Exponential finite element shape functions for a phase field model of brittle fracture

In phase field models for fracture a continuous scalar field variable is used to indicate cracks, i.e. the value 1 of the phase field variable is assigned to sound material, while the value 0 indicates fully broken material. The width of the transition zone where the phase field parameter changes between 1 and 0 is controlled by a regularization parameter. As a finite element discretization of the model needs to be fine enough to resolve the crack field and its gradient, the numerical results are sensitive to the choice of the regularization parameter in conjunction with the mesh size. This is the main challenge and the computational limit of the finite element implementation of phase field fracture models. To overcome this limitation a finite element technique using special shape functions is introduced. These special shape functions take into account the exponential character of the crack field as well as its dependence on the regularization length. Numerical examples show that the exponential shape functions allow a coarser discretization than standard linear shape functions without compromise on the accuracy of the results. This is due to the fact, that using exponential shape functions, the approximation of the surface energy of the phase field cracks is impressively precise, even if the regularization length is rather small compared to the mesh size. Thus, these shape functions provide an alternative to a numerically expensive mesh refinement

varTestnlme: An R Package for Variance Components Testing in Linear and Nonlinear Mixed-Effects Models

The issue of variance components testing arises naturally when building mixed-effects models, to decide which effects should be modeled as fixed or random or to build parsimonious models. While tests for fixed effects are available in R for models fitted with lme4, tools are missing when it comes to random effects. The varTestnlme package for R aims at filling this gap. It allows to test whether a subset of the variances and covariances corresponding to a subset of the random effects, are equal to zero using asymptotic property of the likelihood ratio test statistic. It also offers the possibility to test simultaneously for fixed effects and variance components. It can be used for linear, generalized linear or nonlinear mixed-effects models fitted via lme4, nlme or saemix. Numerical methods used to implement the test procedure are detailed and examples based on different real datasets using different mixed models are provided. Theoretical properties of the used likelihood ratio test are recalled

A Phase Field Modeling Approach of Crack Growth in Materials with Anisotropic Fracture Toughness

Within this contribution, we present a diffuse interface approach for the simulation of crack nucleation and growth in materials, which incorporates an orientation dependency of the fracture toughness. After outlining the basic motivation for the model from an engineering standpoint, the phase field paradigm for fracture is introduced. Further, a specific phase field model for brittle fracture is reviewed, where we focus on the meaning of the auxiliary parameter differentiating between material phases and the coupling of such a parameter to continuum equations in order to obtain the characteristic self organizing model properties. This specific model, as will be explained, provides the phenomenological and methodical basis for the presented enhancement. The formulation of an appropriate evolution equation in terms of a Ginzburg-Landau type equation will be highlighted and several comments on sharp interface models will be made to present a brief comparison. Following up on the basics we then introduce the formulation of a modified version of the model, which additionally to the handling of cracks in linear elastic materials under quasi static loading is also capable of taking into account the effect of resistance variation with respect to the potential crack extension direction. The strong and also the weak forms of the respective governing equations corresponding to the developed anisotropic phase field model are presented. Utilizing the weak formulation as starting point for the discretization of the two fields (displacement field and the phase field), the computational framework in terms of finite elements is introduced. We finally explain several test cases investigated within simulations and discuss the corresponding numerical results. Besides examples, which are set up to illustrate the general model properties, a comparison with crack paths obtained by experimental investigations will be presented in order to show the potential of the developed phase field model

Korrosion von Kulturgut aus Kupferwerkstoffen in Gegenwart von elementarem Schwefel in der Gasphase

