scholarly journals On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures

2020 ◽  
Vol 32 (10) ◽  
pp. 2050033
Author(s):  
Ettore Minguzzi
Keyword(s):  
Level Set ◽  
Spacelike Hypersurface ◽  
Cauchy Hypersurface ◽  
Cone Structure ◽  
Globally Hyperbolic ◽  
Signed Distance ◽  
Density Result ◽  
Separation Result ◽  
Spacelike Hypersurfaces ◽  
Closed Cone

We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance [Formula: see text] from a spacelike hypersurface [Formula: see text] is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface is the level set of a Cauchy temporal (and steep) function of the same regularity as the hypersurface. We also show that in a globally hyperbolic closed cone structure, compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity. We end the work with a separation result and a density result.

Some remarks on the existence of spacelike hypersurfaces of constant mean curvature

1977 ◽  
Vol 82 (3) ◽  
pp. 489-495 ◽  
Author(s):  
A. J. Goddard
Keyword(s):  
Minkowski Space ◽  
Mean Curvature ◽  
General Class ◽  
Constant Mean Curvature ◽  
Spacelike Hypersurface ◽  
De Sitter Space ◽  
De Sitter ◽  
Minimal Graphs ◽  
Complete Spacelike Hypersurface ◽  
Spacelike Hypersurfaces

AbstractBernstein's theorem states that the only complete minimal graphs in R3 are the hyperplanes. We shall produce evidence in favour of some conjectural generalizations of this theorem for the cases of spacelike hypersurfaces of constant mean curvature in Minkowski space and in de Sitter space. The results suggest that the class of asymptotically simple space-times admitting a complete spacelike hypersurface of constant mean curvature may well be considerably smaller than the general class of asymptotically simple space-times.

Application of a Cartesian-Grid Immersed Boundary Method to 3D In-Cylinder Flow

2012 ◽  
Author(s):  
Claudia Günther ◽  
Matthias Meinke ◽  
Wolfgang Schröder
Keyword(s):  
Level Set ◽  
Immersed Boundary Method ◽  
Three Dimensional ◽  
Cartesian Grid ◽  
Distance Functions ◽  
Immersed Boundary ◽  
Splitting Algorithm ◽  
Signed Distance ◽  
Boundary Method ◽  
Cylinder Flow

In this work, a Cartesian-grid immersed boundary method using a cut-cell approach is applied to three-dimensional in-cylinder flow. A hierarchically coupled level-set solver is used to capture the boundary motion by a signed distance function. Topological changes in the geometry due to the opening and closing events of the valves are modeled consistently using multiple signed distance functions for the different components of the engine and taking advantage of a level-set reinitialization method. A continuous discretization of the flow equations in time near the moving interfaces is used to prevent nonphysical oscillations. To ensure an efficient implementation, independent grid adaptation for the flow and the level-set grid is applied. A narrow band approach and an efficient joining/splitting algorithm for the level-set functions minimize the computational overhead to track multiple interfaces. The ability of the current method to handle complex 3D setups is demonstrated for the interface capturing and the flow solution in a three-dimensional piston engine geometry.

UPPER AND LOWER BOUNDS FOR THE VOLUME OF A COMPACT SPACELIKE HYPERSURFACE IN A GENERALIZED ROBERTSON–WALKER SPACETIME

2013 ◽  
Vol 11 (01) ◽  
pp. 1450006 ◽  
Author(s):  
JUAN ÁNGEL ALEDO ◽  
ALFONSO ROMERO ◽  
RAFAEL M. RUBIO
Keyword(s):  
Lower Bounds ◽  
Three Dimensional ◽  
Spacelike Hypersurface ◽  
Upper And Lower Bounds ◽  
Warping Function ◽  
First Eigenvalue ◽  
De Sitter ◽  
Sitter Spacetime ◽  
Angle Function ◽  
Spacelike Hypersurfaces

We provide upper and lower bounds for the volume of a compact spacelike hypersurface in an (n + 1)-dimensional Generalized Robertson–Walker (GRW) spacetime in terms of the volume of the fiber, the hyperbolic angle function and the warping function. Under several geometrical and physical assumptions, we characterize the spacelike slices as the only spacelike hypersurfaces where these bounds are attained. As a consequence of these results, we get an upper bound for the first eigenvalue of a compact spacelike surface in a three-dimensional GRW spacetime whose fiber is a topological sphere, which includes the case of the three-dimensional De Sitter spacetime, and show that the bound is attained if and only if M is a spacelike slice.

scholarly journals Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes

2007 ◽  
Vol 143 (3) ◽  
pp. 703-729 ◽  
Author(s):  
LUIS J. ALÍAS ◽  
A. GERVASIO COLARES
Keyword(s):  
Mean Curvature ◽  
Differential Operators ◽  
Spacelike Hypersurface ◽  
Convergence Condition ◽  
Higher Order ◽  
Spacelike Hypersurfaces ◽  
Newton Transformations ◽  
Higher Order Mean Curvature ◽  
Associated Differential Operators

