A coupled symmetric BE–FE method for acoustic fluid–structure interaction

A coupled symmetric BE–FE method for the calculation of linear acoustic fluid–structure interaction in time and frequency domain is presented. In the coupling formulation a newly developed hybrid boundary element method (HBEM) will be used to describe the behaviour of the compressible fluid. The HBEM is based on Hamilton's principle formulated with the velocity potential. The state variables are separated into boundary variables which are approximated by piecewise polynomial functions and domain variables which are approximated by a superposition of static fundamental solutions. The domain integrals are eliminated, respectively, replaced by boundary integrals and a boundary element formulation with a symmetric mass and stiffness matrix is obtained as result. The structure is discretized by FEM. The coupling conditions fulfil C¹-continuity on the interface. The coupled formulation can also be used for eigenfrequency analyses by transforming it from time domain into frequency domain.     

A new visco- and elastodynamic time domain Boundary Element formulation

The usual time domain Boundary Element Method (BEM) contains fundamental solutions which are convoluted with time-dependent boundary data and integrated over the boundary surface. If the fundamental solution is known, e.g., in Elastodynamics, the temporal convolution can be performed analytically when the boundary data are approximated by polynomial shape functions in time and in the boundary elements. This formulation is well known, but the resulting time-stepping BEM procedure produces instabilities and high numerical damping, when the time step size is chosen too small and too large, respectively. Moreover, in case of viscoelastic or poroelastic domains, the fundamental solution is known only in the frequency domain such that the time history of a response can only be obtained by an inverse transformation of the frequency domain results. Here, a new approach for the evaluation of the convolution integrals, the so-called “Operational Quadrature Methods” developed by LUBICH, is presented. In this formulation, the convolution integral is numerically approximated by a quadrature formula whose weights are determined by the Laplace transform of the fundamental solution and a linear multistep method. Hence, the frequency domain fundamental solution can be used without the need of an inverse transformation. Therefore, the extension to viscoelastic problems succeeds using the elastic-viscoelastic correspondence principle.     

A symmetric and positive definite variational BEM for 2-D free vibration analysis

A general BEM model for structural dynamics is derived by using a symmetric and positive definite variational formulation. The functional employed involves domain displacements and boundary tractions and displacements. These variables are taken to be independent of one another. The boundary variables are expressed in terms of their nodal values while the domain displacement field is approximated by a linear combination of static fundamental solutions. The source point of the latter is located outside the domain. The resolving system is a linear system and for free vibration a classic linear algebraic eigenvalue problem is inferred. The stiffness and mass matrices are symmetric and positive definite and the domain integral, when associated with the inertial term, can be transformed into a boundary integral. Numerical results are presented to prove the efficiency of the method.     

A regular variational boundary model for free vibrations of magneto-electro-elastic structures

In this paper a regular variational boundary element formulation for dynamic analysis of two-dimensional magneto-electro-elastic domains is presented. The method is based on a hybrid variational principle expressed in terms of generalized magneto-electro-elastic variables. The domain variables are approximated by using a superposition of weighted regular fundamental solutions of the static magneto-electro-elastic problem, whereas the boundary variables are expressed in terms of nodal values. The variational principle coupled with the proposed discretization scheme leads to the calculation of frequency-independent and symmetric generalized stiffness and mass matrices. The generalized stiffness matrix is computed in terms of boundary integrals of regular kernels only. On the other hand, to achieve meaningful computational advantages, the domain integral defining the generalized mass matrix is reduced to the boundary through the use of the dual reciprocity method, although this implies the loss of symmetry. A purely boundary model is then obtained for the computation of the structural operators. The model can be directly used into standard assembly procedures for the analysis of non-homogeneous and layered structures. Additionally, the proposed approach presents some features that place it in the framework of the weak form meshless methods. Indeed, only a set of scattered points is actually needed for the variable interpolation, while a global background boundary mesh is only used for the integration of the influence coefficients. The results obtained show good agreement with those available in the literature proving the effectiveness of the proposed approach.     

