Galerkin Boundary Element Method Research Articles

Calculation of T-stress for cracks in two-dimensional anisotropic elastic media by boundary integral equation method

A numerical procedure is proposed to compute the T-stress for two-dimensional cracks in general anisotropic elastic media. T-stress is determined from the sum of crack-face displacements which are computed via an integral equation of the boundary data. To smooth out the data in order to perform accurately numerical differentiation, the sum of crack-face displacement is established in a weak-form integral equation in which the integration domain is simply the crack-tip element. This weak-form integral equation is then solved numerically using standard Galerkin approximation to obtain the nodal values of the sum of crack-face displacements. The procedure is incorporated in a weakly-singular symmetric Galerkin boundary element method in which all integral equations for the traction and displacement on the boundary of the domain and on the crack faces include (at most) weakly-singular kernels. To examine the accuracy and efficiency of the developed method, various numerical examples for cracks in infinite and finite domains are treated. It is shown that highly accurate results are obtained using relatively coarse meshes.

Interfacial debonds of layered anisotropic materials using a quasi-static interface damage model with Coulomb friction

A new quasi-static and energy based formulation of an interface damage model which includes Coulomb friction at the interface between anisotropic solids is provided. The interface traction-relative displacement response is based on an assumption of a thin adhesive layer whose behaviour is analogous to cohesive zone models. The damaged interface is considered, if exposed to a pressure, as a contact zone where Coulomb friction law is also taken into account. As the contacting solids are generally anisotropic, the friction may exhibit some anisotropic behaviour, too, which is included into the proposed model. The solution of the problem is sought numerically by a semi-implicit time-stepping procedure which uses recursive decoupled double minimisation in displacements and damage variables. The spatial discretisation is based on the symmetric Galerkin boundary-element method of a multidomain problem, where the interface variables are calculated by sequential quadratic programming, being a tool for resolving each partial minimisation in the proposed recursive scheme. Sample numerical examples demonstrate applicability of the described model.

A Galerkin BEM for high-frequency scattering problems based on frequency-dependent changes of variables

In this paper we develop a class of efficient Galerkin boundary element methods for the solution of two-dimensional exterior single-scattering problems. Our approach is based upon construction of Galerkin approximation spaces confined to the asymptotic behaviour of the solution through a certain direct sum of appropriate function spaces weighted by the oscillations in the incident field of radiation. Specifically, the function spaces in the illuminated/shadow regions and the shadow boundaries are simply algebraic polynomials whereas those in the transition regions are generated utilizing novel, yet simple, \emph{frequency dependent changes of variables perfectly matched with the boundary layers of the amplitude} in these regions. While, on the one hand, we rigorously verify for smooth convex obstacles that these methods require only an $\mathcal{O}\left( k^{\epsilon} \right)$ increase in the number of degrees of freedom to maintain any given accuracy independent of frequency, and on the other hand, remaining in the realm of smooth obstacles they are applicable in more general single-scattering configurations. The most distinctive property of our algorithms is their \emph{remarkable success} in approximating the solution in the shadow region when compared with the algorithms available in the literature.

Second-Kind Boundary Integral Equations for Scattering at Composite Partly Impenetrable Objects

We consider acoustic scattering of time-harmonic waves at objects composed of several homogeneous parts. Some of those may be impenetrable, giving rise to Dirichlet boundary conditions on their surfaces. We start from the second-kind boundary integral approach of [X. Claeys, and R. Hiptmair, and E. Spindler. A second-kind Galerkin boundary element method for scattering at composite objects. BIT Numerical Mathematics, 55(1):33-57, 2015] and extend it to this setting. Based on so-called global multi-potentials, we derive variational second-kind boundary integral equations posed in $L^2(\Sigma)$, where $\Sigma$ denotes the union of material interfaces. To suppress spurious resonances, we introduce a combined-field version (CFIE) of our new method. Thorough numerical tests highlight the low and mesh-independent condition numbers of Galerkin matrices obtained with discontinuous piecewise polynomial boundary element spaces. They also confirm competitive accuracy of the numerical solution in comparison with the widely used first-kind single-trace approach.

