GRUPO DE GEOFÍSICA MATEMÁTICA Y COMPUTACIONAL MEMORIA Nº 3

Transcription

1 GRUPO DE GEOFÍSICA MATEMÁTICA Y COMPUTACIONAL MEMORIA Nº 3 MÉXICO 2014 MEMORIAS GGMC INSTITUTO DE GEOFÍSICA UNAM

2 MIEMBROS DEL GGMC Dr. Ismael Herrera Revilla Dr. Luis Miguel de la Cruz Salas Dr. Guillermo Hernández García Dra. Graciela Herrera Zamarrón Dr. Norberto Vera Guzmán Dr. Álvaro Alberto Aldama Rodríguez Depto. Recursos Naturales Instituto de Geofísica UNAM Publicado y distribuido por: Grupo de Geofísica Matemática y Computacional GGMC Apartado Postal: CP Unidad de Apoyo Editorial eliedit@geofisica.unam.mx ii

3 Parallel Algorithms for Elastic Systems using Multipliers-Free Domain Decomposition Methods By Ismael Herrera and Iván Contreras 1 Instituto de Geofísica 2 de Octubre del 2012 GRUPO DE GEOFÍSICA MATEMÁTICA Y COMPUTACIONAL GGMC 1 Presented at 19th International Conference on Domain Decomposition Methods, Zhangjiajie, China, August, iii

4 CONTENTS Abstract 1 1. INTRODUCTION 1 2 FEM formulation 2 3. Derived nodes and Discontinuous Vectors 4 4. The matrices t A and A 5. THE MF-DDM FORMULATION AND ITS PARALLEL IMPLEMENTATION 7 6. CONCLUSIONS 8 REFERENCES 9 6 iv

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6 Parallel Algorithms for Elastic Systems using Multipliers-Free Domain Decomposition Methods By Ismael Herrera and Iván Contreras 1 Instituto de Geofísica Universidad Nacional Autónoma de México (UNAM) Apdo. Postal , México, D.F. iherrera@unam.mx Abstract This is the third article of the Minisymposium the Multipliers-free Domain Decomposition Methods for Symmetric and Non-symmetric Matrices. In it, we apply the Multipliers-Free Domain Decomposition Method (MF-DDM) to isotropic problems of Static Elasticity. The purpose of the paper is three-fold: to exhibit the applicability of such methods to systems of equations, confirm some of the advantages that are concomitant to MF-DDMs and to establish simple procedures for developing effective codes for parallel-processing problems of elasticity. Firstly, by means a FEM formulation for Static Elasticity we derive the system-matrix associated with this kind of problems. Once this matrix is available, code development for parallel processing only requires a straightforward application of the MF-DDM matrix formulas that unify non-overlapping DDMs and were presented in paper 1 of this Minisymposium: An Overview. The general procedures that were explained in Paper 2, Implementation Issues, are then applied for developing the parallel-processing codes. Such procedures can be applied straightforwardly whenever the matrix-system is available. In conclusion, this paper supplies a simple manner of developing efficient parallel codes for static elasticity, thereby demonstrating the applicability of MF-DDMs to systems of partial differential equations, and also corroborates in this particular case some of the many advantages that are concomitant to MF-DDMs. 1.- INTRODUCTION Equation Section 1 This is the third article of the Minisymposium that in the framework of DD19 was devoted to present and discuss a new methodology known as the Multipliers-free domain decomposition methods: MF-DDMs, which is equally applicable to symmetric and non-symmetric matrices. Such methodology is based on a direct approach, without recourse to Lagrange multipliers, to non-overlapping domain decomposition methods that has been developed in recent years [1-7]. 1 Presented at 19th International Conference on Domain Decomposition Methods, Zhangjiajie, China, August, 2009.

