Symmetric Boundary Element Method Research Articles

Modelling of the fatigue cracking resistance of grid reinforced asphalt concrete by coupling fast BEM and FEM

We present a computational modelling approach aimed at investigating the effect of fibre grid reinforcement on crack opening displacement and fatigue crack propagation. Grid reinforcements are modelled using elastic membrane finite elements, while the cracked concrete is treated using a symmetric boundary element method (BEM), which in particular allows easy geometrical modelling and meshing of cracks. The BEM is accelerated by the fast multipole method, allowing the handling of potentially large BEM models entailed by three-dimensional configurations hosting multiple cracks. Fatigue crack growth is modelled using the Paris law. The proposed computational approach is first verified on a reinforced cracked beam, and then applied to a three-dimensional configuration featuring a grid-reinforced asphalt pavement.

A dual interpolation Galerkin boundary face method for potential problems

A dual interpolation Galerkin boundary face method (DiGBFM) is applied in this paper by combining the newly developed dual interpolation method with the Galerkin boundary face method (BFM). The dual interpolation method unifies the conforming and nonconforming elements in the BFM implementation. It classifies the nodes of a conventional conforming element into virtual nodes and source nodes. Potentials and fluxes are interpolated using the continuous elements in the same way as conforming BFM, while boundary integral equations (BIEs) are collocated at source nodes, in the same way as nonconforming BFM. In order to arrive at a square linear system, we provide additional constraint equations, which are established by the moving least-squares (MLS) approximation, to condense the degrees of freedom relating to virtual nodes. Compared with the traditional symmetric Galerkin boundary element method (BEM), the symmetry feature of the DiGBFM equations is obtained simply through matrix manipulations, because of the use of the symmetric BEM, and no hypersingular BIE is needed in the DiGBFM. The proposed method has been implemented successfully for solving 2-D steady-state potential problems. Several numerical examples are presented in this paper to show the convergence and accuracy of this new method.

A novel approach of CNN for human motor imagery recognition using the virtual electrode pairs

In order to develop an efficient brain-computer interface system, the brain activity measured by electroencephalography needs to be accurately decoded. In this paper, a motor imagery classification approach is proposed, combining virtual electrodes on the cortex layer with a convolutional neural network; this can effectively improve the decoding performance of the brain-computer interface system. A three layer (cortex, skull, and scalp) head volume conduction model was established by using the symmetric boundary element method to map the scalp signal to the cortex area. Nine pairs of virtual electrodes were created on the cortex layer, and the features of the time and frequency sequence from the virtual electrodes were extracted by performing time-frequency analysis. Finally, the convolutional neural network was used to classify motor imagery tasks. The results show that the proposed approach is convergent in both the training model and the test model. Based on the Physionet motor imagery database, the averaged accuracy can reach 98.32% for a single subject, while the averaged values of accuracy, Kappa, precision, recall, and F1-score on the group-wise are 96.23%, 94.83%, 96.21%, 96.13%, and 96.14%, respectively. Based on the High Gamma database, the averaged accuracy has achieved 96.37% and 91.21% at the subject and group levels, respectively. Moreover, this approach is superior to those of other studies on the same database, which suggests robustness and adaptability to individual variability.

Dynamic Wave Propagation in Fiber Reinforced Piezoelectric Composites with Cracks

This paper presents the transient dynamic analysis of micro-cracks of arbitrary shape in two-dimensional, linear piezoelectric fiber reinforced composite materials. Interface cracks between fiber and matrix as well as cracks inside the matrix and fibers are investigated. For this purpose, a symmetric Galerkin time-domain boundary element method in conjunction with a multi-domain technique is developed. The time-domain fundamental solutions for linear piezoelectric materials are applied. The time discretization is performed by a collocation method. An explicit time-stepping scheme is obtained to compute the discrete boundary data including the generalized crack-opening-displacements (CODs). Iterative solution algorithms are implemented to solve the nonlinear semi-permeable electrical crack-face boundary conditions and for a crack-face contact analysis at time-steps when a physically unacceptable crack-face intersection occurs. Several numerical examples are presented to reveal the influences of the micro-cracks, the material combinations and the transient dynamic loading on the intensity factors and the scattered wave fields.

Source-Modeling Auditory Processes of EEG Data Using EEGLAB and Brainstorm.

