Two-scale Dirichlet-Neumann Preconditioners for Boundary Refinements

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Two-scale Dirichlet-Neumann Preconditioners for

Boundary Refinements

Patrice Hauret, Patrick Le Tallec

To cite this version:

Patrice Hauret, Patrick Le Tallec. Two-scale Dirichlet-Neumann Preconditioners for Boundary

Re-finements. Domain Decomposition Methods in Science and Engineering XVI, 55, pp.447-454, 2007,

Lecture Notes in Computational Science and Engineering, 9783540344681.

�10.1007/978-3-540-34469-8_56�. �hal-00111504�

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for boundary refinements

Patrice Hauret1 _{and Patrick Le Tallec}2

1 _{Graduate Aeronautical Laboratories, MS 205-45,}

California Institute of Technology Pasadena, CA 91125, USA

2 _{Laboratoire de M´ecanique des Solides, CNRS UMR 7649,}

D´epartement de M´ecanique, Ecole Polytechnique, 91128 Palaiseau Cedex, FRANCE

Summary. The present work introduces simple Dirichlet-Neumann

precondition-ers for the resolution of elasticity problems in presence of numerous small disjoint geometric refinements on the boundary of the domain, situation which typically oc-curs in tire industry. Moreover, the condition number of the preconditioned system is proved to be independent of the number and the size of the small details on the boundary. Finally, as an enhancement, the second proposed preconditioner makes use of a coarse space counterbalancing the effect of essential boundary conditions on the small details, and a simple numerical academic test illustrates the increased efficiency. Further details on the motivation as well as complete proofs can be found in [4, 5].

1 Introduction

Let Ω ⊂ Rd_{be the reference configuration of a body, partitioned into a coarse}

region Ω0 where the properties of the material are rather smooth and where

a coarse approximation should be sufficient, and into small disjoint boundary regions denoted by (Ωk)1≤k≤Kwhere a fine discretization is required (e.g.

geo-metrical refinements, fine behavior of the material). Such a situation typically occurs for tires, the internal structure and the surface sculptures playing the role of the coarse and fine zones, respectively. Let us denote by ΓD a part of

the boundary of Ω where displacements are prescribed and by ΓN= ∂Ω \ ΓD

its complementary part. Denoting by H1

∗(Ω) := {v ∈ H1(Ω)d, v|ΓD∩∂Ω = 0}

the space of admissible displacements, our model elastostatic problem consists in finding u ∈ H1 ∗(Ω) such that: a(u, v) := Z Ω Eijklε(u)klε(v)ij = Z Ω f· v + Z ΓN g· v =: l(v), ∀v ∈ H1 ∗(Ω).

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2 Patrice Hauret and Patrick Le Tallec

Here E denotes the fourth order elasticity tensor, f ∈ L2_(Ω)d _{and g ∈}

L2_(Γ

N)d the loading forces, and (v) = 12(∇v + (∇v)

t_{) is the linearized strain}

tensor. Considering that the solution must be computed with a multi-scale approach in order to respect the characteristics of the problem, the strategy proposed in this paper consists in using:

1. mortar formulations [2, 13] on the interfaces Γ0k = ∂Ω0∩ ∂Ωk enabling

to use independent approximations in the coarse and fine regions respec-tively,

2. efficient Dirichlet-Neumann preconditioners [9], which we adapt so that the computational cost of the full algebraic problem remains independent (or at least weakly dependent) of the number and the size of the fine subdomains (Ωk)1≤k≤K.

The sequel is organized as follows. After the introduction of a mortar formula-tion (secformula-tion 2), we propose two possible Dirichlet-Neumann precondiformula-tioners and state their two-scale properties (section 3). In particular, the second en-hanced preconditioner makes use of a coarse space counterbalancing the effect of essential boundary conditions imposed on the boundary sculptures. A sim-ple numerical test shows its increased efficiency for a simsim-ple academic problem. A broader perspective on the subject as well as complete proofs are given in [4, 5].

