Showing papers on "Piecewise published in 1977"
TL;DR: Two methods of constructing piecewise quadratic approximations are described which have the property that, if they are applied on each triangle of a triangulation, then ~(x, y) and its first derivatives are continuous everywhere.
Abstract: The problem of constructing a function 4,(x, y) of two variables on a triangle, such that ~(x, y) and its first derivatives take given values at the vertices, where ¢(x, y) is composed of quadratic pieces, is considered. Two methods of constructing piecewise quadratic approximations are described which have the property that, if they are applied on each triangle of a triangulation, then ~(x, y) and its first derivatives are continuous everywhere.
425 citations
TL;DR: In this article, eight FORTRAN subprograms for dealing computationally with piecewise polynomial functions (of one variable) are presented, built around an algorithm for the stable evaluation of B-sp...
Abstract: Eight FORTRAN subprograms for dealing computationally with piecewise polynomial functions (of one variable) are presented. The package is built around an algorithm for the stable evaluation of B-sp...
198 citations
01 Jun 1977
TL;DR: The paper presents a method of numerically integrating a system of differential equations based on an idea of orthogonal approximation of functions that gives piecewise constant solutions with minimal mean-square error and is computationally similar to the familiar trapezoidal rule of integration.
Abstract: The paper presents a method of numerically integrating a system of differential equations based on an idea of orthogonal approximation of functions. Here, block-pulse functions are chosen as the orthogonal set. The method gives piecewise constant solutions with minimal mean-square error and is computationally similar to the familiar trapezoidal rule of integration. Design of piecewise constant controls or feedback gains for dynamic systems can be simplified following this approach.
160 citations
TL;DR: In this paper, the electron density of an atom in its ground state is piecewise exponentially decreasing as a function of the distance from the nucleus, and it is shown that much improved values for the energy, and electron densities exhibiting shell structure, are obtained from optimization of the density functionals of Thomas-Fermi, Thomas- Fermi-Dirac, and Thomas FermisDirac with inhomogeneity correction.
Abstract: If it is assumed that the electron density of an atom in its ground state is piecewise exponentially decreasing as a function of the distance from the nucleus, then it is shown that much improved values for the energy, and electron densities exhibiting shell structure, are obtained from optimization of the density functionals of Thomas-Fermi, Thomas-Fermi-Dirac, and Thomas-Fermi-Dirac with inhomogeneity correction. An inhomogeneity correction which is one-ninth of the original Weizsacker correction is favored. Numerical results are presented for all first-row atoms and selected second-row atoms, and comparisons are made with results of other methods.
128 citations
TL;DR: In this paper, a finite element formulation is described for problems with solution functions known to have local rλ variation (s), 0<λ<1, and thus singular gradients. But the conditions of continuity, low order solution capability, and accurate numerical integration of the singularity element are discussed with a view towards establishing the general range of applicability.
Abstract: A finite element formulation is described for problems with solution functions known to have local rλ variation (s), 0<λ<1, and thus singular gradients. Special 3-node triangular elements encircle the singularity and focus to share a common node at the singular point. The shape function of each triangle has the appropriate r λ mode and a smooth angular mode expressed in element natural co-ordinates. As with standard elements, the unknowns are the nodal values of the function. Even if the precise angular form of the asymptotic solution is known, the formulation makes no attempt to embed it, but instead piecewise approximates it. This allows assembly of the element coefficient matrix using standard procedures without nodeless variables and bandwidth complications.
The conditions of continuity, low order solution capability, and accurate numerical integration of the singularity element are discussed with a view towards establishing the general range of applicability of the formulation. Numerical applications to the elastic fracture mechanics problems of composite bondline cracking and crack branching are discussed.
109 citations
TL;DR: In this article, it is shown that broken line interpolation as a scheme for piecewise monotone interpolation is hard to improve upon, and that a family of smooth piecewise polynomial interpolants, introduced by Swartz and Varga and noted by Passow to be piecewise non-linear, converges to a piecewise constant interpolant as the degree goes to infinity.
74 citations
01 Jun 1977
TL;DR: Two methods are introduced for reducing additive noise in approximately piecewise constant images and the first is a relaxation labeling formulation which can be psed in cases of more severe noise.
Abstract: Two methods are introduced for reducing additive noise in approximately piecewise constant images. In the first method, the correct gray level of a point P is estimated by a weighted average of the gray levels of P's immediate neighbors. To assign these weights for a given P so that the local averaging operation does not blur the image, the presence of any edges or lines passing through P's neighborhood must be determined. For example, when an edge is present, only those points on P's side of the edge should be used. In general, the distribution of gray levels in P's neighborhood will suggest many possible line and edge configurations which can be used to compute the weight distribution for the averaging operation. The averaging is performed in parallel over the entire image and is iterated for additional smoothing. The second method is a relaxation labeling formulation which can be psed in cases of more severe noise. Each point is now assigned a vector of probabilities over its allowable gray level set. These probabilities are updated as weighted averages of the gray level probabilities at neighboring points. Once again the presence of lines and edges determines how the contribution from each neighboring point is weighted. Examples demonstrating both methods are given.
