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Piecewise Linear Functions (piecewise + linear_function)

Distribution by Scientific Domains
Engineering66%
Mathematics and Statistics33%

Selected Abstracts

Toward faster algorithms for dynamic traffic assignment.

NETWORKS: AN INTERNATIONAL JOURNAL, Issue 1 2003

Abstract Being first in a three-part series promising a practical solution to the user-equilibrium dynamic traffic assignment problem, this paper devises a parametric quickest-path tree algorithm, whose model makes three practical assumptions: (i) the traversal time of an arc i , j is a piecewise linear function of the arrival time at its i -node; (ii) the traversal time of a path is the sum of its arcs' traversal times; and (iii) the FIFO constraint holds, that is, later departure implies later arrival. The algorithm finds a quickest path, and its associated earliest arrival time, to every node for every desired departure time from the origin. Its parametric approach transforms a min-path tree for one departure-time interval into another for the next adjacent interval, whose shared boundary the algorithm determines on the fly. By building relatively few trees, it provides the topology explicitly and the arrival times implicitly of all min-path trees. Tests show the algorithm running upward of 10 times faster than the conventional brute-force approach, which explicitly builds a min-path tree for every departure time. Besides dynamic traffic assignment, other applications for which these findings have utility include traffic control planning, vehicle routing and scheduling, real-time highway route guidance, etc. © 2002 Wiley Periodicals, Inc. [source]

Ground response curves for rock masses exhibiting strain-softening behaviour

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 13 2003
E. Alonso
Abstract A literature review has shown that there exist adequate techniques to obtain ground reaction curves for tunnels excavated in elastic-brittle and perfectly plastic materials. However, for strain-softening materials it seems that the problem has not been sufficiently analysed. In this paper, a one-dimensional numerical solution to obtain the ground reaction curve (GRC) for circular tunnels excavated in strain-softening materials is presented. The problem is formulated in a very general form and leads to a system of ordinary differential equations. By adequately defining a fictitious ,time' variable and re-scaling some variables the problem is converted into an initial value one, which can be solved numerically by a Runge,Kutta,Fehlberg method, which is implemented in MATLAB environment. The method has been developed for various common particular behaviour models including Tresca, Mohr,Coulomb and Hoek,Brown failure criteria, in all cases with non-associative flow rules and two-segment piecewise linear functions related to a principal strain-dependent plastic parameter to model the transition between peak and residual failure criteria. Some particular examples for the different failure criteria have been run, which agree well with closed-form solutions,if existing,or with FDM-based code results. Parametric studies and specific charts are created to highlight the influence of different parameters. The proposed methodology intends to be a wider and general numerical basis where standard and newly featured behaviour modes focusing on obtaining GRC for tunnels excavated in strain-softening materials can be implemented. This way of solving such problems has proved to be more efficient and less time consuming than using FEM- or FDM-based numerical 2D codes. Copyright © 2003 John Wiley & Sons, Ltd. [source]

A new stable space,time formulation for two-dimensional and three-dimensional incompressible viscous flow

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2001
Donatien N'dri
Abstract A space,time finite element method for the incompressible Navier,Stokes equations in a bounded domain in ,d (with d=2 or 3) is presented. The method is based on the time-discontinuous Galerkin method with the use of simplex-type meshes together with the requirement that the space,time finite element discretization for the velocity and the pressure satisfy the inf,sup stability condition of Brezzi and Babu,ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two-dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd. [source]

How to represent continuous piecewise linear functions in closed form

INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, Issue 6 2006
Robert Lum
Abstract This paper presents a simple approach to representing continuous piecewise linear functions in closed form for arbitrary dimensions. The absolute value function is used to embed non-linearity, its usage increasingly nested as the dimension is increased. Copyright © 2006 John Wiley & Sons, Ltd. [source]

On a quadrature algorithm for the piecewise linear wavelet collocation applied to boundary integral equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2003
Andreas Rathsfeld
Abstract In this paper, we consider a piecewise linear collocation method for the solution of a pseudo-differential equation of order r=0, ,1 over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points. For the wavelet basis in the trial space we choose the three-point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. For the corresponding wavelet algorithm, we show that the parametrization can be approximated by low-order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are composite rules of simple low-order quadratures. The whole algorithm for the assembling of the matrix requires no more than O(N [logN]3) arithmetic operations, and the error of the collocation approximation, including the compression, the approximative parametrization, and the quadratures, is less than O(N,(2,r)/2). Note that, in contrast to well-known algorithms by Petersdorff, Schwab, and Schneider, only a finite degree of smoothness is required. In contrast to an algorithm of Ehrich and Rathsfeld, no multiplicative splitting of the kernel function is required. Beside the usual mapping properties of the integral operator in low order Sobolev spaces, estimates of Calderón,Zygmund type are the only assumptions on the kernel function. Copyright © 2003 John Wiley & Sons, Ltd. [source]