Blaž Gotovac

Computational Modeling of Structural Problems Using Atomic Basis Functions

This paper presents the application of the Fup n (x) basis functions in numerical modeling of different engineering problems. Fup n (x) basis functions belong to a class of atomic functions which are infinitely-differentiable functions with compact support. The collocation method has been applied in the development of numerical models. A system of algebraic equations is formed in which the differential equation of the problem is satisfied in collocation points of a closed domain while boundary conditions are satisfied exactly at the domain boundary. Using this way, the required accuracy of approximate solution is obtained simply by an increase in the number of basis functions. So, this concept represents a fully mesh free method. The properties of the atomic basis functions enable a hierarchic expansion of an approximate solution base either in the entire domain or in its segments. Presented numerical models are illustrated by examples of the torsion of prismatic bars, elasto-plastic analyses of beam bending and thin plate bending problems. The results of the analyses are compared with the existing exact and relevant numerical solutions. It can be concluded that the possibility of hierarchically expanding the number of basis functions in the domain significantly accelerates the convergence of a numerical procedure in a simple way. Values of the main solution function, e.g. displacements, and all the values derived from the main solution of the problem such as stresses, bending moments and transversal forces, are calculated in the same points and with the same degree of accuracy since numerical integration is avoided.

Application of the Solution Structure Method in Numerically Solving Poisson’s Equation on the Basis of Atomic Functions

This paper summarizes the main principles of the solution structure method and presents it in combination with atomic basis functions and a collocation technique. The solution of a boundary value p...

Solving the SPRP by using a triple base ABF

Atomske bazne funkcije (ABF) posjeduju svojstvo univerzalnosti vektorskog prostora kao klasične bazne funkcije i svojstvo finitnosti (konačna duljina nosača) kao splineovi te na taj način popunjavaju skup elementarnih funkcija. U ovom radu ukratko su opisana svojstva eksponencijalnih ABF EFupn(x, w), koji za razliku od algebarskih ABF EFupn(x), sadrže parametar ili frekvenciju ω koja im omogućava dodatna aproksimacijska svojstva. Problem izbora vrijednosti parametra ω u numeričkoj analizi riješen je primjenom tzv. trostruke baze. Primjena atomskih baznih funkcija EFupn(x, w) ilustrirana je na primjeru rješavanja singularno perturbiranog rubnog problema (SPRP) i to korištenjem trostruke baze u metodi kolokacije uz primjenu multirezolucijskog postupka.