Rectilinear Case Research Articles

Influence of rectilinear vs radial advection on the yield of A + B → C reaction fronts: A comparison.

In the presence of advection at a constant flow rate in a rectilinear geometry, the properties of planar A + B → C reaction fronts feature the same temporal scalings as in the pure reaction-diffusion case. In a radial injection geometry where A is injected into B radially at a constant flow rate Q, temporal scalings are conserved, but the related coefficients depend on the injection flow rate Q and on the ratio γ of initial concentrations of the reactants. We show here that this dependence of the front properties on the radial velocity allows us to tune the amount of product obtained in the course of time by varying the flow rate. We compare theoretically the efficiency of the rectilinear and radial geometries by computing the amount of product C generated in the course of time or per volume of reactant injected. We show that a curve γc(Q) can be defined in the parameter space (γ, Q) below which, for similar experimental conditions, the total amount of C is larger in the radial case. In addition, another curve γ*(Q) < γc(Q) can be defined such that for γ < γ*, the total amount of C produced is larger in the radial geometry, even if the production of C per unit area of the contact interface between the two reactants is larger in the rectilinear case. This comes from the fact that the length of the contact zone increases with the radius in the radial case, which allows us to produce in fine more product C for a same injected volume of reactant or in reactors of a same volume than in the rectilinear case. These results pave the way to the geometrical optimization of the properties of chemical fronts.

Modeling, Simulation and Control of the Walking of Biped Robotic Devices—Part III: Turning while Walking

In part II of this group of papers, the control of the gait of a biped robot during rectilinear walk was considered. The modeling approach and simulation, using Kane’s method with implementation leveraged by Autolev, a symbolic computational environment that is complementary, was discussed in part I. Performing turns during the walk is technically more complex than the rectilinear case and deserves further investigation. The problem is solved in the present part III as an extension of part II. The robot executes a rectilinear walk on a local reference frame whose progression axis is always tangent, and its origin performs the involute of the path curve. The curve is defined by its curvature (osculating circle) and center of curvature (evolute) along the path. Radius of curvature and center can change continuously (in practice at every sampling time). For postural equilibrium, Center of Gravity and Zero Moment Point (COG/ZMP) follow the same preview reference proposed for rectilinear walk (c o g R e f x ( t ) , c o g ˙ R e f x ( t ), c o g R e f y ( t ) , c o g ˙ R e f y ( t )). The effect of the turn on the sagittal plane is negligible and is ignored, while on the frontal plane it is accounted for by an offset on COG reference to compensate for the centrifugal acceleration. The body trunk and local frame rotation, and the generation of the references on this moving frame of the free foot trajectory during the swing deserve attention.

Stability results for multi-layer radial Hele-Shaw and porous media flows

Motivated by stability problems arising in the context of chemical enhanced oil recovery, we perform linear stability analysis of Hele-Shaw and porous media flows in radial geometry involving an arbitrary number of immiscible fluids. Key stability results obtained and their relevance to the stabilization of fingering instability are discussed. Some of the key results, among many others, are (i) absolute upper bounds on the growth rate in terms of the problem data; (ii) validation of these upper bound results against exact computation for the case of three-layer flows; (iii) stability enhancing injection policies; (iv) asymptotic limits that reduce these radial flow results to similar results for rectilinear flows; and (v) the stabilizing effect of curvature of the interfaces. Multi-layer radial flows have been found to have the following additional distinguishing features in comparison to rectilinear flows: (i) very long waves, some of which can be physically meaningful, are stable; and (ii) eigenvalues can be complex for some waves depending on the problem data, implying that the dispersion curves for one or more waves can contact each other. Similar to the rectilinear case, these results can be useful in providing insight into the interfacial instability transfer mechanism as the problem data are varied. Moreover, these can be useful in devising smart injection policies as well as controlling the complexity of the long-term dynamics when drops of various immiscible fluids intersperse among each other. As an application of the upper bound results, we provide stabilization criteria and design an almost stable multi-layer system by adding many layers of fluid with small positive jumps in viscosity in the direction of the basic flow.

