Giuseppe Viglialoro

A singular elliptic problem related to the membrane equilibrium equations

This paper presents a mathematical problem related to the equilibrium analysis of a membrane with rigid and cable boundaries for the so-called prestressing phase. The idea of using membranes in civil engineering applications such as footbridges, a new technology being developed in Spain, implies higher structural responsibility and more accurate analysis procedures. The membrane is represented by a regular surface with negative gaussian curvature, and its boundary by a set of regular curves the curvature of which depends on the structural elements, rigid or cable. Equilibrium is directly expressed by means of boundary differential problems in terms of the membrane shape and its stress tensor. The membrane–cable interface equilibrium requires taking into account a singular condition that makes the problem more difficult. Therefore, starting from the equilibrium equations, two mathematical approaches can be considered: a direct problem and a dual problem. The direct problem is defined and analysed, and its principal qualitative aspects are studied in this paper. Moreover, a numerical solution procedure is proposed to obtain practical results.

A mixed finite-element finite-difference method to solve the equilibrium equations of a prestressed membrane having boundary cables

ABSTRACT This paper proposes a mathematical method to solve the equilibrium equations of a membrane with rigid and cable boundaries for the so-called prestressing phase. More exactly, no constrained thrust network approach is implemented and the problem is studied starting directly from its continuous formulation. In this sense, the membrane is represented by a regular surface with negative Gaussian curvature, and its boundary by a set of regular curves, whose curvature depends on the structural elements, rigid or cable. The equilibrium is expressed by means of boundary differential problems in terms of the membrane shape and its stress tensor. The membrane-cable interface equilibrium requires taking into account a singular condition which makes the problem more complex. Therefore, the corresponding mathematical problem is defined and analysed, and its principal qualitative aspects are discussed in this paper. Moreover, a numerical resolution procedure is proposed and applied.

Explicit Blow-Up Time for Two Porous Medium Problems with Different Reaction Terms

This paper deals with the blow-up phenomena of classical solutions to porous medium problems, defined in a bounded domain of \(\mathbb{R}^{n}\), with n ≥ 1. We distinguish two situations: in the first case, no gradient nonlinearity is present in the reaction term contrarily to the other case. Specifically, some theoretical and general results concerning the mathematical model, existence analysis and estimates of the blow-up time t∗ of unbounded solutions to these problems are summarized and discussed. More exactly, for both problems, explicit lower bounds of t∗ if blow-up occurs are derived in the case n = 3 and in terms of an auxiliary function. On the other hand, in order to compute the real blow-up times of such blowing-up solutions and discuss their properties, a general resolution method is proposed and used in some two-dimensional examples.

A Free Boundary Approach to Solve the Equilibrium Equations of a Membrane

This chapter deals with a mathematical problem related to the equilibrium analysis of a membrane with rigid and cable boundary. The membrane and its boundary are respectively identified with a regular surface and a set of regular curves. The equilibrium is directly expressed by means of an elliptic problem, in terms of the shape of the membrane and its stress tensor; therefore, a free boundary numerical resolution procedure is presented and applied in a particular case.