Electroencephalography (EEG) source localization approaches are often used to disentangle the spatial patterns mixed up in scalp EEG recordings. However, approaches differ substantially between experiments, may be strongly parameter-dependent, and results are not necessarily meaningful. In this paper we provide a pipeline for EEG source estimation, from raw EEG data pre-processing using EEGLAB functions up to source-level analysis as implemented in Brainstorm. The pipeline is tested using a data set of 10 individuals performing an auditory attention task. The analysis approach estimates sources of 64-channel EEG data without the prerequisite of individual anatomies or individually digitized sensor positions. First, we show advanced EEG pre-processing using EEGLAB, which includes artifact attenuation using independent component analysis (ICA). ICA is a linear decomposition technique that aims to reveal the underlying statistical sources of mixed signals and is further a powerful tool to attenuate stereotypical artifacts (e.g., eye movements or heartbeat). Data submitted to ICA are pre-processed to facilitate good-quality decompositions. Aiming toward an objective approach on component identification, the semi-automatic CORRMAP algorithm is applied for the identification of components representing prominent and stereotypic artifacts. Second, we present a step-wise approach to estimate active sources of auditory cortex event-related processing, on a single subject level. The presented approach assumes that no individual anatomy is available and therefore the default anatomy ICBM152, as implemented in Brainstorm, is used for all individuals. Individual noise modeling in this dataset is based on the pre-stimulus baseline period. For EEG source modeling we use the OpenMEEG algorithm as the underlying forward model based on the symmetric Boundary Element Method (BEM). We then apply the method of dynamical statistical parametric mapping (dSPM) to obtain physiologically plausible EEG source estimates. Finally, we show how to perform group level analysis in the time domain on anatomically defined regions of interest (auditory scout). The proposed pipeline needs to be tailored to the specific datasets and paradigms. However, the straightforward combination of EEGLAB and Brainstorm analysis tools may be of interest to others performing EEG source localization.

Computational Vibroacoustics in Low- and Medium- Frequency Bands: Damping, ROM, and UQ Modeling

Within the framework of the state-of-the-art, this paper presents a summary of some common research works carried out by the authors concerning computational methods for the prediction of the responses in the frequency domain of general linear dissipative vibroacoustics (structural-acoustic) systems for liquid and gas in the low-frequency (LF) and medium-frequency (MF) domains, including uncertainty quantification (UQ) that plays an important role in the MF domain. The system under consideration consists of a deformable dissipative structure, coupled with an internal dissipative acoustic fluid including a wall acoustic impedance, and surrounded by an infinite acoustic fluid. The system is submitted to given internal and external acoustic sources and to prescribed mechanical forces. An efficient reduced-order computational model (ROM) is constructed using a finite element discretization (FEM) for the structure and the internal acoustic fluid. The external acoustic fluid is treated using a symmetric boundary element method (BEM) in the frequency domain. All the required modeling aspects required for the analysis in the MF domain have been introduced, in particular the frequency-dependent damping phenomena and model uncertainties. An industrial application to a complex computational vibroacoustic model of an automobile is presented.

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.

Active macro-zone approach for incremental elastoplastic-contact analysis

SUMMARY The symmetric boundary element method, based on the Galerkin hypotheses, has found an application in the nonlinear analysis of plasticity and in contact-detachment problems, but both dealt with separately. In this paper, we want to treat these complex phenomena together as a linear complementarity problem. A mixed variable multidomain approach is utilized in which the substructures are distinguished into macroelements, where elastic behavior is assumed, and bem-elements, where it is possible that plastic strains may occur. Elasticity equations are written for all the substructures, and regularity conditions in weighted (weak) form on the boundary sides and in the nodes (strong) between contiguous substructures have to be introduced, in order to attain the solving equation system governing the elastoplastic-contact/detachment problem. The elastoplasticity is solved by incremental analysis, called for active macro-zones, and uses the well-known concept of self-equilibrium stress field here shown in a discrete form through the introduction of the influence matrix (self-stress matrix). The solution of the frictionless contact/detachment problem was performed using a strategy based on the consistent formulation of the classical Signorini equations rewritten in discrete form by utilizing boundary nodal quantities as check elements in the zones of potential contact or detachment. Copyright © 2013 John Wiley & Sons, Ltd.

