Posteriori Analysis of a Finite Element Discretization for a Penalized Naghdi Shell

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1 International Journal of Difference Equations ISSN , Volume 8, Number 1, pp (2013) ttp://campus.mst.edu/ijde Posteriori Analysis of a Finite Element Discretization for a Penalized Nagdi Sell Nora Tabouce Laboratoire de Matématiques Appliquées et Modélisation Université 8 Mai 1945 de Guelma, Guelma, Algeria tabmoufida@yaoo.fr Abstract We consider a penalized Nagdi model in Cartesian coordinates for linearly elastic sells wit little regularity. A posteriori analysis of te discrete problem leads to te construction of error indicators, wic satisfy optimal estimates. We describe a mes adaptivity strategy relying on tese indicators and we present a numerical experiment tat confirms its efficiency. AMS Subject Classifications: 74K25, 74S05. Keywords: Nagdi s sell model, finite element approximation, a posteriori error estimates, residuals. 1 Introduction Nagdi s model is a linear elastic sell. Te formulation of te model used ere was introduced in [4, 6]. A posteriori analysis is now an important tool for improving te efficiency of te discretization. We refer to [7, 8] for te first works concerning a plate model and to [2] for sell models. Te first aim of a posteriori analysis is mes adaptivity. Indeed, a muc smaller number of degrees of freedom are needed to obtain a given accuracy wen te final mes is adapted to te solution and te construction of suc mes relies on error indicators, wic only depend on te discrete solution, and ence can be computed in an explicit way and often in a non expensive way. A posteriori estimate proves tat tese indicators provide a good representation of te local error see [11] for a detailed presentation of all tis. Here we perform a posteriori analysis of te discretization, relying on a penalized version studied in [5] and prove upper and lower bounds for te error as a function of residual type indicators. Finally, we describe Received June 29, 2012; Accepted January 27, 2013 Communicated by Sandra Pinelas

2 112 N. Tabouce te strategy tat is used for te adaptivity mes. Numerical experiments are in good agreement wit te analysis. Te article is organized as follows. We first briefly recall te geometry of te midsurface and Nagdi sell formulation given in [4, 6]. Tis formulation involves te infinitesimal rotation vector, a vector unknown tat is tangent to te midsurface. In Section 3, we recall te penalized version of Nagdi s model intended to approximate te above mentioned tangency. Section 4 is devoted to te a posteriori analysis of te finite element discretization. In Section 5 we present te adaptivity strategy and numerical experiments. 2 Presentation of te Model Greek indices and exponents take teir values in te set {1, 2}, wile Latin indices and exponents belong to te set {1, 2, 3}. Let (e 1, e 2, e 3 ) be te canonical ortonormal basis of R 3. We denote by u v te inner product of R 3, u = u u te associated Euclidean norm and u v te vector product of u and v. Let be a bounded connected domain of R 2. We consider a sell of midsurface S = ϕ(), were ϕ W 2, (, R 3 ) is a one-to-one mapping suc tat te two vectors a α (x) = α ϕ(x) are linearly independent at eac point x of. We let a 3 = a 1 a 2 be te unit normal vector on te midsurface a 1 a 2 at point ϕ(x). Te vectors a i (x) define te local covariant basis at point ϕ(x). Te contravariant basis a i (x) is defined by a i a j = δ j i, were δj i is de Kronecker symbol. We let a(x) = a 1 (x) a 2 (x) 2, so tat a(x) is te area element of te midsurface in te cart ϕ. Te first and te second fundamental forms of te surface are given in covariant components by a αβ (x) = a α (x) a β (x) and b αβ (x) = a 3 (x) β a α (x). Te contravariant components of te first fundamental form a αβ (x) = a α (x) a β (x). Te lengt element l on te boundary is given by a αβ τ α τ β, (τ 1, τ 2 ) being te covariant coordinates of a unit vector tangent to. Let a αβρσ L () be te elasticity tensor. We consider ere te case of a omogeneous, isotropic material wit Young modulus E > 0 and Poisson ratio ν, 0 ν 1, were tese components are given by 2 a αβρσ = E 2(1 + ν) (aαρ a βσ + a ασ a βρ ) + Eν 1 ν 2 aαβ a ρσ. (2.1) In tis context, te covariant components of te cange of metric tensor read γ αβ (u) = 1 2 ( αu a β + β u a α ), (2.2) te covariant components of te cange of curvature tensor read χ αβ (u, r) = 1 2 ( αu β a 3 + β u α a 3 + α r a β + β r a α ), (2.3)