A hybrid numerical–asymptotic boundary element method for high frequency scattering by penetrable convex polygons

We present a novel hybrid numerical–asymptotic boundary element method for high frequency acoustic and electromagnetic scattering by penetrable (dielectric) convex polygons. Our method is based on a standard reformulation of the associated transmission boundary value problem as a direct boundary integral equation for the unknown Cauchy data, but with a nonstandard numerical discretization which efficiently captures the high frequency oscillatory behaviour. The Cauchy data is represented as a sum of the classical geometrical optics approximation, computed by a beam tracing algorithm, plus a contribution due to diffraction, computed by a Galerkin boundary element method using oscillatory basis functions chosen according to the principles of the Geometrical Theory of Diffraction. We demonstrate with a range of numerical experiments that our boundary element method can achieve a fixed accuracy of approximation using only a relatively small, frequency-independent number of degrees of freedom. Moreover, for the scattering scenarios we consider, the inclusion of the diffraction term provides an order of magnitude improvement in accuracy over the geometrical optics approximation alone.

Efficient assembly based on B-spline tailored quadrature rules for the IgA-SGBEM

This paper deals with the discrete counterpart of 2D elliptic model problems rewritten in terms of Boundary Integral Equations. The study is done within the framework of Isogeometric Analysis based on B-splines. In such a context, the problem of constructing appropriate, accurate and efficient quadrature rules for the Symmetric Galerkin Boundary Element Method is here investigated. The new integration schemes, together with row assembly and sum factorization, are used to build a more efficient strategy to derive the final linear system of equations. Key ingredients are weighted quadrature rules tailored for B-splines, that are constructed to be exact in the whole test space, also with respect to the singular kernel. Several simulations are presented and discussed, showing accurate evaluation of the involved integrals and outlining the superiority of the new approach in terms of computational cost and elapsed time with respect to the standard element-by-element assembly.

An Improved Assembling Algorithm in Boundary Elements With Galerkin Weighting Applied to Three-Dimensional Stokes Flows

In the boundary element method (BEM), the Galerkin weighting technique allows to obtain numerical solutions of a boundary integral equation (BIE), giving the Galerkin boundary element method (GBEM). In three-dimensional (3D) spatial domains, the nested double surface integration of GBEM leads to a significantly larger computational time for assembling the linear system than with the standard collocation method. In practice, the computational time is roughly an order of magnitude larger, thus limiting the use of GBEM in 3D engineering problems. The standard approach for reducing the computational time of the linear system assembling is to skip integrations whenever possible. In this work, a modified assembling algorithm for the element matrices in GBEM is proposed for solving integral kernels that depend on the exterior unit normal. This algorithm is based on kernels symmetries at the element level and not on the flow nor in the mesh. It is applied to a BIE that models external creeping flows around 3D closed bodies using second-order kernels, and it is implemented using OpenMP. For these BIEs, the modified algorithm is on average 32% faster than the original one.

A non-symmetric coupling of Boundary Elements with the Hybridizable Discontinuous Galerkin method

We propose and analyze a new coupling procedure for the Hybridizable Discontinuous Galerkin Method with Galerkin Boundary Element Methods based on a double layer potential representation of the exterior component of the solution of a transmission problem. We show a discrete uniform coercivity estimate for the non-symmetric bilinear form and prove optimal convergence estimates for all the variables, as well as superconvergence for some of the discrete fields. Some numerical experiments support the theoretical findings.

Second-kind boundary integral equations for electromagnetic scattering at composite objects

We consider electromagnetic scattering of time-harmonic fields in R3 at objects composed of several linear, homogeneous, and isotropic materials. Adapting earlier work on acoustic scattering (Claeys et al., 2015) we develop a novel second-kind direct boundary integral formulation for this scattering problem, extending the so-called Müller formulation for a homogeneous scatterer to composite objects. The new formulation is amenable to Galerkin boundary element discretization by means of discontinuous tangential surface vectorfields. A rigorous proof of its well-posedness is still missing. Yet numerical tests demonstrate excellent stability and competitive accuracy of the new approach compared with a widely used direct Galerkin boundary element method based on a first-kind boundary integral formulation. For piecewise constant approximation our experiments also confirm fast convergence of GMRES iterations independently of mesh resolution.