Axial symmetric stationary heat conduction analysis of non-homogeneous materials by triple-reciprocity boundary element method

The heat conduction problems in homogeneous media can be easily solved by the boundary element method. The spatial variations of heat sources as well as material coefficients gives rise to domain integrals in integral formulations for solution of boundary value problems in functionally gradient materials (FGM), since the fundamental solutions are not available for partial differential equations with variable coefficients, in general. In this paper, we present the development of the triple reciprocity method for solution of axial symmetric stationary heat conduction problems in continuously non-homogeneous media with eliminating the domain integrals. In this method, the spatial variations of domain “sources” are approximated by introducing new potential fields and using higher order fundamental solutions of the Laplace operator.     

Non-singular symmetric boundary element formulation for elastodynamics

In this paper, a non-singular boundary element formulation for 3D-elastostatics and 3D-elastodynamics is presented. The proposed method is based on a generalized variational principle. A weighted superposition of static fundamental solutions is used for the field approximation in the domain, whereas the displacement and stress field on the boundary are interpolated by well-known polynomial shape functions. By separating time- and space-dependence a symmetric equation of motion is derived with time-independent mass and stiffness matrix. The domain integral over inertia terms, leading to the mass matrix, is analytically transformed to the boundary. Thus, a boundary only formulation is derived. Comparing numerical results with analytical solutions clearly shows that the obtained system of equations is well-suited for dynamic problems.     

A meshless hybrid boundary-node method for Helmholtz problems

By coupling the moving least squares (MLS) approximation with a modified functional, the hybrid boundary node-method (hybrid BNM) is a boundary-only, truly meshless method. Like boundary element method (BEM), an initial restriction of the present method is that non-homogeneous terms accounting for effects such as distributed loads are included in the formulation by means of domain integrals, and thus make the technique lose the attraction of its ‘boundary-only’ character.This paper presents a new boundary-type meshless method dual reciprocity-hybrid boundary node method (DR-HBNM), which is combined the hybrid BNM with the dual reciprocity method (DRM) for solving Helmholtz problems. In this method, the solution of Helmholtz problem is divided into two parts, i.e. the complementary solution and the particular solution. The complementary solution is solved by means of hybrid BNM and the particular one is obtained by DRM. The modified variational formulation is applied to form the discrete equations of hybrid BNM. The MLS is employed to approximate the boundary variables, while the domain variables are interpolated by fundamental solutions. The domain integration is interpolated by radial basis function (RBF). The proposed method in the paper retains the characteristics of the meshless method and BEM, which only requires discrete nodes constructed on the boundary of a domain, several nodes in the domain are needed just for the RBF interpolation. The parameters that influence the performance of this method are studied through numerical examples and known analytical fields. Numerical results for the solution of Helmholtz equation show that high convergence rates and high accuracy are achievable.     

Wave propagation in cracked elastic slabs and half-space domains—TBEM and MFS approaches

In this paper, the traction boundary element method (TBEM) and the method of fundamental solutions (MFS), formulated in the frequency domain, are used to evaluate the 3D scattered wave field generated by 2D empty cracks embedded in an elastic slab and a half-space. Both models overcome the thin-body difficulty posed when the classical BEM is applied.The crack exhibits arbitrary cross section geometry and null thickness. In neither model are the horizontal formation surfaces discretized, since appropriate fundamental solutions are used to take them into consideration.The TBEM models the crack as a single line. The singular and hypersingular integrals that arise during the TBEM model's implementation are computed analytically, which overcomes one of the drawbacks of this formulation. The results provided by the proposed TBEM model are verified against responses provided by the classical BEM models derived for the case of an empty cylindrical circular cavity.The MFS solution is approximated in terms of a linear combination of fundamental solutions, generated by a set of virtual sources simulating the scattered field produced by the crack, using a domain decomposition technique. To avoid singularities, these fictitious sources are not placed close to the crack, and the use of an enriched function to model the displacement jumps across the crack is unnecessary.The performances of the proposed models are compared and their limitations are shown by solving the case of a C-shaped crack embedded in an elastic slab and a half-space domain.The applicability of these formulations is illustrated by presenting snapshots from computer animations in the time domain for an elastic slab containing an S-shaped crack, after applying an inverse Fourier transformation to the frequency domain computations.     

