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Effective Transmission Conditions for Thin-Layer
Transmission Problems in Elastodynamics
Marc Bonnet, Aliénor Burel, Patrick Joly
To cite this version:
Marc Bonnet, Aliénor Burel, Patrick Joly. Effective Transmission Conditions for Thin-Layer Trans-mission Problems in Elastodynamics. WAVES 2013 - 11th International Conference on Mathematical an Numerical Aspect of Waves, Jun 2013, Gammarth, Tunisia. �hal-00937962�
Effective Transmission Conditions for Thin-Layer Transmission Problems in Elastodynamics M. Bonnet1 A. Burel1,2∗, P. Joly1
1 POEMS (UMR 7231 CNRS-INRIA-ENSTA), ENSTA Paristech, Palaiseau, France.
2Equipe Analyse Num´erique et Equations aux D´eriv´ees Partielles, Universit´e Paris-Sud XI, Orsay, France. ∗Email: alienor.burel@inria.fr
Introduction
This research is motivated by the numerical model-ing of ultrasonic non-destructive testmodel-ing experiments. Some tested media feature thin layers (made e.g. of resin), which are difficult to handle in numerical com-putations due to the very small element size required for meshing them. Similar issues arise in the treat-ment of thin coatings, see e.g. [2], [3]. To overcome these difficulties, one idea consists in using effective transmission conditions (ETCs) across the two inter-faces bounding the layer. This work aims at estab-lishing such ETCs by means of a formal asymptotic analysis with respect to the (small) layer thickness, in the spirit of [1] for Maxwell’s equations.
1 Problem Setting
We consider the case of a thin layer of an isotropic elastic material occupying the strip Ωi
η = R×[− η 2,
η 2].
In addition, let Ω+η (resp. Ω−η) denote the remaining portions of the propagation domain situated above (resp. below) the thin layer. The upper and lower boundaries of the layer are denoted ∂Ω+η and ∂Ω−η; they are connected to Ω+η and Ω−η, respectively. We assume that the layer material is isotropic and homo-geneous, with mass density ρi and Lam´e coefficients λi, µi. We denote by ρ, λ, and µ the (possibly het-erogeneous) coefficients of the material in Ω±η.
-x Ω+η Ωiη Ω−η ∂Ω+η ∂Ω−η 6 y 0 −η 2 +η 2 6 6 n n
The displacement field uiη in Ωiη, as well as the dis-placement fields u±η inside Ω±η satisfy the elastody-namics equations:
ρi∂t2uiη− div σi(uiη) = 0, in Ωiη, (1) ρ ∂t2u±η − div σ(u±η) = 0, in Ω±η. (2)
where σ(u) and σi(u) respectively denote the stress tensor in the surrounding and layer material, respec-tively, given by Hooke’s law applied to a given dis-placement u. Equations (1) and (2) are coupled with transmission conditions on the interfaces ∂Ω±η (we omit the time variable t for simplicity):
u±ηx, ±η 2 = uiηx, ±η 2 t(u±η)x, ±η 2 = ti(uiη)x, ±η 2 , (3)
where t(u) := σ(u)n and ti(u) := σi(u)n are the trac-tion vectors relative to Ω±η and Ωη and n is the
nor-mal vector to ∂Ω±η (see the figure above).
Eliminating formally uiη, we can write a transmission problem for uη := (u+η, u−η). For any function f :
R2→ Rd, using the notation {f }η = [f ]η, hf iη : R → R2d,
[f ]η(x) := f (x, η/2) − f (x, −η/2),
hf iη(x) := f (x, η/2) + f (x, −η/2)/2,
this transmission condition can be written in the form {t(uη)}η+ Tη{uη}η = 0
where Tη is a (nonlocal) DtN transmission operator
that can easily be defined implicitly from the solu-tion of the interior Dirichlet problem in the strip Ωiη. The next idea is that, when η tends to 0, Tη
be-comes local and that one can get explicit analytical approximations of it.