AbstractIn this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker (GRW) spacetimes. In particular, we consider the following question: under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, essentially, under the so callednull convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.

scholarly journals Generalization of level-set inversion to an arbitrary number of geological units in a regularized least-squares framework

Geophysics ◽  
2021 ◽  
pp. 1-76
Author(s):  
Jérémie Giraud ◽  
Mark Lindsay ◽  
Mark Jessell
Keyword(s):  
Least Squares ◽  
Level Set ◽  
Arbitrary Number ◽  
Tectonic Setting ◽  
Physical Property ◽  
Inversion Method ◽  
Regularized Least Squares ◽  
Signed Distance ◽  
Geological Unit ◽  
Geological Units

We present an inversion method for the recovery of the geometry of an arbitrary number of geological units using a regularized least-squares framework. The method addresses cases where each geological unit can be modeled using a constant physical property. Each geological unit or group assigned with the same physical property value is modeled using the signed-distance to its interface with other units. We invert for this quantity and recover the location of interfaces between units using the level-set method. We formulate and solve the inverse problem in a least-squares sense by inverting for such signed-distances. The sensitivity matrix to perturbations of the interfaces is obtained using the chain rule and model mapping from the signed-distance is used to recover physical properties. Exploiting the flexibility of the framework we develop allows any number of rocks units to be considered. In addition, it allows the design and use of regularization incorporating prior information to encourage specific features in the inverted model. We apply this general inversion approach to gravity data favoring minimum adjustments of the interfaces between rock units to fit the data. The method is first tested using noisy synthetic data generated for a model comprised of six distinct units and several scenarios are investigated. It is then applied to field data from the Yerrida Basin (Australia) where we investigate the geometry of a prospective greenstone belt. The synthetic example demonstrates the proof-of-concept of the proposed methodology, while the field application provides insights into, and potential re-interpretation of, the tectonic setting of the area.

Wavelet SDF-Reps: Solid Modeling With Volumetric Scans

2007 ◽  
Author(s):  
Duane Storti ◽  
Mark A. Ganter ◽  
William R. Ledoux ◽  
Randal P. Ching ◽  
Yangqiu Patrick Hu ◽  
...  
Keyword(s):  
Level Set ◽  
Computer Aided Design ◽  
Grid Point ◽  
Level Set Methods ◽  
Solid Modeling ◽  
Closed Surface ◽  
Scanner Data ◽  
Case Scenario ◽  
Signed Distance ◽  
New Formulation

This paper describes a new formulation of solid modeling that addresses the issue of including parts whose geometry is determined from volumetric scans (CT, MRI, PET, etc.) along with parts whose geometry is designed by traditional computer-aided design (CAD) operations. Such issues arise frequently in the design of medical devices or prostheses where fit and/or interference between man-made artifacts and existing anatomy are essential considerations, but the modeling formulation presented is not limited to medical applications and can be applied to any parts whose volume can be actually or virtually scanned. Scanner data typically comprises a grid of intensity values and segmentation must be performed to determine the extent of the part. In current practice, the segmented scanner data is run through a polygonizer to obtain an approximate tessellation of the object’s surface. Even in the best case scenario where the triangles obtained form a closed surface that accurately approximates the surface of the scanned object, such triangulated models can be problematic due to excessive size. We present an alternative approach based on recent advances in segmentation with level set methods. The output of the level set computation is a grid of approximate values for the signed distance from each grid point to the nearest point on the surface of the scanned object. We propose interpolating the grid of signed distance values to obtain an implicit or function-based representation (f-rep) for the object, and we introduce appropriate wavelets to effectively perform the interpolation while also providing a number of other useful properties including data compression, inherently multi-scale modeling, and capabilities for skeletal-based modeling operations.

A Level Set Method With a Bounded Diffusion for Structural Topology Optimization

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Benliang Zhu ◽  
Rixin Wang ◽  
Hai Li ◽  
Xianmin Zhang
Keyword(s):  
Topology Optimization ◽  
Level Set ◽  
Jacobi Equation ◽  
Structural Topology ◽  
Signed Distance Function ◽  
Diffusion Term ◽  
Signed Distance ◽  
Level Set Function ◽  
Set Function ◽  
Hamilton Jacobi Equation

In level-set-based topology optimization methods, the spatial gradients of the level set field need to be controlled to avoid excessive flatness or steepness at the structural interfaces. One of the most commonly utilized methods is to generalize the traditional Hamilton−Jacobi equation by adding a diffusion term to control the level set function to remain close to a signed distance function near the structural boundaries. This study proposed a new diffusion term and built it into the Hamilton-Jacobi equation. This diffusion term serves two main purposes: (I) maintaining the level set function close to a signed distance function near the structural boundaries, thus avoiding periodic re-initialization, and (II) making the diffusive rate function to be a bounded function so that a relatively large time-step can be used to speed up the evolution of the level set function. A two-phase optimization algorithm is proposed to ensure the stability of the optimization process. The validity of the proposed method is numerically examined on several benchmark design problems in structural topology optimization.