2 Non-conforming formulation

For every 0 ≤ k ≤ K, let (Tk;hk)hk>0 be a sequence of meshes of the

substruc-ture Ωk, hkdenoting the maximal diameter of its elements. The corresponding

finite-element spaces of order q are denoted by (Vk;hk)hk>0 ⊂ H

1

∗(Ωk). As in

[3, 7], for stability purpose when using a discontinuous mortar formulation, terface bubbles can be added on the fine subdomains. As a consequence, we in-troduce the potentially enriched spaces of displacements Xk;hk = Vk;hk⊕Bk;hk

for every 1 ≤ k ≤ K and X0;h0 = V0;h0. For each interface Γ0k, Wk;hk will

stand for the trace of the local space Xk;hk on this interface. In order to

im-pose a weak displacement continuity between Ω0and Ωk, a space of Lagrange

multipliers Mk;hk is introduced on the mesh Tk;hk over Γ0k. Actually, various

choices of continuous or of discontinuous polynomial functions of degree r can be used [2, 10, 12, 8, 6, 7] but in any case, they must satisfy the following fundamental assumptions:

Assumption 1 [Coercivity]. Let u0∈ H1(Ω0)dand uk ∈ H1(Ωk)dbe rigid

motions, i.e. ε(u0) = 0 in L2(Ω0)d×d and ε(uk) = 0 in L2(Ωk)d×d, satisfying

the weak continuity requirement R

Γ0k(u0− uk) · µ = 0 for every µ ∈ Mk;hk.

Then u0= uk almost everywhere on Γ0k.

Assumption 2 [Inf-sup condition]. There exists a mapping πk : L2(Γ0k) →

Wk;hk such that for all v ∈ L

2_(Γ 0k),

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Z Γ0k (πkv) · µ = Z Γ0k v· µ, ∀µ ∈ Mk;hk, satisfying kπkvkk,1 2 ≤ C kvkk, 1

2. The mesh dependent norm k · kk, 1 2 introduced above is defined as in [1, 11] by kvk2 k,1 2 = X K∈Tk;hk diam(K ∩ Γ0k)−1 Z K∩Γ0k v2_.

Assumption 3 [Accuracy]. The total degree r of Lagrange multipliers is bounded from below by r ≥ q − 1, q being the total degree of the displacement shape functions.

Then, the mortar formulation of the problem of interest can be written as finding u = (u0, u1, ..., uK) ∈Qk=0K Xk;hk and λ = (λ1, ..., λK) ∈

QK

k=1Mk;hk

satisfying for every v ∈QK

k=0Xk;hk and µ ∈ QK k=1Mk;hk, a0(u0, v0) + PK k=1b0k(v0, λk) = l0(v0) ak(uk, vk) − bk(vk, λk) = lk(vk), 1 ≤ k ≤ K b0k(u0, µk) − bk(uk, µk) = 0, 1 ≤ k ≤ K. (1)

The above problem uses the obvious notation ak(uk, vk) =

R ΩkEijmnε(uk)mnε(vk)ij, lk(vk) =R_Ωkf· vk+ R ΓN∩∂Ωkg· vk, b0k(v0, µk) = R Γ0kv0· µk and bk(vk, µk) = R Γ0kvk· µk.

3 Two-scale preconditioners

In matrix notation, after elimination of the Lagrange multipliers λk in the

first equation of (1), the system becomes      S0U0= L0, Kk Uk Λk ! = Lk −B0kU0 ! , 1 ≤ k ≤ K, (2)

where S0 = A0−PK_k=1Bt_0kRkK−1k RtkB0k is the Schur complement matrix,

and L0 = L0− K X k=1 Bt 0kRkK−1k Lk 0

the corresponding right hand side. In these definitions, the local stiffness matrix Kk and restriction operator Rkare

given by Kk= Ak −Btk −Bk 0 , Rk Uk Λk = Λk.

