Dual Boundary Element Method Research Articles

Analytical and numerical investigation for true and spurious eigensolutions of an elliptical membrane using the real-part dual BIEM/BEM

The paper performs analytical and numerical investigation of the true and spurious eigensolutions of an elliptical membrane using the real-part boundary integral equation method (BIEM) following the successful work on a circular case by using the dual boundary element method (BEM) (Kuo et al. in Int. J. Numer. Methods Eng. 48:1401–1422, 2000). We extend to the elliptical case in this paper. To analytically study the eigenproblems of an elliptical membrane, the elliptical coordinates and Mathieu functions are adopted. The fundamental solution is expanded into the degenerate kernel by using the elliptical coordinates and the boundary densities are expanded by using the eigenfunction expansion. The Jacobian terms may exist in the degenerate kernel, boundary density and boundary contour integration but they can cancel each other out. Therefore, the orthogonal relations are reserved in the boundary contour integral. It is interesting to find that the BIEM using the real or the imaginary-part kernel to deal with an elliptical membrane yields spurious eigensolutions. This finding agrees with those corresponding to the circular case. The spurious eigenvalues in the real-part BIEM are found to be the zeros of the mth-order (even or odd) modified Mathieu functions of the second kind or their derivatives. To verify this finding, the BEM is implemented. Furthermore, the commercial finite-element code ABAQUS is also utilized to provide eigensolutions for comparisons. It is found that good agreement is obtained.

Dual boundary method for assembled plate structures undergoing large deflection

SUMMARYIn this paper, the dual boundary element method is combined with a multiregion formulation to simulate plate assembly undergoing large deflection. The incremental load approach is used to treat the geometrical nonlinearity, and radial basis functions are used to approximate the derivatives of the large deflection terms. The dual reciprocity method is used to transfer all the domain integrals to the boundary. Once the solution at the boundary is obtained for the assembly, a J −integral for large deflection is implemented to extract the fracture parameters. Copyright © 2011 John Wiley & Sons, Ltd.

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.

Dual BEM analysis of different crack face boundary conditions in 2D magnetoelectroelastic solids

An efficient numerical tool based on the hypersingular formulation of the Boundary Element Method (BEM) for the analysis of different crack face boundary conditions in 2D magnetoelectroelastic media is presented. A new algorithm for the resolution of multiple semipermeable cracks is derived and implemented. The accuracy of the proposed formulation is confirmed by comparison with analytical solutions available in literature. The new results presented in this work may be of interest for the understanding of magnetoelectroelastic cracked solids behavior and the consequent improvements in the design and maintenance of novel devices.

MSD crack propagation by DBEM on a repaired aeronautic panel

► DBEM to investigate damage tolerance performance of repaired aeronautic panels. ► Two-dimensional analysis for efficiently studying varying repair configurations. ► Easy pre- and post-processing, reduced run-times and satisfactory accuracy. ► Run time compatible with the randomization of the crack propagation phenomena. This paper focuses on the use of the Dual Boundary Element Method (DBEM), as implemented in a commercial code (BEASY), to investigate the damage tolerance performance of a riveted repair flat aeronautic panel, realised and tested in the context of the European project “IARCAS” (VI framework). Such panel is assembled in such a way to simulate the in service repairs, with doublers riveted over corresponding cut-out. The panels, repair patches and rivets are modelled in a two-dimensional analysis with no allowance for out-of-plane bending, with edge-cracks initiated from some skin rivet holes and growing due to fatigue load. In the model, the layers representative of each component are overlapped but distinct, providing no allowance for the existing offset. The two-dimensional approximation for this problem is not detrimental to the accuracy of numerical-experimental correlation, so it turn out to be useful to study varying repair configurations, where reduced run times and a lean pre-processing phase are prerequisites.

Improving the stability of time domain dual boundary element method for three dimensional fracture problems: A time weighting approach

This paper deals with the stability of time domain dual boundary element method (DBEM). A time-weighted time domain DBEM is presented in this study and used for the first time in order to improve the stability of the standard time domain dual boundary element method. In this research a time weighting function with a prediction algorithm based on constant velocity algorithm has been utilized. The present approach was tested for three-dimensional fracture problems. The computer cost for the time of the presented approach is very close to the standard form. The results of numerical experiments carried out within this study indicate that the time weighting method, which is suggested for time domain DBEM, has more stability in comparison with the conventional method.

