Simplectic scheme for the corotational formulation

Transcription

1 Simplectic scheme for the corotational formulation Stefan Hüeber, Julien Salomon, Barbara Wohlmuth, Universität Stuttgart, Institut für Angewandte Analysis und Numerische Simulation Pfaffenwaldring Stuttgart Germany Received *****; accepted after revision Presented by Abstract The non linearity involved in Saint Venant-Kirchhoff elastic bodies makes it costly and inefficient compared to the linear case On the other hand a standard linearization of the elastic terms does not provide satisfying numerical results In order to tackle this problem, we propose a scheme that combined the linearized approach with the corotational formulation The resulting algorithm splits the motion of the body into global translation and rotation parts and an internal part involving elastic deformations A first analysis of this scheme is provided, proving that energy, linear and angular momentums are preserved To cite this article: S Hüeber, J Salomon, B Wohlmuth, C R Acad Sci Paris, Ser???? (00? Résumé Schéma simplectique pour la formulation corotationelle La forte non linéarité du modèle de Saint Venant-Kirchhoff des corps élastiques empèche rapidement les simulations trop complexes De plus, une linéarisation standard des termes élastiques ne donne pas des résultats numériques satisfaisants Pour résoudre ce problème, nous proposons un schéma qui combine une linéarisation avec une descritpion corotationnelle du mouvement L algorithme qui en résulte scinde le mouvement en une translation et une rotation globale d une part et en une déformation élastique d autre part Une première analyse prouve que la quantités de mouvement, le moments cinétique ainsi que l énergie sont préservés par ce schéma Pour citer cet article : S Hüeber, J Salomon, B Wohlmuth C R Acad Sci Paris, Ser???? (00? addresses: hueeber@iansuni-stuttgartde (Stefan Hüeber, salomon@iansuni-stuttgartde (Julien Salomon, wohlmuth@iansuni-stuttgartde (Barbara Wohlmuth Preprint submitted to Elsevier Science 8 mars 006

2 Version française abrégée Considérons un corps élastique R dont la déformation ϕ(x, t suit la dynamique (1, où W est l énergie emmagsinée, F la variable de déformation, E est définie par ( et f et g des contraintes extérieures Afin de linéariser cette équation, nous utilisons la formulation corotationnelle qui, suivant (3, décompose le mouvement en une translation T et une rotation R θ d angle θ globales et en une déformation u En s appuyant sur l égalité (4, la linéarisation est alors effectuée suivant u (et non ϕ x et donne lieue à la formulation variationnelle (5 Nous discrétisons alors les équations globales associées a la quantité de mouvement et au moment cinétique selon un schéma de type point milieu suivant (8 et (9 La formulation variationnelle associée à la déformation est discrétisée suivant (10 Nous montrons alors (Lemmes 4 et 43 que si le calcul de l angle est effectué de telle sorte que la condition (13-qui correspond au fait que le moment des forces élastiques soit nul- soit respectée, le moment cinétique ainsi que l énergie E définie par (14 sont conservées par le schéma L existence d une suite d angles vérifiant (13 est enfin démontrée dans le cas d un système de dimension finie (ou, de manière équivalente, après discrétisation en espace sous l hypothèse peu contraignante (15 par un argument de parité (Lemme 54 Throughout this paper with use the notations: a(x, t = t a(x, t, a(x, t = a(x, t t Given a matrix A, we denote by A T its transposed matrix and by tr(a its trace The identity matrix of R is denoted by I, R θ represents the rotation of θ, Π = R π and A : B = tr(at B is the usual matricial scalar product 1 Model 11 Linear elasticity Let R be a bounded domain with a sufficient smooth boundary := describing an elastic body considered to be a Saint Venant-Kirchhoff material of mass density ρ(x The coordinate system is chosen such that the center of gravity has the coordinates (0, 0 Denoting by M the mass of the body, we then have: ρ = M, ρx = 0 The body is subject to some volume forces f(x, t and boundary tractions g(x, t Let ϕ : [0, T ] R be the deformation mapping In the full non linear setting, the evolution of ϕ is described by the weak form of the following equilibrium condition [1]: ρ ϕη + W (E(ϕ x : η = F fη + gη, (1 where η is a function in H 1 (, F is the generic deformation For a given displacement d, the stored energy W ( E(d is: W ( E(d = λ ( tr ( E(d ( + µ tr E(d,

