A Component Mode Synthesis Method of In nite Order

A Component Mode Synthesis Method of In nite Order of Accuracy Using Subdomain Overlapping: Numerical Analysis and Experiments by I. Charpentier1 , F. De Vuyst2 and Y. Maday3 1 Projet

IDOPT (C.N.R.S.-U.F.J.-I.N.P.G.-INRIA), Laboratoire de Modelisation et de Calcul -IMAG, tour IRMA, B.P. 53, FR-38041 Grenoble Cedex 09, France. 2 Laboratoire d'Analyse Numerique, Universite Pierre et Marie Curie, Tour 55-65, 5 etage, 75252 Paris Cedex 05. 3 ASCI, U.P.R. 9029, Bat. 506, Universite Paris Sud, 91405 Orsay Cedex and Analyse Numerique, Universite Pierre et Marie Curie, 75252 Paris Cedex 05.

Abstract. - We present here a new method of component mode synthesis for solving eigenvalue problems of selfadjoint compact partial dierential operators. It consists in a Galerkin method based on discrete functions that are spanned by the solutions of similar eigenvalue problems on smaller overlapping subdomains. From the numerical point of view, the resulting method as other component synthesis methods provides a lot of advantages in terms of storage and computation requirements. Furthermore, as a result of the subdomain overlapping, it is shown that the method has an in nite order of accuracy even if the eigensolutions have singularities at the corners of the boundary. The numerical results con rm the rapid convergence of the method. Key words. - Selfadjoint compact operator; eigenvalue problem; component mode synthesis method; subdomain overlapping; spectral convergence. Resume. - On presente ici une nouvelle methode de synthese modale pour la reso-

lution de problemes aux valeurs propres pour des operateurs aux derivees partielles autoadjoints. Celle-ci consiste en une methode de type Galerkin dont les fonctions de base discretes sont de nies a partir des solutions de problemes aux valeurs propres similaires sur des sous-domaines recouvrants. D'un point de vue numerique, la methode ainsi de nie, comme toutes celles de synthese modale, est interessante pour ses faibles co^uts en termes d'occupation memoire et de calcul. Mais la strategie de recouvrements de sous-domaines nous permet de plus d'obtenir une methode dont la precision est d'ordre in ni m^eme si les solutions propres possedent des singularites aux eventuels coins de la frontiere. Les resultats numeriques con rment la convergence rapide de la methode. Mots cles. - Operateurs compacts auto-adjoints; problemes aux valeurs propres; methode de synthese modale; sous-domaines recouvrants; convergence spectrale.

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A COMPONENT MODE SYNTHESIS METHOD OF INFINITE ORDER OF ACCURACY USING SUBDOMAIN OVERLAPPING: NUMERICAL ANALYSIS AND EXPERIMENTS Isabelle CHARPENTIER1 , Florian DE VUYST2 and Yvon MADAY3 1 Projet IDOPT (C.N.R.S.-U.F.J.-I.N.P.G.-INRIA), Laboratoire de modelisation et de calcul IMAG, tour IRMA, B.P. 53, FR-38041 Grenoble Cedex 09, France. 2 Laboratoire d'Analyse numerique, Universite Pierre et Marie Curie, Tour 55-65, 5 etage, 75252 Paris Cedex 05. 3 ASCI, U.P.R. 9029, Bat. 506, Universite Paris Sud, 91405 Orsay Cedex and Analyse Numerique, Universite Pierre et Marie Curie, 75252 Paris Cedex 05. Abstract. - We present here a new method of component mode synthesis for solving eigenvalue problems of selfadjoint compact partial dierential operators. It consists in a Galerkin method based on discrete functions that are spanned by the solutions of similar eigenvalue problems on smaller overlapping subdomains. From the numerical point of view, the resulting method as other component synthesis methods provides a lot of advantages in terms of storage and computation requirements. Furthermore, as a result of the subdomain overlapping, it is shown that the method has an in nite order of accuracy even if the eigensolutions have singularities at the corners of the boundary. The numerical results con rm the rapid convergence of the method. Key words. - Selfadjoint compact operator; eigenvalue problem; component mode synthesis method; subdomain overlapping; spectral convergence.

Contents

1 Introduction. - Why chosing subdomain overlapping ? 2 2 Implementation of the component mode synthesis method with overlapping. 4 3 Numerical analysis of the method with overlapping. 8 4 Numerical results 11 Acknowledgments 26 References 26 Appendix: component mode analysis with overlapping domain decomposition (Note submitted and accepted to C.R. Acad. Sci. Serie I, 1996) 27

1 Introduction. - Why chosing subdomain overlapping ? Amongst the methods for solving eigenvalue problems of elliptic partial dierential operators, the Component Mode Synthesis (CMS) methods provide a powerful tool for computing the eigenmodes of complex structures because of their eciency in terms of accuracy and because of their relative low cost in memory storage. The basic feature of the CMS methods lies in the decomposition of the global structure in subdomains and the computation (either exactly or approximately) of the eigenmodes of the same operator over each subdomain. These local eigenmodes then serve as basis

2

functions for approximating the global problem. One can easily imagine the natural implementation of these methods in parallel. From the industrial point of view, it can also be noted that, most often, the nal structure under consideration is conceived from a set of component parts that can be sent with precomputed libraries of corresponding (local) eigenmodes. As far as we know, all the variants of the CMS methods deal with nonoverlapping domain decompositions and dier from the de nitions of the \good" coupling modes at the interfaces of the substructures. These modes are necessary to reconstruct the global eigenmodes. Hurty [11], Craig and Bampton [7] have rst proposed some \static" interface modes built from a nite element discretization of the interfaces and a harmonic extension of the interfacial nite element basis. Bourquin [4] has rather proposed to express the interfacial contribution by computing the eigenmodes of the Poincare-Steklov operator on the interface and extend harmonically these modes on the whole domain. The numerical analysis of the CMS methods has been investigated, and error bound estimates on the eigenvalues or the eigenfunctions have been established. This allows us not only to determine which CMS method is the most accurate, but also to understand why the CMS methods work so well. Bourquin has rst exhibited error bounds for both Hurty's, Craig-Bampton methods and his own method in one, two or three space dimensions. We do not intend to present again the numerical analysis of these methods and refer to [4] for more details but we want to explain why the original method is intrinsically of nite order of accuracy even though the basis functions that are used could oer an in nite order of accuracy as it is the case in spectral methods. For the sake of simplicity of our discussion, let us deal with a one-dimensional problem: nd (; u) 2 IR  H01 (]0; 1[) such that  u00 =  u on I =]0; 1[ ; (1) uj@I = 0: It is well-known that this problem has an in nite countable set of solutions fi ; uigi=1;+1 with i > 0 8i  1, arranged in increasing order 1 < 2 < ::: < +1: We even know that i = i2 2 and ui(x) = sin(ix). Of course for this simple problem the method is only relevent to provide a clear explanation on the behaviour of the convergence rate of the approximation. Let us decompose I into two nonoverlapping intervals I 1 =]0; [ and I 2 =]; 1[, 0 <  < 1. Here, Hurty's and Bourquin's methods are equivalent, and, in addition to the local eigenmodes, they involve one only interface mode (x) that is 1 at x = , 0 at both x = 0 and x = 1 and is harmonic | thus linear | over I 1 and I 2 . The local eigenmodes are the solution of problem (1) where I is replaced by I 1 and I 2 and are extended by 0 to be de ned over I. The method then consists in a Galerkin approximation: nd ( ; u ) 2 IR  X such that

8v 2 X ;

Z

Z

I

u0 v0 =  u v I

where X is spanned by the interface mode , the N 1 rst eigenmodes over I 1 and the N 2 rst eigenmodes over I 2 . Bourquin has shown that the relative error in the energy norm made on the ith eigenfunction behaves like C(i; ") (inf k (N k )) 3=2+" , where C(i; ") is a positive constant, bounded for any i, and any " > 0. A way to explain why the convergence is of nite order of accuracy lies in the analysis of the best t of the global mode ui by elements of X . Since the local eigenmodes all vanish at the interface x = , the only way for ui to be well approximated is that the contribution of the \hat"-type function  represents the value ui (). The best t of ui is now obtained from the local best ts of the restriction fik of ui ui () to each I k . Let us consider for instance the function fi1 . It is an easy matter to reproduce it by antisymmetry to get an odd function de ned over ] ; [ and then to extend it by periodicity to get a function over IR, 2{periodic, denoted by Fi1. The best t of fi1 by linear combination of the

3

eigenfunctions u1j = sin(i x ), j = 1; ::; N 1, over I 1 is clearly the restriction of the best t of Fi1 by the truncated Fourier serie of degree N 1. The produced error depends on the regularity of the function Fi1. We know that u lies in C 1 (I). Now, at x = 0, where fi1 has been prolongated, all the odd derivatives of Fi1 are continuous (by construction), and all the even derivatives are 0, because u is solution of problem (1). At x = , Fi1 is continuous, with a continuous derivative (by construction), but its second derivative is discontinuous. This is due to the fact that most generally (u )00 ( ) = u00() 6= 0, then (Fi1)00( ) 6= (Fi1)00(+ )! It is even an easy matter to check that Fi1 lies in H 5=2 (I 1 ) for any  > 0 but not in H 5=2(I 1 ), which explains the error bound found by Bourquin. In opposition to the standard CMS methods, our approach is based on an overlapping domain decomposition I = I 1 [ I 2 . We state that no interfacial mode is necessary because the set fu1i ; u2i gi=1;+1 of the solutions of the eigenproblem de ned on each I k , k = 1; 2 forms a (non orthogonal) basis of H01 (I). Then we can de ne a CMS method through a Galerkin approximation based on the space X spanned here only by the fu1i ; u2j gi=1;N 1 ;j =1;N 2 . The approximation is then of in nite order of accuracy as can tell a close look at the best t of ui . We decompose ui(x)  ui (x) 1(x) + ui (x) 2(x); where fk gk=1;2 is a partition of the unity: 1 

