# Propagation of local alterations using Level-Sets within the Arlequin framework

Olivier Jamond, Hachmi Ben Dhia

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Olivier Jamond, Hachmi Ben Dhia. Propagation of local alterations using Level-Sets within the Arlequin framework. 9e Colloque national en calcul des structures, CSMA, May 2009, Giens, France. hal-01413978

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# Propagation of local alterations using Level-Sets within the Arlequin framework

O. Jamond¹², H. Ben Dhia¹

¹ MSSMAT, UMR 8579 (CNRS, Ecole Centrale Paris)
92295 Chatenay Malabry Cedex France
{olivier.jamond,hachmi.ben-dhia}@ecp.fr

² Centre de Technologie de Ladoux (Michelin)
63000 Clermont-Ferrand

**Résumé** — A Heaviside function is used within the Arlequin method to derive an enriched Arlequin methodology labeled the *Shooting Star Method*. The potential of the latter to propagate cracks with great flexibility and effectiveness is exemplified by a numerical test.

**Mots clés** — Arlequin method, X-FEM, crack propagation.

## 1 Introduction

Generating repetetively meshes that match material discontinuities, propagating cracks or evolving zones with strains or stresses localizations in order to achieve accurate finite element solutions, is a rather tedious and time consuming task. Many flexible numerical methodologies have been developed during the last two decades to tackle efficiently this issue. Among other approaches, let us mention the Partition of Unity Method [8] and the Extended Finite Element Method (X-FEM) (e.g. [9]). These methods, quite reminiscent to Treftz approach, enrich basically the finite element space, associated to a given coarse mesh by a local analytical or semi-analytical knowledge about the globally unknown solution. Typically, Level-Sets functions and asymptotic local behaviours of the solution of the considered problem are used locally in the X-FEM. These methods are particularly relevant for the approximation of problems where the local behaviour of the solution is known. The s-method [6] and the finite element method with patches [7] are full finite element-based approach that enriches the coarse finite element space by a local one.

The Arlequin modeling framework [2, 3, 4, 5]) has a wider range of application and presents as a real flexible multimodel and multiscale paradigm : by superposing a local model (say a patch) to a global model and using a partition of energies, it allows for almost any different representation for the super-imposed models. For the simulation of crack propagation or the evolution of fatigue or damage into a solid that may transform into a macroscopic crack, the project of using the Arlequin framework with possible blending with other partition of unity methods has been initiated few years ago. First steps of this project, using only the Arlequin framework, were achieved and significant numerical results have been reported and/or communicated [10, 11]. Two typical results, extracted from [10, 11], are given in the numerical section. In this work, a step forward is achieved in our project : a new methodology that keep the coarse mesh unchanged during the whole propagation process is developed. For this, the fine Arlequin patch is always relocalized in the near critical crack tip zone and the coarse space is enriched by a Heaviside function, which is
adapted during the propagation process. This approach which could be labeled the *shooting star* method is developed and numerically tested in this work.

## 2 Continuous Arlequin Formulation of a cracked elastic body problem

We consider a bidimensional linear elastic cracked solid, occupying the closure of a bounded domain $\Omega_0$. The boundary of the solid is partitioned into $\Gamma_u$, $\Gamma_g$, $\Gamma$. $\Gamma_u$ is the part where the solid is clamped, and $\Gamma$ defines the crack whose tip is denoted by $o$. Free boundary conditions are applied on the faces of the crack and on $\Gamma_g$. The solid is submitted to a volume density of loads $f$.
Let $\Omega_1$ be a subdomain of $\Omega_0$, located in the near crack tip zone, which is partitioned into two regular non overlapping domains $\Omega_c$ and $\Omega_f$ such that (see figure 1) :

$$o \in \Omega_f \qquad (1)$$

$$\partial\Omega_1 \subset \partial\Omega_c \qquad (2)$$

$$\mathrm{meas}(\Omega_i) > 0, i = c, f \qquad (3)$$

<figure><img src="image_3.png" /><figcaption>Figure 1: Arlequin domains</figcaption></figure>

