Applications of Affine Systems of Walsh Type to Generate Smooth Basis

The question on affine Riesz basis of Walsh affine systems is considered. An affine Riesz basis is constructed, generated by a continuous periodic function that belongs to the space on the real line, which has a derivative almost everywhere; in connection with the construction of this example, we note that the functions of the classical Walsh system suffer a discontinuity and their derivatives almost vanish everywhere. A method of regularization (improvement of differential properties) of the generating function of Walsh affine system is proposed, and a criterion for an affine Riesz basis for a regularized generating function that can be represented as a sum of a series in the Rademacher system is obtained.

TASK ON INDENTATION OF SMOOTH PUNCH OF AN ARBITRARY FORM IN PLANE VIEW IN HARD PLASTIC HALF-SPACE

In the work the general properties of equations of spatial problem of the theory of ideal plasticity in the condition of Treska plasticity and stressed state that correspond to the ridge of surface of fluctuation are viewed. Singular lines in hard plastic space and solutions in the neighborhood of a singular line that are similar to the expansion in beam series are viewed. On the basis of the developed theory the boundary problem appearing at indentation of punch with smooth basis in a certain plastic body which in the simplest case is a half-space is solved.

A locking-free model for Reissner–Mindlin plates: Analysis and isogeometric implementation via NURBS and triangular NURPS

We study a reformulated version of Reissner–Mindlin plate theory in which rotation variables are eliminated in favor of transverse shear strains. Upon discretization, this theory has the advantage that the "shear locking" phenomenon is completely precluded, independent of the basis functions used for displacement and shear strains. Any combination works, but due to the appearance of second derivatives in the strain energy expression, smooth basis functions are required. These are provided by Isogeometric Analysis, in particular, NURBS of various degrees and quadratic triangular NURPS. We present a mathematical analysis of the formulation proving convergence and error estimates for all physically interesting quantities, and provide numerical results that corroborate the theory.

Modelling localized nonlinearities in continuous systems via the method of augmentation by non-smooth basis functions

Existing solutions for continuous systems with localized, non-smooth nonlinearities (such as impacts) focus on exact methods for satisfying the nonlinear constitutive equations. Exact methods often require that the non-smooth nonlinearities be expressed as piecewise-linear functions, which results in a series of mapping equations between each linear regime of the nonlinearities. This necessitates exact transition times between each linear regime of the nonlinearities, significantly increasing computational time, and limits the analysis to only considering a small number of nonlinearities. A new method is proposed in which the exact, nonlinear constitutive equations are satisfied by augmenting the system's primary basis functions with a set of non-smooth basis functions. Two consequences are that precise contact times are not needed, enabling greater computational efficiency than exact methods, and localized nonlinearities are not limited to piecewise-linear functions. Since each nonlinearity requires only a few non-smooth basis functions, this method is easily expanded to handle large numbers of nonlinearities throughout the domain. To illustrate the application of this method, a pinned–pinned beam example is presented. Results demonstrate that this method requires significantly fewer basis functions to achieve convergence, compared with linear and exact methods, and that this method is orders of magnitude faster than exact methods.

Smoothed Linear Modeling for Smooth Spectral Data

Classification and prediction problems using spectral data lead to high-dimensional data sets. Spectral data are, however, different from most other high-dimensional data sets in that information usually varies smoothly with wavelength, suggesting that fitted models should also vary smoothly with wavelength. Functional data analysis, widely used in the analysis of spectral data, meets this objective by changing perspective from the raw spectra to approximations using smooth basis functions. This paper explores linear regression and linear discriminant analysis fitted directly to the spectral data, imposing penalties on the values and roughness of the fitted coefficients, and shows by example that this can lead to better fits than existing standard methodologies.