GenASiS Mathematics : Object-oriented manifolds, operations, and solvers for large-scale physics simulations
Not Available
Cardall, Christian Y.↗
SEARCH · Engineering Papers
Results for “MATHEMATICAL MANIFOLDS”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Nonmaximality of known extremal metrics on torus and Klein bottle
The El Soufi-Ilias theorem establishes a connection between minimal submanifolds of spheres and extremal metrics for eigenvalues of the Laplace-Beltrami operator. Recently, this connection was used to provide several explicit examples of extremal metrics. We investigate the properties of these metrics and prove that none of them is maximal. Bibliography: 24 titles.
97 MATHEMATICS AND COMPUTING↗
On the cylindricity of submanifolds containing a line in Minkowski space
We consider Finsler submanifolds in Minkowski spaces, and in particular, in Randers spaces. We give generalizations of the Toponogov and Cheeger-Gromoll theorems to the case of Randers spaces. Sufficient conditions are obtained for complete Finsler submanifolds in Minkowski spaces to be cylindrical. We also find conditions under which the convexity of a hypersurface in a Randers space implies that the flag curvature is positive. Bibliography: 17 titles. (paper)
97 MATHEMATICS AND COMPUTING↗
Improving Limit Surface Search Algorithms in RAVEN Using Acceleration Schemes: Level II Milestone
The RAVEN code is becoming a comprehensive tool to perform Probabilistic Risk Assessment (PRA); Uncertainty Quantification (UQ) and Propagation; and Verification and Validation (V&V). The RAVEN code is being developed to support the Risk-Informed Safety Margin Characterization (RISMC) pathway by developing an advanced set of methodologies and algorithms for use in advanced risk analysis. The RISMC approach uses system simulator codes applied to stochastic analysis tools. The fundamental idea behind this coupling approach to perturb (by employing sampling strategies) timing and sequencing of events, internal parameters of the system codes (i.e., uncertain parameters of the physics model) and initial conditions to estimate values ranges and associated probabilities of figures of merit of interest for engineering and safety (e.g. core damage probability, etc.). This approach applied to complex systems such as nuclear power plants requires performing a series of computationally expensive simulation runs. The large computational burden is caused by the large set of (uncertain) parameters characterizing those systems. Consequently, exploring the uncertain/parametric domain, with a good level of confidence, is generally not affordable, considering the limited computational resources that are currently available. In addition, the recent tendency to develop newer tools, characterized by higher accuracy and larger computational resources (if compared with the presently used legacy codes, that have been developed decades ago), has made this issue even more compelling. In order to overcome to these limitations, the strategy for the exploration of the uncertain/parametric space needs to use at best the computational resources focusing the computational effort in those regions of the uncertain/parametric space that are “interesting” (e.g., risk-significant regions of the input space) with respect the targeted Figures Of Merit (FOM): for example, the failure of the system, subject of the analysis. These methodologies are named, in the RAVEN environment, adaptive sampling strategies. These methodologies infer system responses from surrogate models constructed from already existing samples (produced using high fidelity simulations) and suggest the most relevant location (coordinate in the input space) of the next sampling point to be explored in the uncertain/parametric domain. When using those methodologies, it is possible to understand features of the system response with a small number of carefully selected samples. This report focuses on the development and improvement of the limit surface search. The limit surface is an important concept in system reliability analysis. Without going into the details, which will be covered later in the report, the limit surface could be briefly described as an hyper-surface in the system uncertainty/parametric space separating the regions leading to a prescribed system outcome. For example, if the uncertainty/parametric space is the one generated by the reactor power level and the duration of the batteries, the system is a nuclear power plant and the system outcome discriminating variable is the clad failure in a station blackout scenario, then the limit surface separates the combinations of reactor power level and battery duration that lead to clad failure from the ones that do not.
21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
Liouville-Type Theorems for Theories of Riemannian Almost Product Structures and Submersions
We prove Liouville type theorems, i.e., non-existence theorems for special classes of Riemannian almost product structures and submersions, that generalize well-known similar results of the theory of compact Riemannian manifolds.
