Embedding Grids Research Articles

On Wavelet Decomposition of the Singular Splines

One of the approaches to the problem of approximating functions with a singularity is the creation of an approximating apparatus based on splines with the same feature. For the wavelet decomposition of spline spaces it is important that the property of the embedding of these spaces is associated with embedding grids. The purpose of this paper is to consider ways of constructing spaces of splines with a predefined singularity and obtain their wavelet decomposition. Here the concept of generalized smoothness is used, within which the mentioned singularity is generalized smooth. This approach leads to the construction of a system of embedded spaces on embedded grids. A spline-wavelet decomposition of mentioned spaces is presented. Reconstruction formulas are done

Computational fluid dynamics of stented intracranial aneurysms using adaptive embedded unstructured grids

AbstractRecently, there has been increased interest in the use of stents as flow diverters in the endovascular treatment of cerebral aneurysms as an alternative to surgical clipping or endovascular embolization with coils. The aim of aneurysm stenting is to block the flow into the aneurysm in order to clot the blood inside the aneurysm and effectively isolate it from the circulation and prevent bleeding from the aneurysm. A hybrid meshing approach that combines body‐fitted grids for the vessels and adaptive embedded grids for the stents is proposed and analyzed. This strategy simplifies considerably the geometry modeling problem and allows accurate patient‐specific hemodynamic simulations with endovascular devices. This approach is compared with the traditional body‐fitted approach in the case of the flow around a circular cylinder at representative Reynolds number and an idealized aneurysm model with a stent. A novel technique to map different stent designs to a given patient‐specific anatomical model is presented. The methodology is demonstrated on a patient‐specific hemodynamic model of an aneurysm of the internal carotid artery constructed from a 3D rotational angiogram and stented with two different stent designs. The results show that the methodology can be successfully used to model patient‐specific anatomies with different stents thereby making it possible to explore different stent designs. Copyright © 2007 John Wiley & Sons, Ltd.

Embedding grids in surfaces

We show that if a very large grid is embedded in a surface, then a large subgrid is embedded in a disc in the surface. This readily implies that: (a) a minor-minimal graph that does not embed in a given surface has no very large grid; and (b) a minor-minimal k-representative embedding in the surface has no very large grid. Similar arguments show (c) that if G is minimal with respect to crossing number, then G has no very large grid. This work is a refinement of Thomassen (J. Combin. Theory Ser. B 70 (1997) 306).

Crossing-number critical graphs have bounded path-width

The crossing number of a graph G , denoted by cr( G ), is defined as the smallest possible number of edge-crossings in a drawing of G in the plane. A graph G is crossing-critical if cr( G − e )<cr( G ) for all edges e of G . We prove that crossing-critical graphs have “bounded path-width” (by a function of the crossing number), which roughly means that such graphs are made up of small pieces joined in a linear way on small cut-sets. Equivalently, a crossing-critical graph cannot contain a subdivision of a “large” binary tree. This assertion was conjectured earlier by Salazar (J. Geelen, B. Richter, G. Salazar, Embedding grids on surfaces, manuscript, 2000).

Efficient embeddings of grids into grids

In this paper we explore one-to-one embeddings of two-dimensional grids into their ideal two-dimensional grids. The presented results are optimal or considerably close to the optimum. For embedding grids into grids of smaller aspect ratio, we prove a new lower bound on the dilation matching a known upper bound. The edge-congestion provided by our matrix-based construction differs from the here presented lower bound by at most one. For embedding grids into grids of larger aspect ratio, we establish five as an upper bound on the dilation and four as an upper bound on the edge-congestion, which are improvements over previous results.

Addendum to "Embedding Grids into Grids: Dilation Four Suffices", Parallel Processing Letters, Vol. 8, No. 2 (1998) 243-250

Addendum to "Embedding Grids into Grids: Dilation Four Suffices", Parallel Processing Letters, Vol. 8, No. 2 (1998) 243-250

Embedding Grids Into Grids: Dilation Four Suffices

We consider those instances of the problem of embedding 2-dimensional grid graphs into other 2-dimensional grids of smaller aspect ratio for which optimal solutions are as yet unknown. For those instances of the problem for which dilation two or dilation three solutions are not known, we show that none requires dilation more than four. The best upper bound previously published is six. We achieve our result by applying previously published methods, namely the so called compression and folding techniques. Optimal solutions (dilation two) are already known for all instances where the compression ratio (the ratio the width of the guest grid to the width of the host grid) is no more than two and for all instances where the compression ratio is sufficiently large. For many of the remaining instances, dilation three solutions are also known. Whether or not any instance really requires dilation greater than two remains an open question.