Fracture in magnetoelectroelastic materials using the extended finite element method

AbstractStatic fracture analyses in two‐dimensional linear magnetoelectroelastic (MEE) solids is studied by means of the extended finite element method (X‐FEM). In the X‐FEM, crack modeling is facilitated by adding a discontinuous function and the crack‐tip asymptotic functions to the standard finite element approximation using the framework of partition of unity. In this study, media possessing fully coupled piezoelectric, piezomagnetic and magnetoelectric effects are considered. New enrichment functions for cracks in transversely isotropic MEE materials are derived, and the computation of fracture parameters using the domain form of the contour interaction integral is presented. The convergence rates in energy for topological and geometric enrichments are studied. Excellent accuracy of the proposed formulation is demonstrated on benchmark crack problems through comparisons with both analytical solutions and numerical results obtained by the dual boundary element method. Copyright © 2011 John Wiley & Sons, Ltd.

A BEM model applied to failure analysis of multi-fractured structures

Due to manufacturing or damage process, brittle materials present a large number of micro-cracks which are randomly distributed. The lifetime of these materials is governed by crack propagation under the applied mechanical and thermal loadings. In order to deal with these kinds of materials, the present work develops a boundary element method (BEM) model allowing for the analysis of multiple random crack propagation in plane structures. The adopted formulation is based on the dual BEM, for which singular and hyper-singular integral equations are used. An iterative scheme to predict the crack growth path and crack length increment is proposed. This scheme enables us to simulate the localization and coalescence phenomena, which are the main contribution of this paper. Considering the fracture mechanics approach, the displacement correlation technique is applied to evaluate the stress intensity factors. The propagation angle and the equivalent stress intensity factor are calculated using the theory of maximum circumferential stress. Examples of multi-fractured domains, loaded up to rupture, are considered to illustrate the applicability of the proposed method.

FML full scale aeronautic panel under multiaxial fatigue: Experimental test and DBEM Simulation

This paper concerns the numerical and experimental characterization of the static and fatigue strength of a flat stiffened panel, designed as a Fibre Metal Laminate (FML) and made of aluminium alloy and Fibre Glass FRP. The panel is full scale and was tested under both static and fatigue bi-axial loads, applied by means of an in house designed and built multiaxial fatigue machine. The strain gauge outcomes from a preliminary static test are compared with the corresponding numerical results, getting a satisfactory correlation. A crack propagation in the FML is simulated by a two dimensional original approach based on the Dual Boundary Element Method (DBEM). To overcome the lack of experimental information on the size of delamination area an “inverse” procedure is applied: the delamination introduced in the DBEM model is calibrated in such a way to minimise the numerical and experimental growth rate differences. This approach aims at providing a general purpose evaluation tool for a better understanding of the fatigue resistance of FML panels, providing a deeper insight into the role of fibre stiffness and of delamination extension on the Stress Intensity Factors. The experimental test was realized in the context of a European research project (DIALFAST).

Dual boundary‐element method: Simple error estimator and adaptivity

AbstractThis paper is concerned with the effective numerical implementation of the adaptive dual boundary‐element method (DBEM), for two‐dimensional potential problems. Two boundary integral equations, which are the potential and the flux equations, are applied for collocation along regular and degenerate boundaries, leading always to a single‐region analysis. Taking advantage on the use of non‐conforming parametric boundary‐elements, the method introduces a simple error estimator, based on the discontinuity of the solution across the boundaries between adjacent elements and implements the p, h and mixed versions of the adaptive mesh refinement. Examples of several geometries, which include degenerate boundaries, are analyzed with this new formulation to solve regular and singular problems. The accuracy and efficiency of the implementation described herein make this a reliable formulation of the adaptive DBEM. Copyright © 2011 John Wiley & Sons, Ltd.