97 MATHEMATICS AND COMPUTING↗
GenASiS Mathematics: Object-oriented manifolds, operations, and solvers for large-scale physics simulations (version 2)
We report GenASiS Mathematics provides modern Fortran classes furnishing extensible object-oriented functionality for the solution of fields governed by selected partial differential equations. The initial release included extensible object-oriented implementations of simple meshes and the evolution of generic conserved currents thereon. This revision - Version 2 of Mathematics - includes significant reorganization and streamlining of these classes, higher-order reconstruction by a different method, a Poisson solver, coarsening to avoid Courant time step limitations near coordinate singularities, and the offloading of computational kernels to GPUs.
97 MATHEMATICS AND COMPUTING↗
The design-to-cost manifold
A mathematical technique which both quantifies the design-to-cost process and the mass/complexity issue is presented. A simplified approximation of the PRICE H production-production cost is used to generate a dual set of differential equations which define the directions of maximum and minimum cost change over (mass, complexity) space. The equations are solved in closed form to obtain the one-dimensional design-to-cost and design-for-cost spaces. Preliminary results indicate that cost is relatively insensitive to changes in mass and that the reduction of complexity, both in the manufacturing process and in the spacecraft itself, is dominant in reducing cost. Two major objections to the design-to-cost method are discussed.
Dean, Edwin B.↗
Topology of codimension-one foliations of nonnegative curvature. II
We prove that a 3-connected closed manifold M of dimension n≥5 does not admit a codimension-one C{sup 2}-foliation of nonnegative curvature. In particular, this gives a complete answer to a question of Stuck on the existence of codimension-one foliations of nonnegative curvature on spheres. We also consider codimension-one C{sup 2}-foliations of nonnegative Ricci curvature on a closed manifold M with leaves having finitely generated fundamental group, and show that such a foliation is flat if and only if M is a K(π,1)-manifold. Bibliography: 13 titles.
97 MATHEMATICS AND COMPUTING↗
Hidden symmetries in Sasaki–Einstein geometries
We describe a method for constructing Killing–Yano tensors on Sasaki spaces using their geometrical properties, without the need of solving intricate generalized Killing equations. We obtain the Killing–Yano tensors on toric Sasaki–Einstein spaces using the fact that the metric cones of these spaces are Calabi–Yau manifolds which in turn are described in terms of toric data. We extend the search of Killing–Yano tensors on mixed 3-Sasakian manifolds. We illustrate the method by explicit construction of Killing forms on some spaces of current interest.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Geodesic least squares regression for scaling studies in magnetic confinement fusion
In regression analyses for deriving scaling laws that occur in various scientific disciplines, usually standard regression methods have been applied, of which ordinary least squares (OLS) is the most popular. However, concerns have been raised with respect to several assumptions underlying OLS in its application to scaling laws. We here discuss a new regression method that is robust in the presence of significant uncertainty on both the data and the regression model. The method, which we call geodesic least squares regression (GLS), is based on minimization of the Rao geodesic distance on a probabilistic manifold. We demonstrate the superiority of the method using synthetic data and we present an application to the scaling law for the power threshold for the transition to the high confinement regime in magnetic confinement fusion devices.
97 MATHEMATICS AND COMPUTING↗
Elliptic G-Operators on Manifolds with Isolated Singularities
In the present work we study elliptic operators on manifolds with singularities in the situation where the manifold is endowed with an action of a discrete group G. As usual in elliptic theory, the Fredholm property of an operator is governed by the properties of its principal symbol. We show that the principal symbol in our situation is a pair consisting of the symbol on the main stratum (interior symbol) and the symbol at the conical point (conormal symbol). The Fredholm property of elliptic elements is obtained.
