Galerkin Boundary Element Method Research Articles

Coupling of BEM subdomains – BETI applied to collocation BEM with mixed basis functions

This work is about the coupling of subdomains for elasticity problems using the boundary element tearing and interconnecting method (BETI). BETI is a substructuring technique which is able to solve large problems involving subdomains efficiently, because its formulation is ideally suited for parallel computation. In contrast to the original formulation of BETI, where the symmetric Galerkin boundary element method (SGBEM) is applied, here the traditional collocation BEM is used. As will be shown the solution algorithm is applied in a very similar way as in the original version. Special attention will be paid to the distribution of tractions along interfaces of BEM subdomains. Especially if the coupling interface is not smooth but has corners and edges erroneous traction results can be obtained by using the non-symmetric approximation of the Dirichlet-to-Neumann map. For a sequential calculation involving several load steps this leads to unacceptable results if a classical continuous approximation of the physical fields is used. Therefore discontinuous basis functions for the displacement and traction field are introduced. As will be shown in this work the quality of results can be improved drastically by this technique.

Analysis of planar cracks in 3D elastic media with consideration of surface elasticity

This paper presents an efficient numerical procedure for analysis of arbitrary shaped, planar cracks in a three-dimensional, linear elastic infinite medium by taking the surface elasticity into account. The concept of surface stresses via Gurtin–Murdoch surface theory is integrated with the classical theory of linear elasticity to form a mathematical model capable of simulating cracks under general loading conditions and nano-scale influences. The key governing equations for the bulk material are established in terms of weakly singular displacement and traction boundary integral equations involving only unknowns on the crack face, whereas that for a zero-thickness material layer on each crack face is established in a weak form following the weighted residual technique. The final governing equations for the bulk material and the two crack-face layers are fully coupled via the continuity condition along the interface. The solution of the resulting coupled system is determined numerically by the coupling of a standard finite element procedure and a symmetric Galerkin boundary element method. Owing to the weakly singular feature of all involved boundary integrals, C\(^{0}\) basis functions are adopted everywhere in the approximation of crack-face data and only the special quadrature for evaluating nearly singular and weakly singular double surface integrals is required. Once the implemented numerical scheme is fully tested with existing reference solutions, it is then applied to investigate the role and influence of surface parameters such as the residual surface tension and the in-plane modulus on the near-front field and the size dependency of predicted solutions. Results of several examples under various scenarios are reported not only to demonstrate the capability and robustness of the proposed method but also to indicate the significance of the surface stresses in enhancing the near-surface stiffness and reducing stresses in a local region ahead of the crack front.

Non-polynomial spline alternatives in Isogeometric Symmetric Galerkin BEM

The application of the Isogeometric Analysis (IgA) paradigm to Symmetric Galerkin Boundary Element Method (SGBEM) is investigated. In order to obtain a very flexible approach, the study is here developed by using non-polynomial spline functions to represent both the domain boundary and the approximate solution. The numerical comparison between IGA-SGBEM and both curvilinear and standard SGBEM approaches shows the general capability of the presented method to produce accurate approximate solutions with less degrees of freedom.

An isogeometric symmetric Galerkin boundary element method for two-dimensional crack problems

The isogeometric analysis is applied for the weakly singular symmetric Galerkin boundary element method (SGBEM) to analyze quasi-static elastic problems including crack problems in two-dimensional domains. This method takes the advantages from the common boundary representation of the isogeometric analysis and the boundary element method. The background of the developed method is to use non-uniform rational B-splines (NURBS) for the Galerkin approximation of both geometry and field variables (i.e.the displacement and traction on the boundary). The basic ingredient of the method is a pair of weakly-singular weak-form integral equations for the displacement and traction on the boundary. These integral equations contain at most weakly-singular kernels of lnr, where r is the distance from a source point to a field point. Various numerical examples are examined to validate the accuracy and efficiency of the proposed method. A model of crack propagation is also discussed to illustrate the use of the method for crack growth simulation. Through the numerical examples, it is observed that the isogeometric SGBEM produces highly accurate results yet it is simple to implement.

A priori error analysis of the BEM with graded meshes for the electric field integral equation on polyhedral surfaces

The Galerkin boundary element discretisations of the electric field integral equation (EFIE) on Lipschitz polyhedral surfaces suffer slow convergence rates when the underlying surface meshes are quasi-uniform and shape-regular. This is due to singular behaviour of the solution to this problem in neighbourhoods of vertices and edges of the surface. Aiming to improve convergence rates of the Galerkin boundary element method (BEM) for the EFIE on a Lipschitz polyhedral closed surface Γ, we employ anisotropic meshes algebraically graded towards the edges of Γ. We prove that on sufficiently graded meshes the h-version of the BEM with the lowest-order Raviart–Thomas elements regains (up to a small order of ε>0) an optimal convergence rate (i.e., the rate of the h-BEM on quasi-uniform meshes for smooth solutions).

