# Modélisation et discrétisation par éléments finis des discontinuités d'interfaces pour les problèmes couplés

Andreas Kölke

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Andreas Kölke. Modélisation et discrétisation par éléments finis des discontinuités d'interfaces pour les problèmes couplés. 8e Colloque national en calcul des structures, CSMA, May 2007, Giens, France. hal-01504128

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# Modélisation et discrétisation par éléments finis des discontinuités d'interfaces pour les problèmes couplés

## Application de XFEM pour les discontinuités mobiles

Andreas Kölke

Assistant Professor for Fluid-Structure Interaction
Institute for Structural Analysis / Computational Sciences in Engineering
Beethovenstraße 51, 38106 Braunschweig, Germany
a.koelke@tu-bs.de

**RÉSUMÉ.** Le traitement numérique des problèmes couplés multiphysiques présentant des changements topologiques d'interfaces fait l'objet de recherches intensives. Afin de suivre de façon Eulérienne ce type de frontières mobiles, une technique basée sur des lignes de niveau est utilisée (Osher et al., 1988). Le domaine, les lignes de niveau ainsi que les différents champs physiques inconnus sont discrétisés par des éléments finis espace-temps (Hughes et al., 1988, Hübner et al., 2004). Ceci conduit à une approche monolytique du système couplé. Les espaces d'approximations des différentes variables physiques sont enrichis afin de pouvoir représenter correctement les discontinuités fortes et faibles dues à la présence des interfaces (méthode des éléments finis espace-temps enrichis). Les variables sont couplées à l'interface par des multi-plicateurs de Lagrange distribués (Kölke et al., 2004).

**ABSTRACT.** The treatment of boundary-coupled multifield problems with topologically changing interfaces is subject of intensive research in computational mechanics. For these types of moving boundaries, the Eulerian interface capturing level set method (Osher et al., 1988) is applied. Space-time finite elements (Hughes et al., 1988, Hübner et al., 2004) are used for discretization of the domain, the level set function and the set of physical unknowns, promoting the monolithic analysis of the coupled system. The choice of extended space-time approximation spaces for physics enables to capture strong and weak discontinuous solutions introduced by interfacial constraints and ensured by distributed Lagrange multipliers (Kölke et al., 2004).

**MOTS-CLÉS:** frontières mobiles, discontinuités d'interfaces, éléments finis espace-temps enrichis

**KEYWORDS:** moving boundaries, interfacial discontinuities, enriched space-time finite elements.
# 1. Introduction

Numerous models in engineering are concerned with the existence of strong and weak discontinuities, occuring with bi-material contact surfaces, chemical reaction fronts, solidification processes, two-fluid flow with surface tension, or propagation of discrete cracks in elastic solids. The motion of these interfaces is governed by the dynamics of the underlying physical process. The shape of physical quantities across and nearby the interface can strongly influence the behavior of the coupled multifield system. The challenge within the numerical solution of such problems is in describing the motion of a topologically changing interface or front and in the treatment of interface induced nonlinearities.

# 2. General Mixed Euler-Lagrange Weighted Residuals Form

The space-time finite element formulation of a general boundary-coupled system, consisting of two compressible and istothermal continua with impermeable interface for the n-th space-time slab $Q^\chi = \Omega^\chi \times [t_n, t_{n+1}]$ is described by mass balance

$$ \sum_{i=1}^{2} \int_{Q_i^\chi} \delta(\rho_i^\chi \mathbf{v}_i^\mathbf{a} \cdot \mathbf{v}_i^\mathbf{a}) \left( \left. \frac{\partial \rho_i^\chi}{\partial t} \right|_\chi + \nabla_\chi \cdot (\rho_i^\chi \mathbf{v}_i^\mathbf{c}) \right) dQ^\chi, \quad [1] $$

and conservation of momentum

$$ + \sum_{i=1}^{2} \int_{Q_i^\chi} \delta(\rho_i^\chi \mathbf{v}_i^\mathbf{a}) \cdot \left( \left. \frac{\partial(\rho_i^\chi \mathbf{v}_i^\mathbf{a})}{\partial t} \right|_\chi + \nabla_\chi \cdot (\rho_i^\chi \mathbf{v}_i^\mathbf{a} \otimes \mathbf{v}_i^\mathbf{c}) - \rho_i^\chi \mathbf{b}_i \right) dQ^\chi \quad [2] $$

