Finite element formulation of exact non‐reflecting boundary conditions for the time‐dependent wave equation

A modified version of an exact Non‐reflecting Boundary Condition (NRBC) first derived by Grote and Keller is implemented in a finite element formulation for the scalar wave equation. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the second‐order local boundary condition derived by Bayliss and Turkel. Two alternative finite element formulations are given. In the first, the boundary operator is implemented directly as a ‘natural’ boundary condition in the weak form of the initial–boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit time‐integration schemes are presented for solution of the finite element semidiscrete equations concurrently with the first‐order differential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the finite element method. The results demonstrate that the finite element formulation of the (modified) NRBC is remarkably robust, and highly accurate. Copyright © 1999 John Wiley & Sons, Ltd.     

The hybrid boundary node method accelerated by fast multipole expansion technique for 3D potential problems

This paper presents a fast formulation of the hybrid boundary node method (Hybrid BNM) for solving problems governed by Laplace's equation in 3D. The preconditioned GMRES is employed for solving the resulting system of equations. At each iteration step of the GMRES, the matrix–vector multiplication is accelerated by the fast multipole method. Green's kernel function is expanded in terms of spherical harmonic series. An oct‐tree data structure is used to hierarchically subdivide the computational domain into well‐separated cells and to invoke the multipole expansion approximation. Formulations for the local and multipole expansions, and also conversion of multipole to local expansion are given. And a binary tree data structure is applied to accelerate the moving least square approximation on surfaces. All the formulations are implemented in a computer code written in C++. Numerical examples demonstrate the accuracy and efficiency of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.     

A SPACE–TIME FINITE ELEMENT METHOD FOR THE EXTERIOR STRUCTURAL ACOUSTICS PROBLEM: TIME-DEPENDENT RADIATION BOUNDARY CONDITIONS IN TWO SPACE DIMENSIONS

A time-discontinuous Galerkin space–time finite element method is formulated for the exterior structural acoustics problem in two space dimensions. The problem is posed over a bounded computational domain with local time-dependent radiation (absorbing) boundary conditions applied to the fluid truncation boundary. Absorbing boundary conditions are incorporated as ‘natural’ boundary conditions in the space–time variational equation, i.e. they are enforced weakly in both space and time. Following Bayliss and Turkel, time-dependent radiation boundary conditions for the two-dimensional wave equation are developed from an asymptotic approximation to the exact solution in the frequency domain expressed in negative powers of a non-dimensional wavenumber. In this paper, we undertake a brief development of the time-dependent radiation boundary conditions, establishing their relationship to the exact impedance (Dirichlet-to-Neumann map) for the acoustic fluid, and characterize their accuracy when implemented in our space–time finite element formulation for transient structural acoustics. Stability estimates are reported together with an analysis of the positive form of the matrix problem emanating from the space–time variational equations for the coupled fluid-structure system. Several numerical simulations of transient radiation and scattering in two space dimensions are presented to demonstrate the effectiveness of the space–time method.     

Plane stress and plate bending coupling in BEM analysis of shallow shells

In this paper the shear deformable shallow shells are analysed by boundary element method. New boundary integral equations are derived utilizing the Betti's reciprocity principle and coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Two techniques, direct integral method (DIM) and dual reciprocity method (DRM), are developed to transform domain integrals to boundary integrals. The force term is approximted by a set of radial basis functions. Several examples are presented to demonstrate the accuracy of the two methods. The accuracy of results obtained by using boundary element method are compared with exact solutions and the finite element method. Copyright © 2000 John Wiley & Sons, Ltd.     

The solution of elastostatic and dynamic problems using the boundary element method based on spherical Hankel element framework

In this paper, new spherical Hankel shape functions are used to reformulate boundary element method for 2‐dimensional elastostatic and elastodynamic problems. To this end, the dual reciprocity boundary element method is reconsidered by using new spherical Hankel shape functions to approximate the state variables (displacements and tractions) of Navier's differential equation. Using enrichment of a class of radial basis functions (RBFs), called spherical Hankel RBFs hereafter, the interpolation functions of a Hankel boundary element framework has been derived. For this purpose, polynomial terms are added to the functional expansion that only uses spherical Hankel RBF in the approximation. In addition to polynomial function fields, the participation of spherical Bessel function fields has also increased robustness and efficiency in the interpolation. It is very interesting that there is no Runge phenomenon in equispaced Hankel macroelements, unlike equispaced classic Lagrange ones. Several numerical examples are provided to demonstrate the effectiveness, robustness and accuracy of the proposed Hankel shape functions and in comparison with the classic Lagrange ones, they show much more accurate and stable results.     

