Boundary Element Method For Problems Research Articles

A high-frequency boundary element method for scattering by a class of multiple obstacles

Abstract We propose a boundary element method for problems of time-harmonic acoustic scattering by multiple obstacles in two dimensions, at least one of which is a convex polygon. By combining a hybrid numerical-asymptotic (HNA) approximation space on the convex polygon with standard polynomial-based approximation spaces on each of the other obstacles, we show that the number of degrees of freedom required in the HNA space to maintain a given accuracy needs to grow only logarithmically with respect to the frequency, as opposed to the (at least) linear growth required by standard polynomial-based schemes. This method is thus most effective when the convex polygon is many wavelengths in diameter and the small obstacles have a combined perimeter comparable to the problem wavelength.

Convergence Properties of Local Defect Correction Algorithm for the Boundary Element Method

Sometimes boundary value problems have isolated regions where the solution changes rapidly. Therefore, when solving numerically, one needs a fine grid to capture the high activity. The fine grid can be implemented as a composite coarse-fine grid or as a global fine grid. One cheaper way of obtaining the composite grid solution is the use of the local defect correction technique. The technique is an algorithm that combines a global coarse grid solution and a local fine grid solution in an iterative way to estimate the solution on the corresponding composite grid. The algorithm is relatively new and its convergence properties have not been studied for the boundary element method. In this paper the objective is to determine convergence properties of the algorithm for the boundary element method. First, we formulate the algorithm as a fixed point iterative scheme, which has also not been done before for the boundary element method, and then study the properties of the iteration matrix. Results show that we can always expect convergence. Therefore, the algorithm opens up a real alternative for application in the boundary element method for problems with localised regions of high activity.

Comparison of finite and boundary element methods in problems of oscillations of a composite shell of revolution with a liquid

The phenomenon of sloshing can be described as the movement of the free surface of a liquid contained in a reservoir under the action of a suddenly applied load. H. Olsen cited the classification of free-surface fluctuations in liquids by identifying three main forms of sloshing: a) longitudinal sloshing, b) vertical sloshing c) rotating sloshing. Sloshing is a phenomenon that is found in a wide range of industrial applications: in containers for storing liquefied gas, fuel tanks of missiles and airplanes, in tanks of cargo tankers. The vibrations of the real tanks are caused by sloshing of the fluid and vibration of the elastic walls. In completely (or almost completely) filled tanks, the free surface cannot experience strong oscillations. This corresponds to the launch of the launch vehicle. However, in further stages of flight, when the level of liquid aggregate falls, the sloshing effect becomes dominant. It was repeatedly noted that powerful sloshing can lead to a violation of the flight trajectory, as happened, for example, during the launch of the Falcon 1 launch vehicle in 2006, 2007 and 2008. The next important problem in the study of the oscillations of the fuel tanks is the study of the associated hydro-elastic oscillations of the fluid interacting with the elastic walls of the tank. New analytical method and computer technology have been developed for analyzing free and forced vibrations of composite fuel tanks of missiles at different stages of flight: during overloads and in microgravity conditions, including sloshing fuel. The proposed method allows us for more accurate analysis of fuel tank oscillations, taking into account the mutual influence of elastic deformations of tank walls and tank filling levels changing during missions and the shape of the free surface of the liquid, the presence of elastic and rigid damping internal partitions, and the change in acceleration of gravity. A mathematical model has been developed for the analysis of fuel sloshing at large amplitudes. The free oscillations of the launch vehicle tanks are considered.

Metric-based anisotropic mesh adaptation for 3D acoustic boundary element methods

This paper details the extension of a metric-based anisotropic mesh adaptation strategy to the boundary element method for problems of 3D acoustic wave propagation. Traditional mesh adaptation strategies for boundary element methods rely on Galerkin discretizations of the boundary integral equations, and the development of appropriate error indicators. They often require the solution of further integral equations. These methods utilize the error indicators to mark elements where the error is above a specified tolerance and then refine these elements. Such an approach cannot lead to anisotropic adaptation regardless of how these elements are refined, since the orientation and shape of current elements cannot be modified.In contrast, the method proposed here is independent of the discretization technique (e.g., collocation, Galerkin). Furthermore, it completely remeshes at each refinement step, altering the shape, size, and orientation of each element according to an optimal metric based on a numerically recovered Hessian of the boundary solution. The resulting adaptation procedure is truly anisotropic and independent of the complexity of the geometry. We show via a variety of numerical examples that it recovers optimal convergence rates for domains with geometric singularities. In particular, a faster convergence rate is recovered for scattering problems with complex geometries.