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An iterative solver can be efficiently used to solve (2) if one is able to define a preconditioner ˜S0 of the exact Schur complement S0 which is spectrally

equivalent to S0, with constants independent of the number and the size of

the small subdomains. When L0, .., LK are given, the application of such a

preconditioner consists in the following operations:

1. Compute L0by solving Dirichlet problems on the small subdomains

pre-scribing zero displacements on the interfaces (Γ0k)1≤k≤K,

2. Solve the extended Neumann problem ˜S0U˜0= L0,

3. Compute ( ˜Uk, ˜Λk) over each Ωk by solving the Dirichlet problem:

Kk _˜ Uk ˜ Λk = Lk −B0kU˜0 .

The most natural -and rather efficient- preconditioner consists in simply using ˜S0 = A0. This is a standard Dirichlet-Neumann preconditioner for

which we prove [5]:

Proposition 1. Assuming that A0 is invertible, i.e. ΓD∩ ∂Ω0has a positive

measure, the following spectral equivalence holds for all U0:

W1,hhS0U0, U0i ≤ hA0U0, U0i ≤ hS0U0, U0i , with: 1 W1,h = 1 + C max k∈I1 Ck c0 + max k∈I2 CkL0 α0Lk ,

where I1 (resp. I2) is the set of indices k ≥ 1 such that Ωk is not fixed on

its boundary (resp. is fixed on a part of its boundary), the positive constants ck and Ck are such that ck|ξ|2 ≤ Eijmnξmnξij ≤ Ck|ξ|2 over Ωk for every

symmetric matrix ξ ∈ Rd×d_{, α}

0 is the coercivity constant of the bilinear form

a0 and Lk = diam(Ωk). The constant C > 0 is independent of the number K

and the size of the subdomains.

This simple choice will lack of efficiency in two simple situations:

1. a fine subdomain Ωk (k ≥ 1) has a small size Lk << L0 and is fixed

on a part of its boundary (k ∈ I2); in this situation, because of its size,

the substructure will have a rather large stiffness to interface rigid body displacements,

2. a fine subdomain Ωk (k ≥ 1) has several stiff modes involving interface

motions (rigid links, incompressibility).

Assuming that these directions of localized interface stiffness be in very small number Nk (this is indeed the case for interface rigid body motions), we

then propose a modification of the previous preconditioner enabling to correct such a lack of efficiency.

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For all k ≥ 1 such that Ωk is fixed on a part of its boundary, we denote by

(ei

k)1≤i≤Nk (with Nk= 6 in general) the interface rigid motions of Γ0k or rigid

links and introduce ˚

Wk = span{eik, i= 1, .., Nk}.

To each interface rigid body motion ei

k, we associate its local ak-harmonic

extension (ui k, λik) ∈ Xk;hk× Mk;δk solution of        ak(v, uik) − Z Γ0k v· λi k = 0, ∀v ∈ Xk;hk, − Z Γ0k ui_k· µ = − Z Γ0k ei_k· µ, ∀µ ∈ Mk;δk. (3)

These solutions span two small local spaces ˚

Xk = span{uik, i= 1, .., Nk} ⊂ Xk;hk,

˚

Mk= span{λik, i= 1, .., Nk} ⊂ Mk;δk.

If k ≥ 1 is such that Ωk is not fixed on its boundary, we adopt

˚

Wk = ˚Mk = {0}.