Sharp Interface Numerical Modeling of Solidification Process of Pure Metal / Sposób Modelowania Numerycznego Procesu Krzepnięcia Z Ostrym Frontem

2012 ◽  
Vol 57 (4) ◽  
pp. 1189-1199 ◽  
Author(s):  
T. Skrzypczak
Keyword(s):  
Mathematical Model ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Level Set ◽  
Distance Function ◽  
Solidification Process ◽  
System Of Equations ◽  
Signed Distance Function ◽  
Interface Motion ◽  
Signed Distance

The paper is focused on the study of the solidification process of pure metals, in which the solidification front is smooth. It has the shape of a surface separating liquid from solid in three dimensional space or a curve in 2D. The location and topology of moving interface change over time and its velocity depends on the values of heat fluxes on the solid and liquid side of it. Such a formulation belongs to a group called Stefan problems. A mathematical model of the Stefan problem is based on differential equations of heat conduction and interface motion. This system of equations is supplemented by appropriate initial and boundary conditions as well as the continuity conditions at the solidification interface. The solution involves the determination of temporary temperature field and interface position. Typically, it is impossible to obtain the exact solution of such problem. This paper presents a mathematical model for the two-dimensional problem. The equation of heat conduction is supplemented with Dirichlet and Neumann boundary conditions. Interface motion is described by the level set equation which solution is sought in the form of temporary distribution of the signed distance function. Zero level of the distance field coincides with the position of the front. Values of the signed distance function obtained from the level set equation require systematic reinitialization. Numerical model of the process based on the finite element method (FEM) is also presented. FEM equations are derived and discussed. The explicit time integration scheme is proposed. It helps to avoid solving the system of equations during each time step. The reinitialization procedure of the signed distance function is described in detail. Examples of numerical analysis of the solidification process of pure copper within the complex geometry are presented. Results obtained from the use of constant material properties are compared with those obtained from the use of temperature dependent properties.

PARABOLICITY OF SPACELIKE HYPERSURFACES IN GENERALIZED ROBERTSON–WALKER SPACETIMES: APPLICATIONS TO UNIQUENESS RESULTS

2013 ◽  
Vol 10 (08) ◽  
pp. 1360014 ◽  
Author(s):  
A. ROMERO ◽  
R. M. RUBIO ◽  
J. J. SALAMANCA
Keyword(s):  
Arbitrary Dimension ◽  
Spacelike Hypersurface ◽  
Universal Covering ◽  
New Technique ◽  
Warping Function ◽  
Complete Spacelike Hypersurfaces ◽  
Uniqueness Results ◽  
Complete Spacelike Hypersurface ◽  
Spacelike Hypersurfaces ◽  
Hyperbolic Angle

We study non-compact complete spacelike hypersurfaces in generalized Robertson–Walker spacetimes of arbitrary dimension whose fiber is parabolic. Under boundedness assumptions on the warping function restricted on a spacelike hypersurface and on the hyperbolic angle of the hypersurface, we prove that a complete spacelike hypersurface is parabolic if the Riemannian universal covering of the fiber is so. As an application of this new technique, several uniqueness results on complete maximal spacelike hypersurfaces are obtained. Also, the corresponding Calabi–Bernstein problems are solved.

scholarly journals Weakly convex hypersurfaces of pseudo-Euclidean spaces satisfying the condition LkHk+1 = λHk+1

2021 ◽  
Vol 40 (3) ◽  
pp. 711-719
Author(s):  
Firooz Pashaie
Keyword(s):  
Mean Curvature ◽  
Linear Part ◽  
Curvature Vector ◽  
Spacelike Hypersurface ◽  
Weakly Convex ◽  
Minkowski Space Time ◽  
Spacelike Hypersurfaces ◽  
The First Variation ◽  
Linearized Operator ◽  
Convex Hypersurfaces

In this paper, we try to give a classification of spacelike hypersurfaces of the Lorentz-Minkowski space-time E1n+1, whose mean curvature vector field of order (k+ 1) is an eigenvector of the kth linearized operator Lk, for a non-negative integer k less than n. The operator Lk is defined as the linear part of the first variation of the (k + 1)th mean curvature of a hypersurface arising from its normal variations. We show that any spacelike hypersurface of E1n+1 satisfying the condition LkHk+1 = λHk+1 (where 0 ≤ k ≤ n − 1) belongs to the class of Lk-biharmonic, Lk-1-type or Lk-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of spacelike hypersurfaces of Lorentz-Minkowski spaces, named the weakly convex hypersurfaces (i.e. on which all of principle curvatures are nonnegative). We prove that, on any weakly convex spacelike hypersurface satisfying the above condition for an integer k (where, 0 ≤ r ≤ n−1), the (k + 1)th mean curvature will be constant. As an interesting result, any weakly convex spacelike hypersurfaces, having assumed to be Lk-biharmonic, has to be k-maximal.

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