Boundary Element Calculations Research Articles

Boundary element computation of partially coated bodies using higher order edge elements

A boundary element method in terms of the field variables is applied to eddy current problems with piecewise constant material properties. In particular, the junction problem in the three-dimensional case is considered. We propose higher order edge elements of quadrilateral shape for the field approximation on the surfaces. The Galerkin method is applied to obtain a set of linear equations. The applicability of the proposed edge elements is investigated by the comparison of numerical results with measured data.

THE SOUND RADIATION EFFICIENCY OF FINITE LENGTH ACOUSTICALLY THICK CIRCULAR CYLINDRICAL SHELLS UNDER MECHANICAL EXCITATION I: THEORETICAL ANALYSIS

The acoustic radiation from circular cylindrical shells is of fundamental and applied interest primarily because cylindrical shells are widely used in industry, and because their acoustic behaviour is different from that of beams and plates due to curvature effects. In previous studies of the subject, cylindrical shells have been categorized into acoustically thin and acoustically thick shells in terms of the ratio between the ring frequency frand the critical frequency fC, i.e., fr/fC<1 for acoustically thin shells, and fr/fC>1 for acoustically thick shells. For acoustically thin shells, it has been found by statistical methods that the radiation efficiency has a peak at the ring frequency. Above the ring frequency, the shells behave like flat plates. For acoustically thick shells, especially with finite length, however, the behavior is not so clear. From the analysis in the wavenumber domain, a formula for calculating the modal radiation efficiency of finite length circular cylindrical shells (immersed in light fluid) under mechanical excitation is obtained analytically. Based on this method, the modal-averaged sound radiation efficiencies of acoustically thick circular cylindrical shells are calculated. It is found that unlike acoustically thin shells, the radiation efficiencies of acoustically thick cylindrical shells very much depend on the acoustic behaviour of each individual vibration mode, and thus on the geometries and the boundary conditions. Results obtained by acoustic boundary element calculations and experiments verify these conclusions.

Hybrid discrete vortex method (DVM)/boundary element method (BEM) calculations for cavity acoustics

A hybrid computational methodology has been developed for cavity acoustics applications. The method utilizes a discrete vortex method coupled to an acoustic boundary element calculation. The discrete vortex method uses a Lagrangian evolution in time and space of a field of Gaussian distributed vortex blobs to simulate a two-dimensional, time-dependent, vorticity dominated shear layer. The method yields accurate and fast unsteady solutions to the time-dependent nonlinear flow equations. The cavity acoustics is modeled using acoustic boundary elements which are distributed on the surface of the cavity geometry. The vortex simulation of the shear layer is coupled to the boundary element calculation through Neumann boundary conditions imposed on the cavity surface. The hybrid method simulates a shear layer interacting with a cavity at low Mach number. In example calculations, acoustic radiation and far-field directivity patterns are calculated for the two-dimensional cavity/shear layer system.

Efficient boundary element calculations using thin-plate acoustic elements

Thin-plate elements are commonly used to represent structural components whose thicknesses are much smaller than the structural wavelength. This simplifies the analysis, because otherwise several elements are required to represent correctly the strain distribution through the thickness, leading to much larger models. Thin-plate elements also have advantages when they are used in acoustic boundary element calculations. By using dipole sources at the midline of the elements to generate the basis functions for the acoustic analysis, the boundary condition can be enforced such that the opposite sides of the plate are forced to move together. The surface velocity boundary condition then only has to be enforced over one side of the elements (even though both sides are in contact with the acoustic medium). Mechanical radiation impedance calculations can also be simplified because the dipole source basis functions create the same pressure field on opposite sides of the plate elements and thus impart no force to the structure. The only exception is for ‘‘self-element’’ terms, where the dipole sources create equal, but opposite, surface pressures. Several example problems are used to validate the calculations, including a vibrating strip, a transversely oscillating disk, and a cantilever plate.

Stress intensity factors for internal circumferential cracks in thin- and thick-walled cylinders

This paper addresses the calculation of mode I stress intensity factors for complete circumferential cracks on the inner surface of a hollow cylinder subjected to axisymmetric loading. The analysis based on the weight function method involves two reference sets of data. These are the stress intensity factor and the crack mouth opening displacement for a crack under uniform loading. Boundary-element computations are performed in order to extend the reference data available in the literature to include the cases of thin-walled cylinders and deep cracks. The derived weight function is valid for a wide range of cylinder inner and outer radii, 0.1≤ R i / R o ≤1, and of the relative crack depth, a/t≤0.8. Its application provides the computation of stress intensity factors with an accuracy of about 3%.