An accurate boundary element method for the exterior elastic scattering problem in two dimensions

This paper is concerned with a Galerkin boundary element method solving the two dimensional exterior elastic wave scattering problem. The original problem is first reduced to the so-called Burton–Miller [1] boundary integral formulation, and essential mathematical features of its variational form are discussed. In numerical implementations, a newly-derived and analytically accurate regularization formula [2] is employed for the numerical evaluation of hyper-singular boundary integral operator. A new computational approach is employed based on the series expansions of Hankel functions for the computation of weakly-singular boundary integral operators during the reduction of corresponding Galerkin equations into a discrete linear system. The effectiveness of proposed numerical methods is demonstrated using several numerical examples.

An isogeometric SGBEM for crack problems of magneto-electro-elastic materials

The isogeometric symmetric Galerkin boundary element method is applied for the analysis of crack problems in two-dimensional magneto-electro-elastic domains. In this method, the field variables of the governing integral equations as well as the geometry of the problems are approximated using non-uniform rational B-splines (NURBS) basis functions. The key advantage of this method is that the isogeometric analysis and boundary element method deal only with the boundary of the domain. To verify the accuracy of the proposed method, numerical examples for crack problems in infinite and finite domains are examined. It is observed that the computed generalized stress intensity factors obtained by the proposed method agree well with the exact solutions and other references.

Isogeometric symmetric Galerkin boundary element method for three-dimensional elasticity problems

The isogeometric analysis (IGA) is applied for the weakly singular symmetric Galerkin boundary element method (SGBEM) to analyzelinear elastostatics problems in three-dimensional domains. The background of the proposed method is to use non-uniform rational B-splines (NURBS) as the basis functions for the approximation of both geometry and field variables (i.e. displacement and traction) of the governing integral equations. Same as weakly singular SGBEM, the basic ingredient of the method is a pair of weakly singular weak-form integral equations for the displacement and traction on the boundary of the domain. These integral equations are solved approximately using standard Galerkin approximation. In addition to the advantages that IGA owned, the proposed method exploits the common boundary representation of CAD model and boundary element method. Various numerical examples of both simple and complex geometries are examined to validate the accuracy and efficiency of the proposed method. Through the numerical examples, it is observed that the IGA–SGBEM produces highly accurate results.

Effective Material Properties in Multiferroic Composite Materials by a Galerkin BEM

In this paper, the effective material parameters of magnetoelectroelastic composites are computed on the base of homogenization techniques performed on the representative volume element (RVE). For this purpose, the resulting boundary value problem is formulated as boundary integral equations (BIEs). The Galerkin method is applied for spatial discretization. The required average stresses and strains are obtained directly from the primary values of the BIEs and therefore additional numerical differentiations as necessary in other methods are avoided. Numerical examples will be presented and discussed to show the efficiency of the present boundary element method (BEM) and the influences of the voids on the effective material properties.

二次元周期境界値問題に対するアイソジオメトリック Galerkin 境界要素法の開発

二次元周期境界値問題に対するアイソジオメトリック Galerkin 境界要素法の開発

Modelling Delamination in Layered Anisotropic Material Structures

A quasi-static model for numerical solution of damage in layered anisotropic material structures has been developed. It invokes a cohesive type response of the interface interpreted as a thin layer of an adhesive. In the model, various types of layered materials can be proposed by using any type of anisotropic materials of the layers with different mechanical properties. The analysis calculates stress and strain quantities in the structure and at the damaged interface. The proposed mathematical approach is based on an energetic formulation looking for a kind of weak solution. The solution is approximated by a time stepping procedure, the Symmetric Galerkin Boundary Element Method (SGBEM), and it utilizes nonlinear programming algorithms.

