Multidimensional Meshes Research Articles

A multislope MUSCL method for vectorial reconstructions

Variables interpolation is one of the key concepts of MUSCL schemes. Originally developed for one-dimensional frameworks, many improvements have been made over the last decades to extend these methods to general unstructured multi-dimensional meshes. It results that scalar interpolation in a finite volume cell-centered framework is already quite well understood. It is known that linear reconstructions have to be limited in order to prevent non-physical oscillations of the solutions while ensuring a spatial second-order accuracy for smooth solutions. This is done thanks to a limiting function which allows the reconstruction to satisfy a monotonicity property, which then ensure the stability of the scheme. Nevertheless, some difficulties arise when we try to extend this process to vectorial variables. Generally, vectorial reconstructions are done componentwise, but this process reveals to be frame-dependent and leads to a loss of precision due to false detection of extrema. In this paper, we present a new method dealing with vectorial reconstructions in a multislope MUSCL context.

Complexity of direct and iterative solvers on space–time formulations and time-marching schemes for h-refined grids towards singularities

We study computational complexity aspects for Finite Element formulations considering hypercubic space–time full and time-marching discretization schemes for h-refined grids towards singularities. We perform a relatively comprehensive study comparing the computational time via time complexities of direct and iterative solvers. We focus on the space–time formulation with refined computational grids and the corresponding time slabs, namely, computational grids obtained by taking the “cross-sections” of the refined space–time mesh. We estimate the computational complexity of solving systems of linear equations for multidimensional meshes with arbitrary dimensional singularities encountered in space–time formulation and time-marching schemes. The choice between space–time formulation and time-marching schemes depends entirely on the problem’s nature and the properties that need to be addressed. Thus, the paper aims to discuss the computational complexities of both approaches rather than suggest a better formulation. Our considerations concern the computational complexity of sequental execution of the multi-frontal solvers, the iterative solvers, and the static condensation. Numerical experiments with Octave confirm our theoretical findings.

Singleton mesh patterns in multidimensional permutations

This paper introduces the notion of mesh patterns in multidimensional permutations and initiates a systematic study of singleton mesh patterns (SMPs), which are multidimensional mesh patterns of length 1. A pattern is avoidable if there exist arbitrarily large permutations that do not contain it. As our main result, we give a complete characterization of avoidable SMPs using an invariant of a pattern that we call its rank. We show that determining avoidability for a d-dimensional SMP P of cardinality k is an O(d⋅k) problem, while determining rank of P is an NP-complete problem. Additionally, using the notion of a minus-antipodal pattern, we characterize SMPs which occur at most once in any d-dimensional permutation. Lastly, we provide a number of enumerative results regarding the distributions of certain general projective, plus-antipodal, minus-antipodal and hyperplane SMPs.

In depth review of the ecology of arenicolous marine fungi

As an ecotone, sandy beaches exist within a multi-dimensional mesh of environmental gradients, shaped by numerous parameters (e.g. temperature, humidity, wave action, sand particle size and salinity). These limit the proliferation of a narrow group of fungal species. Obligate arenicolous marine fungi are an ecological assemblage of sand-associated heterotrophs that inhabit sandy beaches. These organisms have evolved to cope with dynamic beach conditions, having a cosmopolitan distribution across tropical, subtropical and temperate regions. Herein we provide an overview of published works relating to the fungi of sandy beaches, focusing on the past half-century. We outline a broad range of topics in ecology including fungal adaptive traits to intertidal conditions at the morphological and genetic levels, temporal and spatial patterns in community structure, and species variations in substrate preference. Collectively, these concepts should encourage marine mycologists to embrace a holistic set of perspectives to shape the outlook for beach ecology.

Flow and transport in three-dimensional discrete fracture matrix models using mimetic finite difference on a conforming multi-dimensional mesh

We present a comprehensive workflow to simulate single-phase flow and transport in fractured porous media using the discrete fracture matrix approach. The workflow has three primary parts: (1) a method for conforming mesh generation of and around a three-dimensional fracture network, (2) the discretization of the governing equations using a second-order mimetic finite difference method, and (3) implementation of numerical methods for high-performance computing environments. A method to create a conforming Delaunay tetrahedralization of the volume surrounding the fracture network, where the triangular cells of the fracture mesh are faces in the volume mesh, that addresses pathological cases which commonly arise and degrade mesh quality is also provided. Our open-source subsurface simulator uses a hierarchy of process kernels (one kernel per physical process) that allows for both strong and weak coupling of the fracture and matrix domains. We provide verification tests based on analytic solutions for flow and transport, as well as numerical convergence. We also provide multiple expositions of the method in complex fracture networks. In the first example, we demonstrate that the method is robust by considering two scenarios where the fracture network acts as a barrier to flow, as the primary pathway, or offers the same resistance as the surrounding matrix. In the second test, flow and transport through a three-dimensional stochastically generated network containing 257 fractures is presented.