2 X

k=1

k :

One can always assume that is a positive, C 1-function that is nonzero only on I k . Let xI 1 denote the rightward abscissa of the interval I 1 . Then, let us apply on 1ui the procedure of extension by antisymmetry and periodicity as described above. The resulting periodic function has now the regularity of ui itself since at x = 0, the function 1ui has all its derivatives continuous, because 1 ui  ui in a vicinity of zero, the same is true at xI 1 since at this point all derivatives cancel. Therefore, the truncated Fourier serie converges with an error that decays exponentially fast. All the power of the approximation by such nice functions is now at hand. We nish the preliminary discussion by noting that the present method can be introduced in a computional code that already uses a CMS method with nonoverlapping domains. It is enough to cover each interface by an overlapping domain that we call a \sparadrap" (for example, a small rectangle if it is possible). Thus, we can keep the precomputed libraries of local eigenmodes of each component part and take advantage of them. That makes the overlapping CMS methods at least as interesting from the industrial point of view as the standard ones. The outline of the paper is as follows. In section 2 we introduce the basic concepts of the overlapping method and detail its implementation. In section 3 we present the numerical analysis of the method. The main result is that it is in nite order accurate. Numerical results are detailed in section 4 for both one and two dimensional domains. The in nite order of accuracy is veri ed on these numerical examples. A comparison with standard CMS methods con rms the superiority of this new approach. k

2 Implementation of the component mode synthesis method with overlapping. In all that follows, we will only deal with the Laplace operator for the sake of simplicity, but we state that the present method can be used with other selfadjoint compact operators. We consider the following problem: nd the pairs (; u) 2 IR  H01( ) such that 
u =  u in ; (2) u = 0 on @

4

where is a bounded domain. It is known that problem (2) admits an in nite countable set of eigensolutions (i ; ui) with a sequence fi gi=1;+1 of strictly positive eigenvalues conventionally arranged such that 0 < 1  2  ::: < +1: More, the family fuigi=1;+1 can be chosen so as to form an orthogonal basis of both the Hilbert spaces L2 ( ) and H01 ( ) with respect to the standard scalar products (u; v)0; =

Z



uv dx; (u; v)1; =

Z



ru
rv dx:

with norms denoted by k:k0; and j:j1; respectively. We decide to normalize the family fuigi=1;+1 such that kuik0; = 1; 8i  1: To determine the N rst global eigenpairs (i ; ui ), we construct an overlapping domain decomposition of such that

= [Kk=1 k : On each k , we then consider local eigenpairs (ki ; uki )k=1;::;K (ordered with respect to increasing ki ) solutions of the problems 
uk = k uk in k; i i i (3) uki = 0 on @ k : From the numerical point of view, we deal with a nite set of eigensolutions on k : the N k rst eigensolutions on each domain (this choice will be justi ed at the numerical analysis step). In order to work with functions de ned on the whole domain , we extend uki by 0 and denote by ufki this =1;K lies in H 1( ). As explained before, we want to determine extension. It is clear that fufki gki=1 0 ;N k =1;K the approximations (i; ; ui; )i=1;N in the vector space X spanned by the nite basis fufki gki=1 ;N k P of the N (N  N  = Kk=1 N k ) rst global solutions (i ; ui)i=1;N . Variational Galerkin method. - The de nition of these approximations amounts to determine the coecients cki;` with ui; (x) =

K X N X k

k=1 `=1

cki;` ufk` (x) 8 i = 1; ::; N; 8 x 2 :

(4)

This is done through the veri cation of a variational Galerkin approximation of (2) based on the vector space X : nd (i; ; ui; ) in IR  X such that

Z



rui; (x)
r (x)dx = i;

By substituting ui; by its form (4), we obtain

Z



ui; (x) (x)dx 8 2 X :

Z Z X K X N K X N X cki;` rufk` (x)
r (x)dx = i; cki;` ufk` (x) (x)dx 8 2 X : k

k

k=1 `=1



k=1 `=1

(5)

(6)

=1;K allow us to Successive choices for the trial function in the space X of local modes fufki gki=1 ;N k  state the problem into an equivalent matrix form of size N : nd the eigenpairs f(i; ; Ci; )gi=1;N  ,

solutions of the matrix eigenvalue problem

S Ci; = i; M Ci; ;

5

(7)

where S (resp. M) denotes the stiness matrix (resp. the mass matrix), and Ci; contains the coecients cki;` corresponding to the eigenvalue i; . These matrices are made of dierent blocks indexed by pq that express the interaction between the modes of the subdomain p and the modes of the subdomain q : 0 S11 S12 ::: S1K 1 0 Ci1; 1 0 M 11 M 12 ::: M 1K 1 0 Ci1; 1 B B@ S21 S22 ::: S2K C Ci2; C B@ M 21 M 22 ::: M 2K CA BB Ci2; CC (8) B C =  A i ;  @ ::: A @ ::: A ::: ::: ::: ::: ::: ::: ::: ::: S K 1 S K 2 ::: S KK M K 1 M K 2 ::: M KK CiK; CiK; where (Ci1; ; Ci2; ; :::; CiK;)t denotes the partitioned vector Ci; . The dierent blocks S pq and M pq are computed through integrals over p \ q . For instance, one has, for all couple of integers (p; q) in (1; :::; K)2,

R

pq = p q p q S`m

pR\ q ru~` (x)
ru~m (x)dx 8 ` 2 1; ::; N ; 8 m 2 1; ::; N ; pq p q p q M`m = p \ q u~` (x)~um (x)dx 8 ` 2 1; ::; N ; 8 m 2 1; ::; N :

Of course, S pq = M pq = [0]N p N q if p and q do not overlap. Otherwise, due to the orthogonality of the local modes, the former integrals computed for p = q = k have simple values:

R

kk = k rufk(x)
ruf km (x)dx = k` `;m 8 (`; m) 2 (1; ::; N k)2 ; S`m

R ` kk = k u~k (x)~uk (x)dx = `;m 8 (`; m) 2 (1; ::; N k)2 ; M`m m

`

i.e. the blocks fS kk gk=1;K and fM kk gk=1;K are diagonal matrices. It is quite easy to check that matrices S and M are symmetric and positive de nite, since they are obtained from a Galerkin method. From the algorithmic point of view, this remark constitutes the main dierence between our CMS overlapping method and the standard nonoverlapping ones. In fact, the standard methods which involve interface modes lead to a diagonal stiness matrix S whereas the method we propose induces nondiagonal block matrices. This slight dierence and its consequences on the complexity of the algorithm are hugely balanced by the large gain in accuracy as we will see below. Solution of the eigenvalue problem (7) . - Since matrix M is symmetric positive de nite, we can use a Choleski factorization for M: there exists an upper triangular matrix B such that M = tB B where t B is the transposed matrix of B. As a consequence, solving (7) is equivalent to solve the eigenvalue problem: nd all pairs (; x) 2 IR  IRN  solutions of ((tB) 1 S B 1

I) x = 0:

(9)

For the sake of simplicity, we set A = (t B) 1 S B 1 : From the above construction, we see that A is also a symmetric positive de nite matrix. Now, the identi cation of the eigenpairs are rst realized through a Jacobi algorithm which computes all the eigenvalues i for i = 1; ::; N . One can notice that i;  i from the Rayleigh-Ritz principle. We then achieve the computation of the N rst eigenvectors by applying an algorithm of inverse powers realized directly on system (7) to avoid the errors of computation added by the Cholesky factorization and the computation of B 1 . These eigenvectors give us the coecients cki` that determine an approximation in X of ui, thanks to (4). Analytic solutions on rectangles and camembert-shaped domains. - The CMS method can take advantage of the fact that we know the exact solutions of problem (2) when k is a

