Gosling, Peter David (1992) Numerical modelling of stable minimal surfaces. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics T Technology > TA Engineering (General). Civil engineering (General)
Library of Congress Subject Headings (LCSH):	Algorithms, Engineering mathematics, Minimal surfaces, Eigenvalues
Official Date:	September 1992
Dates:	Date Event September 1992 Submitted
Institution:	University of Warwick
Theses Department:	School of Engineering
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Lewis, W. J. (Wanda J.)
Sponsors:	Science and Engineering Research Council (Great Britain) ; University of Warwick ; North Atlantic Treaty Organization
Extent:	xi, 313 leaves : illustrations
Language:	eng

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