A Lagrangian Arlequin formulation of this problem reads as following : (see e.g. [3, 4, 5])

$$\text{Find } (u_0, u_1, \Phi) \in V_0 \times V_1 \times M; \qquad (4)$$

$$\forall v_0 \in V_0, \quad a_0(u_0, v_0) + c(\Phi, v_0) = l_0(v_0) \qquad (5)$$

$$\forall v_1 \in V_1, \quad a_1(u_1, v_1) - c(\Phi, v_1) = l_1(v_1) \qquad (6)$$

$$\forall \Psi \in M, \quad c(\Psi, u_0 - u_1) = 0 \qquad (7)$$

where $V_0 = \{v \in H^1(\Omega_0); v = 0 \text{ on } \Gamma_u\}$ and where $V_1 = H^1(\Omega_1)$. The mediator space $M = H^1(\Omega_c)$.
Furthermore, for $i = 0, 1$, the weighted internal and external virtual works are defined by :

$$\forall u_i, v_i \in V_i, \quad a_i(u_i, v_i) = \int_{\Omega_i} \alpha_i \sigma(u_i) : \varepsilon(v_i) dx \qquad (8)$$

$$\forall v_i \in V_i, \quad l_i(v_i) = \int_{\Omega_i} \beta_i f \cdot v_i dx \qquad (9)$$

The weight parameter functions $(\alpha_i, \beta_i)$, $i = 0, 1$, are defined in $\Omega_0$ and satisfy : ([5])

$$\alpha_i, \beta_i \in [0, 1] \text{ in } \Omega_0 \qquad (10)$$

$$\alpha_0 + \alpha_1 = 1, \beta_0 + \beta_1 = 1 \text{ in } \Omega_0 \qquad (11)$$

$$\alpha_0, \beta_0 = 1 \text{ in } \Omega_0 \setminus \Omega_1 \qquad (12)$$

$$\exists \alpha_0 > 0; \alpha_i \ge \alpha_0 \text{ in } \Omega_f \qquad (13)$$
The coupling operator c is defined by :

$$ \forall (\psi, v) \in M \times V_1, c(\psi, v) = \int_{\Omega_c} \kappa_0 \{\psi \cdot v\} + \kappa_1 \{\varepsilon(\psi) : \varepsilon(v)\} dx \quad (14) $$

This problem has been analyzed mathematically in [5] and proved to be well-posed.

### 3 Discrete formulation : the shooting star method

To solve efficiently the Arlequin problem defined in the previous section, we suggest the following scheme : let $T_h^0$ be a mesh of the unfractured domain $\Omega_0^\Gamma$. Let $T_h^1$ be a mesh of $\Omega_1$, significantly finer than $T_h^0$, especially in the very near crack tip zone. Let $\Omega_o^s$ and $\Gamma_0$ denote the set of elements of $T_h^0$ containing the crack tip o and the part of the crack $\Gamma$ exterior to $\Omega_o^s$, respectively. Let $B_0 = (\phi_{0i}, i = 1, n_0)$ be a classical vector-valued finite element basis associated to the mesh $T_h^0$. We denote by $V_{0h}^r$ the finite element space spanned by $B_0$. This coarse space is enriched as follows. Let $K_0^d$ be the set of indices k such that $\hat{\omega}_k \cap \Gamma_0 \neq \emptyset$, $\hat{\omega}_k$ standing for the interior of the support of the basis function $\phi_{0k}$. Let $\Omega_d$ denote the union of these supports. $\Omega_d$ is divided by the crack $\Gamma$ into two domains, denoted by $\Omega_d^+$ and $\Omega_d^-$. A precise definition of the domain $\Omega_d^+$ (and thus of $\Omega_d^-$) relies on the signed distance function : defining at each point $p \in \Gamma$ a classical positively oriented frame $(\tau_p, \nu_p)$, one can associate to each point $x \in \Omega_d$ (one of) its nearest point(s) $x^*$ on $\Gamma$. The domain $\Omega_d^+$ is then defined by :

$$ \Omega_d^+ = \{x \in \Omega_d ; (x - x^*) \cdot \nu_{x^*} \geq 0\} \quad (15) $$