3 Posteriori Analysis of a Finite Element Discretization 113 and te components of te cange of transverse sear tensor read δ α3 (u, r) = 1 2 ( αu a 3 + r a α ). (2.4) We assume tat te boundary of te cart domain is divided into two parts: γ 0 on wic te sell is clamped and γ 1 = \γ 0 on wic te sell is subjected to applied traction and moment. Let us now consider te function space V() introduced in [4,6]: V() = {V = (v, s) [ H 1 (, R 3 ) ] } 2, s a3 = 0 in, v = s = 0 on γ 0. (2.5) Tis space is endowed wit te natural Hilbert norm V V = ( v 2 H 1 (;R 3 ) + s 2 H 1 (;R 3 ) ) 1 2. (2.6) We now recall te variational formulation of te problem corresponding to te linear Nagdi model wit data (f, N, M) L 2 (; R 3 ) L 2 (γ 1 ; R 3 ) L 2 (γ 1 ; R 3 ): find U = (u, r) V() suc tat V V, a(u, V ) = L(V ), (2.7) were te bilinear form a(, ) is defined by ] a(u, V ) = {ea [γ αβρσ αβ (u)γ ρσ (v) + e2 12 χ αβ(u)χ ρσ (V ) and te linear form L( ) is given by L(V ) = f v a dx + +2e E } a 1 + ν aαβ δ α3 (U)δ β3 (V ) dx, (2.8) γ 1 (N v + M s) l dτ. (2.9) Te data f, N, M represent a given resultant force density, an applied traction density and an applied moment density, respectively. In te above formulas, te tickness of te sell, denoted by e, is assumed to be constant and positive. We refer to [4,6] for te proof of te following results: Te form L is continuous on V() and its norm satisfies ( ) L c f L 2 (;R 3 ) + N L 2 (γ 1 ;R 3 ) + M L 2 (γ 1 ;R 3 ). (2.10) Tere exists a constant c > 0 suc tat V V(), a(v, V ) c V 2 V(). (2.11) Problem (2.7) admits a unique solution U in V().

4 114 N. Tabouce 3 Penalized Version We consider te penalized Nagdi problem introduced in [5], in wic te unknowns are te displacement u and te rotation r, elements of te space H 1 (; R 3 ) witout any ortogonality constraint on r. Te relaxed function space is defined by X = { V = (v, s) H 1 (, R 3 ) 2, v = s = 0 on γ 0 }, (3.1) equipped wit te norm defined in (2.6), wic is now denoted by X(). Teorem 3.1. Let p R be suc tat 0 < p 1. Let f L 2 (, R 3 ), N L 2 (γ 1, R 3 ) and M L 2 (γ 1, R 3 ). Ten tere exists a unique solution to te following problem: find U p = (u p, r p ) X suc tat were Proof. See [5]. V X, a(u p, V ) + 1 p b(r p a 3 ; s a 3 ) = L(V ), (3.2) b(λ; µ) = α λ α µ dx. (3.3) Remark 3.2. We note tat te quantity a(u, V ) can be written in anoter form wic seems more appropriate for implementation, since it uncouples te two components v and s of te test function V = (v, s). Indeed, we introduce te contravariant components of te stress resultant, n ρσ (u) = ea αβρσ γ αβ (u), (3.4) of te stress couple, m ρσ (U) = e3 12 aαβρσ χ αβ (U), (3.5) and of te transverse sear force, We also ave: t β (U) = e E 1 + ν aαβ δ α3 (U). (3.6) χ ρσ (V ) = θ ρσ (v) + γ ρσ (s) wit θ ρσ (v) = 1 2 ( ρv σ a 3 + σ v ρ a 3 ). (3.7) Tus, te bilinear form a (U p, V ) can be rewritten as ( ) a (U p, V ) = n ρσ (u p )γ ρσ (v) + m ρσ (U p )θ ρσ (v) + t β (U p ) β v a 3 a dx ( ) + m ρσ (U p )γ ρσ (s) + t β (U p )s a β a dx. (3.8)