A hybrid boundary node method

A new variational formulation for boundary node method (BNM) using a hybrid displacement functional is presented here. The formulation is expressed in terms of domain and boundary variables, and the domain variables are interpolated by classical fundamental solution; while the boundary variables are interpolated by moving least squares (MLS). The main idea is to retain the dimensionality advantages of the BNM, and get a truly meshless method, which does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regular shaped domains (in general, semi‐sphere in the 3‐D problem) and their boundaries. Numerical examples presented in this paper for the solution of Laplace's equation in 2‐D show that high rates of convergence with mesh refinement are achievable, and the computational results for unknown variables are most accurate. No further integrations are required to compute the unknown variables inside the domain as in the conventional BEM and BNM. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

Analytical treatment of boundary integrals in direct boundary element analysis of plate bending problems

A direct-type Boundary Element Method (BEM) for the analysis of simply supported and built-in plates is employed. The integral equations due to a combined biharmonic and harmonic governing equations are first established. The boundary integrals developed are then evaluated analytically. The domain integrals due to external body forces are also transformed over the boundary and subsequently evaluated analytically. Thus, it needs only the boundary to be discretized. Without loss of generality, the exact expression for the integrals would enhance the solution accuracy of the BEM. This is due to the fact that at locations where the fundamental solutions approach their singular points the value determined by numerical quadrature may be inconsistent and inaccurate. Also, another major advantage of the exact expressions for integrations is the insensitivity to the geometrical location of the source point on the boundary. The distribution of boundary quantities is approximated either over linear or quadratic boundary elements. General type of plate bending problems, with plates of different geometrical shapes supported simply or fixed can be handled. Loading may be applied point concentrated, uniformly distributed within the domain or over the boundary. Also, hydrostatic pressure can be applied. The results obtained by BEM in comparison with those obtained by analytical or other approximate solutions are found to be very accurate and the solution method is efficient.     

Adaptive fast multipole boundary element method for three-dimensional half-space acoustic wave problems

A new adaptive fast multipole boundary element method (BEM) for solving 3-D half-space acoustic wave problems is presented in this paper. The half-space Green's function is employed explicitly in the boundary integral equation (BIE) formulation so that a tree structure of the boundary elements only for the boundaries of the real domain need to be applied, instead of using a tree structure that contains both the real domain and its mirror image. This procedure simplifies the implementation of the adaptive fast multipole BEM and reduces the CPU time and memory storage by about a half for large-scale half-space problems. An improved adaptive fast multipole BEM is presented for the half-space acoustic wave problems, based on the one developed recently for the full-space problems. This new fast multipole BEM is validated using several simple half-space models first, and then applied to model 3-D sound barriers and a large-scale windmill model with five turbines. The largest BEM model with 557470 elements was solved in about an hour on a desktop PC. The accuracy and efficiency of the BEM results clearly show the potential of the adaptive fast multipole BEM for solving large-scale half-space acoustic wave problems that are of practical significance.     

BIEM-based variational principles for elastoplasticity with unilateral contact boundary conditions

The structural step problem for elastic-plastic internal-variable materials is addressed in the presence of frictionless unilateral contact conditions. Basing on the BIEM (boundary integral equation method) and making use of deformation-theory plasticity (through the backward-difference method of computational plasticity), two variational principles are shown to characterize the solution to the step problem: one is a stationarity principle having as unknowns all the problem variables, the other is a saddle-point principle having as unknowns the increments of the boundary tractions and displacements, along with the plastic strain increments in the domain. The discretization by boundary and interior elements transforms the above principles into well-posed mathematical programming formulations belonging to the symmetric Galerkin BEM formulations (with features such as a symmetric sign-definite coefficient matrix, double integrations, and hypersingular integrals).     

Biharmonic analysis of rectilinear plates by the boundary element method

An improved boundary element formulation (BEM) for two-dimensional non-homogeneous biharmonic analysis of rectilinear plates is presented. A boundary element formulation is developed from a coupled set of Poisson-type boundary integral equations derived from the governing non-homogeneous biharmonic equation. Emphasis is given to the development of exact expressions for the piecewise rectilinear boundary integration of the fundamental solution and its derivatives over several types of isoparametric elements. Incorporation of the explicit form of the integrations into the boundary element formulation improves the computational accuracy of the solution by substantially eliminating the error introduced by numerical quadrature, particularly those errors encountered near singularities. In addition, the single iterative nature of the exact calculations reduces the time necessary to compile the boundary system matrices and also provides a more rapid evaluation of internal point values than do formulations using regular numerical quadrature techniques. The evaluation of the domain integrations associated with biharmonic forms of the non-homogeneous terms of the governing equation are transformed to an equivalent set of boundary integrals. Transformations of this type are introduced to avoid the difficulties of domain integration. The resulting set of boundary integrals describing the domain contribution is generally evaluated numerically; however, some exact expressions for several commonly encountered non-homogeneous terms are used. Several numerical solutions of the deflection of rectilinear plates using the boundary element method (BEM) are presented and compared to existing numerical or exact solutions.     