Upscaled discrete fracture matrix model (UDFM): an octree-refined continuum representation of fractured porous media

We develop a computational geometry-based upscaling approach to accurately capture the dynamic processes occurring within and between subsurface fracture networks and the surrounding porous matrix named the Upscaled Discrete Fracture Matrix model (UDFM). Fracture attributes (orientations and apertures) are upscaled and combined with matrix attributes (permeability and porosity) into properties of an unstructured spatially variable Delaunay tetrahedral continuum mesh. The resolution of the mesh depends on proximity to the fractures to preserve the geometric and topological integrity of the underlying fracture network as well as increasing the accuracy of gradients in the dynamics between the fractures and matrix. The UDFM Delaunay mesh and its dual Voronoi mesh can be used in existing multiphysics simulators for flow, solute/heat mass transport, and geomechanics, thereby eliminating the need for the additional development of numerical methods to couple processes in fracture/matrix systems. The model is verified by comparing flow, transport, and coupled heat-mass flow simulations on the provided meshes against analytical and numerical benchmarks. We also provide an additional example to demonstrate the applicability of the UDFM to complex heteregenous fractured media. Overall, the UDFM is accurate in all of the cases considered here and presents an attractive alternative to other modeling strategies that require novel numerical methods or multi-dimensional meshes.

Theoretical background and implementation of the finite element method for multi-dimensional Fokker–Planck equation analysis

Fokker–Planck equation is one of the most important tools for investigation of dynamic systems under random excitation. Finite Element Method represents very effective solution possibility particularly when transition processes are investigated or more detailed solution is needed. However, a number of specific problems must be overcome. They follow predominantly from the large multi-dimensionality of the Fokker–Planck equation, shape of the definition domain and usual requirements on the nature of the solution which are out of a conventional practice of the Finite Element employment. Unlike earlier studies it is coming to light that multi-dimensional simplex elements are the most suitable to be deployed. Moreover, new original algorithms for the multi-dimensional mesh generating were developed as well as original procedure of the governing differential and algebraic systems assembling and subsequent analysis. Finally, an illustrative example is presented together with aspects typical for the problem with large multi-dimensionality.

On optimal simplicial 3D meshes for minimizing the Hessian-based errors

In this paper we derive a multi-dimensional mesh adaptation method which produces optimal meshes for quadratic functions, positive semi-definite. The method generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the tensor metric being computed based on interpolation error estimates. It does not depend, a priori, on the PDEs at hand in contrast to residual methods. The estimated error is then used to steer local modifications of the mesh in order to reach a prescribed level of error in Lp-norm or a prescribed number of elements. The Lp-norm of the estimated error is then minimized in order to get an optimal mesh. Numerical examples in 2D and 3D for analytic challenging problems and an application to a Computational Fluid Dynamics problem are presented and discussed in order to show how the proposed method recovers optimal convergence rates as well as to demonstrate its computational performance.

Embedding Two Disjoint Multi-Dimensional Meshes into Locally Twisted Cubes

Let LTQ_n denote the n-dimensional locally twisted cubes. This paper deals with the problem of how to embed a family of two disjoint multi-dimensional meshes into locally twisted cubes. We develop the following embeddings: for n ≥ 3 and 2 ≤ k ≤ n, two disjoint meshes with size The equation is abbretived 2^(n-k) can be embedded into LTQ_n with unit dilation, unit expansion, and congestion-free. The results obtained are optimal in the sense that the dilations, expansions and congestions of all the embeddings are equal to 1.

A Theoretical Analysis for a New Finite Volume Scheme for a Linear Schrödinger Evolution Equation on General Nonconforming Spatial Meshes

In this article, we shall present in detail the results announced [5]. We consider a model of a linear time dependent Schrödinger equation with a time dependent potential. This model arises, for example, in underwater acoustics and has been studied by Akrivis and Dougalis [3]. We derive a new finite volume scheme on general nonconforming multidimensional spatial meshes introduced recently by Eymard et al. [18] for stationary anisotropic heterogeneous diffusion problems. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. The discretization of the initial condition is performed using a discrete orthogonal projection. A new a priori estimate for the discrete problem is proved. Thanks to this a priori estimate for the discrete problem, we derive error estimates in discrete norms of and W 1, ∞(L 2). Moreover, we establish an error estimate for an approximation for the gradient, in a general framework. We prove that the convergence order is h 𝒟 + k, where h 𝒟 (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption for the exact solution u. These error estimates are useful because they allow us to get error estimates for the approximations of the exact solution and its first derivatives. Results of the present work have been obtained by a comparison with an appropriately chosen auxiliary finite volume scheme.