On the nearest neighbor rule for the metric traveling salesman problem

We present a very simple family of traveling salesman instances with n cities where the nearest neighbor rule may produce a tour that is Θ(logn) times longer than an optimum solution. Our family works for the graphic, the euclidean, and the rectilinear traveling salesman problem at the same time. It improves the so far best known lower bound in the euclidean case and proves for the first time a lower bound in the rectilinear case.

Minimum-link paths revisited

A path or a polygonal domain is C-oriented if the orientations of its edges belong to a set of C given orientations; this is a generalization of the notable rectilinear case (C=2). We study exact and approximation algorithms for minimum-link C-oriented paths and paths with unrestricted orientations, both in C-oriented and in general domains.Our two main algorithms are as follows:A subquadratic-time algorithm with a non-trivial approximation guarantee for general (unrestricted-orientation) minimum-link paths in general domains.An algorithm to find a minimum-link C-oriented path in a C-oriented domain. Our algorithm is simpler and more time-space efficient than the prior algorithm.We also obtain several related results:•3SUM-hardness of determining the link distance with unrestricted orientations (even in a rectilinear domain).•An optimal algorithm for finding a minimum-link rectilinear path in a rectilinear domain. The algorithm and its analysis are simpler than the existing ones.•An extension of our methods to find a C-oriented minimum-link path in a general (not necessarily C-oriented) domain.•A more efficient algorithm to compute a 2-approximate C-oriented minimum-link path.•A notion of “robust” paths. We show how minimum-link C-oriented paths approximate the robust paths with unrestricted orientations to within an additive error of 1.

Capacity inverse minimum cost flow problem

Given a directed graph G = (N,A) with arc capacities u and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector u' for the arc set A such that a given feasible flow x' is optimal with respect to the modified capacities. Among all capacity vectors u' satisfying this condition, we would like to find one with minimum ||u' - u|| value. We consider two distance measures for ||u' - u||, rectilinear and Chebyshev distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP-hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.

Analytical results for 2‐D non‐rectilinear waveguides based on a Green's function

AbstractWe consider the problem of wave propagation for a 2‐D rectilinear optical waveguide which presents some perturbation. We construct a mathematical framework to study such a problem and prove the existence of a solution for the case of small imperfections. Our results are based on the knowledge of a Green's function for the rectilinear case. Copyright © 2008 John Wiley &amp; Sons, Ltd.

A nonlinear fourth order diffusion problem: Convergence to the steady state and non-negativity of solutions

The paper deals with nonlinear diffusion, both time-dependent and time-independent. The spatial terms in the partial differential equation (p.d.e.) contain a second order nonlinear part (where the non-negative diffusivity depends on the dependent variable) and a fourth order linear part. In the context of non-null, time-independent boundary conditions, convergence of the unsteady to the steady state is established. The second part of the paper discusses criteria on data ensuring non-negativity of the solutions. This is done for the steady state irrespective of the spatial dimension; and it is done for the unsteady state for the one-dimensional rectilinear case only, using a result from the first part of the paper.

Speeding up the Dreyfus–Wagner algorithm for minimum Steiner trees

The Dreyfus–Wagner algorithm is a well-known dynamic programming method for computing minimum Steiner trees in general weighted graphs in time O *(3 k ), where k is the number of terminal nodes to be connected. We improve its running time to O *(2.684 k ) by showing that the optimum Steiner tree T can be partitioned into T = T 1∪ T 2 ∪ T 3 in a certain way such that each T i is a minimum Steiner tree in a suitable contracted graph G i with less than $${\frac{k}{2}}$$ terminals. In the rectilinear case, there exists a variant of the dynamic programming method that runs in O *(2.386 k ). In this case, our splitting technique yields an improvement to O *(2.335 k ).

Steiner Minimal Trees in Rectilinear and Octilinear Planes

This paper considers the Steiner Minimal Tree (SMT) problem in the rectilinear and octilinear planes. The study is motivated by the physical design of VLSI: The rectilinear case corresponds to the currently used M-architecture, which uses either horizontal or vertical routing, while the octilinear case corresponds to a new routing technique, X-architecture, that is based on the pervasive use of diagonal directions. The experimental studies show that the X-architecture demonstrates a length reduction of more than 10–20%. In this paper, we make a theoretical study on the lengths of SMTs in these two planes. Our mathematical analysis confirms that the length reduction is significant as the previous experimental studies claimed, but the reduction for three points is not as significant as for two points. We also obtain the lower and upper bounds on the expected lengths of SMTs in these two planes for arbitrary number of points.

