Fictitious Domain Methods for Fracture Models in Elasticity Olivier Bodart, Valérie Cayol, Sébastien Court, and Jonas Koko
Abstract In this paper we are interested in a linear elasticity system modeling the presence of a crack inside a volcano. The traction force on this crack induces discontinuities of the displacement field. The computation of the latter is carried out with a finite element method for which the boundary of the crack is taken into account with a fictitious domain approach; It means that the mesh we consider does not fit to the crack. The interest of this approach lies in a framework where the position and the shape of the crack is lead to evolve, and in that case no re-meshing is required. Keywords Fictitious domain approach • Fracture models
1 Introduction Cracks (or fractures) play a major role in crustal deformations, whether acting in tensile mode or in shear mode. As a consequence, the simulation of displacements produced by inside cracks is an important issue for geological applications. However, when the position and the shape of a crack have to be updated from a step to the next, for instance when studying the propagation of a crack or when inverting surface deformation, such computations are expensive, making the modeling challenging. This study presents a method that efficiently addresses the modeling of cracks, using a fictitious domain approach such that the cracks do not have to fit the mesh. The literature in this field of research is more and more abundant; Let us just cite the
O. Bodart • V. Cayol • S. Court () • J. Koko Université Blaise Pascal - Labex ClerVolc, Campus des Cézeaux 24 avenue des Landais, BP 80026, 63171 Aubière cedex, France e-mail:
[email protected] © Springer International Publishing AG 2016 G. Russo et al. (eds.), Progress in Industrial Mathematics at ECMI 2014, Mathematics in Industry 22, DOI 10.1007/978-3-319-23413-7_80
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original article of the eXtended Finite Element Method [8] for taking into account cracks, or [6] in the context of fluid mechanics. The method we use is inspired from XFEM, but no enrichment of basis functions by singular functions is required near the crack region. The principles of our method are more comparable with the ones of [7] or [3], with the originality that we have to tackle here discontinuous fields of displacement. An other advantage of our method is the simplicity of the implementation. The paper is divided as follows: In Sect. 2 we set the theoretical problem, and transform it into a variational problem. Next in Sect. 3 we explain the discretization we develop here, in particular the way the fictitious domain method is carried out. In the last Sect. 4 we provide numerical experiments, namely convergence tests and also physical experiments.
2 Setting of the Problem Given a domain ˝ of R2 , and a crack T ˝ represented by an injective curve, we consider a steady linear elasticity model governed by the following system: 8 < div L .u/ D f in ˝; u D 0 on @˝; : L .u/n D pn on T : In this system the displacement of the solid is denoted by u, some external forces (like the gravity) by f, and L .u/ D 2".u/ C .div u/iR2 denotes the Lamé stress tensor, with ".u/ D 12 ru C ruT . The coefficients and can be related to the Poisson coefficient and the Young modulus E by the formulas below: D
E ; .1 2/.1 C /
D
E : 2.1 C /
The traction force of value p > 0 is applied on the both sides of the crack T , so we have to make precise the outward normal n on T . Moreover, for giving a sense to the both sides of the crack, we have to be able to determine whether a point of the domain lies on one side or the other of the crack. For that, the most convenient way we have found consists in uncoupling the problem by setting two unknowns displacements instead of a global one (Fig. 1).
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Fig. 1 Splitting of the domain according to the crack
2.1 Uncoupling the Problem from the Discontinuities of the Displacement Field We extend the crack T to , as represented below: The global domain ˝ is now split into two sub-domains ˝ C and ˝ . We have: D 0 [ T ;
˝ D ˝ C [ [ ˝ :
Let us now denote uC D uj˝ C and u D uj˝ C . On the artificial boundary 0 — which is not connected—we have to ensure the continuity of the displacement, that is to say uC u D 0. The system we are now interested in is the following1: 8 ˆ ˆ ˆ ˆ ˆ
0:53;
.x C y/ cos.x/ D1 .x y/ sin.y/ D2
if y 0:53:
In the figures below we compute the relative errors on the displacement, between the exact solution above and the computed one, for different choices of finite elements, and we deduce an approximation of the order of convergence (Figs. 2, 3, 4, and 5).
