Numerical modelling of stable minimal surfaces
This thesis examines the numerical representation of stable minimal surfaces. In particular, the work presented concentrates on the formulation of a finite element, suitable for the analysis of systems subjected to large strains and large displacements. In order to obtain an understanding of the physical properties of a minimal surface, and to verify the proposed numerical solution algorithms, the surfaces developed by several soap-film models are given. The mechanisms involved in the formation of a soap-film (minimal) surface is summarised. Several types of minimal surfaces are investigated, including general surfaces between rigid boundaries, single minimal surfaces between two frames, and those with internal and external flexible boundaries. In addition, the question of the stability of minimal surfaces is discussed, in terms of a finite and an infinitesimal perturbation. The numerical modelling of minimal surfaces is presented, based initially on the discretisation of the form using plane linear (line) and triangular elements. The application of the matrix-based element formulations to the vector-based Dynamic Relaxation solution algorithm is described. The formulations of the elements are assessed in the context of large strains and large displacements. Subsequently, the effects of the violations of the assumptions inherent in the derivation of the element stiffness matrices on the accuracy of the numerical solution are demonstrated, and measures proposed to maintain the stability of the solution algorithm. The numerical solutions to several minimal surfaces are provided, based on the linear and triangular element discretisations respectively. An intended improvement on the plane linear and triangular element formulations is proposed by the derivation of a higher order finite element. A 24 degrees-of-freedom finite element is formulated, representing a general curved elastic (or inelastic) geometrically non-linear continuum, and modelling the condition of plane stress. The element equations are derived with special consideration of the simulation of the effects of large strains and large displacements. An appraisal of the quality of the element formulation is made through the application of the Patch test and the Eigenvalue test. The solutions to several minimal surfaces are presented, from which the effects of the assumptions in the element formulation on the accuracy of the proposed numerical solution algorithm are demonstrated.