much more efficient than previous formulations developed for the analysis of this kind of problems.

BEM modeling of saturated porous media susceptible to damage

This paper deals with the numerical analysis of saturated porous media, taking into account the damage phenomena on the solid skeleton. The porous media is taken into poro-elastic framework, in full-saturated condition, based on Biot’s Theory. A scalar damage model is assumed for this analysis. An implicit boundary element method (BEM) formulation, based on time-independent fundamental solutions, is developed and implemented to couple the fluid flow and two-dimensional elastostatic problems. The integration over boundary elements is evaluated using a numerical Gauss procedure. A semi-analytical scheme for the case of triangular domain cells is followed to carry out the relevant domain integrals. The non-linear problem is solved by a Newton–Raphson procedure. Numerical examples are presented, in order to validate the implemented formulation and to illustrate its efficacy.

Optimal design of tonal noise control inside smart-stiffened cylindrical shells

This paper deals with the abatement of noise inside cylindrical cavities bounded by stiffened shells that are impinged by external tonal sound waves. The problem is analysed in a multidisciplinary context, involving interactions among exterior noise field, elastic shell dynamics, interior acoustics and control system. The actuation in the noise control process is performed through piezoelectric patches embedded in the stiffened shell, driven by a control law obtained from an optimal LQR cyclic control formulation, coupled with a genetic optimization algorithm (GA). A modal approach is applied to describe the smart shell dynamics and the interior acoustics that are coupled in the acoustoelastic plant model, while the exterior acoustic scattering is predicted through a boundary element method formulation. Numerical results deal with an aeronautical problem concerning a general aviation aircraft cabin impinged by the pressure disturbances emitted by a pair of propellers; in particular, the effectiveness and robustness of the active control law synthesized through the GA is investigated.

An ancillary boundary integral equation for magnetostatic analysis

The boundary element method (BEM) has been established as an effective means for magnetostatic analysis. Direct BEM formulations for the magnetic vector potential have been developed over the past 20 years. There is a less well-known direct boundary integral equation (BIE) for the magnetic flux density which can be derived by taking the curl of the BIE for the magnetic vector potential and applying properties of the scalar triple product. On first inspection, the ancillary boundary integral equation for the magnetic flux density appears to be homogeneous, but it can be shown that the equation is well-posed and non-homogeneous using appropriate boundary conditions. In the current research, the use of the ancillary boundary integral equation for the magnetic flux density is investigated as a stand-alone equation and in tandem with the direct formulation for the magnetic vector potential.

Diagonal form fast multipole boundary element method for 2D acoustic problems based on Burton-Miller boundary integral equation formulation and its applications

This paper describes formulation and implementation of the fast multipole boundary element method (FMBEM) for 2D acoustic problems. The kernel function expansion theory is summarized, and four building blocks of the FMBEM are described in details. They are moment calculation, moment to moment translation, moment to local translation, and local to local translation. A data structure for the quad-tree construction is proposed which can facilitate implementation. An analytical moment expression is derived, which is more accurate, stable, and efficient than direct numerical computation. Numerical examples are presented to demonstrate the accuracy and efficiency of the FMBEM, and radiation of a 2D vibration rail mode is simulated using the FMBEM.

Recent Advances and Emerging Applications of the Boundary Element Method

Sponsored by the U.S. National Science Foundation, a workshop on the boundary element method (BEM) was held on the campus of the University of Akron during September 1–3, 2010 (NSF, 2010, “Workshop on the Emerging Applications and Future Directions of the Boundary Element Method,” University of Akron, Ohio, September 1–3). This paper was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts with a brief introduction to the BEM. Then, new developments in Green's functions, symmetric Galerkin formulations, boundary meshfree methods, and variationally based BEM formulations are reviewed. Next, fast solution methods for efficiently solving the BEM systems of equations, namely, the fast multipole method, the pre-corrected fast Fourier transformation method, and the adaptive cross approximation method are presented. Emerging applications of the BEM in solving microelectromechanical systems, composites, functionally graded materials, fracture mechanics, acoustic, elastic and electromagnetic waves, time-domain problems, and coupled methods are reviewed. Finally, future directions of the BEM as envisioned by the authors for the next five to ten years are discussed. This paper is intended for students, researchers, and engineers who are new in BEM research and wish to have an overview of the field. Technical details of the BEM and related approaches discussed in the review can be found in the Reference section with more than 400 papers cited in this review.

