Spherical Harmonics Models and their Application to non-Spherical Shape Particles

The dissertation investigates spherical harmonics method for describing a particle shape. The main object of research is the non-spherical shape particles. The purpose of this dissertation is to create spherical harmonics model for a non-pherical particle. The dissertation also focuses on determining the suitability of the lowresolution spherical harmonics for describing various non-spherical particles. The work approaches a few tasks such as testing the suitability of a spherical harmonics model for simple symmetric particles and applying it to complex shape particles. The first task is formulated aiming to test the modelling concept and strategy using simple shapes. The second task is related to the practical applications, when complex shape particles are considered. The dissertation consists of introduction, 4 chapters, general conclusions, references, a list of publications by the author on the topic of the dissertation, a summary in Lithuanian and 5 annexes. The introduction reveals the investigated problem, importance of the thesis and the object of research, describes the purpose and tasks of the thesis, research methodology, scientific novelty, the practical significance of results and defended statements. The introduction ends in presenting the author’s publications on the topic of the dissertation, offering the material of made presentations in conferences and defining the structure of the dissertation. Chapter 1 revises the literature: the particulate systems and their processes, shapes of the particles and methods for describing the shape, shape indicators. At the end of the chapter, conclusions are drawn and the tasks for the dissertation are reconsidered. Chapter 2 presents the modelling approach and strategies for the points of the particle surface, spherical harmonics, the calculation of the expansion coefficients, integral parameters and curvature and also the conclusions. Chapters 3 and 4 analize the modelling results of the simple and complex particles. At the end of the both chapters conclusions are drawn. 5 articles focusing on the topic of the dissertation have been published: two articles – in the Thomson ISI register, one article – in conference material and scientific papers in Thomson ISI Proceedings data base, one article – in the journal quoted by other international data base, one article – in material reviewed during international conference. 8 presentations on the subject of the dissertation have been given in conferences at national and international levels

Investigation of symmetric non-spherical particle shapes by applying low-resolution spherical harmonics

The issue of mathematical modelling of non-spherical shapes of particles is considered. Thus, application of the spherical harmonics (SH) technique in modelling the simplest symmetric star-shaped particles is demonstrated by applying low-resolution functions. The investigation was restricted to a circular cylinder and a rectangular parallelepiped, geometrically primitive, but widespread oblate industrial shapes. The modelling quality was studied by considering selected error norms and the most important integral characteristics of a particle geometry, including the surface area and volume. The presented results discovered new features of the spherical harmonic technique and enhanced understanding of their applicability to describe non-spherical shapes

Low-resolution spherical harmonics models in application to quasi-quadric particle shapes

In this paper a numerical analysis was performed developing low-resolution spherical harmonics (LRSH) models in order to describe particle shapes. The potential of LRSH, limited by the expansion degree L ≤ 3, to describe quasi-regular particle shapes was explored. The term “quasi” is used hereafter to indicate the monomeric, almost regular shaped, particle described by a single continuous function. This approach reflects the shape of a major part of soil minerals. It was shown, that even the simplest case of the suggested low-resolution harmonics technique with L = 1 showed sufficient accuracy. The main drawback of the suggested approach was that the low-resolution harmonics yield particle shapes with nearly sharp angles, there-fore, enhanced analysis of local surface curvatures becomes necessary. An application using quasi-ellipsoidal particles is enclosed

The application of spherical harmonics for describing a cube-shape particle / Sferinių harmoninių funkcijų taikymas kubo formos dalelei aprašyti

The paper disccusses various models for discrete element methods of modelling particle shapes. The multi-sphere method is most frequently used, though a polyhedral approach and a hybrid sphero-polyhedral model can also be introduced. All above mentioned methods precisely approximate particle shapes, and difficulties in modelling contact between particles arise. Thus, there is a need to have an universal method that gives the analytical expression of the shape. For this purpose, spherical harmonics have been introduced. The article presents the concept of the spherical harmonics model and experimental results describing the cube-shape particle. Santrauka Šiame straipsnyje aptariami įvairūs diskrečiųjų elementų metodo modeliai, taikomi dalelių formoms modeliuoti. Dažniausiai taikomas kelių sferų modelis, bet yra pristatyti ir daugiasienis modelis bei mišrus sferos daugiasienis modelis. Nors aptariami metodai neblogai apkroksimuoja dalelių formą, atsiranda sunkumų skaičiuojant dalelių kontaktą. Todėl reikalingas universalus metodas, leidžiantis gauti analitinę dalelės formos išraišką. Būtent tokiam tikslui naudojamos sferinės harmoninės funkcijos. Straipsnyje pristatomas sferinių funkcijų modelis bei pateikiami eksperimento rezultatai kubo formos dalelei aprašyti. Reikšminiai žodžiai: diskrečiųjų elementų metodas; kubo formos dalelė; sferinės harmoninės funkcijos

Investigation of symmetric non-spherical particle shapes by applying low-resolution spherical harmonics

The issue of mathematical modelling of non-spherical shapes of particles is considered. Thus, application of the spherical harmonics (SH) technique in modelling the simplest symmetric star-shaped particles is demonstrated by applying low-resolution functions. The investigation was restricted to a circular cylinder and a rectangular parallelepiped, geometrically primitive, but widespread oblate industrial shapes. The modelling quality was studied by considering selected error norms and the most important integral characteristics of a particle geometry, including the surface area and volume. The presented results discovered new features of the spherical harmonic technique and enhanced understanding of their applicability to describe non-spherical shapes