4.2 Physical Tests For a rectangle Œ0I 100 Œ0I 50 with a mesh size h D 1, we consider a crack whose position and shape—a segment- can be guessed on the pictures below. In the first test the crack is vertical, in the second one it is inclined, and in the third test it is inclined and touching the surface. On the surface we consider a Neumanntype homogeneous condition: L .u/n D 0. The intensity p > 0 of the traction applied on the both sides of the crack is chosen to be constant. After computation of the displacement, we deform the initial rectangle according to the effects of the displacement, by amplifying the effects of the deformation (Figs. 6, 7, and 8).
Fig. 2 L2 .˝/-relative error (in %)
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Fig. 6 Displacement due to a vertical inside crack applying traction forces
Fig. 7 Displacement due to an inclined inside crack applying traction forces
Fig. 8 Displacement due to an inclined crack intersecting the surface and applying traction forces
5 Conclusion We have solved numerically a crack problem in an elastic medium by a finite element method relying on a fictitious domain approach. Rates of convergence have been computed and physical tests have been performed for underlying the accuracy of our approach. The direct problem we have considered is the first step toward the study of an inverse problem: The goal is to recover information on the crack (position, shape, stress) from surface measurements. The interest of the fictitious domain approach lies in an algorithmic framework in which the position of the sought crack would have to be
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updated; In that case no re-meshing would be needed, only local re-assembling of stiffness matrices would be required, and so we hope to obtain a gain of efficiency in terms of computation time and resources. Acknowledgements This research was financed by the French Government Laboratory of Excellence initiative no. ANR-10-LABX-0006, the Région Auvergne and the European Regional Development Fund. This is Laboratory of Excellence ClerVolc contribution number 139.
References 1. Bodart, O., Cayol, V., Court, S., Koko, J.: Xfem based fictitious domain method for linear elasticity model with crack. SIAM J. Sci. Comput. 38(2), B219–B246 (2016). http://arxiv.org/ abs/1502.03148 2. Choi, Y.J., Hulsen, M.A., Meijer, H.E.H.: Simulation of the flow of a viscoelastic fluid around a stationary cylinder using an extended finite element method. Comput. Fluids 57, 183–194 (2012). doi:10.1016/j.compfluid.2011.12.020. http://dx.doi.org/10.1016/j.compfluid.2011.12. 020 3. Court, S., Fournié, M., Lozinski, A.: A fictitious domain approach for the stokes problem based on the extended finite element method. Int. J. Numer. Methods Fluids 74(2), 73–99 (2014). doi:10.1002/fld.3839. http://dx.doi.org/10.1002/fld.3839 4. Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004) 5. Gerstenberger, A., Wall, W.A.: An extended finite element method/Lagrange multiplier based approach for fluid-structure interaction. Comput. Methods Appl. Mech. Eng. 197(19–20), 1699– 1714 (2008). doi:10.1016/j.cma.2007.07.002. http://dx.doi.org/10.1016/j.cma.2007.07.002 6. Girault, V., Glowinski, R., Pan, T.W.: A fictitious-domain method with distributed multiplier for the Stokes problem. In: Applied Nonlinear Analysis, pp. 159–174. Kluwer/Plenum, New York (1999) 7. Haslinger, J., Renard, Y.: A new fictitious domain approach inspired by the extended finite element method. SIAM J. Numer. Anal. 47(2), 1474–1499 (2009). doi:10.1137/070704435. http://dx.doi.org/10.1137/070704435 8. Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46(1), 131–150 (1999). doi:10.1002/(SICI)10970207(19990910)46:13.0.CO;2-J
