Vector potential methods

Vector potential and related methods, for the simulation of both inviscid and viscous flows over aerodynamic configurations, are briefly reviewed. The advantages and disadvantages of several formulations are discussed and alternate strategies are recommended. Scalar potential, modified potential, alternate formulations of Euler equations, least-squares formulation, variational principles, iterative techniques and related methods, and viscous flow simulation are discussed

Continuous families of isospectral metrics on simply connected manifolds

We construct continuous families of Riemannian metrics on certain simply connected manifolds with the property that the resulting Riemannian manifolds are pairwise isospectral for the Laplace operator acting on functions. These are the first examples of simply connected Riemannian manifolds without boundary which are isospectral, but not isometric. For example, we construct continuous isospectral families of metrics on the product of spheres S^4\times S^3\times S^3. The metrics considered are not locally homogeneous. For a big class of such families, the set of critical values of the scalar curvature function changes during the deformation. Moreover, the manifolds are in general not isospectral for the Laplace operator acting on 1-forms.Comment: 22 pages, published versio

A tree-cotree splitting for the construction of divergence-free finite elements: the high-order case

We extend, to Raviart-Thomas finite elements of any degree, two methods for the construction of basis of the space of divergence-free functions that are well established in the case of degree one. The first one computes directly a basis of the kernel of the divergence operator whereas the second one computes a basis of the image of the curl operator that, if the boundary of the domain is not connected, is completed with a basis of the second de Rham cohomology group (namely, the space of divergence-free functions that are not curls). When using the lower order Whitney elements on a tetrahedral mesh, the degrees of freedom are supported on the vertices, edges, faces and tetrahedra of the mesh respectively and, from Stokes theorem, the matrices describing the differential operators gradient, curl and divergence are the transposed of the connectivity matrices of the mesh. This allows the use of a tree-cotree splitting of associated oriented graphs to efficiently construct a basis of either the kernel of the divergence or the image of the curl operator. We prove that these two properties hold true also for r > 0 when using as degrees of freedom a particular realization, based on Berstein polynomials, of the moments. In this work we analyze in detail the second method, the one based on the identification of a basis of the space of the curls of Nédélec finite elements. (The first one has been analyzed in [5].) Key words. High order Raviart-Thomas finite elements, divergence-free finite elements, spanning tree, oriented graph, incidence matrix AMS subject classifications. 65N30, 05C05 1. Introduction. Two approaches for the construction of a basis of the divergence free Raviart-Thomas finite element space, RT 0 r+1 , in a bounded polyhedral domain, Ω, discretized by a tetrahedral mesh have been presented in the work [6], for the lower order case r = 0. There are not restrictions on the topology of Ω. In the first approach, the authors compute directly a basis of the kernel of the divergence operator. In the second one, the construction starts from a basis of the image of the matrix associated with the curl operator. If the boundary of the domain has p + 1 connected components with p > 0, in this second approach it is necessary to complete the previous set with p discrete representatives of a basis of the second de Rham cohomology group (divergence-free functions that are not curls). These two methods can be extended to the high order case r > 0. The extension of the first one has been analyzed in [5]. In this work we analyze in detail the extension of the second approach. However, for the sake of completeness we include in this introduction a brief description of both methods in the high order case. Let T be a tetrahedral mesh of a bounded polyhedral domain Ω ⊂ R 3. We will denote P − r+1 Λ k (T) the space of Whitney k-differential forms of degree r + 1 (see e.g. [9]). They can be identified with L r+1 , the Lagrange finite elements of degree r + 1, if k = 0, with N r+1 , the first family of Nédélec finite elements of degree r + 1, if k = 1

Doctor of Philosophy

dissertationWith modern computational resources rapidly advancing towards exascale, large-scale simulations useful for understanding natural and man-made phenomena are becoming in- creasingly accessible. As a result, the size and complexity of data representing such phenom- ena are also increasing, making the role of data analysis to propel science even more integral. This dissertation presents research on addressing some of the contemporary challenges in the analysis of vector fields--an important type of scientific data useful for representing a multitude of physical phenomena, such as wind flow and ocean currents. In particular, new theories and computational frameworks to enable consistent feature extraction from vector fields are presented. One of the most fundamental challenges in the analysis of vector fields is that their features are defined with respect to reference frames. Unfortunately, there is no single ""correct"" reference frame for analysis, and an unsuitable frame may cause features of interest to remain undetected, thus creating serious physical consequences. This work develops new reference frames that enable extraction of localized features that other techniques and frames fail to detect. As a result, these reference frames objectify the notion of ""correctness"" of features for certain goals by revealing the phenomena of importance from the underlying data. An important consequence of using these local frames is that the analysis of unsteady (time-varying) vector fields can be reduced to the analysis of sequences of steady (time- independent) vector fields, which can be performed using simpler and scalable techniques that allow better data management by accessing the data on a per-time-step basis. Nevertheless, the state-of-the-art analysis of steady vector fields is not robust, as most techniques are numerical in nature. The residing numerical errors can violate consistency with the underlying theory by breaching important fundamental laws, which may lead to serious physical consequences. This dissertation considers consistency as the most fundamental characteristic of computational analysis that must always be preserved, and presents a new discrete theory that uses combinatorial representations and algorithms to provide consistency guarantees during vector field analysis along with the uncertainty visualization of unavoidable discretization errors. Together, the two main contributions of this dissertation address two important concerns regarding feature extraction from scientific data: correctness and precision. The work presented here also opens new avenues for further research by exploring more-general reference frames and more-sophisticated domain discretizations

Hardy and Lieb-Thirring inequalities for anyons

We consider the many-particle quantum mechanics of anyons, i.e. identical particles in two space dimensions with a continuous statistics parameter $\alpha \in [0,1]$ ranging from bosons ($\alpha=0$) to fermions ($\alpha=1$). We prove a (magnetic) Hardy inequality for anyons, which in the case that $\alpha$ is an odd numerator fraction implies a local exclusion principle for the kinetic energy of such anyons. From this result, and motivated by Dyson and Lenard's original approach to the stability of fermionic matter in three dimensions, we prove a Lieb-Thirring inequality for these types of anyons.Comment: Corrected and accepted version. 30 pages, 4 figure

About the gauge conditions arising in Finite Element magnetostatic problems

International audienceIn this paper, we deal with some magnetostatic models considered in vector potential formulations and solved by a Finite Element solver. In order to ensure the uniqueness of the solution, a gauge condition has to be imposed, and several possibilities occur. Moreover, the source term has to be correctly defined to ensure a physically admissible solution. We show the equivalence between some of these choices for several kinds of boundary conditions. Moreover, we highlight their characteristic behaviors on some numerical benchmarks to illustrate our theoretical results

The Exploratory Role of Idealizations and Limiting Cases in Models

In this article we argue that idealizations and limiting cases in models play an exploratory role in science. Four senses of exploration are presented: exploration of the structure and representational capacities of theory; proof-of-principle demonstrations; potential explanations; and exploring the suitability of target systems. We illustrate our claims through three case studies, including the Aharonov-Bohm effect, the emergence of anyons and fractional quantum statistics, and the Hubbard model of the Mott phase transitions. We end by reflecting on how our case studies and claims compare to accounts of idealization in the philosophy of science literature such as Michael Weisberg’s three-fold taxonomy