arXiv subjects

Selim Esedoglu

Publications and source records attributed to Selim Esedoglu.

17 recordsLinked to original sources

arXiv2026

A Phase-Field Method for Curvature Flow of Networks with Triple Junction Drag

We develop a new phase-field (diffuse interface) approximation of multiphase curvature motion with triple junction drag - an important sharp interface model for the evolution of microstructure in polycrystalline materials during heat treatment. This sharp interface model arises as gradient flow for the total length of the interfacial network with respect to a certain metric. Accordingly, we derive our diffuse interface approximation - a coupled system of partial differential equations that is a variant of the Allen-Cahn system - from a variational perspective, in the style of minimizing movements: starting from a discrete in time approximation that entails a convex optimization problem to advance from one time step to the next. In the process, we propose a simple integral expression that counts the number of junctions using the order parameter that appears to be new even for the standard multiphase Allen-Cahn system. The convergence of the resulting flow to the desired sharp interface limit is then verified via the method of matched asymptotic expansions. Numerical convergence studies against both known exact solutions as well as highly accurate benchmark solutions obtained via front tracking provide clear further evidence for this convergence. Moreover, the method retains the most desirable feature of diffuse interface methods: Automatic handling of topological changes in the network of interfaces.

math.AP↗

arXiv2025

Curvature Flow of Networks with Triple Junction Drag

We consider a PDE system that describes curvature motion of networks with a dynamic boundary condition known as triple junction drag. This model arises in the study of grain boundary evolution in polycrystalline materials. In this system, the surface tension coefficients depend on the crystallographic orientations of the grains, which are allowed to rotate. We prove existence and uniqueness of solutions to this system in the parabolic H\"{o}lder class $C^{2+\alpha,1+\alpha/2}$. Moreover, we extend our existence result to accommodate a wider class of initial networks by relaxing the compatibility conditions on the angles and curvatures at the triple junction. As an important by-product of our result, we demonstrate the possibility of a new type of topological change during the evolution of the network. We also revisit the question of stability of stationary networks and how it is affected by the choice of surface tensions.

math.AP↗

arXiv2024

Median Filters for Anisotropic Wetting / Dewetting Problems

We present new level set methods for multiphase, anisotropic (weighted) motion by mean curvature of networks, focusing on wetting-dewetting problems where one out of three phases is stationary -- a good testbed for checking whether complicated junction conditions are correctly enforced. The new schemes are vectorial median filters: The level set values at the next time step are determined by a sorting procedure performed on the most recent level set values. Detailed numerical convergence studies are presented, showing that the correct angle conditions at triple junctions (which include torque terms due to anisotropy) are indeed indirectly and automatically attained. Other standard benefits of level set methods, such as subgrid accuracy on uniform grids via interpolation and seamless treatment of topological changes, remain intact.

math.NA↗

arXiv2022

On Median Filters for Motion by Mean Curvature

The median filter scheme is an elegant, monotone discretization of the level set formulation of motion by mean curvature. It turns out to evolve every level set of the initial condition precisely by another class of methods known as threshold dynamics. Median filters are, in other words, the natural level set versions of threshold dynamics algorithms. Exploiting this connection, we revisit median filters in light of recent progress on the threshold dynamics method. In particular, we give a variational formulation of, and exhibit a Lyapunov function for, median filters, resulting in energy based unconditional stability properties. The connection also yields analogues of median filters in the multiphase setting of mean curvature flow of networks. These new multiphase level set methods do not require frequent redistancing, and can accommodate a wide range of surface tensions.

math.NA↗

arXiv2022

High Order Schemes for Gradient Flow with Respect to a Metric

New criteria for energy stability of multi-step, multi-stage, and mixed schemes are introduced in the context of evolution equations that arise as gradient flow with respect to a metric. These criteria are used to exhibit second and third order consistent, energy stable schemes, which are then demonstrated on several partial differential equations that arise as gradient flow with respect to the 2-Wasserstein metric.