5 Posteriori Analysis of a Finite Element Discretization 115 To go furter, using tis new form togeter wit te symmetry properties n ρσ (u p ) = n σρ (u p ) and m ρσ (U p ) = m σρ (U p ), and by integration by parts in problem (3.2), we obtain te strong formulation of te penalized problem: ρ ((n ρσ (u p )a σ + m ρσ (U p ) σ a 3 + t ρ (U p )a 3 ) a) = f a in, ρ (m ρσ (U p )a σ a) + t β 1 (U p )a β a p ρρ(r p a 3 )a 3 = 0 in, u p = r p = 0 on γ 0, ν ρ (n ρσ (u p )a σ + m ρσ (U p ) σ a 3 + t ρ (U p )a 3 ) (3.9) a = Nl on γ 1, ν ρ (m ρσ 1 (U p )a σ a + p ρ(r p a 3 )a 3 ) = Ml on γ 1. Te discretization tat we intend to study is constructed by te Galerkin metod from problem (3.2). We refer to [5, 5.1] for more details. 4 A Posteriori Analysis of te Discrete Problem Let (T ) be a regular affine family of triangulations wic covers te domain and P k (K) denote te space of restrictions to K, element of T, of polynomials wit total degree k. Te discrete space is given by X = { V = (v, s ) C 0 (; R 3 ) 2, V K P 1 (K), v = s = 0 on γ 0 }. (4.1) Te discrete problem consists to find U p, = (u p,, r p, ) X suc tat V X, a(u p,, V ) + 1 p b(r p, a 3, s a 3 ) = L(V ). (4.2) Tis problem as a unique solution [5, T. 3.1]. Te a posteriori analysis of problem (4.2) relies on te residual equation a(u p U p,, V ) + 1 p b((r p a 3 r p, a 3 ); s a 3 ) = L(V V ) a(u p,, V V ) 1 p b (r p, a 3 ; (s a 3 s a 3 )), (4.3) valid for all V X() and for all V X. Te construction of error indicators from tese equations requires approximations of te data and of te coefficients [2, 11]. 4.1 Approximation of te Data Let E 1 denote te set of edges of elements of T wic are contained in γ 1. We consider an approximation f of f in Z and approximations N and M of N and M in Z 1, were te spaces Z and Z 1 are defined by Z = { g L 2 () 3 ; K T, g K P 0 (K) 3}, Z 1 = { E L 2 (γ 1 ) 3 ; e E 1, E e P 0 (e) 3}. (4.4)

6 116 N. Tabouce 4.2 Approximation of te Coefficients We denote by a αβ, aαβρσ, ( a) and l, te approximations of a αβ, a αβρσ, a and l, respectively, in te space M wic is given by M = { χ H 1 (); K T, χ K P 1 (K) }. (4.5) Similarly, we consider approximations a k of te vectors a k and d α of te α a 3 in te space M. We also agree to denote by γ αβ( ), χ αβ( ) and δ α3( ) te approximations of te tensors introduced in (2.2) to (2.4). For instance, γ αβ( ) is given by γ αβ(u) = 1 2 ( ) α u a β + β u a α. (4.6) Tis leads to te definition of te approximate linear form L (V ) = f v( a) dx + (N v + M s)l dτ, (4.7) γ 1 and also of approximate bilinear forms { ] a (U, V ) = ea αβρσ [γ αβ (u) γ ρσ (v) + e2 12 χ αβ (U) χ ρσ(v ) +2e E } 1 + ν aαβ δ α3 (U) δβ3 (V ) ( (4.8) a) dx, ( ) b (r a 3, s a 3 ) = α r a 3 + r d α α s a 3 dx. (4.9) It is easy to ceck tat V X() and V X one as L(V V ) a (U p,, V V ) 1 p b (r p, a 3 ; (s a 3 s a 3 )) = (L L )(V V ) + L (V V ) (a a )(U p,, V V ) a (U p,, V V ) 1 p (b b )(r p, a 3 ; (s a 3 s a 3 )) (4.10) 1 p b (r p, a 3 ; (s a 3 s a 3 )). To go furter, we recall some standard notations: (i) E K denotes te set of edges of K wic are not contained in γ 0 and E 1 K te set of elements of E K wic are contained in γ 1 ; (ii) for eac e E K, ν = (ν 1, ν 2 ) is a unit vector normal to e, wit te furter assumption tat, wen e belongs to E 1 K, ν is outward to ;