Frequency-adapted Galerkin boundary element methods for convex scattering problems

We introduce a class of hybrid boundary element methods for the solution of sound soft scattering problems in the exterior of two-dimensional smooth convex obstacles. To facilitate the applicability of our algorithms throughout the entire frequency spectrum, we have enriched our Galerkin approximation spaces, through incorporation of oscillations in the incident field of radiation, into the algebraic and trigonometric polynomial approximation spaces. The resulting methodologies have three distinctive properties. Indeed, from a theoretical point of view (1) they can be tuned to demand only an$$\mathcal {O}(k^{\varepsilon })$$O(kź)increase (for any $$\varepsilon >0)$$ź>0)in the number of degrees of freedom to maintain a fixed accuracy with increasing wavenumberk, owing to the optimal adaptation of approximation spaces to asymptotic stretching (shrinking) of illuminated and deep shadow regions (shadow boundaries), and (2) they are convergent for each fixed wavenumberk, thanks to the additional approximation spaces in the deep shadow region. Perhaps more importantly, from a practical point of view (3) they give rise to linear systems with significantly enhanced condition numbers and this, in turn, allows for more accurate solutions if desired.

Numerical Solution of Frictional Contact Problems for Viscoelastic Solids by SGBEM and Quadratic Programming

The contact problem with Coulomb friction together with a simple Kelvin-Voigt viscoelastic model is studied. The numerical solution is obtained using a time discretization by a semi-implicit formula, the visco-elastic solids in contact being discretized by Symmetric Galerkin Boundary Element Method (SGBEM). The resulting minimization problem with a nonsmooth cost functional is suitably transformed in several ways. Firstly, a transformation is performed to apply SGBEM without any viscoelastic fundamental solution. Secondly, a transformation of contact quantities leads to a minimimization with a quadratic programming structure. Numerical examples show the applicability of the proposed approach to solve rather intricate frictional contact problems.

The influence of friction on delamination in Fibre-Reinforced Composites, An SGBEM implementation.

A new cohesive contact model coupling the interfacial damage and the Coulomb friction is presented from the mathematical and engineering points of view. The formulation of the Cohesive Interface Model (CIM) predicts the interface damage assuming the frictional contact between debonded surfaces. The contribution discusses the role, influence and the intensity of the friction in the process of the interface debonding in two subsequent fracture effects: the crack initiation and the crack propagation. A concept of an energy-based formulation is applied to model the process of interface debonding. The numerical solution is approximated by a time stepping procedure and it has been implemented in the Symmetric Galerkin Boundary Element Method code. The developed contact model has been applied in Fibre-Reinforced Composite (FRC) and the achieved results prove the impact of the friction on the debonding process in FRC.

A quasi-static interface damage model with cohesive cracks: SQP–SGBEM implementation

A quasi-static model for crack-mode sensitive interface damage with linearly elastic bodies at small strains is developed. It invokes a cohesive type response of the interface interpreted as a thin layer of an adhesive. The damage model is defined with the aim to obtain stress–strain relations in the cohesive zone typically employed in engineering interface-models. The weak solution of the problem is sought numerically by a semi-implicit time-stepping procedure which uses recursive double minimization in displacements and damage separately. The spatial discretization is performed by the symmetric Galerkin boundary-element method. Quadratic and sequential quadratic programming are implemented to resolve each partial minimization of the recursive scheme in the computation of the time-discretized solutions. Sample 2D numerical examples demonstrate applicability of the model as well as efficiency of the SGBEM and convex programming numerical implementations.

The panel-clustering method for the wave equation in two spatial dimensions

We consider the numerical solution of the wave equation in a two-dimensional domain and start from a boundary integral formulation for its discretization. We employ the convolution quadrature (CQ) for the temporal and a Galerkin boundary element method (BEM) for the spatial discretization. Our main focus is the sparse approximation of the arising sequence of boundary integral operators by panel clustering. This requires the definition of an appropriate admissibility condition such that the arising kernel functions can be efficiently approximated on admissible blocks. The resulting method has a complexity of O ( N ( N + M ) q 4 + s ) , s ? { 0 , 1 } , where N is the number of time points, M denotes the dimension of the boundary element space, and q = O ( log ? ( N M ) ) is the order of the panel-clustering expansion. Numerical experiments will illustrate the efficiency and accuracy of the proposed CQ-BEM method with panel clustering.