Spanning graph-based nonrectilinear steiner tree algorithms

With advances in fabrication technology of very/ultra large scale integrated circuit (VLSI/ULSI), we must face many new challenges. One of them is the interconnect effects, which may cause longer delay and heavier crosstalk. To solve this problem, many interconnect performance optimization algorithms have been proposed. However, when these algorithms are designed based on rectilinear interconnect architecture, the optimization capability is limited. Therefore, nonrectilinear interconnect architectures become a field of active research in which the octilinear interconnect architecture is the most promising one since it extends from the rectilinear case and greatly shortens the wire length. Meanwhile, an interconnect with less length is helpful to reduce wire capacitance, congestion, and routing area. In an interconnect architecture, the Steiner minimal tree (SMT) construction is one of the key problems. In this paper, we give two practical octilinear Steiner minimal tree (OSMT) construction algorithms based on octilinear spanning graphs (OSGs). The one with edge substitution (OST-E) has a worst-case running time of O(nlogn) and a similar performance as the recent work using batched greedy. The other one with triangle contraction (OST-T) has a small increase in the constant factor of running time and a better performance. These two are the fastest algorithms for octilinear Steiner tree construction so far. Experiments on both industrial and random test cases are conducted to compare with other programs. We also proposed the extension of our algorithms to any /spl lambda/ geometry.

Rotationally optimal spanning and Steiner trees in uniform orientation metrics

We consider the problem of finding a minimum spanning and Steiner tree for a set of n points in the plane where the orientations of edge segments are restricted to λ uniformly distributed orientations, λ=2,3,4,… , and where the coordinate system can be rotated around the origin by an arbitrary angle. The most important cases with applications in VLSI design arise when λ=2 or λ=4. In the former, so-called rectilinear case, the edge segments have to be parallel to one of the coordinate axes, and in the latter, so-called octilinear case, the edge segments have to be parallel to one of the coordinate axes or to one of the lines making 45° with the coordinate axes (so-called diagonals). As the coordinate system is rotated—while the points remain stationary—the length and indeed the topology of the minimum spanning or Steiner tree changes. We suggest a straightforward polynomial-time algorithm to solve the rotational minimum spanning tree problem. We also give a simple algorithm to solve the rectilinear Steiner tree problem in the rotational setting, and a finite time algorithm for the general Steiner tree problem with λ uniform orientations. Finally, we provide some computational results indicating the average savings for different values of n and λ both for spanning and Steiner trees.

Planar cracks running along piecewise linear paths

Consider a crack propagating in a plane according to a loading that results in anti-plane shear deformation. We show here that if the crack path consists of two straight segments making an angle psi not equal 0 at their junction, examples can be provided for which the value of the stress-intensity factor (SIF) actually depends on the previous history of the motion. This is in sharp contrast with the rectilinear case (corresponding to psi = 0), where the SIF is known to have a local character, its value depending only on the position and velocity of the crack tip at any given time.

Stability of screening solitons in photorefractive media.

Normal mode stability of both rectilinear and self-bending photorefractive screening solitons is considered. In each case, the Evans function procedure is used to investigate stability and to search for internal modes. For the rectilinear case, a standard Evans function procedure is applied. However, in the self-bending case the asymptotic form of the eigenvalue problem is a system of Airy equations, instead of the usual system of constant coefficient differential equations. To overcome this difference, a modified version of the Evans function method, using Airy functions rather than exponentials, is implemented and applied. The results confirm stability and give an internal mode pattern in good agreement with full numerical integration.