$$ + \sum_{i=1}^{2} \int_{Q_i^\chi} \delta(\nabla_\chi \cdot \rho_i^\chi \mathbf{v}_i^\mathbf{a}) : {}^x\mathbf{P}^\chi dQ^\chi, \quad [3] $$

on the outer domain boundaries the Dirichlet boundary conditions are defined

$$ + \sum_{i=1}^{2} \int_{P_{D,i}^{\chi}} \delta \mathbf{t}_{i}^{\chi} \cdot (\rho^{\chi} \mathbf{v}_{i}^{\mathbf{a}} - \bar{\rho}^{\chi} \bar{\mathbf{v}}_{i}^{\mathbf{a}}) \, dP^{\chi} - \sum_{i=1}^{2} \int_{P_{D,i}^{\chi}} \delta(\rho_{i}^{\chi} \mathbf{v}_{i}^{\mathbf{a}}) \cdot \mathbf{t}_{i}^{\chi} \, dP^{\chi}, \quad [4] $$

as well as Neumann boundary conditions

$$ - \sum_{i=1}^{2} \int_{P_{N,i}^{\chi}} \delta(\rho_i^\chi \mathbf{v}_i^\mathbf{a}) \cdot \bar{\mathbf{t}}_i^\chi \, dP^\chi. \quad [5] $$

At the common moving coupling boundary Dirichlet-type interface conditions

$$ + \int_{R^\chi} \delta \mathbf{t}_1^\chi \cdot (\rho^\chi \mathbf{v}_1^\mathbf{a} - \rho^\chi \mathbf{v}_2^\mathbf{a}) \, dR^\chi, \quad [6] $$
have to be fulfilled and Neumann interface conditions

$$
- \int_{R^\chi} \delta(\rho_1^\chi \mathbf{v}_1^a) \cdot \mathbf{t}_1^\chi \, dR^\chi + \int_{R^\chi} \delta(\rho_2^\chi \mathbf{v}_2^a) \cdot (\mathbf{t}_1^\chi - \mathbf{N}^T \check{\mathbf{t}}) \, dR^\chi \quad [7]
$$

The initial conditions are imposed in integral form

$$
\begin{align}
&+ \sum_{i=1}^{2} \int_{\Omega^{\chi}} \delta(\rho_{i}^{\chi} \mathbf{v}_{i}^{\mathrm{a}}(t_{a})) \cdot (\rho_{i}^{\chi} \mathbf{v}_{i}^{\mathrm{a}}(t_{a}) - \rho_{i}^{\chi} \mathbf{v}_{i,a}^{\mathrm{a}}) \, d\Omega^{\chi} \tag{[8]} \\
&= 0 \quad \forall \delta\rho_{i}^{\chi}, \delta\mathbf{v}_{i}^{\mathrm{a}}, \delta\mathbf{t}_{i}^{\chi} \nonumber
\end{align}
$$

where $P_D^\chi$ is the outer Dirichlet boundary, $P_N^\chi$ is the outer Neumann boundary of the
domain and $R^\chi$ is the deformable interface in the reference configuration $\mathcal{K}^\chi$ of the
arbitrary Euler-Lagrange description.

Eq. 1 shows the balance of mass after Petrov-Galerkin weighting, that is performed
consistent to physical units. Eqs. 2 and 3 show the weak form of the momentum bal-
lance equation after Bubnov-Galerkin weighting, where $\mathbf{v}_i^a$ is the material velocity and
$\mathbf{v}_i^\zeta$ is the material velocity in reference form, and partial integration of the stress term
with the 1st Piola-Kirchhoff stress tensor$^x\mathbf{P}^\chi$. The boundary condition for mass flux
$\bar{\rho}^\chi\bar{\mathbf{v}}^a$ on the Dirichlet boundary is introduced in integral form with Eq. 4, using bound-
dary tractions as additional variables. The second term in this equation results from
partial integration of the momentum balance equation. A specified traction boundary
condition $\bar{\mathbf{t}}_i^\chi$ on the Neumann boundary takes Eq. 5 into account. The mass conserva-
tion aspect at the common interface $R$ of the continua in $Q_1$ and $Q_2$ represents Eq.
6, while Eq. 7 contains the dynamic interface conditions with $\check{\mathbf{t}}$ as source term of in-
terfacial tractions, e.g. surface tension in two-fluid flow. The local basis $\mathbf{N}$ consists
of the normal and tangential vectors at the interface $R$. In Eq. 8 the initial conditions
are introduced to the variational formulation. This formulation is the basis for further
derivation and simplifictaion to specialized coupled continua, such as two-fluid flow
or fluid-solid systems.