math.NA↗

arXiv2021

A Monotone, Second Order Accurate Scheme for Curvature Motion

We present a second order accurate in time numerical scheme for curve shortening flow in the plane that is unconditionally monotone. It is a variant of threshold dynamics, a class of algorithms in the spirit of the level set method that represent interfaces implicitly. The novelty is monotonicity: it is possible to preserve the comparison principle of the exact evolution while achieving second order in time consistency. As a consequence of monotonicity, convergence to the viscosity solution of curve shortening is ensured by existing theory.

math.NA↗

arXiv2020

High order, semi-implicit, energy stable schemes for gradient flows

We introduce a class of high order accurate, semi-implicit Runge-Kutta schemes in the general setting of evolution equations that arise as gradient flow for a cost function, possibly with respect to an inner product that depends on the solution, and we establish their energy stability. This class includes as a special case high order, unconditionally stable schemes obtained via convexity splitting. The new schemes are demonstrated on a variety of gradient flows, including partial differential equations that are gradient flow with respect to the Wasserstein (mass transport) distance.

math.NA↗

arXiv2019

Second Order Threshold Dynamics Schemes for Two Phase Motion by Mean Curvature

The threshold dynamics algorithm of Merriman, Bence, and Osher is only first order accurate in the two-phase setting. Its accuracy degrades further to half order in the multi-phase setting, a shortcoming it has in common with other related, more recent algorithms such as the equal surface tension version of the Voronoi implicit interface method. As a first, rigorous step in addressing this shortcoming, we present two different second order accurate versions of two-phase threshold dynamics. Unlike in previous efforts in this direction, we present careful consistency calculations for both of our algorithms. The first algorithm is consistent with its limit (motion by mean curvature) up to second order in any space dimension. The second achieves second order accuracy only in dimension two, but comes with a rigorous stability guarantee (unconditional energy stability) in any dimension -- a first for high order schemes of its type.

math.NA↗

arXiv2019

Variational Extrapolation of Implicit Schemes for General Gradient Flows

We introduce a class of unconditionally energy stable, high order accurate schemes for gradient flows in a very general setting. The new schemes are a high order analogue of the minimizing movements approach for generating a time discrete approximation to a gradient flow by solving a sequence of optimization problems. In particular, each step entails minimizing the associated energy of the gradient flow plus a movement limiter term that is, in the classical context of steepest descent with respect to an inner product, simply quadratic. A variety of existing unconditionally stable numerical methods can be recognized as (typically just first order accurate in time) minimizing movement schemes for their associated evolution equations, already requiring the optimization of the energy plus a quadratic term at every time step. Therefore, our approach gives a painless way to extend these to high order accurate in time schemes while maintaining their unconditional stability. In this sense, it can be viewed as a variational analogue of Richardson extrapolation.

math.NA↗

arXiv2019

The Role of Surface Tension and Mobility Model in Simulations of Grain Growth

We explore the effects of surface tension and mobility models in simulations of grain growth using threshold dynamics algorithms that allow performing large scale simulations, while naturally capturing the Herring angle condition at junctions and automatically handling topological transitions. The results indicate that in two dimensions, the different surface tension / mobility models considered do not play a significant role in the stationary grain size distribution. However, in three dimensions, there is a substantial difference between the distributions obtained from the same three models, depending on whether the reduced mobilities are isotropic or anisotropic. Additional results show that in three dimensions, the misorientation distribution function of a grain network with random orientation texture returns to the close vicinity of the Mackenzie distribution even if started very far from it.

cond-mat.mtrl-sci↗

arXiv2018

Statistics of grain growth: experiment versus the Phase-Field-Crystal and Mullins models

We present a detailed comparison of multiple statistical grain metrics for previously reported experimental thin film samples of aluminum with 2D simulations obtained from the Phase-Field-Crystal (PFC) model and a Mullins grain boundary motion model. For all these results, ``universality'' is observed with respect to the dynamics and initial conditions. This comparison reveals that PFC reproduces geometric metrics such as area and perimeter, but does not capture grain shape and topology as accurately. Similarly, the Mullins model captures the number of sides distribution quite well but not other metrics. Our collective comparison of such measurements underscores the critical importance of the use of multiple metrics for comparison of experiments with all present and future models of grain growth in polycrystalline materials.