Boundary-element calculations of electromagnetic band-structure of photonic crystals

A boundary element method is developed for calculating the photon modes of periodic structures whose unit cells consist of piecewise homogeneous dielectric materials of arbitrary shapes. Green’s function techniques are used to derive integral equations for these structures. These equations involve integrals over the boundaries between the regions, which are discretized and solved numerically. Thus the full set of Maxwell’s equations with boundary conditions in d independent variables is changed into an integral equation in d−1 variables. This allows for the calculation of mode frequencies and field patterns for wave vectors throughout the Brillouin zone, thus allowing the determination of photonic band gaps. An illustrative example is given here for a two-dimensional system.

Electrical Double-Layer Interaction between Heterogeneously Charged Colloidal Particles: A Superposition Formulation

The superposition formula for the electrical double-layer interaction between two uniformly charged spherical particles immersed in an unbounded electrolyte has been used extensively since its formulation some 50 years ago. However, the corresponding result for two heterogeneously charged spherical particles has to date remained elusive. In this paper, we present such analytical results and also consider the special case of a heterogeneously charged spherical particle interacting with a uniformly charged plate. In the sphere–plate case we suggest a simple modification to the superposition result, based on the incorporation of image effects, that can dramatically improve its regime of validity. We assess the validity and accuracy of the superposition formulae by comparison with results of a rigorous boundary element calculation of the electrostatic interaction.

Characterization of the computational accuracy in surface crack problems

The paper focuses on the comparison of different methods for calculating stress intensity factors (KI) in surface crack problems based on results of numerical analyses of elastic crack-tip fields. The computational accuracy is quantified by means of the so-called ‘averaged error estimation technique’ which is extended to the evaluation of local errors in the determination of stress intensity factors at characteristic points of the crack front. Numerical data involved in the present study are obtained from boundary-element calculations. Three values of the stress intensity factor, i.e. those defined from nodal tractions, displacements and energy-release rate, are provided. The highest error level is found for the displacement-based data, while the energy-release calculations yield the best accuracy. A considerable increase in the error value is noticed near the intersection of the crack front with a body surface where the conventional assumption on the square-root stress singularity is, in general, not applied. It is shown that the accuracy of stress intensity factor analysis can be improved by eliminating uncertainties associated with the local stress state along the crack front. © 1998 John Wiley Sons, Ltd.

Reduction of free-edge stress intensities in anisotropic bimaterials

We investigate the free-edge stresses in anisotropic bimaterials through the use of the free-edge stress intensity factor, Kf. This requires a determination of the singularity order at the free-surface as well as a calculation of the near-field stresses. We determine the order of the singularity for arbitrary free-surface orientation of the upper material using an eigenvalue analysis for anisotropic bimaterials. The interfacial stresses are determined using a boundary element calculation based on anisotropic, bimaterial Green's functions. The variation of Kf with free-surface orientation is determined. We find that the free-edge singularity vanishes for certain angles dependent on the anisotropic elastic constants.

Three-dimensional wave-envelope elements of variable order for acoustic radiation and scattering. Part II. Formulation in the time domain

A variable-order, infinite “wave-envelope” element scheme is formulated for transient, unbounded acoustical problems. The transient formulation which is local in space and time is obtained by applying an inverse Fourier transformation to a time-harmonic wave-envelope model whose formulation is described in a companion article. This procedure yields a coupled system of second-order differential equations which can be integrated in time to yield transient pressure histories at discrete nodal points. Far-field transient pressures can also be obtained at adjusted times. The method can be applied quite generally to two-dimensional and three-dimensional problems and is compatible with a conventional finite element model in the near field. The utility of the method is confirmed by the presentation of transient solutions for axisymmetric and fully three-dimensional test problems. An implicit time integration scheme is used and computed results are compared to analytic solutions and to solutions obtained from alternative numerical schemes. Close correspondence is demonstrated and the scheme is shown to be stable for the problems which are presented. CPU times for a large three-dimensional problem are shown to compare favorably with those required for an equivalent transient boundary element computation.