"Schwarze Flecken" werden seit den 1970er Jahren immer wieder als Korrosionsproblem in Museen und Sammlungen thematisiert. Die vorliegende Arbeit beinhaltet die erste systematische Untersuchung der Korrosionsprozesse, welche zur Bildung von Kupfersulfiden in Form "Schwarzer Flecken" auf Kupferwerkstoffen führen. Hierbei war es erstmals möglich, die Korrosion mit typischer Morphologie im Laborexperiment reproduzierbar darzustellen. Experimente zur Stabilität der Korrosionsprodukte Chalkosin und Covellin sowie die Dokumentation und Analyse von über 300 betroffenen Objekten und Korrosionsproben liefern ergänzende Informationen. Die charakteristische Morphologie der Produkte wurde im Rahmen der Arbeit lichtmikroskopisch und rasterelektronenmikroskopisch untersucht. Analysen zur chemischen Zusammensetzung der Produkte wurden zerstörungsfrei am Raman-Mikroskop und mit Hilfe von EDX-Analysen durchgeführt. Ergänzend hierzu wurden einzelne Analysen mittels Röntgendiffraktometrie an Pulverproben durchgeführt. Die relative Luftfeuchte stellt bei der Korrosion von Kupferwerkstoffen in Gegenwart von elementarem Schwefel in der Gasphase den wesentlichen Einfluss auf die chemische Zusammensetzung der Korrosionsprodukte dar. Ein systematischer Zusammenhang zwischen der Morphologie, Farbe und chemischen Zusammensetzung der Korrosionsprodukte lässt sich eindeutig nachweisen. Die Temperatur beeinflusst hingegen lediglich die Korrosionsrate. Die Anwesenheit der Legierungspartner Zinn und Zink mit einem Anteil von 8 bzw. 10 % zeigt vergleichsweise geringfügige Auswirkungen auf die Korrosionsrate und die chemische Zusammensetzung der entstehenden Produkte. Die Anwesenheit der Korrosionsprodukte Malachit, Cuprit und Tenorit zeigt ebenfalls keinen bedeutenden Einfluss auf die Korrosion, auch sind die Produkte selbst unter den getesteten Bedingungen deutlich stabiler als Kupfer und die untersuchten Legierungen. Eine deckende Oxidschicht auf der Metalloberfläche schützt bei niedriger relativer Luftfeuchte vor Korrosion. Das Korrosionswachstum an Schwachstellen innerhalb einer Oxidschicht entspricht dem fleckigen Auftreten der Korrosion in der Praxis. Mit steigender relativer Luftfeuchte werden die Kupfersulfide Covellin und Chalkosin, welche als Korrosion in Form "Schwarzer Flecken" auftreten, zunehmend instabil. Covellin wirkt bei steigender relativer Luftfeuchte auf metallisches Kupfer in direktem Kontakt zunehmend korrosiv. Die gewonnenen Erkenntnisse ermöglichen uns ein besseres Verständnis der Korrosionsprozesse und liefern wichtige Informationen für den Umgang mit betroffenen Objekten in der restauratorischen und konservatorischen Praxis

Efficient preconditioned stochastic gradient descent for estimation in latent variable models

Latent variable models are powerful tools for modeling complex phenomena involving in particular partially observed data, unobserved variables or underlying complex unknown structures. Inference is often difficult due to the latent structure of the model. To deal with parameter estimation in the presence of latent variables, well-known efficient methods exist, such as gradient-based and EM-type algorithms, but with practical and theoretical limitations. In this paper, we propose as an alternative for parameter estimation an efficient preconditioned stochastic gradient algorithm. Our method includes a preconditioning step based on a positive definite Fisher information matrix estimate. We prove convergence results for the proposed algorithm under mild assumptions for very general latent variables models. We illustrate through relevant simulations the performance of the proposed methodology in a nonlinear mixed effects model and in a stochastic block model

Fatigue related impairments in oculomotor control are prevented by caffeine

Strenuous exercise can result in an inability of the central nervous system to drive skeletal muscle e ectively, a phenomenon known as central fatigue. The impact of central fatigue on the oculomotor system is currently unexplored. Fatigue that originates in the central nervous system may be related to perturbations in the synthesis and metabolism of several neurotransmitters. In this study we examine central fatigue in the oculomotor system after prolonged exercise. The involvement of central neurotransmission was explored by administering ca eine during exercise. Within a double- blind, randomized, repeated measures, crossover design, 11 cyclists consumed a placebo or ca eine solution during 180 min of stationary cycling. Saccadic eye movements were measured using infra-red oculography. Exercise decreased saccade velocity by 8% (placebo trial). This e ect was reversed by ca eine, whereby velocity was increased by 11% after exercise. A non oculomotor perceptual task (global motion processing) was una ected by exercise. The human oculomotor system is impaired by strenuous exercise of the locomotor system. Ca eine exerts a protective e ect on oculomotor control, which could be related to up-regulated central neurotransmission. In addition, cortical processes supporting global motion perception appear to be robust to fatigue

Numerical homogenization of the Eshelby tensor at small strains

Numerical homogenization methods, such as the FE² approach, are widely used to compute the effective physical properties of microstructured materials. Thereby, the macroscopic material law is replaced by the solution of a microscopic boundary value problem on a representative volume element in conjunction with appropriate averaging techniques. This concept can be extended to configurational or material quantities, like the Eshelby stress tensor, which are associated with configurational changes of continuum bodies. In this work, the focus is on the computation of the macroscopic Eshelby stress tensor within a small-strain setting. The macroscopic Eshelby stress tensor is defined as the volume average of its microscopic counterpart. On the microscale, the Eshelby stress tensor can be computed from quantities known from the solution of the physical microscopic boundary value problem. However, in contrast to the physical quantities of interest, i.e. stress and strain, the Eshelby stress tensor is sensitive to rigid body rotations of the representative volume element. In this work, it is demonstrated how this must be taken into account in the computation of the macroscopic Eshelby stress tensor. The theoretical findings are illustrated by a benchmark simulation and further simulation results indicate the microstructural influence on the macroscopic configurational forces