A robust Nitsche’s formulation for interface problems

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Abstract

In this work, we propose a novel weighting for the interfacial consistency terms arising in a Nitsche variational form. We demonstrate through numerical analysis and extensive numerical evidence that the choice of the weighting parameter has a great bearing on the stability of the method. Consequently, we propose a weighting that results in an estimate for the stabilization parameter such that the method remains well behaved in varied settings; ranging from the configuration of embedded interfaces resulting in arbitrarily small elements to such cases where a large contrast in material properties exists. An important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of (a) elements with arbitrarily small volume fractions, (b) large material heterogeneities and (c) both large heterogeneities as well as arbitrarily small elements. We then highlight the accuracy and efficiency of the proposed formulation through numerical examples, focusing particular attention on interfacial quantities of interest.

Highlights

► Robust Nitsche’s method for “jump”-type constraints developed for embedded finite element methods. ► Established a relationship between weights in the consistency terms and stability. ► Robustness demonstrated through numerical performance with particular emphasis on interfacial quantities.

Introduction

The efficient treatment of interfacial phenomenon has been a challenge facing the computational mechanics community. Be it evolving interface problems such as those encountered in phase transformation/solidification problems, crack propagation problems, fluid–structure interaction problems or frictional contact problems; the key challenge lies in handling the kinematics at the interface efficiently. These problems are often complicated by the presence of discontinuous fields across the interface. Classical Galerkin finite element methods then require the mesh surfaces to align with the said surfaces of discontinuity to preserve optimal convergence behaviour. This adds considerably to the computational expense where the preprocessing step of generating conforming meshes needs to be repeated every time the surface of discontinuity changes orientation.

A discontinuous Galerkin framework, on the other hand, provides a natural way of accommodating these discontinuous fields since it does not assume inter-element continuity a priori as is the case in classical Galerkin approaches. This requirement then, of course, increases the size of the resulting discrete problem. Embedded methods can be seen as a bridge between these two approaches. The key idea in these approaches is to draw from the advantages of continuous Galerkin frameworks away from the discontinuity while following a discontinuous Galerkin approach in the vicinity of the interface. This facilitates a mesh independent representation of the surface of discontinuity without losing optimal convergence rates and thus these approaches have gained in popularity in recent years. However, efficient enforcement of inter-element/interfacial kinematics is critical to preserve the optimal performance of discontinuous Galerkin/embedded methods. As outlined in the seminal review by Arnold et al. [1], depending on the choice of numerical flux, a host of discontinuous Galerkin approaches with varying mathematical properties can be derived.

Similarly, a standard way to account for constraints on embedded surfaces is to build them into the variational statement for the problem through the use of Lagrange mutipliers. However, the stability issues arising in a Lagrange multiplier based approach are well known [2], [3]. This has led researchers to look for alternative ways which circumvent the stability problems arising in a dual Lagrange multiplier based approach while retaining the attractive properties of the method – better constraint enforcement and optimal convergence behaviour.

In this regard, Nitsche’s method [4] has seen a resurgence in recent years. The idea behind a Nitsche based approach is to simply replace the Lagrange multipliers arising in a dual formulation through their physical representation, namely the normal flux at the interface. Nitsche also added an additional penalty like term to restore the coercivity of the bilinear form. In recent years, the flexibility of the approach has resulted in a wide range of applications of the method from symmetric interior penalty formulations in discontinuous Galerkin methods (see Arnold et al. [1] for a detailed review) to embedded meshes (see Laursen et al. [5] and Sanders et al. [6]) and embedded interface methods [7], [11], [12], [13], [14].

Hansbo and Hansbo [7] used Nitsche’s method to model elliptic interface problems with discontinuous coefficients on unfitted meshes. They later extended this method to model both strongly and weakly discontinuous elasticity problems [8]. Sanders et al. [9] also used a Nitsche based extended finite element approach to model problems in elasticity. The presence of a “free” parameter which governed the stability of the method was still a detriment to the method however. Griebel and Schweitzer [10] suggested the solution of a global eigenvalue problem to provide a lower bound on the stabilization parameter.

Mourad et al. [11] and Dolbow and Franca [12] stabilized the unstable Lagrange multiplier space through the use of bubble functions and showed the similarity of the resulting formulation with that of a Nitsche approach. The existence of a lower bound on the stabilization parameter at an element level for these methods suggested that a similar strategy could then be adapted to provide estimates for the stabilization parameter in Nitsche’s method as well. Dolbow and Harari [13] further extended this idea and used numerical analysis to provide estimates for the stabilization parameter. Notably, they showed the existence of closed form algebraic expressions for constant strain triangular and tetrahedral elements while for other finite elements they suggested solving only a local eigenvalue problem resulting in a more computationally efficient method.

