Fig.3.13 – Snapshots at times t = 0.2 and 0.21 s, illustrating the deformation of the ball before energy explosion when using a midpoint time integration scheme.
time (s)
elastic energy (J)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 1
0.5
Energy conserving Euler-Newmark Energy dissipating (alpha = 0.5)
Fig.3.14 – Evolution of elastic energy after impact for first order accuracy Euler-Newmark scheme, and second order accuracy energy conserving and dissipating (α= 0.5) schemes.
enforcing the usual Kuhn-Tucker conditions at entire time steps. The analysis of these techniques is illustrated with numerical simulations. An extension to viscoelasticity is also proposed.
Fig. 3.15 – Contact pressures computed at time t = 0.16 s, when energy penetration is maximal, for Euler-Newmark, midpoint and energy-conserving schemes.
time (s)
penetration energy (J)
0.1 0.2
0.05 0.15 0.25
0 1 2
0.5 1.5 2.5
epsilon = 1.E-5 epsilon = 1.E-4
time (s)
penetration energy (J)
0.15 0.155 0.16 0.165 0.17
0.28 0.29 0.3
0.275 0.285 0.295 0.305
Fig. 3.16 – Evolution of penetration energy during ball impact for η = 1.E-4 and η = 1.E-5 (left), and zoom on the oscillations for= 1.E-5 (right).
time (s)
penetration energy (J)
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
0 100 200 300 400
epsilon = 1.E-4 epsilon = 1.E-3
Fig. 3.17 – Oscillations of penetration energy for a cube impact problem, as the penali- zation coefficient η decrease.
A stabilized discontinuous mortar formulation for elastostatics and elastodynamics problems
R´esum´e
Dans ce chapitre, nous introduisons et analysons une formulation mortier sta- bilis´ee utilisant des multiplicateurs de Lagrange discontinus, pour des approxi- mations aux ´el´ements finis d’ordre 1 et 2 de la solution de probl`emes d’´elas- ticit´e lin´earis´ee. Cette approche s’inscrit dans la continuit´e de Brezzi et Ma- rini [BM00] qui utilisent de tels multiplicateurs pour une formulation dite `a trois champs dans le cadre de probl`emes elliptiques scalaires. Dans le cas d’un grand nombre de sous-domaines, nous montrons en outre l’ind´ependance de la constante de coercivit´e de la forme bilin´eaire associ´ee au probl`eme d’´elas- tostatique non-conforme par rapport au nombre de sous-domaines consid´er´es et `a leur taille, par extension au cas d’interfaces courbes des id´ees de Gopa- lakrishnan et Brenner [Gop99, Bre03, Bre04], et g´en´eralisation de l’op´erateur d’interpolation de Scott et Zhang [SZ90]. De plus, nous rappelons la conver- gence optimale de la m´ethode en ´elastostatique lin´earis´ee en utilisant les outils de Wohlmuth [Woh01], et proc´edons `a une extension au cas de l’´elastodyna- mique. Enfin, des choix concrets d’espaces sont propos´es, les d´etails pratiques de mise en oeuvre sont indiqu´es, et des tests num´eriques viennent illustrer la pr´esente analyse.
103
Abstract
We introduce and analyze first and second order stabilized discontinuous two- field mortar formulations for linearized elasticity problems, following the stabi- lization technique of Brezzi and Marini [BM00] introduced in the scalar elliptic case for a three-field formulation. By extension to the curved interfaces case of the ideas from Gopalakrishnan and Brenner [Gop99, Bre03, Bre04], and from the introduction a generalized Scott and Zhang interpolation operator [SZ90], we prove the independence of the coercivity constant of the broken elasticity bilinear form with respect to the number and the size of the subdomains. Mo- reover, we prove the optimal convergence of the method by mesh refinement by using the tools from Wohlmuth [Woh01] in the elastostatic case, and extend the result to the elastodynamic framework. Finally, we detail practical issues and present numerical tests to illustrate the present analysis.
