Piecewise Smooth Functions Research Articles

Generalized stress intensity factors in the interaction within a rectangular array of rectangular inclusions

To evaluate the mechanical strength of fiber-reinforced composites, it is necessary to consider singular stresses at the end of fibers because they cause crack initiation, propagation, and final failure. A square array of rectangular inclusions under longitudinal tension is considered in this paper. The body-force method is applied to a unit cell region. Then, the problem is formulated as a system of singular integral equations, where the unknown functions are the densities of body forces distributed in infinite plates having the same elastic constants as those of the matrix and inclusions. The unknown functions are expressed as piecewise-smooth functions using power series and two types of fundamental densities which express singular stresses. Generalized stress intensity factors at the corners of inclusions are systematically calculated with varying the shape and spacing of a square array of square and rectangular inclusions.

Controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions

This paper presents a kind of controller synthesis method for fuzzy dynamic systems based on a piecewise smooth Lyapunov function. The basic idea of the proposed approach is to construct controllers for the fuzzy dynamic systems in such a way that a piecewise continuous Lyapunov function can be used to establish the global stability with H/sub /spl infin// performance of the resulting closed loop fuzzy control systems. It is shown that the control laws can be obtained by solving a set of linear matrix inequalities that is numerically feasible with commercially available software. An example is given to illustrate the application of the proposed methods.

Inverse Problems of Boundary Value Problems for Annular Vibrating Membranes with Piecewise Smooth Positive Functions in the Boundary Conditions

The asymptotic expansions of the trace of the heat kernel <artwork name="GAPA31035ei1"> for small positive t, where λν are the eigenvalues of the negative Laplacian <artwork name="GAPA31035ei2"> in Rn (n=2 or 3), are studied for a general annular bounded domain Ω with a smooth inner boundary ∂Ω1 and a smooth outer boundary ∂Ω2 where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the components Γ j (j=1,…,m) of ∂Ω1 and on the components <artwork name="GAPA31035ei3"> of ∂Ω2 are considered such that <artwork name="GAPA31035ei4"> and <artwork name="GAPA31035ei5"> and where the coefficients <artwork name="GAPA31035ei6"> in the Robin boundary conditions are piecewise smooth positive functions. Some applications of Θ (t) for an ideal gas enclosed in the general annular bounded domain Ω are given.

Approximating the jump discontinuities of a function by its Fourier-Jacobi coefficients

In the present paper we generalize Eckhoff's method, i.e., the method for approximating the locations of discontinuities and the associated jumps of a piecewise smooth function by means of its Fourier-Chebyshev coefficients. A new method enables us to approximate the locations of discontinuities and the associated jumps of a discontinuous function, which belongs to a restricted class of the piecewise smooth functions, by means of its Fourier-Jacobi coefficients for arbitrary indices. Approximations to the locations of discontinuities and the associated jumps are found as solutions of algebraic equations. It is shown as well that the locations of discontinuities and the associated jumps are recovered exactly for piecewise constant functions with a finite number of discontinuities. In addition, we study the accuracy of the approximations and present some numerical examples.

Peakon and Foldon Excitations In a (2+1)-Dimensional BreakingSoliton System

Starting from the standard truncated Painlevé expansion and avariable separation approach, a general variable separation solution ofthe breaking soliton system is derived. In addition to the usuallocalized coherent soliton excitations like dromions, lumps, rings,breathers, instantons, oscillating soliton excitations, and previouslyrevealed chaotic and fractal localized solutions, some new types ofexcitations, peakons and foldons, are obtained by introducingappropriate lower-dimensional piecewise smooth functions and multiplevalued functions.

Wavelet footprints: theory, algorithms, and applications

Wavelet-based algorithms have been successful in different signal processing tasks. The wavelet transform is a powerful tool because it manages to represent both transient and stationary behaviors of a signal with few transform coefficients. Discontinuities often carry relevant signal information, and therefore, they represent a critical part to analyze. We study the dependency across scales of the wavelet coefficients generated by discontinuities. We start by showing that any piecewise smooth signal can be expressed as a sum of a piecewise polynomial signal and a uniformly smooth residual (Theorem 1). We then introduce the notion of footprints, which are scale space vectors that model discontinuities in piecewise polynomial signals exactly. We show that footprints form an overcomplete dictionary and develop efficient and robust algorithms to find the exact representation of a piecewise polynomial function in terms of footprints. This also leads to efficient approximation of piecewise smooth functions. Finally, we focus on applications and show that algorithms based on footprints outperform standard wavelet methods in different applications such as denoising, compression, and (nonblind) deconvolution. In the case of compression, we also prove that at high rates, footprint-based algorithms attain optimal performance (Theorem 3).

