Sharp phase-field modeling of isotropic solidification with a super efficient spatial resolution

The numerical resolution efficiency of phase-field models is limited by grid friction, grid anisotropy and pinning. The 1D sharp phase-field model eliminates grid friction and pinning by a global restoration of Translational Invariance (TI) in the discretized phase-field equation (Phys. Rev. Lett. 121, 025501, 2018). In 3D global TI restricts the beneficial modeling properties to a finite number of fixed interface orientations. We propose an accurate scheme to restore TI locally in the local interface normal direction. At one-grid-point interface resolutions, the new model captures the formation of isotropic seaweed structures without spurious dendritic selection by grid anisotropy.Comment: 9 pages, 7 figures, 1 table, 2 supplementary animation

Are We Responsible for Our Emotions and Moods?

The short answer to the question in the title of this paper is yes. Our thesis is that we are indeed responsible for our emotions and moods. We want to help children understand that just as they are responsible for what they do and say, or omit to do or say (along with the consequences of these acts), so are they responsible for much of their affective life. What remains is to explain what we mean by responsibility, emotions, and moods

Frictionless motion of diffuse interfaces by sharp phase-field modeling

Diffuse interface descriptions offer many advantages for the modeling of microstructure evolution. However, the numerical representation of moving diffuse interfaces on discrete numerical grids involves spurious grid friction, which limits the overall performance of the model in many respects. Interestingly, this intricate and detrimental effect can be overcome in finite difference (FD) and fast Fourier transformation (FFT)-based implementations by employing the so-called sharp phase-field method (SPFM). The key idea is to restore the discretization-induced broken translational invariance (TI) in the discrete phase-field equation by using analytic properties of the equilibrium interface profile. We prove that this method can indeed eliminate spurious grid friction in the three-dimensional space. Focusing on homogeneous driving forces, we quantitatively evaluate the impact of spurious grid friction on the overall operational performance of different phase-field models. We show that the SPFM provides superior degrees of interface isotropy with respect to energy and kinetics. The latter property enables the frictionless motion of arbitrarily oriented diffuse interfaces on a fixed 3D grid

Frictionless motion of marginally resolved diffuse interfaces in phase-field modeling

We investigate the influence of artificial grid friction in different phase-field models by considering the stationary motion of an interface between two phases at different bulk free energy levels. Following the striking idea of a Sharp Phase-Field Model (SPFM) from Finel et al., we proof that restoring translational invariance indeed eliminates artificial grid friction effects during stationary interface propagation. Over a largely extended range of possible driving forces the theoretic interface velocities are reproduced by orders of magnitude more accurately, even if the diffuse interface profile is only marginally resolved by just one grid point. We propose a new variant of the SPFM, which restores translational invariance locally in the direction of interface motion. It is shown that, even for marginally resolved interface-profiles, the new SPFM variant provides frictionless motion for arbitrarily oriented planar interfaces. Finally, by considering thermal diffusion limited solidification, we demonstrate the capability of the SPFM-approach to also deal with inhomogeneous driving forces using a one-gird point interface resolution.Comment: 7 pages, 8 figures, 3 supplemental animation

Philosophical Reflection and Cooperative Practices in an Elementary School Mathematics Classroom

Following Matthew Lipman (Lipman, 1991; Lipman, Sharp, & Oscanyan, 1980), we introduced philosophical dialogue (PD) about mathematics in an elementary school to help pupils consider mathematical and meta-mathematical matters. This article describes the social and cognitive activity when pupils engage in PD and some pedagogical conditions necessary to foster the development of PD. Changes in pupils’ discussions from the begin- ning to the end of the test period showed that the dynamic evolved from monological exchanges to dialogical exchanges. Whereas early pupil responses could be characterized mainly as simple answers, later responses displayed more lower-order and even higher- order thinking skills. The data suggest that for PD about mathematics to develop, the teacher must be proficient in the role of mediator. À la suite de Matthew Lipman (Lipman, 1991; Lipman, Sharp et Oscanyan, 1980), les auteurs ont introduit le dialogue philosophique (DP) dans une école primaire afin d’aider les élèves à réfléchir aux questions d’ordre mathématique et méta-mathématique. Cet article décrit l’activité sociale et cognitive à laquelle donne lieu le DP ainsi que certaines des conditions pédagogiques qui doivent être réunies pour favoriser le développement du DP. L’analyse des discussions des élèves du début à la fin de la période d’essai a révélé une évolution de la dynamique, des échanges monologiques aux échanges dialogiques. Si en premier les élèves donnent de simples réponses, par la suite ils démontrent des capa- cités de raisonnement élémentaires et même de plus haut niveau. Ces données semblent indiquer que pour que le DP au sujet de la mathématique se développe, l’enseignant doit devenir un médiateur efficace.