Boundary Element Tearing and Interconnecting Methods

Langer, Ulrich; Steinbach, Olaf

doi:10.1007/s00607-003-0018-2

Cited by 93 publications

(65 citation statements)

References 29 publications

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“…The Boundary Element Tearing and Interconnecting Method (BETI) is a domain decomposition method based on the Symmetric Galerkin BEM [4] similar to the Finite Element Tearing and Interconnecting Method (FETI) for the FEM introduced by [5]. In this work the collocation BEM is used to set up the equation system similar to the original BETI method.…”

Section: Boundary Element Tearing and Interconnecting Methods (Beti)mentioning

confidence: 99%

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Modelling the Process of Sequential Excavation With the Boundary Element Method

Duenser

Beer

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. The Boundary Element Method (BEM) is ideally suited for the simulation of underground constructions like tunnels or caverns. Such structures are modelled with the BEM inside an infinite or semi-infinite domain. As the radiation condition is fulfilled by the BEM no truncation of the domain is necessary. Only the surface of the structure (e.g. tunnel) has to be discretised by boundary elements (BE). An accurate simulation of the tunnelling process has to consider the sequential excavation where parts of the rock mass are excavated at different time and location. This special constructional condition has a direct influence onto the simulation model. In this work different methods are presented which consider the sequential excavation. The first method is the discretisation of the problem by multiple BE regions (MRBEM). Each region, which will be excavated during the excavation process, is discretised by a separate finite BE region. These regions are embedded inside an infinite region which represent the infinite extend of the domain. Thus, a system of BE regions arise which have to be coupled at their common interfaces. Two coupling strategies, the Boundary Element Tearing and Interconnecting Method (BETI) and the method of Interface Coupling (IC) will be presented to solve the sequential tunnel excavation. The second method uses only a single BE region (SRBEM) for every step of excavation. For each load step the geometry/mesh has to be updated. Thus, the mesh of the previous load step will be extended by the surface of the new excavation volume of the current load step. Beside the geometry update an essential part of this method is an accurate evaluation of the excavation loading. The excavation loads for each excavation step are tractions applied at the part of the boundary surface just generated by the geometry update. These tractions depend on all previous load steps and will be evaluated by a calculation of internal results in the interior of the single region. The internal results can be either stresses or displacements. In this work the modelling strategies of the MRBEM and SRBEM approach will be presented. On a realistic tunnel example the accuracy of the results for the mentioned methods will be shown as well as the performance of the calculations. 3168

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Section: Boundary Element Tearing and Interconnecting Methods (Beti)mentioning

confidence: 99%

“…As the stiffness matrix [K] r of a finite region (floating region) is singular [6] special treatment of rigid body motions have to be considered. The whole solution formulation is shown in detail by [4].…”

Section: Boundary Element Tearing and Interconnecting Methods (Beti)mentioning

confidence: 99%

Modelling the Process of Sequential Excavation With the Boundary Element Method

Duenser

Beer

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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“…Appropriate preconditioned iterative strategies were later considered in Carstensen et al [1998], while quite general preconditioners based on operators of the opposite order were introduced in Steinbach and Wendland [1998]. Boundary element tearing and interconnecting (BETI) methods were described in Langer and Steinbach [2003] as counterpart of FETI methods while in Langer et al [2006] these methods were combined with a fast multipole approximation of the local boundary integral operators involved. For an alternative approach to boundary integral domain decomposition methods see also Khoromskij and Wittum [2004].…”

Section: Introductionmentioning

confidence: 99%

Challenges and Applications of Boundary Element Domain Decomposition Methods

Steinbach

2008

Lecture Notes in Computational Science and Engineering

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Boundary integral equation methods are well suited to represent the Dirichlet to Neumann maps which are required in the formulation of domain decomposition methods. Based on the symmetric representation of the local Steklov-Poincaré operators by a symmetric Galerkin boundary element method, we describe a stabilised variational formulation for the local Dirichlet to Neumann map. By a strong coupling of the Neumann data across the interfaces, we obtain a mixed variational formulation. For biorthogonal basis functions the resulting system is equivalent to nonredundant finite and boundary element tearing and interconnecting methods. We will also address several open questions, ideas and challenging tasks in the numerical analysis of boundary element domain decomposition methods, in the implementation of those algorithms, and their applications.

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“…3. Data-sparse BETI preconditioner F for the BETI Schur complements F : Following Langer and Steinbach [2003], we define the inverse BETI preconditioner…”

Section: Solvers and Preconditionersmentioning

confidence: 99%

“…1 Introduction Langer and Steinbach [2003] have recently introduced the BETI methods as boundary element counterparts of the well-established FETI methods which were proposed by Farhat and Roux [1991]. We refer the reader to the monograph by Toselli and Widlund [2005] for more information and references to FETI and FETI-DP methods.…”

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confidence: 99%

Inexact Fast Multipole Boundary Element Tearing and Interconnecting Methods

Langer

Steinbach

Zulehner

Lecture Notes in Computational Science and Engineering

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Summary. The Boundary Element Tearing and Interconnecting (BETI) methods have recently been introduced as boundary element counterparts of the wellestablished Finite Element Tearing and Interconnecting (FETI) methods. In this paper we present inexact data-sparse versions of the BETI methods which avoid the elimination of the primal unknowns and dense matrices. The data-sparse approximation of the matrices and the preconditioners involved is fully based on Fast Multipole Methods (FMM). This leads to robust solvers which are almost optimal with respect to the asymptotic complexity estimates.

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Boundary Element Tearing and Interconnecting Methods

Cited by 93 publications

References 29 publications

Modelling the Process of Sequential Excavation With the Boundary Element Method

Modelling the Process of Sequential Excavation With the Boundary Element Method

Challenges and Applications of Boundary Element Domain Decomposition Methods

Inexact Fast Multipole Boundary Element Tearing and Interconnecting Methods

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