Revisiting Galerkin's method through global piecewise-smooth functions in Christides and Barr's cracked beam theory

Revisiting Galerkin's method through global piecewise-smooth functions in Christides and Barr's cracked beam theory

Free vibrations of multilayered doubly curved shells based on a mixed variational approach and global piecewise-smooth functions

This paper presents the extension of a two-dimensional model that, recently appeared in literature, deals with freely vibrating laminated plates. The extension takes into account the corresponding theory describing the dynamic of freely vibrating multilayered doubly curved shells. The relevant governing differential equations, associated boundary conditions and constitutive equations are derived from one of Reissner’s mixed variational theorems. Both the governing differential equations and the related boundary conditions are presented in terms of resultant stresses and displacements. In spite of the multi-layer nature of the shell, the theory is developed as if the shell were virtually made of a single layer. This choice does not limit the performances of the model, which are comparable to the corresponding three-dimensional theory. This ability is accomplished by an appropriate global expansion of the relevant kinetic and stress quantities, through the thickness of the multilayered shell. The mentioned expansion is realized by a novel selection of global piecewise-smooth functions. Numerical tests illustrate the performance of the model with respect to several elements subjected to a class of simply supported boundary conditions: plates, circular cylindrical shells, spherical and saddle-shape laminates. The model is first tested by comparing its resulting eigen-parameters, with those few existing of exact or approximate three-dimensional models and, finally, new results are provided for several geometrical and material characteristics for plates and shells.

Image denoising: pointwise adaptive approach

A new method of pointwise adaptation has been proposed and studied in Spokoiny [(1998) Ann. Statist.26 1356-1378] in the context of estimation of piecewise smooth univariate functions. The present paper extends that method to estimation of bivariate grey-scale images composed of large homogeneous regions with smooth edges and observed with noise on a gridded design. The proposed estimator $\hat{f}(x)$ at a point x is simply the average of observations over a window $\widehat{U}(x)$ selected in a data-driven way. The theoretical properties of the procedure are studied for the case of piecewise constant images. We present a nonasymptotic bound for the accuracy of estimation at a specific grid point x as a function of the number of pixels n, of the distance from the point of estimation to the closest boundary and of smoothness properties and orientation of this boundary. It is also shown that the proposed method provides a near-optimal rate of estimation near edges and inside homogeneous regions. We briefly discuss algorithmic aspects and the complexity of the procedure. The numerical examples demonstrate a reasonable performance of the method and they are in agreement with the theoretical issues. An example from satellite (SAR) imaging illustrates the applicability of the method.

Higher dimensional inverse problem for a multi-connected bounded domain with piecewise smooth Robin boundary conditions and its physical applications

The asymptotic expansions of the trace of the heat kernel Θ( t)=∑ j=1 ∞exp(− tλ j ) for short-time t, have been derived for a variety of bounded domains, where { λ j } j=1 ∞ are the eigenvalues of the negative Laplace operator −∇ 2=−∑ i=1 3(∂/∂ x i ) 2 in the ( x 1, x 2, x 3)-space. The dependence of Θ( t) on the connectivity of bounded domains and the boundary conditions is analyzed. Particular attention is given for an arbitrary multiply-connected bounded domain Ω in R 3 together with piecewise smooth Robin boundary conditions, where the coefficients in these conditions are assumed to be piecewise smooth positive functions. Some applications of an ideal gas enclosed in the multiply-connected bounded domain Ω are given. We show that the ideal gas cannot feel the shape of its container, although it can feel some geometrical properties of it.

Generalized Stress Intensity Factors at the Fiber End in a Hexagonal Array of Fibers.

To evaluate the mechanical strength of fiber reinforced composites it is necessary to consider singular stresses at the end of fibers because they cause crack initiation, propagation, and final failure. To obtained the magnitude of the singular stress, in this paper, an interaction among a hexagonal array of cylindrical inclusions under longitudinal tension is considered. The body force method is applied to a unit cell region; then, the problem is formulated as a system of singular integral equations, where unknowns are densities of body forces distributed in infinite bodies having the same elastic constants as those of the matrix and inclusions. The unknown functions are expressed as piecewise smooth functions using fundamental densities and power series. Here, the fundamental densities are chosen to represent the symmetric strees singularity, and the skew-symmetric stress singularity. Then, generalized stress intensity factors at the fiber end are systematically calculated with varying the elastic ratio, length, and spacing of fibers. The region when the interaction effect is less than 1% is shown in a figure as a function of fiber length.

A Strongly Semismooth Integral Function and Its Application

As shown by an example, the integral function f : {\bb R}n → {\bb R}, defined by f(x) e ∫ab[B(x, t)]+g(t) dt, may not be a strongly semismooth function, even if g(t) ≡ 1 and B is a quadratic polynomial with respect to t and infinitely many times smooth with respect to x. We show that f is a strongly semismooth function if g is continuous and B is affine with respect to t and strongly semismooth with respect to x, i.e., B(x, t) e u(x)t + v(x), where u and v are two strongly semismooth functions in {\bb R}n. We also show that f is not a piecewise smooth function if u and v are two linearly independent linear functions, g is continuous and g n 0 in [a, b], and n ≥ 2. We apply the first result to the edge convex minimum norm network interpolation problem, which is a two-dimensional interpolation problem.

