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Technische Universit¨At Graz Technische Universitat¨ Graz The All-Floating Boundary Element Tearing and Interconnecting Method G. Of, O. Steinbach Berichte aus dem Institut f ¨ur Numerische Mathematik Bericht 2009/3 Technische Universitat¨ Graz The All-Floating Boundary Element Tearing and Interconnecting Method G. Of, O. Steinbach Berichte aus dem Institut f ¨ur Numerische Mathematik Bericht 2009/3 Technische Universitat¨ Graz Institut f¨ur Numerische Mathematik Steyrergasse 30 A 8010 Graz WWW: http://www.numerik.math.tu-graz.ac.at c Alle Rechte vorbehalten. Nachdruck nur mit Genehmigung des Autors. The All-Floating Boundary Element Tearing and Interconnecting Method G. Of, O. Steinbach Institute of Computational Mathematics Graz University of Technology Steyrergasse 30 A-8010 Graz, Austria {of,o.steinbach}@tugraz.at Abstract The all-floating Boundary Element Tearing and Interconnecting method inco- operates the Dirichlet boundary conditions by additional constraints in the dual formuation of the standard Tearing and Inteconnecting methods. This simplifies the implementation, as all subdomains are considered as floating subdomains. The method shows an improved asymptotic complexity compared to the standard BETI approach. The all-floating BETI method is presented for linear elasticity in this paper. 1 Introduction Domain decomposition methods offer a comfortable treatment of coupled boundary value problems and provide efficient tools for the numerical simulation in particular by paralleliza- tion. The local subproblems can be solved by the most suitable discretization methods, e.g. finite and boundary element methods. The Boundary Element Tearing and Interconnecting (BETI) method was introduced in [16] as the counterpart of the boundary element method to the well–known Finite El- ement Tearing and Interconnecting (FETI) methods [6, 7], which are widely used in en- gineering applications. For a detailed description of several kinds of FETI methods and a wide collection of references see the monograph [30]. The coupling of FETI and BETI methods is discussed in [17]. The main ideas of these methods are the tearing of the primal global degrees of freedom into local ones and the subsequent interconnecting of the local degrees of freedom across the interfaces by means of constraints and Lagrange multipliers as dual variables to enforce the continuity of the local variables across the interfaces. Subdo- mains without sufficient Dirichlet boundary conditions are called floating subdomains and 1 can be treated separately by using a pseudo inverse. The extra degrees of freedom related to the local rigid body motions have to be eliminated by the use of an appropriate orthog- onal projection, see e.g. [2, 11]. In the case of linear elastostatics, this may become rather complicated since the number of involved rigid body motions may differ from subdomain to subdomain. The so–called FETI–DP methods [5] introduce primal variables by global nodes to overcome these difficulties. The selection of this global nodes is important for the performance of the method and seems to be rather involved for linear elastostatics [13]. In [14, 20], we suggested the “all–floating” BETI method. Related numerical results were presented in [21]. In the meantime, the all-floating formulation was also applied for the coupling of FETI and BETI methods [15, 25]. This new version of the BETI method incooperates the Dirichlet boundary conditions by additional constraints in the dual formu- lation. This unifies the treatment of the subdomains, since all subdomains are considered as floating subdomains. Therefore, the implementation is simplified. Independently, the same idea has been introduced in [3] for FETI methods, called Total-FETI. Additionally, the all-floating version utilizes an improved condition number of the preconditioned local Steklov–Poincar´eoperators [19, 29] and shows a better asymptotic complexity. In this paper, we present the all-floating BETI method for linear elasticity. The iterative inexact solution scheme, which we suggested in [14] for standard BETI methods, needs not more than than O(1 + log(H/h)) iterations in the case of the all-floating formulation compared to O((1+log(H/h))2) iterations for the standard BETI method. H denotes the diameter of a subdomain and h is the local meshsize. Therefore the all-floating formulation is asymptotically faster than the standard BETI formulation. The paper is organized as follows: The primal Dirichlet domain decomposition method is described in Sect. 2 as a starting point of the later derivation of the BETI methods. In Sect. 3, we give a short introduction in the boundary element realization of the local Dirichlet to Neumann maps and present a suitable preconditioning strategy and the related spectral equivalence inequalities. The all-floating BETI method is described for linear elasticity in Sect. 4 in details. There, we state our main result on the computational complexity of the proposed method. Finally, two mixed boundary value problems of linear elastostatics are given as numerical examples in Sect. 5 indicating the improved complexity of the all-floating formulation. 