1,409 research outputs found
A Generalized Fast Marching Method for dislocation dynamics
International audienceIn this paper, we consider a Generalized Fast Marching Method (GFMM) as a numerical method to compute dislocation dynamics. The dynamics of a dislocation hyper-surface in (with for physical applications) is given by its normal velocity which is a non-local function of the whole shape of the hyper-surface itself. For this dynamics, we show a convergence result of the GFMM as the mesh size goes to zero. We also provide some numerical simulations in dimension
Comparison principle for a Generalized Fast Marching Method
International audienceIn \cite{CFFM06}, the authors have proposed a generalization of the classical Fast Marching Method of Sethian for the eikonal equation in the case where the normal velocity depends on space and time and can change sign. The goal of this paper is to propose a modified version of the Generalized Fast Marching Method proposed in \cite{CFFM06} for which we state a general comparison principle. We also prove the convergence of the new algorithm
A Non-Local Mean Curvature Flow and its semi-implicit time-discrete approximation
We address in this paper the study of a geometric evolution, corresponding to
a curvature which is non-local and singular at the origin. The curvature
represents the first variation of the energy recently proposed as a variant of
the standard perimeter penalization for the denoising of nonsmooth curves.
To deal with such degeneracies, we first give an abstract existence and
uniqueness result for viscosity solutions of non-local degenerate Hamiltonians,
satisfying suitable continuity assumption with respect to Kuratowsky
convergence of the level sets. This abstract setting applies to an approximated
flow. Then, by the method of minimizing movements, we also build an "exact"
curvature flow, and we illustrate some examples, comparing the results with the
standard mean curvature flow
Advances in the numerical treatment of grain-boundary migration: Coupling with mass transport and mechanics
This work is based upon a coupled, lattice-based continuum formulation that
was previously applied to problems involving strong coupling between mechanics
and mass transport; e.g. diffusional creep and electromigration. Here we
discuss an enhancement of this formulation to account for migrating grain
boundaries. The level set method is used to model grain-boundary migration in
an Eulerian framework where a grain boundary is represented as the zero level
set of an evolving higher-dimensional function. This approach can easily be
generalized to model other problems involving migrating interfaces; e.g. void
evolution and free-surface morphology evolution. The level-set equation is
recast in a remarkably simple form which obviates the need for spatial
stabilization techniques. This simplified level-set formulation makes use of
velocity extension and field re-initialization techniques. In addition, a
least-squares smoothing technique is used to compute the local curvature of a
grain boundary directly from the level-set field without resorting to
higher-order interpolation. A notable feature is that the coupling between mass
transport, mechanics and grain-boundary migration is fully accounted for. The
complexities associated with this coupling are highlighted and the
operator-split algorithm used to solve the coupled equations is described.Comment: 28 pages, 9 figures, LaTeX; Accepted for publication in Computer
Methods in Applied Mechanics and Engineering. [Style and formatting
modifications made, references added.
A Posteriori Error Estimate and Adaptivity for QM/MM Models of Crystalline Defects
Hybrid quantum/molecular mechanics models (QM/MM methods) are widely used in
material and molecular simulations when pure MM models cannot ensure adequate
accuracy but pure QM models are computationally prohibitive. Adaptive QM/MM
coupling methods feature on-the-fly classification of atoms, allowing the QM
and MM subsystems to be updated as needed. The state-of-art "machine-learned
interatomic potentials (MLIPs)" can be applied as the MM models for consistent
QM/MM methods with rigorously justified accuracy. In this work, we propose a
robust adaptive QM/MM method for practical material defect simulation, which is
based on a developed residual-based error estimator. The error estimator
provides both upper and lower bounds for the approximation error, demonstrating
its reliability and efficiency. In particular, we introduce three minor
approximations such that the error estimator can be evaluated efficiently
without losing much accuracy. To update the QM/MM partitions anisotropically, a
novel adaptive algorithm is proposed, where a free interface motion problem
based on the proposed error estimator is solved by employing the fast marching
method. We implement and validate the robustness of the adaptive algorithm on
numerical simulations for various complex crystalline defects
A LEVEL SET APPROACH REFLECTING SHEET STRUCTURE WITH SINGLE AUXILIARY FUNCTION FOR EVOLVING SPIRALS ON CRYSTAL SURFACES
We introduce a new level set method to simulate motion of spirals in a crystal surface governed by an eikonal-curvature ow equation. Our formulation allows collision of several spirals and different strength (different modulus of Burgers vectors) of screw dislocation centers. We represent a set of spirals by a level set of a single auxiliary function u minus a pre-determined multi-valued sheet structure function , which re ects the strength of spirals (screw dislocation centers). The level set equation used in our method for u is the same as that of the eikonal-curvature ow equation. The multi-valued nature of the sheet structure function is only invoked when preparing the initial auxiliary function, which is nontrivial, and in the nal step when extracting information such as the height of the spiral steps. Our simulation enables us not only to reproduce all speculations on spirals in a classical paper by Burton, Cabrera and Frank (1951) but also to nd several new phenomena
- …