Numerical solution of the Poisson-Boltzmann equation using tetrahedral finite-element meshes

The automatic three-dimensional mesh generation system for molecular geometries developed in our laboratory is used to solve the Poisson–Boltzmann equation numerically using a finite element method. For a number of different systems, the results are found to be in good agreement with those obtained in finite difference calculations using the DelPhi program as well as with those from boundary element calculations using our triangulated molecular surface. The overall scaling of the method is found to be approximately linear in the number of atoms in the system. The finite element mesh structure can be exploited to compute the gradient of the polarization energy in 10–20% of the time required to solve the equation itself. The resulting timings for the larger systems considered indicate that energies and gradients can be obtained in about half the time required for a finite difference solution to the equation. The development of a multilevel version of the algorithm as well as future applications to structure optimization using molecular mechanics force fields are also discussed. © 1997 John Wiley & Sons, Inc. J Comput Chem18: 1591–1608, 1997

Predicting the wind noise from the pantograph cover of a train

Finite element and boundary element calculations are combined to predict the flow noise radiated from a 1/10th-scale model of an aerodynamic cover used around the pantograph on a train at 250 km h−1. The solutions of the unsteady air flow over the cover and the resulting sound propagation are divided into two parts in order to keep the problem tractable. First the unsteady fluid flow is solved using large-eddy simulation (LES). The pressure histories on the cover are then used to predict the radiated sound, using a boundary element method to solve the Helmholtz equation. The result thus leans heavily on assumptions about the coupling of the two solutions, the propagation of sound in a disturbed medium and the efficacy of LES. The predicted sound pressure levels are compared with experimental measurements made in an anechoic wind tunnel. © 1997 John Wiley & Sons, Ltd.

Variation in lithospheric stress along ridge‐transform plate boundaries

Three‐dimensional numerical models of asthenospheric flow and deformation in the oceanic lithosphere predict variability in the stress field that reflects the geometry of the ridge‐transform boundary. A series of 3‐D Boundary Element calculations shows how spreading rate, transform offset and segment length each influence the flow and stress fields that develop during plate‐driven asthenospheric flow beneath a ridge‐transform boundary. The predicted patterns of stress‐supported seafloor relief generally follow those observed: median valleys are predicted at slow‐spreading ridges vs no rift valley or, in some cases, a small axial high at fast spreading ridges; nodal deeps occur at ridge‐transform intersections for offsets greater than 25 km; longer segments have more along‐axis deepening than short segments which do not shoal much in their center. The predicted amplitude of the stress supported topography is up to 20% (across axis) and 40% (along axis) of that observed for an assumed asthenospheric viscosity of 5×1019 Pa s and a weak lithosphere (local compensation).

Parallel processing for boundary element computations on distributed systems

A parallel boundary integral algorithm for solving boundary value problems on distributed memory computer systems is presented in this paper. The paper focuses on parallelizations of the influence coefficients matrix generation and the solution of the resulting linear system, which are the two main parts of boundary integral formulations. The distributed parallel boundary integral algorithm presented in this paper generates a part of the influence coefficients matrix on each computation node of a multicomputer platform and stores that part in the local memory. The distributed influence coefficients matrix is then used in its partitioned form by the parallel linear system solver to obtain the solution. The matrix of coefficients is large, dense and nonsymmetric. Three parallel linear system solvers are presented. Among them, Algorithm-1 and Algorithm-2 are based on the conjugate-gradient-squared (CGS) method, Algorithm-3 is based on a direct method. It is found that distributed memory parallel algorithms are very much machine dependent. The performance of a parallel linear system solver depends not only on the method of solution but also on the multicomputer network topology, and its ratio of node computation speed to network data transfer bandwidth. By selecting the algorithm which fits the characteristics of a distributed computer system hardware one can achieve relative maximum performance. The performance characteristics of distributed linear system solvers are studied in order to select an optimum method for the available hardware at minimum development cost. The performance of sequential linear system solvers which have been extensively documented in the literature are not applicable to distributed memory systems. This paper presents results and discussions on the selection of simple robust algorithms, easily implemented on specific hardware topologies and provides information for extending the selected methods to different parallel machines. The results can also serve as a reference for parallel computer benchmarking.

Time-domain approaches to the edge diffraction problem: Applications to boxed loudspeakers

The classical edge diffraction problem has an exact solution for infinite and truncated wedges. There are no explicit solutions for finite wedges but approximate formulations have been presented, many of which are based on the Kirchhoff diffraction approximation. These methods fail at low frequencies and for certain geometries. A method suggested by Medwin etal . [J. Acoust. Soc. Am. 72, 1005–1013 (1982)] for finite wedges, based on the exact Biot–Tolstoy solution, has proven successful when applied to noise barrierlike geometries. They tentatively propose a method for handling multiple diffraction. In the present paper their method is tested for the application of a boxed loudspeaker, a case where multiple diffraction is clearly evident. Comparisons are made with boundary element calculations. Modifications to Medwin’s method are discussed and other methods, for example, as suggested by Vanderkooy [J. Aud. Eng. Soc. 39, 923–933 (1991)], are discussed as well. Time-domain formulations, such as those discussed here, give insight to the polarity, time delay, and time distribution of the higher-order diffraction components.

