Sergey Repin

Stefan A. Sauter

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1039–1043Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comNumerical AnalysisEstimates of the modeling error for the Kirchhoff–Love plate modelEstimation de l’erreur de modélisation pour le modèle de plaque de Kirchhoff–LoveSergey Repin a, Stefan A. Sauter ba St. Petersburg Department of V.A. Steklov Institute of Mathematics, Fontanka 27, 191 023 St. Petersburg, Russiab Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerlanda r t i c l e i n f o a b s t r a c tArticle history:Received 5 November 2009Accepted 3 September 2010Presented by Philippe G. CiarletIn this Note we consider the Kirchhoff–Love model for approximating problems in linearelasticity on thin plates under certain hypotheses. We will present computable errormajorants for the arising modelling error. The majorant for the relative error convergeswith a rate O (h1/2) in the thickness parameter h provided that the KL solution possessesextra reqularity.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éDans ce travail on considère le modèle de Kirchhoff–Love pour approcher les problèmesde plaques minces sous certaines conditions. Nous présentons des majorants d’erreurcalculables. La borne de l’erreur relative converge comme O (h1/2) en terme du paramètred’épaisseur h pourvu que la KL solution ait un peu plus de régularité.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeDans ce travail nous nous intéressons à l’erreur commise lorsque l’on remplace un problème tridimenssionnel d’élasticitélinéaire pour des objets minces par le modèle de Kirchhoff–Love (voir [7]). Plus précisément soit ω̂ le déplacement scalairede la surface moyenne ω d’un corps Ω solution d’une équation biharmonique en dimension 2. Dans le modèle (1,1,2) onreconstruit le déplacement tridimensionnel v112KL en utilisant ŵ et une fonction auxiliaire Ŵ qui incorpore les différentescouches dans v112KL . L’erreur peut être estimée par notre nouveau majorant comme suit. Soit Ω = ω × ]−h/2,h/2[ ⊂ R3,où ω ⊂ R2 est un domaine Lipschitzien et h est supposé petit. Le champ des déplacements est noté u et le tenseur desdéformations par ε(u) := 12 (∇u + (∇u)ᵀ). Dans le cas important de l’élasticité linéaire isotrope, la relation entre le tenseurdes contraintes et des déformations peut s’écrireσ = 2με + λ(traceε)I et ε = 12μσ − ν2μ(1 + ν) (traceσ )I, (1)où λ et μ sont les constantes de Lamé, ν est le coefficient de Poisson et I est l’identité. Soit Γ la partie latérale de Ω etS , S⊕ les parties inférieures et supérieures. On recherche σ , tenseur des contraintes, satisfaisant les conditions d’équilibreDivσ + f = 0 dans Ω, u|Γ = 0, σ · n|S = 0. (2)E-mail addresses: repin@pdmi.ras.ru (S. Repin), stas@math.uzh.ch (S.A. Sauter).1631-073X/$ – see front matter © 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2010.09.0041040 S. Repin, S.A. Sauter / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1039–1043Dans le modèle de Kirchhoff et Love on resout tout d’abord l’équation biharmonique satisfaite par