Dual BEM for crack growth analysis on distributed-memory multiprocessors

The Dual Boundary Element Method (DBEM) has been presented as an effective numerical technique for the analysis of linear elastic crack problems [Portela A, Aliabadi MH. The dual boundary element method: effective implementation for crack problems. Int J Num Meth Engng 1992;33:1269–1287]. Analysis of large structural integrity problems may need the use of large computational resources, both in terms of CPU time and memory requirements. This paper reports a message-passing implementation of the DBEM formulation dealing with the analysis of crack growth in structures. We have analyzed the construction of the system and its resolution. Different data distribution techniques have been studied with several problems. Results in terms of scalability and load balance for these two stages are presented in this paper.     

Numerical–experimental crack growth analysis in AA2024-T3 FSWed butt joints

This paper deals with a numerical and experimental investigation on the influence of residual stresses on fatigue crack growth in AA2024-T3 friction stir welded butt joints. The computational approach is based on the sequential usage of the Finite Element Method (FEM) and the Dual Boundary Element Method (DBEM). Linear elastic FE simulations are performed to evaluate the process induced residual stresses, by means of the contour method. The computed stress field is transferred to a DBEM environment and superimposed to the stress field produced by a remote fatigue traction load applied on a friction stir welded cracked specimen; the crack propagation is then simulated according to a two-parameter growth model. Numerical results have been compared with experimental data showing good agreement and evidencing the predictive capability of the proposed method. The obtained results highlight the influence of the residual stress distribution on crack growth.     

The development of the Trefftzian methodology for engineering design analysis

This paper reviews the developments in the Trefftzian Methodology, which have been undertaken by the authors at the University of Sheffield during the past fifteen years and the application of these developments to engineering design analysis. Initially, in the late 1970s, this work concentrated on the Direct Boundary Element Method (DBEM) and the Indirect Boundary Element Method (IBEM). Unfortunately these methods, as they are normally formulated, give rise to singular integrals, which require special mathematical treatment, when the source and field points coincide on the boundary of the component being analysed. These singular integrals can however be eliminated by placing the source boundary outside the domain of the problem being analysed so that the field and source points never coincide. This technique is known as either the Regular Direct Boundary Element Method (RDBEM) or the Regular Indirect Boundary Element Method (RIBEM) In a further development of the RIBEM, based on the Trefftz Method, the continuous distribution of sources is replaced with sources distributed at discrete points on the source boundary. This modified Trefftz Method eliminates the integrations in the solution procedure, it provides a series solution in terms of the fundamental solution of the problem being analysed and is referred to as the Indirect Discrete Boundary Method (IDBM). The emergence of the IDBM provided the opportunity to develop a combined Boundary Element Finite Element technique which enables these methods to be used simultaneously in a single calculation, thereby exploiting their strengths and minimising their weaknesses. A number of case studies will be discussed in the paper to illustrate the developments in the Trefftzian Methodology and its application to engineering design analysis.     

An object-oriented approach to the Generalized Finite Element Method

The Generalized Finite Element Method (GFEM) is a meshbased approach that can be considered as one instance of the Partition of Unity Method (PUM). The partition of unity is provided by conventional interpolations used in the Finite Element Method (FEM) which are extrinsically enriched by other functions specially chosen for the analyzed problem. The similarities and differences between GFEM and FEM are pointed out here to expand a FEM computational environment. Such environment is an object-oriented system that allows linear and non-linear, static and dynamic structural analysis and has an extense finite element library. The aiming is to enclose the GFEM formulation with a minimum impact in the code structure and meet requirements for extensibility and robustness. The implementation proposed here make it possible to combine different kinds of elements and analysis models with the GFEM enrichment strategies. Numerical examples, for linear analysis, are presented in order to demonstrate the code expansion and to illustrate some of the above mentioned combinations.     