97 MATHEMATICS AND COMPUTING↗
Geodesic least squares regression on information manifolds
We present a novel regression method targeted at situations with significant uncertainty on both the dependent and independent variables or with non-Gaussian distribution models. Unlike the classic regression model, the conditional distribution of the response variable suggested by the data need not be the same as the modeled distribution. Instead they are matched by minimizing the Rao geodesic distance between them. This yields a more flexible regression method that is less constrained by the assumptions imposed through the regression model. As an example, we demonstrate the improved resistance of our method against some flawed model assumptions and we apply this to scaling laws in magnetic confinement fusion.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Affine Kac-Moody symmetric spaces related with A{sub 1}{sup (1)}, A{sub 2}{sup (1)}, A{sub 2}{sup (2)}
Symmetric spaces associated with Lie algebras and Lie groups which are Riemannian manifolds have recently got a lot of attention in various branches of Physics for their role in classical/quantum integrable systems, transport phenomena, etc. Their infinite dimensional counter parts have recently been discovered which are affine Kac-Moody symmetric spaces. In this paper we have (algebraically) explicitly computed the affine Kac-Moody symmetric spaces associated with affine Kac-Moody algebras A{sub 1}{sup (1)},A{sub 2}{sup (1)},A{sub 2}{sup (2)}. We hope these types of spaces will play similar roles as that of symmetric spaces in many physical systems.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Integrable Dynamic Systems with Dissipation and Finitely Many Degrees of Freedom
We establish the integrability for some classes of dynamic systems on the tangent bundles of multidimensional manifolds. We consider the case where the force fields possess variable dissipation. An example of a four-dimensional manifold is discussed in detail.
97 MATHEMATICS AND COMPUTING↗
On a criterion of conformal parabolicity of a Riemannian manifold
The paper relates to the circle of problems concerning the connection between the conformal type of a Riemannian manifold and the canonical form of its isoperimetric function. Two special examples of 2-manifolds are constructed, which explain the meaning, role and importance of the conditions involved in the criterion, previously obtained by the author, which decides whether a noncompact Riemannian n-manifold is conformally parabolic. Bibliography: 8 titles.
97 MATHEMATICS AND COMPUTING↗
On the Stability of Toroidal Manifold for One Class of Dynamical Systems
We study the problem of stability of an invariant toroidal manifold for one class of linear extensions of a dynamical system on a torus. The obtained result is used to investigate the problem of existence of invariant manifold for a nonlinear system of differential equations.
97 MATHEMATICS AND COMPUTING↗
Mixed-mode oscillations in a three-timescale coupled Morris–Lecar system
Mixed-mode oscillations (MMOs) are complex oscillatory behaviors of multiple-timescale dynamical systems in which there is an alternation of large-amplitude and small-amplitude oscillations. It is well known that MMOs in two-timescale systems can arise either from a canard mechanism associated with folded node singularities or a delayed Andronov–Hopf bifurcation (DHB) of the fast subsystem. While MMOs in two-timescale systems have been extensively studied, less is known regarding MMOs emerging in three-timescale systems. In this work, we examine the mechanisms of MMOs in coupled Morris–Lecar neurons with three distinct timescales. We investigate two kinds of MMOs occurring in the presence of a singularity known as canard-delayed-Hopf (CDH) and in cases where CDH is absent. In both cases, we examine how features and mechanisms of MMOs vary with respect to variations in timescales. Our analysis reveals that MMOs supported by CDH demonstrate significantly stronger robustness than those in its absence. Moreover, we show that the mere presence of CDH does not guarantee the occurrence of MMOs. This work yields important insights into conditions under which the two separate mechanisms in two-timescale context, canard and DHB, can interact in a three-timescale setting and produce more robust MMOs, particularly against timescale variations.
97 MATHEMATICS AND COMPUTING↗
The Effective Dynamics of the Volume Preserving Mean Curvature Flow
We consider the dynamics of small closed submanifolds (‘bubbles’) under the volume preserving mean curvature flow. We construct a map from (n+1)-dimensional Euclidean space into a given (n+1)-dimensional Riemannian manifold which characterizes the existence, stability and dynamics of constant mean curvature submanifolds. This is done in terms of a reduced area function on the Euclidean space, which is given constructively and can be computed perturbatively. This allows us to derive adiabatic and effective dynamics of the bubbles. The results can be mapped by rescaling to the dynamics of fixed size bubbles in almost Euclidean Riemannian manifolds.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