Phase-Field Modeling of Hydraulic Fracture

Hydraulically driven fracture has gained more and more research activity in the last few years, especially due to the growing interest of the petroleum industry. Key challenge for a powerful simulation of this scenario is an effective modeling and numerical implementation of the behavior of the solid skeleton and the fluid phase, the mechanical coupling between the two phases as well as the incorporation of the fracture process. Existing models for hydraulic fracturing can be found for example in [1], where the crack path is predetermined, or in [2] who use a phase field fracture model in an elastic framework, however without incorporating the fluid flow. In this work we propose a new compact model structure for the Biot-type fluid transport in porous media at finite strains based on only two constitutive functions, that is the free energy function ψ and a dissipation potential ϕ that includes the incorporation of an additional Poiseuille-type fluid flow in cracks. This formulation is coupled to a phase field approach for fracture and is fully variational in nature, as shown in [3]. In contrast to formulations with a sharp-crack discontinuity, the proposed regularized approach has the main advantage of a straight-forward modeling of complex crack patterns including branching. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

1‐Factor and Cycle Covers of Cubic Graphs

Let G be a bridgeless cubic graph. Consider a list of k 1‐factors of G. Let be the set of edges contained in precisely i members of the k 1‐factors. Let be the smallest over all lists of k 1‐factors of G. Any list of three 1‐factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge‐covers and for the existence of three 1‐factors with empty intersection. Furthermore, if , then is an upper bound for the girth of G. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs. Cubic graphs with have a 4‐cycle cover of length and a 5‐cycle double cover. These graphs also satisfy two conjectures of Zhang 18 . We also give a negative answer to a problem stated in 18 .     

Reasoning about proof and knowledge

In previous work [15], we presented a hierarchy of classical modal systems, along with algebraic semantics, for the reasoning about intuitionistic truth, belief and knowledge. Deviating from Gödel's interpretation of IPC in S4, our modal systems contain IPC in the way established in [13]. The modal operator can be viewed as a predicate for intuitionistic truth, i.e. proof. Epistemic principles are partially adopted from Intuitionistic Epistemic Logic IEL [4]. In the present paper, we show that the S5-style systems of our hierarchy correspond to an extended Brouwer–Heyting–Kolmogorov interpretation and are complete w.r.t. a relational semantics based on intuitionistic general frames. In this sense, our S5-style logics are adequate and complete systems for the reasoning about proof combined with belief or knowledge. The proposed relational semantics is a uniform framework in which also IEL can be modeled. Verification-based intuitionistic knowledge formalized in IEL turns out to be a special case of the kind of knowledge described by our S5-style systems.     

Spike patterns in a reaction–diffusion ODE model with Turing instability

We explore a mechanism of pattern formation arising in processes described by a system of a single reaction–diffusion equation coupled with ordinary differential equations. Such systems of equations arise from the modeling of interactions between cellular processes and diffusing growth factors. We focus on the model of early carcinogenesis proposed by Marciniak‐Czochra and Kimmel, which is an example of a wider class of pattern formation models with an autocatalytic non‐diffusing component. We present a numerical study showing emergence of periodic and irregular spike patterns because of diffusion‐driven instability. To control the accuracy of simulations, we develop a numerical code on the basis of the finite‐element method and adaptive mesh grid. Simulations, supplemented by numerical analysis, indicate a novel pattern formation phenomenon on the basis of the emergence of nonstationary structures tending asymptotically to a sum of Dirac deltas. Copyright © 2013 John Wiley & Sons, Ltd.     

Maximum Δ‐edge‐colorable subgraphs of class II graphs

A graph G is class II, if its chromatic index is at least Δ + 1. Let H be a maximum Δ‐edge‐colorable subgraph of G. The paper proves best possible lower bounds for |E(H)|/|E(G)|, and structural properties of maximum Δ‐edge‐colorable subgraphs. It is shown that every set of vertex‐disjoint cycles of a class II graph with Δ≥3 can be extended to a maximum Δ‐edge‐colorable subgraph. Simple graphs have a maximum Δ‐edge‐colorable subgraph such that the complement is a matching. Furthermore, a maximum Δ‐edge‐colorable subgraph of a simple graph is always class I. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Material parameter identification using model reduction to uniaxial tensile test

Material parameter identification is the necessary link between constitutive modeling and application of material models in complex simulations of real world problems. We present an approach to make use of already developed stress algorithms for three-dimensional finite-element computations in terms of a reduction to uniaxial tensile tests for material parameter identification. Strain-driven stress algorithms lead to the approach of solving differential-algebraic equations. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Continuity of the SLE trace in simply connected domains

We prove that the SLE _κ trace in any simply connected domain G is continuous (except possibly near its endpoints) if κ < 8. We also prove an SLE analog of Makarov’s Theorem about the support of harmonic measure.     

Graph problems arising from parameter identification of discrete dynamical systems

This paper focuses on combinatorial feasibility and optimization problems that arise in the context of parameter identification of discrete dynamical systems. Given a candidate parametric model for a physical system and a set of experimental observations, the objective of parameter identification is to provide estimates of the parameter values for which the model can reproduce the experiments. To this end, we define a finite graph corresponding to the model, to each arc of which a set of parameters is associated. Paths in this graph are regarded as feasible only if the sets of parameters corresponding to the arcs of the path have nonempty intersection. We study feasibility and optimization problems on such feasible paths, focusing on computational complexity. We show that, under certain restrictions on the sets of parameters, some of the problems become tractable, whereas others are NP-hard. In a similar vein, we define and study some graph problems for experimental design, whose goal is to support the scientist in optimally designing new experiments.     

Damage and fracture behavior in dynamic tension tests: Modelling and numerical simulations

The continuum damage model is based on a general thermodynamic framework for the modeling of rate and temperature dependent behavior of anisotropically damaged elastic-plastic materials subjected to fast deformation. The introduction of damaged and fictitious undamaged configurations allows the definition of damage tensors and the corresponding free energy functions lead to material laws affected by damage and temperature. The damage condition and the corresponding damage rule strongly depend on stress triaxiality. Furthermore, the rate and temperature dependence is reflected in a multiplicative decomposition of the plastic hardening and damage softening functions. The macro crack behavior is characterized by a triaxiality dependent fracture criterion. The continuum damage model is implemented into LS-DYNA as user defined material model. Corresponding numerical simulations of unnotched and notched tension tests with high strain rates demonstrate the plastic and damage processes during the deformation leading to final fracture numerically predicted by an element erosion technique. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)