7 Ismael Herrera and Iván Contreras These new methods possess many attractive features; among them: as already mentioned, it yields a framework applicable to symmetric and non-symmetric matrices [7] (indeed, MF-DDMs work nearly as efficiently for non-symmetric matrices as they do for symmetric ones); explicit matrix formulas that unify the different methods (the very same formulas are applicable to both symmetric and non-symmetric problems, independently of whether they originate in a single or a system of differential equations); simplified and very robust codes (for example, a code has been developed that except for a simple change applies to both 2-D and 3-D problems [6], something that is not possible when other approaches are used); and fully parallelizable algorithms. In the case of symmetric matrices, it has been shown that the numerical efficiency of the preconditioned algorithms is at least as good as other state-of-the-art DDMs [6]; albeit, the computational properties of the MF-DDMs are superior. For non-symmetric matrices there is little to compare with, since the literature on DDMs for such matrices is relatively limited [1], but it should be mentioned that the algorithms convergence rates for such cases are nearly as good as those for symmetric matrices [6]. Paper 1 of this minisymposium supplies an overview of the M-F DDMs, while Paper 2 discusses implementation issues from a general perspective. In the present article, we apply the MF-DDM to isotropic problems of Static Elasticity. To this end, we derive the system-matrix associated with this kind of problems using a FEM formulation and, once such a matrix is available, the rest of the treatment is straightforward when the unified matrix MF-DDM formulas are applied. 2.- FEM formulation Equation Section 2 For isotropic materials the differential operator is (see, for example, [8]): L u λ + µ u + µ u (2.1) ( ) May 11 2

8 Parallel Algorithms for Elastic Systems using Multipliers-Free Domain Decomposition Methods Here, the vector-valued function of position, u ( u, u, u ) 1 2 3, is the displacement. In what follows we consider a Dirichlet boundary value problem, in which the displacements are prescribed on the boundary of the elastic body that occupies the domain of the physical space. So: ( λ + µ ) u + µ u = f, in (2.2) u = u, on However, it will be assumed that u = 0 since any problem with non-homogeneous boundary conditions is easily transformed into one with zero boundary conditions. Then, a weak formulation is obtained weighting Eq.(2.2) with a vector-valued function ψ and integrating by parts: A( u, ψ) = {( λ + µ )( ψ )( u) + µ ψ : u} dx= ψ f dx (2.3) Next, a partition of the problem-domain is introduced, whose internal nodes are p = 1,..., N. Sometimes this partition will be referred to as the fine partition because later one more partition, the coarse partition, will be introduced. With each node x p, pi x p we associate a 3D -vector-valued test function to be denoted by ψ, i = 1, 2, 3. Furthermore, let ( pi ψ ) be the j th component of ψ pi. Then, we j choose Here, ( x) p pi ( ψ ( x) ) ( x) = p δ ij j (2.4) is the Lagrange linear interpolate that is characterized by being a piecewise-linear scalar function with the property that p ( q ) δ pq,, = 1,..., x = pq N (2.5) We observe that Eq.(2.5) implies that all the test functions vanish at the nodes located on the domain-external-boundary,. 3 Memorias GMMC

9 Ismael Herrera and Iván Contreras The set of base functions is taken to be the same as the set of test functions. Thus, we define the approximate solution of our boundary-value problem to be: We also define: Eqs.(2.6) and (2.3) together imply that N 3 pi u( x) u pi ψ ( x ) p=1 i=1 (2.6) pi f ψ f d x, p = 1,...,N and i = 1,2,3 (2.7) pi N 3 q= 1 j= 1 A u = f qj, pi qj pi (2.8) Here: qj pi qj pi Aqj, pi {( λ + µ )( ψ )( ψ ) + µ ψ : ψ } dx Using Eq.(2.4), it is seen that 3 p q p q A qj, pi = ( λ + µ ) + µ δij dx xi xj r = 1 xr xr (2.9) (2.10) 3.- Derived nodes and Discontinuous Vectors Equation Section 3,..., P 1 E, the rough partition, and Next, we introduce a second partition of, { } define the set { 1,..., N}. For each α = 1,..., E, the set constituted by the original nodes that belong to the closure of α will be denoted by α and we notice that the union of the family {,..., 1 E} is. At this point we recall the definitions and results presented in paper 1 (see also [#199]); in particular, the derived nodes are pairs p ( p, α ) such that p α P. The whole set of derived nodes is, while for each α { 1,..., E} and each p, we define: {( p, ) ( p, ) } and ( p) ( p, ) ( p, ) { } α α α Ζ α α (3.1) May 11 4