Fast isogeometric boundary element method based on independent field approximation

An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions. All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations. The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis. For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented. The versatility and accuracy of the proposed methodology are demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.

SIMULATION OF THE ACOUSTIC FIELD OF A HORN LOUDSPEAKER BY THE BOUNDARY ELEMENT–RAYLEIGH INTEGRAL METHOD

The boundary element–Rayleigh integral method (BERIM) is developed for the solution of acoustic problems consisting of a cavity with a single opening. The method is a hybrid of the interior boundary element method (BEM) and the Rayleigh integral method for the solution of the Helmholtz equation. FORTRAN codes for expressing the method for axisymmetric problems and general three-dimensional problems are developed. Both methods are applied to the problem of simulating the acoustic field produced by horn loudspeakers. The results are compared with measured data and the traditional exterior BEM. It is shown that the method can have significant computational advantages over the traditional BEM for problems consisting of an open cavity.

Dual boundary element method for problems of the theory of thin inclusions

We have proposed to apply the dual boundary element method in problems of the theory of thin inclusions. The contact conditions on the boundary of a thin inclusion are considered as jumps of displacements and stresses in the body on the median surface of this defect. Thus, the relations between the unknown discontinuities and average values of the displacements and stresses are a model of inclusions. For rectilinear boundary elements, we have constructed models of inclusions, taking into account the tension, shear, and bending of a thin inclusion. Examples for rectilinear and curved inclusions have been considered. Comparison of the results obtained by the proposed technique with data based on the direct approach shows the efficiency of the proposed method.

Coupling of Element Free Galerkin and Natural Boundary Element Method for Problems in Unbounded Domains

Coupling of Element Free Galerkin and Natural Boundary Element Method for Problems in Unbounded Domains

Modified boundary element method for problems on oscillations of flat membranes

A modified boundary element method (MBEM) for hyperbolic problems is exemplified by solving the problem of oscillations of a flat membrane. The modification of the method consists in the analytical computation of the components of the influence vector; the integration is carried out not over all the components of the boundary but only once over a specially chosen base element.

BOUNDARY ELEMENT METHOD IN PROBLEMS OF WATER-SATURATED SOIL THAT HAS CREEP PROPERTY

The aticle offers a calculation algorithm of the tensed & deformed state (TDS) of the water-saturated soil bases, possessing the quality of «creep», by the boundary element method (MBE). The main difference of the recommended modification of MBE from the generally accepted option is a possibility of forecasting the transformation of TDS of ground bases in time with account of their rheological properties.

A simple boundary element method for problems of potential in non‐homogeneous media

AbstractA simple boundary element method for solving potential problems in non‐homogeneous media is presented. A physical parameter (e.g. heat conductivity, permeability, permittivity, resistivity, magnetic permeability) has a spatial distribution that varies with one or more co‐ordinates. For certain classes of material variations the non‐homogeneous problem can be transformed to known homogeneous problems such as those governed by the Laplace, Helmholtz and modified Helmholtz equations. A three‐dimensional Galerkin boundary element method implementation is presented for these cases. However, the present development is not restricted to Galerkin schemes and can be readily extended to other boundary integral methods such as standard collocation. A few test examples are given to verify the proposed formulation. The paper is supplemented by an Appendix, which presents an ABAQUS user‐subroutine for graded finite elements. The results from the finite element simulations are used for comparison with the present boundary element solutions. Copyright © 2004 John Wiley & Sons, Ltd.

An inverse stress reconstruction algorithm for modelling excavation and thermo-mechanical effects of rock structures