Revisited mixed-value method via symmetric BEM in the substructuring approach

Abstract Within the Symmetric Boundary Element Method, the mixed-value analysis is re-formulated. This analysis method contemplates the subdivision of the body into substructures having interface kinematical and mechanical quantities. For each substructure an elasticity equation, connecting weighted displacements and tractions to nodal displacements and forces of the same interface boundary and to external action vector, is introduced. The assembly of the substructures is performed through both the strong and weak regularity conditions of the displacements and tractions. We obtain the solving equations where the compatibility and the equilibrium are guaranteed in the domain Ω for the use of the fundamental solution and at the interface nodes for the strong regularity conditions imposed, whereas the previous quantities are respected in weighted form along the interface boundaries. The mixed-value method leads to a better solution than those obtained through the displacement method of the Symmetric Boundary Element Method, if compared with the analytical solution. By using the Karnak.sGbem program, developed with other researchers and updated through the implementation of the present method, some examples are made which show the advantages related to the computational aspects and to the convergence of the numerical response.

Frictionless contact-detachment analysis: iterative linear complementarity and quadratic programming approaches

The object of the paper concerns a consistent formulation of the classical Signorini's theory regarding the frictionless contact problem between two elastic bodies in the hypothesis of small displacements and strains. The employment of the symmetric Galerkin boundary element method, based on boundary discrete quantities, makes it possible to distinguish two different boundary types, one in contact as the zone of potential detachment, called the real boundary, the other detached as the zone of potential contact, called the virtual boundary. The contact-detachment problem is decomposed into two sub-problems: one is purely elastic, the other regards the contact condition. Following this methodology, the contact problem, dealt with using the symmetric boundary element method, is characterized by symmetry and in sign definiteness of the matrix coefficients, thus admitting a unique solution. The solution of the frictionless contact-detachment problem can be obtained: (i) through an iterative analysis by a strategy based on a linear complementarity problem by using boundary nodal quantities as check quantities in the zones of potential contact or detachment; (ii) through a quadratic programming problem, based on a boundary min-max principle for elastic solids, expressed in terms of nodal relative displacements of the virtual boundary and nodal forces of the real one.

Bioelectromagnetic forward problem: isolated source approach revis(it)ed

Electro- and magnetoencephalography (EEG and MEG) are non-invasive modalities for studying the electrical activity of the brain by measuring voltages on the scalp and magnetic fields outside the head. In the forward problem of EEG and MEG, the relationship between the neural sources and resulting signals is characterized using electromagnetic field theory. This forward problem is commonly solved with the boundary-element method (BEM). The EEG forward problem is numerically challenging due to the low relative conductivity of the skull. In this work, we revise the isolated source approach (ISA) that enables the accurate, computationally efficient BEM solution of this problem. The ISA is formulated for generic basis and weight functions that enable the use of Galerkin weighting. The implementation of the ISA-formulated linear Galerkin BEM (LGISA) is first verified in spherical geometry. Then, the LGISA is compared with conventional Galerkin and symmetric BEM approaches in a realistic 3-shell EEG/MEG model. The results show that the LGISA is a state-of-the-art method for EEG/MEG forward modeling: the ISA formulation increases the accuracy and decreases the computational load. Contrary to some earlier studies, the results show that the ISA increases the accuracy also in the computation of magnetic fields.

On the Computational Aspects of a Symmetric Multidomain Boundary Element Method Approach for Elastoplastic Analysis

The symmetric boundary element method (SBEM) is applied to the elasto-plastic analysis of bodies subdivided into substructures. This methodology is based on the use of: a multidomain SBEM approach, for the evaluation of the elastic predictor; a return mapping algorithm based on the extremal paths theory, for the evaluation of inelastic quantities characterizing the plastic behaviour of each substructure; and a transformation of the domain inelastic integrals of each substructure into corresponding boundary integrals. The elastic analysis is performed by using the SBEM displacement approach, which has the advantage of creating system equations that only consist of nodal kinematical unknowns at the interface boundary among substructures. The elasto-plastic solution utilizes a strain-driven strategy that is characterized by the evaluation of the elastic predictor that is a function of the initial conditions and the load increment. The predictor phase is followed by the use of a return mapping algorithm defined by introducing the extremal paths theory to remove the time integration. Then the computed plastic strains are considered to be constant inelastic actions imposed inside the substructure in a step-by-step procedure. Their presence involves domain integrals with singular kernels. These integrals are considered as Cauchy principal values with which the related free term is associated. In order to compute these domain integrals, the radial integral method is applied to remove the strong singularity.

