Stable phase field approximations of anisotropic solidification

We introduce unconditionally stable finite element approximations for a phase field model for solidification, which take highly anisotropic surface energy and kinetic effects into account. We hence approximate Stefan problems with anisotropic Gibbs{Thomson law with kinetic undercooling, and quasi-static variants thereof. The phase field model is given by #wt + � %(') 't = r: (b(')rw) ; c a � %(')w = " � � �(r') '

The Faith of My Fathers

(Excerpt) In his final years, United States Supreme Court Justice Robert H. Jackson worked on a number of autobiographical writing projects. The previously unknown Jackson text that follows this Introduction is one such writing. Justice Jackson wrote this essay in longhand on thirteen yellow legal pad pages in the early 1950s. It is Jackson’s writing about religion in his life. After Justice Jackson’s death in 1954, his secretary Elsie L. Douglas found the thirteen pages among his papers. She concluded that the pages were “undoubtedly prepared as part of his autobiography,” typed them up, and gave a file folder containing the original pages plus her typescript to Jackson’s son William Eldred Jackson, then a young partner in the Milbank, Tweed, Hope, and Hadley law firm in New York City. Bill Jackson preserved this material carefully for decades but never shared it. Much later, the folder and its contents were entrusted to me. I will be donating it soon to the Library of Congress, for addition to Jackson’s papers there. Justice Jackson’s essay on religion, the material that follows, covers two topics: (1) his religious beliefs and practices plus those of his ancestors, who were farmers in Warren County, Pennsylvania, where young Robert Jackson lived for his first five or so years before moving to and then growing up in adjacent Chautauqua County, New York; and (2) some history on Spiritualist movements in that western Pennsylvania and western New York State region. Unfortunately, Jackson did not explain in his draft how his views on religion were shaped by growing up in a landscape of such varied religious beliefs and practices, including Spiritualism. But the gist of his thinking seems clear enough: there are all kinds of people, religions, and beliefs, and the proper way to live is to give people space and to tolerate what they are and what they choose to believe and to practice in their spaces, so long as they do not intrude unduly on one’s own

Eliminating spurious velocities with a stable approximation of incompressible two-phase flow

We present a parametric finite element approximation of two-phase flow. This free boundary problem is given by the Stokes equations in the two phases, which are coupled via jump conditions across the interface. Using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface. In addition, the mesh quality of the parametric approximation of the interface does not deteriorate, in general, over time; and an equidistribution property can be shown for a semidiscrete continuous-in-time variant of our scheme in two space dimensions. Moreover, on using a simple XFEM pressure space enrichment, we obtain exact volume conservation for the two phase regions. Furthermore, our fully discrete finite element approximation can be shown to be unconditionally stable. We demonstrate the applicability of our method with some numerical results which, in particular, demonstrate that spurious velocities can be avoided in the classical test cases

On stable parametric finite element methods for the Stefan problem and the Mullins-Sekerka problem with applications to dendritic growth

We introduce a parametric finite element approximation for the Stefan problem with the Gibbs–Thomson law and kinetic undercooling, which mimics the underlying energy structure of the problem. The proposed method is also applicable to certain quasi-stationary variants, such as the Mullins–Sekerka problem. In addition, fully anisotropic energies are easily handled. The approximation has good mesh properties, leading to a well-conditioned discretization, even in three space dimensions. Several numerical computations, including for dendritic growth and for snow crystal growth, are presented

Finite-Element Approximation of One-Sided Stefan Problems with Anisotropic, Approximately Crystalline, Gibbs--Thomson Law

We present a finite-element approximation for the one-sided Stefan problem and the one-sided Mullins--Sekerka problem, respectively. The problems feature a fully anisotropic Gibbs--Thomson law, as well as kinetic undercooling. Our approximation, which couples a parametric approximation of the moving boundary with a finite element approximation of the bulk quantities, can be shown to satisfy a stability bound, and it enjoys very good mesh properties which means that no mesh smoothing is necessary in practice. In our numerical computations we concentrate on the simulation of snow crystal growth. On choosing realistic physical parameters, we are able to produce several distinctive types of snow crystal morphologies. In particular, facet breaking in approximately crystalline evolutions can be observed.Comment: 50 pages, 32 figures, 14 tables. Corrected typo

Stable approximations for axisymmetric Willmore flow for closed and open surfaces

For a hypersurface in ${\mathbb R}^3$, Willmore flow is defined as the $L^2$--gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry, image reconstruction and mathematical biology. In this paper, we propose novel numerical approximations for the Willmore flow of axisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants we prove a stability result. We consider both closed surfaces, and surfaces with a boundary. In the latter case, we carefully derive weak formulations of suitable boundary conditions. Furthermore, we consider many generalizations of the classical Willmore energy, particularly those that play a role in the study of biomembranes. In the generalized models we include spontaneous curvature and area difference elasticity (ADE) effects, Gaussian curvature and line energy contributions. Several numerical experiments demonstrate the efficiency and robustness of our developed numerical methods.Comment: 52 pages, 19 figure