Embedding Cartesian Products of Graphs into de Bruijn Graphs

Given a Cartesian productG=G1× … ×Gm(m≥ 2) of nontrivial connected graphsGiand then-dimensional baseBde Bruijn graphD=DB(n), it is investigated whether or notGis a spanning subgraph ofD. Special attention is given to graphsG1× … ×Gmwhich are relevant for parallel computing, namely, to Cartesian products of paths (grids) or cycles (tori). For 2-dimensional de Bruijn graphsD, we present a theorem stating that certain structural assumptions on the factorsG1, …,Gmensure thatG1× … ×Gmis a spanning subgraph ofD. As corollaries, we obtain results improving previous results of Heydemannet al.(J. Parallel Distrib. Comput.23 (1994), 104–111) on embedding grids and tori into de Bruijn graphs. Specifically, we obtain that (i) any gridG=G1× … ×Gmis a spanning subgraph ofD=DB(2) provided that |G| = |D|, and (ii) any torusG=G1× … ×Gmis a spanning subgraph ofD=DB(2) provided that |G| = |D| and that theGiare cycles of even length ≥4. We show that these results have consequences for the casen> 2, too: for evenn, we apply our results to obtain embeddings of grids and toriGinto de Bruijn graphsDB(n) with dilationn/2, where the baseBis a fixed integer ≥2, andnis big enough to ensure |G| ≤ |DB(n)|. We also contrast our results forn= 2 with nonexistence results forn≥ 3 and briefly describe experimental results in the area of parallel image processing.

Embedding grids into grids: Techniques for large compression ratios

We return to the problem of embedding 2-dimensional (h x w) guest grid graphs into 2-dimensional (h' X w') host grid graphs, where w' < w, and h' is the smallest integer such that hw < h'w'. This 2-dimensional problem has many applications in computer science including the simulation of one grid of processors by another of different shape. Also, it can likely be applied to more complex problems such as those involving grids of higher dimension and hypercubes. It is already known that optimal dilation, namely two, is always obtainable whenever w/w', the compression ratio, is no greater than two. Only a restricted set of problem instances with compression ratios greater than two could previously be solved optimally, with respect to dilation. We introduce two new techniques, called folding with compression and folding with extension. By applying folding with compression, we show that optimal solutions always exist if w' ≥ 7h and the remainder of h' integer divided by h is no greater than h' integer divided by h. This condition includes all instances of the problem for which w/w' ≥ h ≥ 7, i.e., all instances for which the compression ratio and h are sufficiently large. There remain instances, among those of intermediate compression ratio, for which we still do not have optimal solutions. We show that some, but not all, of these can be solved by appling folding with extension.

Embedding grids into hypercubes

We consider efficient simulations of mesh connected networks (or good representations of array structures) by hypercube machines. In particular, we consider embedding a mesh or grid G into the smallest hypercube that at least as many points as G, called the optimal hypercube for G. In order to minimize simulation time we derive embeddings which minimize dilation, i.e., the maximum distance in the hypercube between images of adjacent points of G. Our results are: (1) There is a dilation 2 embedding of the [ m × k] grid into its optimal hypercube, provided that ⌈ logm⌉+ log mk 2 ⌈ logm⌉ + logm 2 ⩽⌈ logmk⌉ and (2) For any k < d, there is a dilation k + 1 embedding of a [ a 1 × a 2 × … × a d ] grid into its optimal hypercube, provided that Σ i=1 d − 1 ⌈log B k ⌉ ≤ ⌈ Σ i=1 d log a i ⌉, where B k= a d∏ k i=1a i ∏ k i=12 ⌈ loga i⌉ + ∑ i=1 k ⌊ loga 1⌋ 2 ⌉

Calculation of laminar recirculating flows using a local non‐staggered grid refinement system

AbstractA grid‐embedding technique for the solution of two‐dimensional incompressible flows governed by the Navier‐Stokes equations is presented. A finite volume method with collocated primitive variables is employed to ensure conservation at the interfaces of embedding grids as well as global conservation. The discretized equations are solved simultaneously for the whole domain, providing a strong coupling between regions of different refinement. The formulation presented herein is applicable to uniform or non‐uniform Cartesian meshes. The method was applied to the solution of two scalar transport equations, to cavity flows driven by body and shear forces and to a sudden plane contraction flow. The numerical predictions are compared with the exact solutions when available and with experimental data. The results show that neither the convergence rate nor the stability of the method is affected by the presence of embedded grids. Embedded grids provide a better distribution of grid nodes over the computational domain and consequently the solution accuracy was improved. The grid‐embedding technique proved also that significant savings in computing time could be achieved.

Measurement of three-dimensional Mode-I stress intensity factor using multiple embedded grid moire technique

Multiple embedded grid moire and strain gage techniques are used to calculate the variation in Mode-I stress intensity factor throughout the thickness of ASTM E-399 standard compact tension specimens of Merlon polycarbonate. The specimen grids near the crack tip on the surface and in the interior were recorded for the unloaded condition and for various loaded states using high resolution photographic techniques. Optical processing produced moire patterns from which the change in the crack opening displacement could be determined. From the information about the crack opening displacement in the region of the crack tip, the Mode-I stress intensity factor was calculated for various interior planes and on the surface of the specimen. The stress intensity factor was found to be higher in the midplane than on the surface, and it causes crack initiation to start at the midplane.