6

rectangle or a camembert-shaped domain, i.e. de ned as f(r; ); 0 < r < R; 1 <  < 2 g. One can admit that any two-dimensional polygonal domain can be decomposed as = [k k where the subdomains ( k ; k = 1; K) are either rectangles or camemberts. We directly see the interest of such a method. This allows us to approximate the solutions of eigenproblems on such domains without precomputing any local mode by a standard technique of approximation ( nite dierences, nite elements). The analytic solutions of problem (1) are simply sine functions. The analytic solutions of problem (2) where is a rectangle are then tensorized products of one-dimensional sine solutions due to separation of the variables. The analytic solutions of (2) when k is a camembert-shaped domain can be obtained by solving the Laplace operator in polar coordinates. On such a domain , problem (2) can be reexpressed as: nd (; u) 2 IR  H01( ^k ), ^k =]0; R[]1; 2[ such that @ 2 u 1 @u 1 @ 2 u =  u on ^k ; (10) @r2 r @r r2 @2 u(R; ) = 0 8 2]1 ; 2[; (11) u(r; 1) = u(r; 2) = 0 8r 2]0; R]; (12) so that a singularity appears at r = 0; there is no boundary condition at r = 0. Problem (10)-(12) can be written under a variational form. It is shown to be well-posed in a particular functional context (see [13]) and to possess solutions in H01 ( k ). In (10)-(12), we can separate the variables r and  because of the rectangular domain in the space (r; ). So, we look for an eigensolution (; u) where u is written as u(r; ) = f(r) g(). Then ( ; g()) and (; f(r)) are respectively the solutions of  g00 () =  g() on ]1 ; 2[; (13) g(1 ) = g(2 ) = 0;  r2 f 00(r) + r f 0(r) + ( r2  ) f(r) = 0 on ]0; R[;  (14) f(R) = 0: The eigenfunctions fgi gi1 of p(13) are sine functionspwith eigenvalues i = 2 i2 =(2 1 )2 . Denoting i =  i=(2 1 ), z =  r and fe(z) = f(z= ), problem (14) can be rewritten as

p

e = 0 on ]0;  R[; z 2 fe00 (z) + z fe0 (z) + (z 2 i 2) f(z) p e  R) = 0: f(

(15) (16)

We immediately see that equation (15) is a real Bessel equation of order  = i, and the solutions of (15), (16) are Bessel functions of order i. Furthermore, we see that admissible values of  satisfy an implicit equation (16). But the Bessel functions all admit an in nite countable set of zeros so that such \eigenvalues"  exist. The two kinds of independent Bessel functions solutions of (15), (16) are usually denoted by J (z) and Y (z). The second ones are of no interest in the present context since they do not lie in H01( ): they are too singular in the vicinity of r = 0 and we refere the reader to [14] for more details.p On the other hand, for any admissible value  of (15), (16), the solutions u(r; ) = gi()Ji ( r) always lie in H01( ). More precisely, at xed i, there exists an in nite countable set f(i;j )gj =1;+1 of admissible eigenvalues. We use new indices (`)`=1;+1 in bijection with (i; j) = (i; j)(`) such that the eigenvalues f` g`=1;+1 = fp(i;j )(`)g(i;j )2IN 2 are arranged in increasing values. Then, using this convention, u` (r; ) = gi ()Ji ( (i;j ) r). Note that, in general, the solutions u` are singular at r = 0, and at high energy, i.e. for large values of ` , (@u=@) can strongly oscillate in a vicinity of r = 0.

7

Now, about the CMS method with overlapping, if eigensolutions on camemberts are considered, the computation of the stiness and mass matrices requires the computation of integrals of eigenfunctions on parts of camemberts. It cannot be computed exactly and, because of the oscillatory behaviour of the eigenfunctions on camemberts near a corner, the numerical integration should be made on a polar grid suciently re ned in both the radial and angular direction to catch reasonable gradients. Remark that one could think that the \camembert strategy" is hard to implement because of the p resolution of implicit equations in : Ji ( R) = 0. But, because of properties of invariance of the shape of the eigensolutions by homothety in r and by similitude in , one can once compute the rst Bessel functions of problem (15),(16) for few rst values of i, tabulate the dierent solutions and save it once and for all. The cost of memory or disk occupation is negligeable because we only consider tabulated one-dimensional functions. Let us remark that the family fu` g`=1;+1 is also naturally orthogonal in L2 because it is generated by an eigenvalue problem of a compact operator and because the eigenvalues are not multiple.

3 Numerical analysis of the method with overlapping. The standard tool for the numerical analysis of the approximation of eigenvalue problems is indicated for instance in the synthesis paper of Babuska and Osborn [6] that, in the Hilbert context of selfadjoint operators, states that there exists an index j such that jlambdaj ; i j  C  2 (i ) where, for any eigenvalue  of associated eigenspace V (), we have  () = v2V (max inf kv v k1; : (17) );kvk =1 v 2X 1;

Moreover,





2 i  i;  i + C 0 1max j i  (j ):

(18)

The case of a boundary of class C 1 has already been presented in [8]. Here we want to deal a more general framework. We shall assume that the domain allows for a domain decomposition that is large in the sense that the Hausdor distance between [l6=k l n k and k n([l6=k l ) is strictly positive. If the domain has no corner, such a decomposition is easy to conceive, at least with sparadrap subdomains. In the case where some corners do exist, the use of camembertshaped domains at the level of corners allows us to ful ll the same P assumptions. To such a domain decomposition, we associate a smooth partition of unity 1  Kk=1 k , where (k ; k = 1; K) are positive functions of class C 1 that are zero outside k . The assumption of large decomposition allows us to construct such a partition of unity that satis es, in addition @ r k = 0 on @ k ; for all integer r; r  1: (19) @nr We note X the set of the partitions of the unity that satisfy the above mentioned hypotheses. The existence of corners (each parametrized by ; 1 <  < 2 ) in the domain of computation makes appear singularities in the eigenfunctions. More precisely, the regularity properties of problem (2) are the following ones (see [10]): the application
u 7! u, where u is a solution of problem (1) is continuous from H s 2 ( ) to H s ( ), for any real s  1 satisfying   s < inf 1 +    ; 3 : (20) 2 1 Therefore, as often in numerical analysis of accurate methods, the rate of convergence of the method should be upper bounded by the low regularity of the solutions we want to approximate. This is

8

actually the case if standard nite element or spectral element methods are used to compute the approximations. Here we shall prove that an in nite order of accuracy is nevertheless present even in the case of singular functions. The presence of the camembert shaped domain allows us to capture the singular part of the solution. In this direction the idea of the introduction of the camembert looks like the idea of adding the singular parts of the solution for the approximation of the solution of PDE. Here it appears that not only the rst singularities are captured, but all of them since the approximation is of in nite order of accuracy. In order to prove this result, we shall assume a little more about the choice of the partition of unity. We suppose that, in a vicinity of a corner xk of angle (1k ; 2k ), the associated function k , the support of which Supp(k ) is equal to the camembert f(r; )= 0 < r < Rk ; Rk > 0; 1k <  < 2k g, is identically equal to 1 in a suciently large domain kR : k  1 in kR def = f(r; )= 0 < r < R; 0 < R < Rk ; 1k <  < 2k g. Let us denote by Xc the set of all the partitions of the unity that lie in X and partition the domain corners following these last hypotheses. Our main result of error estimate lies in this essential lemma that we rst state and prove: Lemma 1. - For any partition fk gk=1;K in Xc and any eigensolution ui of problem (2), the following stability result holds:
p(k ui) 2 H01 ( k ) 8p  0; 8k 2 f1; :::; K g; 8i  1 where
p (:) denotes the pth power of the Laplace operator. In addition, let us set



max
p k v H 1 ( k ) : MXc ; (p) = v2V (max min );kvk0; =1 fk gk=1;K 2Xc k21;:::;K

(21)

(22)

there exists a constant c, independent of i such that MXc ;(p)  c((i )p + 1):

(23)

Proof of Lemma 1. - We consider two dierent cases.  If Supp(k ) does not contain a corner, then (k ui) lies in C 1 ( k ) because, on the one hand, ui 2 C 1 ( nfAigi=1;nc), where Ai denotes a corner and nc is the total number of corners, and because, on the other hand, k (Ai
)  0 in a vicinity of Ai for all i in f1; :::; ncg. Now, we have to prove that the trace of
p(k ui) vanishes on @ k for any p  0. It is clear for p = 0, because, u does vanish over @ while (k )j@ kn@ = 0 by de nition of fk gk=1;K . Now, for p = 1, the following equality holds in the sense of distributions:


(k ui) =
k ui + k
ui + 2 rk
rui on k : (24) First, (
k ui )j@ k = 0 because (ui)j@ = 0 and
k = 0 on @ k n(@ ). Second, (k
ui)j@ k = ( i k ui )j@ k = 0 because (ui )j@ = 0 and k = 0 on @ k n(@ ). Finally, (rk
rui)j@ k = 0 because k @ui @k @ui ; rk
rui = @ + @n @n @ @ i where n (resp. ) denotes the normal (resp. tangential) vector to @ k , but also @u @ j@ = (ui )j@ = k @k = 0 on k (formula (19)). The assertion 0, @ = 0, and it has been supposed that @ j@ kn(@ ) @n from p = 2 is proven from the fact that 2 (k )@ 2 (ui ) (25)
2(k ui ) =
2k ui + k
2ui + 4 r(
k )
rui + 4 rk
r(
ui) + 4 @ 