Non-linear MSD crack growth by DBEM for a riveted aeronautic reinforcement

A special specimen is created cutting a rectangular notched area from the surrounding of the upper left corner of a wide body aircraft door. Then a constant amplitude fatigue traction load is applied by a special servo-hydraulic machine, in order to induce a Multi Site Damage (MSD) scenario.The Dual Boundary Element method (DBEM), as implemented in a commercial code, is adopted for a three-dimensional MSD crack growth simulation of such multi-layer and multi-material component. To this aim, the cracked part of a pre-existing global two-dimensional model is extracted and “extruded” in order to generate a three-dimensional submodel, whose boundary conditions are imposed displacements, calculated by the two-dimensional model, along a virtual line corresponding to the submodel boundary. Non-linear contact conditions are applied between the mating plate surfaces in the area surrounding the cracks, in order to precisely model the plate interactions in the area of interest.The three-dimensional approach is aimed to improve, with respect to the two-dimensional approach, the correlation between numerical and experimental results (e.g. by an accurate assessment of the secondary bending effects). The obtained improvements on crack growth rates, in the initial part of the crack propagation, justify the increased computational effort that a three-dimensional non-linear approach involves.The proposed numerical procedure, based on DBEM, is successfully validated for the virtual testing of a complex aeronautic reinforcement.     

A parallel multipolar boundary element method for internal stokes flows

A multipolar expansion technique is applied to the Boundary Element Method (direct and indirect formulation) in order to solve the two-dimensional internal Stokes Flow with first kind boundary conditions. The algorithm is based on a multipolar expansion for the far field and numerical evaluation for the inner field. Due to the nature of the algorithm, it is necessary to resort to the use of an iterative solver for the resulting algebraic linear system of equations. In comparison with the direct BEM formulation, the indirect formulation is more stable with iterative solvers, and does not need to be preconditioned to obtain a fast rate of convergence. A parallel implementation is designed to take advantage of the natural domain decomposition of fast multipolar techniques and bring further improvement. A good result in memory saving and computing time is obtained that enable us to run huge examples which are prohibitive for traditional BEM implementations.     

An integration scheme for electromagnetic scattering using plane wave edge elements

Finite element techniques for the simulation of electromagnetic wave propagation are, like all conventional element based approaches for wave problems, limited by the ability of the polynomial basis to capture the sinusoidal nature of the solution. The Partition of Unity Method (PUM) has recently been applied successfully, in finite and boundary element algorithms, to wave propagation. In this paper, we apply the PUM approach to the edge finite elements in the solution of Maxwell’s equations. The electric field is expanded in a set of plane waves, the amplitudes of which become the unknowns, allowing each element to span a region containing multiple wavelengths. However, it is well known that, with PUM enrichment, the burden of computation shifts from the solver to the evaluation of oscillatory integrals during matrix assembly. A full electromagnetic scattering problem is not simulated or solved in this paper. This paper is an addition to the work of Ledger and concentrates on efficient methods of evaluating the oscillatory integrals that arise. A semi-analytical scheme of the Filon character is presented.     

A boundary element formulation for the substation grounding design

A Boundary Element approach for the numerical computation of substation grounding systems is presented. In this general formulation, several widespread intuitive methods (such as Average Potential Method (APM)) can be identified as the result of specific choices for the test and trial functions and suitable assumptions introduced in the Boundary Element Method (BEM) formulation to reduce computational cost. While linear and parabolic leakage current elements allow to increase accuracy, computing time is drastically reduced by means of new completely analytical integration techniques and semi-iterative methods for solving linear equations systems. This BEM formulation has been implemented in a specific Computer Aided Design system for grounding analysis developed in the last years. The feasibility of this new approach is demonstrated with its application to a real problem.     

基于边界面法的完整实体应力分析理论与应用

提出基于边界面法（Boundary Face Method,BFM）的完整实体应力分析方法.在该分析中,避免对结构作几何上的简化,结构的所有局部细节都按实际形状尺寸作为三维实体处理.以边界积分方程为理论基础的BFM是完整实体应力分析的自然选择.在该方法中,边界积分和场变量插值都在实体边界曲面的参数空间里实现.高斯积分点的几何数据,如坐标、雅可比和外法向量都直接由曲面算得,而不是通过单元插值近似获得,从而避免几何误差.该方法的实现直接基于边界表征的CAD模型,可做到与CAD软件的无缝连接.线弹性问题的应用实例表明,该方法可以简单有效地模拟具有细小特征的复杂结构,并且计算结果的应力精度比边界元法（Boundary Element Method,BEM）和有限元法（Finite Element Method,FEM）高.     