cond-mat.mtrl-sci↗

arXiv2018

On the Voronoi Implicit Interface Method

We present careful numerical convergence studies, using parameterized curves to reach very high resolutions in two dimensions, of a level set method for multiphase curvature motion known as the Voronoi implicit interface method. Our tests demonstrate that in the unequal, additive surface tension case, the Voronoi implicit interface method does not converge to the desired limit. We then present a variant that maintains the spirit of the original algorithm, and appears to fix the non-convergence. As a bonus, the new variant extends the Voronoi implicit interface method to unequal mobilities.

math.NA↗

arXiv2018

A simplified threshold dynamics algorithm for isotropic surface energies

We present a simplified version of the threshold dynamics algorithm given in the work of Esedoglu and Otto (2015). The new version still allows specifying N-choose-2 possibly distinct surface tensions and N-choose-2 possibly distinct mobilities for a network with N phases, but achieves this level of generality without the use of retardation functions. Instead, it employs linear combinations of Gaussians in the convolution step of the algorithm. Convolutions with only two distinct Gaussians is enough for the entire network, maintaining the efficiency of the original thresholding scheme. We discuss stability and convergence of the new algorithm, including some counterexamples in which convergence fails. The apparently convergent cases include unequal surface tensions given by the Read \& Shockley model and its three dimensional extensions, along with equal mobilities, that are a very common choice in computational materials science.

math.NA↗

arXiv2016

Crystallization for a Brenner-like potential

Graphene is a carbon molecule with the structure of a honeycomb lattice. We show how this structure can arise in two dimensions as the minimizer of an interaction energy with two-body and three-body terms. In the engineering literature, the Brenner potential is commonly used to describe the interactions between carbon atoms. We consider a potential of Stillinger-Weber type that incorporates certain characteristics of the Brenner potential: the preferred bond angles are 180 degrees and all interactions have finite range. We show that the thermodynamic limit of the ground state energy per particle is the same as that of a honeycomb lattice. We also prove that, subject to periodic boundary conditions, the minimizers are translated versions of the honeycomb lattice.

cond-mat.mtrl-sci↗

arXiv2013

A Hamilton-Jacobi equation for the continuum limit of non-dominated sorting

We show that non-dominated sorting of a sequence of i.i.d. random variables in Euclidean space has a continuum limit that corresponds to solving a Hamilton-Jacobi equation involving the probability density function of the random variables. Non-dominated sorting is a fundamental problem in multi-objective optimization, and is equivalent to finding the canonical antichain partition and to problems involving the longest chain among Euclidean points. As an application of this result, we show that non-dominated sorting is asymptotically stable under random perturbations in the data. We give a numerical scheme for computing the viscosity solution of this Hamilton-Jacobi equation and present some numerical simulations for various density functions.

math.AP↗

arXiv2013

A PDE-based approach to non-dominated sorting

Non-dominated sorting is a fundamental combinatorial problem in multiobjective optimization, and is equivalent to the longest chain problem in combinatorics and random growth models for crystals in materials science. In a previous work, we showed that non-dominated sorting has a continuum limit that corresponds to solving a Hamilton-Jacobi equation. In this work we present and analyze a fast numerical scheme for this Hamilton-Jacobi equation, and show how it can be used to design a fast algorithm for approximate non-dominated sorting.

math.NA↗

arXiv2012

Colliding Interfaces in Old and New Diffuse-interface Approximations of Willmore-flow

This paper is concerned with diffuse-interface approximations of the Willmore flow. We first present numerical results of standard diffuse-interface models for colliding one dimensional interfaces. In such a scenario evolutions towards interfaces with corners can occur that do not necessarily describe the adequate sharp-interface dynamics. We therefore propose and investigate alternative diffuse-interface approximations that lead to a different and more regular behavior if interfaces collide. These dynamics are derived from approximate energies that converge to the $L^1$-lower-semicontinuous envelope of the Willmore energy, which is in general not true for the more standard Willmore approximation.

math.AP↗