3 where λ and µ are the Lamé parameters and E(d = 1 ( d + dt + d T d ( 1 Corotational description Because of the non linearity involved in the elastic term, the computational cost of the direct resolution of the equation (1 makes the simulation of complex systems costly In order to solve this problem, the corotational approach propose to decompose the deformation as follows [,3]: ϕ(x, t = T (t + R θ (t(x + u(x, t (3 In this equation, T describes the global translation of the body and R θ its global rotation An important property of this decomposition is that: E ( ϕ x = E ( u, (4 which leads to the linearization E(ϕ x ε(u := 1 ( u+ ut After computation of W F, the resulting linearized weak formulation reads: ρ ϕη + R θ σ(u : ε(η = fη + gη, (5 where σ(u := λtr ( ε(u I + µε(u In order to simplify our presentation, we also introduce the relative velocity s(x, t = Rθ T (t ( ϕ(x, t T (t = u(x, t + θ(tπ ( x + u(x, t The acceleration term can then be expressed by ϕ(x, t = T (t + R θ (t (ṡ(x, t + θ(tπs(x, t (6 Time discretization We present in this section the discrete constraints required for our scheme Let us denote by n the time index We use the mid-point rule which has the convenient property: a b + a ḃ = [ab], (7 where we have used the following notations: = n+1 + n, = n+1 n, t with t the step of the time discretization, and is a generic time discretized quantity We keep these notations throughout this section 1 Linear and angular momentum evolution We describe in this section the constrains required for our numerical scheme The global translation and rotation of the body are described the by: ( ( M T d = f + g, dt ( ρ ϕπϕ = fπϕ + gπϕ 3

4 We derived from these equations the following time discretizations (the angular momentum is expressed with respect to the center of gravity: T n+1 = Ṫ n + t ( f + g, (8 M ρs n+1 Π(x + u n+1 = ρs n Π(x + u n ( + t R T fπ(x + u + R T gπ(x + u (9 Note that the elastic terms do not enter explicitly in these equations, so that, in case of achievement of these equations, the motion of the body is correctly split in a global part and an internal part Elastic deformations In order to time discretized (5, we first define iteratively the time discretized relative velocity from s 0 (x and (u n n N : s (x = u (x + θ Π ( x + u (x We then defined a discrete weak formulation corresponding to (5 combined with (6 by: T v + ρ(ṡ + θ Πs v + σ(u : ε(v = Rθ T fv + R T θ where the test function v belongs to H 1 ( R T θ gv, (10 3 Algorithm We derive from the previous constrains the following algorithm Given T 0, θ 0, ϕ 0 ϕ0 and the external forces (f n N and (g n N, θ n and u n, the computation of T n+1, θ n+1 and u n+1 is carried out by the iterative resolution of: T n+1 = Ṫ n + t ( f + g, (11 M R T θ T v + ρ(ṡ + θ Πs v + σ(u : ε(v = Rθ T fv + Rθ T gv (1 s (x = u (x + θ Π ( x + u (x, where θ n+1 is a solution of: σ(u : ε(πu = 0 (13 The non linearity of these equations implies that the condition (13 is achived through an iterative procedure In the next section we prove the existence of u n+1 and θ n+1 fulfilling (1-13 under reasonable conditions in the fully discretized system 4

5 4 Angular momentum and Energy conservation In this section, we prove that the condition (13 implies that energy and the angular are conserved We first note that under the condition (8, the deformation is orthogonal to the translations Lemma 41 Suppose that the translation term is defined by (8, then (u n n N fulfills the conditions: n N, ρu n = 0 It is now possible to prove the conservative properties of our scheme Lemma 4 Suppose that θ n+1 is such that the condition (13 is fulfilled, then the angular momentum is preserved, in the sense that the equation (9 holds Proof: Testing (1 with Π(x + u, we obtain: ρ ( ṡ + θ Πs Π(x + u = Rθ T fπ(x + u + Rθ T gπ(x + u, thanks to (15 and the fact that ε(πx = 0 On the other hand, one has: ρ θ Πs Π(x + u = ρs Π u The result follows from the equation (7 applied with: a n = s n b n = Π(x + u n We finally prove the energy conservation Lemma 43 Suppose that θ n+1 is such that the condition (13 is fulfilled, then the scheme (11-1 is simplectic in the sense that: ( E n+1 E n = t f(ṫ + R θ s + g(ṫ + R θ s where E is the total energy of the body: E n = 1 MṪ n + 1 ρs n + 1 σ(u n : ε(u n Proof: Testing (1 with v = Rθ T T + s, we obtain: ρrθ T T s = 0, ρ ( ṡ + θ Πs T = 0, σ(u : ε(ṫ = 0, by Lemma 41 and the fact that ε (Ṫ = 0 Moreover: ρrθ T T Rθ T T n+1 T = M Ṫ n t ρ ( ṡ + ω Πs s = 1 ( ρs n+1 ρs n t σ(u : ε(s = 1 σ(u n+1 : ε(u n+1 σ(u n : ε(u n t The last equation has been obtained using (13 The result follows 5 (14

