DISCRETE-CONTINUAL BOUNDARY ELEMENT METHODS OF ANALYSIS FOR TWO-DIMENSIONAL AND THREE-DIMENSIONAL STRUCTURES

The aim of this paper is to present so-called discrete-continual boundary element method (DCBEM) of structural analysis. Its field of application comprises buildings constructions, structures and also parts and components for the residential, commercial and un-inhabitant structures with invariability of physical and geometrical parameters in some dimensions. We should mention here in particular such objects as beams, thin-walled bars, strip foundations, plates, shells, deep beams, high-rise buildings, extensional buildings, pipelines, rails, dams and others. DCBEM comes under group of semianalytical methods. Semianalytical formulations are contemporary mathematical models which currently becoming available for realization due to substantial speed-up of computer productivity. DCBEM is based on the theory of the pseudodifferential boundary equations. Corresponding pseudodifferential operators are discretely approximated using Fourier analysis or wavelet analysis. The main DCBEM advantages against the other methods of the numerical analysis is a double reduction in dimension of the problem (discrete numerical division applied not to the full region of the interest but only to the boundary of the region cross section, as a matter of fact one is solving an one-dimensional problem with the finite step on the boundary area of the region), one has opportunities to carrying out very detailed analysis of the specific chosen zones, simplified initial data preparation, simplistic and adaptive algorithms. There are two methods to define and conduct DCBEM analysis developed – indirect (IDCBEM) and direct (DDCBEM), thus indirect like in boundary element method (BEM) applied and used little bit more than direct

Local behavior of distributions and applications

This dissertation studies local and asymptotic properties of distributions (generalized functions) in connection to several problems in harmonic analysis, approximation theory, classical real and complex function theory, tauberian theory, summability of divergent series and integrals, and number theory. In Chapter 2 we give two new proofs of the Prime Number Theory based on ideas from asymptotic analysis on spaces of distributions. Several inverse problems in Fourier analysis and summability theory are studied in detail. Chapter 3 provides a complete characterization of point values of tempered distributions and functions in terms of a generalized pointwise Fourier inversion formula. The relation of the Fourier inversion formula with several summability procedures for divergent series and integrals is established. This work also provides formulas for jump singularities, that is, detection of edges from spectral data, which can be used as effective numerical detectors. Chapters 5 and 6 introduce new summability methods for the determination of jump discontinuities. Estimations on orders of summability are given in Chapter 8. Chapters 4 and 9 give a tauberian theory for distributional point values; this theory recovers important classical tauberians of Hardy and Littlewood, among others, for Dirichlet series. We make a complete wavelet analysis of asymptotic properties of distributions in Chapter 11. This study connects the boundary asymptotic behavior of the wavelet transform with asymptotics of tempered distributions. It is shown that our tauberian theorems become full characterizations. Chapter 10 makes a comprehensive study of asymptotic properties of distributions. Open problems in the area are solved in Chapter 10 and new tools are developed. We obtain a complete structural description of quasiasymptotics in one variable. We introduce the phi-transform for the local analysis of functions, measures, and distributions. In Chapter 7 the transform is used to study distributionally regulated functions (introduced here). Chapter 12 presents a characterization of measures in terms of the boundary behavior of this transform. We characterize the support of tempered distributions in Chapter 13 by various summability means of the Fourier transform