13 research outputs found
Mesh adaptivity for quasi-static phase-field fractures based on a residual-type a posteriori error estimator
In this work, we consider adaptive mesh refinement for a monolithic
phase-field description for fractures in brittle materials. Our approach is
based on an a posteriori error estimator for the phase-field variational
inequality realizing the fracture irreversibility constraint. The key goal is
the development of a reliable and efficient residual-type error estimator for
the phase-field fracture model in each time-step. Based on this error
estimator, error indicators for local mesh adaptivity are extracted. The
proposed estimator is based on a technique known for singularly perturbed
equations in combination with estimators for variational inequalities. These
theoretical developments are used to formulate an adaptive mesh refinement
algorithm. For the numerical solution, the fracture irreversibility is imposed
using a Lagrange multiplier. The resulting saddle-point system has three
unknowns: displacements, phase-field, and a Lagrange multiplier for the crack
irreversibility. Several numerical experiments demonstrate our theoretical
findings with the newly developed estimators and the corresponding refinement
strategy.Comment: This is the preprint version of an accepted article to be published
in the GAMM-Mitteilungen 2019.
https://onlinelibrary.wiley.com/journal/1522260
Some symmetric boundary value problems and non-symmetric solutions
We consider the equation âDelta u = wf âČ(u) on a symmetric bounded domain in Rn with Dirichlet boundary conditions. Here w is a positive function or measure that is invariant under the (Euclidean) symmetries of the domain. We focus on solutions u that are positive and/or have a low Morse index. Our results are concerned with the existence of non-symmetric solutions and the non-existence of symmetric solutions. In particular, we construct a solution u for the disk in R2 that has index 2 and whose modulus |u| has only one reflection symmetry. We also provide a corrected proof of [12, Theorem 1]
-convergence for high order phase field fracture: continuum and isogeometric formulations
We consider second order phase field functionals, in the continuum setting,
and their discretization with isogeometric tensor product B-splines. We prove
that these functionals, continuum and discrete, -converge to a brittle
fracture energy, defined in the space . In particular, in the
isogeometric setting, since the projection operator is not Lagrangian (i.e.,
interpolatory) a special construction is needed in order to guarantee that
recovery sequences take values in ; convergence holds, as expected, if
, being the size of the physical mesh and
the internal length in the phase field energy
A combined finite elementâfinite volume framework for phase-field fracture
Numerical simulations of brittle fracture using phase-field approaches often employ a discrete approximation framework that applies the same order of interpolation for the displacement and phase-field variables. In particular, the use of linear finite elements to discretize both stress equilibrium and phase-field equations is widespread in the literature. However, the use of Lagrange shape functions to model the phase-field is not optimal, as the latter contains cusps for fully developed cracks. These should in turn occur at locations corresponding to Gauss points of the associated FE model for the mechanics. Such a feature is challenging to reproduce accurately with low order elements, and element sizes must consequently be made very small relative to the phase-field regularization parameter in order to achieve convergence of results with respect to the mesh. In this paper, we combine a standard linear FE discretization of stress equilibrium with a cell-centered finite volume approximation of the phase-field evolution equation based on the two-point flux approximation constructed over the same simplex mesh. Compared to a pure FE formulation utilizing linear elements, the proposed framework results in looser restrictions on mesh refinement with respect to the phase-field length scale. This ability to employ coarser meshes relative to the traditional implementation allows for significant reductions on computational cost, as demonstrated in several numerical examples.publishedVersio
The Benefits of Anisotropic Mesh Adaptation for Brittle Fractures Under Plane-Strain Conditions
AbstractWe develop a reliable a posteriori anisotropic first order estimator for the numerical simulation of the Francfort and Marigo model of brittle fracture, after its approximation by means of the Ambrosio-Tortorelli variational model. We show that an adaptive algorithm based on this estimator reproduces all the previously ob-tained well-known benchmarks on fracture development with particular attention to the fracture directionality. Additionally, we explain why our method, based on an extremely careful tuning of the anisotropic adaptation, has the potential of out-performing significantly in terms of numerical complexity the ones used to achieve similar degrees of accuracy in previous studies.
Stochastic phase-field modeling of brittle fracture: computing multiple crack patterns and their probabilities
In variational phase-field modeling of brittle fracture, the functional to be
minimized is not convex, so that the necessary stationarity conditions of the
functional may admit multiple solutions. The solution obtained in an actual
computation is typically one out of several local minimizers. Evidence of
multiple solutions induced by small perturbations of numerical or physical
parameters was occasionally recorded but not explicitly investigated in the
literature. In this work, we focus on this issue and advocate a paradigm shift,
away from the search for one particular solution towards the simultaneous
description of all possible solutions (local minimizers), along with the
probabilities of their occurrence. Inspired by recent approaches advocating
measure-valued solutions (Young measures as well as their generalization to
statistical solutions) and their numerical approximations in fluid mechanics,
we propose the stochastic relaxation of the variational brittle fracture
problem through random perturbations of the functional. We introduce the
concept of stochastic solution, with the main advantage that point-to-point
correlations of the crack phase fields in the underlying domain can be
captured. These stochastic solutions are represented by random fields or random
variables with values in the classical deterministic solution spaces. In the
numerical experiments, we use a simple Monte Carlo approach to compute
approximations to such stochastic solutions. The final result of the
computation is not a single crack pattern, but rather several possible crack
patterns and their probabilities. The stochastic solution framework using
evolving random fields allows additionally the interesting possibility of
conditioning the probabilities of further crack paths on intermediate crack
patterns
A Dimension-Reduction Model for Brittle Fractures on Thin Shells with Mesh Adaptivity
In this paper we derive a new two-dimensional brittle fracture model for thin
shells via dimension reduction, where the admissible displacements are only
normal to the shell surface. The main steps include to endow the shell with a
small thickness, to express the three-dimensional energy in terms of the
variational model of brittle fracture in linear elasticity, and to study the
-limit of the functional as the thickness tends to zero. The numerical
discretization is tackled by first approximating the fracture through a phase
field, following an Ambrosio-Tortorelli like approach, and then resorting to an
alternating minimization procedure, where the irreversibility of the crack
propagation is rigorously imposed via an inequality constraint. The
minimization is enriched with an anisotropic mesh adaptation driven by an a
posteriori error estimator, which allows us to sharply track the whole crack
path by optimizing the shape, the size, and the orientation of the mesh
elements. Finally, the overall algorithm is successfully assessed on two
Riemannian settings and proves not to bias the crack propagation