Interface material failure modeled by the extended finite-element method and level sets

213

223

We focus on developing a computationally efficient finite element method for interface problems. Finite element methods are severely constrained in their ability to resolve interfaces. Many of these limitations stem from their inability in independently representing interface geometry from the underlying discretization. We propose an approach that facilitates such an independent representation by embedding interfaces in the underlying finite element mesh. This embedding, however, raises stability concerns for existing algorithms used to enforce interfacial kinematic constraints. To address these stability concerns, we develop robust methods to enforce interfacial kinematics over embedded interafces.We begin by examining embedded Dirichlet problems -a simpler class of embedded constraints. We develop both stable methods, based on Lagrange multipliers, and stabilized methods, based on Nitsche's approach, for enforcing Dirichlet constraints over three-dimensional embedded surfaces and compare and contrast their performance. We then extend these methods to enforce perfectly-tied kinematics for elastodynamics with explicit time integration. In particular, we examine the coupled aspects of spatial and temporal stability for Nitsche's approach. We address the incompatibility of Nitsche's method for explicit time integration by (a) proposing a modified weighted stress variational form, and (b) proposing a novel mass-lumping procedure.We revisit Nitsche's method and inspect the effect of this modified variational iv form on the interfacial quantities of interest. We establish that the performance of this method, with respect to recovery of interfacial quantities, is governed significantly by the choice for the various method parameters viz. stabilization and weighting. We establish a relationship between these parameters and propose an optimal choice for the weighting. We further extend this approach to handle non-linear, frictional sliding constraints at the interface. The naturally non-symmetric nature of these problems motivates us to omit the symmetry term arising in Nitsche's method.We contrast the performance of the proposed approach with the more commonly used penalty method. Through several numerical examples, we show that with the proposed choice of weighting and stabilization parameters, Nitsche's method achieves the right balance between accurate constraint enforcement and flux recovery -a balance hard to achieve with existing methods. Finally, we extend the proposed approach to intersecting interfaces and conduct numerical studies on problems with junctions and complex topologies.v To Amma (mom) and Nannagaru (dad), vi

Section: Introductionmentioning

confidence: 99%

An efficient finite element method for embedded interface problems

Dolbow

Harari

2008

213

223

“…Now, if changes slightly from * , e.g. = * + with >0, then the source term becomes negative due to the second condition in Equation (12), so that it tends to bring back to * , and the same occurs if <0. This condition is analogous for the roots * = ± ref of Equation (10).…”

Section: Reinitialization Of the Ls Functionmentioning

confidence: 90%

Bounded renormalization with continuous penalization for level set interface‐capturing methods

Battaglia

Storti

D’Elı́a

2010

SUMMARYIn this work, a reinitialization procedure oriented to regularize the level set (LS) function field is presented. In LS approximations for two-fluid flow simulations, a scalar function indicates the presence of one or another phase and the interface between them. In general, the advection of such function produces a degradation of some properties of the LS function, such as the smoothness of the transition between phases and the correct position of the interface. The methodology introduced here, denominated bounded renormalization with continuous penalization, consists of solving by the finite element method a partial differential equation with certain distinguishing properties with the aim of keeping the desirable properties of the LS function. The performance of the strategy is evaluated for several typical cases in one, two and three-dimensional domains, for both the advection and the renormalization stages.

“…The Heaviside function is typically a discontinuous function used to describe the crack surface discontinuity. The methods for determining H (x) usually include: (1) scalar product method noted by Moës et al [10], (2) triangular orientation method noted by Sukumar and Prévost [13], and (3) level set method noted by Belytschko et al [26], Stolarska et al [27], Sukumar et al [28], Nagashima et al [29], and Hettich et al [19,20]. Generally, these methods are mainly focused on the smooth crack geometry, and not well suited for non-smooth crack geometry such as crack kinking.…”

Section: Determination Of the Discontinuous Enrichment Functionsmentioning

confidence: 99%

An enriched element‐failure method (REFM) for delamination analysis of composite structures

Sun

Tan

Liu

et al. 2009

SUMMARYThis paper develops an enriched element-failure method for delamination analysis of composite structures. This method combines discontinuous enrichments in the extended finite element method and element-failure concepts in the element-failure method within the finite element framework. An improved discontinuous enrichment function is presented to effectively model the kinked discontinuities; and, based on fracture mechanics, a general near-tip enrichment function is also derived from the asymptotic displacement fields to represent the discontinuity and local stress intensification around the crack-tip. The delamination is treated as a crack problem that is represented by the discontinuous enrichment functions and then the enrichments are transformed to external nodal forces applied to nodes around the crack. The crack and its propagation are modeled by the 'failed elements' that are applied to the external nodal forces. Delamination and crack kinking problems can be solved simultaneously without remeshing the model or re-assembling the stiffness matrix with this method. Examples are used to demonstrate the application of the proposed method to delamination analysis. The validity of the proposed method is verified and the simulation results show that both interlaminar delamination and crack kinking (intralaminar crack) occur in the cross-ply laminated plate, which is observed in the experiment.