When using boundary element methods for problems with domain effects such as initial/residual stresses, body forces and heating, domain integrals are usually involved. Different numerical algorithms have been developed in the past to deal with these domain integrals, such as interior cell mapping, DRM, MRM, etc., which require considerable code development efforts. To take advantage of the boundary element method (BEM) for simulating fracturing processes, but without additional effort for domain integral calculations or transformations, a hybrid approach is suggested in this paper to simulate the excavation and thermo-mechanical process of rock structures. Equivalent boundary tractions reflecting the combined effects of the initial and redistributed thermo-mechanical stresses in the domain of interest at multiple excavation and heating steps are produced by an algorithm of inverse stress reconstruction, in a sense of least-square optimization. The algorithm utilizes resultant stress fields at each excavation and heating step, either measured or produced by other numerical methods of domain-types such as finite element method/finite difference method (FEM/FDM), and calculates the equivalent boundary tractions for further fracturing analysis. In this paper, the inverse stress reconstruction algorithm developed for this purpose, using a truncated singular value decomposition (SVD) technique, is presented. Three examples of verification, two closed-form solutions of problems of elasticity with simple geometry and one example from a predictive modeling of a planned in situ heating experiment in a rock pillar at the Äspö Hard Rock Laboratory (HRL), Southern Sweden, are presented.

The finite element and boundary element methods in problems of the dynamics of elastic vessels with a liquid

A version of a method of calculating the natural longitudinal vibrations of an elastic axisymmetric vessel, partially filled with liquid, is described. The basis of the method is a combination of boundary elements for the liquid and finite elements for the elastic vessel. Examples which illustrate the effectiveness of the method are given.

Further considerations in modified crack closure integral based computation of stress intensity factor in bem

This paper deals with modified crack closure integral based computation of stress intensity factors in a boundary element method for problems with mechanical loading remote from the crack edges. The modified crack closure integral technique has been coupled with the local smoothing scheme to obtain simple relations for energy release rates for linear, quadratic and quarter point elements around the crack tip. Stress intensity factors calculated through the proposed formulation and the displacement method for a number of examples are compared, wherever possible, with data available in the literature. The results based on the proposed scheme are more accurate than those obtained by the displacement method.

Boundary element methods in problems of thermoelasticity for piecewise-homogeneous bodies

Integral representations of the components of the displacement vector and the stress tensor and the corresponding system of boundary integral equations are derived for a piecewise-homogeneous body. A numerical scheme is developed that allows for the specific behavior of the stress fields in the neighborhood of the corner points of the inclusions. As an example, we consider the thermally stressed state of an elastic half-plane with inclusions of various shapes.

Uncoupled dual formulations of the variational boundary element method in problems of the theory of elasticity

Uncoupled dual formulations (UDFs), different from those considered previously [1–3], are proposed for the boundary functionals of the linear theory of elasticity, in the sense that the displacements and stresses are varied independently, and the equations of state on the boundary are treated as constraints involving Lagrange multipliers. The idea of this device—using Lagrange multipliers to get rid of restrictions in the variational problem, represented by the equations of state—was used previously [4]to formulate dual variational problems of the linear theory of elasticity based on the Lagrange-Castigliano principle. A finite element approximation of the solutions of these problems yields mixed formulations of the finite element method [5]. Thus, the boundary element approximations (BEAs) proposed below for the UDF may be regarded as a special mixed finite element method. Simultaneous BEA of the displacements and stresses extends the applicability of UDFs to cases in which allowance must be made for singularities of the solution, e.g. in contact problems of the theory of elasticity and in fracture mechanics (crack problems).

Computation of stresses and displacements

The problem under consideration involves the elastic analysis of a square plate subjected to a uniform pressure. In the first instance a square section with a central hole is considered using generalized plane strain element. The pressure is applied to the surface of a circular hole located at the centre of the section. This problem is then generalized to that of a square section with nine holes subject to internal pressure. Results involving stresses and displacements are obtained for both cases using the BEASY Boundary Element software package from Computational Mechanics Ltd. A check on the accuracy is obtained by using Lamé thick cylinder theory at selected points. The good agreement obtained gives confidence in the use of the Boundary Element method for problems of this type.

An algorithm based on the boundary element method for problems in engineering mechanics

AbstractThe solution to a two‐dimensional problem using the boundary element method requires the evaluation of a line integral over the boundary. The integrand ot this line integral is a product of a known Green's function and an unknown function. A large number of Green's functions for two‐dimensional problems can be represented by a linear combination of four singular functions. By approximating the unknown function by a linear combination of known polynomials, integrals are generated whose integrand is a product of the polynomiais and one of the four singular functions. To evaluate these integrals analytically, the boundary is approximated by a sum of straight‐line segments. Recursive formulae are established which reduce the generality and the complexity of the integrands to simple expressions. Analytical forms for these simple expressions are found and are used for initiating the algorithm.