Infinite elements for wave problems: a review of current formulations and an assessment of accuracy

Infinite element schemes for unbounded wave problems are reviewed and a procedure is presented forassessing their performance. A general computational scheme is implemented in which orthogonal functions are used for the transverse interpolation within the infinite element region. This is used as a basis for numerical studies of the effectiveness of various combinations of the radial test and trial functions which give rise to different conjugated and unconjugated formulations. Results are presented for the test case of a spherical radiator to which infinite elements are directly attached. Accuracy of the various schemes is assessed for pure multipole solutions of arbitrary order. Previous studies which have indicated that the conjugated and unconjugated schemes are more effective in the far and near fields, respectively, are confirmed by the current results. All of the schemes tested converge to the exact solution as radial order increases. All are however susceptible to ill conditioning. This places practical restrictions on their effectiveness at high radial orders. A close relationship is demonstrated between the discrete equations which arise from first‐order infiniteelement schemes and those derived from the application of more traditional, local non‐reflecting boundary conditions. Copyright © 2000 John Wiley & Sons, Ltd.     

Efficient preconditioners for boundary element matrices based on grey‐box algebraic multigrid methods

This paper is concerned with the iterative solution of the boundary element equations arising from standard Galerkin boundary element discretizations of first‐kind boundary integral operators of positive and negative order. We construct efficient preconditioners on the basis of so‐called grey‐box algebraic multigrid methods that are well adapted to the treatment of boundary element matrices. In particular, the coarsening is based on an auxiliary matrix that represents the underlying topology in a certain sense. This auxiliary matrix is additionally used for the construction of the smoothers and the transfer operators. Finally, we present the results of some numerical studies that show the efficiency of the proposed algebraic multigrid preconditioners. Copyright © 2003 John Wiley & Sons, Ltd.     

A family of second‐order boundary dampers for finite element analysis of two and three‐dimensional problems in transient exterior acoustics

Numerical modelling of exterior acoustics problems involving infinite medium requires truncation of the medium at a finite distance from the obstacle or the structure and use of non‐reflecting boundary condition at this truncation surface to simulate the asymptotic behaviour of radiated waves at far field. In the context of the finite element method, Bayliss–Gunzburger–Turkel (BGT) boundary conditions are well suited since they are local in both space and time. These conditions involve ‘damper’ operators of various orders, which work on acoustic pressure p and they have been used in time harmonic problems widely and in transient problems in a limited way. Alternative forms of second‐order BGT operators, which work on ṗ (time derivative of p) had been suggested in an earlier paper for 3D problems but they were neither implemented nor validated. This paper presents detailed formulations of these second‐order dampers both for 2D and 3D problems, implements them in a finite element code and validates them using appropriate example problems. The developed code is capable of handling exterior acoustics problems involving both Dirichlet and Neumann boundary conditions. Copyright © 2000 John Wiley & Sons, Ltd.     

Improved accuracy for the Helmholtz equation in unbounded domains

Based on properties of the Helmholtz equation, we derive a new equation for an auxiliary variable. This reduces much of the oscillations of the solution leading to more accurate numerical approximations to the original unknown. Computations confirm the improved accuracy of the new models in both two and three dimensions. This also improves the accuracy when one wants the solution at neighbouring wavenumbers by using an expansion in k. We examine the accuracy for both waveguide and scattering problems as a function of k, h and the forcing mode l. The use of local absorbing boundary conditions is also examined as well as the location of the outer surface as functions of k. Connections with parabolic approximations are analysed. Copyright © 2004 John Wiley & Sons, Ltd.     

A Frobenius solution to the scaled boundary finite element equations in frequency domain for bounded media

The scaled boundary finite element method (FEM) is a recently developed semi‐analytical numerical approach combining advantages of the FEM and the boundary element method. Although for elastostatics, the governing homogeneous differential equations in the radial co‐ordinate can be solved analytically without much effort, an analytical solution to the non‐homogeneous differential equations in frequency domain for elastodynamics has so far only been obtained by a rather tedious series‐expansion procedure. This paper develops a much simpler procedure to obtain such an analytical solution by increasing the number of power series in the solution until the required accuracy is achieved. The procedure is applied to an extensive study of the steady‐state frequency response of a square plate subjected to harmonic excitation. Comparison of the results with those obtained using ABAQUS shows that the new method is as accurate as a detailed finite element model in calculating steady‐state responses for a wide range of frequencies using only a fraction of the degrees of freedom required in the latter. Copyright © 2006 John Wiley & Sons, Ltd.     