74 citations
TL;DR: In this paper, the Hartree-Fock energy for the 1s orbital of helium is derived, and the accuracy depends on the number of points N+1 and the polynomial order 2s+1.
Abstract: Piecewise polynomials are examined as basis functions for electronic wavefunctions. The spline function method is a special case, which is shown to be less accurate, for a fixed set of mesh points, than a method based directly on Hermite’s interpolation formula. The determination of a suitable mesh is discussed both inductively and deductively, and a logarithmic formula for the 1s orbital of helium is ’’derived.’’ The accuracy is shown to depend on the number of points N+1 and on the polynomial order 2s+1, approximately according to the formula, δE∼N−4s−2, for appropriate meshes. A striking result is the possibility for systematically increasing the accuracy of the energy by systematically increasing the number of points, without encountering linear dependence problems, is demonstrated by calculations on the helium atom. With a 16‐point theoretically derived mesh, and with seventh order polynomials, we obtain a Hartree–Fock energy for helium of −2.8616799956122 a.u.
66 citations
TL;DR: In this article, an error estimate for an incremental finite element method for plasticity with hardening is presented, where stresses and displacements are approximated by piecewise constant and piecewise linear functions, respectively.
Abstract: We prove an error estimate for an incremental finite element method for plasticity with hardening. Stresses and displacements are approximated by piecewise constant and piecewise linear functions, respectively.
64 citations
TL;DR: The algorithm is shown to be stable when a piecewise cubic polynomial is used which is continuous with its first derivative, while it becomes unstable if the polynometric is made continuous up to the second derivative.
Abstract: A method is described for fitting a piecewise cubic polynomial to a sequence of data by a onepass method. The polynomial pieces are calculated as the data is scanned only once from left to right. The algorithm is shown to be stable when a piecewise cubic polynomial is used which is continuous with its first derivative, while it becomes unstable if the polynomial is made continuous up to the second derivative. The knots of the approximating function are determined successively using a criterion by Powell.
46 citations
TL;DR: In this paper, a new state space for large interconnected power systems is introduced, in which the state of the system can be estimated piecewise by local computations from conventional local measurements.
Abstract: A new state space is introduced for the large interconnected power system in which the state of the system can be estimated piecewise by local computations from conventional local measurements. The new state space is defined, its mathematical equivalence to the conventional state space is proven, and initial illustration is provided of its usefulness.
TL;DR: In this article, a variational approach to whole-Earth dynamics is formulated and implemented with piecewise Hermite cubic splines as trial functions, which can be generalized to include ellipticity and centrifugal coupling, making the full Earth oscillation problem tractable in a single calculation.
Abstract: Summary
A variational approach to whole-Earth dynamics is formulated and implemented with piecewise Hermite cubic splines as trial functions. A fast computational algorithm is developed which allows rotational Coriolis coupling to be taken fully into account for the first time. The method can be generalized to include ellipticity and centrifugal coupling, making the full Earth oscillation problem tractable in a single calculation. This permits the investigation of free elastic modes, fluid core inertial modes and undertone modes associated with any stable regions of the outer core, as well as modes involving wobble, fluctuations in spin rate and displacement of the solid inner core.
The present paper expounds the method and illustrafes its application to the computation of undertone modes previously estimated by direct integration of the equations of motion.
TL;DR: In this paper, the tanh rule for numerical integration is analyzed in the context of the Hardy space, and the optimal parameter choice is determined, and it is shown that the norm of the error functional is asymptotic to $C\exp ( - ({\pi / 2})\sqrt M )$, where M is the number of points used and C is a certain constant.
Abstract: The tanh rule for numerical integration is analyzed in the context of the Hardy space $H^2 $. The optimal parameter choice is determined, and it is shown that the norm of the error functional is asymptotic to $C\exp ( - ({\pi / 2})\sqrt M )$, where M is the number of points used and C is a certain constant. The result is related to recent theorems on approximation of piecewise analytic functions by rational functions.
TL;DR: In this article, a piecewise linear approximation to an arbitrary yield function is constructed in the course of solving a given boundary value problem, and its use is illustrated with some simple examples.
TL;DR: In this paper, a collocation-Galerkin method for the two point boundary value problem based on continuous piecewise polynomial spaces is defined where the collocation points are the roots of a Jacobi Polynomial.