Isogemetric analysis and symmetric Galerkin BEM: A 2D numerical study

Isogeometric approach applied to Boundary Element Methods is an emerging research area (see e.g. Simpson et al. (2012) [33]). In this context, the aim of the present contribution is that of investigating, from a numerical point of view, the Symmetric Galerkin Boundary Element Method (SGBEM) devoted to the solution of 2D boundary value problems for the Laplace equation, where the boundary and the unknowns on it are both represented by B-splines (de Boor (2001) [9]). We mainly compare this approach, which we call IGA-SGBEM, with a curvilinear SGBEM (Aimi et al. (1999) [2]), which operates on any boundary given by explicit parametric representation and where the approximate solution is obtained using Lagrangian basis. Both techniques are further compared with a standard (conventional) SGBEM approach (Aimi et al. (1997) [1]), where the boundary of the assigned problem is approximated by linear elements and the numerical solution is expressed in terms of Lagrangian basis. Several examples will be presented and discussed, underlying benefits and drawbacks of all the above-mentioned approaches.

Computation of Effective Material Properties in Smart Composite Materials by a Symmetric Galerkin BEM

In this paper, the symmetric Galerkin boundary element method (SGBEM) will be developed and applied for boundary value problems with layered and fiber reinforced piezoelectric representative volume elements (RVE) and real macroscopic structures. Mechanical and electric loadings are considered to determine the effective material properties. For this purpose, the resulting boundary value problem is formulated as boundary integral equations (BIEs). The Galerkin method is applied for the spatial discretization of the boundary to solve the BIEs numerically. The required surface derivatives of the generalized displacements are computed directly with a boundary integral equation. Numerical examples will be presented and discussed to show the efficiency of the present SGBEM and the influence of the fiber variation on the effective material properties.

Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations.

We consider the Galerkin boundary element method (BEM) for weakly-singular integral equations of the first-kind in 2D. We analyze some residual-type a posteriori error estimator which provides a lower as well as an upper bound for the unknown Galerkin BEM error. The required assumptions are weak and allow for piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. In particular, our analysis gives a first contribution to adaptive BEM in the frame of isogeometric analysis (IGABEM), for which we formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments underline the theoretical findings and show that the proposed adaptive strategy leads to optimal convergence.

Partitioned solution strategies for coupled BEM–FEM acoustic fluid–structure interaction problems

This paper investigates two FEM–BEM coupling formulations for acoustic fluid–structure interaction (FSI) problems, using the Finite Element Method (FEM) to model the structure and the Boundary Element Method (BEM) to represent a linear acoustic fluid. The coupling methods described interconnect fluid and structure using classical or localized Lagrange multipliers, allowing the connection of non-matching interfaces. First coupling technique is the well known mortar method, that uses classical multipliers and is compared with a new formulation of the method of localized Lagrange multipliers (LLM) for FSI applications with non-matching interfaces. The proposed non-overlapping domain decomposition technique uses a classical non-symmetrical acoustic BEM formulation for the fluid, although a symmetric Galerkin BEM formulation could be used as well. A comparison between the localized methodology and the mortar method in highly non conforming interface meshes is presented. Furthermore, the methodology proposes an iterative preconditioned and projected bi-conjugate gradient solver which presents very good scalability properties in the solution of this kind of problems.

Analysis of cracks in 3D piezoelectric media with various electrical boundary conditions

A weakly singular, symmetric Galerkin boundary element method capable of solving problems of isolated cracks in three-dimensional, linear anisotropic piezoelectric, infinite media with various types of crack-face boundary conditions including impermeable, permeable, semi-permeable, and the energetically consistent boundary condition introduced by Landis (Int J Solids Struct 41:6291–6315, 2004) is established. The key governing boundary integral equation used in the formulation possesses several crucial features including its desirable symmetric weak-form, weakly singular nature, and ability to treat general material anisotropy, arbitrary crack configurations and any type of boundary condition on the crack surface. The positive consequence of utilizing the singularity-reduced integral equations in the modeling, is that all involved singular integrals can be interpreted in the sense of Riemann and their validity requires only continuous crack-face data allowing $$\hbox {C}^{0}$$ -interpolation functions to be employed everywhere in the numerical discretization. Special crack-tip elements with appropriate square-root functions are adopted in a local region along the crack front to accurately approximate the relative crack-face displacement and electric potential. With use of these crack-tip elements, the stress and electric intensity factors can be extracted directly in terms of crack-front nodal data. A system of nonlinear algebraic equations resulting from semi-permeable and energetically consistent boundary conditions is solved by standard Newton–Raphson iterative scheme. Various numerical examples of both planar and non-planar cracks under different types of electrical boundary conditions are considered and the proposed technique is found promising and computationally robust. In addition, it was determined that using crack-tip elements along the crack front significantly enhances the computational performance and that the stress and electric intensity factors can be obtained accurately using relatively coarse meshes.

