Abstract: It is shown that in dimension greater than 4, the minimal area hypersurface separating the faces of a hypercube is the cone over the edges of the hypercube. This constrasts with the cases of two and three dimensions, where the cone is not minimal. For example, a soap film on a cubical frame has a small rounded square in the center. In dimensions over 6, the cone is minimal even if the area separating opposite faces is given zero weight. The proof uses the maximal flow problem that is dual to the minimal surface problem.
Abstract: The Surface Evolver is a computer program that minimizes the energy of a surface subject to constraints. The surface is represented as a simplicial complex. The energy can include surface tension, gravity, and other forms. Constraints can be geometrical constraints on vertex positions or constraints on integrated quantities such as body volumes. The minimization is done by evolving the surface down the energy gradient. This paper describes the mathematical model used and the operations available to interactively modify the surface.
Abstract: It is a classic puzzle to find the shortest set of curves that intersect all straight lines through a square, and the conjectured solution is still unproven. This paper asks the analogous question for a cube, and comes up with the best known solution.
Abstract: Weierstrass representations are given for minimal surfaces that have free boundaries on two planes that meet at an arbitrary dihedral angle. The contact angles of a surface on the planes may be different. These surfaces illustrate the behavior of soapfilms in convex and nonconvex corners. They can also be used to show how a boundary wire can penetrate a soapfilm with a free end, as in the overhand knot surface. They should also cast light on the behavior of capillary surfaces.
Abstract: A new mathematical model of soap films is proposed, called the "covering space model." The two sides of a film are modelled as currents on different sheets of a covering space branching along the film boundary. Hence a film may be seen as the minimal cut separating one sheet of the covering space from the others. The film is thus the oriented boundary of one sheet, which represents the exterior of the film. As oriented boundaries, films may be calibrated with differential forms on the covering space, a version of the min-cut, max-flow duality of network theory. This model applies to unoriented films, films with singularities, films touching only part of a knotted curve, films that deformation retract to their boundaries, and other examples that have proved troublesome for previous soap film models.
Abstract: The soap film problem is to minimize area, and its dual is to maximize the flux of a divergenceless bounded vectorfield. This paper discretizes the continuous problem and solves it numerically. This gives upper and lower bounds on the area of the globally minimizing film. In favorable cases, the method can be used to discover previously unknown films. No initial assumptions about the topology of the film are needed. The paired calibration or covering space model of soap films is used to enable representation of films with singularities.
Abstract: The Surface Evolver has been used to minimise the surface area of various ordered structures for monodisperse foam. Additional features have enabled its application to foams of arbitrary liquid fraction. Early results for the case of dry foam (negligible liquid fraction) produced a structure haveing lower surface area, or energy, than Kelvin's 1887 minimal tetrakaidecahedron. The calculations reported here show that this remains the case when the liquid fraction is finite, up to about 11%, at which point an f.c.c arrangement of the cells becomes preferable.
Abstract: Solder bridging is investigated under the assumption that liquid solder bridges are equilibrium capillary surfaces and that the principal factor that determines whether a bridge will freeze to form a permanent short is its configurational stability. A computational paramemtric bridge stability study is conducted to determine the response of bridging to the system volume, the distance between pads, the contact angle between the liquid metal ant resist surface and the relevant physiochemical properties of the liquid metal.
Abstract: The Surface Evolver is an interactive program for studying the shapes of liquid surfaces. Recently added features permit the calculation of the Hessian matrix of second derivatives of the energy. The Hessian can be used for fast convergence to an equilibrium, and eigenvalue analysis of the stability of that equilibrium. This paper describes the use of the Hessian by the Surface Evolver, presents some sample stability analyses, and gives some numerical results on the accuracy and convergence of the methods. It is also shown how one can evolve unstable surfaces.
Abstract: We consider an eversion of a sphere driven by a gradient flow for
elastic bending energy. We start with a halfway model which is an unstable
Willmore sphere with 4-fold orientation-reversing rotational symmetry. The
regular homotopy is automatically generated by flowing down the gradient of
the energy from the halfway model to a round sphere, using the Surface
Evolver. This flow is not yet fully understood; however, our numerical
simulations give evidence that the resulting eversion is isotopic to one of
Morin's classical sphere eversions. These simulations were presented as
real-time interactive animations in the CAVE automatic virtual environment
at Supercomputing'95, as part of an experiment in distributed, parallel
computing and broad-band, asynchronous networking.
Video available.
Abstract: This paper describes the use of various symmetry features, including periodic boundary conditions, mirror boundaries, and rotational symmetry, in the Evolver. As a test case, we use these features to study foams, in particular the equal-volume foams of Kelvin and Weaire-Phelan. To compute the shape and energy of one of these compound surfaces, it is most efficient to work with only the smallest possible fundamental domain.
Abstract: We consider the problem of estimating stresses in the ascent shape of an elastic high-altitude scientific balloon. The balloon envelope consists of a number of long, flat, tapered sheets of polyethylene called gores that are sealed edge-to-edge to form a complete shape. Because the film is so thin, it has zero bending stiffness and cannot support compressions. In particular, the balloon film forms internal folds of excess material when the volume is not sufficiently large. Because of these factors, a standard finite element approach will have difficulty computing partially inflated balloon shapes. In our approach, we develop a variational principle for computing strained balloon shapes that incorporates regions of folded material as a part of the geometric model. We can apply our model to fully inflated or partially inflated configurations. The equilibrium shape is the solution of minimum energy satisfying a given volume constraint. We apply our model to a design shape representative of those used in scientific ballooning and compute a family of ascent configurations with regions of external contact for a volume as low as 22% of its float value.
Abstract: For idealized, infinitely thin ("dry") soap films, an X is unstable, while for very thick ("wet") soap films it is minimizing. We show that for soap films of relatively small but positive wetness, the X is unstable. Full stability diagrams for the constant liquid fraction case and the constant pressure case are generated. Analogous questions about other singularities remain controversial.
Abstract: Small bubbles in an experimental two-dimensional foam between glass plates regularly undergo a three-dimensional instability as the small bubbles shrink under diffusion or equivalently as the plate separation increases, and end up on one of the plates. The most recent experiments of Cox, Weaire, and Vaz are accompanied by Surface Evolver computer simulations and rough theoretical calculations. We show how a recent second variation formula may be used to perform exact theoretical calculations for infinitesimal perturbations for such a system, and verify results with Surface Evolver simulations.
Abstract: A "dry" conical soap film on a cubical frame is well known not to be stable. Recent experimental evidence seems to indicate that adding liquid to form "Plateau borders" stabilizes the conical film, perhaps to arbitrarily low liquid volumes. This paper presents numerical simulation evidence that the wet cone is unstable for low enough liquid volume, with the critical volume fraction being about 0.000274.
Abstract: The elegant structure of a liquid foam and its constituent parts have fascinated scientists for centuries. A combination of experiments, theory and simulations has elucidated most of its static and quasi-static properties. However, this is only part of a wider subject: dynamic effects remain as a considerable challenge, particularly for wet foams.