Abstract: A collocation–Galerkin method for the two point boundary value problem based on continuous piecewise polynomial spaces is defined where the collocation points are the roots of a Jacobi polynomial.An existence and uniqueness theorem is derived by means of a related variational procedure that is defined using a semi-discrete innerproduct.Optimal rates of convergence are established and $O(h^{2r} )$ order of superconvergence at the nodes is obtained. Additional approximations to the solution and its derivatives that obtain $O(h^{2r} )$ order of superconvergence at any point are described.
TL;DR: In this article, the problem is modified to one in which the minimum is sought of a functional defined on a set of Radon measures, and conditions are given under which a minimizing measurable control exists for the unmodified problem.
Abstract: The existence is considered of a boundary control which drives a system governed by the one-dimensional diffusion equation from the zero state to a given final state, and at the same time minimizes a given functional. The problem is first modified to one in which the minimum is sought of a functional defined on a set of Radon measures. The existence of a minimizing measure is demonstrated, and it is shown that this measure may be approximated by a piecewise constant control. Finally, conditions are given under which a minimizing measurable control exists for the unmodified problem.
01 Jan 1977
TL;DR: A new system of peak component recognition and measurement in digitized waveforms is detailed and an electrocardiogram is employed as an example waveform to demonstrate the bipolar algorithm, which runs with sufficient speed to allow real-time processing.
Abstract: A new system of peak component recognition and measurement in digitized waveforms is detailed. Two input parameters identify waveform context (scale and noise content), and a third specifies baseline tolerance (if applicable). The input waveform is preprocessed by a discrete linear piecewise approximation algorithm yielding a segmentation in endpoint/slope/constant format. Slope values are encoded as symbols of a string which is parsed by a syntax-directed finite-state automaton. One of three different machines and sets of semantic routines is chosen depending upon whether the waveform is baseline-free, unipolar, or bipolar. In the last case, the endpoint values relative to the baseline are encoded as symbols of a second string which modifies the action of the machine. An electrocardiogram is employed as an example waveform to demonstrate the bipolar algorithm, which runs with sufficient speed to allow real-time processing. A proposed on-line implementation of the system is outlined.
TL;DR: In this paper, the authors extended the work of de Boor and Swartz (SIAM J. Numer. Anal., 10 (1973), pp. 582-606) on the solution of two-point boundary value problems by collocation.
Abstract: In this note we extend the work of de Boor and Swartz (SIAM J. Numer. Anal., 10 (1973), pp. 582-606) on the solution of two-point boundary value problems by collocation. In particular, we are concerned with boundary value problems described by integro-differential equations involving derivatives of order up to and including m with m boundary conditions. We study the approximation of (isolated) solutions by means of piecewise polynomial functions of degree less than m + k possessing m -1 continuous derivatives. If the problem is sufficiently smooth and the solution has m +2k continuous derivatives, then one can achieve O(IAlk+m) global convergence by collocating at the zeros of the kth Legendre polynomial relative to each subinterval. At the knots, the approximation and its first m - 1 derivatives are O(IA12k) accurate.
TL;DR: A new descent-type computational algorithm is derived that obtains the unique solution to the minimization of the multihour cost function, which is strictly convex but only piecewise differentiable.
Abstract: Multihour engineering is a technique for designing trunk networks when the hours of peak traffic loads between various pairs of offices do not coincide. A new descent-type computational algorithm for the multihour engineering problem is derived. This algorithm obtains the unique solution to the minimization of the multihour cost function, which is strictly convex but only piecewise differentiable. The noninteger minimum-cost solution is subsequently rounded to the nearest allowable integer solution to give a realizable network. The new algorithm is applied to three numerical examples from the California network. The results are compared with the nonoptimal, nonunique solutions obtained with an earlier algorithm, and with the traditional single-hour solutions.
Book•
01 Jan 1977
TL;DR: In this paper, a valve in which a valve member is mounted for rotary valving motion within a valve casing and an actuator in which at least one pair of expansible bellows engaging opposite sides of a vane and acting in opposition one to the other give rise to rotary motion is described.
Abstract: A valve in which a valve member is mounted for rotary valving motion within a valve casing and an actuator in which at least one pair of expansible bellows engaging opposite sides of a vane and acting in opposition one to the other give rise to rotary motion. The bellows preferably are free from any direct fixation to the vane or the like, and have a particular configuration as described.
TL;DR: In this paper, piecewise polynomial approximations generalizing the Kang and Hansen Hermite methods are applied to the nuclear reactor kinetics equations, and some of the methods proposed are strongly A-stable and therefore quite efficient in the presence of extreme stiffness.
Abstract: Piecewise polynomial approximations generalizing the Kang and Hansen Hermite methods are applied to the nuclear reactor kinetics equations. Some of the methods proposed here are strongly A-stable and therefore quite efficient in the presence of extreme stiffness, as shown by numerical results for some typical test cases.