Numerical approaches for multidimensional simulations of stellar explosions

We introduce numerical algorithms for initializing multidimensional simulations of stellar explosions with 1D stellar evolution models. The initial mapping from 1D profiles onto multidimensional grids can generate severe numerical artifacts, one of the most severe of which is the violation of conservation laws for physical quantities. We introduce a numerical scheme for mapping 1D spherically-symmetric data onto multidimensional meshes so that these physical quantities are conserved. We verify our scheme by porting a realistic 1D Lagrangian stellar profile to the new multidimensional Eulerian hydro code CASTRO. Our results show that all important features in the profiles are reproduced on the new grid and that conservation laws are enforced at all resolutions after mapping. We also introduce a numerical scheme for initializing multidimensional supernova simulations with realistic perturbations predicted by 1D stellar evolution models. Instead of seeding 3D stellar profiles with random perturbations, we imprint them with velocity perturbations that reproduce the Kolmogorov energy spectrum expected for highly turbulent convective regions in stars. Our models return Kolmogorov energy spectra and vortex structures like those in turbulent flows before the modes become nonlinear. Finally, we describe approaches to determining the resolution for simulations required to capture fluid instabilities and nuclear burning. Our algorithms are applicable to multidimensional simulations besides stellar explosions that range from astrophysics to cosmology.

Streamline upwind/Petrov–Galerkin‐based stabilization of proper generalized decompositions for high‐dimensional advection–diffusion equations

SUMMARYThis work is a first attempt to address efficient stabilizations of high dimensional advection–diffusion models encountered in computational physics. When addressing multidimensional models, the use of mesh‐based discretization fails because the exponential increase of the number of degrees of freedom related to a multidimensional mesh or grid, and alternative discretization strategies are needed. Separated representations involved in the so‐called proper generalized decomposition method are an efficient alternative as proven in our former works; however, the issue related to efficient stabilizations of multidimensional advection–diffusion equations has never been addressed to our knowledge. Thus, this work is aimed at extending some well‐experienced stabilization strategies widely used in the solution of 1D, 2D, or 3D advection–diffusion models to models defined in high‐dimensional spaces, sometimes involving tens of coordinates.Copyright © 2013 John Wiley & Sons, Ltd.

Conservative Initial Mapping For Multidimensional Simulations of Stellar Explosions

Mapping one-dimensional stellar profiles onto multidimensional grids as initial conditions for hydrodynamics calculations can lead to numerical artifacts, one of the most severe of which is the violation of conservation laws for physical quantities such as energy and mass. Here we introduce a numerical scheme for mapping one-dimensional spherically-symmetric data onto multidimensional meshes so that these physical quantities are conserved. We validate our scheme by porting a realistic 1D Lagrangian stellar profile to the new multidimensional Eulerian hydro code CASTRO. Our results show that all important features in the profiles are reproduced on the new grid and that conservation laws are enforced at all resolutions after mapping.

A simple weighted essentially nonoscillatory limiter for Runge–Kutta discontinuous Galerkin methods

In this paper, we investigate a simple limiter using weighted essentially non-oscillatory (WENO) methodology for the Runge–Kutta discontinuous Galerkin (RKDG) methods solving conservation laws, with the goal of obtaining a robust and high order limiting procedure to simultaneously achieve uniform high order accuracy and sharp, non-oscillatory shock transitions. The idea of this limiter is to reconstruct the entire polynomial, instead of reconstructing point values or moments in the classical WENO reconstructions. That is, the reconstruction polynomial on the target cell is a convex combination of polynomials on this cell and its neighboring cells and the nonlinear weights of the convex combination follow the classical WENO procedure. The main advantage of this limiter is its simplicity in implementation, especially for multi-dimensional meshes. Numerical results in one and two dimensions are provided to illustrate the behavior of this procedure.