Forward Field Computation with OpenMEEG

To recover the sources giving rise to electro- and magnetoencephalography in individual measurements, realistic physiological modeling is required, and accurate numerical solutions must be computed. We present OpenMEEG, which solves the electromagnetic forward problem in the quasistatic regime, for head models with piecewise constant conductivity. The core of OpenMEEG consists of the symmetric Boundary Element Method, which is based on an extended Green Representation theorem. OpenMEEG is able to provide lead fields for four different electromagnetic forward problems: Electroencephalography (EEG), Magnetoencephalography (MEG), Electrical Impedance Tomography (EIT), and intracranial electric potentials (IPs). OpenMEEG is open source and multiplatform. It can be used from Python and Matlab in conjunction with toolboxes that solve the inverse problem; its integration within FieldTrip is operational since release 2.0.

Transient dynamic crack analysis in piecewise homogeneous, anisotropic and linear elastic composites by a spatial symmetric Galerkin-BEM

Transient elastodynamic analysis of two-dimensional, piecewise homogeneous, anisotropic and linear elastic solids containing interior and interface cracks is presented in this paper. To solve the initial boundary value problem, a spatial symmetric time-domain boundary element method is developed. Stationary cracks subjected to impact loading conditions are considered. Elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids are implemented. The piecewise homogeneous, anisotropic and linear elastic solids are modeled by the multi-domain technique. The spatial discretization is performed by a symmetric Galerkin-method, while a collocation method is utilized for the temporal discretization . An explicit time-stepping scheme is obtained for computing the unknown boundary data. Numerical examples are presented and discussed to show the effects of the interface cracks, the material anisotropy , the material combination and the dynamic loading on the dynamic stress intensity factors .

Computational aspects in 2D SBEM analysis with domain inelastic actions

AbstractThe Symmetric Boundary Element Method, applied to structures subjected to temperature and inelastic actions, shows singular domain integrals.In the present paper the strong singularity involved in the domain integrals of the stresses and tractions is removed, and by means of a limiting operation, this traction is evaluated on the boundary. First the weakly singular domain integral in the Somigliana Identity (S.I.) of the displacements is regularized and the singular integral is transformed into a boundary one using the Radial Integration Method; subsequently, using the differential operator applied to the displacement field, the S.I. of the tractions inside the body is obtained and through a limit operation its expression is evaluated on the boundary. The latter operation makes it possible to substitute the strongly singular domain integral in a strongly singular boundary one, defined as a Cauchy Principal Value, with which the related free term is associated. The expressions thus obtained for the displacements and the tractions, in which domain integrals are substituted by boundary integrals, were utilized in the Galerkin approach, for the evaluation in closed form of the load coefficients connected to domain inelastic actions.This strategy makes it possible to evaluate the load coefficients avoiding considerable difficulties due to the geometry of the solid analyzed; the obtained coefficients were implemented in the Karnak.sGbem calculus code. Copyright © 2009 John Wiley & Sons, Ltd.

Internal springs distribution for quasi brittle fracture via Symmetric Boundary Element Method

Abstract In this paper the symmetric boundary element formulation is applied to the fracture mechanics problems for quasi brittle materials . The basic aim of the present work is the development and implementation of two discrete cohesive zone models using Symmetric Galerkin multi-zone Boundary Elements Method . The non-linearity at the process zone of the crack will be simulated through a discrete distribution of nodal springs whose generalized (or weighted) stiffnesses are obtainable by the cohesive forces and relative displacements modelling. This goal is reached coherently with the constitutive relation σ − Δ u that describes the interaction between mechanical and kinematical quantities along the process zone. The cracked body is considered as a solid having a “particular” geometry whose analysis is obtainable through the displacement approach employed in [Panzeca, T., Salerno, M., 2000. Macro-elements in the mixed boundary value problems. Comp. Mech. 26, 437-446; Panzeca, T., Cucco, F., Terravecchia, S., 2002b. Symmetric Boundary Element Method versus Finite Element Method . Comput. Meth. Appl. Mech. Engrg. 191, 3347-3367] by some of the present authors in the ambit of the Symmetric Galerkin Boundary Elements Method (SGBEM). In this approach the crack edge nodes are considered distinct and the analysis is performed by evaluating all the equation system coefficients in closed form [Guiggiani, M., 1991. Direct evaluation of hypersingular integrals in 2D BEM. In: Proceedings of the 7th GAMM Seminar on Numerical Techniques for Boundary Element Methods. Kiel, Germany; Gray, L.J., 1998. Evaluation of singular and hypersingular Galerkin boundary integrals: direct limits and symbolic computation. In: Sladek, J., Sladek V. (Eds.), Singular Integrals in Boundary Element Methods, Computational Mechanics Publications, Southampton; Panzeca, T., Fujita Yashima, H., Salerno, M., 2001. Direct stiffness matrices of BEs in the Galerkin BEM formulation. Eur. J. Mech. A/Solids 20, 277–298; Terravecchia, S., 2006. Closed form in the Symmetric Boundary Element Approach. Eng. Anal. Bound. Elem. Meth. 30, 479–488]. Some examples show the goodness of the methodology proposed through a comparison with other formulations [Barenblatt, G.I., 1962. Mathematical theory of equilibrium cracks in brittle fracture . Adv. Appl. Mech. 7, 55–129; Saleh, A.L., Aliabadi, M.H., 1995. Crack growth analysis in concrete using Boundary Element Method. Eng. Fract. Mech. 51, 533–545; Aliabadi, M.H., Saleh, A.L., 2002. Fracture mechanics analysis of cracking in plain and reinforced concrete using boundary element method. Eng. Fract. Mech. 69, 267–280]. In these examples the applied loads and the length of the process zone are a priori given and kept fixed during the analysis in order to check the constitutive behavior along the process zone.