3. Evolving and Topologically Changing Interfaces

For computations of multi-field physics with moving boundaries, all interfaces
need to be described in space and time. The level set method (Osher et al., 1988) is
applied to distinguish the two fluid domains $Q_1$ and $Q_2$ using an implicit description
of the interface position in space and time. The method introduces a smooth scalar
function $\phi(\mathbf{x}, t)$, which specifies the signed smallest Euclidean distance between $\mathbf{x}$
and the time dependent interface $R$.
<figure><img src="image_1.png" /><figcaption>**Figure 1.** *Discretization of the space-time domain.*</figcaption></figure>

The motion of the interface $R$ in presence of the fluid velocity $\mathbf{v}$ is described by the solution of a convection equation for $\phi$

$$ \int_Q \delta\phi \left( \frac{\partial\phi}{\partial t} + \mathbf{v} \cdot \nabla\phi \right) dQ = 0 \quad \forall \delta\phi. \qquad [9] $$

Note that the solution of equation (9) will not necessarily remain the distance property of $\phi$. The definition of the level set function $\phi$ allows the identification of each phase at any time.

### 3.1. Enriched Space-Time Finite Element Method

The concept of the space-time formulation (Hughes *et al.*, 1989, Hughes *et al.*, 1988, Tezduyar *et al.*, 1992) is the integration of the model equations simultaneously in space and time in a single step. The domain $Q_n$ is discretized by a set of nodes $\mathcal{N}$ and a triangulation $\mathcal{T}$ is formed for quadrature purposes, leading to space-time finite elements.

The interpolation of physical variables within each domain, e.g. the velocity $\mathbf{v}$, is chosen to be discontinuous in time. Independent degrees of freedom are used at discrete time levels $t_n$ of adjacent time slabs, leading to a time-discontinuous Galerkin formulation, which is of higher order accurate, implicit, and A-stable for integration in time (Johnson, 1993). In case of linear interpolation functions used for first-order ODE's the accuracy is of third order in time. Further, a Galerkin-/Least-squares stabilization is applied for suppression of numerical oscillations in solutions to hyperbolic differential equations (Hughes *et al.*, 1989, Hughes *et al.*, 1988, Tezduyar *et al.*, 1992).
3.2. Local partition of unity FEM

The fundamental idea of XFEM (Belytschko et al., 2001) based on the local par-
tition of unity method (Babuška et al., 1997, Melenk et al., 1996) is to extend the
approximation space V<sub>std</sub> defined by standard ansatz functions N<sub>i</sub> on the support of
discrete nodes i ∈ N, by problem dependent constructed enrichment functions ψ.
These enriching functions incorporate an a priori knowledge about the solution be-
havior of the considered problem. Instead of performing an enrichment of the entire
computational domain, only a sub-domain with nodes j ∈ M that needs special at-
tention is usually enhanced by a local partition of unity

$$
\mathcal{V}_{\text{ext}} = \mathcal{V}_{\text{std}} \otimes \bigcup_{j \in \mathcal{M}} N_j(\mathbf{x}, t)\psi. \quad [10]
$$

This enrichment conception is used to construct the extended approximation for space-
time finite elements

$$
w_h(\mathbf{x}, t) = \sum_{i \in \mathcal{N}} N_i(\mathbf{x}, t) w_i + \sum_{j \in \mathcal{M}} N_j(\mathbf{x}, t) \psi_j(\mathbf{x}, t) w_j^* \quad [11]
$$

with $\psi_j$ the space-time node specific enrichment functions $\psi_j(\mathbf{x}, t) = \phi(\mathbf{x}, t) - \phi(\mathbf{x}_j, t_j)$ to assure the nodal interpolating property of the extended approximation (Chessa et al., 2003). For the treatment of strong and also weak discontinuous solutions in space and time the enrichment function $\psi(\mathbf{x}, t) = \text{sign} \phi(\mathbf{x}, t)$ is constructed with help of the level set value $\phi$ used for an implicit interface description.