10 Parallel Algorithms for Elastic Systems using Multipliers-Free Domain Decomposition Methods We define the multiplicity m( p ) of p, to be the cardinality of ( p) that the family {,..., 1 E} is a partition of. The sets Ι and Ζ and observe Γ are Ι defined by the following conditions: A node p when its multiplicity is one, otherwise it belongs to Γ. A derived node (, ) Ι p α is interior when p, otherwise it an (internal) boundary node ; the sets of internal and boundary nodes are represented by Ι and Γ, respectively. Two finite-dimensional vector spaces D ( ) and D( ) are defined next; their members are the 3 D -vector-valued functions defined in and, respectively. D is generally greater than Clearly, the dimension of D( ) 3N. Any functions u D( ) p and (, ) is 3N, while that of ( ) and u D ( ) when they are evaluated at nodes p α, respectively, yield 3 D -vectors to be denoted by u( p) and u pi, u( p, α ), respectively; furthermore, we will use the notations ( ) and u( piα,, ) for the i th components of such vectors, respectively. A function u D ( ) is continuous, when u( p, α ) is independent of α. The subspace D 12 ( ) D ( ) is made of the continuous functions of D ( ), while the bijection τ : D ( ) D 12 ( ) D ( ) is the natural immersion of D ( ) into D ( ) ; it is defined by: ( τu)( p, α) u( p ), α Ζ( p ) (3.2) The Euclidean inner product in D ( ) is defined to be: N 3 p= 1 i= 1 (, ) (, ),, ( ) uw u pi w pi uw D (3.3) While the Euclidean inner product in D ( ) is: 5 Memorias GMMC

11 Ismael Herrera and Iván Contreras ( ) α Z p p= 1 i= 1 (,, α) w( pi,, α) m( p) N 3 u pi uw, uw, D (3.4) Here Z( p) { 1,..., E} is defined by the condition that ( p, α ) when Z( p) it can be seen that When uw, D( ) ( ) ( ) ( ) α. uw = τ u τ w (3.5) The average matrix, a, is the orthogonal projection, with respect to the Euclidean Inner product, on the subspace D12 ( ) of continuous vectors. Its explicit expression is a 1 = δ α β pqδij (3.6) ( pi,, )( q, j, ) m( p) while the jump matrix, is j I a. Here, I is the identity matrix. 4.- The matrices t A and A Equation Section 4 γ For γ = 1,..., E, we define the linear transformations A : D ( ) D ( ) by A 3 p q p q δαγδβγ ( λ µ ) µ δij dx β α + + γ xi xj r= 1 xr xr γ ( q, j, )( pi,, ) (4.1) t Furthermore, A : D( ) D( ) matrix A: D( ) D( ) is defined by. To this end we choose a set t A E A α α = 1. Next, we define the Γ and define the set of `primal nodes, Γ, to be defined by the condition that a derived node, (, ) D ( ) D( ) p α, belongs to when p. Then, the dual-primal vector subspace,, is defined by requiring that its vector-members be continuous in is defined the set of primal nodes. Then, the transformation a : D( ) D( ) to be the projection of D ( ) on D ( ). It can be seen that [ ]: May 11 6

