Zsolt Gáspár

Mechanical Models for the Subclasses of Catastrophes

First some concepts of the structural stability and the elementary catastrophe theory are shown. A short chapter explains which types of the catastrophes are typical at elastic structures. Hence the load parameter has a special role among the parameters, a subclassification is needed in the stability analysis. The main part of the paper shows this subclassification and illustrates almost every type by simple elastic models.

From spherical circle coverings to the roundest polyhedra

Abstract The problem treated here is: amongst the convex polyhedra that can be circumscribed about the unit sphere and have faces, which has the minimum surface area? A new optimization method based on mechanical analogies is worked out to solve this problem. By using this method, new computer-generated solutions are presented for and . The second of these two conjectured roundest polyhedra has icosahedral symmetry. The relation of the results of this problem to the minimum coverings of the sphere with equal circles is discussed.

Some Basic Issues of Topology Optimization

The aim of this paper is to discuss some issues of pivotal importance in topology optimization, which receive inadequate attention in the literature.

Partial covering of a circle by equal circles. Part I: The mechanical models

How must n equal circles of given radius be placed so that they cover as great a part of the area of the unit circle as possible? To analyse this mathematical problem, mechanical models are introduced. A generalized tensegrity structure is associated with a maximum area configuration of the n circles, whose equilibrium configuration is determined numerically with the method of dynamic relaxation, and the stability of equilibrium is investigated by means of the stiffness matrix of the tensegrity structure. In this Part I, the principles of the models are presented, while an application will be shown in the forthcoming Part II.

The Roundest Polyhedra with Symmetry Constraints

Amongst the convex polyhedra with n faces circumscribed about the unit sphere, which has the minimum surface area? This is the isoperimetric problem in discrete geometry which is addressed in this study. The solution of this problem represents the closest approximation of the sphere, i.e., the roundest polyhedra. A new numerical optimization method developed previously by the authors has been applied to optimize polyhedra to best approximate a sphere if tetrahedral, octahedral, or icosahedral symmetry constraints are applied. In addition to evidence provided for various cases of face numbers, potentially optimal polyhedra are also shown for n up to 132.