Parallel solvers for linear and nonlinear exterior magnetic field problems based upon coupled FE/BE Formulations

An efficient parallel algorithm for solving linear and nonlinear exterior boundary value problems arising, e.g., in magnetostatics is presented. It is based upon the domaindecomposition-(DD)-coupling of Finite Element and Galerkin Boundary Element Methods which results in a unified variational formulation. In this way, e.g., magnetic field problems in an unbounded domain with Sommerfeld's radiation condition can be modelled correctly. The problem of a nonsymmetric system matrix due to Galerkin-BEM is overcome by transforming it into a symmetric but indefinite matrix and applying Bramble/Pasciak's CG for indefinite systems. For preconditioning, the main ideas of recent DD research are being applied. Test computations on a multiprocessor system were performed for two problems of practical interest including a nonlinear example.     

基于边界元素法的柔软物体变形模拟

在计算机动画和虚拟现实技术中，基于物理的建模方法是高真实感地模拟物体受力变形和运动的有效途径．近年来基于边界元的物理模型方法因其简捷的计算模式而受到关注，该文针对当前边界元模型在视觉效果和计算量上的一些缺陷，分别提出了两方面的改进方法，基于LOD的动态自适应多分辨率网格边界元模型和近似的非线性边界元的物理模型，分别用于在不损失视觉效果的前提下减少计算量以及模拟物体大变形，并提出了相应的加速算法，取得了较好的效果．     

Direct Expansion Method of Boundary Condition for solving 3D elliptic equations with small parameters in the irregular domain

In this article, a new methodology, Direct Expansion Method of Boundary Condition (DEMBC), is developed to solve 3D elliptic equations in the irregular domain. First, the previous Rational Differential Quadrature Method (Rational Spectral Collocation Method in (Berrut et al. 2005) [8]), developed by Berrut et al. (2005) [8], has been generalized to solve 3D elliptic equations. Second, it is showed that Direct Expansion Method of Boundary Condition is capable of handling boundary problems with higher efficiency. Finally, with the help of conformal mapping (Tee and Trefethen, 2006) [9] and domain decomposition method, DEMBC and 3D-RDQM are able solve three kinds of 3D elliptic equations with small parameters in the irregular domain. Numerous test results justify the accuracy and efficiency of our approach.     

Transformation Methods for the Numerical Integration of Three-Dimensional Singular Functions

The implementations of the eXtended Finite Element Method and the Boundary Element Method need to face the challenge of integrating singular functions. Since standard quadrature techniques usually produce inaccurate results, a number of specific algorithms have been developed to address this problem. We present a general framework for the systematic formulation of the three-dimensional case. The classical cubic transformation is also considered, including an analytical optimization of its parameters for improved practical efficiency.     

Exact Non-Reflecting Boundary Conditions on Perturbed Domains and <Emphasis Type="Italic">hp</Emphasis>-Finite Elements

For exterior scattering problems one of the chief difficulties arises from the unbounded nature of the problem domain. Inhomogeneous obstacles may require a volumetric discretization, such as the Finite Element Method (FEM), and for this approach to be feasible the exterior domain must be truncated and an appropriate condition enforced at the far, artificial, boundary. An exact, non-reflecting boundary condition can be stated using the classical DtN-FE method if the Artificial Boundary’s shape is quite specific: circular or elliptical. Recently, this approach has been generalized to permit quite general Artificial Boundaries which are shaped as perturbations of a circle resulting in the “Enhanced DtN-FE” method. In this paper we extend this method to a two-dimensional FEM featuring high-order polynomials in order to realize a high rate of convergence. This is more involved than simply specifying high-order test and trial functions as now the scatterer shape and Artificial Boundary must be faithfully represented. This entails boundary elements which conform (to high order) to the true boundary shapes. As we show, this can be accomplished and we realize an arbitrary order FEM without spurious reflections.     