The number of additional degrees of freedom $w_j^*$ depends on the number of nodes $|\mathcal{M}|$ involved in the enrichment and varies from time slab to time slab since the fluid-fluid interface is moving. For two-fluid flows an extended basis (11) is used for all velocity components and the pressure field.

**3.3. Enforcing coupling conditions**

In the variational form, see Eqs. 1 to 8, additional interface tractions t on R are in-
troduced for enforcing the inter-continua coupling conditions, e.g. continuity of velo-
city. Since the interface is moving and changing topologically the discrete description
of these tractions in an explicit manner is not easy to realize. It is promising to apply
an implicit technique for that as it is done for the interface itself with help of the level
set formulation.

The interface traction t(x, t) is expressed by a higher-dimensional function ~t(x, t)
and the iso-surface φ(x, t) = 0 of the level set,

$$
\mathbf{t}(\mathbf{x}, t) = \tilde{\mathbf{t}}(\mathbf{x}, t) \delta_D(\phi(\mathbf{x}, t)), \quad [12]
$$

with $\delta_D$ the Dirac delta function. The embedded Lagrangian multiplier $\tilde{t}$ is defined in the vicinity $Q_e$ of the interface $R$, usually only at nodes whose support is cut by the interface. This approach results in the implicit formulation of the interfacial tractions.
**4. Bibliographie**

Babuška I., Melenk J. M., « The Partition of Unity Method », *Int. J. Numer. Methods Eng.*, vol. 40, p. 727-758, 1997.

Belytschko T., Moës T., Usui S., Parimi C., « Arbitrary Discontinuities in Finite Elements », *Int. J. Numer. Methods Eng.*, vol. 50(4), p. 993-1013, 2001.

Chessa J., Belytschko T., « The Extended Finite Element Method for Two-Phase Fluids », *ASME J. Appl. Mech.*, vol. 70(1), n° 1, p. 10-17, 2003.

Hübner B., Walhorn E., Dinkler D., « A Monolithic Approach to Fluid-Structure Interaction using Space-Time Finite Elements », *Comput. Methods Appl. Mech. Eng.*, vol. 193, n° 23-26, p. 2069-2086, 2004.

Hughes T. J. R., Franca L. P., Hulbert G. M., « A New Finite Element Formulation for Computational Fluid Dynamics : VIII. The Galerkin/Least-Squares Method for Advective-Diffusive Equations », *Comput. Methods Appl. Mech. Eng.*, vol. 73, p. 173-189, 1989.

Hughes T. J. R., Hulbert G. M., « Space-Time Finite Element Methods for Elastodynamics : Formulation and Error Estimates », *Comput. Methods Appl. Mech. Eng.*, vol. 66, p. 339-363, 1988.

Johnson G., « Discontinuous Galerkin Finite Element Methods for Second Order Hyperbolic Problems », *Comput. Methods Appl. Mech. Eng.*, vol. 107, p. 117-129, 1993.

Kölke A., Dinkler D., « Extended Space-Time Finite Elements for Two-Fluid Flows in Fluid-Structure Interaction », in, M. Papadrakis, E. Onate, B. Schrefler (eds), *Proceedings of Sixth World Congress on Computational Mechanics Beijing, WCCM*, 2004.

Melenk J. M., Babuška I., « The Partition of Unity Finite Element Method : Basic Theory and Applications », *Comput. Methods Appl. Mech. Eng.*, vol. 39, p. 289-314, 1996.

Osher S., Sethian J. A., « Fronts Propagating with Curvature-Dependent Speed : Algorithms Based on Hamilton-Jacobi Formulations », *J. Comput. Phys.*, vol. 79, p. 12-49, 1988.

Tezduyar T. E., Behr M., Liou J., « A New Strategy for Finite Element Computations Involving Moving Boundaries and Interfaces - The Deforming-Spatial-Domain/Space-Time Procedure : I. The Concept and the Preliminary Numerical Tests », *Comput. Methods Appl. Mech. Eng.*, vol. 94, p. 339-351, 1992.