Abstract : This minisymposium gathers analysts interested in the evolution of interfaces, singularities and lower dimensional objects, applied to fracture evolution, inverse problems, or shape optimization, and more.
Fracture can be computed by successive minimizations of "free discontinuity" energies, approached with phase-field methods. Evolving or static interfaces, or lower dimensional singularities, can be computed by discretizing geometric measure theoretical objects. Level sets methods can be used to identify defects in conductive media. The speakers address such issues from different point of views or apply similar methods to different problems. This will foster fruitful interaction between the participants and the audience.
[01367] Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems
Format : Talk at Waseda University
Author(s) :
Braxton Osting (University of Utah)
Abstract : Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally. In this talk, I'll discuss numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, $\Sigma$, with genus $\gamma$ and $b$ boundary components, we maximize $\sigma_j(\Sigma,g) \, L(\partial \Sigma, g)$ over a class of smooth metrics, $g$, where $\sigma_j(\Sigma,g)$ is the $j$-th nonzero Steklov eigenvalue and $L(\partial \Sigma, g)$ is the length of $\partial \Sigma$. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. Eigenfunctions corresponding to the extremal Steklov eigenvalues are used to generate a free boundary minimal surface. This is joint work wtih Chiu-Yen Kao and Èdouard Oudet.
[05329] Capturing surfaces using differential forms
Format : Talk at Waseda University
Author(s) :
Stephanie Wang (University of California, San Diego)
Albert Chern (University of California, San Diego)
Abstract : Exterior calculus has been an important tool in solving numerical PDEs by representing physical quantities as differential forms. In this talk we expand the usage of differential forms to a whole new way of representing curves and surfaces. By doing so we reformulate the classical nonconvex Plateau minimal surface problem into a convex optimization problem, and introduce a new implicit surface representation that permits nonempty boundaries.
[05617] Nonlocal approximations of anisotropic surface energies on partitions
Format : Talk at Waseda University
Author(s) :
Selim Esedoglu (University of Michigan)
Abstract : Nonlocal energies approximating the perimeter of sets and, in the multiphase case, partitions, arise in many settings. In particular, they play an important role in the study of threshold dynamics, an algorithm for multiphase mean curvature motion. Ensuring the convergence of nonlocal energies in the multiphase, anisotropic setting turns out to be tricky but essential for the correct behavior of these numerical methods. I will discuss conditions that guarantee convergence of the anisotropic energies.
[05463] On some extensions and applications of thresholding schemes
Format : Talk at Waseda University
Author(s) :
Karel Svadlenka (Kyoto University)
Abstract : In this talk, I will present two examples of application of suitably modified thresholding schemes, which were originally developed by Merriman, Bence and Osher, and later generalized by Esedoglu and Otto. First example concerns elucidating cell pattern formation in morphogenesis of sensory epithelia, and the other example concerns understanding evolution of anisotropic particles on solid substrate. I will also touch on the mathematical background of the schemes and their numerical implementation.
Abstract : Variational phase-field models of fracture have established themselves as a powerful and efficient computational approach in fracture mechanics. They are based on a regularization of Francfort and Marigo variational energy, where the crack geometry and discontinuous displacements are represented by smooth functions The most common implementations are based on finite element discretization via continuous finite elements, and a staggered minimization scheme.
In this talk, we present a new discretization scheme where displacements are discretized using discontinuous Lagrange elements and the phase field variable by Crouzeix-Raviart elements. We compare this scheme to the classical continuous Galerkin scheme and highlight how it leads to a better approximation of the fracture energy.
[01899] Numerical approximation of a viscoelastic Cahn--Hilliard model for tumour growth
Format : Talk at Waseda University
Author(s) :
Dennis Trautwein (University of Regensburg)
Harald Garcke (University of Regensburg)
Abstract : In this talk, we present a phase-field model for tumour growth, where a diffuse interface is separating a tumour from the surrounding host tissue. In our model, we include biological effects like chemotaxis and transport processes by an internal velocity field. We include viscoelastic effects with a general Oldroyd-B type description with stress relaxation and stress generation by growth. We analyze a numerical approximation of the model with a fully-practical, stable and converging discrete scheme, which preserves the physical properties of the model. Finally, we illustrate properties of solutions with the help of numerical simulations.
[02809] A second order Cahn Hilliard model for wetting simulation
Format : Talk at Waseda University
Author(s) :
elie bretin (ICJ & INSA Lyon )
Abstract : We focus in this talk to the approximation of surface diffusion flow using a Cahn–Hilliard-type model.
We introduce a new second order variational phase field model that associates the classical Cahn-Hilliard energy with two degenerate mobilities. We also propose some simple and efficient numerical schemes to approximate the solutions and provide 3D numerical simulations of the wetting of a thin tube on various solid supports. This work was done in collaboration with R. Denis, S. Masnou, G. Terii and A. Sengers
[04707] Finite element minimization of line and surface energies arising in liquid crystals
Format : Talk at Waseda University
Author(s) :
Dominik Stantejsky (McMaster University)
Abstract : Originating in the study of defect structures in nematic liquid crystals, we consider the numerical minimization of an energy posed for two-dimensional surfaces $T$ involving the surface area of $T$ outside an obstacle $E$, as well as the length of the boundary $\partial T$ and a surface integral over the obstacle surface, generalizing both the obstacle- and Plateau problem. We propose a finite element representation of the energy and minimize using an ADMM algorithm.