Improving the performance of the boundary element method with time-dependent fundamental solutions by the use of a wavelet expansion in the time domain

The object of this paper is a wavelet-based formulation of the boundary element method (BEM) for diffusion problems, characterized by time-dependent fundamental solution. While the BEM is a well known and often used technique, its time-dependent formulation for diffusion problems is very rarely used in practical applications, due to the high computational cost which characterizes it. Here, a new formulation is proposed, which, through the use of the wavelet expansion of the time behaviour of the boundary elements, is characterized by a lower CPU time consumption when compared with the standard BEM diffusion formulation. The problem to be solved is transformed into an algebraic system (of higher dimension) and its solution gives the time domain behaviour of the desired quantities; in this way, the time stepping procedure is avoided. Together with the formulation, the analysis of the computational cost, and two examples are given in the paper. Copyright © 2006 John Wiley & Sons, Ltd.     

Transient dynamic elastoplastic analysis by the time-domain BEM formulation

This paper presents a time-domain BEM formulation applied to the solution of transient dynamic elastoplastic problems. The initial stress approach is adopted to solve the elastoplastic problem. Linear time variation is assumed for the displacements and initial stress components whereas traction components are assumed to have a constant time variation. Boundary discretization employs linear elements and the part of the domain where plastic deformation is expected to occur is discretized by employing linear triangular cells. Time integrals are computed analytically, boundary integrals are computed numerically and the domain integrals are computed by following a semi-analytical procedure. A numerical example is presented and the results are compared with another BEM formulation.     

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.     

Green's First Identity Method for Boundary-Only Solution of Self-Weight in BEM Formulation for Thick Slabs

The present paper develops a new technique for treatment of self-weight for building slabs in the boundary element method (BEM). Due to the use of BEM in the analysis, all defined variables are presented on the slab boundary (mesh is defined only along the slab boundary). Self-weight, however, is usually defined over slab domain, hence domain discretisation is required, which spoils the main advantage of the BEM. In this paper a new method is presented to transform self-weight domain integrals to the boundary for such slabs. The proposed method is based on using the so-called Green's first identity. All new kernels for generalized displacements, stress-resultants, and tractions are derived and listed explicitly. The present formulation is implemented into computer code and several examples are tested. Results are compared against results obtained from other numerical method to prove the accuracy and validity of the present formulation.     

Dynamic analysis of shear deformable plates using the Dual Reciprocity Method

The Dual Reciprocity Method is a popular mathematical technique to treat domain integrals in the boundary element method (BEM). This technique has been used to treat inertial integrals in the dynamic thin plate bending analysis using a direct formulation of the BEM based on the elastostatic fundamental solution of the problem. In this work, this approach was applied for the dynamic analysis of shear deformable plates based on the Reissner plate bending theory, considering the rotary inertia of the plate. Three kinds of problems: modal, harmonic and transient dynamic analysis, were analyzed. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed formulation.     

Inelastic transient dynamic analysis of Reissner–Mindlin plates by the D/BEM

A direct domain/boundary element method (D/BEM) for dynamic analysis of elastoplastic Reissner–Mindlin plates in bending is developed. Thus, effects of shear deformation and rotatory inertia are included in the formulation. The method employs the elastostatic fundamental solution of the problem resulting in both boundary and domain integrals due to inertia and inelasticity. Thus, a boundary as well as a domain space discretization by means of quadratic boundary and interior elements is utilized. By using an explicit time‐integration scheme employed on the incremental form of the matrix equation of motion, the history of the plate dynamic response can be obtained. Numerical results for the forced vibration of elastoplastic Reissner–Mindlin plates with smooth boundaries subjected to impulsive loading are presented for illustrating the proposed method and demonstrating its merits. Copyright © 2000 John Wiley & Sons, Ltd.