Parallel implementation of boundary element method with domain decomposition

Algorithms for the parallel boundary element computation of the domain-decomposed problem are considered in this paper. In this method, boundary conditions on virtual internal boundaries are assumed, and then the boundary element analysis for each subdomain is performed in parallel. The assumed conditions are modified according to the numerical results and the process returns to parallel computation. The Uzawa and the Schwarz methods without domain overlap are applied as alternative schemes on the internal boundary. The methods are implemented on the cluster computing system in order to examine the schemes from the view-point of the convergency of the solution.

Some calculations on the stress distribution in an infinite sheet with a single crack and with periodic collinear cracks

In view of the possible occurrence of crack edge buckling in tension loaded sheets with a central crack, some calculations were made on the stress distribution around a central crack. This has been done for a single crack and for periodic collinear cracks. The latter case was explored to obtain an impression on width effects. Elastic behaviour is assumed without crack edge buckling. A comparison is made with boundary element calculation results from the literature.

Calculation of antenna near field reactions on low conducting materials using the finite element method

On account of the progress in using mobile telecommunications systems in the high frequency range, the human body is increasingly exposed to electromagnetic fields. Therefore, the numerical simulation of the power absorbed in human tissue becomes extremely important in order to avoid thermal destruction. A typical example is the strong interaction between the near field of a receiving antenna of a hand held transceiver and the sensitive organs on the head, like the eyes. A finite element formulation is given for calculating 3D electromagnetic scattering problems. The influence of a dipole excited field on low conducting materials situated very close to the antenna is discussed. The backscattering from the conducting object to the antenna can be treated as well. Along the far boundaries of the computational domain, absorbing boundary conditions of second order have been prescribed in order to avoid non-physical reflections. The formulation described is assessed by applying it to a special problem and the solution obtained is compared with a boundary element calculation.

A fast and Space-efficient boundary element method for computing electrostatic and hydration effects in large molecules

At present, there are two widely used approaches for computing molecular hydration and electrostatic effects within the continuum approximation: the finite difference method, in which the electric potential is directly computed on a cubic grid, and the induced polarization charge or boundary element method, in which an induced charge distribution is first computed on the molecular surface and in which solvation effects are then calculated by reference to the reaction field arising from this induced surface charge. While the induced surface charge approach has a number of advantages over finite differences, especially in the computation of hydration forces and solvent stabilization, the applications of this technique have been largely restricted to small molecules. This is primarily due to the very large system of equations that results when the surface of a macromolecule is discretized into elements small enough to ensure an acceptable level of numerical accuracy within the continuum model. This article describes a new algorithm for implementing boundary element calculations within the continuum model. The essence of our approach is only to compute explicitly those interactions between surface elements that are relatively close together and to approximate long-range interactions by grid-based multipole expansion. The resulting system of equations has a relatively sparse coefficient matrix and requires disk storage that increases linearly with molecular surface area. The technique has numerous applications in the analysis of solvation effects in large molecules, especially in the area of conformational analysis, where it is critical to accurately estimate the global hydration energy for the entire structure. © 1996 by John Wiley & Sons, Inc.

Performance evaluation of optimized diffusors

Theoretical boundary element calculations verified that the 2-D Hankel function is a useful and fast approximation for predicting scattering from one-dimensional extruded shapes. Experimental polar response measurements on 1:5 scale models of QRDs and optimized stepped diffusors proposed by Cox [J. Acoust. Soc. Am. 97, 2928–2936 (1995)] verified the increased diffusivity of the stepped diffusors within the frequency range optimized and for the angle of incidence and observer distance used. However, outside this range and at 30° and 60° angles of incidence, the diffusion response for one and two periods of the stepped diffusor is not optimized. Analysis of a faceted half-cylinder, with better diffusivity than the stepped diffusor, reveals the sensitivity and possible limits of the minimization process. Over a broad bandwidth the response for the QRD is better than that of the stepped diffusor at non-normal angles of incidence.