Domain decomposition methods 1 Introduction Domain decomposition techniques are e±cient tools for solving large problems in engineering analysis. Despite their usefulness in dividing large computational tasks in the framework of traditional computer architecture, these methods are particularly attractive regarding parallel computations. These techniques are based on splitting the analysis domain ­ into a number NSD of subdomains s (¯gure 1) such that: NSD[ ­ = ­s (1) s=1 2 Γ2 Γ 4 1 2 Γ 2 2 1 2 Γ3 Γ3 Ω 4 1 Γ 4 Ω 3 Ω 3 Γ3 1 3 Γ1 Ω Γ Γ 3 1 1 3 Γ2 Figure 1: Non overlapping domain decomposition There are several ways of performing domain decompositions. A ¯rst division arises whether the anal- ysis domain is decomposed in overlapping subdomains (Schwarz methods)(Figure 2) or in non overlapping subdomains (i.e. Schur Complement method) (Figure 1). The latter has been widely used in structural mechanics. overlapping region 2 W 1 W Figure 2: Overlapping domain decomposition A second one is related to the discretization technique. A number of nodal points are de¯ned in the domain if the ¯nite di®erence method is employed. In the ¯nite element method, a number of nodal points are de¯ned, but also a number of ¯nite elements connecting those nodes. Thus in ¯nite di®erence methods it may be desirable a partition of the domain like that of ¯gure 3 where nodes are allocated on a given subdomain. This means that there are no nodes on the subdomain boundaries and it is referred to as node-based partition. In ¯nite element methods, a partition where elements be fully contained in a given subdomain is search: an element-based partition (Figure 4). This means that some nodes, the interface nodes, will be located on the subdomain interfaces, and that no elements will be shared among two subdomains. 1 Figure 3: Node-based partition of a ¯nite di®erence grid Figure 4: Element-based partition of a ¯nite element mesh 2 The Schur Complement In this section the ¯nite element method will be supposed to be used for discretizing the continuum problem. Reference will be made to linear elasticity problems where the primal discretized variables are displacements (compatible ¯nite element models). s s Let ­ denote each subdomain and ¡i (i = 1; 3) their boundaries (¯gure 1). ¡1 states for boundaries with kinematical (Dirichlet) conditions , ¡2 for those with mechanical (Neumann) conditions and ¡3 for frontiers with other subdomains. Applying the usual displacement based ¯nite element method to solve a static mechanical problem the following system is get Ku = f (3) where u is a vector containing the discretized unknowns, in this case nodal displacements; f is a vector with the discretized dual variables (nodal forces); and K is the sti®ness matrix. s s s At each subdomain (comprising ­ and ¡3) the subdomain sti®ness matrix K , displacement vector us and force vector f s may be constructed. We can realize partitioning these matrices into a group of internal d.o.f. ºus and a group of interface d.o.f. ¹us. The subdomain sti®ness matrix may be written · ¸ Kºs K~ s Ks = (4) K~ s;T K¹ s 2 and the displacement and force vectors · ¸ · ¸ ºus ºf s us = and f s = (5) ¹us ¹f s The upper symbol º2 means internal d.o.f., 2¹ means interface d.o.f. and 2~ refers to interaction between internal and interface d.o.f.. On the other hand, if we collect the contribution of all subdomains to the interface d.o.f., we can write NSD s KI = A K¹ (6) s=1 as the assembled sti®ness matrix for the whole interface problem, uI being the displacement vector associated to it. Matrices with subscript I have the size of the whole interface problem. Recalling the whole problem, the sti®ness matrix may be written 2 3 Kº1 0 ::: 0 ::: K~ 1 6 I 7 6 0 Kº2 ::: 0 ::: K~ 27 6 I 7 6 . 7 6 . .. 7 K = 6 7 (7) 6 0 0 ::: Kºs ::: K~ s7 6 I 7 6 . 7 4 . .. 