6 5 Analysis of the full-discretized system Our main result is stated in the next theorem Theorem 51 Suppose that (u n n N belongs to a finite dimensional space Under the condition: σ(x : ε(x + u n 0, (15 the equations (11-13 have a solution (u n+1, θ n+1 Note that the condition (15 is not restrictive for small deformations, since σ(x : ε(x = 4(λ + µ > 0 For reasons of conveniency, we consider that the test functions are orthogonal to the translations We first prove that, given θ n+1, the equation (10 has a unique solution Lemma 5 Given θ n, θ n+1, u n H 1 (, f and g, the equation (10 has a unique solution u n+1 Proof: The equation (1 reads : ρm u n+1v + ( t σ(u n+1 ε(v = l n,θn+1 (v, where M = Id + (θ n+1 θ n Π and l n,θn+1 is a continuous linear form on H 1 ( that does not depend on u n+1 The coercivity of right hand side bilinear form is a consequence of the fact that: if θ n+1 = θ n, there exists c > 0 such that: ρm yy + ( t σ(yε(y 4 y L ( + c y L (, y H1 ( and if θ n+1 > θ n, there exists c > 0 such that: ρm Π1/ yπ 1/ y+( t σ(π 1/ yε(π 1/ y 4(θ n+1 θ n y L ( +c y L (, y H1 ( The existence of c and c is a consequence of the Korn s inequality [1] The case θ n+1 θ n can be proved in a similar way Note that the previous lemma holds in the general case, and is based on the fact that M is invertible in the finite dimensional case The next lemma proves in the finite dimensional case, u n+1 is uniformly bounded with respect to θ n+1 Lemma 53 The function θ n+1 R u n+1 is bounded Proof: Testing (1 with v = u n+1 and using Lemma 41, we obtain: ( 4 (θn+1 θ n u n+1 L ( + ( t σ(u n+1 : ε(u n+1 = l n,θn+1 (u n+1 The result then follows from the facts that l n,θn+1 (u is a second order polynomial with respect to θ n+1 and that σ(un+1:ε(un+1 u n+1 L ( is bounded We can now prove the existence of solutions of (15 Lemma 54 Under the condition (15, there exists θn+1 such that: σ(u : ε(πu = 0 Proof: In the finite dimensional case, the equation (1 reads: ρm u + ( t Su = ts n + M ( un (θ n+1 θ n Πx, 6

7 where S is the matrix corresponding to the elastic term It follows that: (I + t ρ M S u = I (θ n+1 θ n Π 4 + (θ n+1 θ n (u n (θ n+1 θ n Πx + t ρ M s n Since for a given v, M v = O( v, we obtain: θn+1 u = x Π(x + u n + O( θ n+1 and: σ(u : ε(πu = u SΠu = 1 xs(x + u n + o(, θ n+1 θ n+1 and the result follows 1 θ n+1, References [1] PG CIARLET, Mathematical Elasticity, Volume I, North-Holland Publishing Co, 1988 [] C FARHAT, K PIERSON, M LESOINNE, The second generation of FETI methods and their application to parallel solution of large-scale linear and geometrically nonlinear structutral analysis problems, Computer methods in Applied Mechanics and Engineering, 184 (000, [3] C GRANDMONT, Y MADAY, P METIER, Existence of a solution for an unsteady elasticity problem in large displacement and small perturbation, C R Acad Sci Paris Sér I Math, 334 (00,