9

so that, it is a sum of contributions that cancel at the boundary, either due to the fact that { u (or
r u) vanishes over @ , { k vanishes over @ k n(@ ) { the term is a product of some derivative of the normal derivative of k thus vanishes over @

from (19) { no derivative of the normal derivative of k appears but then only tangential derivatives of u are present so that the product does also vanish over @ . Finally, the argument can be repeated over for any p.  If k captures a corner, then k ui does not lie in C 1 ( k ). We again distinguish two cases: { over kR , the equality (24) reduces to
(k ui) =
ui = i ui because k is identically equal to 1. Recursively, one nds
p(k ui ) = ( i )p ui that belongs to H 1( kR ). More,
p(k ui )j@ \ kR = ( i )p ui j@ \ kR = 0; { over k n kR , then again (k ui ) 2 C 1 ( k n kR ), so that
p(k ui ) 2 H 1 ( k n kR ) for all p. For the same reasons as those discussed above, we could also show that
p (k ui )j@ k\( k n kR ) = 0 for any p. This ends the proof of lemma 1. We can now easily prove our main result: Theorem 2. - For any integer p > 0, there exists a constant C > 0, such that, for any set of integers fN k gk=1;K ,  ()  C M;(p) p ; (26) k where  = inf k2f1;:::;K g N k +1 . Proof of Theorem 2. - Let us consider an integer ` such that `  inf Kk=1 N k . We de ne a =1;K g de ned as projector P;` from H01 ( ) to X;` def = Spanffuki gki=1 ;` K X k (k w); 8w 2 H 1( ); P;` w def = P;` 0 k=1

fk k de ned as the orthogonal projector from H 1 ( ) to X k def with P;` 0 ;` = Spanffui gi=1;`g with respect to the scalar product (:; :)1; . For any eigenfunction ui of problem (2), the functions k ui are well k (k ui ), so that we have also a good approximation of ui by some functions approximated by P;` of X;` because inf ju vj1;  jui P;` ui j1;

v2X;` i X ! K k   ui k=1 K X k  =

k=1 K X

k=1

K X

k=1

! 1;

k (k ui) P;`



k (k ui)  ui P;` 1;





infk k ui v 1; k : v2X;`

(27)

Because of the orthogonality property of the each family of functions fuki gi=1;` and with the assumption of normalization in L2 ( ), we have for any function ' of H01( k )

j'j21; k = then

+1 X

i=1

Z

ki [

k

' uki ]2

' P k (') 2 = +X1 k [Z ' uk]2: ;` i i 1;

k

i=`+1

10

k

Because uki is a solution of problem (3), it is standard to remark that

Z

k

so that

' uki

Z

k) i = = ' (
u ki

k

1 Z (
') uk ; i ki k

' P k (') 2  ( c )2 j
'j2 : ;` 1;

1;

k`+1 k

k

Iterating p times this argument, in the case where
r ' belongs to H01( k ), for any integer r, 1  r  p, leads to ' P k (') 2  ( c )2p j
p'j2 : ;` 1; k 1; k k`+1 This result can be applied to ' = k ui because of (21) (see Lemma 1), this allows us to obtain, for any p, !2p c 2 k k
p(k ui )k21; k ; infk  ui v 1; k  k v2X;` `+1 and prove theorem 2 from (27).

4 Numerical results

Numerical results on one-dimensional problems. - Several one-dimensional experiments

have already been discussed and detailled in [8]. We give here additional results computed on nonsymmetric subdomains. On table 1 we compare the results given by our method to those obtained by the classical CSM method proposed by Hurty and Craig-Bampton. Our overlapping method is performed on both a \sparadrap" overlapping (I 1 =]0; 0:6[, I 2 =]0:6; 1[, I 3 =]0:3; 0:7[, N 1 = N 2 = N, N 3 = N=2) and a two-subdomains overlapping (I 1 =]0; 0:7[, I 2 =]0:3; 1[). The improvement, when overlapping subdomains are used, is noticiable provided that a sucient number of basis functions is used. During the experiments, when the number of basis functions has achieved a critical level, a problem may occur in the Choleski factorization of the mass matrix (section 2). Indeed, even though the vectors that span X are exactly linearly independent, the determinant of the mass matrix is so small that they are numerically dependent. Other factorization methods should lead to the same instabilities. This is a real problem that can be overcome either by reducing drastically the number of modes on the sparadrap domain or increasing the number of subdomains. We have considered the case of the overlapping domain decomposition using a sparadrap. The number of modes is N 1 = N 2 = N on the extreme intervals and N 3 = L over the sparadrap subdomain. The mass matrix is built up by rst considering the modes on I 1 and I 2 then add the modes of I 3 one by one so that the mass matrix remains numerically invertible. This allows us to overcome the diculty generated by the almost singularity of the matrix on the y when computing the Choleski factorization of the mass matrix. For any N, let us denote by P (N) the maximum number of internal modes. Naturally the curve is decreasing, nevertheless the number of global modes that are well approximated increases with N. For instance, for N = 100, P (N) = 7, the rst modes are computed with an accuracy of the order of 10 10 and the 143 rst modes are computed with an accuracy better than 10 5 . For N = 200, we nd that P (N) = 6 and the 253 rst modes are computed with an accuracy better than 10 5. For an overlapping decomposition without sparadrap, the increase of number of subdomains also allows us to overcome the diculty since for a decomposition in two domains I 1 =]0; 0:75[, I 2 =]0:25; 1[ the maximum number of local modes is 14, with 19 modes approximated with an accuracy better than 10 5 whereas,for a decompostion into 3 subdomains I 1 =]0; 0:4[, I 2 =]0:3; 0:7[,

11

N Craig and Bampton 3 domains overlapping 2 domains overlapping 4 1:850 10 4 0:165 9:6 10 4 5 4 8 2:774 10 7:055 10 1:546 10 6 6 5 12 8:742 10 3:144 10 2:167 10 9 6 9 20 1:984 10 4:086 10 2:643 10 11 6 10 24 1:163 10 2:10910 singular 30 6:086 10 7 singular singular

Table 1: N 1 = N 2 = N , N 3 = N=2. In uence of the number N . Relative error on the rst eigenvalue.  1 2 3 4 8 9

N3 = 1 0:120 1:365 10 2 6:027 10 3 9:516 10 3 1:746 10 3 4:987 10 3

N3 = 2 6:918 10 2 7:197 10 3 3:163 10 3 4:891 10 3 1:108 10 3 3:419 10 3

N3 = 4 7:055 10 4 1:007 10 4 8:063 10 5 2:302 10 4 2:499 10 4 8:956 10 4

N3 = 8 1:049 10 4 1:212 10 5 7:348 10 6 1:602 10 5 6:738 10 6 1:739 10 5

N 3 = 16 3:783 10 5 3:733 10 6 1:801 10 6 3:086 10 6 8:082 10 7 2:067 10 6

Table 2: In uence of the number N 3 of basis functions on the sparadrap interval ]0:3; 0:7[. Relative errors on the eigenvalues. (N 1 = N 2 = 8.)

I

3

=

I 3 =]0:6; 1[,the maximum number of local modes is 46, with 116 modes approximated with an accuracy of the order of 10 5. On table 2, we show the behaviour of the convergence of the method with respect to the number of local modes. We use the same con guration as before but at xed \sparadrap" domain I 3 =]0:3; 0:7[. The numerical analysis suggests a uniform distribution of the number of modes on each subdomain when N 1 and N 2 are not too large. It is con rmed by the numerical experiments. We now adress another limitation of the overlapping method if the overlapping itself is not suf ciently large. For that, with the same decomposition as before, we use a \sparadrap" domain I 3 =]0:6 a; 0:6 + a[ of variable length a. On table 3, we note that the relative errors on the approximate global modes are sensitive to the size of overlapping. The quality of approximation of the rst eigenmodes is largely aected when the overlapping tends to zero. The numerical experiments suggest a reasonable size as it can be easily understood from the theory. Indeed, if the overlapping is too small, the functions arising in the partition of unity have large derivatives that lead to large values of the coecient MXc ; (p).

Numerical results for 2D polygonal domains. - In all that follows, we deal with a L-shaped domain associated to the eigenvalue problem (2). As usual, the eigenpairs are arranged in increasing order of eigenvalues. Dierent domain decompositions on are stressed in order to emphasize the quality of the method and con rm the error estimates of the previous numerical analysis. We essentially discuss the accuracy of the method and examine its ability to capture the corner singularities. Notations. - For the sake of simplicity, we introduce some notations for two particular subdomains: 12











 I 3 = 0:6 I 3 = 0:4 I 3 = 0:2 I 3 = 0:1 I 3 = 0:05 1 7:956 10 4 2:048 10 3 0:210 0:439 0:580 2 3:185 10 5 1:489 10 4 2:529 10 2 8:816 10 2 0:168 5 3:553 10 16 1:776 10 16 1:776 10 16 1:421 10 15 1:350 10 14 7 5:831 10 5 7:067 10 5 1:398 10 3 1:192 10 2 6:650 10 2 8 6:225 10 5 1:688 10 4 2:074 10 3 1:017 10 2 2:910 10 2 13 0:302 0:375 0:457 1:410 2:890 14 0:540 0:733 0:658 1:929 3:792

Table 3: N 1 = N 2 = N 3 = 8. In uence of the length of the sparadrap interval I 3. Relative errors on the eigenvalues.