While this classical form of Nitsche’s method performed optimally in most situations, there were some anomalies. In Mourad et al. [11], Dolbow and Franca [12] and Dolbow and Harari [13], the authors reported high sensitivity of the normal flux when evaluated directly. In fact, Hautefeuille et al. [14] showed non-convergence of the normal flux in L norm when evaluated directly. In addition, Sanders et al. [9] reported mild oscillations in the normal traction and heavy oscillatory behaviour in the tangential traction at the interface. Laursen et al. [5] and Sanders et al. [6] further showed stress locking pattern when tying a soft material with a stiff one. Further, Annavarapu et al. [15] showed the incompatibility of the classical Nitsche formulation with explicit dynamics simulations. Interestingly, similar issues were also reported in discontinuous Galerkin based approximations, for instance, Lew and Negri [16] artificially prevent the boundary from cutting arbitrarily small slices of elements. Finally, we mention Burman and Hansbo’s [17] work where they add a ghost penalty term to improve the conditioning of the discrete system when applying Dirichlet boundary conditions with Nitsche’s method.

In this work, we aim to propose a unified solution to some of those numerical issues. We concentrate on “jump” type constraints in particular and propose a modified numerical flux based on a weighting other than a simple arithmetic average. The idea of using a weighted form has been tried under different contexts before. Notably, Zunino and co-workers [18], [19], [20] proposed and analyzed a stiffness weighted interior penalty approach to model the case of vanishing diffusivity in advection–diffusion-reaction equations. The work of Cai et al. [21] also analyzes a stiffness weighted interior penalty approach for heterogenous problems and establishes robust error estimators. Also in the context of embedded meshes, Sanders et al. [6] successfully used a stiffness weighted approach to alleviate stress locking problems exhibited by classical Nitsche’s method. More recently, Zunino et al. [22] also analyzed an unfitted Nitsche method and proposed an approach that remains robust for the worst case among small cut elements and large heterogeneities.

However, the novelty of our approach lies in (a) establishing a clear relationship between the weights and the stabilization parameter – thereby allowing us to make a choice for the weights which least affects the stability of the method in the face of both small cut elements and large material heterogeneities and (b) numerically demonstrating the critical dependence of the interfacial quantities of interest on the stabilization parameter. In particular, we demonstrate, through numerical examples, that the standard choice of weights results in a stabilization parameter which, although establishes spatial stability, might not necessarily yield a stable interfacial field.

We begin by defining the model problem and the variational form in the next section. In Section 3, we discuss the spatial discretization as well as provide a lower bound on the stabilization parameter. We also show the relationship between the stability of the method and the weighting parameter in this section. In Section 4, we demonstrate the robustness of the proposed method against the classical Nitsche approach on several benchmark examples. Finally, in Section 5, we provide concluding remarks and outlook for our work.

Section snippets

Model problem and variational formulation

We begin by considering a particular Poisson’s problem in the bulk domains Ω1 and Ω2:·κmum=-fminΩm,um=udmonΓdm,κmum·nm=0onΓnm,that are coupled together at the interface by the conditions:κu·n2=j¯onΓ,u=i¯onΓ,where, i¯ and j¯ are sufficiently smooth functions on the interface. The definitions of Ω1,Ω2,Γdm and Γnm are as shown in Fig. 1. The interface Γ acts as a partitioning boundary between the two bulk domains Ω1 and Ω2. The normals nm are considered as outward pointing from their

Spatial discretization

We discretize the bulk domains Ω1 and Ω2 into a set of non-overlapping simplices. The interface Γ is allowed to be embedded in the domain in the sense that it is allowed to cut through the elements. We construct a piecewise planar approximation to the interface and locate it through a zero iso-surface for a signed distance level set function in the domain. The interface is thus naturally discretized as the vertex set of intersection points between the zero iso-surface and the element edges.

Numerical examples

In this section, we revisit several numerical examples studied previously in [13], [9], [5], [14], [6], [12], where classical Nitsche’s method was shown to perform poorly for certain pathological cases. We highlight the robustness of the proposed formulation over those very examples by contrasting its performance with the more conventional form prevalent in literature. Throughout the section and in the figure legends, we denote the weighted form of Nitsche’s method by γ Nitsche and the standard

Conclusion

In this work, we demonstrated the lack of robustness exhibited by the classical Nitsche formulation for a certain class of embedded interface problems. Indeed, the bulk field remains reasonably well behaved even in these pathological cases and it is therefore essential to look at the behaviour of the interfacial field to comment on the robustness of the method. We applied numerical analysis to highlight the possible cause of this lack of robustness and proposed an alternate variational form

Acknowledgements

The support of the Air Force Office of Scientific Research and Sandia National Laboratories to Duke University is gratefully acknowledged. The authors would also like to gratefully acknowledge Professor Isaac Harari, of Tel Aviv University, for his helpful comments.

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