5 ~ 1;T ~ 2;T ~ s;T KI KI ::: KI ::: KI and the vectors 2 3 2 3 ºu1 ºf 1 6 2 7 6º2 7 6ºu 7 6 f 7 6 7 6 7 6::: 7 6::: 7 u = 6 s 7 and f = 6 s 7 (8) 6ºu 7 6ºf 7 4::: 5 4::: 5 uI fI The equilibrium equation (3) may be partitioned into the following systems 8 ºs s ~ s ºs <> K ºu + KI uI = f ; s = 1; NSD NSDX (9) ~ s;T s :> KI ºu + KI uI = fI s=1 Performing gaussian elimination by blocks: 8 ºs s ºs ~ s <> K ºu = f ¡ KI uI ; s = 1; NSD NSD NSD X ¡1 X ¡1 (10) ~ s;T ºs ~ s ~ s;T ºs ºs :>[KI ¡ KI (K ) KI ]uI = fI ¡ KI (K ) f s=1 s=1 The coe±cient matrix of the second equation in (10): NSD X ¡1 ~ s;T ºs ~ s S = KI ¡ KI (K ) KI (11) s=1 is known as the Schur complement matrix or capacitance matrix. The ¯rst group of equations in (10) represents the system of equations for the internal unknowns ºus at each subdomain, resulting from the non penetration character of the decomposition. This part of the solution is perfectly parallelizable. The size of these problems is given by the granularity of the domain decomposition. The second part of (10) represents the interface problem. The size of the Schur complement S is usually much smaller than that of the global matrix K, but it is also more dense than K. The condition number of S is usually much smaller than that of K. On the other hand matrix S may not be explicitly assembled and the solution performed in a subdomain-wise fashion. This part of the solution is coupled for the whole problem. The e±ciency of the global solution is highly tightened to the e±ciency of the solution for the interface problem. 3 3 Direct Solution Method A way of solving problem (10) is by means of direct methods. The interface problem (second equation in (10) ) is solved in the ¯rst place. The Schur complement (11) is computed. It requires that for each subdomain the matrix for internal unknowns Kºs be factorized. Then a substitution is performed using ~ s ~ s;T as rhs vector each column of matrix KI . The matrix so obtained is pre-multiplied by KI and summed over the number of subdomains. This completes the second term of eq. (11). In a similar fashion the second term of the second equation in (10) is formed: solving the internal ºs ~ s;T problem with f as rhs, pre-multiplying the vector obtained by KI and summing over NSD. This completes the second term of the rhs. Once the interface problem (10-b) is solved uI is used to solve the internal problem at each subdomain (eq. 10-a). The algorithm to solve the problem by a direct method is given in Table I. Table I: Solution by direct methods 1. Factorize Kºs, for s = 1; NSD 2. By substitution compute: ¡1 s ºs ~ s A = K KI ¡1 Bs = Kºs ºf s 3. Compute: NSD X ¡1 ~ s;T ºs ºs gI = fI ¡ KI (K ) f s=1 4. Compute: NSD X ¡1 ~ s;T ºs ~ s S = KI ¡ KI (K ) KI s=1 5. Solve S uI = gI 6. For each s = 1; NSD compute s s s ºu = B ¡ A uI If the problem is well posed, the global matrix K is nonsingular. Each of the Kºs, as well as the Schur complement S are also non singular. Furthermore, if matrix K is symmetric and positive de¯nite, as happens in linear elastic problems, so is S. The explicit construction of the Schur complement is no practical for large problems. 4 Iterative Schur Complement Methods The problem (10) may be seen as being solved in two steps: an interface problem and an internal one. A popular strategy is to solve the internal problem (10-a) by direct methods and the interface problem (10-b) by iterative ones. Direct methods are preferred for (10-a) since they lead to close solutions and no error propagation is introduced to the interface problem. Direct methods, however, are not suitable for the interface problems due to their large storage re- quirements. Matrix S is usually full and expensive to construct. Some applications of direct methods are reported in the literature, but they are mainly concerned with special cases (e.g. slender structures) 4 where the interface problem size is limited. (Farhat and Crivelli, 1989; Lessoine et al, 1991). Iterative methods perform well for the interface problem, in particular conjugate gradient techniques with pre- conditioning. As it was already said, the condition number for the Schur complement is less than for 1 1 the whole sti®ness matrix. In the case of a Laplace problem, cond(K) = O( h2 ) while cond(S) = O( h ). Domain decomposition can be seen as a way of preconditioning the whole problem. 5 Solution of the interface problem 5.1 The Preconditioned Conjugate Gradient Method We will focus on the solution of the interface problem (10-b). The interface d.o.f. matrix KI may be written NSDX s KI = KI (12) s=1 and the Schur complement NSDX S = Ss (13) s=1 with s s ~ s;T ºs ¡1 ~ s S = KI ¡ KI (K ) KI (14) Eq. 13 shows that contributions of each subdomain to the matrix S may be computed independently.