 a set of a rectangle and its local modes is denoted by   R ]x1; x2[]y1; y2 [ ; N x ; N y ; where ]x1; x2[]y1 ; y2[ is the rectangle, N x (resp. N y ) characterizes the number of local modes in

the direction x (resp. y);  a set of a camembert-shaped domain and its local modes is de ned by   C (o1 ; o2) ; ]0; R[]1; 2[ ; N r ; N  ; where the camembert is a part of a disk centered in (o1 ; o2) with radius R such that the angular variable  belongs to ]1; 2 [. The number of modes with respect to the polar variable r, (resp. the angular variable ) is N r , (resp. N  ). Criteria of accuracy. - To evaluate the accuracy of the CMS method, we can rst compare the modes for which we know the closed form and their approximate numerical solutions, for both eigenvalues and eigenvectors. The second criterion, when the analytic eigensolutions are not known, consists in comparing the eigenvalues computed from dierent overlapping decompositions. It is known that, because of Poincare's Min-Max principle, the best computed eigenvalues are the lower ones (it is recalled by Babuska-Osborn in [6] for example). Then such a consideration gives an idea of the \best" decomposition. Finally, the local graphic behaviour of the approximate solutions near the corner that generate singularities will give (or not) an idea of the potentiality of the method to capture these singularities. Symmetric L-shaped domain. - In a rst step, we consider as a symmetric L-shaped domain contained in the unit square and decomposed into 3 squared subdomains of size 1=2. This case has become an academic problem discussed e.g. by Babuska and Osborn [6]. Even for such a basic domain, the solutions are not known in closed form except for particular modes. For instance, the third one sin(2x) sin(2y) corresponds to the eigenvalue \8", the 14th one (sin(4x) sin(4y)) corresponds to the eigenvalue \32". Note also that these solutions are analytical on . In general, we only know some local qualitative properties for these solutions : in particular they can be singular at the inner concave corner A(1=2; 1=2). For instance, the rst eigenfunction belongs to any Sobolev space H 5=3 ( ) for  > 0. The local behaviour at A is very singular and drastically limits the convergence rate of the conventional methods. Another particular feature is the presence of multiple eigenvalues associated to independent eigenmodes. It is due to symmetries in the geometry. For this problem, the two rst double eigenmodes (eigenvalue equal to 20) are the 8th and 9th ones. We index the dierent decompositions by means of the following abbreviations: SL (resp. NSL)

13

for a symmetric (resp. non symmetric) L-shaped domain , SD (resp. NSD) for a symmetric (resp. non symmetric) decomposition of and nally CD (resp. NCD) which indicates the use (resp. or not) of camembert subdomains. We introduce the decomposition called SL:NSD:NCD described by the three subdomains





SL:NSD:NCD = R ]0; 0:6[]0;0:5[ ; N x = 4 ; N y = 4     + R ]0:3; 1[]0;0:5[ ; N x = 4 ; N y = 4 + R ]0:5; 1[]0;1[ ; N x = 4 ; N y = 8 :

(28)

For this decomposition, the 1st, 5th , 3rd , 14th, 8th and 9th modes are respectively drawn ( gure 1 to 3.

Figure 1: 1st and 5th eigenmodes on the symmetric L-shaped domain.

Figure 2: 3rd and 14th eigenmodes on the symmetric L-shaped domain. One observes the good results obtained for the regular modes (3 and 14). Modes 8 and 9 associated to the double eigenvalue \20" are also well-computed. But we note that approximate modes 1 and 5 are not accurate in the vicinity of the corner A. In fact, a ner convergence analysis shows a nite order of accuracy for both eigenvalues and eigenmodes linked to the presence of singularity of the eigenfunctions.

14

Figure 3: 8th and 9th eigenmodes on the symmetric L-shaped domain. To overcome this diculty, we propose to use additional subdomains at each corner. The camemberts are well-adapted because we known the eigensolutions in closed form: the Bessel modes. We rst consider following symmetric decomposition





SL:SD:CD = SL:SD:NCD + C (1=2; 1=2) + ]0; 0:5[( ; =2) ; N r = 2 ; N  = 2 ;     SL:SD:NCD = R ]0; 1[]0; 0:5[; N x = 8 ; N y = 4 + R ]0:5; 1[]0; 1[; N x = 4 ; N y = 8 :(29) As expected, the approximation of the singular eigensolutions is better than the one performed without camembert : their eigenvalue are lower than those obtained with SL:SD:CD. Note also that the isocurves that represent the numerical solutions are smoother near the corners when camemberts are used (see gures 4 to 6).

Figure 4: Comparison of the 1st approximated eigenmode using a camembert or not We have also tried a nonsymmetric decomposition that uses a camembert at corner A (SL.NSD.CD). We obtain essentially the same results. Note that the rst eigenvalue is even lower in that case. Note also that we have chosen here a large camembert (R = 0:5), but it is of course possible to deal with smaller radii. On gure 7, we compare the 5th approximate eigenmode obtained on dierent

15

Figure 5: Comparison of the 5th approximated eigenmode using a camembert or not

Figure 6: Comparison of the 6th approximated eigenmode using a camembert or not

16

 3 8 9 1 5

SL.SD.CD SL.SD.NCD SL.NSD.CD SL.NSD.NCD 2:237 10 5 1:510 10 4 1:214 10 4 2:340 10 4 6 6 5 1:324 10 1:112 10 4:957 10 5:856 10 5 4 4 4 2:765 10 4:706 10 5:259 10 6:971 10 4 3:91516() 1:095 10 2 3:295 10 4 1:174 10 2 (  ) 3 5 12:9576 7:208 10 5:402 10 7:548 10 3

Table 4: L-shaped domains. Relative errors on some eigenvalues. subdomain overlappings using camemberts of variable radius: R = 0:5, R = 0:4 and R = 0:2. We still obtain good results for R = 0:3, but the improvement brought by the camembert tends to fade when R tends to zero. For radii smaller than 0:2, the overlapping is not so sucient to capture the singularity as expected.

Figure 7: Comparison of the 5th approximate eigenmode for dierent radii of camemberts: R = 0:5, R = 0:4 and R = 0:2. In table 4 we express relative errors : - between computed eigenvalues and exact ones (3 , 8 and 9 ). - For eigensolutions with singular modes (1 and 5 ), we compare the eigenvalue computed on the decomposition SL:SD:CD to other decompositions. We indicate the reference eigenvalue by (). Additional results. - We present now more results in two space dimensions. The rst serie consists in the computation of the eigenmodes on the previous L-shaped domain, but with a different subdomain decomposition. It is redundant but this is proved to lead to a more accurate computation. The second one consists in the computation of the eigenmodes in a T-shaped domain where the decomposition involves a camembert on one concave corner only. The improvement of the computation is obvious from the dierent plots. The third one consists in the computation of the eigenmodes on a nonsymmetric L-shaped domain.

Eigenmodes on the L-shaped domain with a redundant subdomain decomposition. This decomposition can be resumed into     R ]0; 0:5[]0; 1[; N x = 5 ; N y = 5 + R ]0; 1[]0:5; 1[ ; N x = 5 ; N y = 5     +R ]0; 0:5[]0:1; 0:5[; N x = 4 ; N y = 4 + R ]0:5; 0:9[]0:5; 1[; N x = 4 ; N y = 4   + C (1=2; 1=2) + ]0; 0:5[(0; 3=2) ; N r = 2 ; N  = 2 :

17

(30)

Such a domain decomposition can lead to a dierent bound of the error as far as the norm of the partition of the unity is lower for this case than the classical ones described in this paper. A consequence on the numerical analysis is that the error bound is clearly lower. At the computational point of view, we have compared the results using a classical decomposition and this last one. We can see on gure 9 that the eects of the domain decomposition on the rst eigensolution are much less perceptible using (30). On the rst eigenvalue, a classical decomposition using a camembert would give an \optimal" eigenvalue 1 = 3:917421, even though we compute here a better rst approximate eigenvalue 1 = 3:9123855. The improving is of course paid by the adding of the two small subdomains described in (30) that raises the cost of computation. But, without deteriorating the quality of the results, one can lower the number of basis functions on the two main subdomains in order to have approximately the same number of basis functions and then the same number of unknowns. 5 ’mode_A’ 5.55 5.18 4.81 4.44 4.07 3.7 3.33 2.96 2.59 2.22 1.85 1.48 1.11 0.741 0.37

4 3 2 1 0 -1 -2 -3

x axis

0

0 y axis

Figure 8: First eigenmode on the L-shaped domain for the classical subdomain decomposition using a camembert. We have also drawn the rst eigenmodes computed from the \optimal" subdomain decomposition ( gures 10 to 12).

18

4 ’mode_A’ 4.4 4.1 3.81 3.52 3.22 2.93 2.64 2.35 2.05 1.76 1.47 1.17 0.879 0.586 0.293

3 2 1 0 -1 -2

x axis

0

0 y axis

Figure 9: First eigenmode on the L-shaped domain for the \optimal" subdomain decomposition.

Figure 10: Eigenmodes number 1, 2 and 3 on the L-shaped domain. Optimal subdomain decomposition.

19

Figure 11: Eigenmodes number 4, 5 and 6 on the L-shaped domain. Optimal subdomain decomposition.

Figure 12: Eigenmodes number 7, 8 and 9 on the L-shaped domain. Optimal subdomain decomposition.

20

Eigenmodes on a T-shaped domain. - The subdomain decomposition on this domain consists in the following one:









R ]0; 1[]0:25;0:75[; N x = 8 ; N y = 4 + R ]0:5; 1[]0; 1[; N x = 4 ; N y = 8 :   + C (0:5; 0:75) + ]0; 0:25[( ; =2) ; N r = 3 ; N  = 3 ;

(31)

One can especially appreciate the improvment of the quality of the solution near the corner where a camembert subdomain has been used.