TL;DR: In this paper, the authors analyzed the effect of singular functions in the neighborhood of each angular point for a given geometric configuration on the convergence order for approximate solutions of the boundary value problem.
Abstract: The solution of a two-dimensional elliptic boundary value problem with piecewise smooth external boundaries, interfaces, and diffusion coefficients typical of nuclear reactor structures is known to contain a singular part. The presence of singular functions in the neighborhood of each angular point for a given geometric configuration has important consequences on the convergence orders for approximate solutions of the problem. These consequences are analyzed both theoretically and numerically, in the framework of the finite element method. Some means are described to overcome the damaging effects of the singular points. A thorough numerical study of various reactor configurations extending from liquid-metal fast breeder reactors to pressurized water reactors shows that in the latter case, the use of high-order polynomials is partially unjustified, given the severe limitations on the convergence orders.
TL;DR: In this article, the degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F-spaces Ng-' and two o-results and use pairs of inequalities of Bernsteinand Jackson-type to prove several direct and converse theorems.
Abstract: The degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F-spaces Ng-' (introduced for this purpose in [2]). We obtain two o-results and use pairs of inequalities of Bernsteinand Jackson-type to prove several direct and converse theorems. For f in Ng-' we define a derivative D1 ?f in La, a = (n + p1)-1, which agrees with D f for smoothf, and prove scveral properties of D"?.
TL;DR: In this paper, a simple mechanism is presented that simulates qualitatively the more sophisticated and mathematically complicated methods of Nielson, Cline and Schweikert, who have discussed more rigorously the problem of splines under tension.
Abstract: A simple mechanism is presented that simulates qualitatively the more sophisticated and mathematically complicated methods of Nielson, Cline and Schweikert, who have discussed more rigorously the problem of splines under tension. it is not, however, intended to copy their methods or to yield the same results. Its advantages are its simplicity and its ability to modify the shapes of any piecewise sequence of curve segments, regardless of their nature. It is designed to maintain the order of continuity where the separate curve segments join (at the knots).
TL;DR: In this article, the authors proposed a method that combines the rapid convergence characteristics of the KFM with the ability of the FEM to treat discontinuities without having to determine their exact form.
Abstract: Theme T WO main methods currently are employed in the prediction of the aerodynamic forces acting on lifting surfaces at subsonic flow. These methods inlcude the subsonic kernel function approach, which assumes pressure polynomials to describe the pressure field over the wing, and the finite-element approach, such as the doublet lattice method (DLM) or the vortex lattice method (VLM). The kernel function method (KFM), when based on orthogonal polynomials and carefully determined collocation points, shows a rapid convergence of its solution (with a small number of pressure polynomials) provided that the pressure field over the wing is smooth. Pressure discontinuities, such as those arising from control surface rotations, can be treated successfully using the KFM only when the exact shape of the singularity is known. The overlooking of pressure singularities using the KFM leads to a rapid deterioration in the convergence of the solution and to a general loss in the effectiveness of the method. The finite-element methods (FEM) can cope successfully with unknown pressure singularities provided that their location is known. The FEM, however, require a relatively large number of unknowns for convergence, leading at times to a relatively large residual error at the converged values. In the present paper a method is presented, similar to the one described in Ref. 4 in connection with mixed transonic flow, which combines the rapid convergence characteristics of the KFM with the ability of the FEM to treat discontinuities without having to determine their exact form. The method is tested using a two-dimensional airfoil problem (with control surfaces and gaps) with the intention of establishing its merits before embarking on its extension to the three-dimensional flow case.
TL;DR: State models based on graph-theoretical concepts are presented for the computer-aided analysis of large-scale linear networks by decomposition in time as well as frequency domains to exploit the sparsity and the repetitive structure present in most large- scale networks.
Abstract: State models based on graph-theoretical concepts are presented for the computer-aided analysis of large-scale linear networks by decomposition in time as well as frequency domains. Several examples are analysed which illustrate the validity and efficiency of these models. These models exploit the sparsity and the repetitive structure present in most large-scale networks.
TL;DR: Existence of piecewise optimal control is proved when the cost function includes one or both of a cost of sudden switching of control variables and a cost associated with the maximum rate of variation of the control over segments of the path for which the control is continuous.
Abstract: Abstract Existence of piecewise optimal control is proved when the cost function includes one or both of (a) a cost of sudden switching (discontinuity) of control variables, and (b) a cost associated with the maximum rate of variation of the control over segments of the path for which the control is continuous.
TL;DR: In this paper, it was shown that the stratification? method of solving the inhomogeneous waveguide differential equation is formally equivalent to solution by the Euler method, but with substantially greater computation.
Abstract: By formulating a general framework for the piecewise solution of the inhomogeneous waveguide differential equation, it is shown that the `stratification? method of solving this equation is formally equivalent to solution by the Euler method, but with substantially greater computation.