Stress Intensity Factor Evaluation of Anisotropic Cracked Sheets under Dynamic Loads Using Energy Domain Integral

The aim of this paper is to present a procedure to perform the evaluation of dynamic stress intensity factors of composite cracked sheets. The numerical method that is used to perform the modeling of the crack is the dual boundary element method. The inertial effects are modeled using the dual reciprocity boundary elements method. The Houbolt Method is used to integrate time, and the energy domain integral is used to evaluate stress intensity factors.

FAST MULTIPOLE BOUNDARY ELEMENT METHOD FOR LOW-FREQUENCY ACOUSTIC PROBLEMS BASED ON A VARIETY OF FORMULATIONS

The fast multipole boundary element method (FMBEM), which is an efficient BEM that uses the fast multipole method (FMM), is known to suffer from instability at low frequencies when the well-known high-frequency diagonal form is employed. In the present paper, various formulations for a low-frequency FMBEM (LF-FMBEM), which is based on the original multipole expansion theory, are discussed; the LF-FMBEM can be used to prevent the low-frequency instability. Concrete computational procedures for singular, hypersingular, Burton-Miller, indirect (dual BEM), and mixed formulations are described in detail. The computational accuracy and efficiency of the LF-FMBEM are validated by performing numerical experiments and carrying out a formal estimation of the efficiency. Moreover, practically appropriate settings for numerical items such as truncation numbers for multipole/local expansion coefficients and the lowest level of the hierarchical cell structure used in the FMM are investigated; the differences in the efficiency of the LF-FMBEM when different types of formulations are used are also discussed.

Multiple random crack propagation using a boundary element formulation

This paper proposes a boundary element method (BEM) model that is used for the analysis of multiple random crack growth by considering linear elastic fracture mechanics problems and structures subjected to fatigue. The formulation presented in this paper is based on the dual boundary element method, in which singular and hyper-singular integral equations are used. This technique avoids singularities of the resulting algebraic system of equations, despite the fact that the collocation points coincide for the two opposite crack faces. In fracture mechanics analyses, the displacement correlation technique is applied to evaluate stress intensity factors. The maximum circumferential stress theory is used to evaluate the propagation angle and the effective stress intensity factor. The fatigue model uses Paris’ law to predict structural life. Examples of simple and multi-fractured structures loaded until rupture are considered. These analyses demonstrate the robustness of the proposed model. In addition, the results indicate that this formulation is accurate and can model localisation and coalescence phenomena.

A fast hierarchical dual boundary element method for three‐dimensional elastodynamic crack problems

AbstractIn this work a fast solver for large‐scale three‐dimensional elastodynamic crack problems is presented, implemented, and tested. The dual boundary element method in the Laplace transform domain is used for the accurate dynamic analysis of cracked bodies. The fast solution procedure is based on the use of hierarchical matrices for the representation of the collocation matrix for each computed value of the Laplace parameter. An ACA (adaptive cross approximation) algorithm is used for the population of the low rank blocks and its performance at varying Laplace parameters is investigated. A preconditioned GMRES is used for the solution of the resulting algebraic system of equations. The preconditioners are built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. An original strategy, based on the computation of some local preconditioners only, is presented and tested to further speed up the overall analysis. The reported numerical results demonstrate the effectiveness of the technique for both uncracked and cracked solids and show significant reductions in terms of both memory storage and computational time. Copyright © 2010 John Wiley & Sons, Ltd.

The Boundary Element Method Applied to Visco-Plastic Analysis

This paper presents a new formulation of the Dual Boundary Element Method to visco-plastic problems in a two-dimensional analysis. Visco-plastic stresses and strains around the crack tip are obtained until the visco-plastic strain rate reaches the steady state condition. A perfect visco-plastic analysis is also carried out in linear strain hardening (H’=0) materials. Part of the domain, the part that is susceptible to yield is discretized into quadratic, quadrilateral continuous cells. The loads are used to demonstrate time effects in the analysis carried out. Numerical results are compared with solution obtained from the Finite Element Method (FEM).