FEM-SGBEM coupling for modeling of mode-I planar cracks in three-dimensional elastic media with residual surface tension effects

A computationally efficient numerical technique capable of modeling mode-I planar cracks in three-dimensional linear elastic media by taking the influence of the residual surface tension into account, is presented in this paper. The elastic medium (i.e., the bulk material) is modeled by the classical theory of linear elasticity, whereas the crack surface is treated as a zero-thickness layer perfectly bonded to the bulk material with its behavior governed by the special case of Gurtin–Murdoch surface elasticity model. Governing equations of the bulk material are formulated in terms of weakly singular, weak-form boundary integral equations, whereas those of the surface are cast in a weak form using a weighted residual technique. The solution of the final coupled system of governing equations is subsequently accomplished by using a numerical procedure based primarily on a coupling between standard finite element technique and a weakly singular, symmetric Galerkin boundary element method. Extensive numerical simulations are conducted and the results are compared with available benchmark solutions to verify the formulation and numerical implementation. Applications of the technique to the analysis of nano-crack problems are presented for some selected cases, to study nano-scale influence and size-dependency behavior.

A priori error estimates for a time‐dependent boundary element method for the acoustic wave equation in a half‐space

We investigate a time‐domain Galerkin boundary element method for the wave equation outside a Lipschitz obstacle in an absorbing half‐space. A priori estimates are presented for both closed surfaces and screens, and we discuss the relevant properties of anisotropic Sobolev spaces and the boundary integral operators between them. Copyright © 2014 John Wiley & Sons, Ltd.

Solving multizone and multicrack elastostatic problems: A fast multipole symmetric Galerkin boundary element method approach

Symmetric Galerkin boundary element methods (SGBEMs) for three-dimensional elastostatic problems give rise to fully populated (albeit symmetric) matrix equations, entailing high solution times for large models. This paper is concerned with the formulation and implementation of a multi-level fast multipole SGBEM (FM-SGBEM) for multi-zone elasticity problems with cracks. The subdomain coupling approach is based on a minimal set of interfacial unknowns (i.e. one displacement and one traction vector at any interfacial point) that are defined globally for the complete multizone configuration. Then, unknowns for each subdomain are defined in terms of the global unknowns, with appropriate sign conventions for tractions induced by subdomain numbering. This formulation (i) automatically enforces the perfect-bonding transmission conditions between subdomains, and (ii) is globally symmetric. The subsequent FM-SGBEM basically proceeds by assembling contributions from each subregion, which can be computed by means of an existing single-domain FM-SGBEM implementation such as that previously presented by the authors (Pham et al., Eng Anal Bound Elem 2012;36:1838–47 [36]). Along the way, the computational performance of the FM-SGBEM is enhanced through (a) suitable storage of the near-field contribution to the SGBEM matrix equation and (b) preconditioning by means of nested GMRES. The formulation is validated on numerical experiments for 3D configurations involving many cracks and inclusions, and of sizes up to N≈106.

Mixed-mode fracture & non-planar fatigue analyses of cracked I-beams, using a 3D SGBEM–FEM Alternating Method

In the present paper, computations of mixed mode stress intensity factor (SIF) variations along the crack front, and fatigue-crack-growth simulations, in cracked I-beams, considering different load cases and initial crack configurations, are carried out by employing the three-dimensional SGBEM (Symmetric Galerkin Boundary Element Method)–FEM (Finite Element Method) Alternating Method. For mode-I cracks in the I-beam, the computed SIFs by using the SGBEM–FEM Alternating Method are in very good agreement with available empirical solutions. The predicted fatigue life of cracked I-beams agrees well with experimental observations in the open literature. For mixed-mode cracks in the web or in the flange of the I-beams, no analytical or empirical solutions are available in the literature. Thus mixed-mode SIFs for mixed-mode web and flange cracks are presented, and non-planar fatigue growth simulations are given, as benchmark examples for future studies. Moreover, because very minimal efforts of preprocessing and very small computational burden are needed, the current SGBEM–FEM Alternating Method is very suitable for fracture and fatigue analyses of 3D structures such as I-beams.

Energetic versus maximally-dissipative local solutions of a quasi-static rate-independent mixed-mode delamination model

A quasi-static rate-independent model of delamination of linearly elastic bodies at small strains, sensitive to mode of delamination, using interfacial damage and interfacial plasticity as two internal parameters, is further developed with the aim to extract representations typically employed in engineering interface-models, i.e. fracture envelope and fracture energy dependence on the mode mixity, which are suitable for the model fitting to experimental data. Moreover, two concepts of solutions are implemented: globally stable energy-conserving solutions or stress-driven maximally-dissipative local solutions, arising by the fully implicit or by a semi-implicit time-stepping procedures, respectively, both yielding numerically stable and convergent time-discretizations. Spatial discretization is performed by the symmetric Galerkin boundary-element method (SGBEM). Alternating quadratic programming is implemented to cope with, respectively, global or local, energy-minimizations in the computation of the time-discretized solutions. Sample 2D numerical examples document applicability of the model as well as efficiency of the SGBEM numerical implementation and facilitate comparison of the two mentioned solution concepts.