Multilevel-multiscale Matrix Iteration for Elastic Fractal Structural Analysis

Project: Research

Description

A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,"[1] a property called self-similarity. The term was coined by Benoit Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.[2] A fractal often has the following features:[3] (i)It has a fine structure at arbitrarily small scales. (ii) It is too irregular to be easily described in traditional Euclidean geometric language. (iii) It is self-similar (at least approximately or stochastically). (iv) It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve). (v) It has a simple and recursive definition. Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, sea-shells and snow flakes. Elastic properties of fractal structures have been intensively studied for their responses due to environmental forces since 1984 [4-5]. Elastic fractal structures have been experienced a rapid growth of interest. Since geometric fractals are generated by iterative maps, the researcher proposes to study their elastic properties by translating the iterative maps to matrix iterations in statics and dynamics using multi-level substructure techniques that the researcher originated.

Detail(s)

Project number 7002474 SRG Finished 1/04/09 → 1/03/11