Dual boundary element formulation applied to analysis of multi-fractured domains

This paper deals with analysis of multiple random crack propagation in two-dimensional domains using the boundary element method (BEM). BEM is known to be a robust and accurate numerical technique for analysing this type of problem. The formulation adopted in this work is based on the dual BEM, for which singular and hyper-singular integral equations are used. We propose an iterative scheme to predict the crack growth path and the crack length increment at each time step. The proposed scheme able us to simulate localisation and coalescence phenomena, which is the main contribution of this paper. Considering the fracture mechanics analysis, the displacement correlation technique is applied to evaluate the stress intensity factors. The propagation angle and the equivalent stress intensity factor are calculated using the theory of maximum circumferential stress. Examples of simple and multi-fractured domains, loaded up to the rupture, are considered to illustrate the applicability of the proposed scheme.

A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics

We introduce a novel enriched Boundary Element Method (BEM) and Dual Boundary Element Method (DBEM) approach for accurate evaluation of Stress Intensity Factors (SIFs) in crack problems. The formulation makes use of the Partition of Unity Method (PUM) such that functions obtained from a priori knowledge of the solution space can be incorporated in the element formulation. An enrichment strategy is described, in which boundary integral equations formed at additional collocation points are used to provide auxiliary equations in order to accommodate the extra introduced unknowns. In addition, an efficient numerical quadrature method is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals. Finally, results are shown for mixed mode crack problems; these illustrate that the introduction of PUM enrichment provides for an improvement in accuracy of approximately one order of magnitude in comparison to the conventional unenriched DBEM.

A dual reciprocity BEM approach using new Fourier radial basis functions applied to 2D elastodynamic transient analysis

In this paper, a new boundary element analysis for two-dimensional (2D) transient elastodynamic problems is proposed. The dual reciprocity method (DRM) is reconsidered by employing new radial basis functions (RBFs) to approximate the domain inertia terms. These new RBFs, which are in the form of ζ+ κ sin ( ωr+ α), are called Fourier RBFs hereafter. Using the method of variation of parameters, the particular solution kernels of Fourier RBFs corresponding to displacement and traction, whose a few terms are singular, has been explicitly derived. Therefore, a new simple smoothing trick has been employed to resolve the singularity problem. Moreover, the limiting values of the particular solution kernels have been evaluated. In order to find the unknown parameters of Fourier RBFs, an optimization problem seeking for the optimum value of the Houbolt scheme parameter β that minimizes the mean squared error (MSE) function of the problem is established. Since the MSE function of the proposed RBFs is a function of five unknown parameters (i.e., ζ, κ, ω, α, and β), the genetic algorithm (GA) has been used to solve the necessary optimization problem. In order to illustrate the validity, accuracy, and superiority of the present study, several numerical examples are examined and compared to the results of analytical and other RBFs reported in the literature. Compared to other RBFs, Fourier RBFs show more accurate and stable results. Moreover, these results are obtained using less degree of freedom without any additional internal points that are commonly used to improve the accuracy of the results.

An enriched Dual Boundary Element Method for Fracture Mechanics

The direct collocation Boundary Element Method (BEM) has been enriched in the Partition of Unity sense by a set of functions known to describe the displacement field in the immediate vicinity of a crack tip. The Dual BEM approach is adopted to preclude the need for substructuring. The enrichment introduces a small number of additional degrees of freedom, and this therefore requires a corresponding number of auxiliary integral equations. The paper describes the enrichment strategies that may be employed and the results demonstrate that the enrichment provides for approximately one order of magnitude improvement in accuracy in comparison with the conventional, unenriched Dual BEM.

Development and implementation of some BEM variants—A critical review

Due to rapid development of boundary element method (BEM), this article explores the evolution of BEM over the past half century. We here summarize the overall development and implementation of several well-known BEM variants that includes collocation BEM, galerkin BEM, dual reciprocity BEM, complex variable BEM and analog equation method. Their theoretical and mathematical backgrounds are carefully described and a generalized Laplace’s equation (and Poisson’s equation) is utilized in demonstrating the different approaches involved. An up-to-date review on characteristics and implementation for each of the five variants is presented and also highlighted their significant contributions in boundary element research. In addition, this article tries to cover whole aspect of interests including efficiency, applicability and accuracy in order to give better understanding of BEM evolution. Comparisons and techniques of improvement for these variants are also discussed.