An energy based formulation of a quasi-static interface damage model with a multilinear cohesive law

A new quasi-static and energy based formulation of an interface damage model which provides interface traction-relative displacement laws like in traditional trilinear (with bilinear softening) or generally multilinear cohesive zone models frequently used by engineers is presented. This cohesive type response of the interface may represent the behaviour of a thin adhesive layer. The level of interface adhesion or damage is defined by several scalar variables suitably defined to obtain the required traction-relative displacement laws. 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 variables separately. The symmetric Galerkin boundary-element method is applied for the spatial discretization. Sequential quadratic programming is implemented to resolve each partial minimization in the recursive scheme applied to compute the time-space discretized solutions. Sample 2D numerical examples demonstrate applicability of the proposed model.

A fast hybrid Galerkin method for high-frequency acoustic scattering

We consider the scattering of a time-harmonic acoustic incident plane wave by a smooth convex object. We formulate this problem by the direct boundary integral method, using the classical combined potential approach. Based on the known asymptotics of the solution, we devise particular expansions, valid in various zones of the boundary. To achieve a good approximation at high frequencies with a relative low number of degrees of freedom, we propose a novel Galerkin boundary element method with a hybrid approximation space, consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes: a polynomial grading on the illuminated side and a geometric grading on the shadow side. Using the asymptotic expansions of the solution, we prove that, as , the number of degree of freedom is able to decrease only very modestly to maintain a fixed absolute error bound ( is a typical behavior). Numerical experiments also show that the method achieves a better accuracy as , for a fixed number of degrees.

Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM

The quasistatic normal-compliance contact problem of isotropic homogeneous linear visco-elastic bodies with Coulomb friction at small strains in Kelvin–Voigt rheology is considered. The discretization is made by a semi-implicit formula in time and the Symmetric Galerkin Boundary Element Method (SGBEM) in space, assuming that the ratio of the viscosity and elasticity moduli is a given relaxation-time coefficient. The obtained recursive minimization problem, formulated only on the contact boundary, has a nonsmooth cost function. If the normal compliance responds linearly and the 2D problems are considered, then the cost function is piecewise-quadratic, which after a certain transformation gets the quadratic programming (QP) structure. However, it would lead to second-order cone programming in 3D problems. Finally, several computational tests are presented and analysed, with an additional discussion on numerical stability and convergence of the involved approximated Poincaré–Steklov operators.

The Galerkin boundary element method for transient Stokes flow

Since the fundamental solution for transient Stokes flow in three dimensions is complicated it is difficult to implement discretization methods for boundary integral formulations. We derive a representation of the Stokeslet and stresslet in terms of incomplete gamma functions and investigate the nature of the singularity of the single- and double layer potentials. Further, we give analytical formulas for the time integration and develop Galerkin schemes with tensor product piecewise polynomial ansatz functions. Numerical results demonstrate optimal convergence rates.

Efficient Analysis of 3D Mixed-Mode Cracks of a Pressure Vessel Based on Schwartz-Neuman Alternating Method

The Schwartz-Neuman alternating method is employed to analyze 3D cracks in structural components with complicated geometries.The SIFs of Mode I, II and III for the mixed-mode cracks under complicated stress state are obtained from the alternating computing scheme between finite element method solution for the uncracked body and the symmetric Galerkin boundary element method solution for the crack in an infinite body. The SIFs of the different surface cracks postulated at a regular pipe and elliptical surface cracks at the nozzle-cylinder intersection of a typical reactor pressure vessel are investigated by using the Schwartz-Neuman alternating method. The comparison with other analysis models and results reported in the literature shows that the Schwartz-Neuman alternating method can efficiently and accurately evaluate SIFs of surface cracks of different shapes and depths with much smaller computational models, which indicates that the Schwartz-Neuman alternating method is an efficient method in the evaluation of the SIFs of 3D cracks.