7 Posteriori Analysis of a Finite Element Discretization 117 (iii) for eac e E K, e stands for te lengt of e; (iv) for eac e E K \E 1 K, [ ] e denotes te jump troug e; (v) K is te union of triangles of T tat sare an edge wit K; (vi) K is te union of triangles of T tat intersect K. We recall tat from [3, Teorem IX.3.11 and Corollary IX.3.12] tere exists a Clément type operator R wic maps Hγ 1 0 () into M γ 0 = M Hγ 1 0 and satisfies, for all functions χ Hγ 1 0 (), eac K T and eac e of K wic is not contained in γ 0, χ R χ L 2 (K) + K χ R χ H 1 (K) c K χ H 1 ( K ), χ R χ L 2 (e) c 1 2 χ H 1 ( K ). (4.11) Te idea is to take V equal to (R v, R s) and χ equal to R χ in (4.3). From [2, Lemma 3.3, Lemma 3.4 and Lemma 3.5], we define te quantities linked to te local approximation error on te data: for eac K T, = K f f L 2 (K) 3 + ( ) 1 2 e N N L 2 (e) 3 + M M L 2 (e) 3, (4.12) ε (d) K e E 1 K and also from te global approximation error on te coefficients ε (c) ( = a ( a) L () l l L (γ 1 ) + sup 1 α,β,ρ,σ 2 + sup 1 k 3 a αβρσ a αβρσ ak a k L () L () 3 + sup 1 α 2 + sup 1 α,β 2 a αβ a αβ L () ) α a 3 d α L L. () 3 (4.13) We are now in a position to prove te a posteriori error estimate. In order to state it, we define te error indicators. We use Remark 3.2 to write a(u, V ) and observe tat a similar form olds for a (U, V ), wit te obvious notation for te quantities n ρσ ( ), mρσ ( ), t β ( ) and θ ρσ( ) (in comparison wit (3.4) to (3.7), all coefficients are replaced by teir approximations). For eac K T, te error indicator η K is defined by wit η K = η K1 + η K2 (4.14) η K1 = f K ( ( a) + ρ (n ρσ (u p,)a σ + m ρσ (U p,)d σ + t ρ (U p,) a 3 )( a) ) L 2 (K) [ ( 2 e νρ n ρσ (u p,)a σ + m ρσ (U p,)d σ + t ρ (U ) ] p,)a 3 ( a) e E K \E 1 K + e L 2 (e) e N l ν ρ ( n ρσ (u p,)a σ + m ρσ (U p,)d σ + t ρ (U p,)a 3) ( a) L 2 (e) 3 e E 1 K (4.15)

8 118 N. Tabouce and η K2 = ρ ( K m ρσ (U p,)a σ( ) a) t β (U p,)a β( a) + 1 ( ) ρ ( ρ (r p, a 3 )a 3 ρ (r p, a 3 )d p ρ) L 2 (K) 3 + [ 1 2 e ν ρ m ρσ (U p,)a σ( a) + 1 ] p ν ρ ρ (r p, a 3 )a 3 e E K \EK 1 e L 2 (e) e M l ν ρ m ρσ (U p,)a σ( a) 1 p ν ρ ρ (r p, a 3 )a 3. L 2 (e) 3 e E 1 K (4.16) Note tat tese indicators are of residual type and easy to compute since tey only involve polynomial functions. 4.3 Te Main Results Teorem 4.1. For any data (f, N, M) in L 2 (; R 3 ) L 2 (γ 1 ; R 3 ) L 2 (γ 1 ; R 3 ), te following a posteriori error estimate between te solution U p of problem (3.2) and te solution U p, of problem (4.2) olds: ( ( ) ) 1 2 U p U p, X() c ηk 2 + ε (d)2 K + ε (c). (4.17) K T Proof. We give an abridged proof. From te ellipticity property (2.11), using te residual equation (4.3) by replacing its second member by (4.10), we ten use te triangle inequality tat, combined wit [2, Lemma 3.3, Lemma 3.4 and Lemma 3.5], leads to te bound of te following quantity: L (V V ) a (U p,, V V ) 1 p b (r p, a 3 ; (s a 3 s a 3 )) wit V = (R v, R s). We can write L (V V ) = L ((v R v), 0) + L (0, (s R s)), and a (U p,, V V ) = a (U p,, (v R v, 0)) + a (U p,, (0, s R s), 1 p b (U p,, V V ) = 1 p b (r p, a 3 ; (v R v, 0)) + 1 p b (r p, a 3 ; (0, s a 3 R s a 3 )).