Figure 13: Eigenmodes number 1, 2 and 3 on the T-shaped domain

Figure 14: Eigenmodes number 4, 5 and 6 on the T-shaped domain

21

Figure 15: Eigenmodes number 7, 8 and 9 on the T-shaped domain

Figure 16: Eigenmodes number 10, 11 and 12 on the T-shaped domain

Figure 17: Eigenmodes number 13, 14 and 15 on the T-shaped domain

Figure 18: Eigenmodes number 16, 17 and 18 on the T-shaped domain 22

Eigenmodes on a nonsymmetric L-shaped domain. - The last case we describe is a nonsymmetric L-domain whose eigenmodes do not present any symmetry. We use two-rectangles and a camembert at the inner corner for the subdomain decomposition. More precisely, we use









R ]0; 1[]0; 0:5[ ; N x = 8 ; N y = 4 + R ]0:7; 1[]0:25; 0:8[; N x = 3 ; N y = 4 :   + C (0:7; 0:5) + ]0; 0:3[( ; =2) ; N r = 2 ; N  = 2 :

(32)

Figure 19: Eigenmodes number 1, 2 and 3 on the nonsymmetric L-shaped domain

Figure 20: Eigenmodes number 4, 5 and 6 on the nonsymmetric L-shaped domain

23

Figure 21: Eigenmodes number 7, 8 and 9 on the nonsymmetric L-shaped domain

Figure 22: Eigenmodes number 10, 11 and 12 on the nonsymmetric L-shaped domain

Figure 23: Eigenmodes number 13, 14 and 15 on the nonsymmetric L-shaped domain

Figure 24: Eigenmodes number 16, 17 and 18 on the nonsymmetric L-shaped domain 24

Figure 25: Eigenmodes number 19, 20 and 21 on the nonsymmetric L-shaped domain

25

Acknowledgments. - We would like to acknowledge interesting discussions with M. Costabel that helped us in the proof of Lemma 1. The results of this research, motivated by the G.D.R. \couplage d'equations", have been announced in the note [8] and we thank P. G. Ciarlet for the interest he showed in accepting it.

References

[1] Babuska, I., Guo, B. and Osborn, J.E., 1989 : Regularity and numerical solution of eigenvalue problems with piecewise analytic data, SIAM, J. Numer. Anal.i, 26, No 6, 1534-1560 [2] Bernardi C., Karageorghis A. : Methodes spectrales dans une portion de disque. [Spectral methods in a part of a disk], Publication du Laboratoire d'Analyse numerique, Paris 6. [3] Bernardi, C. and Maday Y., 1992 : Approximations spectrales de problemes aux limites elliptiques, Mathematiques et Applications, Springer-Verlag. [4] Bourquin, F., 1992 : Domain decomposition and eigenvalues of second order operators : convergence analysis, J. Math. Pures Appl. [5] Bourquin, F., 1995 : Error analysis for modal synthesis, a para^tre dans \Actes de la conference sur les methodes de synthese modale", E.C.L., Lyon, France. [6] Babuska, I. and Osborn, J., 1991 : Eigenvalue problems, Handbook of Numerical Analysis (Ciarlet, P.G., Lions, J.L. (ed.)), vol. II, North-Holland, Amsterdam. [7] Craig, R. and Bampton, M.C.C., 1968 : Coupling of substructures for dynamic analysis, AIAA J., 6, 1313-1321 [8] Charpentier, I., De Vuyst, F., Maday, I., 1995 : Methode de synthese modale avec une

decomposition de domaine par recouvrement. [Component mode analysis with overlapping domain decomposition], Note to appear in C.R. Acad. Sci., Serie I, Paris (given in Appendix). [9] Charpentier, I., De Vuyst, F., Maday, I., 1995 : A component mode synthesis method of in nite order of accuracy using subdomain overlapping, a para^tre dans "Actes de la conference

[10] [11] [12] [13] [14]

ENUMATH'95", Paris. Grisvard, P., 1985 : Elliptic problems in nonsmooth domains, Pitman, Boston. Hurty, W.C., 1965 : Dynamic analysis of structural systems using component modes, AIAA J., 4, 678-685 Lions J.L., Magenes E., 1968 : Problemes aux limites et Applications, vol. 1, Dunod, Paris. Mercier B., Raugel G., 1982 : Resolution d'un probleme aux limites dans un ouvert axisymetrique par elements nis en r; z et series de Fourier en , RAIRO 16 4. Nikiforov A. and Vasilii B., 1988 : Special functions of Mathematical Physics, BirkHauser Verlag Basel, Boston.

26

Appendix (Annexe)

Component mode analysis with overlapping domain decomposition (Methode de synthese modale avec une decomposition de domaine par recouvrement) to appear in C. R. Acad. Sci., serie I (France) (note a para^tre dans les C. R. Acad. Sci., serie I)

27

28 Analyse numerique/Numerical analysis

Methode de synthese modale avec une decomposition de domaine par recouvrement par Isabelle Charpentier, Florian De Vuyst et Yvon Maday

Resume { Dans cette Note, on presente une nouvelle methode de calcul de valeurs propres pour certains

operateurs elliptiques par une technique de synthese modale basee sur une decomposition de domaine avec recouvrement. Cette methode s'avere superieure aux techniques classiques dans un certain nombre de cas tant sur le plan de l'analyse theorique que sur le plan des resultats numeriques. On montre en particulier que la convergence de cette methode est d'ordre in ni.

Component mode analysis with overlapping domain decomposition.

Abstract { In this Note, we present a new method for the computation of the eigenmodes of some elliptic operators. This technique is based on the concept of modal component analysis but uses overlapping domain decompositions. This allows us to achieve a better method than the previous ones based on nonoverlapping decompositions both from the theoretical and the numerical point of view. In particular, we are able to derive in nite order convergence for this method. Abridged English Version { The computation of eigenmodes and eigenfunctions of some

partial dierential operators is of major interest in many problems arising for example in structural mechanics and asymptotic analysis of the behaviour in long range of time. When the domain where the eigenvalues are to be computed has a complex shape, the component mode analysis is often used especially in the industrial context. The original method proposed by Craig and Bampton [4] together with the improved version de ned by Bourquin [2] are based on a nonoverlapping decomposition of the domain of computation. The global eigenmodes are approximated by a Galerkin approach based on the space spanned by the rst eigenmodes of the original operator restricted to each subdomain together with some interfacial modes. This method appears as very powerful to approximate the rst global modes. The analysis given in Bourquin [2] allows us to understand why it works well. Nevertheless, the method is of low order accuracy and this is certainly a loss since the eigenmodes should be able to lead to an in nite order of accuracy. The method we propose in this paper is based on a similar idea but is in nite order and appears to be superior to the original one in many situations. This new component mode synthesis method is based on an overlapping domain decomposition written as in (3). Let us assume that the problem we want to solve is to nd u and  such that (1) is satis ed. It is well-known that this problem has a countable set of eigenvalues (2) with corresponding eigenmodes ui . In order to evaluate these modes, we introduce the eigenvalues ki (ordered as in (6)) and the corresponding eigenvectors uki solution of Problem (5) provided with homogeneous Dirichlet conditions over k . We then denote by ufki the extension of the eigenvectors uki by zero over . Our method consists in solving Problem (4) in the space X spanned by the eigenvectors (7). Due to the orthogonality of the family of eigenvectors (ufki )1iN k at xed k, it is an easy matter to note that the method gives rise to matrices with very simple diagonal structures as in the original method and this is certainly a nice feature of this kind of methods. Of course, when it is necessary, the local eigenmodes are computed by solving (e.g.) nite element eigenvalue problems on the (smaller) subdomains k . This leads to an additional error that can be easily taken into account in the analysis by following the same lines as in Bourquin [2]. 3

Note presentee par Philippe G. Ciarlet

29 The numerical analysis of the plain method follows the general framework recalled in e.g. [1] for the approximation of eigenvalues and eigenmodes in the case of Hilbert spaces. It allows us to P state that (8) and (10) holds with " () de ned in (9). Using a partition of unity 1  Kk=1 k associated with the decomposition of the domain satisfying (11), one can state the Theorem. For any positive real number , let M = v2V ()max ;kvk

L2 ( )

=1

kvkH  ( ) :

Then there exists a positive constant C > 0 such that, for any integer N k , one has the error bound (12) with  = inf k kN k +1 .