Three‐dimensional BEM for scattering of elastic waves in general anisotropic media

Based on the full‐space Green's functions, a three‐dimensional time‐harmonic boundary element method is presented for the scattering of elastic waves in a triclinic full space. The boundary integral equations for incident, scattered and total wave fields are given. An efficient numerical method is proposed to calculate the free terms for any geometry. The discretization of the boundary integral equation is achieved by using a linear triangular element. Applications are discussed for scattering of elastic waves by a spherical cavity in a 3D triclinic medium. The method has been tested by comparing the numerical results with the existing analytical solutions for an isotropic problem. The results show that, in addition to the frequency of the incident waves, the scattered waves strongly depend on the anisotropy of the media. Copyright © 2003 John Wiley & Sons, Ltd.     

Trefftz boundary elements—multi‐region formulations

This paper is concerned with an effective numerical implementation of the Trefftz boundary element method, for the analysis of two‐dimensional potential problems, defined in arbitrarily shaped domains. The domain is first discretized into multiple subdomains or regions. Each region is treated as a single domain, either finite or infinite, for which a complete set of solutions of the problem is known in the form of an expansion with unknown coefficients. Through the use of weighted residuals, this solution expansion is then forced to satisfy the boundary conditions of the actual domain of the problem, leading thus to a system of equations, from which the unknowns can be readily determined. When this basic procedure is adopted, in the analysis of multiple‐region problems, proper boundary integral equations must be used, along common region interfaces, in order to couple to each other the unknowns of the solution expansions relative to the neighbouring regions. These boundary integrals are obtained from weighted residuals of the coupling conditions which allow the implementation of any order of continuity of the potential field, across the interface boundary, between neighbouring regions. The technique used in the formulation of the region‐coupling conditions drives the performance of the Trefftz boundary element method. While both of the collocation and Galerkin techniques do not generate new unknowns in the problem, the technique of Galerkin presents an additional and unique feature: the size of the matrix of the final algebraic system of equations which is always square and symmetric, does not depend on the number of boundary elements used in the discretization of both the actual and region‐interface boundaries. This feature which is not shared by other numerical methods, allows the Galerkin technique of the Trefftz boundary element method to be effectively applied to problems with multiple regions, as a simple, economic and accurate solution technique. A very difficult example is analysed with this procedure. The accuracy and efficiency of the implementations described herein make the Trefftz boundary element method ideal for the study of potential problems in general arbitrarily‐shaped two‐dimensional domains. Copyright © 1999 John Wiley & Sons, Ltd.     

Weighted radial basis collocation method for boundary value problems

This work introduces the weighted radial basis collocation method for boundary value problems. We first show that the employment of least‐squares functional with quadrature rules constitutes an approximation of the direct collocation method. Standard radial basis collocation method, however, yields a larger solution error near boundaries. The residuals in the least‐squares functional associated with domain and boundary can be better balanced if the boundary collocation equations are properly weighted. The error analysis shows unbalanced errors between domain, Neumann boundary, and Dirichlet boundary least‐squares terms. A weighted least‐squares functional and the corresponding weighted radial basis collocation method are then proposed for correction of unbalanced errors. It is shown that the proposed method with properly selected weights significantly enhances the numerical solution accuracy and convergence rates. Copyright © 2006 John Wiley & Sons, Ltd.     

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

To simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

Sequentially linearizable initial‐boundary value problems for a neo‐hookean cylinder

Abstract

The sequential linearizabilities of some initial‐boundary value problems for the plane‐strain equations of motion of the neo‐Hookean solid are studied. The initial‐boundary value problems are formulated for a cylinder in plane‐strain condition and subjected to shear and radial deformations or tractions on its inner and outer surfaces. The results obtained extend the previous concept about the sequential linearizability of the governing equations for the neo‐Hookean solid to the sequential linearizability of initial‐boundary value problems for the governing equations.     

Testing complete and compact radial basis functions for solution of eigenvalue problems using the boundary element method with direct integration

This paper analyses the performance of the main radial basis functions in the formulation of the Boundary Element Method (DIBEM). This is an alternative for solving problems modeled by non-adjoint differential operators, since it transforms domain integrals in boundary integrals using radial basis functions. The solution of eigenvalue problem was chosen to performance evaluation. Natural frequencies are calculated numerically using several radial functions and their accuracy is evaluated by comparison with the available analytical solutions and with the Finite Element Method as well. The standard radial basis functions have presented similar performance to compact radial functions, being even slightly superior.     