9 Posteriori Analysis of a Finite Element Discretization 119 Note tat b (r p, a 3 ; (v R v, 0)) = 0. It remains to bound te following two quantities: and A 2 = A 1 = sup s H 1 γ 0 (;R 3 ) L (v R v, 0) a (U p,, (v R v, 0)) sup v Hγ 1 0 (;R 3 ) v H 1 (;R 3 ) 1 s H 1 (;R 3 ) { L (0, s R s) a (U p,, (0, s R s)) 1 p b (r p, a 3 ; (0, (s R s) a 3 )) Setting w = v R v and using te symmetry properties of te n ρσ ( ) and mρσ ( ), we ave L (v R v, 0) a (U p,, (v R v, 0)) = f w( a) dx + N wl dτ n ρσ (u p,) ρ w a σ + m ρσ (U p,) ρ w d σ + t β (U p,) β w a 3)( a) dx. By cutting te integrals on into te sum of integrals on te K in T and integrating by parts on eac K, we derive L (v R v, 0) a (U p,, (v R v, 0)) = f w( a) dx + N w l dτ γ 1 }. γ 1 + K T ( K K ρ ((n ρσ (u p,)a σ + m ρσ (U p,)d σ + t β (U p,)a 3)( a) ) w dx ν ρ (n ρσ (u p,)a σ + m ρσ (U p,)d σ + t β (U p,)a 3)( a) w dτ Using te Caucy Scwarz inequality combined wit (4.11) leads to ). (4.18) A 1 c ( K T η 2 K1 Setting t = s R s, and using te same arguments as previously, we arrive to ) 1 2. Tis concludes te proof. A 2 c ( K T η 2 K2 ) 1 2.

10 120 N. Tabouce Teorem 4.2. For any data (f, N, M) L 2 (; R 3 ) L 2 (γ 1 ; R 3 ) L 2 (γ 1 ; R 3 ), te following bounds old for all indicators defined in (4.15) (4.16): ( ) 1 ( ) 2 2 η Ki c U p U p, X(K ) + ε (d) K + ε (c), i = 1, 2. (4.19) K K Proof. Tis is an abridged proof of te estimate for η K1. We write η K1 in compact form as η K1 = K F L 2 (K) e [G ] e L 2 (e) e N l G L. 2 (e) 3 e E K \E 1 K e E 1 K We take R v = 0 in (4.18) and w = v equal to { F ψ w = K on K, 0 on \K, were ψ K denotes te bubble function on K. Tus, all te terms on te rigt-and side of (4.18) vanis but te integral on K. We observe tat function w (tus F ) on K is a polynomial of degree 3. From te appropriate inverse inequality [3, Proposition VII.4.1] togeter wit (4.10) and (4.3) combined wit [2, Lemma 3.3, Lemma 3.4 and Lemma 3.5], leads to K F L 2 (K) 3 c ( U p U p, X(K) + ε (d) K + ) ε(c) K. (4.20) Similarly, to bound η K1 for any edge e sared by two elements K and K, we take w in (4.18) equal to { P e,k ([G ] e ψ e ) on k K, K }, w = { 0 on \ K K }, were ψ e is te bubble function on e and P e,k is te lifting operator introduced in [3, Lemma XI 2.7] from polynomials on e vanising on e into polynomials on K vanising on K\e, constructed by an affine transformation from a fixed lifting operator on te reference triangle. Finally, for eac e in EK, 1 we take te function w in (4.18) equal to { Pe,k ((N w = l G )ψ e ) on K, 0 on \K, and tis gives te bound for η K1.