le déplacement de lasurface moyenne ŵ , i.e.,∫ω̂ŵ̂ϕ̂ dx̂ =∫ωĝ0ϕ̂ dx̂pour tout ϕ̂ ∈ H20(ω), où ĝ0 désigne une force mise à l’échelle. Ce déplacement est alors utilisé pour reconstruire le champde déplacement tridimensionnel en définissantv112KL :=(−x3 ŵ,1,−x3 ŵ,2, ŵ + x23Ŵ )ᵀ(appelé modèle 112, voir (11)), où Ŵ est une fonction de H10(ω). On pose τ112KL := Lε(v112KL ), où L est le tenseur d’élasticité.Une autre reconstruction possible est donnée parτ imKL(q̂) :=⎡⎢⎢⎣−2μx3(ŵ,11 + ν1−ν ̂ŵ) −2μx3 ŵ,12 12 (x23 − h24 )q̂1−2μx3 ŵ,12 −2μx3(ŵ,22 + ν1−ν ̂ŵ) 12 (x23 − h24 )q̂212 (x23 − h24 )q̂112 (x23 − h24 )q̂2 −(2x33 − 12 x3h2) f̂0⎤⎥⎥⎦ , (3)oùq̂ ∈ Q ĝ0 :={q̂(x̂) ∈ H(ω,div)∣∣∣ div q̂ − 2μ1 − ν ĝ0 = 0, p.p. x̂ ∈ ω}. (4)L’estimation fondamentale d’erreur sur l’énergie ε(u − v112KL ) est la suivante.Les fonctions w ∈ V0 := {w ∈ H1(Ω,R3) | w = 0 on Γ }, resp. τ ∈ Σ := L2(Ω,R3×3sym ), sont munies de la norme‖|ε(w)‖|2 := (Lε(w),ε(w))Ω et ‖|τ‖|2∗ := (L−1τ ,τ )Ω , où (·,·)Ω désigne le produit scalaire dans L2 pour les fonctions sca-laires, vectorielles ou tensorielles. De plus ‖ · ‖Ω := (·,·)1/2Ω .Théorème. Pour tout β > 1 on a∣∣∣∣∣∣ε(u − v112KL )∣∣∣∣∣∣2  β∣∣∣∣∣∣τ 112KL − τ imKL∣∣∣∣∣∣2∗ + 2βC2ωc21(β − 1)2∑i=1∥∥ri(τ imKL)∥∥2Ω, (5)avec ri(τ ), i = 1,2,3, defini en (9). Si ̂ŵ possède des dérivées faibles de carré sommable (ce qui est vrai pour des domaines réguliersou des domaines convexes et polygonaux), alors∣∣∣∣∣∣ε(u − v112KL )∣∣∣∣∣∣2  M2(ŵ, Ŵ , q̂) := β(M1(Ŵ , q̂) + M2(ŵ, Ŵ )) + ββ − 1 M3(ŵ, q̂), (6)pour tout Ŵ ∈ H10(ω) et q̂ ∈ Q ĝ0 (ω), où ρ(ŵ, Ŵ ) := 2Ŵ − ν1−ν ̂ŵ etM1(Ŵ , q̂) := h540∫ω(μ2|∇̂Ŵ |2 + 13(Ŵ ,1q̂1 + Ŵ ,2q̂2) + 13μ|q̂|2)dx̂,M2(ŵ, Ŵ ) := h3420∫ω(70μ(1 − ν)(1 − 2ν) ρ2(ŵ, Ŵ ) − 14ρ(ŵ, Ŵ ) f̂ + f̂2μ(1 + ν))dx̂,M3(ŵ, q̂) := C2ωh36c21∫ω(∣∣∣∣q̂1 − 2μ1 − ν ̂ŵ,1∣∣∣∣2+∣∣∣∣q̂2 − 2μ1 − ν ̂ŵ,2∣∣∣∣2)dx̂.1. Statement of the problemLet Ω := ω × ]−h/2,h/2[ ⊂ R3 where ω ⊂ R2 is a Lipschitz domain and h is small compared to the width of ω. Furthergeometric quantities areS0 := ω × {0}, S⊕ := ω ×{h2}, S := ω ×{−h2}, Γ := ∂ω ×(−h2,h2)and n(x̂) is the normal vector for S := S⊕ ∪ S .The displacement field and the strain tensor are denoted by u and ε(u) := 12 (∇u + (∇u)ᵀ) and coupled with the stresstensor via the Hooke’s law σ := Lε, where the tensor of elasticity constants L satisfies the standard symmetry conditions,S. Repin, S.A. Sauter / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1039–1043 1041is uniformly positive definite and bounded in the L∞-norm. For the case of isotropic material the corresponding relation ispresented by (1). We consider the problem of finding the stress tensor σ such that the equilibrium and boundary conditionsDivσ + f = 0 in Ω, u|Γ = 0, σ · n|S = 0 (7)are satisfied. For the external force, we assume f = (0,0, f̂ (x̂))ᵀ with f̂ ∈ L2(ω) and, for x = (x1, x2, x3), we set x̂ := (x1, x2).A function u ∈ V0 := {w ∈ H1(Ω,R3) | w = 0 on Γ } is a generalized solution of (7) if it satisfies the variational relation∫ΩLε(u) : ε(w)dx =∫Ωf̂ w3(x)dx (8)for all w ∈ V0. Functions w ∈ V0 resp. τ ∈ Σ := L2(Ω,R3×3sym ) are equipped