### Prescribed Scalar Curvature in the Asymptotically Euclidean Setting

David Maxwell
The Yamabe invariant of an asymptotically Euclidean (AE) manifold is defined analogously to that of a compact manifold. Nevertheless, the prescribed scalar curvature problem in the AE setting has features that are quite different from its compact counterpart. For example, a Yamabe positive AE manifold admits a conformally related metric that has a scalar curvature with any desired sign: positive, negative or zero everywhere. In this talk we discuss the resolution of the prescribed nonpositive...

### Chunhyang-ga as pansori-style opera : a guide for performing pansori with classically-trained singers outside of Korea : [supplementary material]

Jason Abram Klippenstein
The attached files are supplements to the author’s doctoral dissertation at https://circle.library.ubc.ca/handle/2429/70913

Tiandong Wang

### Discrete optimal transport: Limits and limitations

Using the finite volume method, one can define a discrete Kantorovich distance with a Riemannian structure based on a Euclidean mesh. We show that in most cases, the limit distance as mesh size tends to zero, in the sense of Gamma- or Gromov-Hausdorff-convergence, is strictly less than the standard Kantorovich distance. This is due to an oscillation effect reminiscent of homogenization. We introduce a geometric condition on the mesh that prevents oscillations and are able...

### Constraints and penalties for phase-field flows in $\mathbb{R}^2$ and $\mathbb{R}^N$

Matteo Negri
We present two gradient flow evolutions, both obtained with alternate schemes for separately-quadratic phase-field energies. The first, in the plane strain setting, features a monotonicity constraint (in time) and a multi-step scheme, for better numerical results. The second, in higher dimension, features instead a penalty method. In this case, strong compactness of the phase-field variable allows to characterize evolutions in terms of curves of maximal slope with respect to the penalty-metric.

### Convergence of alternate minimization algorithms in phase field models of fracture with non-interpenetration

Stefano Almi
In a two dimensional setting, we present a result of convergence of an alternate minimization scheme applied to a phase field model of fracture with non-interpenetration. Our analysis is based on the study of suitable gradient flows of the phase field energy, which connect all the states of the algorithm. The limit evolutions are described in terms of parametrized $BV$-solutions. This is a joint work with M. Negri.

### Globally stable quasistatic evolution for cohesive fracture with fatigue

Giuliano Lazzaroni
In this talk we discuss the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue...

### Phase field methods of fracture in heterogeneous media and structures: a combined bulk-interface-like crack method

Jose Reinoso
Heterogeneity is present in most of natural and engineering systems. In this contribution, I present the recent developments of a combined phase field method for bulk fracture and interface-like cracks. This methodology allows triggering the competition between crack penetration and deflection at an interface, recalling fundamental results from Linear Elastic Fracture Mechanics (LEFM). Following the fundamental developments, I revisit the numerical implementation as well as its application to different systems such as shell-like structures, composite...

### A phase-field approach to quasistatic evolution for a cohesive fracture model

Flaviana Iurlano
In this paper we propose a notion of irreversibility for the evolution of cracks in presence of cohesive forces, which allows for different responses in the loading and unloading processes, motivated by a variational approximation with damage models. We investigate its applicability to the construction of a quasistatic evolution in a simple one-dimensional model. This is a joint work with M. Bonacini and S. Conti.

### Classification of amorphous materials and the dynamic toughness problem

Yasumasa Nishiura
We apply the PCA and topological data analysis to the polymer materials like epoxy resin in order to classify its microstructure depending on the process. Based on this classification, we study the toughness problem via phase field approach.

### Minimisation and Ambrosio-Tortorelli approximation of the Griffith energy with Dirichlet boundary condition

Vito Crismale
I will present recent works about the minimisation of the Griffith energy for brittle fracture in elastic materials, under Dirichlet boundary conditions. Together with Antonin Chambolle (CMAP, Ã cole Polytechnique) we have proven the existence of minimisers and a phase-field approximation Ã la Ambrosio-Tortorelli for this energy.

### A variational phase-field model for hydraulic fracturing in poro-elastic media

Keita Yoshioka
In this talk, we will first go through the construction of a variational phase-field based coupled hydro-mechanical model in poor-elastic media. We will then revisit the problem of a single hydraulic fracture propagating in an infinite impermeable medium in order to justify our coupling strategy. Finally, we will discuss how a phase-field description of a system of cracks can be leveraged to model flow in a fractured porous medium.

Gianni Dal Maso

### Crack bridging and fiber debonding modeling using multiphase continuum and phase-field models

Jeremy Bleyer
In this talk, I will present a particular class of generalized continua called multiphase models which consist of different media possessing their own kinematics and in interaction with each other. This setting is particularly suited to fiber-reinforced media and enables to model in a macroscopic fashion phenomena like bridged cracks or fiber debonding. A variational phase-field combined with a debonding damage law will be proposed for simulating matrix cracks bridged by intact fibers.

### Numerical aspects of phase-field modelling of fracture: ideas, results and challenges

Tymofiy Gerasimov
The irreversibility constraint, the non-convexity of governing energy functional and the intrinsically small length-scale are the main sources of algorithmic and numerical challenges for phase-field models of fracture. The talk aims at summarizing the main ideas, results and challenges that we proposed and encountered in addressing the above issues in the past few years. We highlight - various solution strategies for the discretized coupled problem, such as partitioned (staggered) and frontal (monolithic) schemes, with a...

### Phase-field modelling of ductile failure in fiber-reinforced composites

Giovanni Lancioni
In this talk, a variational model is proposed for the description of ductile failure in composite materials consisting of short strengthening fibers embedded in brittle matrices. The composite is schematized as a mixture of two phases coupled by elastic bonds: a brittle phase and a plastic phase account for matrix and fibers contributions, respectively. Balance and evolution equations are variationally deduced, and the role played by three different internal lengths is discussed. Finally, results of...

### A variational phase-field approach to fatigue in brittle materials

Pietro Carrara
A novel variational framework to model the fatigue behavior of brittle materials based on a phase-field approach to fracture is presented. The standard regularized free energy functional is modified introducing a fatigue degradation function that effectively reduces the fracture toughness as a proper history variable accumulates. This macroscopic approach allows to reproduce the main known features of fatigue crack growth in brittle materials. Numerical experiments show that the WÃ¶hler curve, the crack growth rate curve...

### Mathematical issues in combining evolution and $\Gamma$-convergence: dangers in phase-field dynamics

Christopher Larsen
Mathematicians have generally not emphasized the difference between $\Gamma$-convergence and the convergence necessary for "approximate" dynamic solutions to converge to the correct limiting dynamics. I will discuss what properties limiting dynamic fracture models should have, and how $\Gamma$-convergence can fail to deliver them, with an emphasis on phase-field approximations and some surprising problems.

### Asymptotic geometry of the Hitchin moduli space

Jan Swoboda
In this talk I will explain recent joint work with Rafe Mazzeo, Hartmut Weiss and Frederik Witt on the asymptotics of the natural $L^2$ metric on the Hitchin moduli space of rank-$2$ Higgs bundles. It will be shown that on the regular part of the Hitchin fibration this metric is well-approximated by the so-called semiflat metric coming from the algebraic completely integrable system moduli space is endowed with. This result confirms some aspects of a...

Jo Stratton

Xiang Zhou

### Penalized Latent Dirichlet Allocation Model in Single Cell RNA Sequencing (Abstract ID: A4)

Zhijin Wu

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