12 Parallel Algorithms for Elastic Systems using Multipliers-Free Domain Decomposition Methods a 1 = δpqδpq δαβδpq ( 1 δ α β + pq ) δij ( pi,, )( q, j, ) m( p) (4.2) Here, the symbol δ pq defined by δ pq 1, if p, q 0, if p or q (4.3) Then, the matrix A is defined by A a Aa, the set of primal nodes, is such that A: D ( ) D ( ) Notice that this implies that A: D ( ) D ( ). It will be assumed in what follows that is positive definite. is a bijection. We write The notation here is such that A A A ΠΠ Π A A Π ( ) ( ) ( ) ( ) Π ( Π) ( ) ( ) ( ) A : D Π D Π, A : D D Π ΠΠ A : D D, A : D D Π (4.4) (4.5) Here, Π I and Π. The derived nodes that belong to are referred to as dual nodes. We recall that the notation is such that linear subspace D D is made of the vectors that vanish at every derived node that is not a ( ) ( ) dual node, while D ( Π) is the linear subspace of vectors that vanish at every dual node (i.e., they vanish at ). Furthermore, D = D Π D (4.6) ( ) ( ) ( ) Equation Section THE MF-DDM FORMULATION AND ITS PARALLEL IMPLEMENTATION f D f τ f. Furthermore, We define ( ) by ( ) 1, 2 2 ΠΠ Π Π 2 f f aa f with f = f + f (5.1) 7 Memorias GMMC

13 Ismael Herrera and Iván Contreras Here, we observe that the vectors f D ( Π ) and f D ( ) determined by f D ( ) S, is defined by Then, we seek for a vector u D( ) Π are uniquely Furthermore, the dual-primal Schur-complement matrix, 1 (5.2) Π ΠΠ Π S A A A A such that asu = f and ju = 0 (5.3) 2 Once such a vector has been obtained, the solution of the original continuous problem is 1 1 ( ) 1 u = τ u A A u + A f ΠΠ Π ΠΠ Π (5.4) To obtain the vector u D( ) one can apply any of the MF-DDM basic matrix formulas that unify the non-overlapping domain decomposition methods; they are: The Schur complement method : asu = f and ju = 0 (5.5) The Neumann - Neumann method : as asu = as f and ju = 0 (5.6) The non - preconditioned FETI : S ju = S js f and asu = 0 (5.7) The preconditioned FETI : S js ju = S js js f and asu = 0 (5.8) To implement these algorithms in parallel, we simply apply the procedures of paper 2 [ ], on implementation issues CONCLUSIONS By means a FEM formulation for Static Elasticity we derived the matrix system corresponding to it and, by a straightforward application of the MF-DDM, a simple procedure for developing fully parallelizable computational codes for such problems was obtained. The computational codes so derived are very robust; in particular, using object-oriented programming techniques codes that are applicable May 11 8

14 Parallel Algorithms for Elastic Systems using Multipliers-Free Domain Decomposition Methods to anisotropic materials are easily constructed. Thereby, this paper confirms the applicability of the general MF-DDM matrix formulas and procedures to systems of partial differential equations, as well as other attractive features of the MF-DDMs. REFERENCES 1. Herrera I. and R. Yates, Multipliers-Free Dual-Primal Domain Decomposition Methods for Symmetric and Non-symmetric Matrices: An Overview. In this Volume. 2. Yates R. and Herrera I., Multipliers-Free Dual-Primal Domain Decomposition Methods for Symmetric and Non-symmetric Matrices: Implementation Issues. In this Volume. 3. I. Herrera, Theory of Differential Equations in Discontinuous Piecewise-Defined- Functions, NUMER METH PART D E, 23(3), pp , I. Herrera, New Formulation of Iterative Substructuring Methods without Lagrange Multipliers: Neumann-Neumann and FETI, NUMER METH PART D E 24(3) pp , I. Herrera and R. Yates, Unified Multipliers-Free Theory of Dual Primal Domain Decomposition Methods. NUMER. METH. PART D. E. 25(3) pp , Herrera, I. & Yates R. A., The Multipliers-free Domain Decomposition Methods NUMER. METH. PART D. E (Available Online DOI /num ) 7. Herrera, I. & Yates R. A. The Multipliers-Free Dual Primal Domain Decomposition Methods for Nonsymmetric Matrices NUMER. METH. PART D. E. (submitted), A. Toselli, O. Widlund, Domain decomposition methods- Algorithms and Theory, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2005, 450p. 9 Memorias GMMC