with the corresponding energy norms ‖|ε(w)‖|2 :=(Lε(w),ε(w))Ω and ‖|τ‖|2 := (L−1τ ,τ )Ω , where (·,·)Ω denotes the L2-scalar product for scalar functions, vectors, andtensors. Further ‖ · ‖Ω := (·,·)1/2Ω , and we denote by ‖ · ‖ω , ‖ · ‖S⊕ , ‖ · ‖S the L2-norms on ω, S⊕ , S .In the classical theory of the Kirchhoff–Love plates, the above-described 3D model is replaced by an ansatz via the scalarvalued deflection ŵ of the middle surfaceu1(x) = −x3 ŵ,1, u2(x) = −x3 ŵ,2, u3(x) = ŵ(x̂).We introduce the scaled loads ĝ0 := 6(1−ν)μ f̂0 with f̂ (h) := h2 f̂0 and deduce ŵ as the solution of the problem: find ŵ ∈H20(ω):∫ω̂ŵ̂ϕ̂ dx̂ =∫ωĝ0ϕ̂ dx̂ ∀ϕ̂ ∈ H20(ω).Historically, the subject of error estimation in dimension reduction models was mainly focused on a priori asymptoticerror estimates that evaluate the difference between original and reduced models in terms of small (geometric) parameters.In this context, models in the elasticity theory have been studied by different authors (see e.g., [1,3–6,14] and the referencestherein). Papers [9,10] are devoted to asymptotic consistency of models in the linearized plate theory.2. Error bound for plate type elastic bodiesLet v ∈ V0 denote an approximation of the exact solution u of (8) obtained by some suitable reconstruction of a platemodel. In this section, we present different estimates of the modeling error generated by v. LetΣDiv,n :={τ ∈ Σ ∣∣ Divτ ∈ (L2(Ω))3; τ · n ∈ (L2(S))3}.The following error majorant was introduced in [13] which contains also the proof of the next theorem (see also [11,12]).We use the short notationri(τ ) := div{τi j}3j=1 = τi1,1 + τi2,2 + τi3,3, i = 1,2,3. (9)Theorem 2.1. Assume that r3(τ ) + h2 f̂0 = 0 in Ω and τi,3|S∪S⊕ = 0 for i = 1,2,3. Then,∣∣∣∣∣∣ε(u − v)∣∣∣∣∣∣2 M1(v,τ ) := β∣∣∣∣∣∣Lε(v) − τ ∣∣∣∣∣∣2 + 2βC2ωc21(β − 1)2∑i=1∥∥ri(τ )∥∥2Ω, (10)for any β > 1 and where Cω is the constant in the Friedrichs inequality related to the plain domain ω and c1 > 0 is the lowesteigenvalue of L.3. Error bound for the KL modelLet ŵ = ŵ(x1, x2) be a solution of the Kirchhoff–Love problem. To measure the corresponding modeling error we needto reconstruct 3D displacements and stresses.We employ here the so-called (112)-model for the dimension reconstruction of the displacements, strains, and stresses(see [1,2,8]) and setv112KL := v112KL (ŵ, Ŵ ) :=(−x3 ŵ,1,−x3 ŵ,2, ŵ + x23Ŵ )ᵀ, (11)where Ŵ (x̂) is a specially selected function. Henceforth, we assume that Ŵ ∈ H10(ω). In our analysis, we employ tworeconstructions of 3D stresses based on ŵ and Ŵ . The first one reconstructs stresses in accordance with 3D elasticity1042 S. Repin, S.A. Sauter / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1039–1043relations, i.e., τ 112KL (ŵ, Ŵ ) := Lε(v112KL (ŵ, Ŵ )). Another possible reconstruction (originally introduced in [8]) of the stresstensor uses the KL relations (1) for the components τsk , s,k = 1,2, while the components τi3, i = 1,2,3, are not zero (asin the classical KL theory) and are defined with the help of a correction functions q̂. This improved reconstruction has theform (3), where we assume that q̂ ∈ Q ĝ0 (cf. (4)). This implies that the conditions of Theorem 2.1 are satisfied.3.1. Error estimateFrom (10) we deduce the following theorem (see [13]):Theorem 3.1. For any β > 1 it holds∣∣∣∣∣∣ε(u − v112KL )∣∣∣∣∣∣2  β∣∣∣∣∣∣τ 112KL − τ imKL∣∣∣∣∣∣2∗ + 2βC2ωc21(β − 1)2∑i=1∥∥ri(τ imKL)∥∥2Ω. (12)If ̂ŵ possesses square summable generalized derivatives (which is true for domains with smooth boundaries or for convex polygonaldomains) then ‖|ε(u − v112KL )‖|2  M2(ŵ, Ŵ , q̂) for all Ŵ ∈ H10(ω) and q̂ ∈ Q ĝ0 (ω), where M2 is as in (6).Note that Cω  1πab√a2+b2 holds if ω is contained in a rectangle with side lengths a and b. Thus, M2(ŵ, Ŵ , q̂) containsonly known functions and constants. If ŵ is known, then the majorant is directly computable. By selecting q̂ and Ŵ , wecan (approximately) minimize the value of the majorant. The number obtained presents a guaranteed upper bound of themodeling error encompassed in ŵ .4. Asymptotic behavior of the error majorantIn this section, we will choose Ŵ and q̂ in a concrete way and will get a simplified version of the majorant whichwill allow us to detect the asymptotic behavior of the modelling error. By estimating the products 〈∇̂Ŵ , q̂〉 and ρ(ŵ, Ŵ ) f̂appearing in the definition of M1, M2 by Young’s inequalities we obtain an upper bound of Mi , i = 1,2, which is minimalby choosing Ŵ as the minimizer of the functionalJ : H10(ω) → R, J ( Ẑ) := h2‖∇̂ Ẑ‖2ω +80(1 − ν)3(1 − 2ν)∥∥ρ(ŵ, Ẑ)∥∥2ω.The function q̂ is selected by q̂ := 2μ1−ν ∇̂̂ŵ ∈ Q ĝ0 ; we assume here that ŵ is sufficiently regular. The proof of the followingtheorem is in [13] and uses some regularity results for the minimizer Ŵ (see [1,2]):Theorem 4.1. For any γ > 1, it holds∣∣∣∣∣∣ε(u − v112KL )∣∣∣∣∣∣2  M3(ŵ, Ŵ , q̂0) := μγ80 h5‖∇̂Ŵ ‖2ω +μ(1 − ν)3(1 − 2ν)h3∥∥ρ(ŵ, Ŵ )∥∥2ω+ h5 R(ŵ, f̂0), (13)whereR(ŵ, f̂0) = C Iμ,ν,γ ‖∇̂̂ŵ‖2ω + C IIμ,ν,γ h2‖ f̂0‖2ωand the C I,IIμ,ν,γ are positive constants depending only on μ,ν,γ .By using this theorem the following asymptotic behavior of the error majorant can be derived (see [13])√M3(ŵ, Ŵ , q̂0)‖|ε(v112KL )‖| ch1/2 and√M3(ŵ, Ŵ , q̂0)‖|ε(u)‖|  ch1/2.AcknowledgementsThe authors are grateful to Swiss National Science Foundation for supporting this research under the grant 200021_119809/1.S. Repin, S.A. Sauter / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1039–1043 1043References[1] A.L. Alessandrini, D.N. Arnold, R.S. Falk, A.L. Madureira, Derivation and justification of plate models by variational methods, in: M. Fortin (Ed.), Platesand Shells, Quebec, 1996, in: CRM Proceedings and Lecture Notes, vol. 21, American Mathematical Society, Providence, RI, 1996, pp. 1–20.[2] D. Braess, S. 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Estimates of the modeling error for the Kirchhoff–Love plate model

Repin, Sergey

Sauter, Stefan A.

Numérisation de Documents Anciens Mathématiques

http://www.numdam.org/item/10.1016/j.crma.2010.09.004.pdf

Estimates of the modeling error for the Kirchhoff–Love plate model

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