2 Principle of construction of ETCs
This construction is based on an ansatz for the interior solution uiη of the form
uiη(x, y) = U0 x,y η + η U 1 x,y η + η 2U2 x,y η + ... (4) where Uk : Ωi
1 → R2. This implies in particular a
similar expansion for the traces uiη(x, ±η
2) = u
0
Substituting (4) into (1) allows us to compute explic-itly the Ukfrom the uk± by induction on k: these are polynomial functions in y/η. These expressions lead us to introduce a family of differential operators of order `, A`(∂x, ∂t), ` ≥ 0, such that
A0(∂x, ∂t){u0} = 0 (6) and ti(uiη)x, ±η 2 = t0±(x)+η t1±(x)+η2t2±(x)+... (7) where {tk} = k+1 X j=0 Ak+1−j(∂x, ∂t){uj} (8)
and where we have defined {tk} = [tk], htki, [tk] = tk
+− tk−, htki =
tk++ tk−
2 and the same for {uk}. Note that from (5) and (7)
{uiη}η = {u0} + η {u1} + η2{u2} + ... {ti(uiη)}η = {t0} + η {t1} + η2{t2} + ... (9)
We rewrite the transmission conditions (3) as {t(uη)}η = {ti(uiη)}η, {uη}η = {uiη}η
so that, using (9) and (8), we get
η {t(uη)}η = X k≥0 ηk+1 k+1 X j=0 Ak+1−j(∂x, ∂t){uj}
which, thanks to (6), can be rearranged as η {t(uη)}η = X ` η`A`(∂x, ∂t) X j ηj{uj} = Xk `=0 η`A`(∂x, ∂t) {uη}η+ O(ηk+1)
The transmission condition of order k + 1 is then obtained formally by dropping the O(ηk+1) term. 3 A stable transmission condition
Applying the above method with k = 2 leads to the following transmission conditions
(
A[uη]η = η ht(uη)iη− η BJ ∂xhuηiη,
[t(uη)]η = η (ρ ∂t2− J AJ ∂x2) huηiη− J B ∂x[uη]η,
(10)
where A, B and J are the following 2 × 2 matrices: A =µ i 0 0 λi+2µi , B =µ i 0 0 λi , J =0 1 1 0 . (11) A fundamental point is the well-posedness and uni-form stability in η of the transmission problem (2), (10). This is in fact a consequence of an energy con-servation result: any smooth enough solution of (2), (10) satisfies: d dt Eη + Eηi= 0 (12) where Eη = ρ 2 Z Ωη |∂tuη|2 dx + 1 2 Z Ωη σ(uη) : ε(uη) dx Eηi = η ρ 2 Z R |h∂tuηi|2 dγ + η 2 Z R Qh∂xuηi,[u η] η dγ and Q(x, y) : R2 × R2 → R is the symmetric
quadratic form (JT = J and (BJ )T = J B)
Q(x, y) := xTJ AJ x + 2 xTJ B y + yTA y (13) The well-posedness and stability result is then a con-sequence of the
Theorem. The quadratic form Q(x, y) is positive. At the conference, we shall present various numer-ical simulations to illustrate the accuracy and the ef-ficiency of our approximate model. Moreover, some insights about the error analysis will be given. References
[1] S. Chun, H. Haddar and J.S. Hesthaven, High-order accurate thin layer approximations for time-domain electromagnetics, Part II: Trans-mission layers, in J. Comp. Appl. Math., 8 (2010), pp. 2587–2608.
[2] H. Haddar and P. Joly Stability of thin layer ap-proximation of electromagnetic waves scattering by linear and non linear coatings, in Nonlinear Partial Differential Equations and their Appli-cations Coll`ege de France Seminar Volume XIV, Elsevier, 31 (2002), pp. 415–456.
[3] A. Bendali and K. Lemrabet The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation, in SIAM J. Appl. Math., 6(56) (1996), pp. 1664–1693.