Then, instead of finding U0 such that S0U0 = L0, we propose to compute

u0 ∈ X0;h0, (uk) ∈ ( ˚Xk)1≤k≤K, (λk) ∈ ( ˚Mk)1≤k≤K solution of the coupled

problem                  a0(u0, v0) + K X k=1 Z Γ0k v0· λk = l0(v0), ∀v0∈ X0;h0, ak(uk, vk) − Z Γ0k vk· λk= 0, ∀vk∈ ˚Xk, 1 ≤ k ≤ K, − Z Γ0k uk· µk = − Z Γ0k u0· µk, ∀µk ∈ ˚Mk, 1 ≤ k ≤ K. (4)

This amounts to reduce the local substructure response to the harmonic ex-tension of its stiff interface modes, which belongs to ˚Xk. We introduce the

matrix I0k = ΛtkB0k where Λtk = h Λ1 k, .., Λ Nk k it

is the matrix built with the multipliers computed in (3), and the restriction ˚Ak of the displacement

stiff-ness matrix Ak to the local space ˚Xk

˚_A_k ij = (U i k)tAkUkj = ak(ujk, u i k) = Z Γ0k uj_k· λik, (5)

where (3) has been used. Exploiting (5) to reformulate (4)-2,(4)-3, the system (4) can be rewritten after some algebraic elimination as

˜

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with a new approximate Schur complement given by ˜ S0 = A0+ K X k=1 It 0kA˚ −t k I0k (7) = A0+ K X k=1 Bt 0kΛkA˚−tk ΛtkB0k.

The complexity of its inversion is much smaller than solving S0U0 = L0

because each local problem (3) used in the construction of ˜S0only involves a

subspace of displacements of dimension Nk. Moreover, we prove in [5] that:

Proposition 2. For all U0, the following spectral equivalence holds

W1,hhS0U0, U0i ≤D ˜S0U0, U0 E ≤ hS0U0, U0i , with 1 W1,h = C 1 + max 1≤k≤K Ck c0 .

The constant C > 0 is independent of the number K and the size of the subdomains.

4 Numerical illustration

Let us consider here a two-scale beam (as represented on figure 1) whose both tips are clamped. The material is elastic, isotropic, homogeneous in each substructure, and the displacements under loading are computed by a precon-ditioned conjugate gradient method. Figure 2 illustrates the advantage of the enhanced Dirichlet-Neumann preconditioner when two small substructures are clamped. In conformity with the announced results, the gain in efficiency is independent of the ratio of Young moduli between the fine and coarse zones. Moreover, a factor 3 improvement is achieved in the number of iterations, and roughly speaking in the time of computation. Finally, it is shown in [5] that such a preconditioner can be used as an efficient quasi-tangent operator in the nonlinear framework as soon as boundary geometrical details are sufficiently soft.

5 Conclusion

The domain-decomposition based preconditioners proposed here achieve scale-independent performances. They should be extended to cases where the details overlap the coarse region in the line of the fictitious domains approach, and also to cases where the details are not disjoint but constitute a continuous belt along the boundary.

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Xd3d 8.0.3b (25/09/2003) 2.929397 75.441 147.9526 220.4642 292.9758 365.4874 437.9991 510.5107 583.0223 655.5339 728.0455

Fig. 1. Maximal stress distribution on a deformed configuration of our two-scale

model problem where two of the details are clamped on their lower face.

# iterations L^2 norm of error 0 10 20 30 40 50 60 -25 10 -20 10 -15 10 -10 10 -5 10 0 10 EnhancedSimple r = 10 # iterations L^2 norm of error 0 10 20 30 40 50 60 70 80 90 -25 10 -20 10 -15 10 -10 10 -5 10 0 10 EnhancedSimple r = 100 # iterations L^2 norm of error 0 50 100 -25 10 -20 10 -15 10 -10 10 -5 10 0 10 Enhanced Simple r = 1000 # iterations L^2 norm of error 0 50 100 150 -25 10 -20 10 -15 10 -10 10 -5 10 0 10 5 10 Enhanced Simple r = 1.E6

Fig. 2. Convergence of the simple and enhanced Dirichlet-Neumann algorithms

for different values of the ratio r of Young moduli between the fine and coarse subdomains.

Acknowledgement. The authors gratefully acknowledge the financial support of Michelin Tire Company during the completion of this work at the Center of Applied Mathematics (CMAP), Ecole Polytechnique, and Professor Fran¸cois Jouve for pro-viding his finite element code in which the presented ideas have been implemented.

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