A global arbitrary Lagrangian–Eulerian method for stratified Richtmyer–Meshkov instability

Richtmyer–Meshkov (RM) instability arises when a material interface is accelerated impulsively by shock waves. In this work, an arbitrary Lagrangian–Eulerian method, global ALE method, was proposed for the simulation of stratified RM instability. In the global ALE method, an Eulerian diffusion interface model was implemented based on mass fraction function. Thus all the meshes can be remeshed arbitrarily no matter whether they are material interface or not. Some benchmark problems, such as shock tube problem with different specific ratio, RM instability with small initial perturbation, were computed with the global ALE method, and the numerical results agree well with exact solution or theoretical model. Also, we proposed some stratified RM instability model problems with two or more material interfaces in planar, cylindrical and spherical geometries. Then the stratified RM instabilities were simulated with global ALE method. The interface evolution process was studied and compared in different geometry cases based on simulation results. To overcome the spurious mesh distortion, a sub-zonal Riemann solver method was proposed in appendix part of the paper based on the analysis of the error source of 2D Lagrangian computation due to non-uniform multi-dimensional mesh.

Embedding multi-dimensional meshes into twisted cubes

The twisted cube is an important variant of the most popular hypercube network for parallel processing. In this paper, we consider the problem of embedding multi-dimensional meshes into twisted cubes in a systematic way. We present a recursive method for embedding a family of disjoint multi-dimensional meshes into a twisted cube with dilation 1 and expansion 1. We also prove that a single multi-dimensional mesh can be embedded into a twisted cube with dilation 2 and expansion 1. Our work extends some previously known results.

Heuristic algorithm for generating multilevel Cartesian meshes inmultidimensional regions with moving boundaries

Universal algorithm for fully automated generation of multilevel Cartesian mesh in arbitrary planar regions with moving boundaries is developed and discussed in details. It uses multilevel tree technology and heuristic enabling flexible mesh adaptation to changes of boundaries. Efficient mixed coding for mesh storing is presented. Suggested algorithm generates meshes with properties most desirable for computing various field distribution problems. Main parts of elaborated algorithm are generalized for multidimensional meshes.

Embedding a family of disjoint multi-dimensional meshes into a crossed cube

Crossed cubes are an important class of hypercube variants. This paper addresses how to embed a family of disjoint multi-dimensional meshes into a crossed cube. We prove that for n ⩾ 4 and 1 ⩽ m ⩽ ⌊ n / 2 ⌋ − 1 , a family of 2 m disjoint k-dimensional meshes of size 2 t 1 × 2 t 2 × ⋯ × 2 t k each can be embedded in an n-dimensional crossed cube with unit dilation, where ∑ i = 1 k t i = n − m and max 1 ⩽ i ⩽ k { t i } ⩾ n − 2 m − 1 . This result means that a family of mesh-structured parallel algorithms can be executed on a same crossed cube efficiently and in parallel. Our work extends some recently obtained results.

Inclusion of Sliding Friction in Contact Dynamics Model for Helical Gears

This paper proposes a new analytical model for helical gears that characterizes the contact plane dynamics and captures the velocity reversal at the pitch line due to sliding friction. First, the tooth stiffness density function along the contact lines is calculated by using a finite element code. Analytical formulations are then derived for the multidimensional mesh forces and moments. Contact zones for multiple tooth pairs in contact are identified, and the associated integration algorithms are derived. A new 12-degree-of-freedom, linear time-varying model with sliding friction is then developed. It includes rotational and translational motions along the line-of-action, off-line-of-action, and axial directions. The methodology is also illustrated by predicting time and frequency domain results for several values of the coefficient of friction.

Black hole search in common interconnection networks

AbstractMobile agents operating in networked environments face threats from other agents as well as from the hosts (i.e., network sites) they visit. A black hole is a harmful host that destroys incoming agents without leaving any trace. To determine the location of such a harmful host is a dangerous but crucial task, called black hole search. The most important parameter for a solution strategy is the number of agents it requires (the size); the other parameter of interest is the total number of moves performed by the agents (the cost). It is known that at least two agents are needed; furthermore, with full topological knowledge, Ω(n log n) moves are required in arbitrary networks. The natural question is whether, in specific networks, it is possible to obtain (topology‐dependent but) more cost efficient solutions. It is known that this is not the case for rings. In this article, we show that this negative result does not generalizes. In fact, we present a general strategy that allows two agents to locate the black hole with O(n) moves in common interconnection networks: hypercubes, cube‐connected cycles, star graphs, wrapped butterflies, chordal rings, as well as in multidimensional meshes and tori of restricted diameter. These results hold even if the networks are anonymous. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 47(2), 61–71 2006