Any quasismooth function f (x) in a finite interval [0,x0], which has only a finite number of finite discontinuities and has only a finite number of extremes, can be approximated by a uniformly convergent Fourier series and a correction function. The correction function consists of algebraic polynomials and Heaviside step functions and is required by the aperiodicity at the endpoints (i.e., f (...

As follows from the Schwartz Impossibility Theorem, multiplication of two distributions is in general impossible. Nevertheless, often one needs to multiply a distribution by a discontinuous function, not by an arbitrary distribution. In the present paper we construct a space of distributions where the general operation of multiplication by a discontinuous function is defined, continuous, commut...

We study the regularized, oscillatory, multiscale, quasilinear, elliptic problem in divergence form − div(Qη(x, x , u)∇u) = f, in Ω, (0.1) u = g, on ∂Ω, (0.2) assuming that the conductivity has the form Qη(x, y, r) = a−(x) + (a+(x)− a−(x))βη(r − ψ(x, y)), (0.3) for some domain Ω ⊂ Rn, n ≥ 1, a regularization βη of the Heaviside step function and a locally periodic function ψ(x, y), Y -periodic ...

Abstract This paper presents a comprehensive numerical and experimental study on stress-constrained topology optimization for Fused Deposition Modeling (FDM) additive manufacturing. The qp method is employed to avoid the singularity issue of problems. P-norm function with stability transformation adopted build global stress constraint iterative corrections eliminate gap between maximum local st...

This paper presents and studies a residual a posteriori error estimator for Laplace’s equation in two space dimensions approximated by the eXtended Finite Element Method (XFEM). The XFEM allows to perform finite element computations on multi-cracked domains by using meshes of the non-cracked domain. The main idea consists of adding supplementary basis functions of Heaviside type and singular fu...

Quantitative bounds on rates of approximation by linear combinations of Heaviside plane waves are obtained for sufficiently differentiable functions f which vanish rapidly enough at infinity: for d odd and f ∈ Cd(R), with lower-order partials vanishing at infinity and dth-order partials vanishing as ‖x‖−(d+1+ε), ε > 0, on any domain ⊂ R with unit Lebesgue measure, the L2( )-error in approximati...

Recently a wave equation for unified field theory has been derived [1-3] from a lemma of Cartan-Riemann differential geometry [4, 5] and used to give the first self consistent unified description of the gauge invariant gravitational and electromagnetic fields. The former is the Riemann form and the latter is the torsion form. One form is the Hodge dual of the other. The theory has been shown [1...

This research presents, buckling and free vibration analysis of fiber metal-laminated (FML) plates on a total and partial elastic foundation using the generalized differential quadrature method (GDQM). The partial foundation consists of multi-section Winkler and Pasternak type elastic foundation. Taking into consideration the first-order shear deformation theory (FSDT), FML plate is modeled and...