Boundary element analysis of nanoinhomogeneities of arbitrary shapes with surface and interface effects

In this paper, a boundary element method (BEM) is proposed to analyze the stress field in nanoinhomogeneities with surface/interface effect. To consider this effect, the continuity conditions along the internal interfaces between the matrix and inhomogeneities are modeled by the well-known Gurtin–Murdoch constitutive relation. In the numerical analysis, the interface elastic moduli and the geometry of the nanoscale inhomogeneity are varied to show their influence on the induced stress field. The interaction between nanoscale inhomogeneities and the effect of different geometric shapes of inhomogeneities, including ellipse, triangle, and square are also investigated for different interface material parameters. It is shown that the elastic field can be greatly influenced by the interfacial energy and geometry of nanoscale inhomogeneities. The proposed BEM formulation is very general, including the complete Gurtin–Murdoch model and is further convenient for arbitrary shapes of inhomogeneity.

Coupling of BEM with a large displacement and rotation algorithm

AbstractIn stability analysis of rock blocks, the deformability of the blocks can conveniently be simulated using the boundary element method (BEM). However, all boundary conditions are given as stresses. Thus, the displacement solution is not unique. In this paper, an algorithm is proposed to remove rigid body motions in the solution of the boundary form of Somigliana identity discretized by the direct BEM formulation. The algorithm is applied to the calculation of the normal stiffness of rock blocks and coupled with BS3D, large displacement and rotation algorithm for the general stability of rock blocks. Copyright © 2010 John Wiley & Sons, Ltd.

A boundary elements formulation for 3D fretting-wear problems

Computational wear modeling is an extremely time-consuming problem, especially the 3D cases. In this work, a 3D boundary element method (BEM) formulation for wear modeling is proposed and applied to simulate 3D fretting-wear problems under gross sliding and partial slip conditions. The present formulation applies the BEM to approximate the elastic response of solids, and an augmented Lagrangian formulation to solve the contact problem. Contact restrictions fulfilment is established by a set of projection functions, and wear on contact surfaces is computed using the Archard wear law. The BEM proves to be a very suitable numerical method for this kind of mechanical interaction problems, considering only the boundary degrees of freedom involved in the problem and obtaining a very good approximation of contact tractions with a low number of elements. This is very interesting in terms of computational cost reductions of wear modeling, specially in 3D problems. In that regard, an acceleration strategy is applied to the proposed algorithm. It allows to obtain very important reductions on wear simulation times. The proposed methodology is therefore an efficient numerical tool for 3D fretting-wear problems modeling.

Biomolecular electrostatics using a fast multipole BEM on up to 512 gpus and a billion unknowns

We present teraflop-scale calculations of biomolecular electrostatics enabled by the combination of algorithmic and hardware acceleration. The algorithmic acceleration is achieved with the fast multipole method ( fmm) in conjunction with a boundary element method ( bem) formulation of the continuum electrostatic model, as well as the bibee approximation to bem. The hardware acceleration is achieved through graphics processors, gpus. We demonstrate the power of our algorithms and software for the calculation of the electrostatic interactions between biological molecules in solution. The applications demonstrated include the electrostatics of protein–drug binding and several multi-million atom systems consisting of hundreds to thousands of copies of lysozyme molecules. The parallel scalability of the software was studied in a cluster at the Nagasaki Advanced Computing Center, using 128 nodes, each with 4 gpus. Delicate tuning has resulted in strong scaling with parallel efficiency of 0.8 for 256 and 0.5 for 512 gpus. The largest application run, with over 20 million atoms and one billion unknowns, required only one minute on 512 gpus. We are currently adapting our bem software to solve the linearized Poisson–Boltzmann equation for dilute ionic solutions, and it is also designed to be flexible enough to be extended for a variety of integral equation problems, ranging from Poisson problems to Helmholtz problems in electromagnetics and acoustics to high Reynolds number flow.

Least squares stochastic Boundary Element Method

The main issue in this paper is mathematical formulation and computational implementation of the stochastic Boundary Element Method based on the generalized stochastic perturbation technique. The key feature is a replacement of the given order polynomial response function with the least squares method leading to a numerical determination of this response function. This new approach minimizes the approximation error during the recovery of the structural response indexed with the random input parameter, which is a decisive factor for the entire stochastic method accuracy; contrary to some lower order techniques, numerical implementation of up to the fourth order probabilistic moments is displayed. Computational experiments obey both analyses for the homogeneous and heterogeneous structures with Gaussian random material parameters and also some comparison against the Monte-Carlo simulation and analytical results.