Complex boundary elements for contaminant transport

Recent numerical advances in the Complex Variable Boundary Element Method (CVBEM) provide easier-to-use analysis procedures in the study of advection-dominated contaminant transport of conservative specie migration in a steady groundwater flow field. In this paper, the CVBEM is applied to groundwater advection contaminant transport problems, and the CVBEM numerical error is evaluated by use of the approximate boundary graphical technique. Besides providing actual solutions to many groundwater flow and advective contaminant transport problems, the CVBEM can be used to develop analytic test cases to be used in numerically calibrating other groundwater and contaminant transport numerical models for other classes of problems. In this paper, the CVBEM model is developed by use of a coupled L²/Collocation fit to prescribed boundary conditions.     

MSD crack propagation by DBEM on a repaired aeronautic panel

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.     

Relations between FEM and FVM applied to the poisson equation

The Finite Volume Method (FVM) with Voronoi bosex for discretizing elliptic boundary value problems is discussed. For this method the matrix of the linear system of equations is shown to be equal to the matrix for the Finite Element Method (FEM) iff a Delaunay triangulation is used.     

MHD flow in a rectangular duct with a perturbed boundary

The unsteady magnetohydrodynamic (MHD) flow of a viscous, incompressible and electrically conducting fluid in a rectangular duct with a perturbed boundary, is investigated. A small boundary perturbation $\epsilon$ is applied on the upper wall of the duct which is encountered in the visualization of the blood flow in constricted arteries. The MHD equations which are coupled in the velocity and the induced magnetic field are solved with no-slip velocity conditions and by taking the side walls as insulated and the Hartmann walls as perfectly conducting. Both the domain boundary element method (DBEM) and the dual reciprocity boundary element method (DRBEM) are used in spatial discretization with a backward finite difference scheme for the time integration. These MHD equations are decoupled first into two transient convection–diffusion equations, and then into two modified Helmholtz equations by using suitable transformations. Then, the DBEM or DRBEM is used to transform these equations into equivalent integral equations by employing the fundamental solution of either steady-state convection–diffusion or modified Helmholtz equations. The DBEM and DRBEM results are presented and compared by equi-velocity and current lines at steady-state for several values of Hartmann number and the boundary perturbation parameter.     

Structural design sensitivity using boundary elements and polynomial response function

This main issue of this paper is a conjunction of the structural design sensitivity analysis using the Boundary Element Method with the polynomial response function determination. The procedure is so general that it enables sensitivity analysis for potential and elasticity problems within both homogeneous and heterogeneous plane and 3D problems. The essential difference with respect to the previous approaches like the Direct Differentiation Method or the Adjoint Variable Method is in discrete evaluation of the structural response using the response polynomials of some state parameters and design variable as the independent parameter. Such a determination is carried out via the several solutions of the given boundary value problem, where design parameter mean value is regularly perturbed in each of the solutions to cover the closest neighborhood of this mean value. Those few solutions make it possible to recover the polynomial response function from node-to node within the boundary elements, so that further symbolic differentiation using MAPLE returns the sensitivity gradients particular values. The entire procedure is tested here twice—first example deals with the homogeneous cantilever beam, where comparison against pure analytical differentiation is done and, separately, for two-component composite cantilever, where such a comparison is made against the central difference method linked with the same BEM solution.     

Theoretically supported scalable BETI method for variational inequalities

Summary The Boundary Element Tearing and Interconnecting (BETI) methods were recently introduced as boundary element counterparts of the well established Finite Element Tearing and Interconnecting (FETI) methods. Here we combine the BETI method preconditioned by the projector to the “natural coarse grid” with recently proposed optimal algorithms for the solution of bound and equality constrained quadratic programming problems in order to develop a theoretically supported scalable solver for elliptic multidomain boundary variational inequalities such as those describing the equilibrium of a system of bodies in mutual contact. The key observation is that the “natural coarse grid” defines a subspace that contains the solution, so that the preconditioning affects also the non-linear steps. The results are validated by numerical experiments.