How to learn an unknown environment. I

We consider the problem faced by a robot that must explore and learn an unknown room with obstacles in it. We seek algorithms that achieve a bounded ratio of the worst-case distance traversed in order to see all visible points of the environment (thus creating a map), divided by the optimum distance needed to verify the map, if we had it in the beginning. The situation is complicated by the fact that the latter off-line problem (the problem of optimally verifying a map) is NP-hard. Although we show that there is no such “competitive” algorithm for general obstacle courses, we give a competitive algorithm for the case of a polygonal room with a bounded number of obstacles in it. We restrict ourselves to the rectilinear case, where each side of the obstacles and the room is parallel to one of the coordinates, and the robot must also move either parallel or perpendicular to the sides. (In a subsequent paper, we will discuss the extension to polygons of general shapes.) We also discuss the off-line problem for simple rectilinear polygons and find an optimal solution (in the L 1 metric) in polynomial time, in the case where the entry and the exit are different points.

An optimal algorithm for the rectilinear link center of a rectilinear polygon

The link distance between two points in a polygon P is defined as the minimum number of line segments inside P needed to connect the two points. The link center of a polygon P is the set of points in P which minimizes the maximum link distance to all points in P. The problem of finding the link center of a simple polygon with n edges has been studied extensively in recent years. Several O( nlog n) time algorithms have been given for this problem. We consider the rectilinear case of this problem and give a linear time algorithm to compute the rectilinear link center of a simple rectilinear polygon.

Reductions for the rectilinear steiner tree problem

AbstractThe rectilinear Steiner tree problem asks for a shortest network connecting n terminals in the rectilinear plane (with L1‐metric). The problem can be considered as a special case of the Steiner tree problem in a grid network obtained by drawing horizontal and vertical lines through terminals. The reductions developed for the network variant of the Steiner tree problem can therefore also be used in the rectilinear case. However, computational experience indicates that they are not very powerful in the rectilinear grids. We introduce a limited number of rectilinear reductions based on the geometrical properties of terminals. Experimental results show that these reductions are extremely powerful while being quite simple to verify and easy to implement. For randomly generated problem instances with one terminal per row and column, The number of nonterminals remaining seems to be around 20‐‐25% of their original number. Furthermore, The performance of geometrical reductions improves as the density of terminals in fixed‐size grids grows.

Mixed finite element methods for elastic rods of arbitrary geometry

The boundary-value problem for rods having arbitrary geometry, and subjected to arbitrary loading, is studied within the context of the small-strain theory. The basic assumptions underlying the rod kinematics are those corresponding to the Timoshenko hypotheses in the plane rectilinear case: that is, plane sections normal to the line of centroids in the undeformed state remain plane, but not necessarily normal. The problem is formulated in both the standard and mixed variational forms, and after establishing the existence and uniqueness of solutions to these equivalent problems, the corresponding discrete problems are studied. Finite element approximations of the mixed problem are shown to be stable and convergent. It is shown that the equivalence between the mixed problem and the standard problem with selective reduced integration holds only for the case of rods having constant curvature and torsion, though. The results of numerical experiments are presented; these confirm the convergent behaviour of the mixed problem.

Heuristics and bounds for the travelling salesman location problem on the plane

In this paper we present heuristics for the travelling salesman location problem on the plane with rectilinear or Euclidean distances. For the rectilinear case we show a polynomial heuristic which produces solutions that are at most 50% worse than the optimal solution. For the Euclidean case we present two polynomial heuristics which have a worst-case relative error, respectively, of 0.5 + ϵ and √2 ( 3 2 ) − 1 .

On some union and intersection problems for polygons with fixed orientations

Objects with fixed orientations play an important role in many application areas, for instance VLSI design. Problems involving only rectilinearly oriented (rectangular) objects, as a simplest case, have been studied with the VLSI design application in mind. These objects can be transistors, cells or macros. In reality, they are more suitably represented by polygons rather than just rectangles. In this note we describe how to perform a general decomposition of a set of polygons with fixed orientations in order to solve various computational geometry problems which are important in VLSI design. The decomposition is very simple and efficiently computable, and it allows the subsequent application of algorithms for the rectilinear case, leading to some very efficient and some optimal solutions. We illustrate the technique in detail at the problem of finding the connected components of a set of polygons, for which we derive an optimal solution. The wide applicability of the method is then demonstrated at the problem of finding all pairs of intersecting polygons, yielding an optimal solution.