A Property of Piecewise Smooth Functions

Piecewise smooth equations are increasingly important in the numerical treatment of complementarity problems and models of equilibrium. This note brings out a property of the functions that enter such equations, for instance through penalty expressions.

Generalized H/sub 2/ controller synthesis of fuzzy dynamic systems based on piecewise lyapunov functions

This paper presents a generalized H/sub 2/ controller synthesis method for the Takagi-Sugeno fuzzy dynamic systems based on a piecewise smooth Lyapunov function. The basic idea of the proposed approach is to construct a controller for the fuzzy dynamic systems in such a way that a piecewise continuous Lyapunov function can be used to establish the global stability with generalized H/sub 2/ performance of the resulting closed loop fuzzy control systems. It is shown that the control law can be obtained by solving a set of linear matrix inequalities. An example is given to illustrate the application of the proposed method.

Generalized stress intensity factors of V-shaped notch in a round bar under torsion, tension, and bending

In this study, generalized stress intensity factors K I, λ 1 , K II, λ 2 , and K III, λ 4 are calculated for a V-shaped notched round bar under tension, bending, and torsion using the singular integral equation of the body force method. The body force method is used to formulate the problem as a system of singular integral equations, where the unknown functions are the densities of body forces distributed in an infinite body. In order to analyze the problem accurately, the unknown functions are expressed as piecewise smooth functions using three types of fundamental densities and power series, where the fundamental densities are chosen to represent the symmetric stress singularity and the skew-symmetric stress singularity. Generalized stress intensity factors at the notch tip are systematically calculated for various shapes of V-shaped notches. Normalized stress intensity factors are given by using limiting solutions; they are almost determined by notch depth alone, and almost independent of other geometrical parameters. The accuracy of Benthem–Koiter’s formula proposed for a circumferential crack is also examined through the comparison with the present analysis.

FREE VIBRATIONS OF MULTILAYERED PLATES BASED ON A MIXED VARIATIONAL APPROACH IN CONJUNCTION WITH GLOBAL PIECEWISE-SMOOTH FUNCTIONS

This paper presents a two-dimensional model for the analysis of freely vibrating laminated plates. The governing differential equations, associated boundary conditions and constitutive equations are derived from Reissner's mixed variational theorem. Both the governing differential equations and the related boundary conditions are presented in terms of resultant stresses and displacements. The model is able to provide the results for the corresponding three-dimensional theory. Such a performance is guaranteed from an appropriate expansion of relevant kinetic and stress quantities through the thickness of the multilayered plate. The expansion is realized by using a novel selection of global piecewise-smooth functions (GPSFs). The number of GPSFs can be arbitrarily increased to achieve a two-dimensional plate theory which is, at least, as accurate as that of a full layerwise theory. It is also shown that GPSFs permit to deal a multilayered plate as if it was virtually made of a single layer. Indeed, the theory need not explicitly introduce continuity conditions for both displacements and relevant stresses. The performance of the present two-dimensional model in conjunction with the global piecewise-smooth functions is tested and discussed by comparing its resulting eigen-parameters, for a class of simply supported plates, with those of other two-dimensional models and with those existing of the exact three-dimensional theory.

Stability analysis of piecewise discrete-time linear systems

Presents a stability analysis method for piecewise discrete-time linear systems based on a piecewise smooth Lyapunov function. It is shown that the stability of the system can be established if a piecewise Lyapunov function can be constructed and, moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that is numerically feasible with commercially available software.

The Problem of the Pointwise Fourier Inversion for Piecewise Smooth Functions of Several Variables

The Problem of the Pointwise Fourier Inversion for Piecewise Smooth Functions of Several Variables

The rate of convergence of Fourier expansions in the plane: a geometric viewpoint

Let A be an appropriate planar domain and let f be a piecewise smooth function on ${\mathbb R}^{2}$ . We discuss the rate of convergence of \[ S_{\lambda}f(x)=\int_{\lambda A}\widehat{f}(\xi)\exp(2\pi i\xi\cdot x)d\xi \] in terms of the interaction between the geometry of A and the geometry of the singularities of f. The most subtle case is when x belongs to the singular set of f.

Spectral Reconstruction of Piecewise Smooth Functions from Their Discrete Data

This paper addresses the recovery of piecewise smooth functions from their discrete data. Reconstruction methods using both pseudo-spectral coecients and physical space interpolants have been discussed extensively in the literature, and it is clear that an ap rioriknowledge of the jump discontinuity location is essential for any reconstruction technique to yield spectrally accurate results with high resolution near the discontinuities. Hence detection of the jump discontinuities is critical for all methods. Here we formulate a new localized reconstruction method adapted from the method developed in Gottlieb and Tadmor (1985) and recently revisited in Tadmor and Tanner (in press). Our procedure incorporates the detection of edges into the reconstruction technique. The method is robust and highly accurate, yielding spectral accuracy up to a small neighborhood of the jump discontinuities. Results are shown in one and two dimensions.