Izvorni znanstveni članak
Fractals and formfinding - magic with real numbers
Josip Dvornik
; Department of Technical Mechanics, Faculty of Civil Engineering, University of Zagreb, CROATIA
Damir Lazarević
; Department of Technical Mechanics, Faculty of Civil Engineering, University of Zagreb, CROATIA
Sažetak
The equilibrium shape determination of a prestressed cable net structure termed “formfinding” can be achieved through the classical Newton-Raphson iterative method. The convergence strongly depends on the choice of the starting approximation vector, which must lie within the “ball of convergence” or “basin of attraction” unknown in advance. From experience this basin of attraction is very irregular, hence it was hypothesized that it may be a fractal. For an explanation of this irregularity, a series of numerical experiments was performed with highly simplified cable nets. It was confirmed that basins of attraction were fractals indeed. Cross-sections of these three-dimensional fractals are visualized, and they appear highly interesting and diverse. It is shown that the Newton-Raphson method can also generate fractals with real numbers. Experiments were also performed with other iterative algorithms, with similarly interesting results - both from the equation-solving and from the fractal-producing points of view. It is believed that all discussed methods (except for the force density method which is specific to the formfinding problem) can be also applied to explain the convergence properties of many other nonlinear equations systems.
Ključne riječi
Hrčak ID:
315991
URI
Datum izdavanja:
30.6.2003.
Posjeta: 40 *