A coding error elimination procedure for verifying finite element programs with forced and natural boundary conditions

The general procedure developed by Shih for eliminating computer coding errors has been both simplified for finite element applications and extended for use in the verification of the coding of forced and natural boundary conditions. To accomplish these two objectives simultaneously, use is made of volume weighted residuals for all elements within the solution regime and of area weighted boundary residuals for all elements having natural boundary conditions on one or more of their borders. This procedure can thus be used to verify finite element codes with a combination of both forced (Dirichlet) and natural (Neumann and Robbins) boundary conditions. Depending on the equations being solved and the boundary conditions they are being subjected to, this modification can make possible the implementation of Shih's general procedure into existing finite element codes without having to carry out any additional discretization steps. The employment of this modified procedure is illustrated using two‐dimensional (2‐D) stress analysis and transient heat conduction problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Three-Dimensional Free Vibration Analysis of Sandwich FGM Cylinders with Combinations of Simply-Supported and Clamped Edges and Using the Multiple Time Scale and Meshless Methods

An asymptotic meshless method using the differential reproducing kernel (DRK) interpolation and multiple time scale methods is developed for the three-dimensional (3D) free vibration analysis of sandwich functionally graded material (FGM) circular hollow cylinders with combinations of simply-supported and clamped edge conditions. In the formulation, we perform the mathematical processes of nondimensionalization, asymptotic expansion and successive integration to obtain recurrent sets of motion equations for various order problems. Classical shell theory (CST) is derived as a first-order approximation of the 3D elasticity theory, and the motion equations for higher-order problems retain the same differential operators as those of CST, although with different nonhomogeneous terms. Expanding the primary field variables of each order as the Fourier series functions in the circumferential direction, and interpolating these in the axial direction using the DRK interpolation, we can obtain the leading-order solutions of this analysis. The higher-order modifications can be obtained in a systematic manner, in which the solvability and normality conditions are used to eliminate secular terms and uniquely determine these modifications. Some 3D solutions of the natural frequencies of sandwich FGM cylinders and their corresponding through-thickness distributions of modal variables are given to demonstrate the performance of the asymptotic DRK-based meshless method.     

High‐order Higdon‐like boundary conditions for exterior transient wave problems

Recently developed non‐reflecting boundary conditions are applied for exterior time‐dependent wave problems in unbounded domains. The linear time‐dependent wave equation, with or without a dispersive term, is considered in an infinite domain. The infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC) is imposed on ??. Then the problem is solved numerically in the finite domain bounded by ??. The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. We consider here two reformulations: one that involves high‐order derivatives with a special discretization scheme, and another that does not involve any high derivatives beyond second order. The latter formulation is made possible by introducing special auxiliary variables on ??. In both formulations the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. In contrast to previous papers using similar formulations, here the method is applied to a fully exterior two‐dimensional problem, with a rectangular boundary. Numerical examples in infinite domains are used to demonstrate the performance and advantages of the new method. In the auxiliary‐variable formulation long‐time corner instability is observed, that requires special treatment of the corners (not addressed in this paper). No such difficulties arise in the high‐derivative formulation. Published in 2005 by John Wiley & Sons, Ltd.     

Matched interface and boundary (MIB) for the implementation of boundary conditions in high‐order central finite differences

High‐order central finite difference schemes encounter great difficulties in implementing complex boundary conditions. This paper introduces the matched interface and boundary (MIB) method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high‐order central finite difference schemes. To attain arbitrarily high order, the MIB method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions. The proposed approach is extensively validated via boundary value problems, initial‐boundary value problems, eigenvalue problems, and high‐order differential equations. Successful implementations are given to not only Dirichlet, Neumann, and Robin boundary conditions, but also more general ones, such as multiple boundary conditions in high‐order differential equations and time‐dependent boundary conditions in evolution equations. Detailed stability analysis of the MIB method is carried out. The MIB method is shown to be able to deliver high‐order accuracy, while maintaining the same or similar stability conditions of the standard high‐order central difference approximations. The application of the proposed MIB method to the boundary treatment of other non‐standard high‐order methods is also considered. Copyright © 2008 John Wiley & Sons, Ltd.