INTRODUCTION. - De nombreux problemes issus de la Mecanique font intervenir le calcul

de valeurs propres pour certains operateurs aux derivees partielles, en particulier elliptiques, sur des domaines a geometrie complexe. Ces problemes interviennent dans le calcul de vibration de pieces mecaniques, le calcul de stabilite de structures sous certaines excitations mais aussi dans des problemes de decroissance asymptotique de phenomenes physiques au cours du temps. A n de prevoir le comportement de ces phenomenes les ingenieurs ont de plus en plus recours au calcul numerique de valeurs et de modes propres et, comme c'est le cas pour d'autres problemes, les techniques de decomposition de domaine permettent de faire des calculs plus precis ou d'aborder des problemes de plus grande taille. Dans cette direction la methode de synthese modale (aussi appelee \sous-structuration dynamique" dans la litterature) est tres appreciee. Les travaux recents de F. Bourquin [2] en particulier sur le sujet traitent de l'analyse numerique de ces methodes et introduisent une variante pertinente a la methode initialement proposee par Craig et Bampton [4]. Nous refererons aux articles de Bourquin pour une presentation de la methode. Pour ce qui nous interesse ici, nous pouvons dire seulement que cette methode est basee de facon classique sur une decomposition de domaine sans recouvrement et que l'approximation des modes propres globaux se fait par combinaison lineaire des modes propres locaux, correspondant a chaque sous-domaine, plus quelques fonctions de raccord. L'analyse numerique proposee par Bourquin [2] permet d'expliquer pourquoi la methode marche bien mais fait etat d'un taux de convergence d'ordre ni en fonction du nombre de fonctions locales utilisees. Dans cette note, nous introduisons une methode de synthese modale basee sur une decomposition de domaine avec recouvrement. On montre que la methode presente les m^emes avantages que la methode initiale mais qu'elle est plus performante puisque l'on est a m^eme de montrer que la convergence est d'ordre in ni. Ces conclusions theoriques sont con rmees par des resultats de simulation numerique ou l'on compare, sur quelques cas, notre methode a l'approche classique.

PRE SENTATION DE LA ME THODE. - L'operateur elliptique sur lequel nous allons travailler ici est le Laplacien mais la technique exposee n'est clairement pas liee a la forme de cet

30 operateur. Nous considerons donc le probleme aux valeurs propres suivant: Trouver u et  tels que
u = u; dans

(1)

ou la solution propre u est astreinte a veri er en outre des conditions aux limites homogenes sur la frontiere de . Nous choisirons ici des conditions aux limites de type Dirichlet qui s'avererons ^etre les plus contraignantes au moment de l'analyse numerique. On sait que ce probleme possede une in nite denombrable de valeurs propres positives qui peuvent ^etre rangees par ordre croissant 0 < 1  2  :::  i  :::

(2)

et a qui sont associes a des vecteurs propres ui . On considere maintenant que le domaine est decompose en une suite de sous-domaines avec recouvrement au sens ou il existe des domaines k , 1  k  K tels que

= [Kk=1 k :

(3)

Remarque 1. Cette decomposition peut provenir d'une decomposition sans recouvrement que

l'on a completee par des domaines \sparadraps" contenant en leur interieur les interfaces entre deux sous-domaines non recouvrants.

L'approximation que nous introduisons repose sur une methode variationnelle de type : Trouver u et  tels que Z Z 8v 2 X ; ru rv =  u v ; (4)



ou X est un sous-espace vectoriel de dimension ni de l'espace de Sobolev classique H01 ( ). Dans l'esprit de la methode de synthese modale, l'espace X est engendre par les fonctions propres de l'operateur de Laplace sur chacun des sous-domaines. On considere donc les valeurs propres ki correspondant aux problemes : Trouver uk et k tels que
uk = k uk dans k ; (5) et veri ant des conditions aux limites de type Dirichlet (ou les conditions aux limites originales sur la partie de frontiere de k portee par @ ). La encore, les valeurs propres peuvent ^etre rangees par ordre croissant 0 < k1  k2  :::  ki  ::: (6) On choisit ensuite pour chaque sous-domaine une frequence de coupure, c'est-a-dire un entier N k . L'espace discret X est alors engendre par l'ensemble des vecteurs propres

fufki ; 1  k  K; 1  i  N k g ;

(7)

ou la notation ufk signi e que l'on a etendu par 0 la fonction uk pour qu'elle soit de nie sur tout

.

31

Remarque 2. Il est classique de considerer des approximations des fonctions propres uk par

exemple par une methode d'elements nis. Les K problemes aux valeurs propres (5) etant de plus petite taille que le probleme original, ce calcul est envisageable. Il peut m^eme ^etre envisage qu'il soit eectue en parallele puisque les K problemes sont completement independants. On ne considerera pas ici cette source supplementaire d'erreur dont l'in uence peut facilement ^etre etudiee en combinant l'analyse faite par Bourquin ou celle qui va ^etre detaillee dans la suite. Remarque 3. Il est aussi habituel de considerer des vecteurs propres orthogonaux, soit qu'ils le sont par nature dans le cas de valeurs propres dierentes soit qu'on les orthogonalise s'ils correspondent a des valeurs propres multiples. Cette precaution permet, dans la methode originale, d'obtenir des matrices de masse et de rigidite avec une structure par bloc tres simple. Ici les m^emes conclusions persistent puisque les blocs correspondant a l'in uence de deux fonctions propres provenant d'un m^eme sous-domaine sont diagonaux. On choisira donc une famille orthonormee dans L2 ( k ) et on rappelle au passage que cette famille est totale dans L2 ( k ) et H 1 ( k ).

ANALYSE NUME RIQUE. - L'outil classique pour l'analyse numerique de l'approximation

de problemes de valeurs propres est indique par exemple dans l'article de synthese de Babuska et Osborn [1] qui dans le cadre hilbertien d'operateurs autoadjoints permet d'armer que i  ;i  i + C "2 (i )

(8)

ou pour toute valeur propre  d'espace propre associe V () , on a " () = v2V (max inf kv v kH 1 ( ) : );kvk=1 v 2X 



(9)

De m^eme, l'approximation du (ou d'un) vecteur propre correspondant est optimale puisque, pour une fonction w de l'espace propre correspondant a ;i , on a

kui w kH 1 ( )  C " (i ) :

(10)

Il appara^t ainsi que les qualites d'approximation des valeurs propres sont une consequence des possibilites qu'ont les fonctions de l'espace discret X a approcher les vecteurs propres correspondants. On va se placer dans le cas ou la frontiere du domaine est de regularite C 1 et ou la decomposition en sous-domaines est \large" au sens ou la distance de Hausdor entre [`=6 k ` n k et k n ([`6=k ` ) est strictement positive. Il est alors classique de construire une partition de l'unite reguliere associee a la decomposition en sous-domaine (3), c'est-a-dire des fonctions k positives P et regulieres non nulles seulement sur k telles que 1  Kk=1 k . Il est moins classique (mais neanmoins facile) de construire cette partition de l'unite de sorte que @k = 0 sur @ k : (11) @n Une telle construction de la partition de l'unite permet de demontrer le theoreme suivant :

32

Theoreme. Pour tout reel  > 0 on note M = v2V ();max kvk

L2 ( )

=1

kvkH  ( ) :

Alors il existe une constante C > 0 telle que, pour tout choix d'entiers N k , on ait " ()  C (1 )=2 M ;

(12)

ou  = inf k kN k +1 .

Demonstration. On deduit des hypotheses de regularite sur la frontiere du domaine que les fonctions propres ui , i  0, sont aussi de classe C 1 . On localise ces fonctions au sous-domaine k en

les multipliant par la partition de l'unite. On remarque alors qu'il sut de bien approcher chacune des fonctions uik par des elements vik de l'espace vectoriel engendre par les ukj ; 1  j  N k , note XN k . On obtient ainsi une bonne approximation de ui par des elements de X . En eet, on a K X fk

inf ku vkH 1 ( )  kui v2X i

K X

k=1

vi kH 1 ( ) K X fk

 k(

ui k ) (

 k

(ui k f vik )kH 1 ( )



k=1 K X

k=1

vi )kH 1 ( )

k=1

K X

k=1

kuik vik kH 1 ( k ) :

(13)

Utilisant alors les resultats rappeles dans la remarque 3, on obtient pour toute fonction ' de H 1 ( k ) Z 1 X k'k2H 1 ( k ) = kj [ k 'ukj ]2

j =1

et donc

inf k'

v2XN k

vk2H 1 ( k ) =

1 X j =N +1 k

kj

[

Z

k

'ukj ]2:

On remarque maintenant de facon classique que, si ' 2 H01 ( k ), Z (
ukj) Z k = 'u ' k j

k

k Z j 1 k = kj k
'uj ; ce qui permet d'enoncer inf k' vk2H 1 ( k)  ( kc )2 kN k v2XN k k N

1 Z X

[

j =N k +1

k


'ukj ]2:

33 Appliquant ce resultat a uik , on se trouve dans un cas particulier ou, d'apres (11) et le fait que
ui = i ui s'annule sur @ , on veri e que
(ui k ) est encore un element de H01 ( k ) ce qui permet de reiterer le procede et obtenir, pour tout p inf

v2XN k

kuik

Z

X [ k
p (uik )ukj ]2; vk2H 1 ( k )  ( kc )2p kN k k Nk j =N +1

1

on en deduit le theoreme dans le cas  pair en injectant cette inegalite dans (13) et en utilisant la regularite de . Le cas general se deduit par interpolation. Remarque 4. Le cas ou la decomposition n'est pas large ainsi que le cas de l'approximation de fonctions propres regulieres dans une geometrie singuliere se traite de la m^eme facon en exhibant une partition de l'unite adaptee. Pour ce qui est de l'approximation de vecteurs propres non reguliers, il convient d'assurer dans l'espace discret la presence de fonctions singulieres qui localement vont ^etre capable de tenir compte des singularites de coin presentes dans la solution exacte. Les details de ces applications sont aussi discutes dans [3].