The following finite dimensional space :

$$ V_{0h} = V_{0h}^r + V_{0h}^d = \operatorname{span}\{(\phi_{0i})_{i=1,n_0}, (H\phi_{0j})_{j \in K_0^d}\} \quad (16) $$

is then taken as an approximation of $V_0$, where H refers to the sign-like function, defined by :

$$ H(x) = \begin{cases} +1 & \text{in } \Omega_d^+ \\ -1 & \text{elsewhere} \end{cases} \quad (17) $$

Now, based on the fine mesh $T_h^1$ of the cracked subdomain $\Omega_1$, a classical finite element subspace $V_{1h}$ is constructed for the approximation of the local space $V_1$. The discrete mediator space $M_h$ is taken equal to the space of restriction of $V_{1h}$ to the coupling zone $\Omega_c$ (though other choices are possible, as reported in [4, 5]). Finally, we assume that :

$$ \Omega_o^s \subset \Omega_f \quad (18) $$

The condition (18) is a relaxation of the condition (1) that takes into account the finite scale introduced by the discretization.

With these finite dimensional spaces, our discrete enriched Arlequin approximation of the problem defined by (5)-(14) reads as follows.

$$ \begin{array}{l} \text{Find}(u_{0h}^r, u_{0h}^d, u_{1h}, \phi_h) \in V_{0h}^r \times V_{0h}^d \times V_{1h} \times M_h; \\ \quad \forall (v_{0h}^r, v_{0h}^d, v_{1h}, \psi_h) \in V_{0h}^r \times V_{0h}^d \times V_{1h} \times M_h \\ \quad a_0(u_{0h}^r, v_{0h}^r) + a_0^d(u_{0h}^d, v_{0h}^r) + c(\phi_h, v_{0h}^r) = l_0(v_{0h}^r) \end{array} \quad (19) $$

$$ a_0^d(u_{0h}^r, v_{0h}^d) + a_0(u_{0h}^d, v_{0h}^d) + c^d(\phi_h, v_{0h}^d) = l_0(v_{0h}^d) \quad (20) $$

$$ a_1(u_{1h}, v_{1h}) - c(\phi_h, v_{1h}) = l_1(v_{1h}) \quad (21) $$

$$ c(\psi_h, u_{0h}^r - u_{1h}) + c^d(\psi_h, u_{0h}^d) = 0 \quad (22) $$
where:

$$a_0^d(u_{0h}^r, v_{0h}^d) = \int_{\Omega_0 \cap \Omega_d^+} \alpha_0 \sigma(u_{0h}^r) : \varepsilon(v_{0h}^d) dx + \int_{\Omega_0 \cap \Omega_d^-} \alpha_0 \sigma(u_{0h}^r) : \varepsilon(v_{0h}^d) dx \quad (23)$$

$$c^d(\psi_h, u_{0h}^d) = \int_{\Omega_c \cap \Omega_d^+} \{ \kappa_0 \psi_h \cdot u_{0h}^d + \kappa_1 \varepsilon(\psi_h) : \varepsilon(u_{0h}^d) \} dx \quad (24)$$

$$+ \int_{\Omega_c \cap \Omega_d^-} \{ \kappa_0 \psi_h \cdot u_{0h}^d + \kappa_1 \varepsilon(\psi_h) : \varepsilon(u_{0h}^d) \} dx$$

This system involves terms related to the coupling between the enrichment degrees of freedom of the coarse model and the ones of the fine model. These terms are new when referring to specific computational developments for the standard Arlequin formulation, given in [4] and for the X-FEM, given in [9]. They require additional computational efforts that will be detailed during the conference.

# 4 Numerical results

## 4.1 Previous results

Figure 2, extracted from [10], shows the propagation of damage into a bimaterial solid submitted to a cyclic loading acting on its upper edge. From a numerical point of view, the critical mechanical phenomena in the near crack tip zone are taken into account by a local Arlequin patch, evolving and moving with the evolution of the critical zone during the cyclic loading. Observe that the coarse global mesh is adapted when needed to take into account the part of the crack which is not contained in the Arlequin patch. Observe also that, for the considered test, the direction of propagation of the crack is the (known) straight material interface; which simplifies significantly the simulation and particularly the adaptation of the coarse mesh to match the part of the crack outside of the patch.