11 Posteriori Analysis of a Finite Element Discretization Te Adaptivity Strategy and Numerical Experiment Now we present te adaptivity strategy and some numerical experiments. 5.1 Adaptivity Strategy (i) Construct a first mes T 0. Set i = 0. (ii) Solve te numerical problem on T i. Let u i denote te solution. (iii) Compute η Ki (u i ) on K i T i. (iv) If te global estimator is small enoug, ten stop. (v) Oterwise, compute te new step of mes Ki+1 : Ki+1 = 1 2 K i if η Ki (u i ) T OL, oterwise Ki+1 = l Ki, l 1. Here T OL = 1 n ti K i T i were n ti is te number of triangles of T i. (vi) Generate te new mes T i Numerical Experiment and return to (ii). η Ki (u i ), Te numerical experiment tat we present as been performed on te finite element code FreeFem++, see [10]. Tree-dimensional visualization of te deformed sells uses Medit, a free mes visualization software available at ttp:// jussieu.fr/ frey/logiciels/medit.tml. Hyperbolic paraboloid sell Te reference domain is te square { = (x, y); x + y } 2b, (5.1) as illustrated in [1, and 2.4.2] and te cart ϕ is defined by ( c ) T ϕ(x, y) = x, y, 2b 2 (x2 y 2 ). (5.2) We coose b = 50 cm, c = 10 cm and e = 0.8 cm. We assume tat te sell is clamped on γ 0 = and tat it is subjected to uniform pressure q = 0.01 kp/cm 2. Te mecanical data are E = kp/cm 2, ν = 0.4. Te reference value for tis test is te

12 122 N. Tabouce normal displacement at te center C(0, 0) of te sell. Its value computed by various metods is of cm; see [1]. We take p = 10 3 E, see [5]. 2(1 + ν) Results wit mes adaptation Iteration Degrees of freedom U 3 (C) Results witout mes adaptation Degrees of freedom U 3 (C) Note tat te resolution of te discrete problem wit mes adaptation gives te convergence to te solution after te tird iteration wit a number of degrees of freedom equal to but te resolution of te problem witout mes adaptation gives an approximate solution wit an error of order and a number of degrees of freedom was lower tan calculated by mes adaptation. Figure 5.1 presents te initial and te final adapted meses according to te strategy described above. Figure 5.2 and 5.3 present te over-deformed sell. Figure 5.1: Te initial and adapted meses

13 Posteriori Analysis of a Finite Element Discretization 123 Figure 5.2: Te over-deformed top side Figure 5.3: Te over-deformed bottom side References [1] M. Bernadou, Métodes d éléments finis pour les problèmes de coques minces, Recerces en Matématiques Appliquées 33, Masson, [2] C. Bernardi, A. Blouza, F. Hect and H. Le Dret, A posteriori analysis of finite element discretization of a Nagdi sell model, Laboratoire J-L-Lions de Paris VI, preprint R08048, [3] C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Matématiques & Applications (Berlin), 45, Springer, Berlin, [4] A. Blouza, Existence et unicité pour le modèle de Nagdi pour une coque peu régulière, C. R. Acad. Sci. Paris Sér. I Mat. 324 (1997), no. 7,

14 124 N. Tabouce [5] A. Blouza, F. Hect and H. Le Dret, Two finite element approximations of Nagdi s sell model in Cartesian coordinates, SIAM J. Numer. Anal. 44 (2006), no. 2, [6] A. Blouza and H. Le Dret, Nagdi s sell model: existence, uniqueness and continuous dependence on te midsurface, J. Elasticity 64 (2001), no. 2-3, [7] C. Carstensen, Residual-based a posteriori error estimate for a nonconforming Reissner-Mindlin plate finite element, SIAM J. Numer. Anal. 39 (2002), no. 6, [8] C. Carstensen and J. Scöberl, Residual-based a posteriori error estimate for a mixed Reißner-Mindlin plate finite element metod, Numer. Mat. 103 (2006), no. 2, [9] P. J. Frey and P.-L. George, Maillages, applications aux éléments finis, Hermès, [10] F. Hect and O. Pironneau, FreeFem++, [11] R. Verfürt, A Review of a posteriori error estimation and adaptive mesrefinement tecniques, Wiley and Teubner, 1996.