2 Dirichlet Domain Decomposition Method The mixed boundary value problem of linear elastostatics 3 −div σ(u, x)= 0 for x ∈ Ω ⊂ Ê , γ0u(x)= gD(x) for x ∈ ΓD, (2.1) γ1u(x)= gN (x) for x ∈ ΓN , is considered as model problem where Γ = ∂Ω = ΓD ∪ ΓN . For the sake of simplicity, we assume that ΓD and ΓN are identical for all three components. But all results apply, e.g., 2 for the case of different decompositions for each component or for boundary conditions in normal and tangential directions, too. u(x) denotes the vectorial displacements. The 1 ⊤ relation of the stress tensor σ(u) and the linearized strain tensor e(u)= 2 (∇u + ∇u) is given by Hooke’s law Eν E σ(u)= tr e(u)I + e(u). (1 + ν)(1 − 2ν) (1 + ν) E > 0 denotes the elasticity module and ν ∈ (0, 1/2) is the Poisson ratio. The trace operators are defined by γ0u(x) := lim u(x) for almost all x ∈ Γ Ω∋x→x∈Γ e and e ∂ γ1u(x)= λ div u(x)n(x)+2µ u(x)+ µ n(x) × curl u(x) for almost all x ∈ Γ, ∂nx where n is the exterior unit normal direction. Let the bounded Lipschitz domain Ω be decomposed into p non-overlapping subdomains Ωi, i.e, p Ω= Ωi where Ωi ∩ Ωj = ∅ for i =6 j. i=1 [ Γi := ∂Ωi denotes the Lipschitz boundary of a subdomain. For neighboring subdomains the local coupling interface is given by Γij := Γi ∩ Γj for all i < j. The skeleton of the domain decomposition is defined by p ΓS := Γi =Γ ∪ Γij. i=1 i<j [ [ The material parameters Ei > 0 and νi ∈ (0, 1/2) are assumed to be constant in each subdomain Ωi. Thus, the global boundary value problem (2.1) can be rewritten by local boundary value problems −div σ(ui, x)=0 for x ∈ Ωi, i γ0ui(x)= gD(x) for x ∈ Γi ∩ ΓD, (2.2) i γ1ui(x)= gN (x) for x ∈ Γi ∩ ΓN on the transmission conditions i j i j γ0ui(x)= γ0uj(x) and γ1ui(x)+ γ1uj(x)=0 for x ∈ Γij. i i i γ0 and γ1 denote the trace operators of the subdomain Ωi. Note that γ1 is defined with respect to the exterior normal direction of Ωi. A basic tool is the local Dirichlet to Neumann 3 map including the so-called Steklov–Poincar´eoperator Si which is defined by the traction i Sif i(x)= γ1vi(x) of the solution of the local Dirichlet boundary value problem −div σ(vi, x)=0 for x ∈ Ωi, i γ0vi(x)= f i(x) for x ∈ Γi. 1/2 Next, local transmission problems can be formulated for i = 1,...,p: Find ui ∈ H (Γi) such that i γ0ui(x) = gD(x) for x ∈ Γi ∩ ΓD, i γ1ui(x) = gN (x) for x ∈ Γi ∩ ΓN , i j γ0ui(x) = γ0uj(x) for x ∈ Γij, i j γ1ui(x)+ γ1uj(x) = 0 for x ∈ Γij i γ1ui(x) = (Siui)(x) for x ∈ Γi. Here, a Dirichlet domain decomposition method is considered first. The Neumann i traces are replaced by the local Dirichlet to Neumann maps, γ1ui = Siui, and a global 1/2 3 function u ∈ H (ΓS) is introduced to satisfy the continuity of the local functions ui at the interfaces. In other words, the local functions ui are the restrictions ui = u|Γi of the 1/2 1 global function u to the subdomain boundaries Γi. H (ΓS) is the trace space of H (Ω) on the skeleton ΓS. The Dirichlet boundary conditions on ΓD are considered as constraints of the function u, whereas the Neumann transmission conditions and the Neumann boundary conditions are formulated in a variational sense by 1/2 3 (Siu|Γi )(x)+(Sju|Γj )(x) · vij(x)dsx =0 forall vij ∈ H (Γij) ,i<j ZΓij and 1/2 3 (Siu|Γi )(x) − gN (x) · vN (x)dsx =0 forall vN ∈ H (ΓN ) , i =1,...,p. ZΓi∩ΓN The sum over all coupling interfaces Γij and the Neumann boundary ΓN gives the varia- 1/2 3 tional formulation: Find u ∈ H (ΓS) with u = gD on ΓD and p 3 1/2 (Siu|Γi )(x) · v|Γi (x)dsx = gN (x) · v|ΓN (x)dsx for all v ∈ H0 (ΓS, ΓD) . i=1 Γi ΓN X Z Z h i b 3 b b 1/2 The test space H0 (ΓS, ΓD) is defined by h i 3 1/2 1/2 3 H0 (ΓS, ΓD) := v ∈ H (ΓS) : v(x)= 0 for all x ∈ ΓD . h i n o 1/2 3 1 3 If the function u ∈ H (ΓS) is split by u = u + gD, where gD ∈ [H (Ω)] is a suitable and bounded extension of the given boundary data g on Γ , the variational formulation D S b b b 4 3 1/2 reads as: Find u ∈ H0 (ΓS, ΓD) , such that p h i p u b v g v g v (Si |Γi )(x) · |Γi (x)dsx = N (x) · |ΓN (x)dsx − (Si D|Γi )(x) · |Γi (x)dsx i=1 ZΓi ZΓN i=1 ZΓi X X (2.3) b b 3 b b b 1/2 holds for all v ∈ H0 (ΓS, ΓD) .
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