<figure><img src="image_4.png" /><figcaption>Figure 2: Propagation of damage and creation of a crack within the Arlequin framework (three selected snapshots)</figcaption></figure>

Figure 3, extracted from [11], shows the propagation of a crack in an elastic structure by means of a refined finite element Arlequin patch, adapted to the evolution of the cracked zone, while the coarse mesh is unchanged during the whole process. Data for this problem (called the double cantilever beam test) were taken from the paper by Belytschko and Black [1].
<figure><img src="image_5.png" /><figcaption>Figure 3: Propagation of a crack for the double-cantilever elastic beam</figcaption></figure>

## 4.2 New results

<figure><img src="image_6.png" /><figcaption>Figure 4: geometry for the crack propagation test</figcaption></figure>

We simulate here a quasi-static propagation of a crack in a linear homogeneous and isotropic bidimensional elastic crown (see figure 4) in plane strain conditions. The top of the ring is assigned a displacement $u_0$ in the y direction. A small crack is initiated in the crown. The procedure used for the simulation of the propagation is the following simplified one : for a given size of the crack, the solution is calculated. The polar angle giving the maximum orthonormal stress component on the first crown of elements of the patch surrounding the crack tip is taken as the direction of propagation of the crack. Then a constant increment of 0.5mm is assigned to the crack.

Some steps of the propagation are reported in figures 5-a-b-c. A zoom is given in figure 5-d. These qualitative numerical results aim only at illustrating the effectiveness of our Arelequin/level-set methodology to simulate with enhanced flexibility the propagation of a crack. The main point is that this methodology prevents the remeshing of the whole structure, while capturing the main mechanical features in the critical zones. Only the patch surrounding the crack tip is locally remeshed at each step of the crack propagation.

## 5 Conclusion

A new methodology using the Heaviside enrichment within the multiscale Arelequin framework is developed here. The effectiveness of this methodology to introduce and propagate cracks in sound meshes is exemplified. The extension of this methodology to the simulation of damage propagation is an ongoing work that will be discussed during the conference.

## Références

[1] T. Belytschko, T. Black, Elastic crack growth in finite elements with minimal remeshing Int. J. Numer. Meth. Engrg. 45 (1999) 601-620.
<figure><img src="image_7.png" /><figcaption>Figure 5: propagation of a crack</figcaption></figure>

[2] H. Ben Dhia, Multiscale mechanical problems : the Arlequin method, C. R. Acad. Sci. Série IIb, 326 (1998) 899-904.

[3] H. Ben Dhia, G. Rateau, Application of the Arlequin method to some structures with defects, Finte Element European Review, 11 (2002) 291-304.

[4] H. Ben Dhia, G. Rateau, The Arlequin method as a flexible engineering design tool, Int. J. Numer. Meth. Engrg. 62 (2005) 1442-1462.

[5] H. Ben Dhia, Further insigths by theoretical investigations of the multiscale Arlequin method, Int. J. Multi. Comp. Engrg. 6 (2008) 215-232.

[6] J. Fish, The s-version of the finite element method, Computer and Structures, 43 (1992) 539-547.

[7] R. Glowinski, J. He, J. Rappaz, J. Wagner, Approximation of multi-scale elliptic problems using patches of finite elements, C.R. Acad Sci. Paris, Ser.I. 337 (2003) 779-684.

[8] J.-M. Melenk, I. Babuška, The partition of unity finite element method : Basic theory and applications, Comp. Meth. Appl. Mech. Engrg. 139 (1996) 289-314.

[9] N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without reme-shing, Int. J. Numer. Meth. Engrg. 46 (1999) 131-150.

[10] G. Rateau, H. Ben Dhia, Méthode Arlequin pour la propagation de défauts dans des pneus : Etude de Faisabilité, Post Doctorat Report for Michelin Society, (2005).

[11] Y. Romdhane, H. Ben Dhia, Michelin Society Report, 2006.