4.1 Introduction
In this paper, we introduce, analyze and test a non-conforming formulation using sta- bilized discontinuous mortar elements to find the vector solutionu of linearized elasticity problems such as :
−div(E:ε(u)) =f, Ω⊂Rd,(d= 2,3) u= 0, ΓD,
(E:ε(u))·n=g, ΓN,
(4.1)
where the linearized strain tensor is classically given by : ε(u) = 1
2 ∇u+∇tu ,
and the fourth order elasticity tensorEis assumed to be elliptic over the set of symmetric matrices :
∃α >0,∀ξ ∈Rd×d, ξt =ξ, (E:ξ) :ξ≥α ξ:ξ.
The analysis is also extended to the elastodynamics problem :
ρ∂2u
∂t2 −div(E:ε(u)) =f, [0, T]×Ω, u= 0, [0, T]×ΓD,
(E:ε(u))·n=g, [0, T]×ΓN, u=u0, {0} ×Ω,
∂u
∂t = ˙u0, {0} ×Ω,
(4.2)
and we consider this analysis as a theoretical background for using discontinuous mortar elements in nonlinear elastodynamics.
Mortar methods have been introduced for the first time in [BMP93, BMP94] as a weak coupling between subdomains with nonconforming meshes, or between subproblems solved with different approximation methods. The main purpose was to overcome the very sub-optimal “√
h” error estimate obtained with pointwise matching. The analysis of this method as a mixed formulation was first made in [Bel99].
Nevertheless, in spite of the optimal error convergence obtained with the original mor- tar elements, some numerical difficulties appear. First, the original space of Lagrange multipliers ensuring the weak coupling is rather difficult to build in 3D on the boun- dary of the interfaces when more than two subdomains have a common intersection (see [BM97, BD98]). Moreover, the original constrained space has a non-local basis on the non-conforming artificial interfaces, which may lead to small spurious oscillations of the approximate solution.
To overcome the first difficulty, one idea is given in [Ses98] when displacements are at least approximated by second order polynomials. The introduced Lagrange multipliers have a lower order, still enabling optimal error estimates, and no special treatment is needed on the boundary of the interfaces. To overcome the second difficulty, dual mortar spaces are proposed in [Woh00, Woh01], enabling the localization of the mortar kinematical constraint. In order to benefit from the advantages of these two approaches, we propose to introduce stabilized low order discontinuous mortar elements. This idea has already been introduced for a first order three-field mortar formulation in [BM00], and we exploit it herein in the two-field framework for first and second order elements when dealing with elastostatics and elastodynamics problems.
Mortar formulations also provide a natural framework for domain decomposition, as observed by [Tal93, AKP95, AMW99, AAKP99, Ste99] and the references therein. A large number of subdomains and their small size is therefore a basic difficulty to overcome. To get an optimal use of such domain decomposition methods, it is then crucial that the constants arising in the analysis of the mortar formulation remain independent (or at least weakly dependent) on the number and the size of the subdomains. One can readily check that the only potential dependence on such parameters is hidden in the coercivity constant of the broken bilinear form associated to the linearized elastostatics problem. In the framework of elliptic scalar problems, both [Gop99, Bre03] and [BM00] have shown the independence of the coercivity constant with respect to the number and the size of the subdomains, respectively when considering two and three-field mortar formulations with plane interfaces. An extension to the vector elasticity case has been proposed by [Bre04].
By definition of a generalized Scott and Zhang [SZ90] interpolation operator, we simplify and extend herein the result to potentially curved interfaces.
In section 2, the fundamental assumptions and results arising in mortar element me- thods to approximate the solution of the elastostatics problem (4.1) are recalled. Well- posedness results are recalled in section 3. Moreover, we prove in section 4, the indepen-
dence of the coercivity constant with respect to the number and the size of the subdomains.
In section 5, we recall the optimal convergence of the method by mesh refinement, and generalize the analysis to the elastodynamics problem (4.2) in section 6. We propose in section 7 the analysis of stabilized discontinuous mortar elements, proving the satisfaction of the fundamental assumptions. In section 8, some practical issues are pointed out : the choice of an appropriate penalization term, and the exact integration of the constraint.
We present numerical tests in section 9 to confirm the previous analysis.