RE SULTATS NUME RIQUES. - Les resultats qui suivent montrent la puissance et les limites

de la methode que nous venons de voir. Tout d'abord on a reporte en gure 1 les courbes de logarithme de l'erreur sur les premieres valeurs propres globales pour le probleme u00 = u

(14)

sur le domaine I =]0; 1[. On a utilise la decomposition de domaine sur trois sous-domaines I~1 = ]0; 0:5[, I~2 =]0:5; 1:[ et I~3 =]0:25; 0:75[. Une convergence de type exponentiel est bien notee. En raison de la symetrie de la decomposition, les fonctions propres associees aux valeurs propres paires sont calculees exactement. Le tableau 1 montre la limite de la methode si le recouvrement n'est pas susant. On decompose cette fois-ci I en utilisant les domaines I 1 =]0; 1=2[, I 2 =]1=2; 1[ et le domaine \sparadrap" I 3 =]1=2 a; 1=2 + a[. On voit que pour un recouvrement plus petit que 0; 2 la convergence de la methode est bien alteree. Le tableau 2 montre ensuite le comportement de l'erreur en fonction du nombre de modes utilises sur le domaine \sparadrap"; on note qu'il sut de la moitie des modes sur I 3 =]1=4; 3=4[ pour avoir une approximation coherente de la solution. Le tableau 3 permet de comparer la methode originale de Craig et Bampton [4] par rapport a notre approche sur un cas sparadrap (I 3 =]0:25; 0:75[, N 1 = N 2 = N, N 3 = N=2) et sur un cas a deux sous-domaines recouvrants I~1 =]0; 0:75[, I~2 =]0:25; 1[. On note bien la superiorite de notre approche. Un phenomene interessant sur ce tableau peut ^etre note. On remarque que, pour N = 18 , la methode a deux sous-domaines recouvrants conduit a une matrice de masse numeriquement non inversible ! Ceci peut s'expliquer et se corriger facilement. En eet la 18eme fonction propre ue218 sur I~2 peut ^etre bien approchee par une combinaison lineaire de fonctions ue1i , 1  i  18 et de fonctions ue2i , 1  i  17. Il y a alors \dependance lineaire" numerique entre les 36 vecteurs de nissant V . Ceci peut facilement ^etre teste au cours de l'algorithme et on travaille alors en retirant de la base ces vecteurs presque lies. La derniere experience traite d'une simulation bidimensionnelle sur un cas avec coin, = 1 [ 2 avec 1 =]0; 2[]0; 1[ et

34

log10(erreur rel. sur Lambda) / log10(rel. error on Lambda)

0 Lambda_1 Lambda_3 Lambda_5 Lambda_7

-1

-2

-3

-4

-5

-6

-7

-8 3

4 5 6 7 8 9 Nb. de fonctions par sous-domaine / Nb. of functions per subdomain

10

Figure 26: Synthese modale. Decomposition de domaine monodimensionel. Mesure du logarithme de l'erreur eectue sur les premieres valeurs propres. Figure 26: Component mode synthesis method. One-dimensional domain decomposition. Logarithm of the error on the rst eigenvalues.

 jI3 j = 0:8 jI3 j = 0:4 jI3 j = 0:2 jI3j = 0:1 jI3 j = 0:05 1 :7017 10 4 :1261 10 1 :2418 100 :5375 100 :5430 100 14 14 14 14 2 :1776 10 :1443 10 :3775 10 :7772 10 :5430 10 14 6 3 0 0 3 :7970 10 :8680 10 :2712 10 :1050 10 :5430 100 7 :1358 10 3 :1827 10 3 :2113 10 1 :2965 10 1 :5430 10 1 8 :5551 10 15 :8221 10 15 :6661 10 15 :2220 10 15 :5430 10 14 9 :9746 10 3 :1503 10 2 :5221 10 1 :2848 10 1 :5430 10 1 13 :4082 100 :2158 100 :3935 101 :1393 101 :5430 101 0 0 1 1 14 :8011 10 :8142 10 :6114 10 :1917 10 :5430 101 1 0 1 1 15 :1137 10 :7565 10 :8042 10 :2378 10 :5430 101

Table 5: N 1 = N 2 = N 3 = 5. In uence de la longueur de l'intervalle de raccord I 3. Erreurs relatives sur les valeurs propres. Table 5: N 1 = N 2 = N 3 = 5. In uence of the length of the sparadrap interval I 3 . Relative errors on the eigenvalues.

35

 1 2 3 4 5 6 7 8 9

N3 = 1 :4763 10 1 :3109 10 14 :5851 10 2 :4441 10 11 :2802 10 2 :4441 10 15 :2607 10 2 :3331 10 15 :3404 10 2

N3 = 2 :4763 10 1 :3109 10 14 :5851 10 2 :4441 10 11 :2802 10 2 :4441 10 15 :2607 10 2 :3331 10 15 :3404 10 2

N3 = 4 :3413 10 2 :3775 10 14 :4113 10 4 :3775 10 14 :1679 10 3 :1184 10 14 :7725 10 3 :6661 10 15 :1800 10 2

N3 = 8 :4853 10 4 :4374 10 13 :1015 10 5 :1776 10 14 :7591 10 6 :8882 10 15 :3070 10 5 :1221 10 14 :3542 10 5

N 3 = 16 :1518 10 4 :2065 10 13 :3825 10 6 :5107 10 14 :1376 10 6 :6069 10 14 :8098 10 6 :3331 10 15 :1172 10 5

Table 6: N 1 = N 2 = 8. In uence du nombre N 3 de fonctions de base sur l'intervalle de raccord I 3 = [0:25; 0:75]. Erreurs relatives sur les valeurs propres. Table 6: N 1 = N 2 = 8. In uence of the number N 3 of basis functions on the sparadrap interval I 3 = [0:25; 0:75]. Relative errors on the eigenvalues.

N Craig et Bampton Recouv. 3 domaines Recouv. 2 domaines 4 1:8 10 4 8:8 10 2 4:6 10 4 5 3 6 6:1 10 7:5 10 8:1 10 6 5 3 8 2:7 10 3:4 10 1:3 10 7 5 5 10 1:3 10 8:4 10 2:0 10 9 6 5 12 1:1 10 2:8 10 2:7 10 11 14 7:2 10 6 1:3 10 6 2:9 10 13 6 7 16 6:4 10 5:1 10 2:0 10 13 6 8 18 2:5 10 1:1 10 singul.

Table 7: N 1 = N 2 = 8. In uence du nombre N 3 de fonctions de base sur l'intervalle de raccord I 3 = [0:25; 0:75]. Erreurs relatives sur les valeurs propres. Table 7: N 1 = N 2 = 8. In uence of the number N 3 of basis functions on the sparadrap interval I 3 = [0:25; 0:75]. Relative errors on the eigenvalues.

36

2 =]0; 1[]0; 2[. Dans ce cas, par symetrie, la troisieme fonction propre (voir la gure 2) est analytique et la decomposition en 2 sous-domaines est susante pour approcher ce mode propre. Il en est de m^eme pour la quatorzieme valeur propre. ’res1.txt’ 2.67 2.39 2.11 1.83 1.55 1.27 0.985 0.705 0.425 0.145 -0.136 -0.416 -0.696 -0.976 -1.26 -1.54 -1.82 -2.1 -2.38 -2.66

3eme fonction propre / Third eigenfunction 3 2 1 0 -1 -2 -3

40 35 30 25 0

5

20 10

15

15 20

Axe des x / x Axis

25

Axe des y / y Axis

10 30

5 35

40

0

Figure 27: Calcul des paires propres sur un domaine en \L". Troisieme fonction propre. Figure 27: Computation of the eigenpairs on a \L-shaped" domain. Third eigenfunction.

Remerciements. - Ce travail a bene cie du support scienti que du G.D.R. \Couplage d'equation".

References [1] I. Babuska and J. Osborn, Eigenvalue Problems, Hanbook of numerical analysis, Vol. II, p. 645785, P.G. Ciarlet and J.L. Lions Editions, Elsevier science Publishers B.V. (North-Holland), 1991. [2] F. Bourquin, Component mode synthesis and eigenvalues of second order operators: discretization and algorithm. M2AN, . 26, p. 385-423, 1992. [3] I. Charpentier, F. De Vuyst et Y. Maday, Actes de la Premiere Conference ENUMATH' 1995, a para^tre. [4] R. Craig and M.C.C. Bampton, Coupling of substructures for dynamic ana-lysis, A.I.A.A. Journal, Vol. 6, p. 1313-1321, 1968.

I.C., Projet IDOPT, (C.N.R.S.-U.J.F.-I.N.P.G.-INRIA),

Laboratoire de Modelisation et de Calcul (LMC-IMAG), Tour IRMA, B.P. 53, FR-38041 Grenoble Cedex 09. F.D.V., Analyse Numerique, Univ. Pierre et Marie Curie, Tour 55-65 5e etage, 75252 Paris Cedex 05. Y.M., ASCI, U.P.R. 9029, Bat. 506, Universite Paris Sud, 91405 Orsay Cedex et Analyse Numerique, Univ. Pierre et Marie Curie, 75252 Paris Cedex 05.

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