The symmetric boundary element method for unilateral contact problems

On the basis of the boundary integral equation method, in its symmetric formulation, the frictionless unilateral contact between two elastic bodies has been studied. A boundary discretization by boundary elements leads to an algebraic formulation in the form of a linear complementarity problem. In this paper the process of contact or detachment is obtained through a step by step analysis by using generalized (weighted) quantities as the check elements: the detachment or the contact phenomenon may happen when the weighted traction or the weighted displacement is greater than the weighted cohesion or weighted minimum reference gap, respectively. The applications are performed by using the Karnak.sGbem program made by some of the authors of this paper.

Three-Dimensional Indirect Boundary Element Method Formulation for Dynamic Analysis of Frames Buried in Semiinfinite Elastic Media

This paper presents a formulation of the indirect boundary element method based on the principle of virtual work for the dynamic analysis of frame structures buried in semi-infinite elastic media. The present formulation, which falls in the category of symmetric Galerkin boundary element methods, leads to symmetric stiffness matrices for the continuum that may be defined in terms of conventional structural analysis variables (i.e., generalized displacements and lumped forces). It is shown that, in the context of the present formulation, rotation degrees of freedom may readily be introduced in the interpolation scheme with very little additional computational effort. The consistency of the present formulation with well-established results is assessed by comparing the predictions for the static and dynamic stiffness of single piles with other results from the literature. Finally, the dynamic stiffness of a single buried frame under vertical and horizontal loading is studied. The analysis shows that the stiffness of the full frame may not always be accurately estimated by means of results for single piles, even when dynamic interaction factors are used.

Closed form coefficients in the symmetric boundary element approach

In the area of the structural analysis, the problems connected to the use of the symmetric Galerkin Boundary Element Method (SGBEM) must be investigated especially in the mathematical and computational difficulties that are present in computing the solving system coefficients. Indeed, any coefficient is made by double integrals including often fundamental solutions having a high degree of singularity. Therefore, the related computation proves to be difficult in the solution. This paper suggests a simple computation technique of the coefficients obtained in closed form. Using a particular matrix, called ‘progenitor’ matrix [Panzeca T, Cucco F, Terravecchia S. Symmetric boundary element method versus finite element method. Comp Method Appl Mech Eng 2002;191:3347–67.], it is possible to derive the solving algebraic system made up of a coefficient matrix and of a known term vector. The latter matrices and vectors, arranged in the sub-structure approach, permitted to develop a versatile computation code, called KARNAK.sGbem [Cucco F, Panzeca T, Terravecchia S. The program KARNAK SGBEM release 1.0. Palermo; 2002. Available from: http://www.bemsoft.it.].

Generalized head models for MEG/EEG: boundary element method beyond nested volumes

Accurate geometrical models of the head are necessary for solving the forward and inverse problems of magneto- and electro-encephalography (MEG/EEG). Boundary element methods (BEMs) require a geometrical model describing the interfaces between different tissue types. Classically, head models with a nested volume topology have been used. In this paper, we demonstrate how this constraint can be relaxed, allowing us to model more realistic head topologies. We describe the symmetric BEM for this new model. The symmetric BEM formulation uses both potentials and currents at the interfaces as unknowns and is in general more accurate than the alternative double-layer formulation.